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shishodia_v_Page_007.txt b5a3a54c96bce2a16f775015adda3559 d4bb72797fd8b59de183e5d39cc1af55d3ba7600 1850 F20101111_AAARHL shishodia_v_Page_024.txt 37aa73c1297dd998b5467fef2d235190 e267a22d5d0c21bb9fbcfc043aca43c538d469a4 2913 F20101111_AAARGX shishodia_v_Page_009.txt e7a4bab65bf37ba4cb7191a28edc4cc0 bc73c0c724fd0c293e9871eac8d4e72c0a974a9e 1894 F20101111_AAARHM shishodia_v_Page_025.txt 2be49866294041416490635beee4f90b b582b3f2040a1c730edcf0622e88b6a39928f26b 3271 F20101111_AAARGY shishodia_v_Page_010.txt 9991fdc9587cc8b59d0b78d558fff5df 233ae2e0316a72fd94b4457d53324cd4d919fc4d 2139 F20101111_AAARIA shishodia_v_Page_041.txt 9466b9041fec8479fdbe1d548ba268a5 0766fac53018250042e3f1f4e577ad5ced31f2a7 ENHANCED CONTROL PERFORMANCE AND APPLICATION TO FUEL CELL SYSTEMS By VIKRAM SHISHODIA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 2008 Vikram Shishodia To Carmen ACKNOWLEDGMENTS I would like to express my deepest gratitude to my advisor Dr. O. Crisalle for his support and guidance without which this work would not have been possible. I thank the members of my supervisory committee, Dr. H. Latchman, Dr. G. Hoflund, Dr. W. Lear, and Dr. S. Svoronos, for their guidance and serving on my supervisory committee. I thank my colleagues in the research group who provided insightful conversations on my research topics, besides being great friends. I would especially like to thank Christopher Peek for providing the sample code for ramp tracking which expedited the progress on the problem significantly. I also thank him for all the insightful discussions. I thank my parents for their love, support and encouragement that they have given me throughout my life and during the completion of this work. I would like to express my deepest gratitude to my spiritual teacher Gurumayi Chidvilasananda who has been there for me during every step of my life. Finally, I wish to thank my wife and kids, who have been very supportive, loving and understanding during this journey. TABLE OF CONTENTS page A CK N O W LED G M EN T S ................................................................. ........... ............. ..... L IST O F T A B L E S ...................................................................................................... . 7 L IST O F FIG U R E S ............................................................................... 8 A B S T R A C T ........................................... ................................................................. 1 1 CHAPTER 1 IN T R O D U C T IO N ......................................................................................................13 2 VIRTUAL CONTROL LABORATORY ............................................................... 15 2.1 Introduction ................................................................ ..... ..... ......... 15 2 .2 O bjectiv e ................................................................... ............ 17 2 .3 Inverted P endulum ................................................................17 2.4 C control D design ..................................................................................... ........ 20 2.5 Realization of an InvertedPendulum VCL ...................................... ........... ....25 2 .6 C o n c lu sio n s ............................................................................................................. 2 9 3 PI AND PI2 CONTROLLER TUNING FOR TRACKING THE SLOPE OF A RAMP .......38 3.1 Introduction and B background ................................................................................. ..38 3.2 Problem Statem ent and A pproach......................................... .......................... 40 3.3 Results and Discussion............................................. 43 3.3.1 Tuning Parameters of Controllers ........................... ... 43 3.3.2 Comparison of the Performance of the PI and PI2 Controllers ....................44 3.3.3 Com prison of M etrics..................... ........ .. ...................... ............... 46 3.3.4 Comparison of PI (ITAE) Controller with Literature Precedents .....................47 3.3.5 Local Minima versus Global Minima ........................................................48 3.4 Conclusions ................... ................................... 48 4 GENERALIZED PREDICTIVE CONTROL FOR FUEL CELLS ....................................61 4 .1 Intro du action .......................................................................................................6 1 4.2 Fuel C ell System B ackground...........................................................................61 4.3 O objectives of the R research .......................... ............................. .. ........... .. 64 4.4 Fuel Cell M odel ................................................................................64 4.5 Literature Precedents fo Fuel Cell Control Designs .......................................... 66 4.5.2 F eedforw ard Strategy ........................................... ................... .. ................67 4.5.2.1 Static feedforw ard controller ............... ........... ....................67 4.5.2.2 D ynam ic feedforw ard controller........................................................68 4.5.3 Combination of Static Feedforward with Optimal Feedback Controllers ........69 4.5.3.1 Case where the performance variable is measurable..........................69 4.5.3.2 Case where the performance variable is not measurable....................71 4.6 G eneralized Predictive Control ......................................................... ............... 72 4 .7 B battery of O b servers .......................................................................... .....................79 4.8 Sim ulation Studies and R esults......................................................................... ...... 81 4.8.1 Generalized Predictive Control Results ............................................82 4.8.1.1 Case where the performance variable is measured.............................82 4.8.1.2 Performance variable not measured........................................83 4.8.2 The GPC Approach Evaluated for Robustness...........................................85 4.8.2.1 Case where the performance variable is measured...........................86 4.8.2.2 Case where the performance variable is not measured.....................86 4.8.3 Comparison of the GPC Strategy with Prior Control Designs.....................88 4.8.3.1 Case where all states are measuredsFF with LQR feedback control 88 4.8.3.2 Case where all states are not measuredobserver design....................89 4.8.3.3 Comparison of controller performance with respect to robustness ....90 4.8.4 Feedforw ard Control D esigns ........................................ ....................... 92 4.8.4.1 Case of original m odel......................................... ......................... 92 4.8.4.2 Case of model uncertainty ................. ........................ ............... 94 4.9 Conclusions ........... ............ ........................................... ............... .. 95 5 CONCLUSIONS AND PROPOSITIONS FOR FUTURE WORK................................132 5 .1 C o n c lu sio n s ............. ........... ..... .................. ................. ................ 13 2 5.2 F future W ork ............. ........................................... ............................. 133 APPENDIX A OFFSET BETWEEN AUXILLIARY AND ORIGINAL RAMP................... ............ 134 B OBSERVER DESIGN USING TRANSFER FUNCTION...............................................136 L IST O F R E F E R E N C E S .................................................................................. ..................... 137 B IO G R A PH IC A L SK E T C H ......................................................................... ... ..................... 140 LIST OF TABLES Table page 31 The PI2 controllers optimized tuning parameters linear least square fit equations ..........52 32 The PI controllers optimized tuning parameters linear least square fit equations. ............53 33 Values of the plant parameters used compare the performance of the controllers. ...........53 LIST OF FIGURES Figure page 21 Inverted pendulum ................................... ..... ............. .......... ....... 31 22a Front panel of the VCL where all states are measured. ............ ........... .................32 22b Front panel of the VCL showing observer ............. .......... ..................................33 23 Interaction panel of inverted pendulum VCL. ................ ..... .......... ...............34 24 Controller tab of the navigation panel. ................................................... ............. .....35 25 A analysis tab of the navigation panel .......................................................... ......... .....36 26 Simulation tab of the navigation panel. .................. ..................................................... 37 31 Ramp r and auxiliary ramp ra with constant slope, a. ................................................. 49 32 Closed loop transfer function representation of plant and controller. .......................... 49 33 The PI2 controllers optimal tuning parameters, using ITAE (A, B), IAE (C, D), and ISE (E, F) as the optim izing m etric. ............................................................................50 34 The PI controllers optimal tuning parameters, using ITAE (A, B), IAE (C, D), and ISE (E, F) as the optim izing m etric. ............................................................................51 35 The PI2 and PI controllers' ramp tracking and slope tracking performance using the optim al ITAE control param eters. ............................................ ............................. 54 36 The PI2 and PI controllers' ramp tracking and slope tracking performance using the optim al IAE control param eters ................. ........................................... ............... 55 37 The PI2 and PI controllers' ramp tracking and slope tracking performance using the optim al ISE control param eters. ............................................................. .....................56 38 The ITAE, IAE, and ISE metrics comparison for three plants, using the PI controllers A ), B ), an d C ) ............................................................................. 57 39 The ITAE, IAE, and ISE metrics comparison for three plants, using the PI2 controllers A ), B ), and C). ..................... .... ............ ................. .... ....... 58 310 The PI controllers tuned using the ITAE metric compared with Belanger and Luyben and Peek's controllers for three plants A), B), and C)...................................................59 311 Contour plots for the PI controller tuned using the ITAE metric for the three plants A ), B ), an d C ) ............................................................................. 6 0 41 Schem atic of fuel cell system ................................................ ............................... 96 42a Fuel cell system showing input u, disturbance w, and outputs z1 z2, yz Y2, s3...................97 42b Fuel cell system showing sFF with feedback controller ...................................98 43 Matrices defining the LTI model for the fuel cell model excluding sFF .........................99 44 Matrices defining the LTI model for the fuel cell including sFF. .....................................99 45 The sFF control configurations for fuel cell system. ................ .............................. 100 46 The dFF controller: (a)Schematic diagram, and (b)transfer function representation. .....101 47 The sFF schematic with feedback controller. ...................................... ............... 102 48 The GPC design in feedback block diagram................................... .............103 49 Disturbance profile used for simulation purposes. .................................. .................104 410 The GPC control strategy implementation on the nonlinear fuel cell model in the case when the controlled variable is measured. ........................................ ............... 105 411 The GPC feedback with four observers control scheme implementation on the nonlinear fuel cell m odel. ...................... .. ...................... .. .... .... ............... 106 412 The Norm of errors from the battery of observers.............. .... ............... 107 413 The switching pattern of the battery of observers........... ................... ... .................108 414 Final voltage to the com pressor. .............................................. ............................ 109 415 Observer 1, error between measured and estimated values..........................................110 416 Observer 2, error between measured and estimated values .................. .................111 417 Observer 3, error between measured and estimated values ................... ....................112 418 Observer 4, error between measured and estimated values ................... ....................113 419 The GPC feedback with three observers control scheme implementation on the nonlinear fuel cell m odel. .......... .. ........ ......................................... .............. 114 420 The GPC feedback with two observers control scheme implementation on the nonlinear fuel cell m odel. .......... ...... ........ ... .. ... ... ... .... .............. 115 421 The GPC feedback with one observer control scheme implementation on the nonlinear fuel cell m odel. .......... ...... ........ ... .. ...... ... .... .............. 116 422 The GPC control strategy implementation on the nonlinear fuel cell model with a parameter changed from the value used for control design. ......... ..............................117 423 The GPC controller with the LQG observer control strategy implementation on the altered nonlinear fuel cell m odel......... .................................................. ............... 118 424 The GPC controller with the 4 observers control strategy implementation on the altered nonlinear fuel cell m odel......... .................................................. ............... 119 425 Comparison of the GPC control strategy with the sFF controller combined with LQR feedback strategy on the unaltered nonlinear fuel cell model when the performance variable is m easurable.......... ................................................................... ....... .......... 120 426 The sFF with the LQG observer and LQR feedback, compared to GPC with the LQG Observer control strategy implementation on the unaltered nonlinear fuel cell model when the performance variable is not measurable ..............................................121 427 The sFF with the LQG observer and LQR feedback, compared to GPC with the 4 observers control strategy implementation on the unaltered nonlinear fuel cell model when the performance variable is not measurable ..............................................122 428 The sFF with the LQR feedback, compared to GPC, when performance variable is measurable on the altered nonlinear fuel cell model. ..................................... ........... 123 429 The sFF with the LQG observer and the LQR feedback compared to the GPC with the LQG observer control strategy on the altered nonlinear fuel cell model................... 124 430 The sFF with the LQG observer and the LQR feedback compared to the GPC with the 4 observers control strategy on the altered nonlinear fuel cell model. ....................125 431 The sFF and dFF strategies and the GPC control strategy, performance compared when applied on the unaltered nonlinear fuel cell model. .............................................126 432 The sFF and dFF strategies and the GPC control strategy with the LQG observer, performance compared when applied on the unaltered nonlinear fuel cell model ........127 433 The performances of the sFF and dFF strategies and the GPC with the 4 observers control strategy compared when applied on the unaltered nonlinear fuel cell model. ....128 434 The sFF and dFF strategies and the GPC control strategy, performance compared when applied on the altered nonlinear fuel cell model. ................................................129 435 The sFF and dFF strategies and the GPC control strategy with the LQG observer, performance compared when applied on the altered nonlinear fuel cell model. .............130 436 The performance of sFF and dFF strategies and the GPC with 4 observers control strategy compared when applied on the altered nonlinear fuel cell model................131 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ENHANCED CONTROL PERFORMANCE AND APPLICATION TO FUEL CELL SYSTEMS By Vikram Shishodia May 2008 Chair: Oscar D. Crisalle Major: Chemical Engineering The invertedpendulum virtual control lab, a simulation environment for teaching advanced concepts of process control, is designed using the LabVIEW software tool. Significant advantages of using this simulation tool for pedagogical purposes include avoiding the potential issue of schedule conflicts for securing equipmentaccess time in a physical laboratory and providing a learning resource that becomes accessible to students located in remote geographical places. A set of tuning relationships are proposed for standard proportionalintegral controllers and proportional doubleintegral controllers for the purpose of tracking the slope of a ramp trajectory. Three different performance metrics are investigated to serve as the criteria for optimality, and a numerical optimization procedure is used to minimize each metric over 20,000 different plants. The proportional integral controller with tuning parameters selected to optimize value of the integral of the timeweighted absolute error is recommended for tracking the slope of a ramp traj ectory. A generalized predictive control (GPC) strategy is proposed for a fuel cell system, where the controller incorporates a measured disturbance in the control design. The control objective is to maintain oxygen excess ratio at a prescribed constant value. The performance of the GPC control design is compared with that of the controllers proposed in literature for various scenarios including model uncertainty. The GPC controller has zero offset when the performance variable is measured and performs better than competing designs offered in the literature. The GPC controller is also robust with respect to model uncertainty. A battery of observers with a switching strategy is proposed for estimating the value of the performance variable when it is not measured. The GPC controller with a battery of observers has no offset demonstrating better performance than analogous designs proposed in literature. However, the control performance is not robust when the estimator battery is used and linear models used for observer design are uncertain given that the response offset is not completely eliminated. CHAPTER 1 INTRODUCTION Issues relevant and critical to process control are discussed in this study. Chapter 2 investigates the design and implementation of a virtual control lab (VCL) for "The Inverted Pendulum" problem. The LabVIEW software is used as the platform for simulating the Inverted Pendulum model. The objective is to have a visual computer based application by virtue of which advanced control concepts can be shared and taught to the audience who are primarily students studying control theory and its applications. The VCL is designed in a manner such that various scenarios for the implementation of the controllers can be achieved. The user is given the choice of operating the application with the system in openloop or closedloop configuration. The controller can be tuned manually by the user or use the tuned control parameters computed by the specific control algorithm. The user is allowed to alter the values of the poles for the closed loop system and see its visual impact by simulation performed by the VCL. The VCL provides the opportunity to be operated in the scenarios when all the controlled variables are measurable and also when all of them are not measurable. In the case when the performance variables are not measurable an observer is incorporated in the control design to estimate their value. The impact of all the changes performed in the VCL are displayed visually by the animation of the inverted pendulum system. This key feature of the VCL allows the user to see the visual impact of changing different components of control system and hence facilitating the process of learning. Chapter 3 discusses the problem of controller design tuning for tracking the slope of a ramp. The control objective is to place the output of the system in a linear zone parallel to a ramp trajectory. A first order system with time delay is considered for this study. Two kinds of controllers are used for this study namely, proportionalintegral and proportional doubleintegral. Both the controllers serve the purpose of positioning the system in the desired linear zone which is parallel to a given ramp profile. There is an offset with respect to the ramp observed when only one integrator is used. Zero offset with the ramp is observed when two integrators are used in the controller. In both cases the control objective is met which is to track the slope of the ramp trajectory.. Three different metric are employed to evaluate the performance of the controllers. The MATLAB platform in conjunction with SIMULINK module is used for acquiring the optimized controller parameters. Chapter 4 discusses a generalized predictive control (GPC) strategy proposed for a fuel cell system. The control objective is to regulate the value of the performance variable i.e., the oxygen excess ratio at a desired value. The performance of the GPC control design is compared with that of controllers proposed in prior literature. Various scenarios are considered, including the cases of model uncertainty and unmeasured performance variable. The GPC controller exhibits zero offset in all cases when the performance variable is measured, and also ensure zero or negligible offset when the performance variable is estimated via a battery of estimators. CHAPTER 2 VIRTUAL CONTROL LABORATORY 2.1 Introduction There is a need for the development of internetbased nonconventional pedagogical tools for delivering knowledge to students on various topics of study. The drive stems from the various advantages that these environments offer. First, these applications do not depend on the availability of a physical setup or facility to run experiments [1, 2]. They are also not limited in terms of the number of users who can access the application at any given time, as long as appropriate adjustments are done in the server side of application. There is also no adverse safety issue or concern of damaging expensive equipment when the product is not used correctly. Less training is required for the user to be able to run the tool. Compared to a traditional physical laboratory setup, in these virtual environments, there is more of an opportunity to be able to realize a physical system and introduce more advanced topics and see their effects on the system. A software application that simulates the behavior of a physical system, provides animation to depict how the system behaves, and provides an interface so that the user can observe changes made on the system performance, is highly beneficial from a learning and educational standpoint. The application is referred to as a virtual laboratories since the nature of the "Lab" or the application is "virtual" as it is a software emulator of the physical plant and can be potentially used to remotely control actual physical equipment via web and networking [1, 36]. From the perspective of enhancing the learning experience, the virtual control lab (VCL) supports learning by all three modes, namely active, flexible, and discovery learning. In active learning, tools and material are made available to students so that they can use these resources to actively learn and reinforce the theoretical concepts. Traditionally, physical laboratories, equipment and experimental apparatus are provided to students to reinforce and test the understanding of the student's comprehension of the theory. With limitation of available resources and costs associated with the overall management of logistics, at times this can be a challenging task, which can prove to be not only quite expensive, but also involve safety issues. In those circumstances the VCL can be an excellent solution. It doesn't necessarily need to be a complete replacement, but it can be used in conjunction with existing physical setups to promote active learning. Flexible learning provides opportunities to learn class material when the students or instructor might be having challenges in terms of establishing meeting times, scheduling or location [7]. For instance, if an individual is a parttime student with the obligations of a full time student, that person might have challenges meeting lab times scheduled during regular work hours. The VCL is a most valuable tool to accommodate those circumstances. From the comfort of home and in a more appropriate time, that individual can complete the exercises/material if the VCL is utilized. The VCL is flexible with respect to the schedule and logistics limitations of an individual. The VCL also supports learning via discovery mode. In this scenario, an environment is provided where the student has minimal supervision or instruction [8]. The student is encouraged to learn by making changes and observing the impact of these changes. There are some significant challenges in implementing such a setup in the alternative scenario of a physical laboratory. Due to safety and cost considerations, facilities that promote discovery learning are few. A VCL designed for the purpose, again, proves to be an excellent resource to implement safe and costeffective learning via discovery. There are various implementations of virtual and remote labs reported in the literature. There are several World Wide Web based labs which foster learning by different modes [3, 9 11]. Most of these virtual labs, however, have a few shortcomings in terms of their usage. Many do not provide a sufficiently high level of interactivity with the user. Significant modifications need to be made to the program to implement any changes. Another disadvantage that most of the current virtual labs have is that they are developed on a proprietary software platform. At times significant familiarity with that software is needed to be able to utilize the application. 2.2 Objective The intent is to build a VCL module that treats some advancedlevel control concepts and serves as a pedagogical tool that overcomes shortcomings that existing virtual labs pose from a learning perspective and user interface. The infrastructure created by Peek et al. is used for implementing an Inverted Pendulum VCL [12]. The intent is to develop an animated control module that reinforces advanced control concepts with a friendly user interface. Some examples of key control concepts illustrated in the VCL are linear statespace modeling, controllability, pole placement and observability analysis. Sections 2.3 and 2.4 discuss the Inverted Pendulum system, its dynamics and the associated control concepts. Section 2.5 describes the implementation of the Inverted Pendulum system as a VCL using the LabVIEW software and its animation features [13]. Finally, conclusions from this effort are summarized in Section 2.6. 2.3 Inverted Pendulum The inverted pendulum considered consists of a spherical bob attached to a cart by a rod. A schematic diagram is given in Figure 2.1. The mass of the rod is assumed to be negligible. The rod is mounted by a hinge at the center of the cart. The input of the system is a horizontal force applied to the cart. The cart is free to move only along one coordinate which is the horizontal zaxis. The pendulum is free to rotate 360 degrees with respect to the cart in the xz plane where the xaxis is vertical. It is assumed that there is no friction between the pendulum and the cart at the hinge. The goal is to keep the pendulum in an upright position by manipulating the value of the applied force. The system is inherently nonlinear. To apply linear control theory, the dynamics must be linearized, and represented as a standard statespace realization. Consequently, the system is linearized for small values of the angle that the pendulum makes with the vertical. The nonlinear equations describing the dynamics of the system are M I s +02/sin gsin0cos0) (21) +sin2 0 m 1 f m+M and 0= 1 _cos 02/cosOsinmO+ gsino (22) M sin2 0 m m m where z = dz/dt (23 a) and = dO/dt (23b) where Mis the mass of the cart, m is the mass of the pendulum bob, is the length of the pendulum rod, g is the acceleration due to gravity, z is the horizontal position of the cart, 0 is the angle that the pendulum makes with the vertical, andfis the force (control input) acting on the cart [14]. A linear statespace system is derived from Eqs. 21 23 by linearizing about an operating point ( z, z, 0, ), where z = 0 (24) S= 0 (25) = 0 (26) 0 =0 (27) f =0 (28) The deviation variables for the linear statespace model are X1= Z x, =  x2 = z x = 0  x4 =00 After linearization about the point (z, z, ,, f) = (0, ,0, 0,0) the resulting standard linear statespace model = Ax + bu is given by the equation 0 1 0 0 0 mg = M 0 0 0 0 0 (m+M)g Ml where the elements of the state vector x are distance (xl and angular velocity (x4 = ), and where 0 mg M 0 (m + M)g Ml 0 0 0 1 x+ M u (216) 1 0 0 1 Ml, velocity ( 3 ), z), velocity (x2 z ), angle (x3 0), (217) (29) (210) (211) (212) (213) (214) (215) 0 1 b= M (218) 0 1 MI The control is u, which is the force acting on the cart. The standard output model y=Cx (219) relates the output y to matrix C and state vector x, where the output matrix C is the identity matrix 1000 0100 C = (220a) 0 10 0 001 in the case where all four states are measured. For the case where only one state is measurable, the output matrix C adopts a rowvector form of all zeros, except for one entry that is unity at the location corresponding to the measured state. For the particular case where the state xj, namely the distance of the cart from its original horizontal position, is the only measured state, the output matrix C adopts the form C=[l 0 0 0] (220b) 2.4 Control Design When the system is in the unforced configuration, a stability check done by calculating the eigenvalues of matrix A reveals that there is one eigenvalue that lies in the open right half plane, implying that the system in its unforced state is unstable. This is a regulation problem, as the objective is to make the states evolve towards zero value. For the implementation of the controller, a test for controllability needs to be performed to verify that the system is indeed controllable. The requirement for controllability is that det(Q,) 0 (221) where Qc= b Ab A2b A'b] (222) Using the definitions for A given in Eq. 25 and for b given in Eq. 26, it follows that 1 mg M M212 mg 0 0 Ab= A2b= Ml ,A3b= ( M) (223) 1 (m +M) g 0 MI Mz/22 0 (m + M)g 0 M2/2 01 0mg M M2/2 1 0mg 0 1 (m + M)g Hence, Q, M 1 M21 MM)g (224) 0 0 MI Mz/22 1 (m + M)g 0 0 MI Mz/22 Obviously matrix Qc in Eq. 211 is of full rank, thus det(Q,) 0 (225) which implies that the system Eq. 24 is indeed controllable. The analysis of controllability presented here is found in standard references [3234]. Two scenarios are considered: 1. All states are measured. 2. Some states are not measured. In the case of the first scenario in which all states are measured, a fullstate feedback approach is used in the form of the proportional state feedback control law u = Fx (226) where F is a proportional gain used to address the regulation problem. To determine an appropriate value for matrix F, first substitute the value ofu given by Eq. (226) into the state space equation Eq. (215) leading to x = Ax + b(Fx) (227) The standard solution to Eq. 214 is given by the Variation of Parameters formula as x= x0e(ABF)t (228) where x0 is the vector of the initial value of the state vector x [36]. Ackerman's pole placement algorithm is employed for computing the value of the matrix F that places the poles of the ABF system in the desired location [36]. In the case of the second scenario, in which all states of the system are not measurabed, a Luenberger Observer is incorporated in the controller to estimate the value of the states. Before carrying out an observer design, a check is performed to verify if the system is observable when only one state is measured. The first state, the distance of the cart from the original position, is the only state that is assumed to be measurable. For the system to be observable, the condition det(Q) 0 (229) should be satisfied, where the observability matrix Qo is given by C Q = CA (230) CA2 MCA Matrix A is defined by Eq. (217) and matrix C by Eq. 220b. The expressions CA=[0 1 0 0] (231) CA2 = 0 0mg 0 (232) and CA3 = O 0 0 0 mg (233) M can be used to readily build the observability matrix Qo described by Eq. 217, yielding 1 0 0 0 0 1 0 0 Q, = (234) mg 0 0 m 0 M 0 0 0 mg M Since matrix Qo is diagonal, its determinant is simply the product of the diagonal terms. Hence, 2 2 det(Qo)= mg 0 (235) Given that the determinant is nonzero, it follows that Qo is of full rank, and therefore the system is observable. The analysis of observability presented here is also found in standard references [3436]. The standard equations for observer design x = Ax + bu (236) = Ai +bu + L(y ) (237) produce estimated states i and estimated outputs y = Ci as a function of the measured system output y = Cx, and the Luenberger gain L, in Eq. 237. The control input u = Fi (238) is used to place the poles of Eq. 236 at the desired locations. The error = x i (239) is defined as the difference between the actual values of state vector x and estimated values of state vector i. Hence, the derivative of the error E=xx (240) is computed by differentiating Eq. 239. Substituting Eq. 236 and Eq. 237 in Eq. 240 yields S= (A LC)E (241) Invoking now the Variation of Parameters formula the solution to the differential equation Eq. 241 is = Ee (ALC)t (242) where e0 = x(0) i(0) is the initial value of the error E, i(0) is an initial guess of the value of the estimated state vector, and x(0) is the initial value of the state vector. The poles of matrix ALC should lie on the open half plane for the value of error E to evolve to zero. The poles are placed at the desired location by an appropriate choice of L, which for loworder systems as the one considered here can be easily done via Ackerman's Pole Placement algorithm. Linear quadratic regulator control design. A Linear quadratic control regulator (LQR) control design is implemented in the VCL. The LQR control law u = Kx (243) is obtained by minimizing the cost function J(u)= (xQ x + uTRu + 2xTNu)dt (244) 0 with respect to u. The weighting matrix Q must be symmetric positive semidefinite, and R symmetric positive definite. The weighting function N is specified to be zero. For a linear state space system x = Ax + Bu (245) the solution to the minimization of the cost function results in the steadystate Riccatti equation [36] ATS + SA (SB + N)R1 (BTS + NT) + Q = 0 (246) The acceptable solution to Eq. 246 is a positive definite matrix S which is then used to specify K from the expression K= R(BTS+NT) (247) Since in this case N = 0, therefore K= R1BTS (248) 2.5 Realization of an InvertedPendulum VCL The LabVIEW software and a VCL infrastructure proposed by Peek et al., is used for the implementation of the Inverted Pendulum VCL [12, 13]. Previous softwarebased controltools for the invertedpendulum system reported in the literature have significant value, but the VCL developed in this study has a number of additional desirable pedagogical features [14]. Initially, standalone VIs and subVIs are generated using LabVIEW software for different components of the design before integrating them as a part of a monolithic VCL. There are several reasons why National Instruments' LabVIEW software is used for constructing the VCL. The ease of structuring and maintaining a VCL is significantly high in this software. The LabVIEW software has builtin features for deploying applications on the web. The software also has toolkits specifically designed for control engineering. Implementation of a VCL using LabVIEW does not rely on support from other software packages, as would be the case if some higherlevel language is used to implement the same features found in VCL. Figure 22 shows the front panel of VCL as it appears to a student user. The key elements are the Animation and Interaction Panels, respectively, located on the top and bottomleft areas of the front panel. These two are very critical components of the VCL as the user makes most modifications in the plant and controller setup in the Interaction panel and instantaneously observes an animated result describing the plant and states in the Animation Panel. The Animation Panel has a twodimensional graphic representation of an inverted pendulum. When the VCL is operated, the animated cart responds to the control input by moving to the left or right and causing a pendulum swing. The third panel is the Navigation Panel on the bottomright area of the front panel. The Navigation Panel has five tabs (Information, Plant, Controller, Analysis and Simulation), which provide various pieces of information about the VCL. More information is given about these three panels in the ensuing subsections. Animation, interaction and navigation panel. The animation panel plays the role of providing a visual representation of the plant, namely an inverted pendulum. Any changes that are made to the inverted pendulum mounted on the cart are visually depicted in the Animation Panel. The user has the ability to make changes to the plant and controller in the Interaction Panel. The user can adjust plant parameters, initial condition of the states of the inverted pendulum and assign the different values to control parameters to the controller of choice. The user also has the ability to run the plant in Manual or in Auto mode. Figure 23 depicts some of the various modes that the user can configure parameters in the Interaction Panel. The user has to specify the initial states of the pendulum (position, velocity, angle of the pendulum with the vertical and the angular velocity of the pendulum). The Animation Panel constructs the visual representation of the inverted pendulum based on the information that the user provides. The user has the flexibility of running the VCL in the following two scenarios: (1) all states are measured, or (2) only one state is measured. Based on the choice of the user, an implementation of the corresponding controller is given. When "Manual F" Control is in the "Off' position, the user is allowed to choose the poles for the closed loop matrix ABF and the value of matrix F is calculated from Ackerman's pole placement algorithm. The user can immediately see the impact of poles chosen on the stability of the inverted pendulum in the Navigation Panel under the Analysis tab. When the "Manual F" Control is in the"On" position, the user has the ability to choose the values of the elements of matrix F. When the controller is operated in "Luenberger Observer" mode, as shown in Figure 2, the user has to provide the desired poles for matrix ALC. The only state that can be measured in this mode is the position of the cart. When the controller is in "Off' mode, i.e., the system is in open loop configuration with no feedback, the Analysis tab of the Navigation Panel shows that the pendulum is in an unstable configuration, which is ascertained by the fact that there is one eigenvalue of the system in the open right half plane. The Interaction and Animation panels provide a suite of options and visual representation for the user. The Navigation panel is located on the bottomright area of the front panel of the VCL. The Navigation panel has five tabs entitled: (1) Help, (2) Plant, (3) Controller, (4) Analysis, and (5) Simulation. These tabs provide pertinent and critical information about the VCL to the user. The Help tab, when clicked on, provides general information about the operation of the VCL. The user can access the Help tab without having to leave the VCL. The Help tab displays an embedded PDF file. The Plant and Controller tabs are also embedded PDF files which provide information about the dynamics of the plant (inverted pendulum) and the controller (proportional state feedback and Luenberger Observer). The nonlinear equations and linear state space system for the inverted pendulum are explained in the Plant tab of the Navigation Panel. Figure 23 shows the Plant tab of the Navigation Panel. The Controller tab provides information about pole placement and various other aspects of control design for inverted pendulum VCL, as shown in Figure 24. The Analysis tab, shown in Figure 25, has information about the tools and graphs that are used in control theory. This tab has information about the transfer function, location of poles and zeros in complex plane and Bode plot (frequency response). The Simulation tab, shown in Figure 26, shows plots of the results of numerical simulations describing the states (position, velocity, angle and angular velocity) and inputs as a function of time. Since this is a regulation problem, when stable choices of eigenvalues are given, using the linear state space as the dynamic model, all states converge to the value zero, regardless of the choice of initial state vector. The Simulation tab provides the real time curves of all the states as a function of time. The RungeKutta integration algorithm is employed to compute the numerical response of the plant to the controlling input. When the controller is toggled between "On" and "Off" modes (closedloop and openloop behavior, respectively) in the Interaction panel by the user, the impact of that change on the value of states is depicted immediately in the Simulation tab. The LQR control strategy, as shown in Figure 2, employed in the VCL gives the user the opportunity to implement different control choices, such as varying the weighting on different elements of the cost function and displaying the corresponding LQR gain. 2.6 Conclusions A VCL for the control of an Inverted Pendulum is described. The Inverted Pendulum is a classic example of illustrating statespace model representations and demonstrating the classical control concepts of controllability and observability. The animation features of the VCL provide a visual description of how an inverted pendulum responds as a function of the input force applied. The user is given the opportunity to run the VCL in different modes, such as in open loop and closedloop configurations of the system. The VCL can be utilized as a tool for enhancing learning. The three most widely recognized learning modes (active learning, flexible learning, and learning via discovery) can be easily executed using this VCL module. The module can be used in conjunction with a process control lecture for demonstrating various concepts. The animation capabilities allows the user to see the impact of every change that is made to the control configuration. The Analysis tab also demonstrates that the openloop configuration (unforced system) is unstable, as one of the eigenvalues is in open right half plane. The user is given the choice of choosing the poles for the system and noticing its impact on the plant. The user has the choice of running the VCL in two modes: (1) all states measured or (2) only one state is measured and implementation of a Luenberger Observer. This utilization of the VCL supports active learning of the control material. The VCL can also be presented before or after a conventional lecture. In the first case, the users' motivation to learn about theory presented in class would be enhanced as they have the opportunity to develop and experience with the VCL. In the later case, after the lecture is delivered, an interaction with the VCL would serve as an excellent tool for reinforcing the concepts taught in class. There is significant value in learning an abstract concept taught in lecture and being able to relate to the concept by virtue of seeing the animation and graphs from simulation. In an ideal scenario, the VCL should be used in all three modes (before, during, and after lectures). The tool can also be used to support group activities in a class for learning control theory and doing control homework [15]. The modular feature of the VCL can be taken advantage of by implementing sequential learning. In this mode, different versions of same VCL are progressively given to the user as the class progresses. Each successive version makes more features of the controller available to the user, hence helping students appreciate and learn faster the concepts as they are progressively taught in the class. To validate the benefits and effectiveness, it is proposed that the VCL should be used as a pilot teaching tool in process control classes. The feedback obtained from the students would be highly beneficial in optimizing and incorporating features that could enhance the learning experience. x z Figure 21. Inverted pendulum. () () I I in iMenaM iUain Controller. Of/On LuMnbergerGin Ir T J7. Tim r ijrmrrr I ~I A fl;.. Figure 22a. Front panel of the VCL where all states are measured. I'....' I ~tt ;,i o. na il L ,3 LLL I A .jn ]l L lij. Ti I,, ,, ,, , II i~ IIrr, c..br.iie. I r r . Figure 22b. Front panel of the VCL showing observer. I LpC'rT  .11 III I I r7 r0 S0 17 0 0o .j0 0 ...... .. ........... ........ .............. Figure 23. Interaction panel of inverted pendulum VCL. Help Plant Controller Analysis Simulation TransFer Function Static Stik Feedhnck Controller and Liibenhr  Static state 'Le,;LjL. controller is used for Inverted Mo.'[ei IIllI the pendulum in vertical upright position. The Linear State Space Model is obtained by linearization of the equations that described ti dIn.lmicni (f the inverted pendulum: y= ,+ .L.Sinfg,, r,,in Rm J Th" Lincar State Space Model is: f i 1 1 r ( >I I l < I > PoleZero Map Plot 1 0.75 0.5 g 0.25 0 E 0.25 0.5 0.75 1 0. 1 0,8 till lii] II 0.6 0.4 0.2 0 Time Figure 24. Controller tab of the navigation panel. Bode Magnitude 0 25  2 50 ( 75  o S100 S125  150  175 10Ou Bode Phase 50 C 50 100 1 rn ] III ____ I ____ ] ____ I ____ I ____ lm 10m 100m 1 Frequency (rad/s) Step Response Graph 21 I U 2 U 4 4 .   0 10 k 25 10 100 1k 0 2.5 5 PoleZero Map Plot 'Ft __  JU 200 1 lOOu im I I I I 10m 100m 1 10 Frequency (rad/s) ID 100 1k < Transfer Function 0.0076 s2 + 3.3952E18 s 0.075 s4+2., s3 + 1.6653 s2 + 0.575 s 0.075 0. 0. 0. 0. 0. 0. 0. 0, 7.5 10 12.5 15 17.5 20 Time (s) 22.5 2 22.5 2 6 4 2  0    .... I  4 6 1 0.8 0.6 Input: Output: u I Uiil:!a * Figure 25. ls tb of te n p l Figure 25. Analysis tab of the navigation panel. 02 0.2 0  Help I Plant Controller I Analysis I Simulation I \. / 1.5 1.5 2 2,5 3 3,5 44.1 Time 11:11:1 : AI Figure 26. Simulation tab of the navigation panel. i..i, 51 1 1: 1 I 1:11:1 ,J ,i: _:, ZO:_ :1 : ,JTh r .*,c _,) ________ ______ _ E w pi:'l Run CHAPTER 3 PI AND PI2 CONTROLLER TUNING FOR TRACKING THE SLOPE OF A RAMP 3.1 Introduction and Background In certain applications tracking the slope of the ramp is more critical than tracking the ramp itself. It is not unusual to encounter applications where the set point is the slope of a ramp trajectory. For instance, while growing thin films on a substrate, it is desired that the temperature of the substrate in the reactor increases at a steady rate (i.e., following a trajectory with a specified slope). In these kinds of applications, it becomes crucial to adopt the appropriate choice of controller type with effective values for the tuning parameters and an appropriate metric to ensure adequate performance. Simplicity of tuning relationships plays a critical role in the implementation of a controller. Most successful tuning relationships have been developed via simulation [16, 17]. The cost function normally used involves the feedback error, which is the difference between the set point and output of the plant. A proportionalonly controller leads to a steady state offset with respect to step changes in set point. An integrator needs to be incorporated in the controller to eliminate the offset. Similarly, for a ramp set point, a proportionalintegral (PI) controller is not sufficient to remove the steady state offset. In this case a controller with two integrators, that is a proportional doubleintegral (PI2) controller, is needed to remove the offset [18]. However, a PI controller is sufficient to track the slope of the ramp trajectory. Belanger and Luyben proposed a proportionalintegraldouble integral controller relating the tuning parameters to the ultimate gain and ultimate period of the plant [19]. AlvarezRamirez et al. extended the work of Belanger and Luyben and coined the acronym PI2 [20]. Peek examined three different versions of PI2 controllers for tracking ramp set point [21]. This study investigates the problem of controller tuning for tracking the slope of a ramp signal as depicted in Figure 31. The intent is to design a controller that leads the output trajectory to follow a line parallel to the ramp. It is also critical to minimize transients. The control performance is deemed as poor when the system experiences large deviations during the transients. To investigate the problem, a first order system with time delay is adopted as the plant model. Since the set point is a ramp, integral action needs to be incorporated in the control design. Two kinds of controllers are used for this study, namely a proportionalintegral (PI) controller and proportional doubleintegral (PI2) controller. Both controllers serve the purpose of placing the system output on the desired line, parallel to the ramp set point. However, there is an offset with respect to the original ramp trajectory when PI controllers are used because only one integrator is included in the control scheme. To identify the appropriate indicator to measure the performance of the controller with respect to the system and objective in question, three different metrics are used: integral of the timeweighted absolute error (ITAE), integral of the absolute error (IAE), and integral of the square of error (ISE). Each of the controllers is tuned to minimize the metric value for a given set of plant parameters. The Simplex optimization routine is used to acquire tuning parameters via the minimization of the metric adopted. The MATLAB platform in conjunction with the SIMULINK module is used for conducting the simulations. A set of optimal tuning relationships for controllers to track the slope of a ramp for 20,000 different plants are presented. The performance of all the controllers is evaluated and results are compared with the prior work of Belanger and Luyben. The performance of the controllers is also compared with that of those proposed by Peek. 3.2 Problem Statement and Approach The objective is to use PI and PI2 controller to make a firstorder plant with time delay follow the slope of a ramp. The plant is represented by the transfer function G,(s)= K e (31) TS+l where K is the gain, r is the time constant, and 0 is the time delay of the plant. The input to the plant is denoted as u and the output asy. The transfer function representation of the PI controller Ge(s)= K 1+ (32) i's and the transfer function for the PI2 controller investigated here is G,(s)= KK 1+ (33) 7s where K, is the proportional gain, and r, is the integralaction time constant. The plant and the controller are configured in the closedloop arrangement shown in Figure 32. The ramp function serving as the set point in Figure 32 is given by r(t) = at (34) where t is the time and the constant slope a is taken as a = 1. Three error metrics are used for tuning the controllers, namely, the integral of the time weighted absolute error (ITAE), the integral of the absolute error (IAE), and the integral of the square of error (ISE), respectively defined by the integral equations ITAE = lim t(35) ITAE = lim \t r(t) y(t) dt (35) tf > 0 0 IAE = lim f r(t) y(t) dt (36) tf >CO 0 t! ISE = lim f(r(t) y(t))2 dt (37) tf >C 0 0 where y(t) is the plant output and tf is the extent of time over which the metric is computed. When the closed loop uses a PI controller, the output of the system exhibits a steady state offset es with respect to the original ramp characterized through the analytical expression e = r (38) KK, the derivation of which is given in the APPENDIX using Final Value Theorem. A modified or auxiliary ramp ra (t) is defined via the relationship ra (t) = r(t) e, (39) Figure 31 shows the ramp and the auxiliary ramp trajectories. The corresponding auxiliary error metrics for PI controller, modified from Eq. 34, Eq. 35 and Eq. 36 are given by tf ITAE= lim it r (t) y(t) dt (310) tf *co 0 tf AE = lim f (t) y(t) dt (311) tf ,co 0 0 tI ISE lim (r(t) y(t))2dt (312) tf "*o 0 where the ITAE, is the auxiliary integral of the timeweighted absolute error, the AE, is the auxiliary integral of the absolute error, and the ISE, is the integral of the square of error. It is to be noted that in the case of PI2 controllers, the value of e, is zero i.e., r, (t) = r(t). Hence, the ITAE, IAE and ISE error metrics for PI2 retain the original form of Eq. 35, Eq. 36, and Eq. 37, respectively, even when all computations are carried out using their auxiliary counterparts. The ITAE expression (Eq. 35) has been a popular choice for control parameter optimization, as it assigns less weight to errors occurring in the initial times and more weight to error at longer times. Error is defined as the closedloop feedback difference between the auxiliary set point and the output. This is traditionally a useful measure to adopt, as for a step response it is inevitable that there is a relatively large error during initial times, which needs to be given less significance compared to the error that is encountered at later times. The ITAE may not be the best metric for the problem in question, however. The desired trajectory of the output is the one which tracks the slope of the ramp without abrupt deviations in trajectory. The ITAE is forgiving of aggressive output values at initial times as it gives less significance to error at early times. Sometimes, it is desired to adopt a metric which gives an equal importance to errors occurring at initial times as well. From that perspective, the two other metrics are also used in this work for optimization purposes, namely, the IAE (Eq. 36) and the ISE (Eq. 37). A routine in MATLAB is written to simulate the plant output for a given controller with its parameters specified [22]. The simplex optimization routine is employed for tuning the control parameters. For a fixed value of plant parameters (gain, time constant and time delay), the parameters of the controller (proportional gain and integral action) are altered to minimize the (ITAE, IAE or ISE) value of the cost function. For a PI controller, where there is offset with respect to the ramp, an auxiliary ramp parallel to original one is constructed. The steady state offset is calculated analytically using Eq. 38. The three error metrics (ITAE, IAE and ISE) are minimized using the new ramp. These metrics are defined over an infinitely long time; however, for practical purposes the final time is chosen to be finite and defined by the formula tf =15 max(r,0) (313) The rationale behind this choice is that by this extent of final time, any reasonably performing controller should make the value of error significantly small. The optimized tuning parameters are nondimensionalized by combining with plant parameters, and plots of optimized tuning parameters were constructed. Time responses are constructed with the optimized values to verify the responses of the process with the controlling action incorporated. Peek analyzes the performance of three different configurations of PI2 controllers [21]. That study concludes that there is no significant difference in the performance of the three configurations. The transfer function for PI2 controller given in Eq. 32 is used in this study as it is the easiest to tune because it involves only two parameters, namely the proportional gain K, and the integralaction time constant r, . 3.3 Results and Discussion 3.3.1 Tuning Parameters of Controllers Tuning parameters are calculated for both controllers PI and PI2 using the ITAE, IAE, and ISE as the optimizing metric. The optimized controller parameters are nondimesionalised using the plant gain and time constant [17]. The resulting plots of KKc versus 0/r, and r /r versus 0/r are shown in Figures 33 and 34 for the PI2 and PI controllers. The tuning parameters obtained using ITAE, IAE and ISE as the metric are shown in the first, second, and third row of each figure, respectively. The plant parameters K, 0, and r are selected such that K and r range from 0.1 to 50. Twenty logarithmically equallyspaced points are considered for both K and r in their specified ranges. After each time constant r is defined, the values of the delay parameter 0 is set by defining the ratio of 0/r to range from 0.1 to 100, with 50 logarithmically equallyspaced points inside the range. For a fixed value of 0/r ratio, the value of 0 is computed from the value of r and of the fixed 0/r ratio. Hence, tuning parameters were obtained for 20,000 different plants for each controller and for each metric. The graphs in Figures 33 and 34 show that, in general, as the 0 r ratio increases, the value of the optimal KKc product and of the Tr/r ratio decrease. The value of the KKc product represents the proportional control action on the closed loop system, and it is expected to vary inversely to the 0 r ratio. In other words, the control action will be higher for smaller values of / r ratio, and smaller as 0/r increases. This is qualitatively reflected in Figures 33 and 34 for all three optimizing metrics and for the two controllers considered. On the same vein, the Tr/ ratio is indicative of integral action for the system and it is expected to behave analogously to the proportional control action. The integral action should be more aggressive for smaller values of / r ratio compared to higher values of the ratio. This is, indeed, observed from Figures 33 and 34. Leastsquare fits for the optimized control parameters are given in Tables 31 and 32 for the PI2 and PI controller, respectively. The least square fit relates the optimal KKc product with the 0 r ratios and the optimal r/r ratio with the 0/r ratio. If the KKc versus / r curve and/or Tr/r versus 0/r curve is significantly nonlinear, a break point at a certain value of 0/r ratio is identified and two leastsquare fits are presented for the same curve, one above and one below the 0 r ratio breakpoint value. The results are for 0/r values ranging between 101 and 102 only. 3.3.2 Comparison of the Performance of the PI and PI2 Controllers The performance of the PI and PI2 controllers is characterized for three different plants (plant parameters given in Table 33) using the optimal parameters prescribed by each of the three optimizing metrics. The time response curves for each plant are constructed, for both the controllers and the tuning parameters prescribed by the respective optimizing metric, to assess the timedomain performance of each tuning prescription. The proportional gain of the three plants was taken to be the same value, namely K = 1.0. Three different values of the 0 r ratio are taken from the domain of the values for 20,000 different plants. The ratio values of 0.1, 3.0 and 100 (minimum, middle and maximum of the range considered) are selected. Several combinations of 0/r can satisfy each value of the ratio. The value of r is selected such that it covers the domain of the different values of r selected for all the plants in this study. The values of 0.1, 1.9 and 50 are selected for r. From these values of r the value of 0 is computed for each value of the ratio. Figures 35, 36 and 37, illustrate the time responses of the three plants for the ITAE, IAE and ISE metrics, respectively, using PI2 and PI controllers. Each figure demonstrates the performance of PI2 versus PI controller for the three plants. When a PI2 controller is used, the output follows the original ramp, whereas with a PI controller an offset is introduced in relation to original ramp and the output follows the auxiliary ramp parallel to the original ramp. The original and auxiliary ramp are plotted in each time response curve as well. With increasing values of the 0 r ratio, the offset between the original and auxiliary ramp increases. This is expected as the offset is directly proportional to r, and inversely proportional to the product of KKc as shown by Eq. 38. With increasing value of 0/r, r, increases and the product ofKKc decreases, hence the offset increases. It is also observed that it takes longer for steady state to be reached with increasing values of the 0/r ratio. The output of each plant tracks the slope of the original ramp for each controller and every optimizing metric. As discussed earlier, there is offset with respect to the original ramp when PI controller is used and there is no offset when PI2 controller is used. Regardless of the fact of whether offset is introduced or not, as long as the output is parallel to the original ramp, the control objective is satisfied. The better performing controller is the one that reaches faster the original or auxiliary ramp, depending on the controller adopted, and with minimal transient values. It is observed that for all the three plants and optimizing metrics considered, PI controller performs better than PI2 controller. Output of each of the plant, on using PI controller, tracks the auxiliary ramp much faster compared to the output when PI2 controller is used. It is observed that the plant output is more oscillatory during transient time for ISE prescribed tuning compared to those obtained from the ITAE and IAE criteria. This is an expected result as in the ISE metric the square of the error is used. Even though both PI2 and PI controllers exhibit oscillatory behavior, the phenomenon is more prominent in the case of PI2 controller. From these observations, it is concluded that the PI controllers are better performing than PI2, regardless of the optimizing metric adopted to tune, for the three plants considered, and provided that the unavoidable resulting offset is acceptable to the user. 3.3.3 Comparison of Metrics After it is established that PI is a better performing controller, the next step is to identify the optimizing metric with which the PI controller gives the best performance. Time responses for the same three plants are constructed using the optimizing metrics. Figures 38 and 39 shows the time responses for the three metrics. For the sake of comparison, even though it is established that the PI is a better performing controller, time response curves are generated for the PI2 controller as well. Figure 38 shows the PI controller time responses and Figure 39 shows the PI controller time responses, for the three plants. Note that the offset introduced while using PI controller, is a function of the tuning parameters Kc and rt. Since the values of these parameters are different for each metric, the output using a PI controller follows a different auxiliary ramp, depending on which optimizing metric was used. All the auxiliary ramps, however, are parallel to the original ramp. It is observed that ISE is the worst optimizing metric, as the output of the plant is most oscillatory and has larger deviation from the auxiliary ramp as shown in Figure 38 and 39. Also, it takes longer in the case of the ISE to reach steady state. The other two metrics, ITAE and IAE, are quite close in their performance. The time response curves suggest that for tuning purposes the ITAE is a better metric for the PI controller and the IAE is better metric for the PI2 controller. For the PI controller, the IAE demonstrates more oscillatory output compared to the ITAE metric. Also, the output of the plants reaches steady state sooner when the ITAE is used as the optimizing metric compared to when the IAE is adopted. 3.3.4 Comparison of PI (ITAE) Controller with Literature Precedents After determining that the PI controller using the ITAE as the optimizing metric exhibits highly desirable performance, the next step is to compare its performance with controllers proposed in the literature. Peek suggests a PI2 controller with ITAE as the optimizing metric for tracking a ramp set point [21]. Though the objective is slightly different than the one in this study, the controller recommended by Peek does satisfy the control objective of this work [21]. Belanger and Luyben recommend a double integrator controller for tracking a ramp, which satisfies the control objective of this study as well [19]. Figure 310 shows the performance of the three different controllers for the three plants considered. It is observed that the PI controllers tuned using the ITAE metric gives the best results followed by the controllers recommended by Peek, and then by those proposed by Belanger and Luyben. As the 0/r ratio increases for the plants, it becomes increasingly obvious that the PI controller gives the best results. For the plant with the ratio 0/r = 100, the highest value, the PI controller makes the output reach steady state much faster and with least transients compared to the other two controllers. 3.3.5 Local Minima versus Global Minima Figure 311 shows the contour plots for the ITAE metric using PI controller for the three plants. It is observed from Figure 311 (A) that there is more than one minima for the ITAE metric and different values of control parameters Kc and r, for each minima. The goal is to obtain global minima for the optimizing metric and the corresponding optimal control parameters. If local minima is reached as opposed to global minima, that would potentially lead to scatter in the optimal control parameters curves as shown in Figures 33 and 34. Hence, the optimization work described here is conducted with care to avoid local minima results. The measure adopted consists of utilizing different initial guesses for each optimization routine execution, leading to the identification of different local minima, when they exist. The smallest of such minima is then accepted as the best approximation to the global minimum. Although this approach is neither rigorous nor exhaustive, it provides excellent practical results in the context of this study. 3.4 Conclusions Optimal tuning relationships for the PI and PI2 controllers using the optimizing metrics ITAE, IAE and ISE are presented for 20,000 firstorder plants with time delay. The plants considered have the 0/r ratio value varying from 101 to 102. The validation and comparison of the controllers performance is done by their deployment on three different plants. On the basis of results obtained, the PI controllers using ITAE as the optimizing metric are the best performing controllers for the purpose of tracking the slope of a ramp trajectory. They perform better than controllers proposed in prior literature. r 1 I* auxiliary ramp r. Figure 31. Ramp r and auxiliary ramp r, with constant slope, a. Figure 32. Closed loop transfer function representation of plant and controller. 01 10 101 102 V0/ Ir:  0/r 101 102 0 101 10 101 1C O/T B 10 0 .............. ................... +t ...,......:.i. +.9...... O/r D 100 o.. I _  .. . !. 9. " 0/r 100 " 10 102 101 4. 101 102 O/r Figure 33. The PI2 controllers optimal tuning parameters, using ITAE (A, B), IAE (C, D), and ISE (E, F) as the optimizing metric. 101 O/ 0/r 0 1 0 101 10 101 1C O/T 100 101 0/r Figure 34. The PI controllers optimal tuning parameters, using ITAE (A, B), IAE (C, D), and ISE (E, F) as the optimizing metric. ,1 101 O/ ii 101 0/r Table 31. The PI2 Criterion ITAE IAE ISE controllers optimized tuning parameters linear least square fit equations. Optimal PI2 Parameters 1.26(O/r)076 /r < 2.5 K 0.74(0/r)004 /r >2.5 0{.48(/r)050 0/T < 2.5 0 .74(0/r)092 O/r >2.5 S1.26(0/r)076 KK o0.78(0/r)006 0.48(0/r)050 T/7I 0.69(0/r)089 KKc ,. 1.58(/r)076 0.94(0/r) 0o8 0.45(O/r)052 0.65(0/r)0 88 <2.5 > 2.5 <2.5 > 2.5 <2.5 > 2.5 <2.5 > 2.5 Table 32. The PI controllers optimized tuning parameters linear least square fit equations. Criterion Optimal PI Parameters ITAE K 0.73(/r)073 O/r 2.5 c 0.41(0/r)006 O/r >2.5 0.82(/r) 013 0/r < 2.5 1.58(0/r) 088 O/T > 2.5 IAE 1.07(0/r)00 O/r < 2.5 c 0.57(0/r)008 O/r > 2.5 0.73(0/r) 026 O/r 2.5 1.42(O/r)088 /r > 2.5 ISE KK 1.40(/r)080 O/r < 2.5 C 0.74(/r) 008 O/r > 2.5 0 .75(0/r)036 O/r < 2.5 1.42(0/r)088 0/r > 2.5 Table 33. Values of the plant parameters used compare the performance of the controllers. Parameters Ratio Plant K r 0 0/r 1 1 0.1 0.01 0.1 2 1 1.9 5.6 3.0 3 1 50 50000 100 0 18 37 t x 104 I ' 56 75 0 1.9 3.8 t 7.5 x 104 Figure 35. The PI2 and PI controllers' ramp tracking and slope tracking performance using the optimal ITAE control parameters. 56 3 37 18 0 7.5 C 0 18 37 t x 104 I 1 56 75 0 1.9 3.8 t 7.5 x 104 Figure 36. The PI2 and PI controllers' ramp tracking and slope tracking performance using the optimal IAE control parameters. 56 3 37 18 0 7.5 C 0 18 37 t x 104 I 1 56 75 0 1.9 3.8 t 7.5 x 104 Figure 37. The PI2 and PI controllers' ramp tracking and slope tracking performance using the optimal ISE control parameters. 56 3 37 18 0 7.5 C 37L 0 18 37 56 75 t x 104 7.5 C ITAE 5 6 IAE 5.6 ISE . .3.8 R am p ... S K= 1.9 = 50 7 = 5000 0 0 1.9 3.8 5.6 7.5 t x 104 Figure 38. The ITAE, IAE, and ISE metrics comparison for three plants, using the PI controllers A), B), and C). ITAE . IAE ..  ISE .. Ramp K= S=1.9 / 0 = 5.6 0 18 37 t x 104 I 1 56 75 0 1.9 3.8 t 7.5 x 104 Figure 39. The ITAE, IAE, and ISE metrics comparison for three plants, using the PI2 controllers A), B), and C). 56 3 37 18 0 7.5 C 0.4 0.8 t x 104 7.5 F 5.6 . Pec .. Rai S3.8 Au: 1.9 , L./ 0 0 1.9 3.8 5.6 t Figure 310. The PI controllers tuned using the ITAE metric compared with Belanger and Luyben and Peek's controllers for three plants A), B), and C). 1.2 1.5 7.5 x 104 3.5 3 8 0.4031  0.403 0.4029 0.2908 0.3175 S0.3174 0.3174 5.0454 10 1/t i 0.2908 1 5.0505 l/1 0.2908 5.0556 x 104 Figure 311. Contour plots for the PI controller tuned using the ITAE metric for the three plants A), B), and C). CHAPTER 4 GENERALIZED PREDICTIVE CONTROL FOR FUEL CELLS 4.1 Introduction A generalized predictive control (GPC) strategy is designed and implemented for a polymer electrolyte membrane (PEM) fuel cell system. It is vital for efficient performance of the fuel cell to ensure the robust and precise control of the performance variable which is oxygen excess ratio. A review of prior control strategies developed for this fuel cell system, is presented and compared to the new GPC scheme proposed. The performance of the different controllers is evaluated in the case where the model available for design suffers from uncertainty. 4.2 Fuel Cell System Background Fuel cell are electrochemical devices that directly convert the chemical energy of gaseous reactants to electrical energy. They are widely considered as an alternative to fossil fuels which are limited in supply. For a typical fuel cell, water and heat are byproducts generated as a result of operation. The reactants needed are hydrogen and oxygen. Both of these reactants are widely available, and a proliferation of applications based on these fuels would tremendously reduce our dependence on fossil fuels. This is an additional motivation for developing and engineering fuel cell system as it is friendly to the environment. Fuelcell based automobiles have no harmful emissions, such as CO2 which combustionengine automobiles contribute significantly to the environment [23]. Fuel cells are an efficient and clean source of energy production. William R. Grove discovered the principle of operation of fuel cells in 1839 [24]. From a classical standpoint, a fuel cell is comprised of two electrodes with an electrolyte located between them. The electrolyte has the special property that it allows only protons (positively charged hydrogen atoms) to pass through it. In contrast, the membrane does not allow electrons to pass through. Hydrogen gas passes over the anode electrode, and with the assistance of a catalyst, breaks down into protons and electrons. The protons selectively pass through the membrane to reach opposing cathode electrode. The membrane is an electronic insulator. The membrane is comprised of fluorocarbon chain to which the sulfonic acid groups are attached. On hydration of the membrane the hydrogen ions become mobile. The electrons flow through an external circuit, creating a current flow. An oxygen flow passes over the cathode and combines with the protons and electrons to generate water. The reaction at the anode is 2H2 4H+ + 4e (41) and at the cathode is 02 + 4H+ + 4e 2H20 (42) hence, the overall reaction is 2H2 + 02 2H20 (43) Several kinds of fuel cell designs have been developed and are currently being studied [25 30]. A schematic diagram of an automotive fuel cell system including the structural relationships among the input, outputs, and disturbance signals is given in Figure 41. A compressor and pressurized hydrogen tank are used to provide the reactants oxygen and hydrogen, respectively [31]. The compressor plays a crucial role as it ensures that the desired air flow rate reaches the cathode based on power demands. The supply and return manifold models based on thermodynamic consideration provide information about desired air flow rate needed. A nonlinear curve fitting method is used to describe the compressor behavior [31]. The net power delivered by the fuel cell system is the difference between the power generated and power consumed to run the compressor motor to deliver a particular air flow rate. An excess amount of air flow provided to the cathode is referred to as the oxygen excess ratio Al2 which is defined as the ratio of the rate of oxygen supplied to rate of oxygen consumed. With increase in oxygen excess ratio, resulting in high oxygen partial pressure, there is increase in power delivered. However, it is at the cost of increased power consumption by the compressor to deliver a higher air flow rate. Beyond a particular value of the oxygen excess ratio there is loss in net power as a result of increased power consumption by the compressor. It has been shown in literature that having oxygen excess ratio in the vicinity of 2, the fuel cell system delivers highest net power. A humidifier, in fuel cell system, is used to add water to the reactants to avoid dehydration of the membrane. A water separator is used to extract water from the air, leaving the fuel cell stack, which is recycled back to the humidifier via the water tank. The voltage generated by the fuel cell needs to be conditioned before it is fed to the traction motor. Appropriate usage of an external battery with the fuel cell power supply helps in minimizing the transient responses and delivering better system efficiency. Polymer electrolyte membrane (PEM) fuel cells, also referred to as proton exchange membrane fuel cells, are recognized particularly promising for utilization in automobiles as a substitute for the internal combustion engine. This is because of the fact that the PEM fuel cells have high power density, long cell life, low corrosion, and use a solid electrolyte. For effective utilization of fuel cell technology, it is vital that that the fuel cell system is accurately understood, monitored and that its process variables be held under tight control for various operating conditions. From a control engineering perspective the fuel cell can be divided into four subsystems, namely (1) supply of the reactants air and hydrogen, (2) humidification of the reactants and of the membrane, (3) heat management, and (4) power management. The model used in this study assumes a perfect humidifier and coolers for the reactants and the membrane. The model assumes a perfect power management system which controls the power drawn from the fuel cell stack. A fast proportional controller is implemented on the hydrogen flow that tracks cathode pressure [31]. This reduces the control problem to regulating the air supply, as the oxygen level varies on the cathode side, due to varying power demands. 4.3 Objectives of the Research The objectives of this study are the following: 1. Propose a systematic design solution for treating the of fuelcell control problem of regulating the oxygen excess ratio (performance variable) at a value of 2 by synthesizing a GPC strategy. 2. Develop a systematic GPC design procedure for a fuel cell model in the scenario where all states are measurable. 3. Develop a systematic GPC design procedure for the fuel cell model when selected states are not measurable by incorporating an observer in the controller. 4. Evaluate the robustness of the GPC controller with respect to model uncertainty. 5. Rederive and correct, as needed, linear models obtained from nonlinear formulations proposed in the prior literature for the fuelcell model used for this study [31]. 6. Retune and redesign, as needed, all modelbased controllers proposed in the prior literature [31] to take into account the corrected linear models [31]. 7. Compare by means of simulations the performance of the different controllers proposed in prior literature [31] with the GPC strategy. 4.4 Fuel Cell Model The fuel cell model proposed by Pukrushpan et al. is the basis for this study [31]. The state equation for the model of a fuel cell system is of the form x = f(x,u,w) z = g(x,u, w) y = h(x, u, w) (44) (45) (46) where x represents the states of the fuel cell system, u is the input, w is the disturbance, and z and y are the outputs. The state vector is given by the expression x=[ 0m2 mH2 2 0c Psm sm mw,n rm ]T (47) where the vector elements are the mass of oxygen m02 in cathode volume, the mass of hydrogen mH2 inside the anode volume, the mass of nitrogen m,2 in cathode volume, the rotational speed of the compressor o)c the pressure of the supply manifold ps,, the mass of air in the supply manifold mm, the mass of water at anode m ,a, and the pressure of the return manifold prm. The outputs used as the performance variables are organized in the vector z = [eP, A02 ] (48) where e = P"" pdel is the difference between the desired power P"' and the actual power delivered pd", and A02 is the oxygen excess ratio. The control objective in this study is confined to regulating the value of A2 at the desired value Aet = 2 [31]. Three additional measured outputs are organized in the vector y=[WcP psm Vst ] (49) where the vector elements are the mass flow rate of the air from the compressor Wc, the pressure of the supply manifold ps and the voltage of the stack v,,. The control input is given by U = Vcm (410) where vcm is the voltage signal sent to the compressor which in turn delivers a corresponding flow rate of air to the cathode. Finally, the disturbance w= It (411) is the stack current I,t. The nonlinear model described in Eqs. 44 46 and shown in Figure 42a is linearized about a nominal operating point at which the fuel cell model generates a net power of P"d =40kW and sustains an oxygen excess ratio o2 = 2. These conditions are realized when I,, = 191 A and v, = 164 V. The corresponding values of the states are x =[ moi2 H2 W cp m sn m iwan Prm (4 [2.0e 3 5.6e 4 1.3e 2 8.2e3 2.2e5 3.8e 2 1.le 3 1.8e5] The physical units of the states x are [kgs kgs kgs rad/s pascal kgs kgs pascal] (411b) respectively. Hence the nominal operating point is given by the input u = 164V, disturbance w = 191A, performance variable z2 = 2, and vector x as described by Eq. 41 la. Two linearization cases considered in this study are the following: 1. The direct linearization of the fuel cell model Eqs. 44 46 and shown in Figure 4 2a [31]. 2. The linearization of a combined system consisting of a static feedforward controller, connected to the fuel cell model Eqs. 44 46 shown in Figure 42b. The static feedforward control law used in case 2 is described in Section 4.5.2.1. The state, input, output, and feedthrough matrices of the resulting linear timeinvariant models for cases 1 and 2 are shown in Figures 43 and 44, respectively. Through personal communication with the members of the research group at University of Michigan we learned that the matrices reported in [31] are affected by errors, hence these matrices are rederived and found to differ slightly from the ones given by Pukrushpan et al. [31] 4.5 Literature Precedents fo Fuel Cell Control Designs Pukrushpan et al. investigate the design and application of two main control strategies along with 2 major variations of each of these strategies, as indicated in the following list [31]: 1. Feedforward strategy a. Static feedforward (sFF) b. Dynamic feedforward (dFF) 2. Combination of static feedforward strategy with optimal feedback control a. Linear quadratic regulator (LQR) b. Linear quadratic gaussian observer (LQG) in combination with LQR feedback. A succinct discussion of these strategies is given in the ensuing sections 4.5.2 Feedforward Strategy The two feedforward control strategies proposed in the literature are a static feedforward (sFF) and a dynamic feedforward (dFF) scheme. The sFF controller is derived from simulations and using the results of substantial experimental work. The dFF controller is based on a linear model, and thus its performance on the nonlinear model is dependent on the non linear model's proximity to the nominal operating point. The models used for control design are discussed in greater detail in the next subsections. 4.5.2.1 Static feedforward controller Pukrushpan et al. propose the sFF control law [31] u = 123w +378 (412) where the stack current w is the measured disturbance impacting the fuel cell model. The control input u is the voltage vcm applied to the compressor. A schematic of the sFF controller is shown in Figure 45. The derivation of Eq. 412 consists of first seeking a function relating the disturbance I,, to the required air mass flow rate Wcp, in such a fashion that the flow rate, achieved by invoking thermodynamics principles, negates the effect of the disturbance on the performance variable A0 A resulting static function is obtained by means of simulations and experimental work which corelates the control input vcm to the required air mass flow rate Wp for a particular value of disturbance I,, maintaining the desired value of the performance variable A0 This static function is implemented as a static sFF controller in the form of Eq. 4 12. 4.5.2.2 Dynamic feedforward controller Pukrushpan et al. propose the dynamic feedforward (dFF) control law du = Kudw (413) where 9u = u u and 6w = w w are the deviation values from the nominal control input u and the nominal disturbance w respectively, and Kuw is the transfer function K ideal Kw = (414) (1+ )(1+)(1+) a1 a2 a3 where s is the Laplace variable, a1, a2, and a3 are filter constants, and K'de"l = G ,,G2 where G2,, and G,2, are transfer functions describing the map Z, = Gwt + G2,,ut (415) where z2 = z2 Z2 is the deviation of the performance variable (oxygen excess ratio) from its nominal value z2 [31]. The schematic and transferfunction representations of the dynamic feedforward controller dFF are given in Figure 46. The derivation of the dFF control transfer function begins by considering the transfer function &Z2 = GwaW + Gz,, K w, (416) obtained by substituting Eq. 413 into Eq. 415. The expression 416 should become identically zero for an effective dFF controller K,,. The expression K,:al =G1 ,G (417) is obtained from Eq. 416 after equating 2 = 0. However, G is not a proper function. Low pass filters are added to implement a causal controller, resulting in 1I K G z2uG2w (418) (1+ )(1+ )(1+) al a2 a3 The authors of this derivation propose the filterconstant values a, = 80, a2 = 120, and a3 =120 [31]. 4.5.3 Combination of Static Feedforward with Optimal Feedback Controllers Feedforward controllers lack robustness to disturbance and model variations. To address this issue, feedback controllers are added in conjunction with feedforward controllers. The following two scenarios are considered for feedback control design, in conjunction with static feedforward control [3234]: a. The performance variable is measurable. b. The performance variable is not measurable, leading to the introduction of a state observer to estimate its value. The model used for designing the optimal feedback controller is defined in such a fashion that it includes the static feedforward control relationship. A schematic of the resulting feedforward and feedback control is shown in Figure 47. 4.5.3.1 Case where the performance variable is measurable Pukrushpan et al. propose the linear quadratic regulator (LQR) feedback control law: u= up KPx KIq (419) where up is a precompensator, iu = u u and & = x x are the deviation values of the feedback control input u and the state vector x from their nominal points u and x, respectively, and q is the integral defined through the differential equation ,Yeq (420) S= y,1" y, (420) where y1"e = Wc" is the analytically calculated value of the air mass flow rate needed for a particular value of the disturbance It to attain the desired oxygen excess ratio, and y, = Wcp is the measured value of the air mass flow rate. In the control law given by Eq. 419, gain matrices K, and KI refer to the optimal gains resulting from minimizing the quadratic cost function J= j(x'C'T C& +qTQq+ uT'R5u)dt (421) 0 where, Q,, Q1, and R are weighting matrices for the performance variable, state q and control input u, respectively. The term C, refers to the second row of the C matrix, given in Figure 44 describing the linear state space model. Note that the second row is associated with the performance variable Ai The precompensator up in the feedback control law Eq. 419 ,used to take into account the disturbance effect on the performance variable, is given by up =[ (A B,K,) B, [D2" C2 (A B,Kp)1 B ] (422) where A, B,,B and D Z2 are elements of the linear state space model given in Figure 44. In particular Bw and B, are the first and second column of the B matrix, respectively and D 2W is the second element of the first column of the D matrix. A schematic diagram of the sFF scheme supplemented with the feedback controller is shown in Figure 47. The optimal linear quadratic regulator 419 is implemented to prescribe part of the feedback control input. The additional state q, as described in Eq. 420, is introduced to minimize offset. Using Qz = 10000 and Q, = 0.001 in Eq. 421 and minimizing the cost function 421 the optimal values of controller gains obtained are K =[23.03 1.4e17 25.46 5.2e5 2.0e6 2.64 1.9e17 1.le6] (423) and K, = 0.001 (424) Note that 423 and 424 differ from the values reported by Pukrushpan et al. [31]. This discrepancy is a consequence of the difference, reported in Section 4.4, with the state matrices used by Pukrushpan et al.. 4.5.3.2 Case where the performance variable is not measurable Optimal observer. When selected states are not measurable, Pukrushpan et al. propose the following modification of the control law 419: u = up K,& Kq (425) where 5x = x x is the deviation of an estimated state vector k from its nominal value x The estimated state is computed from the Kalman based observer S= A + Bu + L(y ) (426) 5 = C + Du (427) where L is the optimal observer gain calculated using the linear quadratic gaussian (LQG) method, and 5 is the estimated values of the measured outputs [31]. Three measurable outputs, namely the compressor mass flow rate W the pressure of the supply manifold ps, and the stack voltage v,,, are used as inputs to the observer to estimate the value of the states [31]. The optimal gain of the Kalman observer is 7.43e015 12.22e009 417.04e015 18.54e015 31.69e009 2.18e+000 2.54e015 25.06e009 349.02e015 578.06e012 4.22e003 49.54e009 L = (428) 332.44e 009 460.51e 003 4.60e 006 (4 78.31e015 124.80e009 143.89e015 270.13e018 9.95e009 1.85e+000 486.77e 009 527.22e 003 5.97e 006 Note that 428 differs from the value reported by Pukrushpan et al. [31]. This is a consequence of using different state matrices than what were used by Pukrushpan et al. as explained in Section 4.4. In addition, we also made an approximation to the noise variance matrices used for LQG design, since only partial information is available in [31], hence adding another source of discrepancy. 4.6 Generalized Predictive Control The generalized predictive control (GPC) law is given by the discretetime law [35, 36] Au(t) = k T (m(t) yOL(t)) (429) where k is the GPC control gain vector, m(t) = A"' is the set point, yOL (t) is the vector of the constant forcing values of the performance variable, u is the prescribed control input, and the symbol A represents the difference operator. The vectors m(t), yOL (t),and the gain k have R elements each, representing their respective values upto the prediction horizon R. The process involved for obtaining the GPC control law Eq. 429 is discussed in the following sections. Generalized predictive control design. The first step leading to the derivation ofEq. 4 29 is to acquire a DARMA deterministicc autoregressive moving average) model from continuous state space model [3538]. To accomplish this, a discrete version of the linear state space model is obtained. The sampling period Tis chosen such that about 1020 samples are taken during the transient time. The ztransform is performed on the discrete model and the z variable is replaced with the q forwardshift operator. The numerator and the denominator are divided with the highest power of q to obtain a denominator polynomial A and a numerator polynomial B as functions of the backward shift operator q 1. Polynomial B is defined such that it does not include the unit time delay introduced by sample and hold operation. Finally, the DARMA model for the fuel cell model is obtained in the form A(q')y(t) = B(q')u(t 1) (430) A(q ')= 1+aq' +.. +a,q' +.. +a, q (431) B(q') = b, +bq' +..+bq +..+b qb (432) and where A and B are polynomials in the backward shift operator q'1. The factors a, and b, are the coefficients of the powers q' in polynomials A and B, respectively. The symbol y represents the output (performance variable), i.e. the oxygen excess ratio in the case of the fuel cell, and u is the control input, i.e., the voltage supplied to the compressor motor Vcm to deliver the desired air mass flow rate [29, 30]. Second, a predictor for the performance variable is designed using the Diophantine equations 1= EAA(q )+q F,(q ') (433) where E, = 1+eq 1 +..+ejq J +..+e, q (434) S= f + fq +..+fjq +..+fq "n (435) i = 1,2,..,R (436) and where R is the prediction horizon, A = 1 q and E and F are polynomials in powers of the backward shift operator q'1. The factors e, andf are the coefficients of the powers q1 in polynomials E, and F,, respectively [30, 32]. Next, multiplying both sides of the DARMA model (Eq. 430) by the polynomial q'E,A and invoking Eqs. 433436 yields y(t + i) = E, (q1)B(q1)Au(t + i 1) + F (q')y(t) (437) A where y(t + i) is the predicted value of the output at the instant t + i. Next, the constant forcing value of the performance variable is calculated. The product of polynomials E, and B can be decomposed in the form E, (q )B(q ) = G,(q )+ q 'F,(q ) (438) where G, and F, are operator polynomials given by F, (q) = Yo +Y1ql +..+yJqJ +..+nb + l + (439) G,(q)= go +glql1 +..+g,q +..+g lq '+ (440) and where y, and g, are the coefficients of powers of q1 in the respective polynomials. Substituting Eq. 438 into Eq. 437 yields y(t +i)= G,(q)Au(t + i 1)+,(q)Au(t 1)+ F(q )y(t) (441) A constant forcing Au(t + i 1) = 0 for i = 1,2,.., produces a constantforcing output in Eq. 441 of the form yOL (t + ) = ,(q1)Au(t 1) + F,(q1)y(t) (442) where OL (t + i) represents the constantforcing response of y at instants t + i, i= 1,2,..,. Third, an objective/costfunction J is established and minimized with respect to the controlinput increment Au to obtain the GPC control law. The appropriate cost function adopted is defined by J(Au)= (m(t) y(t))' a(m(t) y(t)) + AuTAu (443) where o is a weighting matrix, m and y are vectors of future set point and predicted future output respectively, u is the vector of control inputs, and h is the weighting matrix for Au. The vectors m and y have R elements each, representing their respective future values upto the prediction horizon R. Vectors u and Au has L elements, where L is the control horizon defined as the instant where the control design specifies that u(t+i)=u(t+L ), i=L,L +,...,R 1 (443a) In the case of the fuel cell, the performance variable y is composed of a contribution from the disturbance variable d (namely, the stack current Ist), and a contribution from the manipulated variable u (namely, the voltage to the compressor v,, ) leading to its definition y = d+ = yOL + GdAd+ yL +G Ai/ (444) where Yd is the component of the performance variable contributed by disturbance d. Note that the notation for the disturbance is changed from w Sections 4.4 to 4.6 to din Sections 4.7 which involve the GPC strategy. Signal y, is the component of the performance variable y contributed by manipulated variable u. The terms yOL and yOL represent the constantforcing values of Yd and y, respectively. The terms Gd and G, are the associated dynamic matrix polynomials for Yd and y, respectively. The termGdAd is set as GdAd = 0, as Ad = 0 since d is a disturbance and its unknown future values are assumed to be equal to the current value, i.e., d(t,) is assumed to be constant and equal to d(t), i = 1,2,... This reduces Eq. 444 y = y+y = yOL +yOL +G At, (445) Hence, since y = yL when Au = 0, it follows that yO = YdL y (446) Substituting the value ofy described by Eq. 445 in the cost function Eq. 443 yields J(Au)= (m(t) yOL(t) GAu)TO (m(t) yOL(t) GAu) + AuTAu (447) which when minimized with respect to Au yields Au = (GuoTGu) + I)(GTuT)X (448) where X = m(t) yOL(t) The control law is extracted from Eq. 448, takes the form Au(t)= kTX (449) where kT is the first row of the matrix (G TGo) + I) (GuoT), and u(t) = u(t 1)+kTX by the GPC algorithm. Eq. 449 can be easily rewritten via a simple substitution of factor X to reduce to Eq. 429. This completes the derivation of the GPC control law Eq. 429. For stability analysis and simulation purpose it is important to develop a closedloop transfer function for the GPC loop. The GPC control law described by Eq. 449 in summation form is given by R Au(t) = k,[m, yr o ] (450) where the subscript i denotes the values of the respective variables at the time instance t+i. Then using Eq. 442 ,(t + i) = Fd, (q1)Ad(t 1)+ F, (q1)yd() (451) S(t + i) = (q)Au(t 1) + F, (q')y, (t) (452) Substituting the values of y, and y, given by the Eqs. 451 and 452 into the GPC control law given by Eq. 450 and rearranging terms yields R R R (I+q kjF,,)Au(t) = k q'm, A(ql Y krd,,)d(t) =1 R1 ,1 R (453) (_kFd, (q 1)yd(t)(_ k qF,, )y (t) z=1 z1i R Let R, =(l+q k, F,,)A (454) i=1 R Rd =A(q k d,,) (455) i=1 T = kq' (456) i=1 R Sd = IkFd,,(q1) (457) 1=1 R and S= = kF,,(q1) (458) i=1 Inserting in Eqs. 454 458 into Eq. 453 yields Ruu(t) = Tm(t) Rdd(t) Sdyd (t) Su yu (t) (459) Since y, (t) = y(t) yd (t) (460) Eq. 459 can be rewritten as Ruu(t) = Tm(t) Rdd(t) Sdy ( (t) (t) yd (t)) (461) which after rearranging terms results in Ru(t) = Tm(t) Rdd(t) (Sd S. )yd (t) Sy(t) (462) Also, since Yd 1 d d(t) (463) Ad Eq. 461 can be expressed in the form B, R u(t) = Tm(t) Rdd(t) (Sd S )q 1 d d(t) Sy(t) (464) Ad which after a trivial series of algebraic operations can be written as AdRu(t) = AdTm(t)(AdRd +(Sd S )q'Bd)d(t) AdSy(t) (465) Now, let R = AdR (466) T= AdTm(t) (467) S = AdRd + (Sd S,)qBd (468) and S, = AdSu (469) Substituting Eqs. 466 469 into Eq. 465 yields Ru(t) = Tm(t) Sdd(t) S,y(t) (470) which finally leads to the closedloop established by the GPC control strategy Ru(t) = Tm(t) S(t) (471) where S= (Sd Sy) (472) and y(t) ) (473) a y(t)) Figure 48 shows a closed loop schematic of the GPC controller for the augmented model which incorporates the disturbance. To obtain the closedloop transfer functions relating the input u(t) and the outputy(t) to the set point m(t) and the disturbance d(t), first consider the openloop relationships yd(t) = Bd d(t) (474) Ad B and y (t) = q1 u(t) (475) A Let q'Bd = Bd and q'B = B,. Substituting Eqs. 474 and 475 into Eq. 459 and rearranging terms yields u(t) r A.R T + (t) Ad) ARd + SdBd d(t) (476) A, R, + S,) Ad A, R, + S, )B which is the closed loop transfer function from the set point m(t) and the disturbance d(t) to the control input u(t). Inserting into Eq. 465 the expression y(t)= KBd d(t) + \ u2(t)A (477) obtained from Eqs. 445, 474, and 475 yields y(t)= A SBT m(t ) + B(A R B(AdR d(t) (478) AR+SB Ad(A R, +SB) which is the closed loop transfer function from the set point m(t) and the disturbance d(t) to the output y(t). 4.7 Battery of Observers In this scheme, a parallel battery of four Kalman based observers designed using four different nominal points, respectively, are deployed for estimating the value of the controlled variable where the equations for each observer i is given by Ix = A I + Bu+L (y ) (479) y, = C,,, + D,u (480) where i, and y, are the estimated state vector and estimated output vectors from observer i, respectively. The state matrices A,, B C, and D, correspond to the nominal point for which the observer i is designed. The observer which delivers the best performance is used for providing its estimated value of the controlled variable to the GPC controller. The observers are based on linear models derived from the nonlinear model at different nominal values of the disturbance. The observer is defined by Eqs. 479 480. The observer gain L is computed from LQG principles as defined in previous section for each of the observers. The determination of the best performing observer is done by comparing the norm of the error in the measured outputs produced by each observer. The errornorm N, for observer i is defined by N, = e', +e,, +e3,, (481) where e, = W (482) e2, = Psm sm, (483) e3, = vt ~t, (484) are the errors defined by the difference between the measured value of the outputs (Wc psm, and v, ) from the nonlinear model and their estimated values (Wp, ,, sm and t,,,) from the observer i, respectively. The observer that delivers the least value of the norm is selected as the best current observer and is implemented to deliver its estimated value of the controlled variable to the controller. The nominal value of the disturbances for which the observers are designed are w=100, 191, 125, 22 A. The values are chosen to account for the range of disturbance values that the fuel cell model experiences. 4.8 Simulation Studies and Results This section presents simulations results obtained from various control strategies implemented on the nonlinear fuel cell model. The MATLAB and SIMULINK software computational tools are used for simulation purposes and for assessing the controllers' performance [22]. The nonlinear fuel cell model created by Pukrushpan et al., in the SIMULINK environment is used for simulations [31]. The performance of the generalized predictive control strategy is presented first. The scenario where the performance variable is measured is discussed, followed by the case when it is not measured. In the latter case an observer is incorporated in the control design to estimate the unmeasured value. The robustness of the GPC control design is assessed by examining its performance on an altered fuel cell model obtained by modifying a parameter of the original nonlinear model. Finally, a comparison of the GPC controller's performance with that of controllers proposed in literature is conducted. For all the simulation studies carried out in this work, the disturbance d ( i.e., the stack current It ) profile shown in Figure 49 is used, which is identical to the one used by Pukrushpan et al [31]. The yaxis denotes the value of the disturbance and the xaxis is time. The trajectory of the disturbance is implemented as step changes. In Figures 410 419, showing the simulation results for various control configurations, the yaxis denotes the value of the performance variable oxygen excess ratio Al2 and the xaxis is depicts the time t. The dotted black line indicates the value of the desired set point i.e., m(t) = 2 for the performance variable. 4.8.1 Generalized Predictive Control Results As discussed in Section 4.4, the GPC approach is implemented on the fuel cell model to attain the desired objective of regulating the performance variable, i.e., keeping the oxygen excess ratio at a value of 2. The following scenarios are considered for evaluating the performance of the GPC strategy: 1. The controlled variable is measured. 2. The controlled variable is not measured. A battery of observers based on Kalman filtering is included to estimate the value of performance variable. 3. The design model is uncertain. The robustness of the GPC controllers under model uncertainty for cases (1) and (2) above is evaluated by examining the performance on a modified fuel cell model obtained by varying a parameter of the original model. 4. The control is benchmarked against literature precedents. The performance of the GPC controller is compared to control designs proposed for the fuel cell model in prior literature. 4.8.1.1 Case where the performance variable is measured The performance of the GPC controller when the controlled variable is measured is discussed in this section. Since in this case the controlled variable (oxygen excess ratio) is measured, its value is directly available to the GPC controller. The GPC algorithm described in Section 4.6 dynamically prescribes the value of the control input (the voltage to the compressor motor vt) which in turn delivers the appropriate compressor mass flow rate (Wcp) to maintain the value of the controlled variable at the desired setpoint value of 2. Figure 410 shows the results of a simulation study where the GPC control design is implemented on the nonlinear fuel cell model. The xaxis in Figure 410 denotes the time and the yaxis depicts the value of the controlled variable, i.e., the oxygen excess ratio measured from the nonlinear fuel cell. The initial value of states, disturbance and control input for the model are chosen as discussed in Section 4.4. The dotted line indicates the set point of the controlled variable, which in this study is constant and equal to 2. The black solid line depicts the measured value of the controlled variable under conditions where the fuel cell is subjected to the disturbance profile (values of stack current It) shown in Figure 49 and the input is prescribed by the GPC algorithm. Figure 410 shows that the GPC controller successfully returns the controlled variable at the desired value without steadystate offset. Only small deviations from the set point are observed during transients. The spikes in the signal plot reflect instances where the disturbance suddenly changes values in a stepwise fashion, as documented in Figure 49 and hence they are unavoidable. A value of the controlled variable below 2 indicates that a positive step change in disturbance took place, with the effect of depleting the oxygen concentration in the cathode and consequently reducing the oxygen excess ratio before the control action could remedy the problem. Analogously, a transient value of the controlled variable lying above 2 reflects an opposite scenario of an occurrence of a negative step change in disturbance. The asymptotic offsetfree behavior observed is expected as the GPC approach incorporates an integrator as an essential component of its structure. 4.8.1.2 Performance variable not measured Battery of four observers. Figure 411 shows the time response of the performance variable when a battery of four observers are included in the controller to account for the situation where the controlled variable is not measured. The observer design steps are discussed in Section 4.7. In Figure 411, the solid line indicates the performance variable trajectory as a function of time. The dotted line is the desired set point value of 2 for the performance variable. The measured outputs from the fuel cell model are (1) the compressor mass flow rate Wp, (2) the stack voltage vt ,and (3) the supply manifold pressurepsm. These measurements are provided to the observers to estimate the value of the performance variable. The observers incorporated in the control strategy provide estimates for the performance variable and the norm error value that they generate. According to the switching algorithm the best performing observer, that generates the least value of the normerror value, feeds its estimated value of oxygen excess ratio to the GPC controller. The GPC strategy attempts to regulate the value of the estimated performance variable at 2. Excellent results are obtained by using a battery of observers which is evident by zerooffset/near zerooffset of the performance variable value from the setpoint. The normerror values of the four observers are shown in Figure 412. The switching pattern indicating the observer used for estimating the value of oygen excess ratio is shown in Figure 413. The value of the control input, the voltage prescribe to the compressor motor, is shown in Figure 414. The deviation from the estimated and measured three outputs for the observers 1, 2, 3, and 4 are shown in Figures 415, 416, 417, and 418, respectively. Battery of three observers. An identical strategy described above is used for observer design except that a battery of three observers instead of four is implemented. Observer 3 is excluded from the battery of observers. Figure 419 shows the time response of the performance variable. The solid line indicates the performance variable trajectory as a function of time. The dotted line is the desired set point value of 2 for the performance variable. A deterioration in control performance is observed compared to the case when four observers are used. This result is expected as there are fewer observes available that are designed in the vicinity of the different operating points of the nonlinear fuel cell model. As a result, the estimated value of the performance variable provided by the observers do not coincide with the actual value resulting in some deviation from the set point because the observer design is based on a linear model. The GPC strategy in turn attempts to regulate the erroneous estimated value of the performance variable and consequently a certain degree of deterioration in controller performance is expected. Battery of two observers. Figure 420 shows the time response of the performance variable when only two observers are employed in battery of observers. The solid line indicates the performance variable trajectory as a function of time. The dotted line is the desired set point value of 2 for the performance variable. Observers 1 and 3 are excluded from the battery of observers. A further deterioration in controller performance is noted compared to the case where three observers are employed. This is an expected result as discussed earlier. Employing only one observer. Figure 421 shows the time response of the performance variable when only observer 2 is used to estimate the value of the performance variable. The solid line indicates the performance variable trajectory as a function of time and the dotted line is the set point value of 2. This scenario exhibits the worst controller performance compared to the cases where batteries of observers having 2, 3, or 4 observers are used. The GPC controller with one observer supplement fails to deliver offset free regulation. This is an expected result, as discussed earlier. 4.8.2 The GPC Approach Evaluated for Robustness For robustness considerations, the GPC strategy was evaluated for model uncertainty. The GPC controller designed for the original fuel cell model is used on a modified model which is acquired by changing a parameter of the original model. More specifically, the return manifold throttle area changed from 0.0020 m2 to 0.0023 m2. The two scenarios considered for robustness analysis are 1. The performance variable is measured. 2. The performance variable is not measured. An LQG observer is included in the control strategy to estimate the value of performance variable. 4.8.2.1 Case where the performance variable is measured Figure 422 shows the time response of the performance variable from the modified nonlinear fuel cell model when the GPC control law is designed for the original model. The solid black line indicates the time response of the performance variable from the modified nonlinear fuel cell model. The intention is to examine the performance of the GPC controller in a scenario of model uncertainty. In this case the performance variable is measured and is fed as a direct input to the GPC controller. The simulation conditions are identical to the scenario discussed in Section 4.7.1.1, except that the modified non linear model is used. As observed from Figure 422, the GPC controller displays robustness to model uncertainty because it ensures effective rejection of the effect of different values of disturbance, and produces offsetfree steadystate responses. The robustness of the GPC algorithm to model uncertainty is a key advantage of the controller. Even though the GPC strategy is designed for the original model, the feedback control of the performance variable produced by the modified model gives the controller the opportunity to make the necessary changes to minimize deviations from the set point. There is zero offset as the integrator in the GPC controller adjusts its output appropriately to eliminate the error of the performance variable with respect to the set point. 4.8.2.2 Case where the performance variable is not measured Figure 423 shows the time response of the performance variable from the modified nonlinear fuel cell model when it cannot be directly measured. An LQG observer is incorporated in the control design to estimate the value of the performance variable by utilizing the values of the measured outputs (1) compressor mass flow rate Wp, (2) stack voltage vt, and (3) pressure of supply manifold ps. The simulation conditions are identical to those given in Section 4.7.1.2, except that the altered model is used. The GPC controller and LQG observer designed for the original fuel cell model are used. The objective is to study the controller performance for model uncertainty when the performance variable is not directly measurable. The solid line is the time response of the performance variable from the modified model. Figure 423 shows that when the LQG observer in conjunction with the GPC controller is applied to the modified model, significant degradation in performance is observed compared to the case when the performance variable is directly measured. This can be attributed to the fact that the GPC controller is attempting to regulate at the desired set point value of 2 the value of performance variable estimated by the LQG observer. However, the LQG observer is optimally designed for the original model at a nominal operating point which is different from the actual operating point and hence delivers an erroneous estimated value of the performance variable. In the case when the performance variable is directly measured, previously discussed in Section 4.7.1.1 the GPC controller successfully makes adjustments to minimize a correct value of the error. In the current case when the LQG observer is included, the GPC algorithm makes an effort to minimize an incorrect error. The LQG observer is optimally designed for the original linear model at the nominal point. In the current case not only is the nonlinear model used but the problem is amplified further by the fact that there is a deviation from the original nonlinear model by virtue of the uncertainty in one of its parameters. Figure 424 shows the time response of the performance variable from the modified nonlinear fuel cell model when a battery of 4 observers, as described in Section 4.7, are employed to estimate the value of the performance variable. The solid line is the time response of the performance variable from the modified model. Due to the reasons, as discussed above, no significant improvement in controller performance is observed. 4.8.3 Comparison of the GPC Strategy with Prior Control Designs In this section comparison of the GPC controller's performance with controllers proposed in prior literature for fuel cell is conducted. Pukrushpant et al. propose controllers for the following scenarios [31]: 1. All states are measured, leading to the implementation of the sFF controller with an LQR strategy for feedback. 2. All states are not measured, leading to the implementation of an LQG observer to estimate the value of the performance variable in addition to the sFF controller with the LQR strategy for feedback. 4.8.3.1 Case where all states are measuredsFF with LQR feedback control Figure 425 shows the time response of the performance variable for the GPC strategy and the sFF with LQR feedback control strategy. The approach of sFF with LQR feedback control strategy is discussed in Section 4.5.3.1. In Figure 425, the solid line is a plot of the performance variable as a function of time in the case where the GPC controller is used. The dashed line represents the performance variable trajectory when the sFF with the LQR feedback strategy is applied. The dotted line is for the set point, fixed at value 2. It is noted from the figure that the GPC controller delivers better performance as the sFF with the LQR feedback control strategy is not able to eliminate the steadystate offset. The GPC algorithm is able to eliminate offset as discussed in Section 4.7.1.1. The LQR controller prescribes an optimal control input based on the linear model at the nominal point. However, the controller's performance is being examined then applied on a nonlinear model. Consequently, as there is deviation of the disturbance from the nominal operating point the performance of the LQR controller degrades. 4.8.3.2 Case where all states are not measuredobserver design Figure 426 shows the time response of the performance variable when an LQG observer is incorporated in the controller in the scenario where the performance variable is not measured. The dashed line indicates the time response of the performance variable when the static controller sFF with the LQG observer and the LQR feedback controller is applied. The controller strategy is discussed in Section 4.5.3.2. The solid line is the value of the performance variable when the GPC feedback with an observer is applied as the control strategy. It is noted from the time response curves that there is deterioration in both controllers performance compared to the case reported in Figure 425 where the performance variable is measured. Note, however, that the LQG observer with the GPC feedback controller exhibits better performance compared to the sFF with the LQG observer and the LQR feedback controller. The LQG observer component is identical for both the controllers, consequently the strategy for estimating the value of the performance variable is alike. The GPC algorithm is able to regulate the estimated value of the performance variable in a superior fashion. The sFF with LQR feedback controller lacks the same level of dynamic ability, and is not able to deliver the same level of regulatory performance that the GPC approach realizes. The LQR controller computes an optimal gain for the linear model at the nominal point. The implementation of the LQR strategy at nonnominal point i.e., on the nonlinear model degrades its performance. Figure 427 shows the time response of the performance variable when a battery of 4 observers, as described in Section 4.7, are employed to estimate the value of the performance variable. The solid line is the time response of the performance variable. Due to the reasons, as discussed earlier, excellent results are obtained by using a battery of observers which is evident by zerooffset/near zerooffset of the performance variable value from the setpoint. 4.8.3.3 Comparison of controller performance with respect to robustness The controllers discussed in Sections 4.7.3.1 and 4.7.3.2 (the sFF with LQR feedback and the sFF with LQG observer and LQR feedback), are evaluated and compared with the GPC controller for their robustness to model uncertainty. An approach identical to that of Section 4.7.2 is adopted. The modified model is obtained from the original nonlinear model by altering the value of the return manifold throttle area from 0.0020 m2 to 0.0023 m2. The two scenarios considered are: 1. All states measured i.e., performance variable is measured. 2. All states not measured, i.e., an observer is included to estimate the value of performance variable. All states measurable, sFF with LQR feedback. The performance of the GPC controller and the sFF with LQR feedback controller is compared in the scenario of model uncertainty. The robustness of the controllers is examined by implementing them on models for which they were not originally designed. In the current case the performance variable is measured. The control strategies for the GPC and the sFF with LQR feedback controllers described in Section 4.6 and Section 4.5.3.1, respectively, are implemented on the modified nonlinear fuel cell model. The difference in the simulation scenario compared to the one presented in Section 4.7.3.1 is that the model used to describe the nonlinear fuel cell dynamics has a different return manifold throttle area. Figure 428 shows the time responses of the two control strategies. The dashed line indicates the performance variable response to the sFF with LQR feedback control strategy. The solid line is the performance variable response to the GPC control strategy. The GPC controller delivers better performance by eliminating offset as discussed in Section 4.7.2.1. The sFF with LQR feedback does not deliver the same level of performance as an offset with respect to the set point is observed. The relatively poor performance of the sFF with LQR feedback strategy is expected as the feedforward and optimal gain are designed for a linear nominal point of the unmodified plant model. Performance variable not measured, sFF with LQG observer and LQR feedback. Figure 429 shows the time response of the performance variable for the two control strategies (the LQG observer with GPC feedback and the sFF with LQG observer and LQR feedback) in the case where the performance variable is not directly measured and an LQG observer is incorporated in the control design to estimate its value. The LQG observer is identical to the sFF with LQR feedback and the GPC control strategies. The dashed line indicates the time response of the performance variable when the sFF with LQG observer and LQR feedback controller is applied. The solid line is the performance variable timeresponse when the GPC approach with LQG observer is implemented. The simulation conditions are identical to the one discussed in Section 4.7.3.2, except that the modified model as discussed in Section 4.7.2 is used. In the case when the performance variable is not measured and the GPC strategy with LQG observer is implemented on the modified model, degradation in controller performance is observed as compared to when the performance variable is measurable as discussed in Section 4.7.2.2. However, the GPC with an LQG observer control design delivers better performance (smaller offset) than that of the sFF with LQG observer and LQR feedback control design. This can be attributed to the fact that the erroneous estimated value of the performance variable is fed as an input for the LQR feedback controller which itself is being implemented on a model it was not designed for. Consequently, a relatively higher degradation in controller performance is seen on incorporation of the observer. The GPC design performs comparatively better though it also fails to deliver offset free behavior. As mentioned before, the observer is identical for the two controllers. In the case of the sFF with LQR feedback strategy, the situation is compounded further by the fact that the LQR gain is not only being calculated at the nonnominal point (nonlinear model) but on a modified nonlinear model. Figure 430 shows the time response of the performance variable from the modified nonlinear fuel cell model when a battery of 4 observers, as described in Section 4.7, are employed to estimate the value of the performance variable. The solid line is the time response of the performance variable from the modified model. Due to the reasons, as discussed above, no significant improvement in controller performance is observed. 4.8.4 Feedforward Control Designs Two exclusively feedforward control designs, namely (1) static feedforward (sFF) and (2) dynamic feedforward (dFF) without feedback components are proposed by Pukruspan et. al. [31]. The two controllers' performance is compared with the GPC strategy for the following two cases: 1. Application on the original model 2. Application on the modified model to assess and compare robustness to model uncertainty. 4.8.4.1 Case of original model Figures 431 and 432 compares the performance of the sFF and dFF controllers with the GPC algorithm on the unaltered original nonlinear fuel cell model for the following two scenarios: (1) the performance variable is measured, and (2) the case when it is not. In the scenario where the performance variable is not measurable, an LQG observer is included in the GPC control design. The sFF and dFF controllers are discussed in Sections 4.5.2.1 and 4.5.2.2, respectively. The GPC approach and the GPC with the LQG observer strategies are implemented as described in Sections 4.6.1 and 4.5.3.2, respectively, and the simulations are conducted as discussed in Section 4.7.1. The solid and dashed black lines represent the time responses of the performance variable in the cases when the sFF and dFF controller are applied, respectively. The red and blue lines indicate the time responses of the performance variable when the GPC strategy is adopted for the scenarios where the performance variable is measured and not measured, respectively. In the case when the performance variable is not measured, an LQG observer is included in the control design to estimate its value. In the case when the performance variable is measurable, the GPC controller delivers the best result by eliminating offset with respect to the set point as discussed in Section 4.7.1.1. In the case when the performance variable is not measurable and an LQG observer is incorporated in the control design there is a relative degradation in control performance, compared to the case when it is measured, as discussed in Section 4.7.1.2. The performance of the dFF controller is the worst observed, as expected because the controller is designed for the linear model of a fuel cell at the nominal point. As the deviation from the nominal point increases, the controller dynamics of the dFF design do not correspond to the model it was designed for. The sFF strategy performs better compared to dFF approach as the deviation from nominal point increases. This behavior is expected as the sFF algorithm is derived from simulations and experimental results based on the nonlinear model. The sFF, dFF and GPC with observer strategies are not able to eliminate offset. Figure 433 shows the time response of the performance variable when a battery of 4 observers, as described in Section 4.7, are employed to estimate the value of the performance variable. The estimated value is fed to the GPC controller. The solid line is the time response of the performance variable. Due to the reasons, as discussed earlier, excellent results are obtained by using a battery of observers which is evident by zerooffset/near zerooffset of the performance variable value from the setpoint. 4.8.4.2 Case of model uncertainty The robustness of the sFF and dFF controllers to model uncertainty is assessed by implementing them on a modified model derived by modifying a parameter in the original model, as explained in Section 4.7.2. The performance of the sFF and dFF controller is compared to that of the GPC approach. The two cases for the GPC strategy considered are: (1) the performance variable is measured, and (2) the performance variable is not measured in which case an LQG observer is included in the control design to estimate its value, as explained in Section 4.5.3.2. The sFF, dFF, GPC (with and without observer) control strategies designed for the original model are applied to the modified nonlinear model. The simulations are performed in an identical manner to that explained in Section 4.7.4.1, except that the modified nonlinear model was adopted. Figures 434 and 435 shows the results of the simulations. The solid lines are the traces for the performance variable in the case when GPC approach is employed, respectively implemented for the scenarios (1) and (2) described above. The solid and dashed lines represent the time responses of the performance variable for the sFF and dFF control strategies, respectively. Figure 434 shows that in the case where the performance variable is measured the GPC controller delivers the best result by eliminating offset of the performance variable with respect to the set point, as discussed in Section 4.7.2.1. However, due to reasons discussed in Section 4.7.2.2, in the case when the performance variable is not measured, there is a relative degradation in the GPC controller performance. The performance of the sFF and dFF controllers degrades as well when they are implemented on the modified fuel cell model. The dFF controller performs the worst due to reasons discussed in Section 4.7.4.1. Additionally, in the case of dFF controller the problem is further compounded by the fact that, besides being applied at a nonnominal operating point (adopting nonlinear model), a modified model is used. Figure 436 shows the time response of the performance variable from the modified nonlinear fuel cell model when a battery of 4 observers, as described in Section 4.7, are employed to estimate the value of the performance variable. The estimated value of the performance variable is fed to the GPC controller. The solid line is the time response of the performance variable from the modified model. Due to the reasons, as discussed earlier, no significant improvement in controller performance is observed. 4.9 Conclusions An elegant solution is proposed for the problem of fuel cell control by implementing the GPC scheme. The GPC approach employs the augmented model which incorporates the measured disturbance in its algorithm. The GPC controller demonstrates better performance compared to the control strategies proposed in literature for various scenarios. The GPC strategy is the best performing controller regulating the value of the performance variable at the desired set point of 2 with zero offset in all cases when it is measurable. The GPC controller demonstrates the highest level of robustness towards the issue of model uncertainty by exhibiting zero offset of the performance variable with respect to the set point when applied to a modified fuel cell model. H2 Tank Vcm Motor Compressor PS WP Humidifier Power battery Vst 2 Fuel cell stack IWater Separator Water Tank Figure 41. Schematic of fuel cell system. Fuel Cell System Z1 = et Z2 = 202 Y1 = Wp Y2 Psm Y3 = Vst Figure 42a. Fuel cell system showing input u, disturbance w, and outputs zl z2, yi, Y2, y3. 2W = Ist U = Vcm ^ > W 'Is I Fuel Cell System static S4Feedback Figure 42b. Fuel cell system showing sFF with feedback controller.                                 Z 12.62 0 10.95 0 8.4e7 6e18 0 2.4e7 0 315.8 0 0 1.004e6 0 35.34 0 37.57 0 46.31 8.6e24 2.76e6 2e17 0 1.58e6 0 0 0 17.19 0.2032 0 0 0 2.6e8 0 2.97e8 379.4 38.7 1.06e7 0 0 33.28 0 38.03 4.834e5 4.8e6 0 0 0 0 295.6 0 0 9.33e7 0 63.61 0 4.045e8 0 4.621e8 0 0 0 0 51.22 A 3.16e5 3.98e6 0 0 0 0 5.245e5 0 0 0 0 405.1 0 0 0 0 4.94e6 1.967e6 1.089e5 2.066 0 0 0 0 180.2 165.7 1273 0 1454 4.3e22 1.388e4 9.94e16 0 0 0.01049 0 0 0 0 4.834e5 1.16e6 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 2.59e4 1.03e4 569.9 0 0 0 0 0 0.2965 0 C D Figure 43. Matrices defining the LTI model for the fuel cell model excluding sFF. 12.62 0 10.95 0 8.4e7 6e18 0 2.4e7 0 315.8 0 0 1.004e6 0 35.34 0 37.57 0 46.31 8.6e24 2.76e6 2e17 0 1.58e6 0 0 0 36.8 0.2032 0 0 0 2.6e8 0 2.97e8 379.4 38.7 1.06e7 0 0 33.28 0 38.03 4.834e5 4.8e6 0 0 0 0 295.6 0 0 9.33e7 0 63.61 0 4.045e8 0 4.621e8 0 0 0 0 51.22 A 4.94e6 1.967e6 1.089e5 4.131 0 0 0 0 1273 0 1454 4.3e22 1.388e4 9.94e16 0 0 0 0 0 4.834e5 1.16e6 0 0 0 0 0 0 0 1 0 0 0 2.59e4 1.03e4 569.9 0 0 0 0 0 3.16e5 0 3.98e6 0 0 0 716.4 1065 0 0 0 0 5.245e5 0 0 0 B 112.9 435.7 0.01049 0 0 0 0 0 0.2965 0 D Figure 44. Matrices defining the LTI model for the fuel cell including sFF. U) , U = Vcm static > Z1 =ep Z2 = AZ0, Y, = Wp y1 p Y2 Psm 73 = st > Figure 45. The sFF control configurations for fuel cell system. 100 Fuel Cell System U = Vcm  dynamic = Z1= epFn Y, = W > Z2 sm2 2= Psm 3 V=st Figure 46. The dFF controller: (a)Schematic diagram, and (b)transfer function representation. 101 w = Ist Fuel Cell System w Ist z Fuel Cell System sFF \ /  Feedback < Figure 47. The sFF schematic with feedback controller.                                 z d(t) u(Q  ) B+ ^ + YO( yJ Fu T 1/ R y) ya(t) Figure 48. The GPC design in feedback block diagram. Figure 48. The GPC design in feedback block diagram. Id(t) 285 E i I 220 0 155 901 0 5 10 15 20 25 30 Time (sec) Figure 49. Disturbance profile used for simulation purposes. F^ 2 1.8 O 1.6 1.4 1.2 1 I I I 0 5 10 15 20 25 30 Time (sec) Figure 410. The GPC control strategy implementation on the nonlinear fuel cell model in the case when the controlled variable is measured. .. setpoint  GPC with 4 observers . . 15 Time (sec) Figure 411. The GPC feedback with four observers control scheme implementation on the nonlinear fuel cell model. 1.8 1.6 1.4 1.2 x 104 .. Observer 1  Observer 2 Observer 3 Observer 4 14 I ' A / ' N 15 Time (sec) Figure 412. The Norm of errors from the battery of observers. 107  . . Obs e SObs 3 j Obs 2 Obs 1  0 5 10 15 20 25 30 Time (sec) Figure 413. The switching pattern of the battery of observers. 200 S180 E 160 0 140 > 120 100 801 0 5 10 15 20 25 30 Time (sec) Figure 414. Final voltage to the compressor. 0.04 0.02 0 0.02 0.04 10 20 30 Time (sec) x 104 10 20 Time (sec) Time (sec) Figure 415. Observer 1, error between measured and estimated values. zero error  Observer 1 b n  0.01 zero error 0.005 Observer 2 0 S0.005 0.01 0.015 0 10 20 30 Time (sec) 15000 . zero error  Observer 2 G 10000 S5000 5000 0 10 20 30 Time (sec) 2 0 o 1 2 ....... zero error S Observer 2 3 4 0 10 20 30 Time (sec) Figure 416. Observer 2, error between measured and estimated values. 0.02 Szero error  Observer 3 S0.01 0 0.01 0.02 0 10 20 30 Time (sec) x 104 5 ....... zero error S Observer 3 10 0 10 20 30 Time (sec) CA 0 . 10Observer 3 10 15 20 0 10 20 30 Time (sec) Figure 417. Observer 3, error between measured and estimated values. 105 20 Figure 417. Observer 3, error between measured and estimated values. 10 20 Time (sec) 10 20 Time (sec) 10 " Time (sec) Figure 418. Observer 4, error between measured and estimated values. 0.01 0 0.01 0.02 0.03 x 104 0 "tv zero error Observer 4 .. setpoint  GPC with 3 observers . . . . . . . . 15 Time (sec) Figure 419. The GPC feedback with three observers control scheme implementation on the nonlinear fuel cell model. 1.8 1.6 1.4 1.2 S 2 < 1.8 1.6 1.4 1.2 1 I I 0 5 10 15 20 25 30 Time (sec) Figure 420. The GPC feedback with two observers control scheme implementation on the nonlinear fuel cell model. C [ L 2 1.8 1.6 1.4 1.2 1 0 5 10 15 20 25 30 Time (sec) Figure 421. The GPC feedback with one observer control scheme implementation on the nonlinear fuel cell model. FL 2 1.8 O 1.6 1.4 1.2 1 I I I 0 5 10 15 20 25 30 Time (sec) Figure 422. The GPC control strategy implementation on the nonlinear fuel cell model with a parameter changed from the value used for control design. S 2 1.8 O 1.6 1.4 1.2 1I I I I I 0 5 10 15 20 25 30 Time (sec) Figure 423. The GPC controller with the LQG observer control strategy implementation on the altered nonlinear fuel cell model. 5 10 15 20 25 30 Time (sec) Figure 424. The GPC controller with the 4 observers control on the altered nonlinear fuel cell model. strategy implementation 2 C 2 0 V 1.8 1.6 1.4 1.2 0 5 10 15 20 25 30 Time (sec) Figure 425. Comparison of the GPC control strategy with the sFF controller combined with LQR feedback strategy on the unaltered nonlinear fuel cell model when the performance variable is measurable. 2 N.K i  I il ~ 1.8 O 1.6  1.4 1.2 0 5 10 15 20 25 30 Time (sec) Figure 426. The sFF with the LQG observer and LQR feedback, compared to GPC with the LQG Observer control strategy implementation on the unaltered nonlinear fuel cell model when the performance variable is not measurable. setpoint 2.6 GPC with 4 observers sFF with LQG observer and LQR 2.4 2.2 1.6 1.4 1.2 I I I I 0 5 10 15 20 25 30 Time (sec) Figure 427. The sFF with the LQG observer and LQR feedback, compared to GPC with the 4 observers control strategy implementation on the unaltered nonlinear fuel cell model when the performance variable is not measurable. 0 2 S1.8  1.6 1.4 1.2 0 5 10 15 20 25 30 Time (sec) Figure 428. The sFF with the LQR feedback, compared to GPC, when performance variable is measurable on the altered nonlinear fuel cell model. S 2 .. ... .. 1.8 1.6 1.4 1.2 1 I I I I I 0 5 10 15 20 25 30 Time (sec) Figure 429. The sFF with the LQG observer and the LQR feedback compared to the GPC with the LQG observer control strategy on the altered nonlinear fuel cell model. S 2 S1.8  O 1.6 1.4 1.2 1 I 0 5 10 15 20 25 30 Time (sec) Figure 430. The sFF with the LQG observer and the LQR feedback compared to the GPC with the 4 observers control strategy on the altered nonlinear fuel cell model. C 2 1.8  1.6 1.4 1.2 1 I I 0 5 10 15 20 25 30 Time (sec) Figure 431. The sFF and dFF strategies and the GPC control strategy, performance compared when applied on the unaltered nonlinear fuel cell model. 1.8 O 1.6 1.4 1.2 1I I I I I 0 5 10 15 20 25 30 Time (sec) Figure 432. The sFF and dFF strategies and the GPC control strategy with the LQG observer, performance compared when applied on the unaltered nonlinear fuel cell model. 2 1.8 1.6 1.4 1.2 1I I I I I 0 5 10 15 20 25 30 Time (sec) Figure 433. The performances of the sFF and dFF strategies and the GPC with the 4 observers control strategy compared when applied on the unaltered nonlinear fuel cell model. 1.8  2 1.6 1.4 1.2 0 5 10 15 20 25 30 Time (sec) Figure 434. The sFF and dFF strategies and the GPC control strategy, performance compared when applied on the altered nonlinear fuel cell model. 1.8 O 1.6 1.4 1.2 1 I I I I 0 5 10 15 20 25 30 Time (sec) Figure 435. The sFF and dFF strategies and the GPC control strategy with the LQG observer, performance compared when applied on the altered nonlinear fuel cell model. 2 .. . 1.8 O 1.6 1.4 1.2 1 I I I I I 0 5 10 15 20 25 30 Time (sec) Figure 436. The performance of sFF and dFF strategies and the GPC with 4 observers control strategy compared when applied on the altered nonlinear fuel cell model. CHAPTER 5 CONCLUSIONS AND PROPOSITIONS FOR FUTURE WORK 5.1 Conclusions A Virtual Control Lab (VCL) with an inverted pendulum that can be utilized as a tool for enhanced learning is described. The VCL can be used in conjunction with a process control lecture for demonstrating various advanced concepts or can be used by students located in geographically remote places. The animation module of the VCL allows the user to visually observe the impact of the control design. The VCL can potentially minimize the problem of scheduling laboratory time for physical equipment. The problem of tracking the slope of ramp is examined. On the basis of results obtained, a PI controller using the ITAE as the optimizing metric demonstrates the best results. The controller performance is better than that of controllers proposed in literature. Tuning relationships for PI and PI2 controllers, using three different optimizing metrics and 20,000 different plants is presented. An elegant solution is proposed for the problem of fuel cell control by implementing a GPC strategy which incorporates disturbance measurements to produce a manipulated variable. The GPC design demonstrates better performance compared to control strategies available in prior literature. The GPC controller results in zero offset in the performance variable in the nonlinear model when the performance variable is measured. The GPC controller is also the best performing controller when evaluated for model uncertainty. The controller exhibits zero offset, showing its strength from a viewpoint of robustness, when employed on a modified model. In the case when the performance variable is not measurable, a battery of observers is implemented to estimate the value of the performance variable deliver the best result. However, the robustness problems still exist. 5.2 Future Work To validate the benefits and effectiveness, it is proposed that the VCL is tested as a pilot teaching tool in process control classes taught at both undergraduate and graduate levels. The feedback obtained from the instructors and students will be highly beneficial in improving and incorporating features that could enhance the learning experience. It would be desirable to add to the VCL a few more advanced control strategies such as the GPC approach. For future work regarding the ramp tracking problem it is proposed to validate the simulation results for physical setups that can take advantage of the tuning relationships presented. It is also suggested to conduct a more comprehensive study to minimize the scatter in the optimizing metric error for low 0/r values of the ratio, that is, for systems with little dead time. For future work on the fuel cell control problem it is proposed to evaluate the design of observers which take model uncertainty into account. A battery of observers can be designed at each operating point, with each observer corresponding to a linearized model obtained from different values of parameters. Then a bumpless switching strategy could be designed to select an appropriate state estimate to feed to the controller. The experimental implementation of the control scheme in a physical fuel cell may be of significant value to confirm or refine the results discussed in this dissertation. APPENDIX A OFFSET BETWEEN AUXILLIARY AND ORIGINAL RAMP The final value theorem is used to calculate the shift between the original ramp and the modified ramp. The transfer function for the process with time delay is given by K G = e (Al) and the transfer function for PI controller is given by G =K, 1+ (A2) and the equation for ramp in the time domain is r(t) = at (A3) Performing the Laplace transform of Eq. A3, with the initial condition r(t = 0) = 0, yields r(s) = (A4) s Using Eqs. Ai and A2, it is possible to derive the following standard closed loop relationship between the output y and the ramp r : GGc y(s) = r(s) (A5) I+GGC Now consider the error e(t) = r(t) y(t) (A6) defined as the difference between the set point and output. Applying final value theorem lime(t) = lims(r(s) y(s)) (A7) t i E s and inserting Eq. A4 and A5 in Eq. A7 yields lime(t)= lims a t> s>o S2 GpGc a 1+GpG s2 (A8) a 1 or lime(t) = lim ( t)co so s 1 + GpG, Inserting Eq. Ai and Eq. A2 in Eq. A9 results in lime(t) = lim ( t s sKO K 1 1T+l e 1 Tj Simplifying terms and applying the limits on the right hand side of Eq. A10 yields 1 lime(t) = a (1 t)o 1 KKC TI Hence, lime(t) = ( (t KKC which is an analytical expression for the offset between the auxiliary and the original ramp. A9) A10) A11) A12) APPENDIX B OBSERVER DESIGN USING TRANSFER FUNCTION The observer equations are given by S= A + Bu + L(y C) = C + Du Eq. B1 can be rewritten as = (A LC) + Bu + Ly Performing Laplace Transform sX(s) i(t = 0) = (A LC)X(s) + BU(s) + LY(s) This results (s A + LC)X(s) = BU(s) + LY(s) + (t = 0) On further manipulation i(s) = MBu(s) +MLy(s) + M(t = 0) where M = (sI A +LC)1 Now if S= C_ Z(s) = C.fBU(s) + C.LY(s) + CAf/(t = 0) Z(s) = GMU(s) + GCY(s) + GjCoX(0) where G,, = CB, G, = C,ML, and Ga O = CM. (Bl) (B2) (B3) (B4) (B5) (B6) (B7) (B8) (B9) LIST OF REFERENCES 1. T. F. Junge and C. Schmid, Webbased remote experimentation using a laboratoryscale optical tracker, Proceedings of the American Control Conference, Chicago Illinois, June 2000. 2. D. Gillet, C. Salzmann, R. Longchamp, and D. Bonvin, Telepresence: an opportunity to develop practical experimentation in automatic control education, European Controls Conference, Brussels Belgium, July 1997. 3. D. Gillet, F. Geoffroy, K. Zeramdini, A. V. Nguyen, Y. Rekik and Y. Piguet, The cockpit: An effective metaphor for Webbased experimentation in engineering education, International Journal of Engineering Education, 19(3), August 2002, pp. 389397. 4. D. Gillet, G. Fakas, eMersion: a new paradigm for Webbased training in engineering education, International Conference on Engineering Education, Oslo, Norway, 2001, 84B10. 5. A. Bhandari, M. H. Shor, Access to an instructional control laboratory experiment through the World Wide Web, Proceedings of the American Control Conference, Philadelphia, June 1998. 6. Ch. Schmid, T.I. Eikass, B. Foss and D. Gillet, A remote laboratory experimentation network, 1st IFAC Conference on Telematics Applications in Automation and Robotics, Weingarten, Germany, July 2426, 2001. 7. H. Latchman, C. Salzmann, S. Thotapilly and H. Bouzekri, Hybrid asynchronous and synchronous learning networks in distance education, International Conference on Engineering Education, Rio de Janeiro, Brazil, 1998. 8. B. Armstrong and R. Perez, controls laboratory program with an accent on discovery learning, IEEE Control Systems Magazine, February 2001, pp. 1420. 9. J. W. Overstreet and A. Tzes, Internetbased client/server virtual instruments designs for real time remoteaccess control engineering laboratory, Proceedings of the American Control Conference, San Diego, 1999. 10. C. Schmid, The virtual control lab VCLab for education on the Web, Proceedings of the American Control Conference, Philadelphia, June 1998. 11. O. D. Crisalle and H. A. Latchman, Virtual control laboratory for multidisciplinary engineering education, NSF Award No. DUE9352523, proposal funded by the National Science Foundation under the Instrumentation and Laboratory Improvement Program, (1993). 12. C. S. Peek; O. D. Crisalle; S. Depraz and D. Gillet, The virtual control laboratory paradigm: Architectural design requirements and realization through a DCmotor example, International Journal of Engineering Education, 21(6), 2005, pp. 11341147 13. National Instruments Corp., 11500 N. Mopac Expressway, Austin, TX 787593504, USA, www.ni.com. 14. M. W. Dunnigan, Enhancing statespace control teaching with a computerbased assignment, IEEE Transactions on Education,44(2), May 2001, pp. 129136. 15. P. C. Wankat and F. S. Oreovicz, Teaching Engineering, McGrawHill, (1993). 16. A. M. Lopez; J. A. 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Peek, Development of virtual control laboratories for control engineering education, double integral controllers for ramp tracking and predictive control for robust control and luenberger observers, PhD Candidacy Exam, Dept. of Chemical Engineering, College of Engineering, University of Florida; 2006. 22. Mathworks Inc. MATLAB. Version 7.2.0.232, R2006a (2006). 23. S. Davis, Transportation Energy Data Book, 20th edn, U. S. Department of Energy, (2000). 24. W. Grove, A small voltaic battery of great energy, Philosophical Magazine, 15, 1839, pp. 287293. 25. J. Amphlett, R. Baumert, R. Mann, B. Peppley, P. Roberage and A. Rodrigues, Parametric modelling of the performance of a 5kW protonexchange membrane fuel cell stack, Journal of Power Sources, 49, 1994, pp. 349356. 26. J. Amphlett, R. Baumert, R. Mann, B. Peppley and P. Roberage, Performance modeling of the Ballard Mark IV solid polymer electrolyte fuel cell, Journal of Electrochemical Society, 142(1), 1995, pp. 915. 27. D. Bemardi, Waterbalance calculations for solidpolymerelectrolyte fuel cells, Journal ofElectrochemical Society, 137(11), 1990, pp. 33443350. 28. D. Bernardi and M. Verbrugge, A mathematical model of the solidpolymerelectrolyte fuel cell, Journal ofElectrochemical Society, 139(9), 1992, pp. 24772491. 29. J. Baschuk and X. Li, Modelling of polymer electrolyte membrane fuel cells with variable degrees of water flooding, Journal of Power Sources, 86, 2000, pp. 186191. 30. J. Eborn, L. Pedersen, C. Haugstetter and S. Ghosh, System level dynamic modeling of fuel cell power plants, Proceedings of the American National Conference, 2003, pp. 20242029. 31. J. T. Pukrushpan, A. G. Stefanopoulou and H. Peng, Control of fuel cell power systems: principles, modeling, analysis and feedback design, Advances in Industrial Control. SpringerVerlag London Limited, (2004). 32. D. Seborg, T. Edgar and D. Mellichamp, Process Dynamics and Control, 2nd edn, Wiley, (2004). 33. K. Ogata, Modern Control Engineering, 4th edn, Prentice Hall, (2001). 34. R. Stefani, B. Shahian, C. Savant and G. Hostetter, Design of Feedback Control Systems, 4th edn, Oxford University Press, (2002). 35. O. D. Crisalle, D. E. Seborg and D. A. Mellichamp, Theoretical analysis of longrange predictive controllers, Proceedings of the American Control Conference, Pittsburgh, Pennsylvania, 1989. 36. D. W. Clarke, C. Mohtadi and P. S. Tufts, Generalized predictive control part I. The basic algorithm, Automatica, 23(2), 1987, pp. 137148. 37. D. W. Clarke, C. Mohtadi and P. S. Tufts, Generalized predictive control part II. Extensions and interpretations, Automatica, 23(2), 1987, pp. 149160. 38. D. W. Clarke and C. Mohtadi, Properties of generalized predictive control, Automatica, 25(6), 1989, pp. 859875. BIOGRAPHICAL SKETCH Vikram Shishodia was born in Delhi, India. He graduated with a B.Tech. in chemical engineering from the Indian Institute of Technology Delhi in 1992. Mr. Vikram Shishodia joined the graduate program at the University of Florida in 1993. He graduated with an M.S. in materials science and engineering in 1996. After developing a successful career as a process engineer with Intel Corporation for nine years he returned to academia to pursue higher degrees in chemical engineering at the University of Florida. From 2005 to 2008 he was a graduate student and simultaneously worked as Assistant Director for the Division of Student Affairs at the College of Engineering of University of Florida. He graduated with an M.S. in chemical engineering in 2003 and a Ph.D. in chemical engineering from the University of Florida in 2008. PAGE 1 1 ENHANCED CONTROL PERFORMANCE AND APPLICATION TO FUEL CELL SYSTEMS By VIKRAM SHISHODIA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 PAGE 2 2 2008 Vikram Shishodia PAGE 3 3 To Carmen PAGE 4 4 ACKNOWLEDGMENTS I would like to express my deep est gratitude to m y advisor Dr O. Crisalle for his support and guidance without which this work would not ha ve been possible. I thank the members of my supervisory committee, Dr. H. Latchman, Dr. G. Hoflund, Dr. W. Lear, and Dr. S. Svoronos, for their guidance and serving on my supervisory committee. I thank my colleagues in the research group who provided insightful conversations on my research topics, besides being gr eat friends. I would especially like to thank Christopher Peek for providing the sample code for ramp track ing which expedited the progress on the problem significantly. I also thank him fo r all the insightful discussions. I thank my parents for their love, support and encouragement that they ha ve given me throughout my life and during the completion of this work. I would like to express my deepest gratit ude to my spiritual teacher Gurumayi Chidvilasananda who has been there for me during ev ery step of my life. Finally, I wish to thank my wife and kids, who have been very supportiv e, loving and understandi ng during this journey. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4LIST OF TABLES................................................................................................................. ..........7LIST OF FIGURES.........................................................................................................................8ABSTRACT...................................................................................................................................11CHAPTER 1 INTRODUCTION..................................................................................................................132 VIRTUAL CONTROL LABORATORY............................................................................... 152.1Introduction................................................................................................................... 152.2Objective.......................................................................................................................172.3Inverted Pendulum........................................................................................................172.4Control Design..............................................................................................................202.5Realization of an InvertedPendulum VCL.................................................................. 252.6Conclusions...................................................................................................................293 PI AND PI2 CONTROLLER TUNING FOR TRAC KING THE SLOPE OF A RAMP....... 383.1Introduction and Background........................................................................................383.2Problem Statement and Approach................................................................................. 403.3Results and Discussion.................................................................................................. 433.3.1Tuning Parameters of Controllers..................................................................... 433.3.2Comparison of the Performance of the PI and PI2 Controllers......................... 443.3.3Comparison of Metrics...................................................................................... 463.3.4Comparison of PI (ITAE) Controller with Literature Precedents..................... 473.3.5Local Minima versus Global Minima...............................................................483.4Conclusions...................................................................................................................484 GENERALIZED PREDICTIVE CONTROL FOR FUEL CELLS ....................................... 614.1Introduction................................................................................................................... 614.2Fuel Cell System Background....................................................................................... 614.3Objectives of the Research............................................................................................644.4Fuel Cell Model............................................................................................................644.5Literature Precedents fo Fuel Cell Control Designs..................................................... 664.5.2Feedforward Strategy........................................................................................674.5.2.1Static feedforward controller..............................................................674.5.2.2Dynamic feedforward controller......................................................... 68 PAGE 6 6 4.5.3Combination of Static Feedforward with Optimal Feedback Controllers........ 694.5.3.1Case where the performance variable is measurable.......................... 694.5.3.2Case where the performance variable is not measurable.................... 714.6Generalized Predictive Control..................................................................................... 724.7Battery of Observers.....................................................................................................794.8Simulation Studies and Results..................................................................................... 814.8.1Generalized Predictive Control Results............................................................ 824.8.1.1Case where the performance variable is measured............................. 824.8.1.2Performance variable not measured.................................................... 834.8.2The GPC Approach Evaluated for Robustness................................................. 854.8.2.1Case where the performance variable is measured............................. 864.8.2.2Case where the performance variable is not measured....................... 864.8.3Comparison of the GPC Strategy with Prior Control Designs.......................... 884.8.3.1Case where all states are measuredsFF with LQR feedback control 884.8.3.2Case where all states are not measuredobserver design.................... 894.8.3.3Comparison of controller performance with respect to robustness....904.8.4Feedforward Control Designs...........................................................................924.8.4.1Case of original model........................................................................ 924.8.4.2Case of model uncertainty.................................................................. 944.9Conclusions...................................................................................................................955 CONCLUSIONS AND PROPOSITI ONS FOR FUTURE WORK ..................................... 1325.1Conclusions.................................................................................................................1325.2Future Work................................................................................................................133APPENDIX A OFFSET BETWEEN AUXILLIAR Y AND ORIGINAL RAMP ........................................ 134B OBSERVER DESIGN USI NG TRANSFER FUNCTION ..................................................136LIST OF REFERENCES.............................................................................................................137BIOGRAPHICAL SKETCH.......................................................................................................140 PAGE 7 7 LIST OF TABLES Table page 31 The PI2 controllers optimized tuning parameters linear least square fit equations............ 5232 The PI controllers optimized tuning para meters linear least square fit equations............. 5333 Values of the plant parameters used co mpare the performance of the controllers............ 53 PAGE 8 8 LIST OF FIGURES Figure page 21 Inverted pendulum.......................................................................................................... ...3122a Front panel of the VCL wh ere all states are measured...................................................... 3222b Front panel of the VCL showing observer......................................................................... 3323 Interaction panel of inverted pendulum VCL....................................................................3424 Controller tab of the navigation panel...............................................................................3525 Analysis tab of the navigation panel..................................................................................3626 Simulation tab of the navigation panel.............................................................................. 3731 Ramp r and auxiliary ramp ra with constant slope, ........................................................4932 Closed loop transfer function repr esentation of plant and controller................................4933 The PI2 controllers optimal tuning parameters using ITAE (A, B), IAE (C, D), and ISE (E, F) as the optimizing metric................................................................................... 5034 The PI controllers optimal tuning parame ters, using ITAE (A, B), IAE (C, D), and ISE (E, F) as the optimizing metric................................................................................... 5135 The PI2 and PI controllers ramp tracking a nd slope tracking performance using the optimal ITAE control parameters...................................................................................... 5436 The PI2 and PI controllers ramp tracking a nd slope tracking performance using the optimal IAE control parameters......................................................................................... 5537 The PI2 and PI controllers ramp tracking a nd slope tracking performance using the optimal ISE control parameters......................................................................................... 5638 The ITAE, IAE, and ISE metrics comparison for three plants, usi ng the PI controllers A), B), and C).....................................................................................................................5739 The ITAE, IAE, and ISE metrics co mparison for three plants, using the PI2 controllers A), B), and C).................................................................................................. 58310 The PI controllers tuned using the ITAE metric compared with Belanger and Luyben and Peeks controllers for three plants A), B), and C)....................................................... 59311 Contour plots for the PI c ontroller tuned using the ITAE metric for the three plants A), B), and C).....................................................................................................................60 PAGE 9 9 41 Schematic of fuel cell system............................................................................................ 9642a Fuel cell system showing input u, disturbance w and outputs z1, z2, y1, y2, y3...................9742b Fuel cell system showing sFF with feedback controller.................................................... 9843 Matrices defining the LTI model fo r the fuel cell model excluding sFF........................... 9944 Matrices defining the LTI model for the fuel cell including sFF...................................... 9945 The sFF control configura tions for fuel cell system........................................................10046 The dFF controller: (a)Schematic diagra m, and (b)transfer function representation...... 10147 The sFF schematic with feedback controller................................................................... 10248 The GPC design in feedback block diagram....................................................................10349 Disturbance profile used for simulation purposes........................................................... 104410 The GPC control strategy implementati on on the nonlinear fuel cell model in the case when the controlled va riable is measured................................................................ 105411 The GPC feedback with four observe rs control scheme implementation on the nonlinear fuel cell model................................................................................................. 106412 The Norm of errors from the battery of observers........................................................... 107413 The switching pattern of the battery of observers............................................................ 108414 Final voltage to the compressor....................................................................................... 109415 Observer 1, error between m easured and estimated values............................................. 110416 Observer 2, error between m easured and estimated values............................................. 111417 Observer 3, error between m easured and estimated values............................................. 112418 Observer 4, error between m easured and estimated values............................................. 113419 The GPC feedback with three observers control scheme implementation on the nonlinear fuel cell model................................................................................................. 114420 The GPC feedback with two observers control scheme implementation on the nonlinear fuel cell model................................................................................................. 115421 The GPC feedback with one observe r control scheme implementation on the nonlinear fuel cell model................................................................................................. 116 PAGE 10 10 422 The GPC control strategy implementati on on the nonlinear fuel ce ll model with a parameter changed from the value used for control design............................................. 117423 The GPC controller with the LQG obser ver control strategy implementation on the altered nonlinear fuel cell model...................................................................................... 118424 The GPC controller with the 4 observers control strategy implementation on the altered nonlinear fuel cell model...................................................................................... 119425 Comparison of the GPC control strategy w ith the sFF controller combined with LQR feedback strategy on the unaltered nonlinea r fuel cell model when the performance variable is measurable...................................................................................................... 120426 The sFF with the LQG observer and LQR feedback, compared to GPC with the LQG Observer control strategy implementation on the unaltered nonlin ear fuel cell model when the performance variab le is not measurable........................................................... 121427 The sFF with the LQG observer and LQR feedback, compared to GPC with the 4 observers control strategy implementation on the unaltered nonlin ear fuel cell model when the performance variab le is not measurable........................................................... 122428 The sFF with the LQR feedback, compared to GPC, when performance variable is measurable on the altered nonlinear fuel cell model....................................................... 123429 The sFF with the LQG observer and the LQR feedback compared to the GPC with the LQG observer control strategy on th e altered nonlinear fuel cell model................... 124430 The sFF with the LQG observer and the LQR feedback compared to the GPC with the 4 observers control strategy on the altered nonlinear fuel cell model....................... 125431 The sFF and dFF strategies and the GPC control strategy, performance compared when applied on the unaltered nonlinear fuel cell model................................................ 126432 The sFF and dFF strategies and the GPC control strategy with the LQG observer, performance compared when applied on the unaltered nonlinear fuel cell model.......... 127433 The performances of the sFF and dFF st rategies and the GPC with the 4 observers control strategy compared when applied on the unaltered nonlinear fuel cell model..... 128434 The sFF and dFF strategies and the GPC control strategy, performance compared when applied on the altered nonlinear fuel cell model.................................................... 129435 The sFF and dFF strategies and the GPC control strategy with the LQG observer, performance compared when applied on the altered nonlinear fuel cell model.............. 130436 The performance of sFF and dFF strategi es and the GPC with 4 observers control strategy compared when applied on th e altered nonlinear fuel cell model...................... 131 PAGE 11 11 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ENHANCED CONTROL PERFORMANCE AND APPLICATION TO FUEL CELL SYSTEMS By Vikram Shishodia May 2008 Chair: Oscar D. Crisalle Major: Chemical Engineering The invertedpendulum virtual control la b, a simulation environment for teaching advanced concepts of process control, is desi gned using the LabVIEW soft ware tool. Significant advantages of using this simulation tool for pe dagogical purposes include avoiding the potential issue of schedule conflicts for securing equi pmentaccess time in a physical laboratory and providing a learning resource that becomes accessible to students lo cated in remote geographical places. A set of tuning relationships are proposed for standard proportionalin tegral controllers and proportional doubleintegral controllers for the purpos e of tracking the slope of a ramp trajectory. Three different performance metrics are investigated to serve as the criteria for optimality, and a numerical optimization procedure is used to minimize each metric over 20,000 different plants. The proportional integral controlle r with tuning parameters select ed to optimize value of the integral of the timeweighted absolute error is recommended for tracking the slope of a ramp trajectory. A generalized predictive contro l (GPC) strategy is proposed for a fuel cell system, where the controller incorporates a measured disturbanc e in the control design. The control objective is to maintain oxygen excess ratio at a prescribed constant value. The performance of the GPC PAGE 12 12 control design is compared with that of the controllers propo sed in literature for various scenarios including model uncertainty. The GPC controller has zero offset when the performance variable is measured and performs better than competing designs offered in the literature. The GPC controller is also robust w ith respect to model uncertainty. A battery of observers with a switching stra tegy is proposed for estimating the value of the performance variable when it is not measure d. The GPC controller with a ba ttery of observers has no offset demonstrating better performance than analogous de signs proposed in literature. However, the control performance is not robust when the estimat or battery is used and linear models used for observer design are uncertain give n that the response offset is not completely eliminated. PAGE 13 13 CHAPTER 1 INTRODUCTION Issues relevant and critical to process c ontrol are discussed in this study. Chapter 2 investigates the design and im plementation of a virtual control lab (VCL) for The Inverted Pendulum problem. The LabVIEW software is used as the platform for simulating the Inverted Pendulum model. The objective is to have a vi sual computer based application by virtue of which advanced control concepts can be shared and taught to the audience who are primarily students studying control th eory and its applic ations. The VCL is designed in a manner such that various scenarios for the implemen tation of the controllers can be achieved. The user is given the choice of operating the application with the system in openloop or closedloop configuration. The controller can be tuned manually by the us er or use the tuned control parameters computed by the specific control algorithm. The user is allowed to alter the values of the poles for the closed loop system and see it s visual impact by simulation performed by the VCL. The VCL provides the opportunity to be ope rated in the scenarios when all the controlled variables are measurable and also when all of them are not measurable. In the case when the performance variables are not measurable an observer is incorporated in the control design to estimate their value. The impact of all the cha nges performed in the VCL are displayed visually by the animation of the inverted pendulum system. This key feature of the VCL allows the user to see the visual impact of changing differe nt components of control system and hence facilitating the process of learning. Chapter 3 discusses the problem of controll er design tuning for tr acking the slope of a ramp. The control objective is to place the outpu t of the system in a linear zone parallel to a ramp trajectory. A first order system with time de lay is considered for this study. Two kinds of controllers are used for this study namely, proportio nalintegral and propor tional doubleintegral. PAGE 14 14 Both the controllers serve the purpose of positioning the system in the desired linear zone which is parallel to a given ramp prof ile. There is an offset with respect to the ramp observed when only one integrator is used. Zero offset with th e ramp is observed when two integrators are used in the controller. In both cases the control objective is met which is to track the slope of the ramp trajectory.. Three different metric are employed to evaluate the performance of the controllers. The MATLAB pl atform in conjunction with SIMULINK module is used for acquiring the optimized controller parameters. Chapter 4 discusses a generalized predictive co ntrol (GPC) strategy prop osed for a fuel cell system. The control objective is to regulate the value of the pe rformance variable i.e., the oxygen excess ratio at a desired value. The perf ormance of the GPC control design is compared with that of controllers proposed in prior literature. Various scenarios are considered, including the cases of model uncertainty and unmeasured performance variable. The GPC controller exhibits zero offset in all cases when the perfor mance variable is measured, and also ensure zero or negligible offset when the performance vari able is estimated via a battery of estimators. PAGE 15 15 CHAPTER 2 VIRTUAL CONTROL LABORATORY 2.1 Introduction There is a need for the developm ent of in ternetbased nonconventional pedagogical tools for delivering knowledge to students on various topics of study. The drive stems from the various advantages that these environments offer. First, these applications do not depend on the availability of a physical setup or facility to run experiments [1, 2]. They are also not limited in terms of the number of users who can access the application at any given time, as long as appropriate adjustments are done in the server side of application. There is also no adverse safety issue or concern of damaging expensive equipmen t when the product is not used correctly. Less training is required for the user to be able to run the tool. Compared to a traditional physical laboratory setup, in these virtual environments, there is more of an opportunity to be able to realize a physical system and introduce more advanced topics a nd see their effects on the system. A software application that simulates the behavior of a physical system, provides animation to depict how the system behaves, and provides an interface so that the us er can observe changes made on the system performance, is highly beneficial from a lear ning and educational standpoint. The application is referred to as a virt ual laboratories since the nature of the Lab or the application is virtual as it is a softwa re emulator of the physical plant and can be potentially used to remotely control actual phy sical equipment via web and networking [1, 36]. From the perspective of enhancing the learni ng experience, the virtual control lab (VCL) supports learning by all three modes, namely activ e, flexible, and discovery learning. In active learning, tools and material are made available to students so that they can use these resources to actively learn and reinforce th e theoretical concepts. Trad itionally, physical laboratories, equipment and experimental apparatus are provid ed to students to reinforce and test the PAGE 16 16 understanding of the students comprehension of the theory. With limitation of available resources and costs associated with the overall management of logistics, at times this can be a challenging task, which can prove to be not only quite expensive, but also involve safety issues. In those circumstances the VCL can be an excellent solution. It doesnt n ecessarily need to be a complete replacement, but it can be used in conj unction with existing physical setups to promote active learning. Flexible learning provides opportunities to l earn class material when the students or instructor might be having challenges in terms of establishing meeting times, scheduling or location [7]. For instance, if an individual is a parttime student with the obligations of a fulltime student, that person might have challenges meeting lab times scheduled during regular work hours. The VCL is a most valuable tool to acco mmodate those circumstances. From the comfort of home and in a more appropriate time, that indi vidual can complete the exercises/material if the VCL is utilized. The VCL is flexible with respec t to the schedule and logistics limitations of an individual. The VCL also supports learning vi a discovery mode. In this scenario, an environment is provided where the student has minimal supervis ion or instruction [8]. The student is encouraged to learn by making changes and observ ing the impact of these changes. There are some significant challenges in implementing such a setup in the alternative scenario of a physical laboratory. Due to safety and cost consideratio ns, facilities that promote discovery learning are few. A VCL designed for the purpose, again, prove s to be an excellent resource to implement safe and costeffective learning via discovery. There are various implementations of virtual an d remote labs reported in the literature. There are several World Wide Web based labs which foster learning by different modes [3, 9 PAGE 17 17 11]. Most of these virtual labs, however, have a few shortcomings in terms of their usage. Many do not provide a sufficiently high level of interact ivity with the user. Significant modifications need to be made to the program to implement a ny changes. Another disa dvantage that most of the current virtual labs have is that they are developed on a proprietary software platform. At times significant familiarity with th at software is needed to be able to utilize the application. 2.2 Objective The intent is to build a VCL m odule that treat s some advancedlevel control concepts and serves as a pedagogical tool that overcomes shor tcomings that existing virtual labs pose from a learning perspective and user interface. The infrastructure created by Peek et al. is used for implementing an Inverted Pendulum VCL [12]. The inte nt is to develop an animated control module that reinforces advanced control concepts with a friendly user interface. Some examples of key control concepts illustrated in the VC L are linear statespace modeling, controllability, pole placement and observability analysis. Sec tions 2.3 and 2.4 discuss the Inverted Pendulum system, its dynamics and the associated c ontrol concepts. Sec tion 2.5 describes the implementation of the Inverted Pendulum system as a VCL using the LabVIEW software and its animation features [13]. Finally, conclusions fr om this effort are summarized in Section 2.6. 2.3 Inverted Pendulum The inverted pendulum considered consists of a spherical bob attached to a cart by a rod. A schematic diagram is given in Figure 2.1. The mass of the rod is assumed to be negligible. The rod is mounted by a hinge at the center of the cart. The input of the system is a horizontal force applied to the cart. The cart is free to move only along one coor dinate which is the horizontal z axis. The pendulum is free to rotate 360 de grees with respect to the cart in the xz plane where the x axis is vertical. It is assumed that there is no friction between the pendulum and the cart at the hinge. PAGE 18 18 The goal is to keep the pendulum in an uprig ht position by manipulating the value of the applied force. The system is inherently nonlinea r. To apply linear c ontrol theory, the dynamics must be linearized, and represented as a standa rd statespace realizat ion. Consequently, the system is linearized for small values of the a ngle that the pendulum ma kes with the vertical. The nonlinear equations describing the dynamics of the system are cossinsin sin 1 z2 2g l m f m M (21) and sin sincos cos sin 12 2g m Mm l m f m M l (22) where dz/dt z (23a) and /dt d (23b) where M is the mass of the cart, m is the mass of the pendulum bob, l is the length of the pendulum rod, g is the acceleration due to gravity, z is the horizontal position of the cart, is the angle that the pendulum make s with the vertical, and f is the force (contro l input) acting on the cart [14]. A linear statespace system is derived from Eqs. 21 23 by linearizing about an operating point ( ),,,, fzz where 0 z (24) 0 z (25) 0 (26) 0 (27) 0 f (28) PAGE 19 19 The deviation variables for the linear statespa ce model are zzx 1 (29) zzx 2 (210) 3x (211) 4x (212) ffu (213) After linearization about the point )0,0,0,0,0(),,,,( fzz (214) the resulting standard linear statespace model u bAxx (215) is given by the equation u Ml M Ml M)g(m M mg 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 x x (216) where the elements of the state vector x are distance (x1 = z ), velocity (x2 = z ), angle ( x3 = ), and angular velocity ( x4 = ), and where 0 )( 0 0 1 0 0 0 0 0 0 0 0 1 0 Ml gMm M mg A (217) PAGE 20 20 Ml M 1 0 1 0 b (218) The control is u, which is the force acting on the cart. The standard output model xCy (219) relates the output y to matrix C and state vector x, where the output matrix C is the identity matrix 1000 0100 0010 0001C (220a) in the case where all four states are measured. For the case where only one state is measurable, the output matrix C adopts a rowvector form of all zeros, except for one entry that is unity at the location corresponding to the measured state. For the particular case where the state x1, namely the distance of the cart from its original horizontal position, is the only meas ured state, the output matrix C adopts the form 0001 C (220b) 2.4 Control Design When the system is in the unforced configura tion, a stability check done by calculating the eigenvalues of matrix A reveals that there is one eigenvalue th at lies in the open right half plane, implying that the system in its unforced state is unstable. PAGE 21 21 This is a regulation problem, as the objective is to make the states evolve towards zero value. For the implementation of the controller, a test for controllabilit y needs to be performed to verify that the system is indeed controllab le. The requirement for controllability is that 0Qc ) det( (221) where bAbAAbbQ3 2 c (222) Using the definitions for A given in Eq. 25 and for b given in Eq. 26, it follows that 0 1 0 1 Ml M Ab 22 2 2)( 0 0 lM gMm lM mg bA 0 )( 022 22 3lM gMm lM mg bA (223) Hence, 0 )( 0 1 )( 0 1 0 0 0 1 0 1 022 22 2 22lM gMm Ml lM gMm Ml lM mg M lM mg McQ (224) Obviously matrix cQ in Eq. 211 is of full rank, thus 0 )det( cQ (225) which implies that the system Eq. 24 is indeed controllable. The analysis of controllability presented here is found in standard references [3234]. Two scenarios are considered: 1. All states are measured. 2. Some states are not measured. PAGE 22 22 In the case of the first scenario in which all states are measured, a fullstate feedback approach is used in the form of th e proportional state feedback control law Fx u (226) where F is a proportional gain used to address the regulation problem. To determine an appropriate value for matrix F first substitute the value of u given by Eq. (226) into the state space e quation Eq. (215) leading to ) ( FxbAxx (227) The standard solution to Eq. 214 is given by the Variation of Parameters formula as te)( 0 BFAxx (228) where 0x is the vector of the initial value of the state vector x [36]. Ackermans pole placement algorithm is employed for computing the value of the matrix F that places the poles of the ABF system in the desired location [36]. In the case of the second scenario, in which a ll states of the system are not measurabed, a Luenberger Observer is incorporated in the contro ller to estimate the value of the states. Before carrying out an observer design, a check is performed to verify if the system is observable when only one state is measured. The first state, the di stance of the cart from the original position, is the only state that is assumed to be measurable. For the system to be observable, the condition 0 )det( oQ (229) should be satisfied, where the observability matrix oQ is given by 3 2CA CA CA C Qo (230) Matrix A is defined by Eq. (217) and matrix C by Eq. 220b. The expressions PAGE 23 23 0010 CA (231) 0 002M mg CA (232) and M mg 0003CA (233) can be used to readily build the observability matrix oQ described by Eq. 217, yielding M mg M mgo000 0 00 0010 0001 Q (234) Since matrix oQ is diagonal, its determinant is simply the product of the diagonal terms. Hence, 0 det2 22 M gmoQ (235) Given that the determinant is nonzero, it follows that oQ is of full rank, and therefore the system is observable. The analysis of observability presented here is also found in standard references [3436]. The standard equations for observer design u bAxx (236) ) ( yyLbxAx u (237) produce estimated states x and estimated outputs xCy as a function of the measured system output Cxy, and the Luenberger gain L in Eq. 237. The control input PAGE 24 24 xF u (238) is used to place the poles of Eq. 236 at the desired locations. The error xx (239) is defined as the difference between the actual values of state vector x and estimated values of state vector x Hence, the derivative of the error xx (240) is computed by differentiating Eq. 239. Substitu ting Eq. 236 and Eq. 237 in Eq. 240 yields LC) (A (241) Invoking now the Variation of Parameters form ula the solution to th e differential equation Eq. 241 is te)( 0 LCA (242) where ) 0( )0(0xx is the initial value of the error ) 0( x is an initial guess of the value of the estimated state vector, and )0( x is the initial value of the state vector The poles of matrix ALC should lie on the open half pl ane for the value of error to evolve to zero. The poles are placed at the desired location by an appropriate choice of L, which for loworder systems as the one considered here can be easily done via Ackermans Pole Placement algorithm. Linear quadratic regulator control design. A Linear quadratic control regulator (LQR) control design is implemented in the VCL. The LQR control law Kx u (243) is obtained by minimizing the cost function dtuRuu uJT T T)2 ()(0 Nx xQx (244) PAGE 25 25 with respect to u. The weighting matrix Q must be symmetric positive semidefinite, and R symmetric positive definite. The weighting function Nis specified to be zero. For a linear state space system u BAxx (245) the solution to the minimization of the cost f unction results in the steadystate Riccatti equation [36] 0Q)NS(BN)R(SBSASAT T1 T (246) The acceptable solution to Eq. 246 is a positive definite matrix S which is then used to specify K from the expression )NS(BRKT T1 (247) Since in this case 0 N, therefore SBRKT1 (248) 2.5 Realization of an InvertedPendulum VCL The LabVIEW software and a VCL infrastructure proposed by Peek et al., is used for the implementation of the Inverted Pe ndulum VCL [12, 13]. Previous softwarebased controltools for the invertedpendulum system reported in the literature have significant value, but the VCL developed in this study has a number of additional desirable pedagogical featur es [14]. Initially, standalone VIs and subVIs are generated using LabVIEW software for different components of the design before integrating them as a part of a monolithic VCL. There are several reasons why National Instruments LabVIEW software is us ed for constructing the VCL. The ease of structuring and maintaining a VCL is significantly high in this software. The LabVIEW software has builtin features for deploying applications on the web. The software also has toolkits specifically designed for control e ngineering. Implementation of a VCL using PAGE 26 26 LabVIEW does not rely on support from other soft ware packages, as would be the case if some higherlevel language is used to implem ent the same features found in VCL. Figure 22 shows the front panel of VCL as it ap pears to a student user. The key elements are the Animation and Interaction Panels, respec tively, located on the top and bottomleft areas of the front panel. These two are very critical components of the VCL as the user makes most modifications in the plant and controller set up in the Interaction pa nel and instantaneously observes an animated result describing the plan t and states in the Animation Panel. The Animation Panel has a twodimensional graphic re presentation of an inve rted pendulum. When the VCL is operated, the animated cart responds to the control input by moving to the left or right and causing a pendulum swing. The third panel is the Navigation Panel on the bottomright area of the front panel. The Navigation Panel ha s five tabs (Information, Plant, Controller, Analysis and Simulation), which provide various pieces of information about the VCL. More information is given about these thre e panels in the ensuing subsections. Animation, interaction and navigation panel. The animation panel plays the role of providing a visual representation of the plant, namely an inverted pendulum. Any changes that are made to the inverted pendulum mounted on th e cart are visually depicted in the Animation Panel. The user has the ability to make changes to the plant and controller in the Interaction Panel. The user can adjust plant parameters, in itial condition of the states of the inverted pendulum and assign the different values to control parameters to the controller of choice. The user also has the ability to run the plant in Manu al or in Auto mode. Figure 23 depicts some of the various modes that the user can confi gure parameters in the Interaction Panel. PAGE 27 27 The user has to specify the initial states of the pendulum (position, velocity, angle of the pendulum with the vertical and the angular velo city of the pendulum). The Animation Panel constructs the visual representa tion of the inverted pendulum ba sed on the information that the user provides. The user has th e flexibility of running the VCL in the following two scenarios: (1) all states are measured, or (2) only one state is measured. Based on the choice of the user, an implementation of the correspondi ng controller is given. When Manual F Control is in the Off position, the user is allowed to c hoose the poles for the closed loop matrix ABF and the value of matrix F is calculated from Ackermans pole placement algorithm. The user can immediately see the impact of poles chosen on the stability of the inverted pendulum in the Navigation Panel under the Analysis tab. When the Manual F Control is in theOn position, the user has the ability to choose the values of the elements of matrix F. When the controller is operated in Luenberger Observer mode, as show n in Figure 2, the user has to provide the desired poles for matrix ALC. The only state that can be measured in this mode is the position of the cart. When the cont roller is in Off mode, i.e., the system is in open loop configuration with no feedback, the Analysis tab of the Navigation Panel show s that the pendulum is in an unstable configuration, which is as certained by the fact that there is one eigenvalue of the system in the open right half plane. Th e Interaction and Animation panels provide a suite of options and visual representation for the user. The Navigation panel is located on the bottomri ght area of the front panel of the VCL. The Navigation panel has five tabs entitled: (1) Help, (2) Plant, (3) Controller, (4) Analysis, and (5) Simulation. These tabs provide pertinent and critical information about the VCL to the user. The Help tab, when clicked on, provides genera l information about the op eration of the VCL. The user can access the Help tab without having to leave the VCL. The Help tab displays an PAGE 28 28 embedded PDF file. The Plant and Controller ta bs are also embedded PDF files which provide information about the dynamics of the plant (inve rted pendulum) and the controller (proportional state feedback and Luenberger Observer). The nonlinear equations and linear state space system for the inverted pendulum are explained in the Plant tab of the Navigation Panel. Figure 23 shows the Plant tab of the Navigation Panel. Th e Controller tab provides information about pole placement and various other aspects of control de sign for inverted pendulum VCL, as shown in Figure 24. The Analysis tab, shown in Figure 25, has information about the tools and graphs that are used in control theory. This tab has information about the transfer function, location of poles and zeros in complex plane and Bode plot (freque ncy response). The Simulation tab, shown in Figure 26, shows plots of the re sults of numerical simulations describing the states (position, velocity, angle and angular veloci ty) and inputs as a function of tim e. Since this is a regulation problem, when stable choices of eigenvalues ar e given, using the lin ear state space as the dynamic model, all states converge to the value zero, regardless of the choice of initial state vector. The Simulation tab provides the real time curves of all the states as a function of time. The RungeKutta integration algor ithm is employed to compute the numerical response of the plant to the controlling input. When the controller is toggled between On and Off modes (closedloop and openloop behavior, respectivel y) in the Interaction panel by the user, the impact of that change on the value of states is depicted immediately in the Simulation tab. The LQR control strategy, as shown in Figure 2, employed in the VCL gives the user the opportunity to implement different control choices, such as va rying the weighting on different elements of the cost function and displaying the corresponding LQR gain. PAGE 29 29 2.6 Conclusions A VCL for the control of an Inverted Pendulum is described. The Inverted Pendulum is a classic example of illustrating statespace model representations and demonstrating the classical control concepts of controllabili ty and observability. The anima tion features of the VCL provide a visual description of how an inverted pendul um responds as a function of the input force applied. The user is given the opportunity to ru n the VCL in different modes, such as in openloop and closedloop configur ations of the system. The VCL can be utilized as a tool for e nhancing learning. The three most widely recognized learning modes (active le arning, flexible learning, and le arning via discovery) can be easily executed using this VCL module. The mo dule can be used in conjunction with a process control lecture for demonstrating various concepts. The animation capabilities allows the user to see the impact of every change that is made to the control configuration. The Analysis tab also demonstrates that the openloop configuration (u nforced system) is unstable, as one of the eigenvalues is in open right half plane. The user is given the c hoice of choosing the poles for the system and noticing its impact on the plant. Th e user has the choice of running the VCL in two modes: (1) all states measured or (2) only one state is meas ured and implementation of a Luenberger Observer. This utilization of th e VCL supports active learning of the control material. The VCL can also be presented before or after a conventional lectur e. In the first case, the users motivation to learn about theory presente d in class would be enhanced as they have the opportunity to develop and experi ence with the VCL. In the later case, af ter the lecture is delivered, an interaction with the VCL would se rve as an excellent tool for reinforcing the concepts taught in class. There is significant value in learning an abst ract concept taught in lecture and being able to relate to the concept by virtue of seeing the animation and graphs from PAGE 30 30 simulation. In an ideal scenario the VCL should be used in all three modes (before, during, and after lectures). The tool can also be used to support group activ ities in a class for l earning control theory and doing control homeworks [15]. The modular f eature of the VCL can be taken advantage of by implementing sequential learni ng. In this mode, different versions of same VCL are progressively given to the user as the class progresses. Each successive version makes more features of the controller available to the user, hence helping students appr eciate and learn faster the concepts as they are progr essively taught in the class. To validate the benefits and effectiveness, it is proposed that the VCL should be used as a pilot teaching tool in process cont rol classes. The feedback obtai ned from the students would be highly beneficial in optimizing and incorpora ting features that could enhance the learning experience. PAGE 31 31 mg l f z Mg x Figure 21. Inverted pendulum. PAGE 32 32 Figure 22a. Front panel of the VC L where all states are measured. PAGE 33 33 Figure 22b. Front panel of the VCL showing observer. PAGE 34 34 Figure 23. Interaction pane l of inverted pendulum VCL. PAGE 35 35 Figure 24. Controller ta b of the navigation panel. PAGE 36 36 Figure 25. Analysis ta b of the navigation panel. PAGE 37 37 Figure 26. Simulation tab of the navigation panel. PAGE 38 38 CHAPTER 3 PI AND PI2 CONTROLLER TUNING FOR TRAC KING THE SLOPE OF A RAMP 3.1 Introduction and Background In certain applications tracking the slope of the ramp is more critical than tracking the ramp itself. It is not unusual to encounter applications where the set point is the slope of a ramp trajectory. For instance, while growing thin films on a substrate, it is desired that the temperature of the substrate in the reactor increases at a steady rate (i.e., following a trajectory with a specified slope). In these kinds of applications, it becomes crucial to adopt the appropriate choice of controller type with effective values fo r the tuning parameters and an appropriate metric to ensure adequate performa nce. Simplicity of tuni ng relationships plays a critical role in the implementation of a controlle r. Most successful t uning relationships have been developed via simulation [16, 17]. The cost function normally used involves the feedback error, which is the difference between the set point and out put of the plant. A proportionalonly controller lead s to a steady state offset with respect to step changes in set point. An integrator needs to be incorporated in the cont roller to eliminate the offset. Similarly, for a ramp set point, a proportionalinte gral (PI) controller is not sufficient to remove the steady state offset. In this case a contro ller with two integrators, that is a proportional doubleintegral (PI2) controller, is needed to remove the o ffset [18]. However, a PI controller is sufficient to track the slope of the ramp trajectory. Belanger and Luyben proposed a proportionalintegraldouble integral controller relating the tuning parameters to the ultimate gain and ultimate period of th e plant [19]. AlvarezRamirez et al. extended the work of Belanger and Luyben and coined the acronym PI2 [20]. Peek examined three different versions of PI2 controllers for tracking ramp set point [21]. PAGE 39 39 This study investigates the problem of controller tuning for tracking the slope of a ramp signal as depicted in Figure 31. The intent is to design a controller that leads the output trajectory to follow a line parallel to the ramp. It is also critical to minimize transients. The control performance is deemed as poor when the system experiences large deviations during the transients. To investigate the problem, a first order syst em with time delay is adopted as the plant model. Since the set point is a ramp, integral action needs to be incorporated in the control design. Two kinds of controllers are used for this study, namely a proportionalintegral (PI) controller and proportion al doubleintegral (PI2) controller. Both contro llers serve the purpose of placing the system output on the desired line, parallel to the ramp set point. However, there is an offset with respect to the original ramp trajectory when PI controllers are used because only one integrator is included in the control scheme. To identify the appropriate indicator to meas ure the performance of the controller with respect to the system and objective in question, th ree different metrics are used: integral of the timeweighted absolute error (ITAE), integral of the absolute error (IAE), and integral of the square of error (ISE). Each of the controllers is tuned to minimize the metric value for a given set of plant parameters. The Simplex optimizati on routine is used to acq uire tuning parameters via the minimization of the metric adopted. The MATLAB platform in conjunction with the SIMULINK module is used for conducting the simula tions. A set of optimal tuning relationships for controllers to track the slope of a ramp for 20,000 different plants are presented. The performance of all the controllers is evaluated an d results are compared with the prior work of Belanger and Luyben. The performance of the cont rollers is also compared with that of those proposed by Peek. PAGE 40 40 3.2 Problem Statement and Approach The objective is to use PI and PI2 controller to make a firs torder plant with time delay follow the slope of a ramp. The plant is represented by the transfer function s pe s K sG 1 ) ( (31) where K is the gain, is the time constant, and is the time delay of the plant. The input to the plant is denoted as u and the output as y. The transfer function representation of the PI controller is s KsGi c c1 1)( (32) and the transfer function for the PI2 controller investigated here is 21 1)( s KsGi c c (33) where cK is the proportional gain, andi is the integralaction time constant. The plant and the controller are configured in the closedloop arrangement shown in Figure 32. The ramp function serving as the set poi nt in Figure 32 is given by ttr )( (34) where t is the time and the constant slope is taken as 1 Three error metrics are used for tuning the c ontrollers, namely, the integral of the timeweighted absolute error (ITAE), the integral of the absolute error ( IAE ), and the integral of the square of error (ISE ), respectively defined by the integral equations f ft tdttytrt ITAE0)()( lim (35) PAGE 41 41 f ft tdttytr IAE0)()( lim (36) f ft tdttytr ISE0 2)()( lim (37) where ) (ty is the plant output and ft is the extent of time over which the metric is computed. When the closed loop uses a PI controller, the output of the system exhibits a steady state offset ssewith respect to the original ramp charact erized through the analytical expression c i ssKK e (38) the derivation of which is given in the APPEND IX using Final Value Theorem. A modified or auxiliary ramp ) (tra is defined via the relationship ss aetrtr )() ( (39) Figure 31 shows the ramp and th e auxiliary ramp trajectories. The corresponding auxiliary error metrics for PI controller, modi fied from Eq. 34, Eq. 35 and Eq. 36 are given by f ft a t adttytrt ITAE0)()( lim (310) f ft a t adttytr IAE0)()( lim (311) f ft a t adttytr ISE0 2)()( lim (312) where the aITAE is the auxiliary integral of the timeweighted absolute error, theaIAE is the auxiliary integral of the absolute error, and the aISE is the integral of the square of error. It is to PAGE 42 42 be noted that in the case of PI2 controllers, the value of sse is zero i.e., ) ()(trtra Hence, the ITAE, IAE and ISE error metrics for PI2 retain the original form of Eq. 35, Eq. 36, and Eq. 37, respectively, even when all computations are carried out using their auxiliary counterparts. The ITAE expression (Eq. 35) has been a popular choice for control parameter optimization, as it assigns less weight to errors occurring in the initial times and more weight to error at longer times. Error is defined as the closedloop feedback difference between the auxiliary set point and th e output. This is tradit ionally a useful measure to adopt, as for a step response it is inevitable that there is a relativel y large error during initia l times, which needs to be given less significance compared to the error that is encountered at later times. The ITAE may not be the best metric for the problem in que stion, however. The desired trajectory of the output is the one which tracks the slope of the ramp without abrupt deviations in trajectory. The ITAE is forgiving of aggressive output values at initial times as it gives less significance to error at early times. Sometimes, it is desired to adopt a metric which gives an equal importance to errors occurring at initial times as well. From that perspective, the two other metrics are also used in this work for optimization purposes, na mely, the IAE (Eq. 36) and the ISE (Eq. 37). A routine in MATLAB is written to simulate the plant output fo r a given controller with its parameters specified [22]. The simplex optimi zation routine is employed for tuning the control parameters. For a fixed value of plant parame ters (gain, time constant and time delay), the parameters of the controller (pr oportional gain and integral action) are altered to minimize the (ITAE, IAE or ISE) value of the co st function. For a PI controlle r, where there is offset with respect to the ramp, an auxiliary ramp parallel to original one is constr ucted. The steady state offset is calculated analytically using Eq. 38. PAGE 43 43 The three error metrics (ITAE, IAE and ISE) are minimized using the new ramp. These metrics are defined over an infinitely long time; however, for practical purposes the final time is chosen to be finite and defined by the formula ) ,max(15 ft (313) The rationale behind this choice is that by this extent of fina l time, any reasonably performing controller should make the value of error significantly small. The optimized tuning parameters are nondimensionalized by combining with plan t parameters, and plots of optimized tuning parameters were constructed. Time responses are constructed with the optimized values to verify the responses of the process with the controlling action incorporated. Peek analyzes the performance of th ree different configurations of PI2 controllers [21]. That study concludes that there is no significant difference in the performance of the three configurations. The transfer function for PI2 controller given in Eq. 32 is used in this study as it is the easiest to tune becaus e it involves only two parameters, namely the proportional gain cK and the integralaction time constant i 3.3 Results and Discussion 3.3.1 Tuning Parameters of Controllers Tuning parameters are calculated for both controllers PI and PI2 using the ITAE, IAE, and ISE as the optimizing metric. The optimized c ontroller parameters are nondimesionalised using the plant gain and time constant [17]. The resulting plots of KKc versus / and / I versus / are shown in Figures 33 and 34 for the PI2 and PI controllers. The tuning parameters obtained using ITAE, IAE and ISE as the metric are shown in the first, second, and third row of each figure, respectively. The plant parameters K, and are selected such that K and range from 0.1 to 50. Twenty logarithmically equallyspaced points are considered for both K and in their PAGE 44 44 specified ranges. After each time constant is defined, the values of the delay parameter is set by defining the ratio of / to range from 0.1 to 100, with 50 logarithmically equallyspaced points inside the range. For a fixed value of / ratio, the value of is computed from the value of and of the fixed / ratio. Hence, tuning parameters were obtained for 20,000 different plants for each controller and for each metric. The graphs in Figures 33 and 34 show that, in general, as the / ratio increases, the value of the optimal KKc product and of the / i ratio decrease. The value of the KKc product represents the proportional control action on the closed loop sy stem, and it is expected to vary inversely to the / ratio. In other words, the control ac tion will be higher for smaller values of / ratio, and smaller as / increases. This is qualitatively reflected in Figures 33 and 34 for all three optimizing metrics and for the two controllers considered. On the same vein, the / i ratio is indicative of integral action for the system and it is expected to behave analogously to the proportional control action. The in tegral action should be more aggr essive for smaller values of / ratio compared to higher values of the ratio. This is, ind eed, observed from Figures 33 and 34. Leastsquare fits for the optimized control pa rameters are given in Tables 31 and 32 for the PI2 and PI controller, respectively. The least square fit relates the optimal KKc product with the / ratios and the optimal / i ratio with the / ratio. If the KKc versus / curve and/or / i versus / curve is significantly nonlinear, a br eak point at a certain value of / ratio is identified and two leastsquare fits are presented for the same curve, one above and one below the / ratio breakpoint value. The results are for / values ranging between 101 and 102 only. 3.3.2 Comparison of the Performance of the PI and PI2 Controllers The performance of the PI and PI2 controllers is characterize d for three different plants (plant parameters given in Table 33) using th e optimal parameters prescribed by each of the PAGE 45 45 three optimizing metrics. The time response curves for each plant are constructed, for both the controllers and the tuning parameters prescribed by the respective optimizing metric, to assess the timedomain performance of each tuning pres cription. The proportiona l gain of the three plants was taken to be the same value, namely K = 1.0. Three different values of the / ratio are taken from the domain of the values for 20,000 differe nt plants. The ratio values of 0.1, 3.0 and 100 (minimum, middle and maximum of the ra nge considered) are selected. Several combinations of / can satisfy each value of the ratio. The value of is selected such that it covers the domain of the different values of selected for all the plants in this study. The values of 0.1, 1.9 and 50 are selected for From these values of the value of is computed for each value of the ratio. Figures 35, 36 and 37, illustrate the time re sponses of the three plants for the ITAE, IAE and ISE metrics, respectively, using PI2 and PI controllers. Each figure demonstrates the performance of PI2 versus PI controller for the three plants. When a PI2 controller is used, the output follows the original ramp, whereas with a PI controller an offset is introduced in relation to original ramp and the output follows the auxiliary ramp parallel to the original ramp. The original and auxiliary ramp are plotted in each time response curve as well. With increasing values of the / ratio, the offset between the original a nd auxiliary ramp increases. This is expected as the offset is directly proportional to i and inversely proporti onal to the product of KKc as shown by Eq. 38. With increasing value of / i increases and the product of KKc decreases, hence the offset increases. It is also ob served that it takes longer for steady state to be reached with increasing values of the / ratio. The output of each plant tracks the slope of the original ramp for each controller and every optimizing metric. As discussed earlier, there is offset with respect to the original ramp when PI PAGE 46 46 controller is used and there is no offset when PI2 controller is used. Re gardless of the fact of whether offset is introduced or not, as long as the output is para llel to the original ramp, the control objective is satisfied. The better performing controller is the one that reaches faster the original or auxiliary ramp, depending on the controller adopted, and with minimal transient values. It is observed that for all the thr ee plants and optimizing metrics considered, PI controller performs better than PI2 controller. Output of each of the plant, on using PI controller, tracks the auxiliary ramp much fast er compared to the output when PI2 controller is used. It is observed that the plant output is more oscillato ry during transient time for ISE prescribed tuning compared to those obtained from the ITAE and IAE cr iteria. This is an expected result as in the ISE metric the square of the er ror is used. Even though both PI2 and PI controllers exhibit oscillatory behavior, the phenomenon is more prominent in the case of PI2 controller. From these observations, it is concluded that the PI controllers are better performing than PI2, regardless of the optimizing metric adopted to tu ne, for the three plants considered, and provided that the unavoidable resulting offs et is acceptable to the user. 3.3.3 Comparison of Metrics After it is established that PI is a better performing controller, the next step is to identify the optimizing metric with which the PI controlle r gives the best performance. Time responses for the same three plants are constructed using the optimizing metrics. Figures 38 and 39 shows the time responses for the three metrics. For the sake of comparison, even though it is established that the PI is a be tter performing controller, time response curves are generated for the PI2 controller as well. Figure 38 shows the PI controller time responses and Figure 39 shows the PI controller time respons es, for the three plants. Note th at the offset introduced while using PI controller, is a func tion of the tuning parameters Kc and I Since the values of these parameters are different for each metric, the ou tput using a PI controller follows a different PAGE 47 47 auxiliary ramp, depending on which optimizing me tric was used. All the auxiliary ramps, however, are parallel to the original ramp. It is observed that ISE is the worst optimizi ng metric, as the output of the plant is most oscillatory and has larger deviation from the auxiliary ramp as shown in Figure 38 and 39. Also, it takes longer in the case of the ISE to reach steady state. The other two metrics, ITAE and IAE, are quite close in their performance. The time response curves suggest that for tuning purposes the ITAE is a better metric for the PI co ntroller and the IAE is better metric for the PI2 controller. For the PI controller, the IAE demonstrates mo re oscillatory output co mpared to the ITAE metric. Also, the output of the plants reaches steady state sooner when the ITAE is used as the optimizing metric compared to when the IAE is adopted. 3.3.4 Comparison of PI (ITAE) Co ntroller with Literature Precedents After determining that the PI controller using the ITAE as the optimizing metric exhibits highly desirable performance, the next step is to compare its performance with controllers proposed in the literatu re. Peek suggests a PI2 controller with ITAE as the optimizing metric for tracking a ramp set point [21]. Though the objective is slightly di fferent than the one in this study, the controller recommended by Peek does satisfy the control ob jective of this work [21]. Belanger and Luyben recommend a double integrator controller for tracking a ramp, which satisfies the control objective of this study as well [19]. Figure 310 shows the performance of the three different controllers for the three plants considered. It is observed that the PI contro llers tuned using the ITAE metric gives the best results followed by the controllers recomme nded by Peek, and then by those proposed by Belanger and Luyben. As the / ratio increases for the plants it becomes increasingly obvious that the PI controller gives the best results. For the plant with the ratio / = 100, the highest PAGE 48 48 value, the PI controller makes the output reach st eady state much faster and with least transients compared to the other two controllers. 3.3.5 Local Minima versus Global Minima Figure 311 shows the contour plots for the ITAE metric using PI controller for the three plants. It is observed from Figure 311 (A) that there is more than one minima for the ITAE metric and different values of control parameters Kc and i for each minima. The goal is to obtain global minima for the optimizing me tric and the corresponding optimal control parameters. If local minima is reached as oppos ed to global minima, that would potentially lead to scatter in the optimal control parameters curves as shown in Figures 33 and 34. Hence, the optimization work described here is conducted wi th care to avoid local minima results. The measure adopted consists of utilizing different initial guesses for each optimization routine execution, leading to the identification of different local minima, when they exist. The smallest of such minima is then accepted as the best ap proximation to the global minimum. Although this approach is neither rigorous nor exhaustive, it provides excellent practical results in the context of this study. 3.4 Conclusions Optimal tuning relationships for the PI and PI2 controllers using th e optimizing metrics ITAE, IAE and ISE are presented for 20,000 firstorder plants with time delay. The plants considered have the / ratio value varying from 101 to 102. The validation and comparison of the controllers performance is done by thei r deployment on three different plants. On the basis of results obtained, the PI contro llers using ITAE as the optimizing metric are the best performing controllers for the purpose of tracking the slope of a ramp trajectory. They perform better than controllers proposed in prio r literature. PAGE 49 49 Figure 31. Ramp r and auxiliary ramp ra with constant slope, Figure 32. Closed loop transfer functi on representation of plant and controller. PAGE 50 50 Figure 33. The PI2 controllers optimal tuning parameters using ITAE (A, B), IAE (C, D), and ISE (E, F) as the optimizing metric. PAGE 51 51 Figure 34. The PI controllers optimal tuning parameters, using ITAE (A, B), IAE (C, D), and ISE (E, F) as the optimizing metric. PAGE 52 52 Table 31. The PI2 controllers optimized tuning paramete rs linear least square fit equations. Criterion Optimal PI2 Parameters ITAE 5.2 74.0 5.2 48.0 5.2 74.0 5.2 26.1 92.0 50.0 04.0 76.0 i cKK IAE 5.2 69.0 5.2 48.0 5.2 78.0 5.2 26.1 89.0 50.0 06.0 76.0 i cKK ISE 5.2 65.0 5.2 45.0 5.2 94.0 5.2 58.1 88.0 52.0 08.0 76.0 i cKK PAGE 53 53 Table 32. The PI controllers optimized tuni ng parameters linear least square fit equations. Criterion Optimal PI Parameters ITAE 5.2 58.1 5.2 82.0 5.2 41.0 5.2 73.0 88.0 13.0 06.0 73.0 i cKK IAE 5.2 42.1 5.2 73.0 5.2 57.0 5.2 07.1 88.0 26.0 08.0 80.0 i cKK ISE 5.2 42.1 5.2 75.0 5.2 74.0 5.2 40.1 88.0 36.0 08.0 80.0 i cKK Table 33. Values of the plant parameters us ed compare the performance of the controllers. Parameters Ratio Plant K / 1 1 0.1 0.01 0.1 2 1 1.9 5.6 3.0 3 1 50 50000 100 PAGE 54 54 0 0.4 0.8 1.2 1.5 0 0.4 0.8 1.2 1.5 ty A K = 1 = 0.1 = 0.01 PI2 Ramp PI Aux 0 18 37 56 75 0 18 37 56 75 ty B K = 1 = 1.9 = 5.6 PI2 Ramp PI Aux 0 1.9 3.8 5.6 7.5 x 10 4 0 1.9 3.8 5.6 7.5 x 10 4 ty C K = 1 = 50 = 5000 PI2 Ramp PI Aux Figure 35. The PI2 and PI controllers ramp tracking a nd slope tracking performance using the optimal ITAE control parameters. PAGE 55 55 0 0.4 0.8 1.2 1.5 0 0.4 0.8 1.2 1.5 ty A K = 1 = 0.1 = 0.01 PI2 Ramp PI Aux 0 18 37 56 75 0 18 37 56 75 ty B K = 1 = 1.9 = 5.6 PI2 Ramp PI Aux 0 1.9 3.8 5.6 7.5 x 10 4 0 1.9 3.8 5.6 7.5 x 10 4 ty C K = 1 = 50 = 5000 PI2 Ramp PI Aux Figure 36. The PI2 and PI controllers ramp tracking a nd slope tracking performance using the optimal IAE control parameters. PAGE 56 56 0 0.4 0.8 1.2 1.5 0 0.4 0.8 1.2 1.5 ty A K = 1 = 0.1 = 0.01 PI2 Ramp PI Aux 0 18 37 56 75 0 18 37 56 75 ty B K = 1 = 1.9 = 5.6 PI2 Ramp PI Aux 0 1.9 3.8 5.6 7.5 x 10 4 0 1.9 3.8 5.6 7.5 x 10 4 ty C K = 1 = 50 = 5000 PI2 Ramp PI Aux Figure 37. The PI2 and PI controllers ramp tracking a nd slope tracking performance using the optimal ISE control parameters. PAGE 57 57 0 0.4 0.8 1.2 1.5 0 0.4 0.8 1.2 1.5 ty A K = 1 = 0.1 = 0.01 ITAE IAE ISE Ramp 0 18 37 56 75 0 18 37 56 75 ty B K = 1 = 1.9 = 5.6 ITAE IAE ISE Ramp 0 1.9 3.8 5.6 7.5 x 10 4 0 1.9 3.8 5.6 7.5 x 10 4 ty C K = 1 = 50 = 5000 ITAE IAE ISE Ramp Figure 38. The ITAE, IAE, and ISE metric s comparison for three plants, using the PI controllers A), B), and C). PAGE 58 58 0 0.4 0.8 1.2 1.5 0 0.4 0.8 1.2 1.5 ty A K = 1 = 0.1 = 0.01 ITAE IAE ISE Ramp 0 18 37 56 75 0 18 37 56 75 ty B K = 1 = 1.9 = 5.6 ITAE IAE ISE Ramp 0 1.9 3.8 5.6 7.5 x 10 4 0 1.9 3.8 5.6 7.5 x 10 4 ty C K = 1 = 50 = 5000 ITAE IAE ISE Ramp Figure 39. The ITAE, IAE, and ISE metric s comparison for three plants, using the PI2 controllers A), B), and C). PAGE 59 59 0 0.4 0.8 1.2 1.5 0 0.4 0.8 1.2 1.5 ty A K = 1 = 0.1 = 0.01 BL PI Peek Ramp Aux 0 18 37 56 75 0 18 37 56 75 ty B K = 1 = 1.9 = 5.6 BL PI Peek Ramp Aux 0 1.9 3.8 5.6 7.5 x 10 4 0 1.9 3.8 5.6 7.5 x 10 4 ty C K = 1 = 50 = 5000 BL PI Peek Ramp Aux Figure 310. The PI controllers tuned using th e ITAE metric compared with Belanger and Luyben and Peeks controllers for three plants A), B), and C). PAGE 60 60 1/iKc A K = 1 = 0.1 = 0.01 8 10 12 3 3.5 4 1/iKcK = 1 = 1.9 = 5.6 0.2908 0.2908 0.2908 0.4029 0.403 0.4031 1/iKcK = 1 = 50 = 5000 5.0454 5.0505 5.0556 x 10 0.3174 0.3174 0.3175 Figure 311. Contour plots for the PI controller tuned using the ITAE metric for the three plants A), B), and C). PAGE 61 61 CHAPTER 4 GENERALIZED PREDICTIVE CONTROL FOR FUEL CELLS 4.1 Introduction A generalized predictive cont rol (GPC) strategy is designed and implemented for a polymer electrolyte membrane (PEM) fuel cell system It is vital for efficient performance of the fuel cell to ensure the robust and precise control of the performance variable which is oxygen excess ratio. A review of prior control strategies developed for this fuel cell system, is presented and compared to the new GPC scheme proposed. The performance of the di fferent controllers is evaluated in the case where the model avai lable for design suffers from uncertainty. 4.2 Fuel Cell System Background Fuel cell are electrochemical devices that dire ctly convert the chemical energy of gaseous reactants to electrical energy. Th ey are widely considered as an alternative to fossil fuels which are limited in supply. For a typical fuel cell, water and heat are byproducts generated as a result of operation. The reactants need ed are hydrogen and oxygen. Both of these reactants are widely available, and a proliferation of applications based on these fuels would tremendously reduce our dependence on fossil fuels. This is an additiona l motivation for developing and engineering fuel cell system as it is friendly to the environmen t. Fuelcell based automobiles have no harmful emissions, such as CO2 which combustionengine automobiles contribute significantly to the environment [23]. Fuel cells are an effi cient and clean source of energy production. William R. Grove discovered the principle of op eration of fuel cells in 1839 [24]. From a classical standpoint, a fuel cell is comprised of two electrodes with an electrolyte located between them. The electrolyte has the special property that it allows only protons (positively charged hydrogen atoms) to pass through it. In contrast, the membrane does not allow electrons to pass through. Hydrogen gas passes over the an ode electrode, and with the assistance of a PAGE 62 62 catalyst, breaks down into protons and electr ons. The protons sele ctively pass through the membrane to reach opposing cathode electrode. Th e membrane is an electronic insulator. The membrane is comprised of fluorocarbon chain to which the sulfonic acid groups are attached. On hydration of the membrane the hydrogen ions b ecome mobile. The electrons flow through an external circuit, creating a curr ent flow. An oxygen flow passe s over the cathode and combines with the protons and electrons to generate water. The reaction at the anode is 2H2 4H+ + 4e(41) and at the cathode is O2 + 4H+ + 4e2H2O (42) hence, the overall reaction is 2H2 + O2 2H2O (43) Several kinds of fuel cell designs have been developed and are curren tly being studied [2530]. A schematic diagram of an automotive fuel cell system including the structural relationships among the input, outputs, and distur bance signals is give n in Figure 41. A compressor and pressurized hydrogen tank are used to provide the reactants oxygen and hydrogen, respectively [31]. The compressor plays a crucial role as it ensure s that the desired air flow rate reaches the cathode based on power dema nds. The supply and return manifold models based on thermodynamic consideration provide inform ation about desired air flow rate needed. A nonlinear curve fitting method is used to desc ribe the compressor behavior [31]. The net power delivered by the fuel cell system is the difference between the power generated and power consumed to run the compressor motor to deliver a particular air flow rate An excess amount of air flow provided to the cathode is referred to as the oxygen excess ratio 2O which is defined as the ratio of the rate of oxygen supplied to rate of oxygen consumed. With increase in oxygen PAGE 63 63 excess ratio, resulting in high oxygen partial pressure, there is increase in power delivered. However, it is at the cost of increased power consumption by the compressor to deliver a higher air flow rate. Beyond a particular value of the oxygen excess ratio there is loss in net power as a result of increased power consumption by the comp ressor. It has been shown in literature that having oxygen excess ratio in the vi cinity of 2, the fuel cell sy stem delivers highest net power. A humidifier, in fuel cell system, is used to add water to the reactan ts to avoid dehydration of the membrane. A water separator is used to extract water from the ai r, leaving the fuel cell stack, which is recycled back to the humidifier via the water tank. The voltage generated by the fuel cell needs to be conditioned before it is fed to the traction motor. Appropriate usage of an external battery with th e fuel cell power supply helps in mi nimizing the transient responses and delivering better system efficiency. Polymer electrolyte membrane (PEM) fuel ce lls, also referred to as proton exchange membrane fuel cells, are recognized particularly promising for ut ilization in automobiles as a substitute for the internal combustion engine. This is because of the fact that the PEM fuel cells have high power density, long cell life, low corros ion, and use a solid electrolyte. For effective utilization of fuel cell technology, it is vital that that the fuel cell system is accurately understood, monitored and that its process variab les be held under tight control for various operating conditions. From a control engineering perspective the fuel cell can be divided into four subsystems, namely (1) supply of the reactants air and hydroge n, (2) humidification of the reactants and of the membrane, (3) heat management, and (4) power management. The model used in this study assumes a perfect humidifier and coolers for the reactants and the membrane. The model assumes a perfect power management system whic h controls the power drawn from the fuel cell PAGE 64 64 stack. A fast proportional controller is impl emented on the hydrogen flow that tracks cathode pressure [31]. This reduces the control problem to regulating the air suppl y, as the oxygen level varies on the cathode side, due to varying power demands. 4.3 Objectives of the Research The objectives of this study are the following: 1. Propose a systematic design solution for trea ting the of fuelcell control problem of regulating the oxygen excess ratio (perform ance variable) at a value of 2 by synthesizing a GPC strategy. 2. Develop a systematic GPC design procedure for a fuel cell model in the scenario where all states are measurable. 3. Develop a systematic GPC design procedure for the fuel cell model when selected states are not measurable by incorpora ting an observer in the controller. 4. Evaluate the robustness of the GPC contro ller with respect to model uncertainty. 5. Rederive and correct, as needed, lin ear models obtained from nonlinear formulations proposed in the prior literature for the fuel cell model used for this study [31]. 6. Retune and redesign, as needed, all mode lbased controllers pr oposed in the prior literature [31] to take into account the corrected linear models [31]. 7. Compare by means of simulations the perf ormance of the different controllers proposed in prior literature [31] with the GPC strategy. 4.4 Fuel Cell Model The fuel cell model proposed by Pukrushpan et al. is the basis for this study [31]. The state equation for the model of a fuel cell system is of the form ),,( wuxfx (44) ),,( wuxgz (45) ),,( wuxhy (46) where x represents the states of the fuel cell system, u is the input, w is the disturbance, and z and y are the outputs. The state vector is given by the expression PAGE 65 65 T rm anwsm sm cp N H Opmmp mmm ] [,2 2 2 x (47) where the vector elements are the mass of oxygen 2Om in cathode volume, the mass of hydrogen 2Hm inside the anode volume, the mass of nitrogen 2Nm in cathode volume, the rotational speed of the compressor cp the pressure of the supply manifoldsmp, the mass of air in the supply manifold smm, the mass of water at anode anwm,, and the pressure of the return manifold rmp. The outputs used as the performance va riables are organized in the vector T O Pnete ][2 z (48) where del set PPPenetis the difference between the desired power set P and the actual power delivereddel P and 2O is the oxygen excess ratio. The contro l objective in this study is confined to regulating the value of 2O at the desired value 22set O [31]. Three additional measured outputs are organized in the vector T stsm cpvpW ] [ y (49) where the vector elements are the mass fl ow rate of the air from the compressor cpW, the pressure of the supply manifoldsmp, and the voltage of the stack stv. The control input is given by cmvu (410) where cmvis the voltage signal sent to the compressor which in turn delivers a corresponding flow rate of air to the cathode. Finally, the disturbance stIw (411) is the stack current Ist. PAGE 66 66 The nonlinear model described in Eqs. 44 46 and shown in Figur e 42a is linearized about a nominal operating point at which th e fuel cell model generates a net power of del P =40kW and sustains an oxygen excess ratio 22O. These conditions are realized when 191stIA and 164cmvV. The corresponding values of the states are T T rm anwsm sm cp N H Oeeeeeeee pmmp mmm]58.131.128.352.232.823.146.530.2[ ] [,2 2 2 x (411a) The physical units of the states x are [kgs kgs kgs rad/s pasc al kgs kgs pascal] (411b) respectively. Hence the nominal ope rating point is gi ven by the input 164 uV, disturbance 191 wA, performance variable 22 z, and vector x as described by Eq. 411a. Two linearization cases considered in this study are the following: 1. The direct linearization of the fuel cell m odel Eqs. 44 46 and shown in Figure 42a [31]. 2. The linearization of a combined system consisting of a static feedforward controller, connected to th e fuel cell model Eqs. 44 46 shown in Figure 42b. The static feedforward control law used in case 2 is described in Section 4.5.2.1. The state, input, output, and feedthrough matrices of the re sulting linear timeinvariant models for cases 1 and 2 are shown in Figures 43 and 44, respec tively. Through personal communication with the members of the research group at University of Michigan we lear ned that the matrices reported in [31] are affected by errors, hence these matr ices are rederived and found to differ slightly from the ones given by Pukrushpan et al. [31] 4.5 Literature Precedents fo Fuel Cell Control Designs Pukrushpan et al. investigate the design and applicati on of two main control strategies along with 2 major variations of each of these strategies, as indi cated in the following list [31]: PAGE 67 67 1. Feedforward strategy a. Static feedforward (sFF) b. Dynamic feedforward (dFF) 2. Combination of static feedforward st rategy with optimal feedback control a. Linear quadratic regulator (LQR) b. Linear quadratic gaussian observe r (LQG) in combination with LQR feedback. A succinct discussion of these strategies is given in the ensuing sections 4.5.2 Feedforward Strategy The two feedforward control strategies proposed in the literature are a static feedforward (sFF) and a dynamic feedforward (dFF) scheme. Th e sFF controller is de rived from simulations and using the results of substant ial experimental work. The dFF controller is based on a linear model, and thus its performance on the nonlinea r model is dependent on the non linear models proximity to the nominal operating point. The models used for control design are discussed in greater detail in th e next subsections. 4.5.2.1 Static feedforward controller Pukrushpan et al. propose the sFF control law [31] 378123 wu (412) where the stack current w is the measured disturbance impacting the fuel cell model. The control input u is the voltage cmv applied to the compressor. A schematic of the sFF controller is shown in Figure 45. The derivation of Eq. 412 cons ists of first seeking a function relating the disturbance stIto the required air mass flow rate cpW, in such a fashion that the flow rate, achieved by invoking thermodynamics principles, negates the effect of the disturbance on the performance variable2O A resulting static function is obt ained by means of simulations and experimental work which co relates the control input cmv to the required air mass flow ratecpW PAGE 68 68 for a particular value of disturbance stI, maintaining the desired value of the performance variable 2O This static function is implemented as a static sFF controller in the form of Eq. 412. 4.5.2.2 Dynamic feedforward controller Pukrushpan et al. propose the dynamic feedforward (dFF) control law wKuuw (413) where uuu and www are the deviation values fr om the nominal control input u and the nominal disturbance w, respectively, and uwK is the transfer function )1)(1)(1(3 2 1sss K Kideal uw uw (414) where s is the Laplace variable, 1 2 and 3 are filter constants, and wzuz ideal uwGGK2 1 2 where uzG2 and wzG2are transfer functions describing the map uGwGzuz wz 2 22 (415) where 222zzz is the deviation of the performance variable (oxygen excess ratio) from its nominal value 2z [31]. The schematic and transferf unction representations of the dynamic feedforward controller dFF are given in Figure 46. The derivation of the dFF control transfer function begins by considering the transfer function wKGwGzuwuz wz 2 22 (416) obtained by substituting Eq. 413 into Eq. 415. The expression 416 should become identically zero for an effective dFF controller uwK. The expression PAGE 69 69 wzuz ideal uwGGK2 1 2 (417) is obtained from Eq. 416 after equating 02 z However,1 2wzGis not a proper function. Low pass filters are added to implement a causal controller, resulting in wzuz uwGG sss K2 1 2 3 2 1)1)(1)(1( 1 (418) The authors of this derivation pr opose the filterconstant values 801 1202 and 1203 [31]. 4.5.3 Combination of Static Feedforwar d with Optimal Feedback Controllers Feedforward controllers lack r obustness to disturbance and model variations. To address this issue, feedback controllers are added in conjunction with f eedforward controllers. The following two scenarios are considered for f eedback control design, in conjunction with static feedforward control [3234]: a. The performance variable is measurable. b. The performance variable is not measurab le, leading to the introduction of a state observer to estimate its value. The model used for designing the optimal feedba ck controller is defined in such a fashion that it includes the static f eedforward control relationship. A schematic of the resulting feedforward and feedback cont rol is shown in Figure 47. 4.5.3.1 Case where the performance variable is measurable Pukrushpan et al. propose the linear quadr atic regulator (LQR) f eedback control law: qKxKuuI pp (419) PAGE 70 70 where pu is a precompensator, uuu and xxx are the deviation values of the feedback control input u and the state vector x from their nominal points u and x respectively, and q is the integral defined th rough the differential equation 11yyqreq (420) where req cp reqWy 1is the analytically calculated value of the air mass flow rate needed for a particular value of the disturbance Ist to attain the desire d oxygen excess ratio, and cpWy1 is the measured value of the air mass flow rate. In the control law given by Eq. 419, gain matrices pK and IK refer to the optimal gains resulting fr om minimizing the quadratic cost function 0) (2 2dtuRuqQqxCQCxJT I T zz T z T (421) where, zQ, IQ, and R are weighting matrices for the performance variable, stateq and control input u, respectively. The term2zC refers to the second row of the C matrix, given in Figure 44 describing the linear state space model. Note that the second row is associated with the performance variable 2O The precompensator pu in the feedback control law Eq. 419 ,used to take into account the disturbance effect on the performance variable, is given by wBKBACDBKBACuwpu zwzupu zp1 1 1)( )(2 2 2 (422) where A, wB,uB, and wzD2 are elements of the linear state sp ace model given in Figure 44. In particular wB and uB are the first and second column of the B matrix, respectively and wzD2 is the second element of th e first column of the D matrix. A schematic diagram of the sFF scheme supplemented with the feedback cont roller is shown in Figure 47. PAGE 71 71 The optimal linear quadratic regulator 419 is implemented to prescribe part of the feedback control input. The additional state q, as described in Eq. 420, is introduced to minimize offset. Using 10000 zQand 001 .0 iQ in Eq. 421 and minimizing the cost function 421 the optimal values of c ontroller gains obtained are 61.1e171.9e 2.64 62.0e 55.2e 25.46171.4e23.03pK (423) and 001 .0 iK (424) Note that 423 and 424 differ from the values reported by Pukrushpan et al. [31]. This discrepancy is a consequence of the difference, reported in Secti on 4.4, with the state matrices used by Pukrushpan et al.. 4.5.3.2 Case where the performan ce variable is not measurable Optimal observer. When selected states are not measurable, Pukrushpan et al. propose the following modification of the control law 419: qKxKuuI pp (425) where xxx is the deviation of an estimated state vector x from its nominal value x The estimated state is computed from the Kalman based observer ) ( yyLBuxAx (426) DuxCy (427) where L is the optimal observer gain calculated using the linear quadratic gaussian (LQG) method, and y is the estimated values of the measured outputs [31]. Three measurable outputs, namely the compressor mass flow rate cpW, the pressure of the supply manifold smpand the stack voltage stv, are used as inputs to the observer to estimate the valu e of the states [31]. The optimal gain of the Kalman observer is PAGE 72 72 0065.97e003527.22e 009486.77e000+1.85e 0099.95e018270.13e 015143.89e009124.80e 01578.31e0064.60e003460.51e 009332.44e00949.54e0034.22e 012578.06e 015349.02e00925.06e 0152.54e000+2.18e 00931.69e 01518.54e015417.04e 00912.22e0157.43e L (428) Note that 428 differs from the value reported by Pukrushpan et al. [31]. This is a consequence of using different state matrices than what were used by Pukrushpan et al. as explained in Section 4.4. In addition, we also made an appr oximation to the noise variance matrices used for LQG design, since only partial in formation is available in [31], hence adding another source of discrepancy. 4.6 Generalized Predictive Control The generalized predictive control (GPC) law is given by the discre tetime law [35, 36] (t)ym(t)kOL T)(tu (429) where k is the GPC control gain vector, set O2m(t) is the set point, (t)yOL is the vector of the constant forcing values of the performance variable, u is the prescribed control input, and the symbol represents the difference operator. The vectors m(t), (t)yOL,and the gain k have R elements each, representing their respective values upto the prediction horizon R. The process involved for obtaining the GPC c ontrol law Eq. 429 is discusse d in the following sections. Generalized predictive control design. The first step leading to the derivation of Eq. 429 is to acquire a DARMA (deterministic au toregressive moving av erage) model from continuous state space model [3538]. To accomplish this, a discrete versi on of the linear state space model is obtained. The sampling period T is chosen such that about 1020 samples are PAGE 73 73 taken during the transient time. The ztransform is performed on the discrete model and the z variable is replaced with the q forwardshift operator. The nu merator and the denominator are divided with the highest power of q to obtain a denominator polynomial A and a numerator polynomial B as functions of the backward shift operator 1 q. Polynomial B is defined such that it does not include the unit time delay introduced by sample and hold operation. Finally, the DARMA model for the fuel cell model is obtained in the form ) 1()()()(1 1 tuqBtyqA (430) a an n i iqaqaqaqA ....1)(1 1 1 (431) b bn n i iqbqbqbbqB .... )(1 10 1 (432) and where A and B are polynomials in the backward shift operator 1q. The factors ia and ib are the coefficients of the powers iq in polynomials A and B, respectively. The symbol y represents the output (performance variable), i.e. the oxygen excess ratio in the case of the fuel cell, and u is the control input, i.e., the voltage supplied to the compressor motor vcm to deliver the desired air mass flow rate [29, 30]. Second, a predictor for the performance va riable is designed using the Diophantine equations ) ()(11 1qFqqAEi i i (433) where 1 1 1 1.. ..1 i i j j iqeqeqeE (434) a an n j j iqfqfqffF .. ..1 10 (435) Ri,..,2,1 (436) PAGE 74 74 and where R is the prediction horizon, 11q, and E and F are polynomials in powers of the backward shift operator 1q. The factors ej and fj are the coefficients of the powers jq in polynomials Ei and Fi respectively [30, 32]. Next, multiplying both sides of the DARMA model (Eq. 430) by the polynomial i iEq and invoking Eqs. 43336 yields )()()1()()()(1 11tyqFituqBqEityi i (437) where ) (ity is the predicted value of the output at the instant it Next, the constant forcing value of the performance variable is calculated. The product of polynomials Ei and B can be decomposed in the form )()()()(1 1 1 1 qqqGqBqEi i i i (438) where Gi and i are operator polynomials given by 1 1 1 10 1.. .. )( b bn n j j iq q q q (439) 1 1 1 1 1.. .. )( i i j j o iqgqgqggqG (440) and where j and jgare the coefficients of powers of jq in the respective polynomials. Substituting Eq. 438 into Eq. 437 yields )()()1()()1()()(1 1 1tyqFtuqituqGityi i i (441) A constant forcing 0 )1(itu for ,.., 2,1 i produces a constantforcing output in Eq. 441 of the form ) ()()1()()(1 1tyqFtuqityi i OL (442) where ) (ityOL represents the consta ntforcing response of y at instants it, ,.., 2,1i. PAGE 75 75 Third, an objective/costfunction J is established and minimized with respect to the controlinput increment uto obtain the GPC control law. The appropriate cost function adopted is defined by u uy(t)m(t) y(t)m(t)T )()(TuJ (443) where is a weighting matrix, m and y are vectors of future set point and predicted future output respectively, u is the vector of control inputs, and is the weighting matrix for u. The vectors m and y have R elements each, representing their respective future values upto the prediction horizon R. Vectors u and u has L elements, where L is the control horizon defined as the instant where the c ontrol design specifies that ) 1()( Ltuitu, 1 ,...,1, RLLi (443a) In the case of the fuel cell, the performance variable y is composed of a contribution from the disturbance variable d (namely, the stack current Ist), and a contribution from the manipulated variable u (namely, the voltage to the compressorcmv) leading to its definition uGydGyyyyu OL u d OL dud (444) where dy is the component of th e performance variable y contributed by disturbance d Note that the notation for the di sturbance is changed from w Sections 4.4 to 4.6 to d in Sections 4.7 which involve the GPC strategy. Signal yu is the component of the performance variable y contributed by mani pulated variable u. The termsOL dy and OL uy represent the constantforcing values of dy and uy respectively. The termsdG and uG are the associated dynamic matrix polynomials for dy and uy respectively. The termdGd is set as 0 dGd, as0 d since d is a disturbance and its unknown future values ar e assumed to be equal to the current value, i.e., )(itd is assumed to be constant and equal to ) (td, ,.. 2,1 i. This reduces Eq. 444 PAGE 76 76 uGyyyyyu OL u OL dud (445) Hence, since OLyy when 0 u, it follows that OL u OL d OLyyy (446) Substituting the value of y described by Eq. 445 in the cost function Eq. 443 yields u u u)G(t)y(m(t) u)G(t)y(m(t)T u OL T u OL uJ (447) which when minimized with respect to u yields )X (G I))G (G uTT u 1 u TT u (448) where (t)ym(t)XOL. The control law is extracted from Eq. 448, takes the form XkT)(tu (449) where T k is the first row of the matrix ) (G I))G (GTT u 1 u TT u and XkT)1()(tutu by the GPC algorithm. Eq. 449 can be easily rewr itten via a simple substitution of factor X to reduce to Eq. 429. This completes the de rivation of the GPC c ontrol law Eq. 429. For stability analysis and si mulation purpose it is importa nt to develop a closedloop transfer function for the GPC loop. The GPC control law described by Eq. 449 in summation form is given by ] [)(,, 1OL iu OL idi R i iyymktu (450) where the subscript i denotes the values of the respectiv e variables at the time instance t+i. Then using Eq. 442 )()()1()()(1 1 ,tyqFtdqityd id id OL id (451) )()()1()()(1 1 ,tyqFtuqityu iu iu OL iu (452) PAGE 77 77 Substituting the values of OL idy, and OL iuy, given by the Eqs. 451 and 452 into the GPC control law given by Eq. 450 and rearranging terms yields R i u iui R i d idi R i idi R i i i i iu R i ityqFktyqFk tdkqmqktukq1 1 1 1 1 1 1 1 1)()(()()(( )() ( )() 1( (453) Let ) 1(, 1 1 iu R i i ukqR (454) R i idi dkqR1 1) ( (455) R i i iqkT1 (456) R i idi dqFkS1 1 ,) ( (457) and R i iui uqFkS1 1 ,) ( (458) Inserting in Eqs. 454 458 into Eq. 453 yields ) ()()()()( tyStyStdRtTmtuRuu dd d u (459) Since ) ()()( tytytyd u (460) Eq. 459 can be rewritten as )) ()(()()()()( tytyStyStdRtTmtuRd u dd d u (461) which after rearranging terms results in ) ()()()()()( tyStySStdRtTmtuRu dud d u (462) Also, since )(1td A B qyd d d (463) PAGE 78 78 Eq. 461 can be expressed in the form )()()()()()(1tyStd A B qSStdRtTmtuRu d d ud d u (464) which after a trivial series of algebraic operations can be written as ) ()())(()()(1tySAtdBqSSRAtTmAtuRAud d uddd d ud (465) Now, let udRAR ~ (466) )( ~ tTmATd (467) d uddddBqSSRAS1)( ~ (468) and udySAS ~ (469) Substituting Eqs. 466 469 into Eq. 465 yields ) ( ~ )( ~ )( ~ )( ~ tyStdStmTtuRy d (470) which finally leads to th e closedloop established by the GPC control strategy )( ~ ~ ~ )( ~ )( ~ tyStmTtuR (471) where ydSSS ~ ~ ~ ~ (472) and )( )( )( ~ ty td ty (473) Figure 48 shows a closed loop schematic of the GPC controller for the augmented model which incorporates the disturbance. To obtain the closedloop transf er functions relating the input u(t) and the output y(t) to the set point m(t) and the disturbance d(t) first consider the op enloop relationships )( )(1td A B qtyd d d (474) PAGE 79 79 and )( )(1tu A B qtyu u u (475) Let ddBBq 1 and uuBBq 1. Substituting Eqs. 474 and 475 into Eq. 459 and rearranging terms yields )( )( )(td BSRA BSRA A A tm BSRA TA tuuuuu dddd d u uuuu u (476) which is the closed loop transfer function from the set point m(t) and the disturbance d(t) to the control input u(t) Inserting into Eq. 465 the expression )()()(tu A B td A B tyu u d d (477) obtained from Eqs. 445, 474, and 475 yields )( ) ( ) () ( )( )(td BSRAA BSRABBSRAB tm BSRA TB tyuuuud dddduuuuud uuuu u (478) which is the closed loop transfer function from the set point m(t) and the disturbance d(t) to the output ) ( ty 4.7 Battery of Observers In this scheme, a parallel ba ttery of four Kalman based observers designed using four different nominal points, respectively, are deployed for estimating the value of the controlled variable where the equations for each observer i is given by ) ( i iiiiiyyLuBxAx (479) uDxCyiiii (480) PAGE 80 80 where ix and iy are the estimated state vector and es timated output vectors from observer i respectively. The state matrices iiiCBA ,and iD correspond to the nominal point for which the observer i is designed. The observer which delivers the best performa nce is used for providing its estimated value of the controlled variable to the GPC controller. The observers are based on linear models derived from the nonlinear model at different nomina l values of the disturbance. The observer is defined by Eqs. 479 480. The observer gain L is computed from LQG principles as defined in previous section for each of the observers. The determination of the best performing observer is done by comparing the norm of the error in the measured outputs produced by each observer. The errornorm iN for observer i is defined by 2 ,3 2 ,2 2 ,1iii ieeeN (481) where icp cpiWWe, ,1 (482) ism smippe, ,2 (483) iststivve, ,3 (484) are the errors defined by the difference be tween the measured value of the outputs (cpW ,smp and stv ) from the nonlinear model and their estimated values (icpW, ,ismp, and istv, ) from the observer i respectively. The observer that delivers th e least value of the norm is selected as the best current observer and is implemented to delive r its estimated value of the controlled variable to the controller. The nominal value of the dist urbances for which the observers are designed are w=100, 191, 125, 22 A. The values are chosen to account for the range of disturbance values that the fuel cell model experiences. PAGE 81 81 4.8 Simulation Studies and Results This section presents simulations results obtained from various control strategies implemented on the nonlinear fuel cell model. The MATLAB and SIMULINK software computational tools are used for simulation purposes and for assessing the controllers performance [22]. The nonlinear fu el cell model created by Pukrushpan et al., in the SIMULINK environment is used for simulations [31]. The performance of the generali zed predictive control strategy is presented first. The scenario where the performance variable is meas ured is discussed, followed by the case when it is not measured. In the latter case an observer is incorporated in the control design to estimate the unmeasured value. The robustness of the GPC control design is assessed by examining its performance on an altered fuel cell model obtained by modifying a parameter of the original nonlinear model. Finally, a comparison of the GPC controllers perf ormance with that of controllers proposed in literature is conducted. For all the simulation studies carried out in this work, the disturbance d ( i.e., the stack current Ist ) profile shown in Figure 49 is used, whic h is identical to the one used by Pukrushpan et al [31]. The yaxis denotes the value of th e disturbance and the xaxis is time. The trajectory of the disturbance is implemented as step ch anges. In Figures 410 419, showing the simulation results for various control configurations, the yaxis denotes the value of the performance variable oxygen excess ratio 2O and the xaxis is depicts the time t The dotted black line indicates the value of the desired set point i.e., 2 )( tm for the performance variable. PAGE 82 82 4.8.1 Generalized Predictive Control Results As discussed in Section 4.4, the GPC approach is implemented on the fuel cell model to attain the desired objective of regulating the performance variable, i.e., keeping the oxygen excess ratio at a value of 2. The following scenarios are considered for evaluating the performance of the GPC strategy: 1. The controlled variable is measured. 2. The controlled variable is not measured. A battery of observers based on Kalman filtering is included to estimate the value of performance variable. 3. The design model is uncertain. The robustness of the GPC controllers under model uncertainty for cases (1) and (2) above is evaluated by examining the performance on a modified fuel cell model obtained by varying a parameter of the original model. 4. The control is benchmarked against literature precedents. The performance of the GPC controller is compared to control de signs proposed for the fuel cell model in prior literature. 4.8.1.1 Case where the performance variable is measured The performance of the GPC controller when the controlled variable is measured is discussed in this section. Sin ce in this case the controlled variable (oxygen excess ratio) is measured, its value is directly available to the GPC controller. The GPC algorithm described in Section 4.6 dynamically prescribes the value of the control input (the vol tage to the compressor motor vst) which in turn delivers the approp riate compressor mass flow rate ( Wcp) to maintain the value of the controlled variable at the desired setpoint value of 2. Figure 410 shows the results of a simula tion study where the GPC control design is implemented on the nonlinear fuel cell model. The xaxis in Figure 410 denotes the time and the yaxis depicts the value of th e controlled variable, i.e., the oxygen excess ratio measured from the nonlinear fuel cell. The initial value of states, distur bance and control input for the model are chosen as discussed in Section 4.4. The dotted line indicates the set point of the PAGE 83 83 controlled variable, which in this study is consta nt and equal to 2. The black solid line depicts the measured value of the controlled variable unde r conditions where the fu el cell is subjected to the disturbance profile (v alues of stack current Ist) shown in Figure 49 and the input is prescribed by the GPC algorithm. Figure 410 shows that the GPC controller suc cessfully returns the controlled variable at the desired value without steadystate offset. Only small deviations from the set point are observed during transients. The spikes in the sign al plot reflect instances where the disturbance suddenly changes values in a stepwise fashion, as documented in Figure 49 and hence they are unavoidable. A value of the controlled variable be low 2 indicates that a po sitive step change in disturbance took place, with the effect of depleting the oxygen concentration in the cathode and consequently reducing the oxygen excess ratio before the control action could remedy the problem. Analogously, a transient value of the controlled variable lying above 2 reflects an opposite scenario of an occurrence of a negative step change in disturba nce. The asymptotic offsetfree behavior observed is expected as the GPC approach incorporates an integrator as an essential component of its structure. 4.8.1.2 Performance variable not measured Battery of four observers. Figure 411 shows the time response of the performance variable when a battery of four observers ar e included in the controller to account for the situation where the controlled variable is not measured. The observer desi gn steps are discussed in Section 4.7. In Figure 411, the solid line indicates the performance variable trajectory as a function of time. The dotted line is the desired set point value of 2 for the performance variable. The measured outputs from the fuel cell mo del are (1) the compressor mass flow rate Wcp, (2) the stack voltage vst ,and (3) the supply manifold pressure psm. These measurements are provided to the observers to estimate the valu e of the performance variable. The observers PAGE 84 84 incorporated in the control strategy provide es timates for the performance variable and the norm error value that they generate. According to the switching algorithm the best performing observer, that generates the least value of the normerror value, feeds its estimated value of oxygen excess ratio to the GPC controller. The GP C strategy attempts to regulate the value of the estimated performance variable at 2. Excel lent results are obtained by using a battery of observers which is evident by zerooffset/near zero offset of the performance variable value from the setpoint. The normerror values of the four observers are shown in Figure 412. The switching pattern indicating the observer used for estima ting the value of oygen ex cess ratio is shown in Figure 413. The value of the control input, th e voltage prescribe to the compressor motor, is shown in Figure 414. The deviation from th e estimated and measured three outputs for the observers 1, 2, 3, and 4 are shown in Figu res 415, 416, 417, and 418, respectively. Battery of three observers. An identical strategy described above is used for observer design except that a battery of three observers in stead of four is implemented. Observer 3 is excluded from the battery of observers. Figure 419 shows the time response of the performance variable. The solid line indicates the performance variable traject ory as a function of time. The dotted line is the desired set point value of 2 for the performance variable. A deterioration in control performance is observed compared to the case when four observers are used. This result is expected as there are fewer obs erves available that are designed in the vicinity of the different operating points of the nonlinear fuel cell model. As a result, the estimated value of the performance variable provided by the observers do not coincide with the actual value resulting in some deviation from the set point because the observer design is based on a linear model. The PAGE 85 85 GPC strategy in turn attempts to regulate the erroneous estimated value of the performance variable and consequently a certain degree of dete rioration in controller pe rformance is expected. Battery of two observers. Figure 420 shows the time response of the performance variable when only two observers are employed in battery of observers. The solid line indicates the performance variable trajectory as a function of time. The dotted line is the desired set point value of 2 for the performance variable. Observ ers 1 and 3 are excluded from the battery of observers. A further deterioratio n in controller performance is no ted compared to the case where three observers are employed. This is an expected result as discussed earlier. Employing only one observer. Figure 421 shows the time response of the performance variable when only observer 2 is used to estimate the value of the performance variable. The solid line indicates the performance variable traj ectory as a function of ti me and the dotted line is the set point value of 2. This scenario exhibits the worst controller perf ormance compared to the cases where batteries of observers having 2, 3, or 4 observers are used. The GPC controller with one observer supplement fails to deliver offset fr ee regulation. This is an expected result, as discussed earlier. 4.8.2 The GPC Approach Evaluated for Robustness For robustness considerations, the GPC strategy was evaluated for model uncertainty. The GPC controller designed for the original fuel cell model is used on a modified model which is acquired by changing a parameter of the original mo del. More specifically, the return manifold throttle area changed from 0.0020 m2 to 0.0023 m2. The two scenarios considered for robustness analysis are 1. The performance variable is measured. 2. The performance variable is not measured An LQG observer is included in the control strategy to estimate the value of performance variable. PAGE 86 86 4.8.2.1 Case where the performance variable is measured Figure 422 shows the time response of th e performance variable from the modified nonlinear fuel cell model when the GPC control law is designed for the original model. The solid black line indicates the time response of the performance variable from the modified nonlinear fuel cell model. The intention is to ex amine the performance of the GPC controller in a scenario of model uncertainty. In this case the performance vari able is measured and is fed as a direct input to the GPC controller. The simulation conditions are identical to the scenario discussed in Section 4.7.1.1, except that the modified non linear model is used. As observed from Figure 422, the GPC controller displays robustness to model uncertainty because it ensures effect ive rejection of the effect of different values of disturbance, and produces offsetfree steadystate responses. The robustness of the GPC algorithm to model uncertainty is a key advantage of the controller. Even though the GPC strategy is designed for the original model, the feedback control of th e performance variable produced by the modified model gives the controller the opportunity to ma ke the necessary changes to minimize deviations from the set point. There is ze ro offset as the integrator in the GPC controller adjusts its output appropriately to eliminate the e rror of the performance variable with respect to the set point. 4.8.2.2 Case where the performance variable is not measured Figure 423 shows the time response of th e performance variable from the modified nonlinear fuel cell model when it cannot be directly measured. An LQG observer is incorporated in the control design to estimate the value of the performance variable by utilizing the values of the measured outputs (1) compressor mass flow rate Wcp, (2) stack voltage vst, and (3) pressure of supply manifold psm. The simulation conditions are identical to those given in Section 4.7.1.2, except that the altered model is used. The GP C controller and LQG observer designed for the original fuel cell model are used. The objective is to study the controller performance for model PAGE 87 87 uncertainty when the performance variable is not directly measurable. The solid line is the time response of the performance variable from the modified model. Figure 423 shows that when the LQG observer in conjunction with the GPC controller is applied to the modified model, significant degr adation in performance is observed compared to the case when the performance variable is directly measured. This can be attributed to the fact that the GPC controller is attempting to regulate at the desired set point value of 2 the value of performance variable estimated by the LQG observer. However, the LQG observer is optimally designed for the original model at a nominal ope rating point which is different from the actual operating point and hence delivers an erroneous estimated value of the performance variable. In the case when the performance variable is direct ly measured, previously discussed in Section 4.7.1.1 the GPC controller successfully makes adjustments to minimize a correct value of the error. In the current case when the LQG observer is included, the GPC algorithm makes an effort to minimize an incorrect error. The LQG observer is optimally designed for the original linear model at the nominal point. In the curren t case not only is the nonlinear model used but the problem is amplified further by the fact that there is a deviation from the original nonlinear model by virtue of the uncertainty in one of its parameters. Figure 424 shows the time response of th e performance variable from the modified nonlinear fuel cell model when a battery of 4 ob servers, as described in Section 4.7, are employed to estimate the value of the performanc e variable. The solid line is the time response of the performance variable from the modified mo del. Due to the reasons, as discussed above, no significant improvement in controller performance is observed. PAGE 88 88 4.8.3 Comparison of the GPC Strategy with Prior Control Designs In this section comparison of the GPC controll ers performance with controllers proposed in prior literature for fuel cell is conducted. Pukrushpant et al propose controllers for the following scenarios [31]: 1. All states are measured, leading to the implementation of the sFF controller with an LQR strategy for feedback. 2. All states are not measured, leading to the implementation of an LQG observer to estimate the value of the performance variab le in addition to the sFF controller with the LQR strategy for feedback. 4.8.3.1 Case where all states are measuredsFF with LQR feedback control Figure 425 shows the time response of the performance variable for the GPC strategy and the sFF with LQR feedback control strategy. Th e approach of sFF with LQR feedback control strategy is discussed in Section 4.5.3.1. In Figur e 425, the solid line is a plot of the performance variable as a function of time in the case wher e the GPC controller is used. The dashed line represents the performance variable trajectory wh en the sFF with the LQR feedback strategy is applied. The dotted line is for the set point, fixed at value 2. It is noted from the figure that the GPC cont roller delivers better performance as the sFF with the LQR feedback control strategy is not ab le to eliminate the steadystate offset. The GPC algorithm is able to eliminate offset as discus sed in Section 4.7.1.1. The LQR controller prescribes an optimal control input based on the li near model at the nominal point. However, the controllers performance is being examined then applied on a nonlinear model. Consequently, as there is deviation of the disturbance from th e nominal operating point the performance of the LQR controller degrades. PAGE 89 89 4.8.3.2 Case where all states are not measuredobserver design Figure 426 shows the time response of the performance variable when an LQG observer is incorporated in the controller in the scenario wh ere the performance variable is not measured. The dashed line indicates the time response of the performance variable when the static controller sFF with the LQG observer and the LQR feedback controller is applied. The controller strategy is disc ussed in Section 4.5.3.2. The solid line is the value of the performance variable when the GPC feedback with an observer is applied as the control strategy. It is noted from the time response curves that there is deterioration in both controllers performance compared to the case reported in Figure 425 where the performance variable is measured. Note, however, that the LQG observer with the GPC feedback controller exhibits better performance compared to the sFF with the LQG observer and the LQR feedback controller. The LQG observer component is identical for both th e controllers, consequently the strategy for estimating the value of the performance variable is alike. The GPC algorithm is able to regulate the estimated value of the performance variable in a superior fashion. The sFF with LQR feedback controller lacks the same level of dyna mic ability, and is not able to deliver the same level of regulatory performance that the GPC ap proach realizes. The LQR controller computes an optimal gain for the linear model at the nominal point. The implementation of the LQR strategy at nonnominal point i.e., on the nonlinear model degrades its performance. Figure 427 shows the time response of th e performance variable when a battery of 4 observers, as described in Section 4.7, are employed to estimate the value of the performance variable. The solid line is the time response of the performance variable. Due to the reasons, as discussed earlier, excellent results are obtained by using a battery of obs ervers which is evident by zerooffset/near zerooffset of the performance variable value from the setpoint. PAGE 90 90 4.8.3.3 Comparison of controller performance with respect to robustness The controllers discussed in Sections 4.7.3.1 and 4.7.3.2 (the sFF with LQR feedback and the sFF with LQG observer and LQR feedback), are evaluated and compared with the GPC controller for their robustness to model uncertainty. An approach identical to that of Section 4.7.2 is adopted. The modified model is obtained from the original nonlinear model by altering the value of the return manifo ld throttle area from 0.0020 m2 to 0.0023 m2. The two scenarios considered are: 1. All states measured i.e., performance variable is measured. 2. All states not measured, i.e., an observer is included to estimate the value of performance variable. All states measurable, sFF with LQR feedback. The performance of the GPC controller and the sFF with LQR feedback controller is co mpared in the scenario of model uncertainty. The robustness of the controllers is examined by implementing them on models for which they were not originally designed. In the current case the performance variable is measured. The control strategies for the GPC and the sFF with LQR feedback controllers described in Section 4.6 and Section 4.5.3.1, respectively, are implemented on the modified nonlinear fuel cell model. The difference in the simulation scenario compared to the one presented in Section 4.7.3.1 is that the model used to describe the no nlinear fuel cell dynamics has a different return manifold throttle area. Figure 428 shows the ti me responses of the two control strategies. The dashed line indicates the performance variable response to the sFF with LQR feedback control strategy. The solid line is the performance va riable response to the GPC control strategy. The GPC controller delivers better performance by eliminating offset as discussed in Section 4.7.2.1. The sFF with LQR feedback does not deliver the same level of performance as an offset with respect to the set point is ob served. The relatively poo r performance of the sFF PAGE 91 91 with LQR feedback strategy is expected as the feedforward and optimal gain are designed for a linear nominal point of the unmodified plant model. Performance variable not measured, sFF with LQG observer and LQR feedback. Figure 429 shows the time response of the perf ormance variable for the two control strategies (the LQG observer with GPC feedback and th e sFF with LQG observer and LQR feedback) in the case where the performance variable is no t directly measured and an LQG observer is incorporated in the control design to estimate its value. The LQG observer is identical to the sFF with LQR feedback and the GPC control strategies The dashed line indicates the time response of the performance variable when the sFF with LQG observer and LQR feedback controller is applied. The solid line is the performance vari able timeresponse when the GPC approach with LQG observer is implemented. The simulation co nditions are identical to the one discussed in Section 4.7.3.2, except that the modified m odel as discussed in Section 4.7.2 is used. In the case when the performance variable is not measured and the GPC strategy with LQG observer is implemented on the modified model, degradation in contro ller performance is observed as compared to when the performance va riable is measurable as discussed in Section 4.7.2.2. However, the GPC with an LQG observer control design delivers better performance (smaller offset) than that of the sFF with LQG observer and LQR feedback control design. This can be attributed to the fact th at the erroneous estimated value of the performance variable is fed as an input for the LQR feedback controller wh ich itself is being implemented on a model it was not designed for. Consequently, a relatively higher degradation in controller performance is seen on incorporation of the observer. The GPC desi gn performs comparatively better though it also fails to deliver offset free behavior. As mentio ned before, the observer is identical for the two controllers. In the case of the sFF with LQR feedback strategy, the situation is compounded PAGE 92 92 further by the fact that the LQR gain is not only being calculated at the nonnominal point (nonlinear model) but on a modified nonlinear model. Figure 430 shows the time response of th e performance variable from the modified nonlinear fuel cell model when a battery of 4 ob servers, as described in Section 4.7, are employed to estimate the value of the performanc e variable. The solid line is the time response of the performance variable from the modified mo del. Due to the reasons, as discussed above, no significant improvement in controller performance is observed. 4.8.4 Feedforward Control Designs Two exclusively feedforward control designs, namely (1) static feedforward (sFF) and (2) dynamic feedforward (dFF) without feedback components are proposed by Pukruspan et. al. [31]. The two controllers performance is comp ared with the GPC strategy for the following two cases: 1. Application on the original model 2. Application on the modified model to assess and compare robustness to model uncertainty. 4.8.4.1 Case of original model Figures 431 and 432 compares the performance of the sFF and dFF controllers with the GPC algorithm on the unaltered original nonlinear fuel cell model for the following two scenarios: (1) the perfor mance variable is measured, and (2) the case when it is not. In the scenario where the performance variable is not m easurable, an LQG observer is included in the GPC control design. The sFF and dFF controllers are discussed in Se ctions 4.5.2.1 and 4.5.2.2, respectively. The GPC approach and the GPC with the LQG obse rver strategies are implemented as described in Sections 4.6.1 and 4.5.3.2, respectively, a nd the simulations are conducted as discussed in PAGE 93 93 Section 4.7.1. The solid and dashed black line s represent the time responses of the performance variable in the cases when the sFF and dFF controll er are applied, respectiv ely. The red and blue lines indicate the time responses of the perfor mance variable when the GPC strategy is adopted for the scenarios where the performance variable is measured and not measured, respectively. In the case when the performance variable is not measured, an LQG observer is included in the control design to estimate its value. In the case when the performance variable is measurable, the GPC controller delivers the best result by eliminating offset with respect to the set point as discussed in Section 4.7.1.1. In the case when the performance variable is not m easurable and an LQG observer is incorporated in the control design there is a relative degradatio n in control performance, compared to the case when it is measured, as discussed in Section 4.7. 1.2. The performance of the dFF controller is the worst observed, as expected because the contro ller is designed for the linear model of a fuel cell at the nominal point. As the deviation fr om the nominal point increases, the controller dynamics of the dFF design do not correspond to the model it was designed for. The sFF strategy performs better compared to dFF a pproach as the deviation from nominal point increases. This behavior is expected as th e sFF algorithm is derived from simulations and experimental results based on the nonlinear model. The sFF, dFF and GPC with observer strategies are not able to eliminate offset. Figure 433 shows the time response of th e performance variable when a battery of 4 observers, as described in Section 4.7, are employed to estimate the value of the performance variable. The estimated value is fed to the GPC controller. The solid line is the time response of the performance variable. Due to the reasons, as discussed earlier, excellent results are obtained PAGE 94 94 by using a battery of observers which is evid ent by zerooffset/near zerooffset of the performance variable value from the setpoint. 4.8.4.2 Case of model uncertainty The robustness of the sFF and dFF controll ers to model uncertainty is assessed by implementing them on a modified model derive d by modifying a parameter in the original model, as explained in Section 4.7.2. The performance of the sFF and dFF controller is compared to that of the GPC approach. The tw o cases for the GPC strategy considered are: (1) the performance variable is measured, and (2) the performance variable is not measured in which case an LQG observer is included in the control design to estimate its value, as explained in Section 4.5.3.2. The sFF, dFF, GPC (with and without observer) co ntrol strategies designed for the original model are applied to the modified nonlinear model. The simulations are performed in an identical manner to that explained in Section 4.7.4.1, except that the modified nonlinear model was adopted. Figures 434 and 435 shows th e results of the simulations. The solid lines are the traces for the performance variable in the case wh en GPC approach is employed, respectively implemented for the scenarios (1) and (2) describe d above. The solid and dashed lines represent the time responses of the performance variab le for the sFF and dFF control strategies, respectively. Figure 434 shows that in the case where th e performance variable is measured the GPC controller delivers the best result by eliminating offset of the performance variable with respect to the set point, as discussed in Section 4.7.2.1. However, due to reasons discussed in Section 4.7.2.2, in the case when the performance variable is not measured, there is a relative degradation in the GPC controller performance. The performance of the sFF an d dFF controllers degrades as PAGE 95 95 well when they are implemented on the modified fu el cell model. The dFF controller performs the worst due to reasons discussed in Section 4.7. 4.1. Additionally, in the case of dFF controller the problem is further compounde d by the fact that, besides being applied at a nonnominal operating point (adopting nonlinear model), a modified model is used. Figure 436 shows the time response of th e performance variable from the modified nonlinear fuel cell model when a battery of 4 ob servers, as described in Section 4.7, are employed to estimate the value of the perfor mance variable. The estimated value of the performance variable is fed to the GPC controller. The solid line is the time response of the performance variable from the modified model. Due to the reasons, as discussed earlier, no significant improvement in contro ller performance is observed. 4.9 Conclusions An elegant solution is proposed for the problem of fuel cell control by implementing the GPC scheme. The GPC aproach employs the augmented model which incorporates the measured disturbance in its algorithm. The GPC controller demonstrates better performance compared to the control strategies proposed in literature for various scenarios. The GPC strategy is the best performing controller regulating the value of the performance variable at the desired set point of 2 with zero offset in all cases when it is measurable. The GPC controller demonstrates the highest level of robustness towa rds the issue of model uncertainty by exhibiting zero offset of the performance variable with resp ect to the set point when applied to a modified fuel cell model. PAGE 96 96 battery Fuel cell stack Motor Compressor Humidifier Water Tank Water Separator2O cmvstvH2 Tank PowersmpcpWstI Figure 41. Schematic of fuel cell system. PAGE 97 97 Fuel Cell System stIw cmvu netPez 122 Oz cpWy 1smpy 2stvy 3 Figure 42a. Fuel cell system showing input u disturbance w and outputs z1, z2, y1, y2, y3. PAGE 98 98 Fuel Cell System stIw cmvu z y static Feedback + Figure 42b. Fuel cell system showing sFF with feedback controller. PAGE 99 99 A 12.62 0 10.95 0 8.4e7 6e18 0 2.4e7 0 315.8 0 0 1.004e6 0 35.34 0 37.57 0 46.31 8.6e24 2.76e6 2e17 0 1.58e6 0 0 0 17.19 0.2032 0 0 0 2.6e8 0 2.97e8 379.4 38.7 1.06e7 0 0 33.28 0 38.03 4.834e5 4.8e6 0 0 0 0 295.6 0 0 9.33e7 0 63.61 0 4.045e8 0 4.621e8 0 0 0 0 51.22 B 3.16e5 0 3.98e6 0 0 0 0 405.1 0 0 0 0 5.245e5 0 0 0 C 4.94e6 1.967e6 1.089e5 2.066 0 0 0 0 1273 0 1454 4.3e22 1.388e4 9.94e16 0 0 0 0 0 4.834e5 1.16e6 0 0 0 0 0 0 0 1 0 0 0 2.59e4 1.03e4 569.9 0 0 0 0 0 D 180.2 165.7 0.01049 0 0 0 0 0 0.2965 0 Figure 43. Matrices defini ng the LTI model for the fuel cell model excluding sFF. A 12.62 0 10.95 0 8.4e7 6e18 0 2.4e7 0 315.8 0 0 1.004e6 0 35.34 0 37.57 0 46.31 8.6e24 2.76e6 2e17 0 1.58e6 0 0 0 36.8 0.2032 0 0 0 2.6e8 0 2.97e8 379.4 38.7 1.06e7 0 0 33.28 0 38.03 4.834e5 4.8e6 0 0 0 0 295.6 0 0 9.33e7 0 63.61 0 4.045e8 0 4.621e8 0 0 0 0 51.22 B 3.16e5 0 3.98e6 0 0 0 716.4 1065 0 0 0 0 5.245e5 0 0 0 C 4.94e6 1.967e6 1.089e5 4.131 0 0 0 0 1273 0 1454 4.3e22 1.388e4 9.94e16 0 0 0 0 0 4.834e5 1.16e6 0 0 0 0 0 0 0 1 0 0 0 2.59e4 1.03e4 569.9 0 0 0 0 0 D 112.9 435.7 0.01049 0 0 0 0 0 0.2965 0 Figure 44. Matrices defining the LTI model for the fuel cell including sFF. PAGE 100 100 Fuel Cell System stIw cmvu netPez 122 Oz cpWy 1smpy 2stvy 3 static Figure 45. The sFF control conf igurations for fuel cell system. PAGE 101 101 (a) Fuel Cell System stIw cmvu netPez 122 Oz cpWy 1smpy 2stvy 3 dynamic (b) + + u uwKw wzG2uzG22z Figure 46. The dFF controller: (a)Schematic diagram, and (b)transfer function representation. PAGE 102 102 Fuel Cell System stIw cmvu z y sFF Feedback + Figure 47. The sFF schematic with feedback controller. PAGE 103 103 )( tu )( td )( tm)( ~ ty)( tyR ~ /1S ~ ~ T ~ )( td + )( )( ty td )( tyu)( tyd + + d dA B u uA B Figure 48. The GPC design in feedback block diagram. PAGE 104 104 0 5 10 15 20 25 30 90 155 220 285 350 Time (sec)Stack Current (Amp) Figure 49. Disturbance profile used for simulation purposes. PAGE 105 105 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC Figure 410. The GPC control strategy implementation on the nonlinear fuel cell model in the case when the controlled variable is measured. PAGE 106 106 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC with 4 observers Figure 411. The GPC feedback with four observers control scheme implementation on the nonlinear fuel cell model. PAGE 107 107 0 5 10 15 20 25 30 0 2 4 6 8 10 12 x 104 Time (sec)Normerror Observer 1 Observer 2 Observer 3 Observer 4 Figure 412. The Norm of errors from the battery of observers. PAGE 108 108 0 5 10 15 20 25 30 Obs 1 Obs 2 Obs 3 Obs 4 Time (sec)Observer selected Figure 413. The switching patter n of the battery of observers. PAGE 109 109 0 5 10 15 20 25 30 80 100 120 140 160 180 200 220 240 Time (sec)Voltage to compressor (volts) Figure 414. Final voltage to the compressor. PAGE 110 110 0 10 20 30 .04 .02 0 0.02 0.04 Time (sec)Wcp error (kg/s) zero error Observer 1 0 10 20 30 0 2 4 6 8 x 10 4 Time (sec)Psm error (pascal) zero error Observer 1 0 10 20 30 0 5 10 15 20 Time (sec)Vst error (volt) zero error Observer 1 Figure 415. Observer 1, error between measured and estimated values. PAGE 111 111 0 10 20 30 .015 .01 .005 0 0.005 0.01 Time (sec)Wcp error (kg/s) zero error Observer 2 0 10 20 30 0 5000 10000 15000 Time (sec)Psm error (pascal) zero error Observer 2 0 10 20 30 0 1 2 Time (sec)Vst error (volt) zero error Observer 2 Figure 416. Observer 2, error between measured and estimated values. PAGE 112 112 0 10 20 30 .02 .01 0 0.01 0.02 Time (sec)Wcp error (kg/s) zero error Observer 3 0 10 20 30 0 5 x 10 4 Time (sec)Psm error (pascal) zero error Observer 3 0 10 20 30 0 5 Time (sec)Vst error (volt) zero error Observer 3 Figure 417. Observer 3, error between measured and estimated values. PAGE 113 113 0 10 20 30 .03 .02 .01 0 0.01 Time (sec)Wcp error (kg/s) zero error Observer 4 0 10 20 30 0 5 x 10 4 Time (sec)Psm error (pascal) zero error Observer 4 0 10 20 30 0 10 Time (sec)Vst error (volt) zero error Observer 4 Figure 418. Observer 4, error between measured and estimated values. PAGE 114 114 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC with 3 observers Figure 419. The GPC feedback with thre e observers control scheme implementation on the nonlinear fuel cell model. PAGE 115 115 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC with 2 observers Figure 420. The GPC feedback with two observers control scheme implementation on the nonlinear fuel cell model. PAGE 116 116 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC with 1 observer Figure 421. The GPC feedback with one observer control scheme implementation on the nonlinear fuel cell model. PAGE 117 117 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC Figure 422. The GPC control strategy implementation on the nonlinear fuel cell model with a parameter changed from the value used for control design. PAGE 118 118 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC with observer Figure 423. The GPC controller with the LQG observer control strategy implementation on the altered nonlinear fuel cell model. PAGE 119 119 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC with 4 observers Figure 424. The GPC controller with the 4 observers control strategy implementation on the altered nonlinear fuel cell model. PAGE 120 120 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC sFF with LQR Figure 425. Comparison of the GPC c ontrol strategy with the sFF controller combined with LQR feedback strategy on the unaltered nonlinear fuel cell model when the performance variable is measurable. PAGE 121 121 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC with observer sFF with LQG observer and LQR Figure 426. The sFF with the LQG observer and LQR feedback, compared to GPC with the LQG Observer control stra tegy implementation on the unaltered nonlinear fuel cell model when the perf ormance variable is not measurable. PAGE 122 122 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC with 4 observers sFF with LQG observer and LQR Figure 427. The sFF with the LQG observer and LQR feedback, compared to GPC with the 4 observers control strategy implementation on the unaltered nonlinear fuel cell model when the perf ormance variable is not measurable. PAGE 123 123 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC sFF with LQR Figure 428. The sFF with the LQR feedback, compared to GPC, when performance variable is measurable on the altered nonlinear fuel cell model. PAGE 124 124 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC with observer sFF with LQG observer and LQR Figure 429. The sFF with the LQG observer and the LQR feedback compared to the GPC with the LQG observer control stra tegy on the altered nonlinear fuel cell model. PAGE 125 125 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC with 4 observers sFF with LQG observer and LQR Figure 430. The sFF with the LQG observer and the LQR feedback compared to the GPC with the 4 observers control stra tegy on the altered nonlinear fuel cell model. PAGE 126 126 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC dFF sFF Figure 431. The sFF and dFF strategies and the GPC control st rategy, performance compared when applied on the una ltered nonlinear fuel cell model. PAGE 127 127 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC with observer dFF sFF Figure 432. The sFF and dFF strategies and the GPC control strategy with the LQG observer, performance compared when applied on the unaltered nonlinear fuel cell model. PAGE 128 128 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC with 4 observers dFF sFF Figure 433. The performances of the sFF and dFF strategies and the GPC with the 4 observers control strategy compar ed when applied on the unaltered nonlinear fuel cell model. PAGE 129 129 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC dFF sFF Figure 434. The sFF and dFF strategies and the GPC control st rategy, performance compared when applied on the al tered nonlinear fuel cell model. PAGE 130 130 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC with observer dFF sFF Figure 435. The sFF and dFF strategies and the GPC control strategy with the LQG observer, performance compared when applied on the altered nonlinear fuel cell model. PAGE 131 131 0 5 10 15 20 25 30 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Time (sec)O2 Excess Ratio setpoint GPC with 4 observers dFF sFF Figure 436. The performance of sFF and dFF strategies and the GPC with 4 observers control strategy compared when applie d on the altered nonlinear fuel cell model. PAGE 132 132 CHAPTER 5 CONCLUSIONS AND PROPOSITIONS FOR FUTURE WORK 5.1 Conclusions A Virtual Control Lab (VCL) with an inverted pe ndulum that can be utilized as a tool for enhanced learning is described. The VCL can be used in conjunction with a process control lecture for demonstrating various advanced conc epts or can be used by students located in geographically remote places. The animation modul e of the VCL allows the user to visually observe the impact of the control design. The VCL can potentially minimize the problem of scheduling laboratory time for physical equipment. The problem of tracking the slope of ramp is ex amined. On the basis of results obtained, a PI controller using the ITAE as the optimizing metric demonstrates the best results. The controller performance is better than that of controllers proposed in literature. Tuning relationships for PI and PI2 controllers, using three differe nt optimizing metrics and 20,000 different plants is presented. An elegant solution is proposed for the prob lem of fuel cell control by implementing a GPC strategy which incorporates disturbance m easurements to produce a manipulated variable. The GPC design demonstrates better performance co mpared to control strategies available in prior literature. The GPC controller results in zero offset in the performance variable in the nonlinear model when the performance variable is m easured. The GPC controller is also the best performing controller when evalua ted for model uncertainty. The c ontroller exhibits zero offset, showing its strength from a view point of robustness, when employed on a modified model. In the case when the performance variable is not m easurable, a battery of observers is implemented to estimate the value of the performance variable deliver the best result. However, the robustness problems still exist. PAGE 133 133 5.2 Future Work To validate the benefits and effectiveness, it is proposed that the VCL is tested as a pilot teaching tool in process control classes taught at both undergraduate and graduate levels. The feedback obtained from the instructors and students will be highly beneficial in improving and incorporating features that coul d enhance the learning experience. It would be desirable to add to the VCL a few more advanced control strategies such as the GPC approach. For future work regarding the ramp tracking problem it is proposed to validate the simulation results for physical setups that can take advantage of the tuning relationships presented. It is also suggested to conduct a mo re comprehensive study to minimize the scatter in the optimizing metric error for low / values of the ratio, that is, for systems with little dead time. For future work on the fuel cell control problem it is proposed to evaluate the design of observers which take model uncertainty into accoun t. A battery of observers can be designed at each operating point, with each observer corres ponding to a linearized model obtained from different values of parameters. Then a bumpless switching strategy could be designed to select an appropriate state estimate to feed to the co ntroller. The experimental implementation of the control scheme in a physical fuel cell may be of si gnificant value to confirm or refine the results discussed in this dissertation. PAGE 134 134 APPENDIX A OFFSET BETWEEN AUXILLIARY AND ORIGINAL RAMP The final value theorem is used to calculate the shift between the original ramp and the modified ramp. The transfer function for the process with time delay is given by s pe s K G 1 (A1) and the transfer function for PI controller is given by s KGI cc1 1 (A2) and the equation for ramp in the time domain is ttr ) ( (A3) Performing the Laplace transform of Eq A3, with the initial condition 0 )0( tr yields 2)( s sr (A4) Using Eqs. A1 and A2, it is possible to de rive the following standard closed loop relationship between the output y and the ramp r : )( 1 )( sr GG GG sycp cp (A5) Now consider the error )()()(tytrte (A6) defined as the difference between the set point and output. Applying final value theorem )()(lim)(lim0sysrstes t (A7) and inserting Eq. A4 and A5 in Eq. A7 yields PAGE 135 135 2 2 01 lim)(lim s GG GG s stecp cp s t (A8) or cp s tGGs te 1 1 lim)(lim0 (A9) Inserting Eq. A1 and Eq. A2 in Eq. A9 results in s Ke s K s teI c s s t 1 1 1 1 1 lim)(lim0 (A10) Simplifying terms and applying the limits on the right hand side of Eq. A10 yields I c tKK te 1 1 )(lim (A11) Hence, c I tKK te)(lim (A12) which is an analytical expression for the offset between the auxiliary and the original ramp. PAGE 136 136 APPENDIX B OBSERVER DESIGN USING TRANSFER FUNCTION The observer equations are given by ) ( xCyLBuxAx (B1) DuxCy (B2) Eq. B1 can be rewritten as LyBuxLCAx )( (B3) Performing Laplace Transform )()()( )()0( )( sLYsBUsXLCAtxsXs (B4) This results ) 0( )()()( ) ( txsLYsBUsXLCAsI (B5) On further manipulation ) 0( )()()( txMsMLysMBusx (B6) where 1) ( LCAsIM Now if xCzz (B7) ) 0( )()()( txMCsMLYCsMBUCsZz z z (B8) ) 0( )()()( )( XGsYGsUGsZoxz yz uz (B9) where MBCGzuz, MLCGzyz, and MCGz xz )0( PAGE 137 137 LIST OF REFERENCES 1. T. F. Junge and C. Schmid, Webbased remote experimentation us ing a laboratoryscale optical tracker, Proceedings of the American Control Conference Chicago Illinois, June 2000. 2. D. Gillet, C. Salzmann, R. Longchamp, a nd D. Bonvin, Telepresence: an opportunity to develop practical experimentation in automatic control education, European Controls Conference, Brussels Belgium July 1997. 3. D. Gillet, F. Geoffroy, K. Zeramdini, A. V. Nguyen, Y. Rekik and Y. 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Clarke, C. Mohtadi and P. S. Tu fts, Generalized predictiv e control part II. Extensions and interpretations, Automatica 23(2), 1987, pp. 149160. 38. D. W. Clarke and C. Mohtadi, Prop erties of generalized predictive control, Automatica, 25(6), 1989, pp. 859875. PAGE 140 BIOGRAPHICAL SKETCH Vikram Shishodia was born in Delhi, India. He graduated with a B.Tech. in chemical engineering from the Indian Institute of Technology Delhi in 1992. Mr. Vikram Shishodia joined the graduate program at the University of Florida in 1993. He graduated with an M.S. in materials science and engineering in 1996. After developing a successful career as a process engineer with Intel Corporation for nine years he returned to academia to pursue higher degrees in chemical engineering at the University of Florida. From 2005 to 2008 he was a graduate student and simultaneously worked as Assistant Director for the Division of Student Affairs at the College of Engineering of University of Flor ida. He graduated with an M.S. in chemical engineering in 2003 and a Ph.D. in chemical engineering from the University of Florida in 2008. 