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## Material Information- Title:
- Applications of Linear Parameter-Varying Control for Aerospace Systems
- Creator:
- FITZPATRICK, KRISTIN LEE (
*Author, Primary*) - Copyright Date:
- 2003
## Subjects- Subjects / Keywords:
- Aircraft ( jstor )
Aircraft pitch ( jstor ) Altitude ( jstor ) Hypersonic aircraft ( jstor ) Hypersonics ( jstor ) Mach number ( jstor ) Modeling ( jstor ) Parametric models ( jstor ) Simulations ( jstor ) Velocity ( jstor )
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- University of Florida
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- Copyright Kristin Lee Fitzpatrick. Permission granted to University of Florida to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 12/20/2003
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- 54506274 ( OCLC )
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APPLICATIONS OF LINEAR PARAMETER-VARYING 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 PARAMETER-VARYING CONTROL THEORY .......... 5 3 LINEAR PARAMETER-VARYING CONTROL FOR AN F/A-18 ....... 10 3.1 Problem Statement ................... ....... 10 3.2 Open-loop Dynamics .......... ......... ....... 12 3.3 Control Objectives ................... ...... 14 3.4 Synthesis ...... .......... ........... 15 3.5 Simulation .......... ......... ........... 17 3.6 Conclusion .......... ........... .......... 20 4 LINEAR PARAMETER-VARYING 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 Parameter-Varying System ........ ............ 27 4.5 Control Design .......... ....... ............. 30 4.6 Simulation ................... ............ 33 4.6.1 Open-Loop Simulation ................... 33 4.6.2 Closed-Loop Simulation ......... ........ ... 35 4.7 Conclusion .................. ............ 38 5 LINEAR PARAMETER-VARYING 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 Reduced-Order 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 Open-Loop Simulation ............... .. .. 54 5.9.2 Reduced-Order Closed-Loop Simulation . . .... 57 5.9.3 Full-Order Closed-Loop Simulation . . . 60 5.10 Conclusion. .................. . . ......63 6 CONCLUSION ................... . . .. 64 REFERENCES ............. ........ . . 65 BIOGRAPHICAL SKETCH .............. . . ..... 70 LIST OF TABLES Table page 3-1 Original Design Points ............. . . ...... 12 3-2 Frequency and Damping Ratio of Design and Analysis Models ...... 14 3-3 Frequencies and Damping Ratios of the Target Model . . ... 16 3-4 Induced Norms of Closed-Loop System .................. ..17 3-5 Numerical Results ............... . . .... 19 4-1 Model Dimensions and Flight Conditions ................. ..27 4-2 Modes of the Hypersonic Model .................. ...... 30 4-3 Modes of the Target Model .................. ....... .. 32 4-4 Open-Loop Synthesis Norms .................. ...... 33 4-5 Point Design Norms .............. . . .... 33 5-1 Induced Norms of Closed-Loop System .................. ..53 LIST OF FIGURES Figure page 2-1 Jo_ Block Diagram (Gain-Scheduled) ......... ........ .... 7 3-1 F/A-18 ............ .... ....................... 11 3-2 Flight Envelope/Parameter Space .................. .... .. 12 3-3 Synthesis Block Diagram .................. ........ .. 15 3-4 Closed-loop Block Diagram .................. ..... 18 3-5 Pitch Rate for Design Points .................. ..... .. 18 3-6 Pitch Rate for Analysis Point .................. ...... .. 19 3-7 Controller Elevator Deflection .................. ..... .. 19 4-1 Inner-Loop/Outer-Loop Design .................. ..... .. 25 4-2 Simplified Model of a Generic Hypersonic Vehicle . . .... 25 4-3 Synthesis Block Diagram .................. ........ .. 31 4-4 Open-Loop Transfer Functions ................ . ..34 4-5 Open-Loop Angle of Attack Result .............. .... .. 35 4-6 Input Elevon Deflection ............ . . ...... 35 4-7 Closed-loop Design ............ . . ... 36 4-8 Closed-Loop Transfer Functions ................ .... .. 36 4-9 Closed-Loop Angle of Attack Result .... . .... 37 4-10 Elevon Deflection Command ............... ..... .. 38 4-11 Elevon Deflection Rate .......... . . ... 38 5-1 Stokes Driven Cavity Flow Problem .... . .... 41 5-2 Controller Block Diagram .................. .. ...... 51 5-3 Open-Loop Flow Velocities for Full-Order Model . . .... 55 5-4 Open-Loop Flow Velocities for Reduced-Order Model with 1650 Phase Differential .................. . . ...... 56 5-5 Open-Loop Flow Velocities for Reduced-Order Model with 2100 Phase Differential .................. . . ...... 56 5-6 Trajectory of Phase Differential ................ .... .. 56 5-7 Open-Loop Flow Velocities for Reduced-Order Model over a Trajectory of Phase Differentials ............ . . .... 57 5-8 Closed-loop System .............. . . .... 58 5-9 Closed-Loop Flow Velocities for Reduced-Order Model with 1650 Phase Differential .................. . . ...... 59 5-10 Closed-Loop Flow Velocities for Reduced-Order Model with 2100 Phase Differential .................. . . ...... 59 5-11 Closed-Loop Flow Velocities for Reduced-Order Model over a Trajec- tory of Phase Differentials .................. ...... .. 60 5-12 Closed-Loop Flow Velocities for Full-Order Model . . .... 61 5-13 Closed-Loop Flow Velocities for Full-Order Model with Controller As- sociated with 1650 Phase Differential . . . .... 61 5-14 Closed-Loop Flow Velocities for Full-Order Model with Controller As- sociated with 2100 Phase Differential . . . .... 62 5-15 Closed-Loop Flow Velocities for Full-Order 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 PARAMETER-VARYING CONTROL FOR AEROSPACE SYSTEMS By Kristin Lee Fitzpatrick December 2003 Chair: Richard C. Lind, Jr. Major Department: Mechanical and Aerospace Engineering Gain-scheduling 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 gain-scheduling techniques have been researched for the control of systems that vary with time-varying 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 time-varying system. The gain-scheduling technique known as linear parameter-varying 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 closed-loop 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/A-18, 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 parameter-varying controller created for each system to show the usefulness of a linear parameter-varying framework as a gain-schedule 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). Gain-scheduling 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 gain-scheduling 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 closed-loop 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 gain-scheduled controller that can be treated as a single entity. This technique achieves gain-scheduling with a parameter-dependent 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 gain-scheduled 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/A-18 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 Gain-scheduling has evolved hand in hand with the progress of mechanical systems. Gain-scheduling techniques are currently used for control design for both linear and nonlinear systems. This study focused on systems pertaining to aerospace applications. Gain-scheduling in aerospace applications came about during WWII as autopi- loting became necessary with the birth of jet aircraft and guided missiles [1]. When gain-scheduling was first conceived for the military, it was created through hardware and was quite costly. Gain-scheduling was not adopted commercially until the cre- ation of digital control, nearly a quarter-century after military use. The development of gain-scheduling over past decades led to several design techniques and the use of gain-scheduling for many different aerospace systems. Several gain-scheduling 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 time-invariant (LTI), linear time-varying (LTV), and linear parameter- varying (LPV). Gain-scheduling is most often applied to linear parameter-varying systems, which are affine functions of parameters that affect their operation. Gain-scheduling can also be applied to the control of nonlinear systems. Several linearization techniques can be used for nonlinear systems before a gain-scheduled 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 gain-scheduling controllers have also been applied to nonlinear systems [4, 5, 6]; however, nonlinear systems do not necessarily have to be linearized. Set-valued methods for LPV systems have also been applied to nonlinear systems with quasi-LPV representations [7]. Linearization errors were accommodated as linear state-dependent disturbances; constraints on systems' states and controls were specified; rates of transitions among operating regions were addressed, which allows even the local-point designs to be nonlinear. The classical gain-scheduling 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 D-K 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 gain-scheduling law is developed. A study also used a design algorithm for a state feedback law based on gain-scheduling for an LPV multi-input multi-output system [11]. The state feedback control law places the system's poles in a neighborhood of desired locations and stabilizes the closed-loop system. Though this classical approach has worked well for many applications, there is no guarantee of robustness or stability of the closed-loop 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 closed-loop robustness and stability with the LPV system). The method of D-K 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 F-16 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 lateral-directional LPV controllers were designed for the F/A-18 [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 gain-scheduling 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 parameter-varying control theory is discussed in more detail in the next section. CHAPTER 2 LINEAR PARAMETER-VARYING CONTROL THEORY Linear parameter-varying controller synthesis is a gain-scheduling 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 parameter-varying system. A linear parameter-varying system depends affinely on a set of norm-bounded time-varying operating parameters. It considers linear systems whose open-loop 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 multi-input multi-output discrete-time linear parameter-varying 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 parameter-varying 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 time-varying 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 parameter-dependent 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 state-space 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 _1-like synthesis problem relative to the interconnection in Figure 2-1. If there exists a continuous differentiable function X(0) d e K(., 0) Figure 2-1: X Block Diagram (Gain-Scheduled) defined on Rn where X(0) > 0, (2.9) the worst-case closed-loop 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) 6eReoldl|2#O,deL2 ld|12 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 D-K 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 X-33 is presented in [28]. The main benefit of using the LPV framework is that it allows gain-scheduled controllers to be treated as a single entity, with the gain-scheduling being achieved with the parameter-dependent controller and automatic interpolation law, which removes the ad-hoc scheduling schemes that were necessary in the past. CHAPTER 3 LINEAR PARAMETER-VARYING CONTROL FOR AN F/A-18 3.1 Problem Statement Several control designs have been applied to the control of F-18 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 gain-scheduling is necessary for controlling the F/A-18 throughout its operating domain. A longitudinal variable structure controller was created for an F-18 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 lateral-directional 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 gain-scheduling the flight envelope used for the project had to be small, M = [0.35,0.55] and altitude = [20,28]kft. Gain-scheduled approximations to H_ controllers for the F/A-18 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/A-18 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/A-18 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 parameter-varying controller for the longitudinal dynamics of an F/A-18 over a chosen flight envelope. The F/A-18, shown in Figure 3-1, 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 3-1: F/A-18 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 two-dimensional parameter space were to be used to design the LPV controller and are listed in Table 3-1. 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 3-1: 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 3-2. 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 3-2: Flight Envelope/Parameter Space 3.2 Open-loop Dynamics The F/A-18 models used for this project are longitudinal short-period 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/A-18 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 F-18 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 S-1.8415 P2= -0.0192 The model for the F-18 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 3-2. All of the damping ratios are greater than zero, which affirms that the models are stable. Table 3-2: 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 parameter-varying 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/A-18 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 parameter-varying 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 3-3 incorporates all the necessary elements needed to create the controller which will accomplish the controller objectives. This system contains the open-loop 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 3-3: 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/A-18 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 3-3. -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 3-3: 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 3-3, robust H_ controllers were designed for the models at each of the design points and a linear parameter-varying 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 3-4. Table 3-4: Induced Norms of Closed-Loop System Open-Loop Model _12-norm P1 0.891 P2 0.775 P3 0.775 P1 P3 0.971 It is important to note that all of the closed-loop 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 3-4 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 time-varying nature of Mach number and altitude without excessive loss of performance. 3.5 Simulation The closed-loop 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 closed-loop system for the models can be seen in Figure 3-4. The simulations use the same open-loop dynamics but include the linear parameter-varying 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 3-5. The point design K(O) '-- q, Figure 3-4: Closed-loop 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 3-5: 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 3-6. 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 3-5. 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 3-6: Table Rise Time Settling Time Peak Overshoot 4 6 8 10 Time (s) Pitch Rate for Analysis Point 3-5: 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 3-7 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 3-7: 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/A-18 aircraft with a linear parameter-varying controller. This type of controller was chosen because the change in dynamics of the aircraft could be modeled with a system whose state-space 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/A-18 model presented in this project. CHAPTER 4 LINEAR PARAMETER-VARYING 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 parameter-varying 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 linear-parameter varying controller for the hypersonic vehicle ignored the mode shape of the vehicle and separated the rigid-body 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 pre-specified 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 Multi-Model Eigenstructure, which designs a robust fixed-gain 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 gain-scheduling 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 gain-scheduling 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 inner-loop controller and an outer-loop controller. The inner-loop 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 rigid-body and structural dynamics will not be separated. The outer-loop controller of the aircraft will be a rigid-body controller which will work as a traditional flight controller for rigid aircraft and will be designed in a future project. The inner-loop structural damping controller is the focus of this project. A diagram of the inner-loop/outer-loop control design is shown if Figure 4-1. The K(O) controller is the linear parameter-varying inner-loop controller and the Kut controller is the rigid-body outer-loop controller. The P is the hypersonic plant model. input--- output Figure 4-1: Inner-Loop/Outer-Loop 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 X-30 vehicles. A generalized shape can be seen in Figure 4-2. Figure 4-2: 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 4-1. Table 4-1: 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 Parameter-Varying System The time-varying 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 X-30 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 1.5026E-4 -3.2374E-3 -5.2818E1 -3.2200E1 2.3762E-2 5.7314E-1 7.5583E-3 1.1744E-7 -3.1848E-7 -3.3921E-2 0 0 -5.7586E-6 0 -7.4858E-1 0 9.6079E-6 0 1.0158E-1 0 1.5833E0 0 2.4280E3 1 1.4681E-4 2.8801E-6 0 1 0 0 -5.1609E-2 9.2411E-2 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 -1.596E-2 0 8.111E- 1 0 -2.925E1 0 1.261E3 -2.253E 2 0 5.190E0 0 2.209E2 C(O) ==[ ] Co ] 0 -1.8285E-4 1 -9.4975E-1 (4.2) (4.3) (4.4) (4.5) 0 0 1 0 0 1.7453E-2 0 0 0 0 0 1 0 1.7453E-2 0 0 0 0 1 0 -1.7453E-2 C= -4.6971E-5 2.0641E-4 6.2428E0 0 -1.0921E-2 1.0896E-1 0 -5.3709E-6 1.0095E-3 0 0 0 0 0 -3.5754E-1 6.0213E-1 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 parameter-varying 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 4-2 The table shows the frequency of each of the modes and the damping corresponding to the frequency. The four modes of the open-loop dynamics are (i) a height mode, (ii) an unstable phugoid-like 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 inner-loop LPV controller. Table 4-2: 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 parameter-varying 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 4-3. 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 4-3: 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 open-loop 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 4-3. 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 4-3: 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 rigid-body 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 rigid-body dynamics. Stabilizing the rigid-body 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 open-loop 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 4-4. 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 4-4: Open-Loop 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 4-5. Table 4-5: 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 time-varying 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 Open-Loop Simulation The frequency response of the open-loop transfer function between the elevon defection and the angle of attack for the target model, cold model and hot model is shown in Figure 4-4. 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 4-4: Open-Loop Transfer Functions The simulation of an angle of attack response to an eleven deflection input for the open-loop hypersonic models at both the hot and cold temperatures and the target model is shown in Figure 4-5. The eleven deflection used for the following simulation of is shown in Figure 4-6. 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 open-loop 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 closed-loop simulation. 3 target -40 -2 0 5 10 15 Time (s) Figure 4-5: Open-Loop Angle of Attack Result Input Elevon Deflection 5 C4 3- 52 0 0 5 10 15 Time(s) Figure 4-6: Input Elevon Deflection 4.6.2 Closed-Loop Simulation The closed-loop dynamics are simulated to demonstrate the performance of the controller for the hypersonic models at both the hot and cold temperature. The closed-loop system for both models can be seen in Figure 4-7. The system shown in Figure 4-7 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 4-7: Closed-loop Design vehicle for the open-loop 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 closed-loop system for both the cold and hot temperature hypersonic models is shown in Figure 4-8. 102 target cold 101 -- hot 100 10 10 104 10 10 102 Frequency (rad/sec) Figure 4-8: Closed-Loop 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 closed-loop simulation of the angle of attack response to the same elevon deflection used in the open-loop simulation is shown in Figure 4-9. 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 4-9: Closed-Loop Angle of Attack Result As can be seen, the oscillations that were apparent in the open-loop 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 4-10 is plotted for the closed-loop 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 4-11 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 4-11: Elevon Deflection Rate 4.7 Conclusion This project considered the control of the structural dynamics of a hypersonic vehicle with a linear parameter-varying controller. This type of controller was chosen because the change in the dynamics of the hypersonic vehicle could be modeled with a system whose state-space 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 PARAMETER-VARYING 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 open-loop 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 open-loop 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 parameter-varying 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 open-loop simulations show a rise in the horizontal velocity along the centerline as the phase differential is increased. The closed-loop 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 Navier-Stokes 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 reduced-order 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 Navier-Stokes 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 5-1, 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 no-slip 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 5-1: 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 2-D 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 Navier-Stokes 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. Non-dimensional variables are defined as x* = x/L = y/L p* = p/pLV, t* =tf The resulting non-dimensional Navier-Stokes equation can be written as Re- St- -+Re V** = -Re-StV*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 Reduced-Order Linear Dynamics For the geometry depicted in Figure 5-1, 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, two-dimensional 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 V-V=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 spatially-varying 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 reduced-order models at specific phase differentails. Similarly, we assume that spatially-varying 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) V-gi = 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 reduced-order 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 reduced-order 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 time-domain 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 frequency-domain 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) s-AI(6) s-A-1() 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 reduced-order model described by P is a fairly accurate representation of the open-loop 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 Open-loop 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 Navier-Stokes 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 non-dimensional 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 reduced-order models associated with each phase differential, which contain three states, can actually be considered as subspaces of the full-order 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 reduced-order model represents a subspace of the modes observed in that full-order 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 5-2. This model contains the open-loop 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 closed-loop 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 5-2: 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 open-loop dynamics consist of the cavity's velocity at the top, a control basis and flow shape basis. The part of the open-loop 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 loop-shaping 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 closed-loop 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 non-dimensional system. The weighting is chosen as the inverse so that Wk = 5. The creation of a linear parameter-varying 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 = s-I 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 parameter-varying controller have been designed for the system in Figure 5-2. 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 open-loop 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 5-1. Table 5-1: Induced Norms of Closed-Loop System Phase Differential HI norm 1500 7.475 1650 8.177 1800 7.647 1950 10.143 2100 10.829 1500-2100 12.785 The closed-loop 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 closed-loop 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 closed-loop simulations. Also, the values in Table 5-1 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 5-1 is the norm associated with the LPV controller. Allowing the phase differential to be time-varying 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 time-varying nature of the phase differential without excessive loss of performance. 5.9 Simulation 5.9.1 Open-Loop Simulation A series of open-loop simulations are performed to demonstrate the fluid qual- itative response resulting from the disturbance for both full-order and reduced-order 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 reduced-order 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 3-dimensional 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 non-dimensional. This characteristic applies to both the time and velocity component so no units are noted for the simulations. The open-loop flow for the full-order model is used as a comparison for the reduced-order model simulations (Figure 5-3). 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 5-3: Open-Loop Flow Velocities for Full-Order Model The flow for the reduced-order model with a phase differential of 1650 is shown in Figure 5-4. This plot also demonstrates a sinusoidal nature, but has a smaller open-loop magnitude compared to the full-order flow with the highest velocity being 0.07. The flow for the reduced-order model with an phase differential of 2100 is shown in Figure 5-5. The flow again demonstrates a sinusoidal nature and the velocities are slightly larger than those of the full-order model. A sinusoidal trajectory of phase differentials shown in Figure 5-6 is used in a simulation which shows the open-loop characteristics of the reduced-order flow as phase differential changes. 0.21 -0.24 20 15 \ 1 10 0.5 5 Point Index 0 Time Figure 5-4: Open-Loop Flow Velocities for Reduced-Order Model with 1650 Phase Differential 0.2- J 0- -0.22 20 15 1 10 0.5 5 Point Index 0 Time Figure 5-5: Open-Loop Flow Velocities for Reduced-Order Model with 210 Phase Differential 210 200 '2190 180 1 70 160 Time Figure 5-6: Trajectory of Phase Differential Ilb The flow velocities for the reduced-order model throughout the time-varying phase trajectory are shown in Figure 5-7. The sinusoidal nature that is apparent in all of the other open-loop flows is slightly different for this open-loop flow. This difference is due to the changing of the parameter through the trajectory. The full-order flow does not have a dependence on phase differential, therefore, the velocities for the full-order model's flow over the phase differential trajectory are the same as those plotted in Figure 5-3. 0.2 20 15 1 10 0.5 5 Point Index 0 Time Figure 5-7: Open-Loop Flow Velocities for Reduced-Order Model over a Trajectory of Phase Differentials An interesting feature to note is that the flow for each reduced-order 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 Reduced-Order Closed-Loop Simulation The closed-loop dynamics are also simulated to demonstrate the performance of the controller for the reduced-order models, in this section, and the full-order model, in the next section. The diagram of the closed-loop system for both the reduced-order models and the full-order model can be seen in Figure 5-8. These simulations use the same open-loop dynamics but include the linear parameter-varying 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 reduced-order models for two specific cases of phase differential and over a time-varying trajectory of phase differentials. h Figure 5-8: Closed-loop System The measured velocities for the reduced-order model with a phase differential of 1650 in response to the LPV controller with a phase differential of 1650 is shown in Figure 5-9. The comparison of these velocities with the open-loop measurements in Figure 5-4 demonstrate a reduction of velocity along the center of the cavity, where the velocity is greatest, of roughly 70%. The measured velocities for the reduced-order model with a phase differential of 2100 in response to the LPV controller with a phase differential of 2100 is shown in Figure 5-10. The reduction in velocities is apparent by comparing the closed-loop velocities in Figure 5-10 with the open-loop velocities in Figure 5-5, which shows a reduction along the center of the cavity of roughly 90%. The closed-loop simulation of the reduced-order models over the phase differential trajectory, whcih also effects the controller, is shown in Figure 5-11. The velocity magnitude shows a clear reduction in magnitude compared to the open-loop simulation 0.2 -0.2 L 20 15 \ 1 10 0.5 5 Point Index 0 Time Figure 5-9: Closed-Loop Flow Velocities for Reduced-Order Model with 1650 Phase Differential 0.2 "0 - -0.2< 20 15 5 1 10 0.5 5 Point Index 0 Time Figure 5-10: Closed-Loop Flow Velocities for Reduced-Order Model with 2100 Phase Differential of flow over the phase differential trajectory, which was shown in Figure 5-7. 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 reduced-order models at specific phase differentials but also over a time-varying trajectory of phase differentials. The simulations did show some differences between each of the reduced-order models. In particular, the amount of attenuation was slightly less for the reduced-order model with a phase differential of 1650 but much higher for the reduced-order 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 5-11: Closed-Loop Flow Velocities for Reduced-Order Model over a Trajectory of Phase Differentials seems almost contradictory considering that the open-loop simulations showed a decrease in flow velocities for the same models. 5.9.3 Full-Order Closed-Loop Simulation The simulations that were performed for the reduced-order models were repeated using the full-order model. The reduced-order models are subspaces of this full-order model so the performance of the controllers on the full-order model is actually of predominant interest. The measured velocities in response to an HoO controller, created specifically for the full-order model, are shown in Figure 5-12. Clearly, the magnitude of the velocity is dramatically decreased below the open-loop level. The velocities in Figure 5-12 are several orders of magnitude less than the corresponding open-loop velocities in Figure 5-3. This response will be used as a comparison for the responses from the full-order model controlled by the LPV controller. The velocities for the full-order model in response to the LPV controller with a phase differential of 1650 is shown in Figure 5-13. Though the velocities were not reduced to the extent of the full-order simulation in Figure 5-12, they were reduced by an amount comparable to the response shown by the reduced-order model at a 1650 0.2 0- -0.2< 20 15 \ 1 10 0.5 5 Point Index 0 Time Figure 5-12: Closed-Loop Flow Velocities for Full-Order 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 5-13: Closed-Loop Flow Velocities for Full-Order Model with Controller Asso- ciated with 1650 Phase Differential The velocity magnitudes for the full-order model in response to the LPV controller for a phase differential of 2100 is shown in Figure 5-14. The reduction in velocity compared to the open-loop flow of the full-order flow in Figure 5-3 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 5-12, the velocities were reduced by an amount comparable to the reduced-order model at a 2100 phase differential. 10 Point Index 0.5 0 Time Figure 5-14: Closed-Loop Flow Velocities for Full-Order Model with Controller Asso- ciated with 210 Phase Differential The closed-loop simulation of the full-order model controlled by the LPV controller over the phase differential trajectory is shown in Figure 5-15. The velocities show a clear reduction in magnitude compared to the open-loop full-order flow in Figure 5-3. The flow along the center of the cavity is reduced by 66%. 0.2- IU 0.5 5 Point Index 0 Time Figure 5-15: Closed-Loop Flow Velocities for Full-Order Model over a Trajectory of Phase Differentials The disturbance rejection is significant for both the reduced-order models and the full-order model. These reductions confirm that the LPV controller, created for a phase differential parameter, will work not only for the reduced-order models, which are dependent on phase differential, but also for the full-order 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 state-space systems and a controller can be designed using the linear parameter-varying framework. The resulting controller is shown to significantly decrease the flow velocities within the cavity for both the reduced-order subspaces and also the full-order flow. CHAPTER 6 CONCLUSION Practically all mechanical systems that involve motion need to be controlled with a gain-scheduling 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/A-18, 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/A-18 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. This paper has introduced a gain-scheduled control methodology, which uses Xo synthesis to create a linear parameter-varying controller, that is suitable for such systems. The LPV controller created for the F/A-18 longitudingal dynamics proved to induce a pitch rate for the aircraft that was similar to a designated target pitch rate. The LPV controller created for the structural dynamics of a hypersonic aircraft successfully damped out the vibrations induced by a temperature change. The LPV controller for the fluid dynamics within a driven cavity significantly decreased the horizontal component of the flow velocities along the centerline of the cavity for both the reduced-order subspaces and the full order flow. The results of the control methodology to create proficient controllers for three very different aerospace applications leads to the conclusion that this methodology could be useful for other aerospace applications. REFERENCES [1] W. Rugh and J. 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Lumley, "The Structure of Inhomogeneous Turbulent Flows," Atmopsheric Turbulence and Radio Wave Propagation, edited by A.M. Yaglom and V.I. Tararsky, Nauka, Moscow, 1967, pp. 166-178. [56] J.A. Burns and O. Yuh-Roung, "Feedback Control of the Driven Cavity Problem Using LQR Designs," IEEE Conference on Decision and Control, Vol. 1, 1994, pp. 289-294. [57] L. Baramov, O. Tutty and E. Rogers, "Robust Control of Linearized Poiseuille Flow," Journal of Guidance, Control and Dynamics, Vol. 25, No. 1, 2002, pp. 145-151. [58] H.T. Banks, K. Ito, "Structural Actuator Control of Fluid/Structure Interactions," IEEE Conference on Decision and Control, Vol. 1, 1994, pp. 283-288. [59] O.K. Rediniotis, J. Ko, X. Yue, A.J. Kurdila, "Synthetic Jets, their Reduced Order Modeling and Applications to Flow Control," AIAA Aerospace Sciences Meeting and Exhibit, AIAA-99-1000, 1999. [60] S.S. Ravindran, "Reduced-Order Adaptive Controllers for MHD Flows Using Proper Orthogonal Decomposition," IEEE Conference on Decision and Control, Vol. 3, 2001, pp. 2454-2459. [61] L. Ji and J. Zhou, "The Boundary Element Method For Boundary Control of The Linear Stokes Flow," IEEE Conference on Decision and Control, Vol. 3, 1990, pp. 1192-1194. [62] C.F.M. Coimbra and R.H. Rangel, "Spherical Particle Motion in Unsteady Viscous Flows," AIAA Aerospace Science Meeting and Exhibit, 1999, AIAA-99-1031. [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. |

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PAGE 1 APPLICA TIONS OF LINEAR P ARAMETER-V AR YING CONTR OL FOR AER OSP A CE SYSTEMS By KRISTIN LEE FITZP A TRICK A THESIS PRESENTED T O THE GRADU A TE SCHOOL OF THE UNIVERSITY OF FLORID A IN P AR TIAL FULFILLMENT OF THE REQ UIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORID A 2003 PAGE 2 T ABLE OF CONTENTS pageLIST OF T ABLES. . . . . . . . . . . . . . . . . . i vLIST OF FIGURES. . . . . . . . . . . . . . . . . . vABSTRA CT. . . . . . . . . . . . . . . . . . . . viiCHAPTER 1 INTR ODUCTION. . . . . . . . . . . . . . . . 11.1 Ov ervie w. . . . . . . . . . . . . . . . . 11.2 Background. . . . . . . . . . . . . . . . 22 LINEAR P ARAMETER-V AR YING CONTR OL THEOR Y. . . . . 53 LINEAR P ARAMETER-V AR YING CONTR OL FOR AN F/A-18. . . . 103.1 Problem Statement. . . . . . . . . . . . . . 103.2 Open-loop Dynamics. . . . . . . . . . . . . . 123.3 Control Objecti v es. . . . . . . . . . . . . . 143.4 Synthesis. . . . . . . . . . . . . . . . . 153.5 Simulation. . . . . . . . . . . . . . . . . 173.6 Conclusion. . . . . . . . . . . . . . . . . 204 LINEAR P ARAMETER-V AR YING CONTR OL FOR A HYPERSONIC AIRCRAFT. . . . . . . . . . . . . . . . . . 224.1 Problem Statement. . . . . . . . . . . . . . 224.2 Generic Hypersonic V ehicle. . . . . . . . . . . . 254.3 Hypersonic Model. . . . . . . . . . . . . . . 264.4 Linear P arameter -V arying System. . . . . . . . . . 274.5 Control Design. . . . . . . . . . . . . . . 304.6 Simulation. . . . . . . . . . . . . . . . . 334.6.1 Open-Loop Simulation. . . . . . . . . . . 334.6.2 Closed-Loop Simulation. . . . . . . . . . . 354.7 Conclusion. . . . . . . . . . . . . . . . . 385 LINEAR P ARAMETER-V AR YING CONTR OL FOR A DRIVEN CA VITY. 395.1 Problem Statement. . . . . . . . . . . . . . 395.2 Background. . . . . . . . . . . . . . . . 40 ii PAGE 3 5.3 Dri v en Ca vity Geometry . . . . . . . . . . . . 41 5.4 Go v erning Equations of Motion . . . . . . . . . . 42 5.5 Reduced-Order Linear Dynamics . . . . . . . . . . 44 5.6 Creeping Flo w in a Dri v en Ca vity . . . . . . . . . . 47 5.7 Excitation Phase Dif ferential . . . . . . . . . . . 49 5.8 Control Design . . . . . . . . . . . . . . . 50 5.8.1 Control Objecti v es . . . . . . . . . . . . 50 5.8.2 Synthesis . . . . . . . . . . . . . . . 53 5.9 Simulation . . . . . . . . . . . . . . . . 54 5.9.1 Open-Loop Simulation . . . . . . . . . . . 54 5.9.2 Reduced-Order Closed-Loop Simulation . . . . . . 57 5.9.3 Full-Order Closed-Loop Simulation . . . . . . . 60 5.10 Conclusion . . . . . . . . . . . . . . . . 63 6 CONCLUSION . . . . . . . . . . . . . . . . . 64 REFERENCES . . . . . . . . . . . . . . . . . . 65 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . 70 iii PAGE 4 LIST OF T ABLES T able page 3Â–1 Original Design Points . . . . . . . . . . . . . . 12 3Â–2 Frequenc y and Damping Ratio of Design and Analysis Models . . . 14 3Â–3 Frequencies and Damping Ratios of the T ar get Model . . . . . . 16 3Â–4 Induced Norms of Closed-Loop System . . . . . . . . . 17 3Â–5 Numerical Results . . . . . . . . . . . . . . . 19 4Â–1 Model Dimensions and Flight Conditions . . . . . . . . . 27 4Â–2 Modes of the Hypersonic Model . . . . . . . . . . . 30 4Â–3 Modes of the T ar get Model . . . . . . . . . . . . . 32 4Â–4 Open-Loop Synthesis Norms . . . . . . . . . . . . 33 4Â–5 Point Design Norms . . . . . . . . . . . . . . . 33 5Â–1 Induced Norms of Closed-Loop System . . . . . . . . . 53 i v PAGE 5 LIST OF FIGURES Figure page 2Â–1 H Block Diagram (Gain-Scheduled) . . . . . . . . . . 7 3Â–1 F/A-18 . . . . . . . . . . . . . . . . . . 11 3Â–2 Flight En v elope/P arameter Space . . . . . . . . . . . 12 3Â–3 Synthesis Block Diagram . . . . . . . . . . . . . 15 3Â–4 Closed-loop Block Diagram . . . . . . . . . . . . 18 3Â–5 Pitch Rate for Design Points . . . . . . . . . . . . 18 3Â–6 Pitch Rate for Analysis Point . . . . . . . . . . . . 19 3Â–7 Controller Ele v ator Deection . . . . . . . . . . . . 19 4Â–1 Inner -Loop/Outer -Loop Design . . . . . . . . . . . . 25 4Â–2 Simplied Model of a Generic Hypersonic V ehicle . . . . . . 25 4Â–3 Synthesis Block Diagram . . . . . . . . . . . . . 31 4Â–4 Open-Loop T ransfer Functions . . . . . . . . . . . . 34 4Â–5 Open-Loop Angle of Attack Result . . . . . . . . . . 35 4Â–6 Input Ele v on Deection . . . . . . . . . . . . . . 35 4Â–7 Closed-loop Design . . . . . . . . . . . . . . . 36 4Â–8 Closed-Loop T ransfer Functions . . . . . . . . . . . 36 4Â–9 Closed-Loop Angle of Attack Result . . . . . . . . . . 37 4Â–10 Ele v on Deection Command . . . . . . . . . . . . 38 4Â–11 Ele v on Deection Rate . . . . . . . . . . . . . . 38 5Â–1 Stok es Dri v en Ca vity Flo w Problem . . . . . . . . . . 41 5Â–2 Controller Block Diagram . . . . . . . . . . . . . 51 5Â–3 Open-Loop Flo w V elocities for Full-Order Model . . . . . . . 55 v PAGE 6 5Â–4 Open-Loop Flo w V elocities for Reduced-Order Model with 165 o Phase Dif ferential . . . . . . . . . . . . . . . . 56 5Â–5 Open-Loop Flo w V elocities for Reduced-Order Model with 210 o Phase Dif ferential . . . . . . . . . . . . . . . . 56 5Â–6 T rajectory of Phase Dif ferential . . . . . . . . . . . 56 5Â–7 Open-Loop Flo w V elocities for Reduced-Order Model o v er a T rajectory of Phase Dif ferentials . . . . . . . . . . . . . . 57 5Â–8 Closed-loop System . . . . . . . . . . . . . . . 58 5Â–9 Closed-Loop Flo w V elocities for Reduced-Order Model with 165 o Phase Dif ferential . . . . . . . . . . . . . . . . 59 5Â–10 Closed-Loop Flo w V elocities for Reduced-Order Model with 210 o Phase Dif ferential . . . . . . . . . . . . . . . . 59 5Â–11 Closed-Loop Flo w V elocities for Reduced-Order Model o v er a T rajectory of Phase Dif ferentials . . . . . . . . . . . . 60 5Â–12 Closed-Loop Flo w V elocities for Full-Order Model . . . . . . 61 5Â–13 Closed-Loop Flo w V elocities for Full-Order Model with Controller Associated with 165 o Phase Dif ferential . . . . . . . . . 61 5Â–14 Closed-Loop Flo w V elocities for Full-Order Model with Controller Associated with 210 o Phase Dif ferential . . . . . . . . . 62 5Â–15 Closed-Loop Flo w V elocities for Full-Order Model o v er a T rajectory of Phase Dif ferentials . . . . . . . . . . . . . . 62 vi PAGE 7 Abstract of Thesis Presented to the Graduate School of the Uni v ersity of Florida in P artial Fulllment of the Requirements for the De gree of Master of Science APPLICA TIONS OF LINEAR P ARAMETER-V AR YING CONTR OL FOR AER OSP A CE SYSTEMS By Kristin Lee Fitzpatrick December 2003 Chair: Richard C. Lind, Jr Major Department: Mechanical and Aerospace Engineering Gain-scheduling control has been an engineering practice for decades and can be described as the linear re gulation of a system whose parameters are changed as a function of v arying operating conditions. Se v eral gain-scheduling techniques ha v e been researched for the control of systems that v ary with time-v arying parameters. These techniques create controllers at v arious points within the parameter space of the system and use an interpolation la w to change controllers as the parameter changes with time. The process of creating such an interpolation la w can be v ery rigorous and timeconsuming and the resulting controller is not guarnateed to stablize the time-v arying system. The gain-scheduling technique kno wn as linear parameter -v arying control, ho we v er solv es a linear matrix inequality con v e x problem to create a single controller that has an automatic interpolation la w and is guaranteed to stabilize the closed-loop system. This paper demonstrates the use of this technique to create controllers for three aerospace systems. The rst system is the longitudinal dynamics of the F/A-18, the second system is the structural dynamics of a hypersonic v ehicle and the third system is the o w dynamics within a dri v en ca vity Simulations are performed using the linear vii PAGE 8 parameter -v arying controller created for each system to sho w the usefulness of a linear parameter -v arying frame w ork as a gain-schedule design technique. viii PAGE 9 CHAPTER 1 INTR ODUCTION 1.1 Ov ervie w The dynamics of aerospace systems that deal with ight and process control are af fected by v ariations in the parameters that mak e up their operating space (i.e., altitude, Mach number temperature). Gain-scheduling techniques are used to create a controlling scheme that will w ork throughout the system' s operating space. The resulting controller will v ary based on the same parameters as the system' s plant model. The traditional gain-scheduling technique can be brok en do wn into three major steps. The rst step in v olv es 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 de vised by linearly interpolating between these re gional controllers as the v ehicle mo v es through its operating range. This technique w orks well for some systems; ho we v er it does not guarantee stability and rob ustness of the closed-loop system. Another disadv antage of this method is the possibility of a skipping beha vior due to the switch between controllers. This thesis presents the technique of creating a single gain-scheduled controller that can be treated as a single entity This technique achie v es gain-scheduling with a parameter -dependent controller that will w ork throughout an operating range or ight en v elope. Benets of this technique are that it remo v es the need of creating se v eral controllers for dif ferent parameters within an operating domain and remo v es the need for the creation of a gain-scheduled control la w 1 PAGE 10 2 This study applies this method to three specic aerospace systems. The rst application of this technique is creation of a controller for the F/A-18 longitudinal ax es o v er a specic ight en v elope. The second application is control of the structural dynamics of a hypersonic aircraft o v er a temperature range. The third application is control of the v elocity along the center of a dri v en ca vity o w o v er a range of phase dif ferentials within the o w 1.2 Background Gain-scheduling has e v olv ed hand in hand with the progress of mechanical systems. Gain-scheduling techniques are currently used for control design for both linear and nonlinear systems. This study focused on systems pertaining to aerospace applications. Gain-scheduling in aerospace applications came about during WWII as autopiloting became necessary with the birth of jet aircraft and guided missiles [ 1 ]. When gain-scheduling w as rst concei v ed for the military it w as created through hardw are and w as quite costly Gain-scheduling w as not adopted commercially until the creation of digital control, nearly a quarter -century after military use. The de v elopment of gain-scheduling o v er past decades led to se v eral design techniques and the use of gain-scheduling for man y dif ferent aerospace systems. Se v eral gain-scheduling methods ha v e been de v eloped for designing controllers for linear systems. The three main classes of linear systems that apply to aerospace systems are linear time-in v ariant (L TI), linear time-v arying (L TV), and linear parameter v arying (LPV). Gain-scheduling is most often applied to linear parameter -v arying systems, which are af ne functions of parameters that af fect their operation. Gain-scheduling can also be applied to the control of nonlinear systems. Se v eral linearization techniques can be used for nonlinear systems before a gain-scheduled controlling scheme is de v eloped. The most common approach is based on Jacobian linearization of the nonlinear plant about a f amily of operating points (i.e., equilibrium PAGE 11 3 points) [ 2 ]. The system can also be linearized along a trajectory in the e v ent the linearized dynamics do not e xhibit good performance or stability a w ay from equilibrium points [ 3 ]; ho we v er the trajectory must be kno wn in adv ance to perform the control design. Once the control scheme is created, it can be applied to controlling the nonlinear system. Simulations of gain-scheduling controllers ha v e also been applied to nonlinear systems [ 4 5 6 ]; ho we v er nonlinear systems do not necessarily ha v e to be linearized. Set-v alued methods for LPV systems ha v e also been applied to nonlinear systems with quasi-LPV representations [ 7 ]. Linearization errors were accommodated as linear state-dependent disturbances; constraints on systems' states and controls were specied; rates of transitions among operating re gions were addressed, which allo ws e v en the local-point designs to be nonlinear The classical gain-scheduling 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 w as demonstrated for a missile autopilot that uses synthesis with D-K iteration to create controllers; and an iteration scheme is designed o v er the operating domain [ 8 ]. Another method for a missile autopilot creates controllers at distinct operating conditions using H control synthesis; and then creates a schedule for the controllers by remo ving coupling terms [ 9 ]. Another project in v olv ed creating H point design controllers at specic equilibrium points [ 10 ]. That project reduces the controllers to second order which are then realized in a feedback path conguration for which a gain-scheduling la w is de v eloped. A study also used a design algorithm for a state feedback la w based on gain-scheduling for an LPV multi-input multi-output system [ 11 ]. The state feedback control la w places the system' s poles in a neighborhood of desired locations and stabilizes the closed-loop system. Though this classical approach has w ork ed well for man y applications, there is no guarantee of rob ustness or stability of the closed-loop systems. PAGE 12 4 A more recent approach, that appeared in the late 1990' s, in v olv es creating an LPV controller that uses an automatic interpolation la w o v er the operating domain (which has guaranteed closed-loop rob ustness and stability with the LPV system). The method of D-K iteration with synthesis w as 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 F-16 aircraft [ 13 ], and a hypersonic aircraft [ 14 ], an LPV controller can also be created by letting the controller ha v e the same linear fractional relationship with the v arying parameters as the system [ 15 ] while attempting to minimize the H norm. This technique is further e xpanded with the controlling of the longitudinal ax es of an F16 aircraft in a project that breaks a parameter space into tw o smaller o v erlapping parameter spaces, synthesizes an LPV controller for each space, and then uses blending functions to form a single LPV controller [ 16 ]. An LPV controller w as created [ 17 ] for an LPV system, where parameter dependent feedback control la ws are constructed after transforming the original LPV system into canonical form. Separate longitudinal and lateral-directional LPV controllers were designed for the F/A-18 [ 18 ]. The original controllers were formed using H synthesis and then rob ustness w as increased to meet military standards by using synthesis. Other recent ef forts at using realtime parameter information in control strate gies included minimizing linear matrix inequalities [ 19 20 ]. This thesis presents one of the more recent gain-scheduling techniques for creating an LPV controller using H synthesis, which is designed to w ork for the LPV system' s entire operating domain. The operating domain of an LPV system is also kno wn as the system' s parameter space. Linear parameter -v arying control theory is discussed in more detail in the ne xt section. PAGE 13 CHAPTER 2 LINEAR P ARAMETER-V AR YING CONTR OL THEOR Y Linear parameter -v arying controller synthesis is a gain-scheduling technique for designing one controller that will w ork o v er a range of parameters without ha ving to create a scheduling scheme. In order to use the LPV frame w ork the plant model must be created as a linear parameter -v arying system. A linear parameter -v arying system depends af nely on a set of norm-bounded time-v arying operating parameters. It considers linear systems whose open-loop dynamics are af ne functions of the operating parameters. A method of identifying multi v ariable LPV state space systems that are based on local parameterization and gradient search in the resulting parameter space is presented in [ 21 ]. T w o identication methods were purposed in [ 22 ] for a class of multi-input multi-output discrete-time linear parameter -v arying systems. Both methods are based on the subspace state space method, which w as suggested by [ 23 ] in the early 1990s. LPV modeling of aircraft dynamics, kno wn as the bounding box approach and the small hull approach [ 24 ]. A general case of a linear parameter -v arying plant, whose dynamical equations depend on physical coef cients that v ary during operation, has the form Pq n xAqxB 1qdB 2qu eC 1qxD 11qdD 12qu yC 2qxD 21qdD 22qu (2.1) where qt q 1t rq nt q iq it q i (2.2) 5 PAGE 14 6 is a time-v arying v ector of physical parameters (i.e., v elocity angle of attack, stif fness); A B C D are af ne functions of qt, x is the state v ector y is the measured outputs, e is the re gulated outputs or errors, d is the e xogenous disturbances, and u is the controlled input. When the coef cients under go lar ge v ariations it is often impossible to achie v e high performance o v er the entire operating range with a single rob ust L TI controller When parameter v alues are measured in real time controllers that incorporate such measurements to adjust to current operating conditions w ould be benecial. These controllers are said to be scheduled by the parameter measurements. This control theory typically achie v es higher performance when considering lar ge v ariations in operating conditions. In the e v ent that dif ferent parameters ef fect the system dif ferently weighting functions can be used to compensate for the dif ferences. If the parameter v ector qttak es v alues within a geometric shape of R n with cornersP iNi1N2 n, the plant system matrix Sq: x ey AqtBqtrCqtrDqtr x d u (2.3) ranges in a matrix polytope with v ertices SP i. Gi v en con v e x decomposition qta 1 P 1 a N P Na i0N i1 a i1 (2.4) of q o v er the corners of the parameter re gion, the system matrix is gi v en by Sqa 1 SP 1 a N SP N (2.5) This suggests seeking parameter -dependent controllers with equations Kq n zA KqzB Kqy uC KqzD Kqy (2.6) PAGE 15 7 and with a v erte x property where a gi v en con v e x decomposition qt niN a i P i of the current parameter v alue qt. The v alues of A Kq, B Kq, C Kq, D Kqare deri v ed from the v alues A KP i, B KP i, C KP i, D KP iat the corners of the parameter re gion by !A KqB KqC KqD Kq "$# % N iN a i !A KP iB KP iC KP iD KP i "$# % (2.7) In other w ords, the controller state-space matrices at the operating point qtare obtained by con v e x interpolation of the L TI v erte x controllers K i : !A KP iB KP iC KP iD KP i "$# % (2.8) This yields a smooth scheduling of the controller matrices by the parameter measurements qt. As an e xample, consider the follo wing H -lik e synthesis problem relati v e to the interconnection in Figure 2Â–1 If there e xists a continuous dif ferentiable function Xq Kq Pq &d &e u & y 'Figure 2Â–1: H Block Diagram (Gain-Scheduled) dened on R n where Xq)(0(2.9) PAGE 16 8 the w orst-case closed-loop RMS gain from d to e does not e xceed some le v el g(0, and !I 0 AqBq 0 I CqDq "$# # # # # # # % XqXq 0 0 Xq0 0 0 0 0 +g I 0 0 0 0 1 g I"$# # # # # # # % !I 0 AqBq 0 I CqDq "$# # # # # # # % ,0 (2.10) hold for all q-R n then the system is quadratically stable and the L 2 norm from d to e is smaller than g The quadratic stability of a system allo ws the parameter to change with arbitrary speed without threatening stability of the system and is dened as e xisting if there e xists a real positi v e denite matrix PP T(0 such that A TqPPq ,0.qt)-R q(2.11) The induced L 2 norm of a quadratically stable LPV system G is dened as/G/ su p q0R q su p1d122 03d0L 2/e/2 /d/2 (2.12) with R q 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, synthesis design [ 8 ], Linear Quadratic Gaussian (LQG) control design [ 25 ], and H control design [ 26 ]. The synthesis technique attempts to minimize the v alue o v er stabilizing the controller K, and diagonal, D, while D-K iteration is used to reduce the cost function. The LQG controller design method synthesizes a controller which is optimal with respect to a specied quadratic performance inde x and tak es into account the Gaussian white noise disturbances acting on the system. The technique used for the projects presented in this paper is the H control synthesis technique which uses the linear fractional form of the LPV system and creates the controller while attempting to minimize the H norm. By letting the controller ha v e the same linear PAGE 17 9 fractional relationship with the v arying parameters as the LPV system the H control problem is formulated using linear matrix inequalities (LMI). The appearance of LMIs in the control synthesis sho ws ho w the control problem is a con v e x optimization problem [ 27 ], as w as described in the pre vious e xample. Another e xample of creating a con v e x optimization problem with LMI e xpressions for the use of nding an LPV controller for the attitude control of an X-33 is presented in [ 28 ]. The main benet of using the LPV frame w ork is that it allo ws gain-scheduled controllers to be treated as a single entity with the gain-scheduling being achie v ed with the parameter -dependent controller and automatic interpolation la w which remo v es the ad-hoc scheduling schemes that were necessary in the past. PAGE 18 CHAPTER 3 LINEAR P ARAMETER-V AR YING CONTR OL FOR AN F/A-18 3.1 Problem Statement Se v eral control designs ha v e been applied to the control of F-18 aircraft. A controller w as designed using H and synthesis techniques for a single ight condition [ 29 ]. Though this technique w orks well for a single point in the ight en v elope a type of gain-scheduling is necessary for controlling the F/A-18 throughout its operating domain. A longitudinal v ariable structure controller w as created for an F-18 model with parameter perturbations [ 30 ]. Though this technique can attain the con v entional goals of stability and tracking for uncertain nonlinear plants, a reference trajectory for tracking control must be specied, which indicates that the controller cannot operate o v er a lar ge ight en v elope. A lateral-directional controller w as created using synthesis with parametric uncertainty to account for gain dif ferences between a nominal model and trim models and multiplicati v e uncertainty to account for changes between a nominal model and other trim models within the chosen ight en v elope [ 31 ]. Because this technique uses constant blocks of uncertainty instead of gain-scheduling the ight en v elope used for the project had to be small, M540350556and altitude7420286k f t Gain-scheduled approximations to H controllers for the F/A-18 Acti v e Aeroelastic W ing, located at N ASA Langle y Research Center were de v eloped within another project [ 32 ]. Point design controllers were created within a small ight en v elope and then a scheduling scheme of the gains had to be formed. A multi v ariable LPV controller w as designed using H synthesis for the F/A-18 System Research Aircraft (SRA), located at N ASA Langle y Research Center in [ 33 ]. Though this technique is also chosen for control synthesis in the project presented in this chapter the ight en v elope that the controller had to operate within is smaller in [ 33 ], with 10 PAGE 19 11 M740350706and altitude8415326k f t A similar project in v olving an LPV controller for the F/A-18 SRA, in [ 18 ], uses the same synthesis technique b ut uses an e v en smaller ight en v elope, M940450556and altitude:420256k f t than [ 33 ]. The project presented in this study discusses the formation and simulations of a linear parameter -v arying controller for the longitudinal dynamics of an F/A-18 o v er a chosen ight en v elope. The F/A-18, sho wn in Figure 3Â–1 has a ceiling of 50,000+ f t and a speed of M=1.7+. As the aircraft' s altitude v aries so does the air density which af fects the aircraft' s response to control surf ace deections. Furthermore, the amount of deection necessary for a particular maneuv er v aries as the Mach number v aries. These aerodynamic changes that occur with the lar ge range in altitude and Mach number mak e it necessary to incorporate a gain scheduling technique for control. The ight en v elope for this project is limited to Mach numbers from 0.4 to 0.8, which includes both incompressible and compressible subsonic o ws, and an altitude range from 10,000 f t to 30,000 f t which includes a density change of roughly 0.9 E;3 sl ug PAGE 20 12 pressure, q for the model at Mach=0.40 at an altitude of 30 kft P 4 w as too lo w to control and therefore the model w as discarded. T able 3Â–1: Original Design Points Design Point Mach Number Altitude ( f t ) P 1 0.4 10,000 P 2 0.8 10,000 P 3 0.8 30,000 P 4 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 ight en v elope which represents the parameter space and the placement of the models used for this project are sho wn in Figure 3Â–2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5 10 15 20 25 30 35 P 1 P 2 P 3 P A Mach NumberAltitude (kft) Design PointDesign PointDesign PointAnalysis Point Figure 3Â–2: Flight En v elope/P arameter Space 3.2 Open-loop Dynamics The F/A-18 models used for this project are longitudinal short-period approximations that were de v eloped with tw o states, one input and one output. The states include angle of attack ( d e g ) and pitch rate ( d e g PAGE 21 13 The model for the F/A-18 at Mach=0.40 at an altitude of 10 kft is gi v en as P 1 such that qP 1 d P 1 +07433 4256200 +05642+00022+04064 +00662 0 573 0 (3.1) The model for the F-18 at Mach=0.80 at an altitude of 10 kft is gi v en as P 2 such that qP 2 d P 2 +18415 8531909 +20292+00192+09431 +02568 0 573 0 (3.2) The model for the F-18 for Mach=0.80 at an altitude of 30 kft is gi v en as P 3 such that qP 3 d P 3 +08399 7911313 +09314+00075+04499 +01190 0 573 0 (3.3) The model of the analysis point with an altitude of 20,000 ft and Mach=0.6 is gi v en as P A such that qP A d P A +08280 6170114 +08269+00075+04499 +00994 0 573 0 (3.4) The frequenc y and damping ratio for the each of the models were determined and are sho wn in T able 3Â–2 All of the damping ratios are greater than zero, which af rms that the models are stable. PAGE 22 14 T able 3Â–2: Frequenc y and Damping Ratio of Design and Analysis Models Model w z P 1 1.113 0.5166 P 2 4.257 0.3271 P 3 2.512 0.2567 P A 2.288 0.2764 The linear parameter -v arying model for the parameter space is gi v en as Pqand is gi v en as qPqd P=q>@?BAP 1CED F G G G G HJI1K0982 427K57 I1K465I0K017I0K5367 I0K1906 0 0 0LNM M M M Oq 1=t> D F G G G G H1K0016I62K06 1K0978 0K0117 0K4932 0K1378 0 0 0LNM M M M Oq 2=t>(3.5) Where q q 1 0 0 q 2 and where q 1-P4016represents the systems dependence on Mach number and q 2-P4016represents the systems dependence on altitude. The aircraft ying at a Mach number of 0.4 corresponds to a q 10 and at a Mach number of 0.8 corresponds to a q 11. The aircraft ying at an altitude of 10,000 f t corresponds to a q 20 and at an altitude of 30,000 f t corresponds to a q 21. 3.3 Control Objecti v es The control objecti v e for the F/A-18 longitudinal ight controller is to track a gi v en pitch rate command to within certain tolerances of a tar get response generated by a tar get model that has desirable dynamics. The commanded pitch rate is a step input which be gins at zero magnitude and becomes 10 d e g PAGE 23 15 3.4 Synthesis The system sho wn in Figure 3Â–3 incorporates all the necessary elements needed to create the controller which will accomplish the controller objecti v es. This system contains the open-loop dynamics as described byPand a tar get modelTused for model follo wing. The system also incorporates weighting functions used for loop shaping, which are gi v en asW pW nW kW u. u & W u & W k &e 2 & Pq && W p &e 1 q c & R S T Rn & W n &+ &&y Figure 3Â–3: Synthesis Block Diagram The system has 2 disturbances, 1 control, 2 errors and 1 measurement, which were referred to asdueyin the LPV Control section. The disturbances are random noise n-R af fecting the sensor measurement and the commanded pitch rate q c The control input is u-R which af fects the ele v ator deection. The sensor measurement of pitch rate, y is used for feedback to the controller The errors are, e 1 the error between the tar get pitch rate response and the LPV model response and the weighted control ef fort, e 2 The tar get model T describes an F/A-18 model that has dynamics which outputs a desirable pitch rate response. The tar get model is used for model follo wing to aid in obtaining the LPV controller The tar get model is not a function of parameters in the operation space. The frequenc y and damping ratio of the tar get model are presented in PAGE 24 16 T able 3Â–3 T +2 0 0 +198 0+3+3 +278 0 3+3 218 +079+047 179 0 (3.6) T able 3Â–3: Frequencies and Damping Ratios of the T ar get Model T ar get w z 2.0 1.0 4.2426 0.7071 The performance lter gi v en as W p serv es to normalize the error in the model follo wing between the tar get model and the LPV model. The lter W p is chosen to meet design specications in accordance to pitch rates of the aircraft in response to a commanded pitch rates. The actual lter is realized as W p70. The lter W k is used to normalize the penalty placed on the amount of actuation commanded by the controller This lter reects the capabilities of the actuation system. The weighting is chosen as the in v erse of the actuator' s magnitude of motion, W k05. The input matrix v aries from model to model within the parameter space. This v ariation necessitates an input lter within the synthesis. The lter sho wn as W u is used as the input lter and has a v alue of 1 E 5 sT1 E 5 Also, the lter W n is included to account for noise that corrupts measurements by the sensor The inclusion of noise is needed to pro vide a minimal le v el of penalty on the sensors, which will gi v e rob ustness to the controller The design did not w ant to consider a lar ge amount of noise so the lter w as chosen as W n001. Using the system in Figure 3Â–3 rob ust H controllers were designed for the models at each of the design points and a linear parameter -v arying controller w as designed for the entire parameter space. The techniques of H control are used to PAGE 25 17 reduce the induced norm from the input to the weighted errors. The softw are from the Analysis and Synthesis ToolboxforMatlabis used for the actual computation of the controller [ 34 ]. The same weightings are used to create the controllers in order to achie v e the same performance le v el for all of the points in the parameter space. The resulting induced norms achie v ed by the indi vidual controllers and the LPV controller are sho wn in T able 3Â–4 T able 3Â–4: Induced Norms of Closed-Loop System Open-Loop Model H -norm P 1 0.891 P 2 0.775 P 3 0.775 P 1+P 3 0.971 It is important to note that all of the closed-loop norms are less than unity These magnitudes indicate that the controllers are able to achie v e the desired performance and rob ustness objecti v es. The last entry in T able 3Â–4 is the norm associated with the LPV controller Allo wing the altitude and Mach number to v ary with time increases the norm as e xpected. Ho we v er this norm did not raise much abo v e the norm associated with an y of the point designed H controllers and stayed belo w unity This condition indicates that the LPV controller is capable of accounting for the time-v arying nature of Mach number and altitude without e xcessi v e loss of performance. 3.5 Simulation The closed-loop dynamics are simulated with a 10 d e g PAGE 26 18 d & Kq Pq &q Sq c ''Figure 3Â–4: Closed-loop Block Diagram responses only v ary roughly 0.2% from the tar get 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 a w ay from the v ertices points of the parameter space that were used to create the controller 0 2 4 6 8 10 0 2 4 6 8 10 12 Time (s)Pitch Rate (deg/sec) CommandTargetP 1 P 2 P 3 Figure 3Â–5: Pitch Rate for Design Points The analysis point w as chosen to be the f arthest from the v ertices of the parameter space which results in a Mach number of 0.6 and an altitude of 20,000 f t The responses of the analysis model and the tar get model, using the same step command that w as used for the point design simulation, are sho wn in Figure 3Â–6 The results appear to be quite close to the tar get response. Numerical results were pulled from the plot to mak e a closer comparison and are sho wn in T able 3Â–5 The same time response and delay time are apparent for both the analysis model and tar get model responses. The settling time of the analysis model response lags the tar get response by 0.5 seconds, which is within the control objecti v es. The maximum PAGE 27 19 0 2 4 6 8 10 0 2 4 6 8 10 12 Time (s)Pitch Rate (deg/sec) CommandTargetP A Figure 3Â–6: Pitch Rate for Analysis Point T able 3Â–5: Numerical Results T ar get Model Analysis Model Rise T ime 0.21 sec 0.21 sec Settling T ime 1.63 sec 2.13 sec Peak Ov ershoot 5 % 1.2 % o v ershoot of the analysis model response w as less than that of the tar get model response and remains within the bounds of the controller objecti v e. The controller commanded ele v ator deection from the simulations is sho wn in Figure 3Â–7 and is used to determine if the actuation of the ele v ator is reasonable for each of the tested models. All of the v alues are ne gati v e because a ne gati v e ele v ator 0 2 4 6 8 10 -20 -15 -10 -5 0 Time (s)Commanded Elevator Deflection (deg) P A P 1 P 2 P 3 Figure 3Â–7: Controller Ele v ator Deection PAGE 28 20 deection 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 follo ws is to maintain the pitch rate command. The peak of the ele v ator command for the system with the model associated with a point in the parameter space ha ving a Mach number of 0.4 and an altitude of 10,000 f t P 1 has the greatest v alue, -17 o The v alue is reasonable because at lo wer speeds a greater angle is needed to get the same response as ying at f aster speeds and there is a less chance that the control surf ace will be damaged by the slo wer airo w The ele v ator command peak for the system using the model with a Mach number of 0.8 and an altitude of 10,000 f t P 2 has the smallest v alue, -4.5 o This lo wer v alue is all that is needed for the desired pitch rate due to the speed of the airo w around the control surf ace at the higher Mach number which decreases the time the maneuv er requires. The peak v alue is -9.5 o for the ele v ator command associated with the system using the model that has a Mach number of 0.8 and an altitude of 30,000 f t P 3 which is an acceptable magnitude. This v alue being higher than the v alue corresponding to the same Mach number b ut with a lo wer altitude of 10,000 f t is e xpected because at a higher altitude the density is less and so fe wer air particles are present to be af fected by the deection, therefore a lar ger angle is necessary The peak commanded ele v ator deection for the system using the analysis model, P A is -12 o which is also acceptable. The magnitude is reasonable because it is less than the v alue commanded for P1 due to the higher Mach number of the analysis model and is not too small that the increase in altitude w ould ha v e an adv erse af fect. 3.6 Conclusion This project considered the control of the pitch rate of an F/A-18 aircraft with a linear parameter -v arying controller This type of controller w as chosen because the change in dynamics of the aircraft could be modeled with a system whose state-space matrix and input matrix were af ne functions of the parameters, Mach number and PAGE 29 21 altitude. Once the controller w as created, it w as tested at certain points within the parameter space using a step pitch rate input. The results allo w for the conclusion that the LPV controller performed the specied objecti v es and is therefore a suf cient controller for the F/A-18 model presented in this project. PAGE 30 CHAPTER 4 LINEAR P ARAMETER-V AR YING CONTR OL FOR A HYPERSONIC AIRCRAFT 4.1 Problem Statement All aircraft o wn today y within the subsonic, transonic and supersonic ight re gimes. The push to w ard f aster and higher ying aircraft has mo v ed the en v elope into the hypersonic re gime. This push comes from both military and commercial groups. The military w ants a bomber that can y at high altitude, o v er a long range and at high speeds, so that the v ehicle is nearly impossible to shoot do wn. Commercial groups w ould lik e to ha v e a more reliable w ay of sending satellites into lo w earth orbit. The major problem with the use of rock ets is that if something goes wrong during ascension into orbit the car go will most lik ely be destro yed along with the rock et. The use of a hypersonic aircraft presents a more reliable transportation for the satellite because if an error did occur during the ight there w ould be a chance that the aircraft could maneuv er to a landing area. Though the concept of hypersonic ight 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 v ehicles must operate. This obstacle may ha v e slo wed the creation of such v ehicles b ut se v eral control theories ha v e still been created. The more popular control theories include H [ 35 ], synthesis [ 36 ] and linear parameter -v arying control [ 37 ]. The theories in v olving H and synthesis, ho we v er only considered a single ight condition for the hypersonic v ehicle. Also, the pre vious project that used a linear -parameter v arying controller for the hypersonic v ehicle ignored the mode shape of the v ehicle and separated the rigid-body dynamics and the structural dynamics of the hypersonic model. A scheduled longitudinal control scheme w as 22 PAGE 31 23 created which incorporated a set of parameter controllers, where the parameters were Mach number and dynamic pressure, and w as 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 ef fect of structural modes on the response of the hypersonic v ehicle. Rob ust ight control systems are synthesized for the longitudinal motion of a hypersonic v ehicle using stochastic cost functions and ten design parameters [ 39 ]. That project also focused on the control of the rigid body dynamics of the hypersonic v ehicle without addressing structural dynamics. The control of the longitudinal motion of a hypersonic v ehicle w as also addressed, where rob ust ight control systems with a nonlinear dynamic in v ersion structure were synthesized [ 40 ]. Nonlinear control la ws were designed so the control systems w ould operate o v er a chosen ight en v elope. Again, the rigid body dynamics were the focus of control. A dual neural netw ork structure w as de v eloped that serv ed as feedback control and optimized the v ehicles trajectory to pre-specied b urnout conditions in v elocity ight path angle and altitude [ 41 ]. That project serv es more as an aid in the study of trajectory optimization than as a control theory for hypersonic v ehicles. Another project applied a hierarchical inte grated control methodology to a hypersonic v ehicle to reduce stabilizing control po wer required for specic ight conditions [ 42 ]. That methodology decomposes the hypersonic model into decoupled subsystems, creates a controller for each subsystem and a control la w for each subsystem controller is deri v ed. The decoupling of a hypersonic system may not be feasible due to the lar ge de gree of coupling between the physical structure and propulsion system of the v ehicle. Also, the creation of separate control la ws is laborious compared to the LPV method which forms an automatic interpolation la w The control of the lateral dynamic stability characteristics of a hypersonic v ehicle for a specied Mach number and altitude trajectory has also been detailed in a project [ 43 ]. The controller w as designed using Multi-Model Eigenstructure, which designs a rob ust x ed-gain PAGE 32 24 controller that guarantees rob ust stability and desired ight qualities along a specied reference trajectory The controller w ould need to be altered if the v ehicle de viated from the preset trajectory or if a ight en v elope w as to be considered. The same model of the longitudinal dynamics of a typical hypersonic v ehicle were used, where a unied approach to H 2 and H optimal control w as used to design a controller for a specic ight condition [ 44 ]. A unied approach alle viates dif culties with the Â“o v er cro wdingÂ” of a system' s roots inside the unit circle along with other numerical dif culties. Using the technique in that project w ould require more controllers to be created at other operating conditions along with a gain-scheduling la w if the v ehicle' s operating range spanned more than a single condition. Some of the challenges of hypersonic ight include the v arying of the hypersonic v ehicle' s dynamic characteristics due to a wide range of operating conditions and mass distrib utions for which a type of gain-scheduling technique appears to be essential [ 45 46 ]. Further discussion of a typical hypersonic v ehicle' s dynamics addresses ho w the combination of the propulsion system and aeroelastic ef fects contrib ute to the o v erall dynamic character of the v ehicle, which presents the need of structural dynamic controller [ 47 ]. This necessity is the moti v ation for the project presented in this chapter The controller designed for the hypersonic v ehicle for this project w as split into an inner -loop controller and an outer -loop controller The inner -loop controller is an LPV controller which must acti v ely damp the structural modes across a temperature range. Unlik e pre vious hypersonic controls, this controller will focus on the damping of the mode shape that is associated with the structural dynamics of the v ehicle, which will operate throughout a range of a specic operating parameter and for which the hypersonic model' s rigid-body and structural dynamics will not be separated. The outer -loop controller of the aircraft will be a rigid-body controller which will w ork as PAGE 33 25 a traditional ight controller for rigid aircraft and will be designed in a future project. The inner -loop structural damping controller is the focus of this project. A diagram of the inner -loop/outer -loop control design is sho wn if Figure 4Â–1 The Kqcontroller is the linear parameter -v arying inner -loop controller and the K ou t controller is the rigid-body outer -loop controller The P is the hypersonic plant model. K ou t Kq P & & &in pu t &ou t pu t R R 'Figure 4Â–1: Inner -Loop/Outer -Loop Design 4.2 Generic Hypersonic V ehicle The intended uses of hypersonic v ehicles ranges from putting satellites into lo w earth orbit to being the ne xt stealth bomber These missions require the v ehicle to tra v el through each ight re gime: subsonic, transonic, supersonic, hypersonic and orbital. This project will only consider the aircraft' s ight within the hypersonic re gime. Se v eral hypersonic designs ha v e been created which attempt to maximize aerodynamic and propulsi v e ef cienc y while still ha ving enough controllability Most of these designs incorporate the ele v ator and aileron into one structure kno wn as the ele v on. The form of hypersonic v ehicle used for this project is similar to the N ASP and X-30 v ehicles. A generalized shape can be seen in Figure 4Â–2 Figure 4Â–2: Simplied Model of a Generic Hypersonic V ehicle PAGE 34 26 This conguration of a hypersonic v ehicle combines the fuselage with the propulsion system. This combination greatly af fects the ight dynamics of the v ehicle. The forebody of the v ehicle acts as the compressor for the engine. The air o w through this compressor creates a pitch up moment. The aftbody of the v ehicle acts as the e xit nozzle for the engine. The airo w through the e xit nozzle creates a pitch do wn moment. Also, a change in angle of attack or sideslip af fects the engine inlet conditions which changes the propulsion performance. T o create a controller for this type of v ehicle the angle of attack, pitch angle and pitch rate are measured for feedback to the controller Another area of hypersonic ight that must be considered when creating a controller is the speed, and consequently temperature, at which the v ehicle ies. As the v ehicle enters the hypersonic re gime, the strength of shock w a v es increase and lead to higher temperatures in the re gion between the shock and the body As Mach number increases further the shock layer temperature becomes lar ge enough that chemical reactions occur in the air Also, an increase in temperature ef fects the structural dynamics of the v ehicle in that there is a reduction in the frequenc y of the structural modes. Therefore, the controller created in this project will consider temperature as the ight parameter 4.3 Hypersonic Model The hypersonic model [ 48 ] used for this project w as limited to the longitudinal motion and w as de v eloped with se v en states, three inputs and six outputs. The states include altitude, v elocity angle of attack, pitch angle, pitch rate, and tw o elastic states for the fuselage bending mode. The inputs include ele v on deection, dif fuser area ratio and fuel o w ratio. The outputs include angle of attack, pitch rate at forebody pitch rate at aftbody comb ustor inlet pressure, Mach and thrust which will be used as feedback to the controller Only the angle of attack and the tw o pitch rates are to be used as feedback to the controller due to their strong dependence on the structural PAGE 35 27 dynamics. Aerodynamic, inertial, propulsi v e, and elastic forces were used to deri v e the equations of motion for the hypersonic v ehicle [ 37 ]. The model dimensions and ight conditions are sho wn in T able 4Â–1 T able 4Â–1: Model Dimensions and Flight Conditions Length 150 f t Mass 300,000 l b Height 100,000 f t Mach 8 Dynamic Pressure 1017 ps f 4.4 Linear P arameter -V arying System The time-v arying operating parameters, q are ight parameters which af fect the aircraft during ight. These parameters are measured by sensors on the aircraft and are sent to the controller This project tak es into account only one ight parameter temperature, due to the lar ge af fect that temperature has on a hypersonic v ehicle' s structural dynamics. This parameter will ha v e a range from (0 o F to 5000 o F) to match the temperature ranges noted for the hypersonic ight of the X-30 and the HyperX v ehicles [ 49 ]. The parameter dependence of the model is sho wn in the matrices belo w q0 for the coldest temperature and q1 for the hottest temperature within the range. As the ight parameter temperature, changes during ight so does the amount it af fects changes in the aircraft. This problem can be compensated with the use of weighting functions which will be discussed in the ne xt section. AqVUAW qUA qW(4.1) PAGE 36 28 AX Y Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z [0 0;7\9248 E 3 7\9248 E 3 0 0 0 1\5026 E;4;3\2374 E;3;5\2818 E 1;3\2200 E 1 2\3762 E;2 5\7314 E;1 7\5583 E;3 1\1744 E;7;3\1848 E;7;3\3921 E;2 0 1 1\4681 E;4 2\8801 E;6 0 0 0 0 1 0 0;5\7586 E;6 9\6079 E;6 1\5833 E 0 0;5\1609 E;2 9\2411 E;2;1\8285 E;4 0 0 0 0 0 0 1;7\4858 E;1 1\0158 E;1 2\4280 E 3 0;7\4847 E 0;3\1086 E 2;9\4975 E;1] ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ _(4.2) A q 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0+25 E 2+02 (4.3) B 0 0 0+6435 E 1+7462 E 1 1261 E 3+1448 E+2+1596 E+2+2253 E+2 0 0 0+2455 E 0 8111 E+1 5190 E 0 0 0 0 6740 E 2+2925 E 1 2209 E 2 (4.4) Cq`UCW qUC qW(4.5) PAGE 37 29 Ca b c c c c c c c c c c c c c c d0 0 1 0 0 1e7453 Ef2 0 0 0 0 0 1 0 1e7453 Ef2 0 0 0 0 1 0f1e7453 Ef2f4e6971 Ef5 2e0641 Ef4 6e2428 E 0 0f1e0921 Ef2 1e0896 Ef1 0f5e3709 Ef6 1e0095 Ef3 0 0 0 0 0f3e5754 Ef1 6e0213 Ef1 1e8399 E 4 0f3e2185 E 1 3e2112 E 2 0gih h h h h h h h h h h h h h j(4.6) C q 0 0 0 0 0 C16lk105 0 0 0 0 0 0 0 C17mk005 0 0 0 0 0 0 C 227lk005 0 0 0 0 0 C 236lk005 0 0 0 0 0 0 0 0 0 0 0 0 0 C 256lk005 0 (4.7) D 0 0 0 0 0 0 0 0 0 0+7229 0 0 0 0 0+3158 E 4 5995 E 5 (4.8) As seen in the linear parameter -v arying matrices abo v e, both the state matrix4A6and the observ ation matrix4C6change with temperature. It is common for the state matrix to change as operating parameters change, b ut it is not common, in traditional aircraft, for the observ ation matrix to change. This change in the observ ation matrix accounts for the mode shape changes of the hypersonic v ehicle. The modes of the hypersonic model are sho wn for dif ferent temperatures in T able 4Â–2 The table sho ws the frequenc y of each of the modes and the damping PAGE 38 30 corresponding to the frequenc y The four modes of the open-loop dynamics are ( i ) a height mode, ( ii ) an unstable phugoid-lik e 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 frequenc y than the structural mode at the hot temperature. Minimizing the af fect that the temperature has on this mode is the objecti v e of the inner -loop LPV controller T able 4Â–2: Modes of the Hypersonic Model Cold Hot Mode wr ad PAGE 39 31 u & W k &e k R+ d & & Pq S S S W n &n &+ S &+ S &+ Sy & T &R & W p &e p X SFigure 4Â–3: Synthesis Block Diagram y-R 3 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 open-loop dynamics of the LPV system is described by Pq. Where, Pq Aq B Cq D (4.9) A tar get model, T is created to describe a hypersonic model with desirable structural damping and therefore incorporates the controller objecti v e. The tar get model w as used for model follo wing to aid in obtaining the LPV controller The tar get model modes and corresponding damping are sho wn in T able 4Â–3 The tar get model has a lar ge 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 lter W p w ould normally be used to dene the design specications in the frequenc y domain. F or this synthesis W p w as made equal to 1.5 which PAGE 40 32 T able 4Â–3: Modes of the T ar get Model T ar get Mode wr ad PAGE 41 33 dra ws from the connection of the rst input, q to the rst output, the e p meaning that the lar gest error comes from the performance of the angle of attack meeting the ele v on deection command. T able 4Â–4: Open-Loop Synthesis Norms H nor m wr ad PAGE 42 34 is located in the high frequenc y re gion is the structural mode. The tar get model' s structural mode peak sho ws a damped response, which is desirable. The hot and cold hypersonic models' structural mode peaks, ho we v er are v ery sharp which implies that there is v ery 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. 10 2 10 0 10 2 10 4 10 3 10 2 10 1 10 0 10 1 10 2 Frequency (rad/sec)Open Loop Transfer Function targetcoldhot Figure 4Â–4: Open-Loop T ransfer Functions The simulation of an angle of attack response to an ele v on deection input for the open-loop hypersonic models at both the hot and cold temperatures and the tar get model is sho wn in Figure 4Â–5 The ele v on deection used for the follo wing simulation of is sho wn in Figure 4Â–6 The angle magnitude is small due to the speed at which the v ehicle ies, a lar ge angle w ould be harmful at high speeds. Unlik e the tar get model response the open-loop model response at both the hot and cold temperatures sho w an oscillation for approximately v e seconds. This oscillation is due to the lack of structural damping and should be remo v ed by the controller during the closed-loop simulation. PAGE 43 35 0 5 10 15 4 3 2 1 0 1 2 3 Time (s)Angle of Attack (deg) targetcoldhot Figure 4Â–5: Open-Loop Angle of Attack Result 0 5 10 15 0 1 2 3 4 5 6 Time(s)Input Elevon Deflection (deg) Input Elevon Deflection Figure 4Â–6: Input Ele v on Deection 4.6.2 Closed-Loop Simulation The closed-loop dynamics are simulated to demonstrate the performance of the controller for the hypersonic models at both the hot and cold temperature. The closed-loop system for both models can be seen in Figure 4Â–7 The system sho wn in Figure 4Â–7 has one input signal and six output signals. The input signal d remains the ele v on deection. The outputs include angle of attack ( a ), pitch rate at forebody ( q f ), pitch rate at aftbody ( q a ), comb ustor inlet pressure ( P i ), Mach ( M ) and thrust ( T h ). The X is the same simple controller used to stabilize the PAGE 44 36 Kq R d & & Pq &T h &M &P i &q a &q f &a ' ' X SFigure 4Â–7: Closed-loop Design v ehicle for the open-loop synthesis. The Kqis the LPV controller that w as created using the synthesis model. The frequenc y response of the transfer function between the ele v on deection and the angle of attack for the closed-loop system for both the cold and hot temperature hypersonic models is sho wn in Figure 4Â–8 10 2 10 0 10 2 10 4 10 3 10 2 10 1 10 0 10 1 10 2 Frequency (rad/sec)Transfer Function targetcoldhot Figure 4Â–8: Closed-Loop T ransfer Functions Again, the peaks in the high frequenc y re gion correspond to the structural mode of the tar get 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 f alls within the temperature PAGE 45 37 range that a similar damped peak w ould result. So the control objecti v e of damping the structural mode w as fullled by the LPV controller The closed-loop simulation of the angle of attack response to the same ele v on deection used in the open-loop simulation is sho wn in Figure 4Â–9 The results are again presented for the system at both the hot and cold temperatures and for the tar get model. 0 5 10 15 4 3 2 1 0 1 2 3 Time (s)Angle of Attack (deg) targetcoldhot Figure 4Â–9: Closed-Loop Angle of Attack Result As can be seen, the oscillations that were apparent in the open-loop simulation ha v e been remo v ed by the controller This response is due to the damping which the controller imposed on the system. The hypersonic models' responses also follo w the tar get model response more closely throughout the simulation. The controller commanded ele v on deection in Figure 4Â–10 is plotted for the closed-loop simulation in order to v erify that the motion commanded did not violate the limited motion due to the high Mach number Because the command ne v er e xceeds a magnitude of 5 o the command does not violate the constraint associated with the ele v on actuator The corresponding deection rate in Figure 4Â–11 is plotted to v erify that the command does not violate the motion tolerances of the ele v on actuator The magnitude of the deection rate is within the limits associated with the actuator PAGE 46 38 0 5 10 15 4 3 2 1 0 1 2 3 4 5 Time(s)Elevon Deflection (deg) cold commandhot command Figure 4Â–10: Ele v on Deection Command 0 5 10 15 60 40 20 0 20 40 60 80 Time(s)Elevon Deflection Rate (deg/s) coldhot Figure 4Â–11: Ele v on Deection Rate 4.7 Conclusion This project considered the control of the structural dynamics of a hypersonic v ehicle with a linear parameter -v arying controller This type of controller w as chosen because the change in the dynamics of the hypersonic v ehicle could be modeled with a system whose state-space matrix and observ ation matrix were af ne functions of the parameter temperature. Once this controller w as created, it w as tested o v er a temperature range with an ele v on deection input. The results allo wed for the conclusion that the LPV controller performed the specied objecti v e and is therefore a suf cient controller for the hypersonic model presented in this project. PAGE 47 CHAPTER 5 LINEAR P ARAMETER-V AR YING CONTR OL FOR A DRIVEN CA VITY 5.1 Problem Statement Research into o w control techniques has been continually e v olving as related technologies mature. These technologies include hardw are de v elopment, such as sensors and actuators [ 51 ], and softw are de v elopment, such as models and simulations [ 52 ], associated with uid dynamics. In each case, the technologies are being de v eloped with careful consideration of the requirements for control design and implementation [ 53 ]. A particular challenge for o w control has been the de v elopment of open-loop models for which controllers can be designed. The equations of motion for such dynamics are well kno wn 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 sho wn that models can indeed be generated that are amenable to control a specic type of o w [ 54 ]. The system in that study is restricted to creeping o w in a dri v en ca vity Specically the left and right sides of the ca vity ha v e zero o w v elocity whereas the top and bottom boundaries are dri v en by e xogenous o w with x ed v elocity and frequenc y Models are generated by considering the linearized dynamics associated with modes obtained via proper orthogonal decomposition [ 55 ]. These modes were used to deri v e controllers for disturbance rejection. The deri v ed controllers were able to k eep the o w nearly stationary at v arious points throughout the ca vity for v arying o w re gime despite the e xogenous input[ 56 57 ]. This project e xtends the w ork of Feng [ 54 ] to consider dif ferent o w conditions for the dri v en ca vity Specically the open-loop models are generated by considering 39 PAGE 48 40 the phase dif ferential between the disturbances at the top and bottom of the ca vity Feng' s study only used models of the o w resulting from upper and lo wer o w boundaries that were 180 o dif ferent in phase. This project will consider using models of the o w resulting from upper and lo wer o w boundary conditions that are 210 o 195 o 180 o 165 o and 150 o dif ferent in phase. A range of phase dif ferentials w as used to deri v e controllers that accommodate the unpredictable phase dif ference that w ould be associated with a real dri v en ca vity A linear parameter -v arying controller is designed for the models corresponding to the range of v arious phase dif ferentials. The group of models can be considered as lying within a parameter space with the parameter being the phase dif ferential between the e xcitation and disturbance. The open-loop simulations sho w a rise in the horizontal v elocity along the centerline as the phase dif ferential is increased. The closed-loop simulations sho w a distinct reduction in the centerline v elocity and therefore pro v e the ef fecti v eness of the LPV controller 5.2 Background The de v elopment of control algorithms has made much progress in the last fe w decades for aerospace applications. The control of uid o w ho we v er is one of the most dif cult applications and remains mostly unresolv ed. The primary dif culty lies with the inherent nonlinearity of the Na vier -Stok es equations which are the go v erning equations of the uid o w Ev en approximations of this equation can ha v e lar ge dimensionality W ithin the last decade studies of o w control ha v e addressed specic o w re gimes. These re gimes include the control of a dri v en ca vity using LQR control [ 56 ], using piezoceramic actuators to control a channel o w [ 58 ] and the use of synthetic jets for o w control [ 59 ]. In more recent years, much research has been done to deri v e methods to generate reduced-order o w models. T w o specic techniques that emplo y a reduced basis are proper orthogonal decomposition (POD) [ 60 ] and uid mode methods [ 53 ]. Proper PAGE 49 41 orthogonal decomposition is a model reduction technique in which the most ener getic modes are systematically e xtracted from numerical simulations. This method of reduction w as used to create the models used in this project. The uid mode method uses basis functions which are closely related to the physics of the problem being solv ed. Another area of interest for this project is what is kno wn as Stok es or creeping o w The limitations of using Stok es o w are that the o w must be incompressible and ha v e a Re ynolds number less than one. One side ef fect of lo wering a o w' s Re ynolds number is that the acceleration term within the Na vier -Stok es go v erning equation becomes small compared to the viscous force term. This change allo ws the equation to be simplied into the linear Stok es equation [ 61 62 ]. 5.3 Dri v en Ca vity Geometry This project will in v estigate o w control for the ca vity sho wn in Figure 5Â–1 where h 0tis the v elocity along the top of the ca vity btis the v elocity along the bottom of the ca vity and GG LnG RnG TnG B is the boundary of the domain. This ca vity is enclosed by rigid w alls with no-slip boundary conditions on the right and left sides. The top and bottom, ho we v er ha v e nonzero boundary conditions in general. Figure 5Â–1: Stok es Dri v en Ca vity Flo w Problem The o w at the top and bottom boundaries ha v e uniform spatial distrib ution. This restriction implies that the o w at an y point along the upper boundary is identical PAGE 50 42 to the o w at an y other point along the upper boundary Similarly the o w at an y point along the bottom boundary is identical to the o w at an y other point along the bottom boundary Such a perfect distrib ution is not possible because of the singularity at the points on the corners where the o w is mo ving on the horizontal boundary b ut stationary on the v ertical boundary Such a situation is ob viously an approximation, b ut this e xample does serv e as an initial problem to demonstrate the methodology The approximation within the 2-D ca vity is based on a grid with an inde x of 21x21 points. It is assumed that the measurements of the o w v elocity are tak en at 19 points along the horizontal centerline of the ca vity with the outer points lying one grid point a w ay from the closest boundary w all. These measurements only pro vide the horizontal v elocity of the o w Also, the sensors generating these measurements are assumed to e xist within the ca vity without altering the o w Again, such a situation is ob viously an approximation, b ut the e xample serv es to demonstrate the methodology 5.4 Go v erning Equations of Motion Consider rst the unsteady Na vier -Stok es equations r oV troVpoV:+ p DoV (5.1) subject to boundary conditions described in the past section. The parameteroV is the v elocity eld, p is the pressure, r is the density and is the viscosity of the uid. The constants that will be used to nondimensionalize the problem include a characteristic dimension L, characteristic v elocity V s and characteristic PAGE 51 43 frequenc y f Non-dimensional v ariables are dened as xq)x PAGE 52 44 while k eeping in mind that these are nondimensional v ariables. 5.5 Reduced-Order Linear Dynamics F or the geometry depicted in Figure 5Â–1 dene the o w domain as W The boundary of the uid o w domain is gi v en by WG TnG BnG LnG R (5.13) It is assumed that the input v elocity imparted by the mo ving w all can be represented in term of N c functionsog ii1rrn c Each of the functionsog i are dened on the entire domain W b ut are assumed to e xhibit specic properties on the boundary It is required thatog ixWxy n 1 f or x-G B 0 f or x-WzG B (5.14) Additionally it is required that{Wog ipÂˆ n d S0 (5.15) for i1rN c This last condition is required to guarantee compatibility of the o w eld with the continuity equation. The dynamics for incompressible, tw o-dimensional creeping o ws can be represented in the strong form of Stok es equation by re writing the Equation 5.12 oV t+qDoV p of (5.16) Lik e wise, the strong form of the continuity equation for incompressible o w is p oV0 (5.17) In these equations,oV is the o w v elocity p is the pressure andof is the body force. It is assumed that spatially-v arying functionsof i for i1rrN s ha v e been determined from the POD procedure which will be discussed in detail later The functionsof i constitute the reduced basis used to represent the N s states in the control model, PAGE 53 45 which include phase dif ferential information. There is a dif ferent set ofof i functions associated with each parameter within the phase dif ferential subspace d =[150,210], which is used to create reduced-order models at specic phase dif ferentails. Similarly we assume that spatially-v arying functionsog i for i1rrN c ha v e lik e wise been deri v ed from a simulation or e xperiment. The functionsog i comprise the inuence functions that determine the controls acting on the uid o w It is assumed that these functions satisfy the follo wing conditions which are con v entional in man y reduced basis formulations : p of i0 for i1rrN s (5.18) p og i0 for i1rN c (5.19)of ixW0 for i1rrN s (5.20)og i0 for x< -G B (5.21) In terms of these reduced basis functions, the v elocity appearing in the Stok es equations is assumed to tak e the formoVxyt oV Mt|N C i1og ixyb it}N S i1of ixya it(5.22) T o deri v e a reduced-order model appropriate for control synthesis, it is necessary to con v ert the strong form of the go v erning equations to weak form. The inner product, bilinear form a p p~are dened, respecti v ely as ou ov 4L 2WÂ€ 62 D2 i1{W u i v i d x (5.23) a ou ovDq 2 i1 2 j1{W u i x k v i x k d x (5.24) By substituting the v elocity into the strong form of the go v erning equations, taking the inner product of the resulting e xpression with an arbitrary basis functionof i and PAGE 54 46 inte grating o v er the domain W we obtainÂN C i1og ixy b it}N S i1of ixy a it of jÂ‚ 4L 2WÂ€ 62aÂ oV MN C i1og ixyb itÂƒN C i1of ixya it of jÂ‚ Â„r of of jt 4L 2WÂ€ 62 (5.25) The terms are gathered and put the equations in a canonical form appropriate for control synthesis. N C i1 og i of j 4L 2WÂ€ 62 b it}N S i1 of i of j 4L 2WÂ€ 62 a it a oV M of jN C i1 a og i of jb itrN S i1 a of i of ja it r of of jt 4L 2WÂ€ 62 (5.26) A ne w set of states that include both the original reduced state v ariables and the amplitude of the controls is dened as XtÂ… n atbtBÂ† Â‡ Âˆ(5.27) The ne w set of controls is dened to be the time deri v ati v e of the original controls utÂŠÂ‰ btÂŒÂ‹(5.28) W ith these denitions of the state Xtand controls ut, it is possible to write the weak form of the go v erning equations asÂ MÂŽ Xt Â AÂŽXt| Â BÂŽut Â CtÂÂŽ(5.29) It is important to note that the matrix4 M6will be diagonal if the reduced basis v ectors are deri v ed from a proper orthogonal decomposition. The nal form of the reduced PAGE 55 47 state equations can be obtained by premultiplying by the in v erse of the matrix4 M6 Xt Â MÂŽÂŒÂ1Â AÂŽXt Â MÂŽsÂ1 QXtr Â MÂŽÂ‘Â1Â BÂŽut(5.30) which can be re written in the nal form as Xt94AqÂ6Xt|P4B6ut|Â’4EqtÂ6(5.31) where4AqÂ6is a linear function of q ,4B6is a constant matrix, and4EqtÂ6is a function of h 0t. 5.6 Creeping Flo w in a Dri v en Ca vity Since the dynamical model is linear we can constructoV MtasoV MtÂ“oV 0ph 0t(5.32) whereoV 0 is the v elocity eld of creeping o w when constant horizontal v elocity is imposed on the top of the ca vity It is required thatoV 0xWxy n 1 f or x-G T 0 f or x-WzG T (5.33) Also, we choose N c1 andog as the v elocity eld that corresponds to a constant horizontal v elocity imposed on the bottom w all of the ca vity It is required thatoV 0xWxy n 1 f or x-G B 0 f or x-WzG B (5.34) The v elocity can subsequently be decomposed asoVxytÂ” oV 0 h 0t| ogxybtN S i1of ixya it(5.35) PAGE 56 48 Thus, the reduced-order model in Equation 5.31 can be re written e xplicitly as n a bÂ† Â‡ Âˆ 4A 1qÂ6Â•4A 2qÂ60 0 n a bÂ† Â‡ Âˆ 4E 1qÂ6Â•4E 260 0 n h 0 h 0Â† Â‡ Âˆ 4B 161 b (5.36) Consider just the partition associated with the time-domain equation for a aA 1qat|A 2qbt|B 1 bt E 1qh 0tE 2 h 0t(5.37) F ormulate the equi v alent frequenc y-domain e xpression s asÂ–A 1qasA 2qbs|sB 1 bs E 1qhs|sE 2 hs(5.38) PAGE 57 49 A transfer function representation of the inputs to outputs can no w be solv ed. asVsE 2E 1q s+A 1qh 0ssB 1A 2q s+A 1qbs A 1qE 1q|A 1qE 2 I E 2 hs A 1 A 2A 1 B 1 I B 1 bs A 1qE 1qA 2qI E 2 B 1 n hsbs Â† Â‡ Âˆ Ps n hsbs Â† Â‡ Âˆ(5.39) The reduced-order model described by P is a f airly accurate representation of the open-loop dynamics for Stok es o w in the dri v en ca vity with qwO1. All of the models will be based on a q1 in this project. There is a decrease in accurac y as q changes from unity because of unmodeled nonlinearities. The linear parameter v arying controller is designed and tested for models o v er the range of phase dif ferentials dÂ“41502106. 5.7 Excitation Phase Dif ferential Open-loop models of the o w dynamics are generated by analyzing simulated responses of the dri v en ca vity This simulation uses the Stok es o w as the uid dynamics. Separate models are generated for each set of o w conditions that corresponded to dif ferent relationships between the upper and lo wer boundaries. Each of these models ha v e physical limitations associated with them. The Re ynolds number for each model is Re=0.1. The combination of the lo w Re ynolds number and the approximation of the Na vier -Stok es equation creates a o w which is dominated by viscous ef fects. PAGE 58 50 The uid at the upper and lo wer boundaries w as constrained to mo v e at a sinusoidal frequenc y with constant amplitude. The frequenc y of this sinusoidal is essentially considered a non-dimensional unity because the time scales in the dynamics are all normalized. This e xcitation acts as a disturbance that af fects the entire o w within the ca vity The v ariations between the o w conditions used for model de v elopment were phase relationships between the upper and lo wer boundaries. Specically the sinusoidal o ws dif fered in phase by 210 o 195 o 180 o 165 o and 150 o between the upper and lo wer boundaries. These phase dif ferences induced dif ferent modal structures within the o w so the v arious models were generated to represent a basis for this range of o w conditions. The reduced-order models associated with each phase dif ferential, which contain three states, can actually be considered as subspaces of the full-order o w which contains 623 states. The e xogenous disturbances will, in general, not ha v e a constant phase dif ferential so the o w eld will contain modes associated with man y phase differentials throughout an y time e v olution. As such, each reduced-order model represents a subspace of the modes observ ed in that full-order o w eld. Thus, synthesizing controllers for these models with phase dif ferentials is essentially designing compensators that are optimal with respect to each subspace. 5.8 Control Design 5.8.1 Control Objecti v es The objecti v e of o w control in this project is to reject the ef fects of the e xogenous disturbance at the top of the dri v en ca vity Physically the control seeks to minimize the horizontal component of the uid v elocity at a set of sensor locations. These sensors are located at 19 sites e v enly distrib uted along the horizontal centerline of the ca vity The end sensors are located 1 grid point a w ay from the boundary w all. PAGE 59 51 The controller is designed for disturbance rejection using the model sho wn in Figure 5Â–2 This model contains the open-loop dynamics as described byV mFYP. The weighting functions used for loop shaping are gi v en asW pW nW kW yW hW uW d. The mathematical objecti v e of the control design is to choose a K such that the closed-loop transfer function from disturbances to errors has an induced norm less than unity for all plants within the parameter space. u R & W k R W a R D a '+ & &e 2 W u b &h & W h & Pd & Y R & F a &+ & W p &e 1 & V m S R W y R+ W n &n & &y Figure 5Â–2: Controller Block Diagram The system has 3 input v ectors and 3 output v ectors. The inputs are the random noise n-R 19 af fecting the sensor measurements, the e xogenous disturbance h-R af fecting the upper boundary of the ca vity and the control input u-R af fecting the lo wer boundary of the ca vity The outputs are the (frequenc y domain) weighted horizontal v elocity measurements e 1-R 19 the (frequenc y domain) weighted control ef fort e 2-R and the sensor measurements y-R 19 used for feedback to the controller The open-loop dynamics consist of the ca vity' s v elocity at the top, a control basis and o w shape basis. The part of the open-loop dynamics kno wn as V m is the mean v elocity along the top of the ca vity and has the equation V mh ot oV o where h o is the initial top disturbance and V o is a static Stok es o w along the top. The control basis for the plant tak es the form Ybt, which w as depicted asogxybtin the equations presented in the pre vious sections, and coincides with the controlled v elocity along the PAGE 60 52 bottom of the ca vity The shape of the uid o w within the ca vity is determined by the o w shape basis F The o w shape can tak e se v eral forms; for e xample, the o w could be one lar ge v orte x circling the entire ca vity or tw o v ortices of equal size with opposite rotation that meet along the horizontal center line of the ca vity The lter gi v en as W p serv es to normalize the measurement of o w v elocity collected by the sensors. This lter achie v es loop-shaping that denes the design specications in the frequenc y domain. The lter W p is chosen to reect the in v erse of acceptable v elocities in response to unity disturbances. The actual lter is realized as W p100 which implies the o w v elocities should be less than 0.01 in the closed-loop system. The lter W k is used to normalize the penalty placed on the amount of actuation commanded by the controller This lter reects the capabilities of the actuation system. The magnitude of motion for the control actuator is allo wed to be 02 in the non-dimensional system. The weighting is chosen as the in v erse so that W k5. The creation of a linear parameter -v arying controller requires that an output lter be used because a certain part of the observ ation matrix and feedthrough matrix v ary with the dif ferent plants within the parameter space. The lter W y is used as the output lter for this synthesis. The lter w as chosen as W y1 E 5 sT1 E 5 Also, the input matrix v aries among the dif ferent plant models within the parameter space. This v ariation necessitates input lters within the synthesis. The lters sho wn as W u and W h are used as input lters and ha v e the v alue of 1 E 5 sT1 E 5 Multiplicati v e uncertainty w as introduced into the control actuation in order to increase the system' s rob ustness with respect to the unmodeled dynamics and disturbances. The uncertainty is represented in the synthesis as D a such that/D a/1. A weighting w as used to limit the amount of uncertainty allo wed into the control actuation and w as chosen to be W a02, which allo wsQ20% uncertainty in the input u PAGE 61 53 Also, the lter W n is included to account for noise that corrupts measurements by the sensors. The inclusion of noise is needed to pro vide a minimal le v el penalty on the sensors. The design did not w ant to consider a lar ge amount of noise so the lter w as chosen as W n0001. 5.8.2 Synthesis Rob ust controllers and a linear parameter -v arying controller ha v e been designed for the system in Figure 5Â–2 The techniques of H control are used to reduce the induced norm from e xogenous inputs to weighted errors [ 63 ]. The softw are from the Analysis and Synthesis ToolboxforMatlabis used for the actual computation of the controller [ 34 ]. Separate controllers are synthesized for each of the open-loop models based on e xcitation phase dif ferentials. F or each model, the same weightings are used to reect the desire of achie ving the same performance le v el for each phase dif ferential. The resulting induced norms achie v ed by the controllers are sho wn in T able 5Â–1 T able 5Â–1: Induced Norms of Closed-Loop System Phase Dif ferential H norm 150 o 7.475 165 o 8.177 180 o 7.647 195 o 10.143 210 o 10.829 150 o+210 o 12.785 The closed-loop norms are all greater than unity Intuiti v ely these magnitudes imply the controller is not able to achie v e the desired performance and rob ustness objecti v es. Realistically it must be k ept in mind that there are twenty inputs and twenty outputs creating a lar ge number of transfer functions. This f act suggests that the magnitude of the norms is not unreasonable. The resulting closed-loop properties are studied in more detail shortly It is sho wn that the lar ge norms are caused by e xcessi v e control actuation. Essentially the controller is not able to achie v e the PAGE 62 54 desired disturbance attenuation without e xceeding the actuation limits. F ortunately this violation is at lo w frequencies and is not e xpected to ha v e a dramatic impact on the closed-loop simulations. Also, the v alues in T able 5Â–1 are interesting in the sense that the induced norms increase as the le v el of phase dif ferential increases. Such beha vior indicates that the e xcitation phase dif ferential does indeed ha v e a lar ge impact on the uid dynamics. The increasingly poor performance of the controllers demonstrates that the o w modes for a phase dif ferential of 210 o ha v e properties that are more dif cult to control than those for a phase dif ferential of 150 o for e xample. The last entry in T able 5Â–1 is the norm associated with the LPV controller Allo wing the phase dif ferential to be time-v arying increases the norm as e xpected. What is important to note is that this norm did not raise much abo v e the norm associated with the H controller for the 210 o phase dif ferential model. This condition indicates that the LPV controller is able account for the time-v arying nature of the phase dif ferential without e xcessi v e loss of performance. 5.9 Simulation 5.9.1 Open-Loop Simulation A series of open-loop simulations are performed to demonstrate the uid qualitati v e response resulting from the disturbance for both full-order and reduced-order models. These simulations are similar in the sense that the same magnitude of disturbance is used for the boundary conditions on the top of the domain. Con v ersely the simulations in v olving the reduced-order models dif fer in that the o w on the bottom boundary has dif ferent v alues of phase lag with respect to the o w on the top boundary A series of plots will be sho wn to visualize the o w conditions. In each, the v alue of horizontal v elocity will be sho wn as a function of time. The plots are 3-dimensional because the v elocity measured at each of the 19 sensors is sho wn as a function of time. PAGE 63 55 Again, it is important to note that all measurements are non-dimensional. This characteristic applies to both the time and v elocity component so no units are noted for the simulations. The open-loop o w for the full-order model is used as a comparison for the reduced-order model simulations (Figure 5Â–3 ). This plot clearly sho ws the sinusoidal nature of the o w that results from the top e xogenous disturbance changing with the sine function, h osin2 p t. The o w near the center of the ca vity near point 11, sho ws the lar gest v elocity with a magnitude near -0.2 at t03 to +0.2 at t07. 0 0.5 1 5 10 15 20 -0.2 0 0.2 Time Point Index Velocity Figure 5Â–3: Open-Loop Flo w V elocities for Full-Order Model The o w for the reduced-order model with a phase dif ferential of 165 o is sho wn in Figure 5Â–4 This plot also demonstrates a sinusoidal nature, b ut has a smaller open-loop magnitude compared to the full-order o w with the highest v elocity being 0.07. The o w for the reduced-order model with an phase dif ferential of 210 o is sho wn in Figure 5Â–5 The o w again demonstrates a sinusoidal nature and the v elocities are slightly lar ger than those of the full-order model. A sinusoidal trajectory of phase dif ferentials sho wn in Figure 5Â–6 is used in a simulation which sho ws the open-loop characteristics of the reduced-order o w as phase dif ferential changes. PAGE 64 56 0 0.5 1 5 10 15 20 -0.2 0 0.2 Time Point Index Velocity Figure 5Â–4: Open-Loop Flo w V elocities for Reduced-Order Model with 165 o Phase Dif ferential 0 0.5 1 5 10 15 20 -0.2 0 0.2 Time Point Index Velocity Figure 5Â–5: Open-Loop Flo w V elocities for Reduced-Order Model with 210 o Phase Dif ferential 0 0.2 0.4 0.6 0.8 1 150 160 170 180 190 200 210 TimePhase Differential Figure 5Â–6: T rajectory of Phase Dif ferential PAGE 65 57 The o w v elocities for the reduced-order model throughout the time-v arying phase trajectory are sho wn in Figure 5Â–7 The sinusoidal nature that is apparent in all of the other open-loop o ws is slightly dif ferent for this open-loop o w This dif ference is due to the changing of the parameter through the trajectory The full-order o w does not ha v e a dependence on phase dif ferential, therefore, the v elocities for the full-order model' s o w o v er the phase dif ferential trajectory are the same as those plotted in Figure 5Â–3 0 0.5 1 5 10 15 20 -0.2 0 0.2 Time Point Index Velocity Figure 5Â–7: Open-Loop Flo w V elocities for Reduced-Order Model o v er a T rajectory of Phase Dif ferentials An interesting feature to note is that the o w for each reduced-order model with phase dif ferential has a similar shape b ut signicantly dif ferent magnitude. The maximum v elocity measured at the center of the ca vity is smaller in magnitude for the models with phase dif ferentials located at the be ginning of the range than the models with phase dif ferentials near the end of the range. This feature indicates the o w is indeed strongly dependent on phase dif ferential and should be considered for control design. 5.9.2 Reduced-Order Closed-Loop Simulation The closed-loop dynamics are also simulated to demonstrate the performance of the controller for the reduced-order models, in this section, and the full-order model, in the ne xt section. The diagram of the closed-loop system for both the reduced-order PAGE 66 58 models and the full-order model can be seen in Figure 5Â–8 These simulations use the same open-loop dynamics b ut include the linear parameter -v arying controller that w as synthesized o v er the range of phase dif ferentials, which contains 24 states. In each simulation, the o w on the upper boundary is the same, b ut no w the o w on the lo wer boundary results only from the commands issued by the controller In this section, the controller w as tested with reduced-order models for tw o specic cases of phase dif ferential and o v er a time-v arying trajectory of phase dif ferentials. K &h &b Pd & Y R & F a & & V m SFigure 5Â–8: Closed-loop System The measured v elocities for the reduced-order model with a phase dif ferential of 165 o in response to the LPV controller with a phase dif ferential of 165 o is sho wn in Figure 5Â–9 The comparison of these v elocities with the open-loop measurements in Figure 5Â–4 demonstrate a reduction of v elocity along the center of the ca vity where the v elocity is greatest, of roughly 70%. The measured v elocities for the reduced-order model with a phase dif ferential of 210 o in response to the LPV controller with a phase dif ferential of 210 o is sho wn in Figure 5Â–10 The reduction in v elocities is apparent by comparing the closed-loop v elocities in Figure 5Â–10 with the open-loop v elocities in Figure 5Â–5 which sho ws a reduction along the center of the ca vity of roughly 90%. The closed-loop simulation of the reduced-order models o v er the phase dif ferential trajectory whcih also ef fects the controller is sho wn in Figure 5Â–11 The v elocity magnitude sho ws a clear reduction in magnitude compared to the open-loop simulation PAGE 67 59 0 0.5 1 5 10 15 20 -0.2 0 0.2 Time Point Index Velocity Figure 5Â–9: Closed-Loop Flo w V elocities for Reduced-Order Model with 165 o Phase Dif ferential 0 0.5 1 5 10 15 20 -0.2 0 0.2 Time Point Index Velocity Figure 5Â–10: Closed-Loop Flo w V elocities for Reduced-Order Model with 210 o Phase Dif ferential of o w o v er the phase dif ferential trajectory which w as sho wn in Figure 5Â–7 The reduction along the center is roughly 80%. The disturbance rejection is signicant for the LPV controller with the reducedorder models. These reductions conrm that the LPV controller will w ork not only for reduced-order models at specic phase dif ferentials b ut also o v er a time-v arying trajectory of phase dif ferentials. The simulations did sho w some dif ferences between each of the reduced-order models. In particular the amount of attenuation w as slightly less for the reduced-order model with a phase dif ferential of 165 o b ut much higher for the reduced-order model with a phase dif ferential of 210 o This decrease in attenuation PAGE 68 60 0 0.5 1 5 10 15 20 -0.2 0 0.2 Time Point Index Velocity Figure 5Â–11: Closed-Loop Flo w V elocities for Reduced-Order Model o v er a T rajectory of Phase Dif ferentials seems almost contradictory considering that the open-loop simulations sho wed a decrease in o w v elocities for the same models. 5.9.3 Full-Order Closed-Loop Simulation The simulations that were performed for the reduced-order models were repeated using the full-order model. The reduced-order models are subspaces of this full-order model so the performance of the controllers on the full-order model is actually of predominant interest. The measured v elocities in response to an H controller created specically for the full-order model, are sho wn in Figure 5Â–12 Clearly the magnitude of the v elocity is dramatically decreased belo w the open-loop le v el. The v elocities in Figure 5Â–12 are se v eral orders of magnitude less than the corresponding open-loop v elocities in Figure 5Â–3 This response will be used as a comparison for the responses from the full-order model controlled by the LPV controller The v elocities for the full-order model in response to the LPV controller with a phase dif ferential of 165 o is sho wn in Figure 5Â–13 Though the v elocities were not reduced to the e xtent of the full-order simulation in Figure 5Â–12 the y were reduced by an amount comparable to the response sho wn by the reduced-order model at a 165 o PAGE 69 61 0 0.5 1 5 10 15 20 -0.2 0 0.2 Time Point Index Velocity Figure 5Â–12: Closed-Loop Flo w V elocities for Full-Order Model phase dif ferential. The reduction in the v elocity magnitude is e vident along the center of the ca vity and is roughly 80%. 0 0.5 1 5 10 15 20 -0.2 0 0.2 Time Point Index Velocity Figure 5Â–13: Closed-Loop Flo w V elocities for Full-Order Model with Controller Associated with 165 o Phase Dif ferential The v elocity magnitudes for the full-order model in response to the LPV controller for a phase dif ferential of 210 o is sho wn in Figure 5Â–14 The reduction in v elocity compared to the open-loop o w of the full-order o w in Figure 5Â–3 is v ery clear The v elocities along the centerline of the ca vity were reduced by 60%. Though the v elocities were not as reduced as much as those in the simulation in Figure 5Â–12 the v elocities were reduced by an amount comparable to the reduced-order model at a 210 o phase dif ferential. PAGE 70 62 0 0.5 1 5 10 15 20 -0.2 0 0.2 Time Point Index Velocity Figure 5Â–14: Closed-Loop Flo w V elocities for Full-Order Model with Controller Associated with 210 o Phase Dif ferential The closed-loop simulation of the full-order model controlled by the LPV controller o v er the phase dif ferential trajectory is sho wn in Figure 5Â–15 The v elocities sho w a clear reduction in magnitude compared to the open-loop full-order o w in Figure 5Â–3 The o w along the center of the ca vity is reduced by 66%. 0 0.5 1 5 10 15 20 -0.2 0 0.2 Time Point Index Velocity Figure 5Â–15: Closed-Loop Flo w V elocities for Full-Order Model o v er a T rajectory of Phase Dif ferentials The disturbance rejection is signicant for both the reduced-order models and the full-order model. These reductions conrm that the LPV controller created for a phase dif ferential parameter will w ork not only for the reduced-order models, which are dependent on phase dif ferential, b ut also for the full-order model. PAGE 71 63 5.10 Conclusion Flo w control is an e xceedingly dif cult challenge because of the nonlinearities and time v ariations inherent to o w elds. These inherent dif culties can be a v oided when restricting the o w to creeping Stok es o w within a dri v en ca vity This project has introduced a control methodology suitable for such a system. In particular the controllers are designed by considering subspaces of the o w eld that describe modes associated with phase dif ferential between e xogenous disturbances. The models of these subspaces are realized as state-space systems and a controller can be designed using the linear parameter -v arying frame w ork. The resulting controller is sho wn to signicantly decrease the o w v elocities within the ca vity for both the reduced-order subspaces and also the full-order o w PAGE 72 CHAPTER 6 CONCLUSION Practically all mechanical systems that in v olv e motion need to be controlled with a gain-scheduling technique. Aerospace systems in particular ha v e the possibility to ha v e v ery e xtensi v e operating domains. Three specic aerospace systems were discussed in this paper the longitudinal dynamics of an F/A-18, the structural dynamics of a hypersonic v ehicle and the o w dynamics of a dri v en ca vity The parameters that depicted the operating domain of the F/A-18 problem were altitude and Mach number The parameter that depicted the operating domain of the structure of the hypersonic v ehicle w as temperature and the operating domain of the dri v en ca vity w as depicted by the phase dif ferential within the uid. This paper has introduced a gain-scheduled control methodology which uses H synthesis to create a linear parameter -v arying controller that is suitable for such systems. The LPV controller created for the F/A-18 longitudingal dynamics pro v ed to induce a pitch rate for the aircraft that w as similar to a designated tar get pitch rate. The LPV controller created for the structural dynamics of a hypersonic aircraft successfully damped out the vibrations induced by a temperature change. The LPV controller for the uid dynamics within a dri v en ca vity signicantly decreased the horizontal component of the o w v elocities along the centerline of the ca vity for both the reduced-order subspaces and the full order o w The results of the control methodology to create procient controllers for three v ery dif ferent aerospace applications leads to the conclusion that this methodology could be useful for other aerospace applications. 64 PAGE 73 REFERENCES [1] W Rugh and J. Shamma, Â“Research on Gain Scheduling, Â” A utomatica V ol. 36, Issue 10 John Hopkins Uni v ersity Baltimore and UCLA, Los Angeles, October 2000. [2] D.J. Leith and W .E. Leithead, Â“Surv e y of Gain-Scheduling Analysis and Design, Â” International J ournal of Contr ol V ol. 73, 2000, pp. 1001-1025. [3] B. Bamieh and L. 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PAGE 78 BIOGRAPHICAL SKETCH Kristin Fitzpatrick w as born in Blue Hill, Maine on March 26, 1980. Her f amily mo v ed to Florida in 1988 after the death of her mother She recei v ed her high school diploma from the Center for Adv anced T echnologies, a magnet program in St. Petersb ur g, Florida. She then attended the Uni v ersity of Florida and recei v ed a de gree in Aerospace Engineering with Honors in December 2002. She has w ork ed with the aerospace dynamics and control research group under the direction of Dr Rick Lind and Dr Andy K urdila and is projected to recei v e her Master of Science de gree in aerospace engineering in December 2003. She will stay at the Uni v ersity of Florida to pursue a doctorate in aerospace engineering with the focus in dynamics and control. 70 |