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APPLICATIONS OF LINEAR PARAMETERVARYING CONTROL FOR AEROSPACE SYSTEMS By KRISTIN LEE FITZPATRICK A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2003 TABLE OF CONTENTS page LIST OF TABLES ................... ............ iv LIST OF FIGURES ................... ............... v ABSTRACT. ..................... ................. vii CHAPTER 1 INTRODUCTION ................... ......... 1 1.1 Overview ................... ............ 1 1.2 Background ................... .......... 2 2 LINEAR PARAMETERVARYING CONTROL THEORY .......... 5 3 LINEAR PARAMETERVARYING CONTROL FOR AN F/A18 ....... 10 3.1 Problem Statement ................... ....... 10 3.2 Openloop Dynamics .......... ......... ....... 12 3.3 Control Objectives ................... ...... 14 3.4 Synthesis ...... .......... ........... 15 3.5 Simulation .......... ......... ........... 17 3.6 Conclusion .......... ........... .......... 20 4 LINEAR PARAMETERVARYING CONTROL FOR A HYPERSONIC AIRCRAFT .............. .. ............. ...... 22 4.1 Problem Statement ......... .......... ...... 22 4.2 Generic Hypersonic Vehicle ......... .............. 25 4.3 Hypersonic Model .................. ....... 26 4.4 Linear ParameterVarying System ........ ............ 27 4.5 Control Design .......... ....... ............. 30 4.6 Simulation ................... ............ 33 4.6.1 OpenLoop Simulation ................... 33 4.6.2 ClosedLoop Simulation ......... ........ ... 35 4.7 Conclusion .................. ............ 38 5 LINEAR PARAMETERVARYING CONTROL FOR A DRIVEN CAVITY 39 5.1 Problem Statement ................... ....... 39 5.2 Background ................... ........... 40 5.3 Driven Cavity Geometry .................. ... .. 41 5.4 Governing Equations of Motion .............. .... .. 42 5.5 ReducedOrder Linear Dynamics ..... . .... 44 5.6 Creeping Flow in a Driven Cavity ...... . . .. 47 5.7 Excitation Phase Differential .................. .. .. 49 5.8 Control Design ............... . . . 50 5.8.1 Control Objectives ......... . . . ......50 5.8.2 Synthesis. .............. . . .. 53 5.9 Simulation ........... ....... . . ..54 5.9.1 OpenLoop Simulation ............... .. .. 54 5.9.2 ReducedOrder ClosedLoop Simulation . . .... 57 5.9.3 FullOrder ClosedLoop Simulation . . . 60 5.10 Conclusion. .................. . . ......63 6 CONCLUSION ................... . . .. 64 REFERENCES ............. ........ . . 65 BIOGRAPHICAL SKETCH .............. . . ..... 70 LIST OF TABLES Table page 31 Original Design Points ............. . . ...... 12 32 Frequency and Damping Ratio of Design and Analysis Models ...... 14 33 Frequencies and Damping Ratios of the Target Model . . ... 16 34 Induced Norms of ClosedLoop System .................. ..17 35 Numerical Results ............... . . .... 19 41 Model Dimensions and Flight Conditions ................. ..27 42 Modes of the Hypersonic Model .................. ...... 30 43 Modes of the Target Model .................. ....... .. 32 44 OpenLoop Synthesis Norms .................. ...... 33 45 Point Design Norms .............. . . .... 33 51 Induced Norms of ClosedLoop System .................. ..53 LIST OF FIGURES Figure page 21 Jo_ Block Diagram (GainScheduled) ......... ........ .... 7 31 F/A18 ............ .... ....................... 11 32 Flight Envelope/Parameter Space .................. .... .. 12 33 Synthesis Block Diagram .................. ........ .. 15 34 Closedloop Block Diagram .................. ..... 18 35 Pitch Rate for Design Points .................. ..... .. 18 36 Pitch Rate for Analysis Point .................. ...... .. 19 37 Controller Elevator Deflection .................. ..... .. 19 41 InnerLoop/OuterLoop Design .................. ..... .. 25 42 Simplified Model of a Generic Hypersonic Vehicle . . .... 25 43 Synthesis Block Diagram .................. ........ .. 31 44 OpenLoop Transfer Functions ................ . ..34 45 OpenLoop Angle of Attack Result .............. .... .. 35 46 Input Elevon Deflection ............ . . ...... 35 47 Closedloop Design ............ . . ... 36 48 ClosedLoop Transfer Functions ................ .... .. 36 49 ClosedLoop Angle of Attack Result .... . .... 37 410 Elevon Deflection Command ............... ..... .. 38 411 Elevon Deflection Rate .......... . . ... 38 51 Stokes Driven Cavity Flow Problem .... . .... 41 52 Controller Block Diagram .................. .. ...... 51 53 OpenLoop Flow Velocities for FullOrder Model . . .... 55 54 OpenLoop Flow Velocities for ReducedOrder Model with 1650 Phase Differential .................. . . ...... 56 55 OpenLoop Flow Velocities for ReducedOrder Model with 2100 Phase Differential .................. . . ...... 56 56 Trajectory of Phase Differential ................ .... .. 56 57 OpenLoop Flow Velocities for ReducedOrder Model over a Trajectory of Phase Differentials ............ . . .... 57 58 Closedloop System .............. . . .... 58 59 ClosedLoop Flow Velocities for ReducedOrder Model with 1650 Phase Differential .................. . . ...... 59 510 ClosedLoop Flow Velocities for ReducedOrder Model with 2100 Phase Differential .................. . . ...... 59 511 ClosedLoop Flow Velocities for ReducedOrder Model over a Trajec tory of Phase Differentials .................. ...... .. 60 512 ClosedLoop Flow Velocities for FullOrder Model . . .... 61 513 ClosedLoop Flow Velocities for FullOrder Model with Controller As sociated with 1650 Phase Differential . . . .... 61 514 ClosedLoop Flow Velocities for FullOrder Model with Controller As sociated with 2100 Phase Differential . . . .... 62 515 ClosedLoop Flow Velocities for FullOrder Model over a Trajectory of Phase Differentials .................. ........ .. .. 62 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science APPLICATIONS OF LINEAR PARAMETERVARYING CONTROL FOR AEROSPACE SYSTEMS By Kristin Lee Fitzpatrick December 2003 Chair: Richard C. Lind, Jr. Major Department: Mechanical and Aerospace Engineering Gainscheduling control has been an engineering practice for decades and can be described as the linear regulation of a system whose parameters are changed as a function of varying operating conditions. Several gainscheduling techniques have been researched for the control of systems that vary with timevarying parameters. These techniques create controllers at various points within the parameter space of the system and use an interpolation law to change controllers as the parameter changes with time. The process of creating such an interpolation law can be very rigorous and time consuming and the resulting controller is not guarnateed to stablize the timevarying system. The gainscheduling technique known as linear parametervarying control, however, solves a linear matrix inequality convex problem to create a single controller that has an automatic interpolation law and is guaranteed to stabilize the closedloop system. This paper demonstrates the use of this technique to create controllers for three aerospace systems. The first system is the longitudinal dynamics of the F/A18, the second system is the structural dynamics of a hypersonic vehicle and the third system is the flow dynamics within a driven cavity. Simulations are performed using the linear parametervarying controller created for each system to show the usefulness of a linear parametervarying framework as a gainschedule design technique. CHAPTER 1 INTRODUCTION 1.1 Overview The dynamics of aerospace systems that deal with flight and process control are affected by variations in the parameters that make up their operating space (i.e., altitude, Mach number, temperature). Gainscheduling techniques are used to create a controlling scheme that will work throughout the system's operating space. The resulting controller will vary based on the same parameters as the system's plant model. The traditional gainscheduling technique can be broken down into three major steps. The first step involves separating the operating range into subspaces and creating a parameterized model of each subspace. In the second step, controllers are created for each of these models. Finally, in the third step, a scheduling scheme is devised by linearly interpolating between these regional controllers as the vehicle moves through its operating range. This technique works well for some systems; however, it does not guarantee stability and robustness of the closedloop system. Another disadvantage of this method is the possibility of a skipping behavior due to the switch between controllers. This thesis presents the technique of creating a single gainscheduled controller that can be treated as a single entity. This technique achieves gainscheduling with a parameterdependent controller that will work throughout an operating range or flight envelope. Benefits of this technique are that it removes the need of creating several controllers for different parameters within an operating domain and removes the need for the creation of a gainscheduled control law. This study applies this method to three specific aerospace systems. The first application of this technique is creation of a controller for the F/A18 longitudinal axes over a specific flight envelope. The second application is control of the structural dynamics of a hypersonic aircraft over a temperature range. The third application is control of the velocity along the center of a driven cavity flow over a range of phase differentials within the flow. 1.2 Background Gainscheduling has evolved hand in hand with the progress of mechanical systems. Gainscheduling techniques are currently used for control design for both linear and nonlinear systems. This study focused on systems pertaining to aerospace applications. Gainscheduling in aerospace applications came about during WWII as autopi loting became necessary with the birth of jet aircraft and guided missiles [1]. When gainscheduling was first conceived for the military, it was created through hardware and was quite costly. Gainscheduling was not adopted commercially until the cre ation of digital control, nearly a quartercentury after military use. The development of gainscheduling over past decades led to several design techniques and the use of gainscheduling for many different aerospace systems. Several gainscheduling methods have been developed for designing controllers for linear systems. The three main classes of linear systems that apply to aerospace sys tems are linear timeinvariant (LTI), linear timevarying (LTV), and linear parameter varying (LPV). Gainscheduling is most often applied to linear parametervarying systems, which are affine functions of parameters that affect their operation. Gainscheduling can also be applied to the control of nonlinear systems. Several linearization techniques can be used for nonlinear systems before a gainscheduled controlling scheme is developed. The most common approach is based on Jacobian linearization of the nonlinear plant about a family of operating points (i.e., equilibrium points) [2]. The system can also be linearized along a trajectory in the event the lin earized dynamics do not exhibit good performance or stability away from equilibrium points [3]; however, the trajectory must be known in advance to perform the control design. Once the control scheme is created, it can be applied to controlling the non linear system. Simulations of gainscheduling controllers have also been applied to nonlinear systems [4, 5, 6]; however, nonlinear systems do not necessarily have to be linearized. Setvalued methods for LPV systems have also been applied to nonlinear systems with quasiLPV representations [7]. Linearization errors were accommodated as linear statedependent disturbances; constraints on systems' states and controls were specified; rates of transitions among operating regions were addressed, which allows even the localpoint designs to be nonlinear. The classical gainscheduling approach is to create a number of controllers within the operating domain; and then, using a scheduling scheme, to switch between them as the system parameters change. One method that uses this approach was demonstrated for a missile autopilot that uses p synthesis with DK iteration to create controllers; and an iteration scheme is designed over the operating domain [8]. Another method for a missile autopilot creates controllers at distinct operating conditions using J'& control synthesis; and then creates a schedule for the controllers by removing coupling terms [9]. Another project involved creating HJ_ point design controllers at specific equilibrium points [10]. That project reduces the controllers to second order which are then realized in a feedback path configuration for which a gainscheduling law is developed. A study also used a design algorithm for a state feedback law based on gainscheduling for an LPV multiinput multioutput system [11]. The state feedback control law places the system's poles in a neighborhood of desired locations and stabilizes the closedloop system. Though this classical approach has worked well for many applications, there is no guarantee of robustness or stability of the closedloop systems. A more recent approach, that appeared in the late 1990's, involves creating an LPV controller that uses an automatic interpolation law over the operating domain (which has guaranteed closedloop robustness and stability with the LPV system). The method of DK iteration with p synthesis was used to create an LPV controller for a missile whose operating parameters are angle of attack and Mach number [6]. As demonstrated in the creation of controllers for a tailless aircraft [12], an F16 aircraft [13], and a hypersonic aircraft [14], an LPV controller can also be created by letting the controller have the same linear fractional relationship with the varying parameters as the system [15] while attempting to minimize the _o norm. This technique is further expanded with the controlling of the longitudinal axes of an F 16 aircraft in a project that breaks a parameter space into two smaller overlapping parameter spaces, synthesizes an LPV controller for each space, and then uses blending functions to form a single LPV controller [16]. An LPV controller was created [17] for an LPV system, where parameter dependent feedback control laws are constructed after transforming the original LPV system into canonical form. Separate longitudinal and lateraldirectional LPV controllers were designed for the F/A18 [18]. The original controllers were formed using JL synthesis and then robustness was increased to meet military standards by using p synthesis. Other recent efforts at using real time parameter information in control strategies included minimizing linear matrix inequalities [19, 20]. This thesis presents one of the more recent gainscheduling techniques for creating an LPV controller using Ho synthesis, which is designed to work for the LPV system's entire operating domain. The operating domain of an LPV system is also known as the system's parameter space. Linear parametervarying control theory is discussed in more detail in the next section. CHAPTER 2 LINEAR PARAMETERVARYING CONTROL THEORY Linear parametervarying controller synthesis is a gainscheduling technique for designing one controller that will work over a range of parameters without having to create a scheduling scheme. In order to use the LPV framework the plant model must be created as a linear parametervarying system. A linear parametervarying system depends affinely on a set of normbounded timevarying operating parameters. It considers linear systems whose openloop dynamics are affine functions of the operating parameters. A method of identifying multivariable LPV state space systems that are based on local parameterization and gradient search in the resulting parameter space is presented in [21]. Two identification methods were purposed in [22] for a class of multiinput multioutput discretetime linear parametervarying systems. Both methods are based on the subspace state space method, which was suggested by [23] in the early 1990s. LPV modeling of aircraft dynamics, known as the bounding box approach and the small hull approach [24]. A general case of a linear parametervarying plant, whose dynamical equations depend on physical coefficients that vary during operation, has the form x =A(O)x+ Bi()d +B2(0)u P(.,O)= e=Ci(O)x+Dl(O)df+D12(O)u (2.1) y= C2(0)x+D21(0)d+D22(0)u where 0(t) = (01(t),..., (t)), 9i < Oi(t) < Oi (2.2) is a timevarying vector of physical parameters (i.e., velocity, angle of attack, stiffness); A, B, C, D are affine functions of 0(t), x is the state vector, y is the measured outputs, e is the regulated outputs or errors, d is the exogenous disturbances, and u is the controlled input. When the coefficients undergo large variations it is often impossible to achieve high performance over the entire operating range with a single robust LTI controller. When parameter values are measured in real time controllers that incorporate such measurements to adjust to current operating conditions would be beneficial. These controllers are said to be scheduled by the parameter measurements. This control theory typically achieves higher performance when considering large variations in operating conditions. In the event that different parameters effect the system differently weighting functions can be used to compensate for the differences. If the parameter vector 0(t) takes values within a geometric shape of Rn with corners {Hi}N1 (N = 2n), the plant system matrix S(:= e A()) B((t)) d (2.3) C(O(t)) D(O(t)) y u ranges in a matrix polytope with vertices S(FIi). Given convex decomposition N 0(t) = al + ... + N, ai > 0, i= 1 (2.4) i=1 of 0 over the comers of the parameter region, the system matrix is given by S(0) = aiS(ni) +... + UNS(IN). (2.5) This suggests seeking parameterdependent controllers with equations { = AK(9)+BK(9)y K(., ) = K() + BK(O)y (2.6) u= CK(0) + DK()y and with a vertex property where a given convex decomposition 0(t) = C=N oni; of the current parameter value 0(t). The values of AK(O),BK(O),CK(O),DK(O) are derived from the values AK(FIi),BK(Fi),CK(Fi),DK(Fni) at the comers of the parameter region by AK (0) BK(0) N AK() BK() (2.7) = a U (2.7) CK(0) DK(0) i=N CK( i) DK Ji) In other words, the controller statespace matrices at the operating point 0(t) are obtained by convex interpolation of the LTI vertex controllers Ki := AK(1i) BK(fi) (2.8) \CK(1Ii) DK(1i) This yields a smooth scheduling of the controller matrices by the parameter measure ments 0(t). As an example, consider the following _1like synthesis problem relative to the interconnection in Figure 21. If there exists a continuous differentiable function X(0) d e K(., 0) Figure 21: X Block Diagram (GainScheduled) defined on Rn where X(0) > 0, (2.9) the worstcase closedloop RMS gain from d to e does not exceed some level y > 0, and I 0 X(e) X(e) 0 0 ( I 0 A(O) B(O) X(O) 0 0 0 A(O) B(O) < 0 (2.10) 0 I 0 0 y 0 0 I C(0) D(0) 0 0 0 I C(0) D(0) hold for all 0 E Rn, then the system is quadratically stable and the L2 norm from d to e is smaller than y. The quadratic stability of a system allows the parameter to change with arbitrary speed without threatening stability of the system and is defined as existing if there exists a real positive definite matrix P = pT > 0 such that AT(O)P+P(O) <0 V 0(t) E Re. (2.11) The induced L2 norm of a quadratically stable LPV system G is defined as IGII =sup sup e2 (2.12) 6eReoldl2#O,deL2 ld12 with Re being a set of feasible parameter trajectories. There is more than one synthesis technique for designing an LPV controller once the LPV model is formed. Currently there are three predominant synthesis techniques, p synthesis design [8], Linear Quadratic Gaussian (LQG) control design [25], and X control design [26]. The p synthesis technique attempts to minimize the p value over stabilizing the controller, K, and diagonal, D, while DK iteration is used to reduce the cost function. The LQG controller design method synthesizes a controller which is optimal with respect to a specified quadratic performance index and takes into account the Gaussian white noise disturbances acting on the system. The technique used for the projects presented in this paper is the Hto control synthesis technique which uses the linear fractional form of the LPV system and creates the controller while attempting to minimize the to norm. By letting the controller have the same linear fractional relationship with the varying parameters as the LPV system the Hio control problem is formulated using linear matrix inequalities (LMI). The appearance of LMIs in the control synthesis shows how the control problem is a convex optimization problem [27], as was described in the previous example. Another example of creating a convex optimization problem with LMI expressions for the use of finding an LPV controller for the attitude control of an X33 is presented in [28]. The main benefit of using the LPV framework is that it allows gainscheduled controllers to be treated as a single entity, with the gainscheduling being achieved with the parameterdependent controller and automatic interpolation law, which removes the adhoc scheduling schemes that were necessary in the past. CHAPTER 3 LINEAR PARAMETERVARYING CONTROL FOR AN F/A18 3.1 Problem Statement Several control designs have been applied to the control of F18 aircraft. A controller was designed using XJI and p synthesis techniques for a single flight condition [29]. Though this technique works well for a single point in the flight envelope a type of gainscheduling is necessary for controlling the F/A18 throughout its operating domain. A longitudinal variable structure controller was created for an F18 model with parameter perturbations [30]. Though this technique can attain the conventional goals of stability and tracking for uncertain nonlinear plants, a reference trajectory for tracking control must be specified, which indicates that the controller cannot operate over a large flight envelope. A lateraldirectional controller was created using p synthesis with parametric uncertainty to account for gain differences between a nominal model and trim models and multiplicative uncertainty to account for changes between a nominal model and other trim models within the chosen flight envelope [31]. Because this technique uses constant blocks of uncertainty instead of gainscheduling the flight envelope used for the project had to be small, M = [0.35,0.55] and altitude = [20,28]kft. Gainscheduled approximations to H_ controllers for the F/A18 Active Aeroelastic Wing, located at NASA Langley Research Center, were developed within another project [32]. Point design controllers were created within a small flight envelope and then a scheduling scheme of the gains had to be formed. A multivariable LPV controller was designed using H_ synthesis for the F/A18 System Research Aircraft (SRA), located at NASA Langley Research Center, in [33]. Though this technique is also chosen for control synthesis in the project presented in this chapter, the flight envelope that the controller had to operate within is smaller in [33], with M = [0.35,0.70] and altitude = [15,32]kft. A similar project involving an LPV controller for the F/A18 SRA, in [18], uses the same synthesis technique but uses an even smaller flight envelope, M = [0.45,0.55] and altitude = [20,25]kft, than [33]. The project presented in this study discusses the formation and simulations of a linear parametervarying controller for the longitudinal dynamics of an F/A18 over a chosen flight envelope. The F/A18, shown in Figure 31, has a ceiling of 50,000+ft and a speed of M=1.7+. As the aircraft's altitude varies so does the air density which affects the aircraft's response to control surface deflections. Furthermore, the amount of deflection necessary for a particular maneuver varies as the Mach number varies. These aerodynamic changes that occur with the large range in altitude and Mach number make it necessary to incorporate a gain scheduling technique for control. The flight envelope for this project is limited to Mach numbers from 0.4 to 0.8, which includes both incompressible and compressible subsonic flows, and an altitude range from 10,000ft to 30,000ft, which includes a density change of roughly 0.9 E 3 slug/ft3. Figure 31: F/A18 The flight envelope can also be considered the parameter space for which the LPV controller will be designed. The parameter space is two dimensional with the first parameter dimension being Mach number and the second parameter dimension being altitude. Originally four points within this twodimensional parameter space were to be used to design the LPV controller and are listed in Table 31. However, the dynamic pressure, q, for the model at Mach=0.40 at an altitude of 30 kft, P4, was too low to control and therefore the model was discarded. Table 31: Original Design Points Design Point Mach Number Altitude (ft) P1 0.4 10,000 P2 0.8 10,000 P3 0.8 30,000 P4 0.4 30,000 The controller performance is tested with each of the remaining models and with a model whose dynamics represent the aircraft at a Mach number of 0.6 and at an altitude of 20,000 ft. A depiction of the flight envelope which represents the parameter space and the placement of the models used for this project are shown in Figure 32. 35 o Design Point o Design Point 30 o Design Point 0 3 x Analysis Point 25 S20 P <15 10 o P o 2 .3 0.4 0.5 0.6 0.7 0.8 0.9 Mach Number Figure 32: Flight Envelope/Parameter Space 3.2 Openloop Dynamics The F/A18 models used for this project are longitudinal shortperiod approxima tions that were developed with two states, one input and one output. The states include angle of attack (deg) and pitch rate (deg/sec). The input is the elevator deflection and the output is pitch rate. The model for the F/A18 at Mach=0.40 at an altitude of 10 kft is given as P1 such that q = P1 6. 0.7433 Pl= 0.0022 The model for the F18 that q = P2 8. 425.6200 0.4064 0.5642 0.0662 (3.1) S 0 57.3 0 at Mach=0.80 at an altitude of 10 kft is given as P2 such S1.8415 P2= 0.0192 The model for the F18 that q = P3 8. 853.1909 0.9431 2.0292 0.2568 (3.2) S 0 57.3 0 for Mach=0.80 at an altitude of 30 kft is given as P3 such 0.8399 791.1313 0.9314 P3= 0.0075 0.4499 0.1190 0 57.3 0 The model of the analysis point with an altitude of 20,000 ft and Mach=0.6 is given as PA such that q = PA 8. PA= 0.8280 617.0114 0.8269 0.0075 0.4499 0.0994 (3.3) (3.4) S 0 57.3 0 The frequency and damping ratio for the each of the models were determined and are shown in Table 32. All of the damping ratios are greater than zero, which affirms that the models are stable. Table 32: Frequency and Damping Ratio of Design and Analysis Models Model o) P1 1.113 0.5166 P2 4.257 0.3271 P3 2.512 0.2567 PA 2.288 0.2764 The linear parametervarying model for the parameter space is given as P(O) and is given as q = P(O) 8. 1.0982 427.57 1.465 1.0016 62.06 1.0978 P(0)= [Pi]+ 0.017 0.5367 0.1906 01(t)+ 0.0117 0.4932 0.1378 02(t) 0 0 0 0 0 0 (3.5) 1i 0 Where 0 = 02 and where 01 E [0, 1] represents the systems dependence 0 92 on Mach number and 02 E [0, 1] represents the systems dependence on altitude. The aircraft flying at a Mach number of 0.4 corresponds to a 01 = 0 and at a Mach number of 0.8 corresponds to a 01 = 1. The aircraft flying at an altitude of 10,000ft corresponds to a 02 = 0 and at an altitude of 30,000ft corresponds to a 2 = 1. 3.3 Control Objectives The control objective for the F/A18 longitudinal flight controller is to track a given pitch rate command to within certain tolerances of a target response generated by a target model that has desirable dynamics. The commanded pitch rate is a step input which begins at zero magnitude and becomes l0deg/sec at the time of one second and remains that magnitude until the simulation ends at ten seconds. The response of the system with the linear parametervarying controller to the commanded pitch rate must have a rise time within 0.05 sec of the target rise time, an overshoot within 4% of the target overshoot, and a settling time within 0.6 sec of the target settling time. The controller should also have a level of robustness to account for errors in the signal. 3.4 Synthesis The system shown in Figure 33 incorporates all the necessary elements needed to create the controller which will accomplish the controller objectives. This system contains the openloop dynamics as described by {P} and a target model {T} used for model following. The system also incorporates weighting functions used for loop shaping, which are given as {Wp, Wn, Wk, Wu}. n n y qc  u P(0) Wp e1 Figure 33: Synthesis Block Diagram The system has 2 disturbances, 1 control, 2 errors and 1 measurement, which were referred to as {d, u,e,y} in the LPV Control section. The disturbances are random noise n E R affecting the sensor measurement and the commanded pitch rate qc. The control input is u E R, which affects the elevator deflection. The sensor measurement of pitch rate, y, is used for feedback to the controller. The errors are, el, the error between the target pitch rate response and the LPV model response and the weighted control effort, e2. The target model T describes an F/A18 model that has dynamics which outputs a desirable pitch rate response. The target model is used for model following to aid in obtaining the LPV controller. The target model is not a function of parameters in the operation space. The frequency and damping ratio of the target model are presented in Table 33. 2 0 0 1.98 0 3 3 2.78 T = (3.6) 0 3 3 2.18 0.79 0.47 1.79 0 Table 33: Frequencies and Damping Ratios of the Target Model Target 2.0 1.0 4.2426 0.7071 The performance filter given as Wp serves to normalize the error in the model following between the target model and the LPV model. The filter Wp is chosen to meet design specifications in accordance to pitch rates of the aircraft in response to a commanded pitch rates. The actual filter is realized as Wp = 70. The filter Wk is used to normalize the penalty placed on the amount of actuation commanded by the controller. This filter reflects the capabilities of the actuation system. The weighting is chosen as the inverse of the actuator's magnitude of motion, Wk = 0.5. The input matrix varies from model to model within the parameter space. This variation necessitates an input filter within the synthesis. The filter shown as Wu is used as the input filter and has a value of 1E5 s+lE5" Also, the filter W, is included to account for noise that corrupts measurements by the sensor. The inclusion of noise is needed to provide a minimal level of penalty on the sensors, which will give robustness to the controller. The design did not want to consider a large amount of noise so the filter was chosen as W, = 0.01. Using the system in Figure 33, robust H_ controllers were designed for the models at each of the design points and a linear parametervarying controller was designed for the entire parameter space. The techniques of JH. control are used to reduce the induced norm from the input to the weighted errors. The software from the p Analysis and Synthesis Toolbox for Matlab is used for the actual computation of the controller [34]. The same weightings are used to create the controllers in order to achieve the same performance level for all of the points in the parameter space. The resulting induced norms achieved by the individual controllers and the LPV controller are shown in Table 34. Table 34: Induced Norms of ClosedLoop System OpenLoop Model _12norm P1 0.891 P2 0.775 P3 0.775 P1 P3 0.971 It is important to note that all of the closedloop norms are less than unity. These magnitudes indicate that the controllers are able to achieve the desired performance and robustness objectives. The last entry in Table 34 is the norm associated with the LPV controller. Allowing the altitude and Mach number to vary with time increases the norm as expected. However, this norm did not raise much above the norm associated with any of the point designed 12 controllers and stayed below unity. This condition indicates that the LPV controller is capable of accounting for the timevarying nature of Mach number and altitude without excessive loss of performance. 3.5 Simulation The closedloop dynamics are simulated with a l0deg/s pitch rate step input to demonstrate the performance of the controller for each of the design models and for the analysis model. The diagram of the closedloop system for the models can be seen in Figure 34. The simulations use the same openloop dynamics but include the linear parametervarying controller that was synthesized over the parameter space. The response to the step input of the LPV controller with the point design models and the response of the target model are shown in Figure 35. The point design K(O) ' q, Figure 34: Closedloop Block Diagram responses only vary roughly 0.2% from the target model response. This characteristic is due to the LPV controller being created with the models at those points. The performance of the controller must also be tested with a model that lies away from the vertices points of the parameter space that were used to create the controller. 12 a 6 8 4  Command Target SPl 2 P 0 2 4 6 8 10 Time (s) Figure 35: Pitch Rate for Design Points The analysis point was chosen to be the farthest from the vertices of the parameter space which results in a Mach number of 0.6 and an altitude of 20,000ft. The responses of the analysis model and the target model, using the same step command that was used for the point design simulation, are shown in Figure 36. The results appear to be quite close to the target response. Numerical results were pulled from the plot to make a closer comparison and are shown in Table 35. The same time response and delay time are apparent for both the analysis model and target model responses. The settling time of the analysis model response lags the target response by 0.5 seconds, which is within the control objectives. The maximum 8 6 2 00 2 34 Figure 36: Table Rise Time Settling Time Peak Overshoot 4 6 8 10 Time (s) Pitch Rate for Analysis Point 35: Numerical Results Target Model Analysis Model 0.21 sec 0.21 sec 1.63 sec 2.13 sec 5 % 1.2% overshoot of the analysis model response was less than that of the target model response and remains within the bounds of the controller objective. The controller commanded elevator deflection from the simulations is shown in Figure 37 and is used to determine if the actuation of the elevator is reasonable for each of the tested models. All of the values are negative because a negative elevator 0 2 4 6 8 10 Time (s) Figure 37: Controller Elevator Deflection deflection causes a pitch up in the aircraft, which is the commanded input of the simulations. The peaks appear at the time when the input pitch rate command is initiated and the command response that follows is to maintain the pitch rate command. The peak of the elevator command for the system with the model associated with a point in the parameter space having a Mach number of 0.4 and an altitude of 10,000ft, PI, has the greatest value, 170. The value is reasonable because at lower speeds a greater angle is needed to get the same response as flying at faster speeds and there is a less chance that the control surface will be damaged by the slower airflow. The elevator command peak for the system using the model with a Mach number of 0.8 and an altitude of 10,000ft, P2, has the smallest value, 4.5. This lower value is all that is needed for the desired pitch rate due to the speed of the airflow around the control surface at the higher Mach number, which decreases the time the maneuver requires. The peak value is 9.50 for the elevator command associated with the system using the model that has a Mach number of 0.8 and an altitude of 30,000ft, P3, which is an acceptable magnitude. This value being higher than the value corresponding to the same Mach number but with a lower altitude of 10,000ft is expected because at a higher altitude the density is less and so fewer air particles are present to be affected by the deflection, therefore a larger angle is necessary. The peak commanded elevator deflection for the system using the analysis model, PA, is 120, which is also acceptable. The magnitude is reasonable because it is less than the value commanded for P1 due to the higher Mach number of the analysis model and is not too small that the increase in altitude would have an adverse affect. 3.6 Conclusion This project considered the control of the pitch rate of an F/A18 aircraft with a linear parametervarying controller. This type of controller was chosen because the change in dynamics of the aircraft could be modeled with a system whose statespace matrix and input matrix were affine functions of the parameters, Mach number and 21 altitude. Once the controller was created, it was tested at certain points within the parameter space using a step pitch rate input. The results allow for the conclusion that the LPV controller performed the specified objectives and is therefore a sufficient controller for the F/A18 model presented in this project. CHAPTER 4 LINEAR PARAMETERVARYING CONTROL FOR A HYPERSONIC AIRCRAFT 4.1 Problem Statement All aircraft flown today fly within the subsonic, transonic and supersonic flight regimes. The push toward faster and higher flying aircraft has moved the envelope into the hypersonic regime. This push comes from both military and commercial groups. The military wants a bomber that can fly at high altitude, over a long range and at high speeds, so that the vehicle is nearly impossible to shoot down. Commercial groups would like to have a more reliable way of sending satellites into low earth orbit. The major problem with the use of rockets is that if something goes wrong during ascension into orbit the cargo will most likely be destroyed along with the rocket. The use of a hypersonic aircraft presents a more reliable transportation for the satellite because if an error did occur during the flight there would be a chance that the aircraft could maneuver to a landing area. Though the concept of hypersonic flight has been discussed since the 1950s the mass construction of hypersonic aircraft has been hindered by the necessity of the technology and the price of materials that are able to withstand the elements in which the vehicles must operate. This obstacle may have slowed the creation of such vehicles but several control theories have still been created. The more popular control theories include /L1 [35], p synthesis [36] and linear parametervarying control [37]. The theories involving Xto and p synthesis, however, only considered a single flight condition for the hypersonic vehicle. Also, the previous project that used a linearparameter varying controller for the hypersonic vehicle ignored the mode shape of the vehicle and separated the rigidbody dynamics and the structural dynamics of the hypersonic model. A scheduled longitudinal control scheme was created which incorporated a set of parameter controllers, where the parameters were Mach number and dynamic pressure, and was determined from linear designs using analytic functions of the parameters [38]. That project focused on the control of the rigid body dynamics and did not recognize the effect of structural modes on the response of the hypersonic vehicle. Robust flight control systems are synthesized for the longitudinal motion of a hypersonic vehicle using stochastic cost functions and ten design parameters [39]. That project also focused on the control of the rigid body dynamics of the hypersonic vehicle without addressing structural dynamics. The control of the longitudinal motion of a hypersonic vehicle was also addressed, where robust flight control systems with a nonlinear dynamic inversion structure were synthesized [40]. Nonlinear control laws were designed so the control systems would operate over a chosen flight envelope. Again, the rigid body dynamics were the focus of control. A dual neural network structure was developed that served as feedback control and optimized the vehicles trajectory to prespecified burnout conditions in velocity, flight path angle and altitude [41]. That project serves more as an aid in the study of trajectory optimization than as a control theory for hypersonic vehicles. Another project applied a hierarchical integrated control methodology to a hypersonic vehicle to reduce stabilizing control power required for specific flight conditions [42]. That methodology decomposes the hypersonic model into decoupled subsystems, creates a controller for each subsystem and a control law for each subsystem controller is derived. The decoupling of a hypersonic system may not be feasible due to the large degree of coupling between the physical structure and propulsion system of the vehicle. Also, the creation of separate control laws is laborious compared to the LPV method which forms an automatic interpolation law. The control of the lateral dynamic stability characteristics of a hypersonic vehicle for a specified Mach number and altitude trajectory has also been detailed in a project [43]. The controller was designed using MultiModel Eigenstructure, which designs a robust fixedgain controller that guarantees robust stability and desired flight qualities along a specified reference trajectory. The controller would need to be altered if the vehicle deviated from the preset trajectory or if a flight envelope was to be considered. The same model of the longitudinal dynamics of a typical hypersonic vehicle were used, where a unified approach to H2 and Xo optimal control was used to design a controller for a specific flight condition [44]. A unified approach alleviates difficulties with the "over crowding" of a system's roots inside the unit circle along with other numerical difficulties. Using the technique in that project would require more controllers to be created at other operating conditions along with a gainscheduling law if the vehicle's operating range spanned more than a single condition. Some of the challenges of hypersonic flight include the varying of the hypersonic vehicle's dynamic characteristics due to a wide range of operating conditions and mass distributions for which a type of gainscheduling technique appears to be essential [45, 46]. Further discussion of a typical hypersonic vehicle's dynamics addresses how the combination of the propulsion system and aeroelastic effects contribute to the overall dynamic character of the vehicle, which presents the need of structural dynamic controller [47]. This necessity is the motivation for the project presented in this chapter. The controller designed for the hypersonic vehicle for this project was split into an innerloop controller and an outerloop controller. The innerloop controller is an LPV controller which must actively damp the structural modes across a temperature range. Unlike previous hypersonic controls, this controller will focus on the damping of the mode shape that is associated with the structural dynamics of the vehicle, which will operate throughout a range of a specific operating parameter and for which the hypersonic model's rigidbody and structural dynamics will not be separated. The outerloop controller of the aircraft will be a rigidbody controller which will work as a traditional flight controller for rigid aircraft and will be designed in a future project. The innerloop structural damping controller is the focus of this project. A diagram of the innerloop/outerloop control design is shown if Figure 41. The K(O) controller is the linear parametervarying innerloop controller and the Kut controller is the rigidbody outerloop controller. The P is the hypersonic plant model. input output Figure 41: InnerLoop/OuterLoop Design 4.2 Generic Hypersonic Vehicle The intended uses of hypersonic vehicles ranges from putting satellites into low earth orbit to being the next stealth bomber. These missions require the vehicle to travel through each flight regime: subsonic, transonic, supersonic, hypersonic and orbital. This project will only consider the aircraft's flight within the hypersonic regime. Several hypersonic designs have been created which attempt to maximize aerody namic and propulsive efficiency while still having enough controllability. Most of these designs incorporate the elevator and aileron into one structure known as the eleven. The form of hypersonic vehicle used for this project is similar to the NASP and X30 vehicles. A generalized shape can be seen in Figure 42. Figure 42: Simplified Model of a Generic Hypersonic Vehicle This configuration of a hypersonic vehicle combines the fuselage with the propulsion system. This combination greatly affects the flight dynamics of the vehicle. The forebody of the vehicle acts as the compressor for the engine. The air flow through this compressor creates a pitch up moment. The aftbody of the vehicle acts as the exit nozzle for the engine. The airflow through the exit nozzle creates a pitch down moment. Also, a change in angle of attack or sideslip affects the engine inlet conditions which changes the propulsion performance. To create a controller for this type of vehicle the angle of attack, pitch angle and pitch rate are measured for feedback to the controller. Another area of hypersonic flight that must be considered when creating a controller is the speed, and consequently temperature, at which the vehicle flies. As the vehicle enters the hypersonic regime, the strength of shock waves increase and lead to higher temperatures in the region between the shock and the body. As Mach number increases further, the shock layer temperature becomes large enough that chemical reactions occur in the air. Also, an increase in temperature effects the structural dynamics of the vehicle in that there is a reduction in the frequency of the structural modes. Therefore, the controller created in this project will consider temperature as the flight parameter. 4.3 Hypersonic Model The hypersonic model [48] used for this project was limited to the longitudinal motion and was developed with seven states, three inputs and six outputs. The states include altitude, velocity, angle of attack, pitch angle, pitch rate, and two elastic states for the fuselage bending mode. The inputs include eleven deflection, diffuser area ratio and fuel flow ratio. The outputs include angle of attack, pitch rate at forebody, pitch rate at aftbody, combustor inlet pressure, Mach and thrust which will be used as feedback to the controller. Only the angle of attack and the two pitch rates are to be used as feedback to the controller due to their strong dependence on the structural dynamics. Aerodynamic, inertial, propulsive, and elastic forces were used to derive the equations of motion for the hypersonic vehicle [37]. The model dimensions and flight conditions are shown in Table 41. Table 41: Model Dimensions and Flight Conditions Length 150 ft Mass 300,000 lb Height 100,000 ft Mach 8 Dynamic Pressure 1017 psf 4.4 Linear ParameterVarying System The timevarying operating parameters, 0, are flight parameters which affect the aircraft during flight. These parameters are measured by sensors on the aircraft and are sent to the controller. This project takes into account only one flight parameter, temperature, due to the large affect that temperature has on a hypersonic vehicle's structural dynamics. This parameter will have a range from (0F to 50000F) to match the temperature ranges noted for the hypersonic flight of the X30 and the HyperX vehicles [49]. The parameter dependence of the model is shown in the matrices below, 0 = 0 for the coldest temperature and 0 = 1 for the hottest temperature within the range. As the flight parameter, temperature, changes during flight so does the amount it affects changes in the aircraft. This problem can be compensated with the use of weighting functions which will be discussed in the next section. A() = [A +0 [Ae ] (4.1) 0 0 7.9248E3 7.9248E3 0 0 0 1.5026E4 3.2374E3 5.2818E1 3.2200E1 2.3762E2 5.7314E1 7.5583E3 1.1744E7 3.1848E7 3.3921E2 0 0 5.7586E6 0 7.4858E1 0 9.6079E6 0 1.0158E1 0 1.5833E0 0 2.4280E3 1 1.4681E4 2.8801E6 0 1 0 0 5.1609E2 9.2411E2 0 0 0 0 7.4847E0 3.1086E2 0 0 0 0 0 0 2.5E2 0 0 0 0 0 0 0.2 0 6.435E1 1.448E 2 0 2.455E0 0 6.740E2 0 7.462E1 1.596E2 0 8.111E 1 0 2.925E1 0 1.261E3 2.253E 2 0 5.190E0 0 2.209E2 C(O) ==[ ] Co ] 0 1.8285E4 1 9.4975E1 (4.2) (4.3) (4.4) (4.5) 0 0 1 0 0 1.7453E2 0 0 0 0 0 1 0 1.7453E2 0 0 0 0 1 0 1.7453E2 C= 4.6971E5 2.0641E4 6.2428E0 0 1.0921E2 1.0896E1 0 5.3709E6 1.0095E3 0 0 0 0 0 3.5754E1 6.0213E1 1.8399E4 0 3.2185E1 3.2112E2 0 (4.6) 0 0 0 0 0 C(1,6)*1.05 0 0 0 0 0 0 0 C(1,7) 0.05 0 0 0 0 0 0 C2(2,7)*0.05 Co = (4.7) 0 0 0 0 0 C2(3,6)*0.05 0 0 0 0 0 0 0 0 0 0 0 0 0 C2(5,6)*0.05 0 0 0 0 0 0 0 0 0 0 D = (4.8) 0 7.229 0 0 0 0 0 3.158E4 5.995E5 As seen in the linear parametervarying matrices above, both the state matrix [A] and the observation matrix [C] change with temperature. It is common for the state matrix to change as operating parameters change, but it is not common, in traditional aircraft, for the observation matrix to change. This change in the observation matrix accounts for the mode shape changes of the hypersonic vehicle. The modes of the hypersonic model are shown for different temperatures in Table 42 The table shows the frequency of each of the modes and the damping corresponding to the frequency. The four modes of the openloop dynamics are (i) a height mode, (ii) an unstable phugoidlike mode, (iii) an unstable pitch mode and (iv) the structural mode. As can be seen in the table, the structural mode for the model at the cold temperature has a higher frequency than the structural mode at the hot temperature. Minimizing the affect that the temperature has on this mode is the objective of the innerloop LPV controller. Table 42: Modes of the Hypersonic Model Cold Hot Mode o(rad/sec) ( m(rad/sec) i 0.0024 1.00 0.0024 1.00 ii 0.1666 1.00 0.1790 1.00 0.1677 1.00 0.1804 1.00 iii 1.462 1.00 1.518 1.00 1.554 1.00 1.608 1.00 iv 17.65 0.0268 15.84 0.0062 4.5 Control Design The control objective of the linear parametervarying controller is to damp out the structural mode in order to minimize the affect that temperature has on the model. The controller should also contain a level of robustness to account for errors in signals. A system was created that incorporated the necessary elements to accomplish these objectives. The first step in the finding the LPV controller was to create a synthesis model shown in Figure 43. The system has 2 disturbance inputs, 1 control input, 2 error outputs and 1 measurement output. The disturbance vector n E R3 is random noise which affects sensor measurements. The incorporation of noise creates a small level of robustness within the controller. The disturbance 8 E R is a commanded eleven deflection. The control input u E R is the excitation from the controller affecting the control actuators. The error ep E R is the weighted measurements of the angle of attack by the sensors. The error ek E R is the error of the control actuation. The measurements in the vector 5 t P(9) n Wn u ek Figure 43: Synthesis Block Diagram y E R3 are the sensor measurements of angle of attack, pitch rate at the forebody and pitch rate at the aftbody which will be used for feedback to the controller. The openloop dynamics of the LPV system is described by P(O). Where, P() = A() (4.9) C(9) D A target model, T, is created to describe a hypersonic model with desirable structural damping and therefore incorporates the controller objective. The target model was used for model following to aid in obtaining the LPV controller. The target model modes and corresponding damping are shown in Table 43. The target model has a large magnitude of damping corresponding to its structural mode compared to the damping found in the hot and cold temperature models. It is this amount of damping that the controller must impose upon the hypersonic model throughout the temperature range. The performance filter, Wp, would normally be used to define the design specifi cations in the frequency domain. For this synthesis Wp was made equal to 1.5 which Table 43: Modes of the Target Model Target Mode om(rad/sec) i 0.0024 1.00 ii 0.1728 1.00 0.1735 1.00 iii 1.478 1.00 1.590 1.00 iv 16.75 0 .2381 allows measurements through all frequencies to pass through with only a slight de crease in gain. This passage throughout all frequencies was allowed because of the simple controller X which was incorporated into the system to stabilize the vehicle. A simple HJ_ controller, X, is created in order to stabilize the rigidbody dynamics of the hypersonic vehicle without an affect on the structural mode. This small con troller was implemented so that the structural dynamics controller would not try to alter the rigidbody dynamics. Stabilizing the rigidbody of the model allows the creation of the LPV controller for the structural dynamics. The filter, W, passes an allowed amount of noise to the sensors. W, = 0.01 because only a small amount of noise was needed to pass into the system to ensure that the controller would be robust. The filter, Wk, is used to normalize the restriction on the amount of actuation the controller commands. Wk was chosen so that the weighting is the inverse of the actuators' magnitudes of motion, Wk = s+180 The results of the openloop synthesis were then used to create the LPV controller, K(O), using the LMI ControlToolbox [50]. To determine how well the controller would work the H_ norm was found for the system throughout the temperature range, along with the H_ norm for the system at the cold temperature and at the hot temperature. The frequencies at which the HJ_ norm occurred for the model at the hot and cold temperatures were also found. The results of this test are shown in Table 44. The magnitude of the XJo norms of the model at the hot and cold temperatures mainly draws from the connection of the first input, q, to the first output, the ep, meaning that the largest error comes from the performance of the angle of attack meeting the elevon deflection command. Table 44: OpenLoop Synthesis Norms Snorm (o(rad/sec) System 0.9386 Cold 0.9159 17.90 Hot 0.91196 19.81 Xo controllers were made specifically for the model at the cold temperature and the model at the hot temperature. The H_ norms of these point designs were found and used to compare to those found for the full LPV system. The results are shown in Table 45. Table 45: Point Design Norms Xo norm Cold 0.1476 Hot 0.1679 Compared to the norms of the system with the LPV controller at the hot and cold temperatures and the norms of XJI controllers at the point designs, the norm of the LPV system is relatively high. This difference results from the timevarying nature of the parameters of the system. Despite this increase in magnitude the LPV system's Snorm is still less than one, showing that the LPV controller that was created is capable of controlling the system. 4.6 Simulation 4.6.1 OpenLoop Simulation The frequency response of the openloop transfer function between the elevon defection and the angle of attack for the target model, cold model and hot model is shown in Figure 44. The plot of the response in the frequency domain demonstrates the need for the control of the structural dynamics. The peak in the response that is located in the high frequency region is the structural mode. The target model's structural mode peak shows a damped response, which is desirable. The hot and cold hypersonic models' structural mode peaks, however, are very sharp which implies that there is very little damping. These peaks in magnitude correspond to a bending of the aircraft at the frequencies at which the peaks occur, which brings the desire for the controller to be able to damp structural mode. 102 target cold 101  hot O 10 10 104 102 10 102 Frequency (rad/sec) Figure 44: OpenLoop Transfer Functions The simulation of an angle of attack response to an eleven deflection input for the openloop hypersonic models at both the hot and cold temperatures and the target model is shown in Figure 45. The eleven deflection used for the following simulation of is shown in Figure 46. The angle magnitude is small due to the speed at which the vehicle flies, a large angle would be harmful at high speeds. Unlike the target model response the openloop model response at both the hot and cold temperatures show an oscillation for approximately five seconds. This oscillation is due to the lack of structural damping and should be removed by the controller during the closedloop simulation. 3 target 40 2 0 5 10 15 Time (s) Figure 45: OpenLoop Angle of Attack Result Input Elevon Deflection 5 C4 3 52 0 0 5 10 15 Time(s) Figure 46: Input Elevon Deflection 4.6.2 ClosedLoop Simulation The closedloop dynamics are simulated to demonstrate the performance of the controller for the hypersonic models at both the hot and cold temperature. The closedloop system for both models can be seen in Figure 47. The system shown in Figure 47 has one input signal and six output signals. The input signal 8 remains the eleven deflection. The outputs include angle of attack (a), pitch rate at forebody (qf), pitch rate at aftbody (qa), combustor inlet pressure (Pi), Mach (M) and thrust (Th). The X is the same simple controller used to stabilize the Th Figure 47: Closedloop Design vehicle for the openloop synthesis. The K(O) is the LPV controller that was created using the synthesis model. The frequency response of the transfer function between the eleven deflection and the angle of attack for the closedloop system for both the cold and hot temperature hypersonic models is shown in Figure 48. 102 target cold 101  hot 100 10 10 104 10 10 102 Frequency (rad/sec) Figure 48: ClosedLoop Transfer Functions Again, the peaks in the high frequency region correspond to the structural mode of the target model and the hot and cold temperature hypersonic models. The structural mode is clearly damped by the controller for the hypersonic models. It is important to note that these responses correspond to the end points of the temperature range, which implies that if the model were tested at a temperature that falls within the temperature range that a similar damped peak would result. So the control objective of damping the structural mode was fulfilled by the LPV controller. The closedloop simulation of the angle of attack response to the same elevon deflection used in the openloop simulation is shown in Figure 49. The results are again presented for the system at both the hot and cold temperatures and for the target model. target cold 2 hot <2 3 4 0 5 10 15 Time (s) Figure 49: ClosedLoop Angle of Attack Result As can be seen, the oscillations that were apparent in the openloop simulation have been removed by the controller. This response is due to the damping which the controller imposed on the system. The hypersonic models' responses also follow the target model response more closely throughout the simulation. The controller commanded eleven deflection in Figure 410 is plotted for the closedloop simulation in order to verify that the motion commanded did not violate the limited motion due to the high Mach number. Because the command never exceeds a magnitude of 50 the command does not violate the constraint associated with the eleven actuator. The corresponding deflection rate in Figure 411 is plotted to verify that the command does not violate the motion tolerances of the eleven actuator. The magnitude of the deflection rate is within the limits associated with the actuator. Scold command 4 hot command 2 80 cold g 40 20 20 40 60 0 5 10 15 Time(s) Figure 411: Elevon Deflection Rate 4.7 Conclusion This project considered the control of the structural dynamics of a hypersonic vehicle with a linear parametervarying controller. This type of controller was chosen because the change in the dynamics of the hypersonic vehicle could be modeled with a system whose statespace matrix and observation matrix were affine functions of the parameter, temperature. Once this controller was created, it was tested over a temperature range with an eleven deflection input. The results allowed for the conclusion that the LPV controller performed the specified objective and is therefore a sufficient controller for the hypersonic model presented in this project. CHAPTER 5 LINEAR PARAMETERVARYING CONTROL FOR A DRIVEN CAVITY 5.1 Problem Statement Research into flow control techniques has been continually evolving as related technologies mature. These technologies include hardware development, such as sensors and actuators [51], and software development, such as models and simula tions [52], associated with fluid dynamics. In each case, the technologies are being developed with careful consideration of the requirements for control design and implementation [53]. A particular challenge for flow control has been the development of openloop models for which controllers can be designed. The equations of motion for such dynamics are well known and detailed computational simulations are routinely performed. Unfortunately, the equations of motion are highly nonlinear and no methods are currently practical that can directly utilize them for feedback control synthesis. A recent study has shown that models can indeed be generated that are amenable to control a specific type of flow [54]. The system in that study is restricted to creep ing flow in a driven cavity. Specifically, the left and right sides of the cavity have zero flow velocity whereas the top and bottom boundaries are driven by exogenous flow with fixed velocity and frequency. Models are generated by considering the linearized dynamics associated with modes obtained via proper orthogonal decomposition [55]. These modes were used to derive controllers for disturbance rejection. The derived controllers were able to keep the flow nearly stationary at various points throughout the cavity for varying flow regime despite the exogenous input[56, 57]. This project extends the work of Feng [54] to consider different flow conditions for the driven cavity. Specifically, the openloop models are generated by considering the phase differential between the disturbances at the top and bottom of the cavity. Feng's study only used models of the flow resulting from upper and lower flow boundaries that were 1800 different in phase. This project will consider using models of the flow resulting from upper and lower flow boundary conditions that are 210, 1950, 1800, 1650 and 1500 different in phase. A range of phase differentials was used to derive controllers that accommodate the unpredictable phase difference that would be associated with a real driven cavity. A linear parametervarying controller is designed for the models corresponding to the range of various phase differentials. The group of models can be considered as lying within a parameter space with the parameter being the phase differential between the excitation and disturbance. The openloop simulations show a rise in the horizontal velocity along the centerline as the phase differential is increased. The closedloop simulations show a distinct reduction in the centerline velocity and therefore prove the effectiveness of the LPV controller. 5.2 Background The development of control algorithms has made much progress in the last few decades for aerospace applications. The control of fluid flow, however, is one of the most difficult applications and remains mostly unresolved. The primary difficulty lies with the inherent nonlinearity of the NavierStokes equations which are the governing equations of the fluid flow. Even approximations of this equation can have large dimensionality. Within the last decade studies of flow control have addressed specific flow regimes. These regimes include the control of a driven cavity using LQR control [56], using piezoceramic actuators to control a channel flow [58] and the use of synthetic jets for flow control [59]. In more recent years, much research has been done to derive methods to generate reducedorder flow models. Two specific techniques that employ a reduced basis are proper orthogonal decomposition (POD) [60] and fluid mode methods [53]. Proper orthogonal decomposition is a model reduction technique in which the most energetic modes are systematically extracted from numerical simulations. This method of reduction was used to create the models used in this project. The fluid mode method uses basis functions which are closely related to the physics of the problem being solved. Another area of interest for this project is what is known as Stokes or creeping flow. The limitations of using Stokes flow are that the flow must be incompressible and have a Reynolds number less than one. One side effect of lowering a flow's Reynolds number is that the acceleration term within the NavierStokes governing equation becomes small compared to the viscous force term. This change allows the equation to be simplified into the linear Stokes equation [61, 62]. 5.3 Driven Cavity Geometry This project will investigate flow control for the cavity shown in Figure 51, where ho(t) is the velocity along the top of the cavity, 3(t) is the velocity along the bottom of the cavity and F = FL U FR U FT U FB is the boundary of the domain. This cavity is enclosed by rigid walls with noslip boundary conditions on the right and left sides. The top and bottom, however, have nonzero boundary conditions in general. u = l(x)ho(t) fixed input: v= 0 TF {v=0 rL R v=0 rB control: {u=1(x)P(t) [ v=0 Figure 51: Stokes Driven Cavity Flow Problem The flow at the top and bottom boundaries have uniform spatial distribution. This restriction implies that the flow at any point along the upper boundary is identical to the flow at any other point along the upper boundary. Similarly, the flow at any point along the bottom boundary is identical to the flow at any other point along the bottom boundary. Such a perfect distribution is not possible because of the singularity at the points on the corners where the flow is moving on the horizontal boundary but stationary on the vertical boundary. Such a situation is obviously an approximation, but this example does serve as an initial problem to demonstrate the methodology. The approximation within the 2D cavity is based on a grid with an index of 21x21 points. It is assumed that the measurements of the flow velocity are taken at 19 points along the horizontal centerline of the cavity, with the outer points lying one grid point away from the closest boundary wall. These measurements only provide the horizontal velocity of the flow. Also, the sensors generating these measurements are assumed to exist within the cavity without altering the flow. Again, such a situation is obviously an approximation, but the example serves to demonstrate the methodology. 5.4 Governing Equations of Motion Consider first the unsteady NavierStokes equations p + pV VV = Vp +uAV (5.1) subject to boundary conditions described in the past section. The parameter V is the velocity field, p is the pressure, p is the density and p is the viscosity of the fluid. The constants that will be used to nondimensionalize the problem include a characteristic dimension L, characteristic velocity Vs, and characteristic frequency f. Nondimensional variables are defined as x* = x/L = y/L p* = p/pLV, t* =tf The resulting nondimensional NavierStokes equation can be written as Re St +Re V** = ReStV*p*+A** where St is Strouhal number defined as Lf St = V, Define S= Re St =pL il This project will consider flow such that X ~ 0(1) and the Reynolds number is Re=0.1. In this case, the terms on the right hand side will balance the first term on the left hand side and the second term will be neglected. Define the constant 1 1 0 X Re St Finally, the governing equations for the approximation of Stokes flow is achieved AP* + V = 0 at* For convenience, Equation 5.11 is rewritten as aV oAP + Vp = 0 at (5.10) (5.11) (5.12) (5.2) (5.3) (5.4) (5.5) (5.6) (5.7) (5.8) (5.9) while keeping in mind that these are nondimensional variables. 5.5 ReducedOrder Linear Dynamics For the geometry depicted in Figure 51, define the flow domain as i. The boundary of the fluid flow domain is given by On = Fr U F U FLU FR (5.13) It is assumed that the input velocity imparted by the moving wall can be repre sented in term of Nc functions gi, i = 1...nc. Each of the functions gi are defined on the entire domain i, but are assumed to exhibit specific properties on the boundary. It is required that 1 for x E FB i a\(x) = fr(5.14) 0 forxEOn\FB Additionally, it is required that Sag fidS= 0 (5.15) for i = 1...Nc. This last condition is required to guarantee compatibility of the flow field with the continuity equation. The dynamics for incompressible, twodimensional creeping flows can be repre sented in the strong form of Stokes equation by rewriting the Equation 5.12.  AV + Vp= f (5.16) Likewise, the strong form of the continuity equation for incompressible flow is VV=0 (5.17) In these equations, V is the flow velocity, p is the pressure and f is the body force. It is assumed that spatiallyvarying functions ); for i = 1...Ns have been determined from the POD procedure which will be discussed in detail later. The functions )i constitute the reduced basis used to represent the Ns states in the control model, which include phase differential information. There is a different set of )i functions associated with each parameter within the phase differential subspace 8=[150,210], which is used to create reducedorder models at specific phase differentails. Similarly, we assume that spatiallyvarying functions gi, for i = 1...Nc, have likewise been derived from a simulation or experiment. The functions gi comprise the influence functions that determine the controls acting on the fluid flow. It is assumed that these functions satisfy the following conditions which are conventional in many reduced basis formulations : V i = 0 for i= 1...Ns (5.18) Vgi = 0 for i = 1... Nc (5.19) ji an = 0 for i= 1...N, (5.20) i = 0 for x V FB (5.21) In terms of these reduced basis functions, the velocity appearing in the Stokes equa tions is assumed to take the form Nc Ns (x,y, t) = (t) + gi(x,y) fi(t) + s Oi(x,y)ci(t) (5.22) i=1 i=1 To derive a reducedorder model appropriate for control synthesis, it is necessary to convert the strong form of the governing equations to weak form. The inner product, bilinear form a(,) are defined, respectively, as A2 2 (#,9) [L2(c)] 2 vd (5.23) By substituting the velocity into the strong form of the governing equations, taking the inner product of the resulting expression with an arbitrary basis function Oi and integrating over the domain ni, we obtain i1 i=1 [L2()]2 Nc Nc (5.25) +a M + L i(x, Y) i(t) + i(x, y)ai(t), j i=1 i= [L2(n)]2 The terms are gathered and put the equations in a canonical form appropriate for control synthesis. Nc Ns __+ j) ++j) 2 __+ i i' [L2( )]2 i(t I [L2 )] Nc Ns +a(yM,) + a(fi j)(t)(+ a j)i(t) (5.26) i=1 i=1 = [L2 ()]2 A new set of states that include both the original reduced state variables and the amplitude of the controls is defined as X(t)= P( (5.27) The new set of controls is defined to be the time derivative of the original controls u(t) = {(t) (5.28) With these definitions of the state X(t) and controls u(t), it is possible to write the weak form of the governing equations as [AR] X(t) = [A] X(t) + [P] u(t) + [((t)] (5.29) It is important to note that the matrix [A] will be diagonal if the reduced basis vectors are derived from a proper orthogonal decomposition. The final form of the reduced state equations can be obtained by premultiplying by the inverse of the matrix [M] X(t) = [] [A] X(t) + [M]1 (X(t)) + [i 1 [B] u(t) (5.30) which can be rewritten in the final form as X(t) = [A(0)]X(t) + [B]u(t) + [E(0, t)] (5.31) where [A(0)] is a linear function of 0, [B] is a constant matrix, and [E(0, t)] is a function of ho(t). 5.6 Creeping Flow in a Driven Cavity Since the dynamical model is linear, we can construct VM(t) as VM(t) = Vo. ho(t) (5.32) where Vo is the velocity field of creeping flow when constant horizontal velocity is imposed on the top of the cavity. It is required that 1 forxE r Vo la(x) = fr (5.33) 0 forxEn\FrT Also, we choose Nc = 1 and g as the velocity field that corresponds to a constant horizontal velocity imposed on the bottom wall of the cavity. It is required that I 1 forx EF Vo la(x) = fr B (5.34) 0 forxEOn\FB The velocity can subsequently be decomposed as Ns P(x,y,t) = Voho(t) +(x,y) (t)+ i(x,y)ai(t) (5.35) i=1 Thus, the reducedorder model in Equation 5.31 can be rewritten explicitly as { } [Ai(e)] [A2( )] { [E1(0)] [E2] ho 0 0 ho + [BI] 3 (5.36) 1 Consider just the partition associated with the timedomain equation for a 6 = Ai(O)a(t)+A2(O) 1(t)+BI3(t) + Ei(O)ho(t)+ E2ho(t) (5.37) Formulate the equivalent frequencydomain expression sa(s) = Ai(O)a(s)+A2(0)3(S)+SB (s) + El ()h(s) + sE2h(s) (5.38) A transfer function representation of the inputs to outputs can now be solved. sE2 +E (0) sB1 +A2() a(s) (0) ho(s) + P(s) sAI(6) sA1() Ai(o) Ei(O)+Ai(O)E2 h() I E2 + A A2+AIB () I B1 Ai(9) Ei(9) A2(0) h(s) I E2 B1 P (s) = P(s) h(s) (5.39) P(s) J The reducedorder model described by P is a fairly accurate representation of the openloop dynamics for Stokes flow in the driven cavity with 0 ~ 0(1). All of the models will be based on a 0 = 1 in this project. There is a decrease in accuracy as 0 changes from unity because of unmodeled nonlinearities. The linear parameter varying controller is designed and tested for models over the range of phase differentials =[150,210]. 5.7 Excitation Phase Differential Openloop models of the flow dynamics are generated by analyzing simulated responses of the driven cavity. This simulation uses the Stokes flow as the fluid dynamics. Separate models are generated for each set of flow conditions that corre sponded to different relationships between the upper and lower boundaries. Each of these models have physical limitations associated with them. The Reynolds number for each model is Re=0.1. The combination of the low Reynolds number and the approximation of the NavierStokes equation creates a flow which is dominated by viscous effects. The fluid at the upper and lower boundaries was constrained to move at a sinusoidal frequency with constant amplitude. The frequency of this sinusoidal is essentially considered a nondimensional unity because the time scales in the dynamics are all normalized. This excitation acts as a disturbance that affects the entire flow within the cavity. The variations between the flow conditions used for model development were phase relationships between the upper and lower boundaries. Specifically, the sinu soidal flows differed in phase by 210, 1950, 180, 1650, and 150 between the upper and lower boundaries. These phase differences induced different modal structures within the flow so the various models were generated to represent a basis for this range of flow conditions. The reducedorder models associated with each phase differential, which contain three states, can actually be considered as subspaces of the fullorder flow, which contains 623 states. The exogenous disturbances will, in general, not have a constant phase differential so the flow field will contain modes associated with many phase dif ferentials throughout any time evolution. As such, each reducedorder model represents a subspace of the modes observed in that fullorder flow field. Thus, synthesizing con trollers for these models with phase differentials is essentially designing compensators that are optimal with respect to each subspace. 5.8 Control Design 5.8.1 Control Objectives The objective of flow control in this project is to reject the effects of the ex ogenous disturbance at the top of the driven cavity. Physically, the control seeks to minimize the horizontal component of the fluid velocity at a set of sensor locations. These sensors are located at 19 sites evenly distributed along the horizontal centerline of the cavity. The end sensors are located 1 grid point away from the boundary wall. The controller is designed for disturbance rejection using the model shown in Fig ure 52. This model contains the openloop dynamics as described by {Vm,, ,,P}. The weighting functions used for loop shaping are given as {Wp, W,, Wk, Wy, Wh, Wu, Wd}. The mathematical objective of the control design is to choose a K such that the closedloop transfer function from disturbances to errors has an induced norm less than unity for all plants within the parameter space. n W 1 y h el P(8) + WWp e Aa Wa e2 U Figure 52: Controller Block Diagram The system has 3 input vectors and 3 output vectors. The inputs are the random noise n E R19 affecting the sensor measurements, the exogenous disturbance h E R affecting the upper boundary of the cavity, and the control input u E R affecting the lower boundary of the cavity. The outputs are the (frequency domain) weighted horizontal velocity measurements el E R19, the (frequency domain) weighted control effort e2 E R, and the sensor measurements y E R19 used for feedback to the controller. The openloop dynamics consist of the cavity's velocity at the top, a control basis and flow shape basis. The part of the openloop dynamics known as Vm is the mean velocity along the top of the cavity and has the equation Vm = h(t)Vo where ho is the initial top disturbance and Vo is a static Stokes flow along the top. The control basis for the plant takes the form YP(t), which was depicted as g(x,y)3(t) in the equations presented in the previous sections, and coincides with the controlled velocity along the bottom of the cavity. The shape of the fluid flow within the cavity is determined by the flow shape basis D. The flow shape can take several forms; for example, the flow could be one large vortex circling the entire cavity or two vortices of equal size with opposite rotation that meet along the horizontal center line of the cavity. The filter given as Wp serves to normalize the measurement of flow velocity collected by the sensors. This filter achieves loopshaping that defines the design specifications in the frequency domain. The filter Wp is chosen to reflect the inverse of acceptable velocities in response to unity disturbances. The actual filter is realized as Wp = 100 which implies the flow velocities should be less than 0.01 in the closedloop system. The filter Wk is used to normalize the penalty placed on the amount of actuation commanded by the controller. This filter reflects the capabilities of the actuation system. The magnitude of motion for the control actuator is allowed to be 0.2 in the nondimensional system. The weighting is chosen as the inverse so that Wk = 5. The creation of a linear parametervarying controller requires that an output filter be used because a certain part of the observation matrix and feedthrough matrix vary with the different plants within the parameter space. The filter Wy is used as the output filter for this synthesis. The filter was chosen as Wy = sI Also, the input matrix varies among the different plant models within the param eter space. This variation necessitates input filters within the synthesis. The filters shown as Wu and Wh are used as input filters and have the value of 1E5 Multiplicative uncertainty was introduced into the control actuation in order to increase the system's robustness with respect to the unmodeled dynamics and disturbances. The uncertainty is represented in the synthesis as Aa such that  Aa oo< 1. A weighting was used to limit the amount of uncertainty allowed into the control actuation and was chosen to be Wa = 0.2, which allows 20% uncertainty in the input U. Also, the filter Wn is included to account for noise that corrupts measurements by the sensors. The inclusion of noise is needed to provide a minimal level penalty on the sensors. The design did not want to consider a large amount of noise so the filter was chosen as W = 0.001. 5.8.2 Synthesis Robust controllers and a linear parametervarying controller have been designed for the system in Figure 52. The techniques of ,X control are used to reduce the induced norm from exogenous inputs to weighted errors [63]. The software from the p Analysis and Synthesis Toolbox for Matlab is used for the actual computation of the controller [34]. Separate controllers are synthesized for each of the openloop models based on excitation phase differentials. For each model, the same weightings are used to reflect the desire of achieving the same performance level for each phase differential. The resulting induced norms achieved by the controllers are shown in Table 51. Table 51: Induced Norms of ClosedLoop System Phase Differential HI norm 1500 7.475 1650 8.177 1800 7.647 1950 10.143 2100 10.829 15002100 12.785 The closedloop norms are all greater than unity. Intuitively, these magnitudes imply the controller is not able to achieve the desired performance and robustness objectives. Realistically, it must be kept in mind that there are twenty inputs and twenty outputs creating a large number of transfer functions. This fact suggests that the magnitude of the norms is not unreasonable. The resulting closedloop properties are studied in more detail shortly. It is shown that the large norms are caused by excessive control actuation. Essentially, the controller is not able to achieve the desired disturbance attenuation without exceeding the actuation limits. Fortunately, this violation is at low frequencies and is not expected to have a dramatic impact on the closedloop simulations. Also, the values in Table 51 are interesting in the sense that the induced norms increase as the level of phase differential increases. Such behavior indicates that the excitation phase differential does indeed have a large impact on the fluid dynamics. The increasingly poor performance of the controllers demonstrates that the flow modes for a phase differential of 210 have properties that are more difficult to control than those for a phase differential of 150, for example. The last entry in Table 51 is the norm associated with the LPV controller. Allowing the phase differential to be timevarying increases the norm as expected. What is important to note is that this norm did not raise much above the norm associated with the Ho_ controller for the 210 phase differential model. This condition indicates that the LPV controller is able account for the timevarying nature of the phase differential without excessive loss of performance. 5.9 Simulation 5.9.1 OpenLoop Simulation A series of openloop simulations are performed to demonstrate the fluid qual itative response resulting from the disturbance for both fullorder and reducedorder models. These simulations are similar in the sense that the same magnitude of dis turbance is used for the boundary conditions on the top of the domain. Conversely, the simulations involving the reducedorder models differ in that the flow on the bot tom boundary has different values of phase lag with respect to the flow on the top boundary. A series of plots will be shown to visualize the flow conditions. In each, the value of horizontal velocity will be shown as a function of time. The plots are 3dimensional because the velocity measured at each of the 19 sensors is shown as a function of time. Again, it is important to note that all measurements are nondimensional. This characteristic applies to both the time and velocity component so no units are noted for the simulations. The openloop flow for the fullorder model is used as a comparison for the reducedorder model simulations (Figure 53). This plot clearly shows the sinusoidal nature of the flow that results from the top exogenous disturbance changing with the sine function, ho = sin(27t). The flow near the center of the cavity, near point 11, shows the largest velocity with a magnitude near 0.2 at t = 0.3 to +0.2 at t = 0.7. 0.2 20 15 .1 10 0.5 5 Point Index 0 Time Figure 53: OpenLoop Flow Velocities for FullOrder Model The flow for the reducedorder model with a phase differential of 1650 is shown in Figure 54. This plot also demonstrates a sinusoidal nature, but has a smaller openloop magnitude compared to the fullorder flow with the highest velocity being 0.07. The flow for the reducedorder model with an phase differential of 2100 is shown in Figure 55. The flow again demonstrates a sinusoidal nature and the velocities are slightly larger than those of the fullorder model. A sinusoidal trajectory of phase differentials shown in Figure 56 is used in a simulation which shows the openloop characteristics of the reducedorder flow as phase differential changes. 0.21 0.24 20 15 \ 1 10 0.5 5 Point Index 0 Time Figure 54: OpenLoop Flow Velocities for ReducedOrder Model with 1650 Phase Differential 0.2 J 0 0.22 20 15 1 10 0.5 5 Point Index 0 Time Figure 55: OpenLoop Flow Velocities for ReducedOrder Model with 210 Phase Differential 210 200 '2190 180 1 70 160 Time Figure 56: Trajectory of Phase Differential Ilb The flow velocities for the reducedorder model throughout the timevarying phase trajectory are shown in Figure 57. The sinusoidal nature that is apparent in all of the other openloop flows is slightly different for this openloop flow. This difference is due to the changing of the parameter through the trajectory. The fullorder flow does not have a dependence on phase differential, therefore, the velocities for the fullorder model's flow over the phase differential trajectory are the same as those plotted in Figure 53. 0.2 20 15 1 10 0.5 5 Point Index 0 Time Figure 57: OpenLoop Flow Velocities for ReducedOrder Model over a Trajectory of Phase Differentials An interesting feature to note is that the flow for each reducedorder model with phase differential has a similar shape but significantly different magnitude. The maximum velocity measured at the center of the cavity is smaller in magnitude for the models with phase differentials located at the beginning of the range than the models with phase differentials near the end of the range. This feature indicates the flow is indeed strongly dependent on phase differential and should be considered for control design. 5.9.2 ReducedOrder ClosedLoop Simulation The closedloop dynamics are also simulated to demonstrate the performance of the controller for the reducedorder models, in this section, and the fullorder model, in the next section. The diagram of the closedloop system for both the reducedorder models and the fullorder model can be seen in Figure 58. These simulations use the same openloop dynamics but include the linear parametervarying controller that was synthesized over the range of phase differentials, which contains 24 states. In each simulation, the flow on the upper boundary is the same, but now the flow on the lower boundary results only from the commands issued by the controller. In this section, the controller was tested with reducedorder models for two specific cases of phase differential and over a timevarying trajectory of phase differentials. h Figure 58: Closedloop System The measured velocities for the reducedorder model with a phase differential of 1650 in response to the LPV controller with a phase differential of 1650 is shown in Figure 59. The comparison of these velocities with the openloop measurements in Figure 54 demonstrate a reduction of velocity along the center of the cavity, where the velocity is greatest, of roughly 70%. The measured velocities for the reducedorder model with a phase differential of 2100 in response to the LPV controller with a phase differential of 2100 is shown in Figure 510. The reduction in velocities is apparent by comparing the closedloop velocities in Figure 510 with the openloop velocities in Figure 55, which shows a reduction along the center of the cavity of roughly 90%. The closedloop simulation of the reducedorder models over the phase differential trajectory, whcih also effects the controller, is shown in Figure 511. The velocity magnitude shows a clear reduction in magnitude compared to the openloop simulation 0.2 0.2 L 20 15 \ 1 10 0.5 5 Point Index 0 Time Figure 59: ClosedLoop Flow Velocities for ReducedOrder Model with 1650 Phase Differential 0.2 "0  0.2< 20 15 5 1 10 0.5 5 Point Index 0 Time Figure 510: ClosedLoop Flow Velocities for ReducedOrder Model with 2100 Phase Differential of flow over the phase differential trajectory, which was shown in Figure 57. The reduction along the center is roughly 80%. The disturbance rejection is significant for the LPV controller with the reduced order models. These reductions confirm that the LPV controller will work not only for reducedorder models at specific phase differentials but also over a timevarying trajectory of phase differentials. The simulations did show some differences between each of the reducedorder models. In particular, the amount of attenuation was slightly less for the reducedorder model with a phase differential of 1650 but much higher for the reducedorder model with a phase differential of 2100. This decrease in attenuation 0.2 A 0.2 20 15 \ 1 10 0.5 5 Point Index 0 Time Figure 511: ClosedLoop Flow Velocities for ReducedOrder Model over a Trajectory of Phase Differentials seems almost contradictory considering that the openloop simulations showed a decrease in flow velocities for the same models. 5.9.3 FullOrder ClosedLoop Simulation The simulations that were performed for the reducedorder models were repeated using the fullorder model. The reducedorder models are subspaces of this fullorder model so the performance of the controllers on the fullorder model is actually of predominant interest. The measured velocities in response to an HoO controller, created specifically for the fullorder model, are shown in Figure 512. Clearly, the magnitude of the velocity is dramatically decreased below the openloop level. The velocities in Figure 512 are several orders of magnitude less than the corresponding openloop velocities in Figure 53. This response will be used as a comparison for the responses from the fullorder model controlled by the LPV controller. The velocities for the fullorder model in response to the LPV controller with a phase differential of 1650 is shown in Figure 513. Though the velocities were not reduced to the extent of the fullorder simulation in Figure 512, they were reduced by an amount comparable to the response shown by the reducedorder model at a 1650 0.2 0 0.2< 20 15 \ 1 10 0.5 5 Point Index 0 Time Figure 512: ClosedLoop Flow Velocities for FullOrder Model phase differential. The reduction in the velocity magnitude is evident along the center of the cavity and is roughly 80%. 0.2 0.2 20 15 \ 1 10 0.5 5 Point Index 0 Time Figure 513: ClosedLoop Flow Velocities for FullOrder Model with Controller Asso ciated with 1650 Phase Differential The velocity magnitudes for the fullorder model in response to the LPV controller for a phase differential of 2100 is shown in Figure 514. The reduction in velocity compared to the openloop flow of the fullorder flow in Figure 53 is very clear. The velocities along the centerline of the cavity were reduced by 60%. Though the velocities were not as reduced as much as those in the simulation in Figure 512, the velocities were reduced by an amount comparable to the reducedorder model at a 2100 phase differential. 10 Point Index 0.5 0 Time Figure 514: ClosedLoop Flow Velocities for FullOrder Model with Controller Asso ciated with 210 Phase Differential The closedloop simulation of the fullorder model controlled by the LPV controller over the phase differential trajectory is shown in Figure 515. The velocities show a clear reduction in magnitude compared to the openloop fullorder flow in Figure 53. The flow along the center of the cavity is reduced by 66%. 0.2 IU 0.5 5 Point Index 0 Time Figure 515: ClosedLoop Flow Velocities for FullOrder Model over a Trajectory of Phase Differentials The disturbance rejection is significant for both the reducedorder models and the fullorder model. These reductions confirm that the LPV controller, created for a phase differential parameter, will work not only for the reducedorder models, which are dependent on phase differential, but also for the fullorder model. AW 5.10 Conclusion Flow control is an exceedingly difficult challenge because of the nonlinearities and time variations inherent to flow fields. These inherent difficulties can be avoided when restricting the flow to creeping Stokes flow within a driven cavity. This project has introduced a control methodology suitable for such a system. In particular, the controllers are designed by considering subspaces of the flow field that describe modes associated with phase differential between exogenous disturbances. The models of these subspaces are realized as statespace systems and a controller can be designed using the linear parametervarying framework. The resulting controller is shown to significantly decrease the flow velocities within the cavity for both the reducedorder subspaces and also the fullorder flow. CHAPTER 6 CONCLUSION Practically all mechanical systems that involve motion need to be controlled with a gainscheduling technique. Aerospace systems in particular have the possibility to have very extensive operating domains. Three specific aerospace systems were discussed in this paper, the longitudinal dynamics of an F/A18, the structural dynamics of a hypersonic vehicle and the flow dynamics of a driven cavity. The parameters that depicted the operating domain of the F/A18 prob lem were altitude and Mach number. The parameter that depicted the operating domain of the structure of the hypersonic vehicle was temperature and the operating domain of the driven cavity was depicted by the phase differential within the fluid. 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Zhou, "The Boundary Element Method For Boundary Control of The Linear Stokes Flow," IEEE Conference on Decision and Control, Vol. 3, 1990, pp. 11921194. [62] C.F.M. Coimbra and R.H. Rangel, "Spherical Particle Motion in Unsteady Viscous Flows," AIAA Aerospace Science Meeting and Exhibit, 1999, AIAA991031. [63] K. Zhou, J.C. Doyle and K. Glover, "Robust and Optimal Control," Prentice Hall, 1st Edition, 1997. BIOGRAPHICAL SKETCH Kristin Fitzpatrick was born in Blue Hill, Maine on March 26, 1980. Her family moved to Florida in 1988 after the death of her mother. She received her high school diploma from the Center for Advanced Technologies, a magnet program in St. Petersburg, Florida. She then attended the University of Florida and received a degree in Aerospace Engineering with Honors in December 2002. She has worked with the aerospace dynamics and control research group under the direction of Dr. Rick Lind and Dr. Andy Kurdila and is projected to receive her Master of Science degree in aerospace engineering in December 2003. She will stay at the University of Florida to pursue a doctorate in aerospace engineering with the focus in dynamics and control. 