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NONLINEAR CONTROL OF LINEAR PARAMETER VARYING SYSTEMS WITH APPLICATIONS TO HYPERSONIC VEHICLES By ZACHARY DONALD WILCOX A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 2010 Zachary Donald Wilcox This work is dedicated to my parents, family, friends, and advisor, who have provided me with support during the challenging moments in this dissertation work. ACKNOWLEDGMENTS I would like to express sincere gratitude to my advisor, Dr. Warren E. Dixon, who is a person with remarkable affability. As an advisor, he provided the necessary guidance and allowed me to develop my own ideas. As a mentor, he helped me understand the intricacies of working in a professional environment and helped develop my professional skills. I feel fortunate in getting the opportunity to work with him. TABLE OF CONTENTS page ACKNOWLEDGMENTS ..................... ............ 4 LIST OF TABLES ...................... ............... 7 LIST OF FIGURES ..................... ............... 8 A BSTRACT . . .... .. 10 CHAPTER 1 INTRODUCTION ..................... ............ 12 1.1 Motivation and Problem Statement ..................... .. .. 12 1.2 Outline and Contributions ................... ....... 16 2 LYAPUNOVBASED EXPONENTIAL TRACKING CONTROL OF LPV SYS TEMS WITH AN UNKNOWN SYSTEM MATRIX, UNCERTAIN INPUT MA TRIX VIA DYNAMIC INVERSION ................... ..... 19 2.1 Introduction ...................... ............ 19 2.2 Linear Parameter Varying Model ..................... ... .. 21 2.3 Control Development .................... ......... 23 2.3.1 Control Objective .................... ........ 23 2.3.2 OpenLoop Error System ................... ..... 24 2.3.3 ClosedLoop Error System ................. ....... 25 2.4 Stability Analysis .................... ............ 27 2.5 Conclusions . ... 30 3 HYPERSONIC VEHICLE DYNAMICS AND TEMPERATURE MODEL ... 32 3.1 Introduction .................... .............. 32 3.2 Rigid Body and Elastic Dynamics ................. ....... 32 3.3 Temperature Profile Model ................... ....... 33 3.4 Conclusion ..................... ............... 38 4 LYAPUNOVBASED EXPONENTIAL TRACKING CONTROL OF A HY PERSONIC AIRCRAFT WITH AEROTHERMOELASTIC EFFECTS ..... 39 4.1 Introduction .................... .............. 39 4.2 HSV Model ..................... .............. 41 4.3 Control Objective ..................... .......... 42 4.4 Simulation Results .................... ........... 44 4.5 Conclusion .................................... 48 5 CONTROL PERFORMANCE VARIATION DUE TO NONLINEAR AEROTHER MOELASTICITY IN A HYPERSONIC VEHICLE: INSIGHTS FOR STRUC TURAL DESIGN .................... .............. 53 5.1 Introduction .................... .............. 53 5.2 Dynamics and Controller .................... ........ 54 5.3 Optimization via Random Search and Evolving Algorithms ........ 55 5.4 Example Case .................... .............. 57 5.5 Results ..................... ................. 61 5.6 Conclusion ...................... .............. 73 6 CONCLUSIONS AND FUTURE WORK ......................... 75 6.1 Conclusions ..................... .............. 75 6.2 Contributions .................... .............. 76 6.3 Future W ork ..................... ............. 77 APPENDIX A OPTIMIZATION DATA .................... .......... 79 REFERENCES ..................... ................. 89 BIOGRAPHICAL SKETCH .................... ............ 94 LIST OF TABLES Table page 31 Natural frequencies for 5 linear temperature profiles (Nose/Tail) in degrees F. Percent difference is the difference between the maximum and minimum fre quencies divided by the minimum frequency. ..... 36 51 Optimization Control Gain Search Statistics ..... 73 A1 Total cost function, used to generate Figure 511 and 512 (Part 1) ...... ..80 A2 Total cost function, used to generate Figure 511 and 512 (Part 2) ...... ..80 A3 Control input cost function, used to generate Figure 57 and 58 (Part 1) 81 A4 Control input cost function, used to generate Figure 57 and 58 (Part 2) 81 A5 Error cost function, used to generate Figure 59 and 510 (Part 1) ... 82 A6 Error cost function, used to generate Figure 59 and 510 (Part 2) ... 82 A7 Pitch rate, peaktopeak error, used to generate Figure 513 and 514 (Part 1) .83 A8 Pitch rate, peaktopeak error, used to generate Figure 513 and 514 (Part 2) .83 A9 Pitch rate, steadystate peaktopeak error, used to generate Figure 521 and 5 22 (Part 1) ....................................... 84 A10 Pitch rate, steadystate peaktopeak error, used to generate Figure 521 and 5 22 (Part 2) .... ................ ................ 84 A11 Pitch rate, time to steadystate, used to generate Figure 517 and 518 (Part 1) 85 A12 Pitch rate, time to steadystate, used to generate Figure 517 and 518 (Part 2) 85 A13 Velocity, peaktopeak error, used to generate Figure 515 and 516 (Part 1) .86 A14 Velocity, peaktopeak error, used to generate Figure 515 and 516 (Part 2) .86 A15 Velocity, steadystate peaktopeak, used to generate Figure 523 and 524 (Part 1) ......................................... 87 A16 Velocity, steadystate peaktopeak, used to generate Figure 523 and 524 (Part 2 ) . . 8 7 A17 Velocity, time to steadystate, used to generate Figure 519 and 520 (Part 1) 88 A18 Velocity, time to steadystate, used to generate Figure 519 and 520 (Part 2) 88 LIST OF FIGURES Figure page 31 Modulus of elasticity for the first three dynamic modes of vibration for a free free beam of titanium ................... ............... 34 32 Frequencies of vibration for the first three dynamic modes of a freefree tita nium beam .................... ................. .. 35 33 Nine constant temperature sections of the HSV used for temperature profile modeling. ......................................... 35 34 Linear temperature profiles used to calculate values shown in Table 31. 37 35 Asymetric mode shapes for the hypersonic vehicle. The percent difference was calculated based on the maximum minus the minimum structural frequencies divided by the minimum.. ............................ 37 41 Temperature variation for the forebody and aftbody of the hypersonic vehicle as a function of time ................... ............. 45 42 In this figure, fi denotes the ith element in the disturbance vecor f. Disturbances from top to bottom: velocity fy, angle of attack fa, pitch rate fQ, the 1st elas tic structural mode i)1, the 2nd elastic structural mode i)2, and the 3rd elastic structural mode i)3, as described in (411). ..... .. 46 43 Reference model ouputs y,, which are the desired trajectories for top: velocity Vm (t), middle: angle of attack a, (t), and bottom: pitch rate Qm (t). 47 44 Top: velocity V (t), bottom: velocity tracking error ev (t). .. 48 45 Top: angle of attack a (t), bottom: angle of attack tracking error e (t). 49 46 Top: pitch rate Q (t), bottom: pitch rate tracking error CQ (t) .. 49 47 Top: fuel equivalence ratio f. Middle: elevator deflection 6,. Bottom: Canard deflection 6. .. ............... ..................... 50 48 Top: altitude h (t), bottom: pitch angle 0 (t) . 50 49 Top: 1st structural elastic mode p7. Middle: 2nd structural elastic mode rl2. Bot tom: 3rd structural elastic mode r3. . .. 51 51 HSV surface temperature profiles. T0os, E [450F, 900F], and Ttail E [100F, 800F]. 54 52 Desired trajectories: pitch rate Q (top) and velocity V (bottom). ... 58 53 Disturbances for velocity V (top), angle of attack a (second from top), pitch rate Q (second from bottom) and the 1st structural mode (bottom). ...... ..58 54 Tracking errors for the pitch rate Q in degrees/sec (top) and the velocity V in ft/sec (bottom)....................... .............. 59 55 Control inputs for the elevator 6, in degrees (top) and the fuel ratio Of (bottom). 60 56 Cost function values for the total cost Qtot (top), the input cost ,co, (middle) and the error cost err, (bottom). ......................... .. 60 57 Control cost function ,co, data as a function of tail and nose temperature pro files. ...................................... ... .. 62 58 Control cost function Qco, data (filtered) as a function of tail and nose temper ature profiles .................... ................ .. 62 59 Error cost function Q er data as a function of tail and nose temperature profiles. 63 510 Error cost function Q er, data (filtered) as a function of tail and nose tempera ture profiles .. . . ..... 63 511 Total cost function Qtot data as a function of tail and nose temperature profiles. 64 512 Total cost function Qtot data (filtered) as a function of tail and nose tempera ture profiles .. . . ..... 65 513 Peaktopeak transient error for the pitch rate Q (t) tracking error in deg./sec.. 66 514 Peaktopeak transient error (filtered) for the pitch rate Q (t) tracking error in deg./sec.. ..................... .................. 66 515 Peaktopeak transient error for the velocity V (t) tracking error in ft/sec.. ... 67 516 Peaktopeak transient error (filtered) for the velocity V (t) tracking error in ft./sec.. .................... ................ 67 517 Time to steadystate for the pitch rate Q (t) tracking error in seconds. ....... 68 518 Time to steadystate (filtered) for the pitch rate Q (t) tracking error in seconds. 68 519 Time to steadystate for the velocity V (t) tracking error in seconds. ...... ..69 520 Time to steadystate (filtered) for the velocity V (t) tracking error in seconds. .69 521 Steadystate peaktopeak error for the pitch rate Q (t) in deg./sec.. ...... 70 522 Steadystate peaktopeak error (filtered) for the pitch rate Q (t) in deg./sec.. 71 523 Steadystate peaktopeak error for the velocity V (t) in ft./sec.. ... 71 524 Steadystate peaktopeak error (filtered) for the velocity V (t) in ft./sec. 72 525 Combined optimization p chart of the control and error costs, transient and steadystate values.... ................ ............. .. 73 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NONLINEAR CONTROL OF LINEAR PARAMETER VARYING SYSTEMS WITH APPLICATIONS TO HYPERSONIC VEHICLES By Zachary Donald Wilcox August 2010 Chair: Warren E. Dixon Major: Aerospace Engineering The focus of this dissertation is to design a controller for linear parameter varying (LPV) systems, apply it specifically to airbreathing hypersonic vehicles, and examine the interplay between control performance and the structural dynamics design. Specifically a Lyapunovbased continuous robust controller is developed that yields exponential tracking of a reference model, despite the presence of bounded, nonvanishing disturbances. The hypersonic vehicle has time varying parameters, specifically temperature profiles, and its dynamics can be reduced to an LPV system with additive disturbances. Since the HSV can be modeled as an LPV system the proposed control design is directly applicable. The control performance is directly examined through simulations. A wide variety of applications exist that can be effectively modeled as LPV systems. In particular, flight systems have historically been modeled as LPV systems and associated control tools have been applied such as gainscheduling, linear matrix inequalities (LMIs), linear fractional transformations (LFT), and /types. However, as the type of flight environments and trajectories become more demanding, the traditional LPV controllers may no longer be sufficient. In particular, hypersonic flight vehicles (HSVs) present an inherently difficult problem because of the nonlinear aerothermoelastic coupling effects in the dynamics. HSV flight conditions produce temperature variations that can alter both the structural dynamics and flight dynamics. Starting with the full nonlinear dynamics, the aerothermoelastic effects are modeled by a temperature dependent, parameter varying statespace representation with added disturbances. The model includes an uncertain parameter varying state matrix, an uncertain parameter varying nonsquare (column deficient) input matrix, and an additive bounded disturbance. In this dissertation, a robust dynamic controller is formulated for a uncertain and disturbed LPV system. The developed controller is then applied to a HSV model, and a Lyapunov analysis is used to prove global exponential reference model tracking in the presence of uncertainty in the state and input matrices and exogenous disturbances. Simulations with a spectrum of gains and temperature profiles on the full nonlinear dynamic model of the HSV is used to illustrate the performance and robustness of the developed controller. In addition, this work considers how the performance of the developed controller varies over a wide variety of control gains and temperature profiles and are optimized with respect to different performance metrics. Specifically, various temperature profile models and related nonlinear temperature dependent disturbances are used to characterize the relative control performance and effort for each model. Examining such metrics as a function of temperature provides a potential inroad to examine the interplay between structural/thermal protection design and control development and has application for future HSV design and control implementation. CHAPTER 1 INTRODUCTION 1.1 Motivation and Problem Statement Recent research on nonlinear inversion of the input dynamics based on Lyapunov stability theory provides a stepping stone to LPV dynamic inversion. In [27, 28], dynamic inversion techniques are used to design controllers that can adaptively and robustly stabilize statespace systems with uncertain constant parameters and additive unknown bounded disturbances. However, this work is limited to timeinvarient parameters and therefore is not applicable to LPV systems. The work presented in this chapter is an extension of the work in [27, 28], and provides a continuous robust controller that is able to stabilize general perturbed LPV systems with disturbances, when both the state, input matrices, timevarying parameters, and disturbances are unknown. The design of guidance and control systems for airbreathing HSV is challeng ing because the dynamics of the HSV are complex and highly coupled as in [10], and temperatureinduced stiffness variations impact the structural dynamics such as in [21]. Much of this difficulty arises from the aerodynamic, thermodynamic, and elastic coupling (aerothermoelasticity) inherent in HSV systems. Because HSV travel at such high veloc ities (in excess of Mach 5) there are large amounts of aerothermal heating. Aerothermal heating is nonuniform, generally producing much higher temperatures at the stagnation point of airflow near the front of the vehicle. Coupled with additional heating due to the engine, HSVs have large thermal gradients between the nose and tail. The structural dynamics, in turn, affect the aerodynamic properties. Vibration in the forward fuselage changes the apparent turn angle of the flow, which results in changes in the pressure distribution over the forebody of the aircraft. The resulting changes in the pressure dis tribution over the aircraft manifest themselves as thrust, lift, drag, and pitching moment perturbations as in [10]. To develop control laws for the longitudinal dynamics of a HSV capable of compensating for these structural and aerothermoelastic effects, structural temperature variations and structural dynamics must be considered. Aerothermoelasticity is the response of elastic structures to aerodynamic heating and loading. Aerothermoelastic effects cannot be ignored in hypersonic flight, because such ef fects can destabilize the HSV system as in [21]. A loss of stiffness induced by aerodynamic heating has been shown to potentially induce dynamic instability in supersonic/hypersonic flight speed regimes as in [1]. Yet active control can be used to expand the flutter bound ary and convert unstable limit cycle oscillations (LCO) to stable LCO as shown in [1]. An active structural controller was developed in [26], which accounts for variations in the HSV structural properties resulting from aerothermoelastic effects. The control design in [26] models the structural dynamics using a LPV framework, and states the benefits to using the LPV framework are twofold: the dynamics can be represented as a single model, and controllers can be designed that have affine dependency on the operating parameters. Previous publications have examined the challenges associated with the control of HSVs. For example, HSV flight controllers are designed using genetic algorithms to search a design parameter space where the nonlinear longitudinal equations of motion contain uncertain parameters as in [4, 30, 49]. Some of these designs utilize Monte Carlo simulations to estimate system robustness at each search iteration. Another approach [4] is to use fuzzy logic to control the attitude of the HSV about a single low end flight condition. While such approaches as in [4, 30, 49] generate stabilizing controllers, the procedures are computationally demanding and require multiple evaluation simulations of the objective function and have large convergent times. An adaptive gainscheduled controller in [55] was designed using estimates of the scheduled parameters, and a semi optimal controller is developed to adaptively attain Ho control performance. This controller yields uniformly bounded stability due to the effects of approximation errors and algorithmic errors in the neural networks. Feedback linearization techniques have been applied to a controloriented HSV model to design a nonlinear controller as in [32]. The model in [32] is based on a previously developed HSV longitudinal dynamic model in [8]. The control design in [32] neglects variations in thrust lift parameters, altitude, and dynamic pressure. Linear output feedback tracking control methods have been developed in [44], where sensor placement strategies can be used to increase observability, or reconstruct full state information for a statefeedback controller. A robust output feedback technique is also developed for the linear parameterizable HSV model, which does not rely on state observation. A robust setpoint regulation controller in [17] is designed to yield asymptotic regulation in the presence of parametric and structural uncertainty in a linear parameterizable HSV system. An adaptive controller in [19] was designed to handle (linear in the parameters) modeling uncertainties, actuator failures, and nonminimum phase dynamics as in [17] for a HSV with elevator and fuel ratio inputs. Another adaptive approach in [41] was recently developed with the addition of a guidance law that maintains the fuel ratio within its choking limits. While adaptive control and guidance control strategies for a HSV are investigated in [17, 19, 41], neither addresses the case where dynamics include unknown and unmodeled disturbances. There remains a need for a continuous controller, which is capable of achieving exponential tracking for a HSV dynamic model containing aerothermoelastic effects and unmodeled disturbances (i.e., nonvanishing disturbances that do not satisfy the linear in the parameters assumption). In the context of the aforementioned literature, a contribution of this dissertation (and in the publications in [51] and [52]) is the development of a controller that achieves exponential model reference output tracking despite an uncertain model of the HSV that includes nonvanishing exogenous disturbances. A nonlinear temperaturedependent parametervarying statespace representation is used to capture the aerothermoelastic ef fects and unmodeled uncertainties in a HSV. This model includes an unknown parameter varying state matrix, an uncertain parametervarying nonsquare (column deficient) input matrix, and a nonlinear additive bounded disturbance. To achieve an exponential tracking result in light of these disturbances, a robust, continuous Lyapunovbased controller is developed that includes a novel implicit learning characteristic that compensates for the nonvanishing exogenous disturbance. That is, the use of the implicit learning method enables the first exponential tracking result by a continuous controller in the presence of the bounded nonvanishing exogenous disturbance. To illustrate the performance of the developed controller, simulations are performed on the full nonlinear model given in [10] that includes aerothermoelastic model uncertainties and nonlinear exogenous disturbances whose magnitude is based on airspeed fluctuations. In addition to the control development, there exists the need to understand the interplay of a control design with respect to the vehicle dynamics. A previous control oriented design analysis in [6] states that simultaneously optimizing both the structural dynamics and control is an intractable problem, but that controloriented design may be performed by considering the closedloop performance of an optimal controller on a series of different openloop design models. The best performing design model is then said to have the optimal dynamics in the sense of controllability. Knowledge of the optimal thermal gradients will provide insight to engineers on how to properly weight the HSV's thermal protection system for both steadystate and transient flight. The preliminary work by authors in [6] provides a controloriented design architecture by investigating control performance variations due to thermal gradients using an Hoo controller. Chapter 5 seeks to extend the control oriented design concept to examine control performance variations for HSV models that include nonlinear aerothermoelastic disturbances. Given these disturbances, Chapter 5 focuses on examining control performance variations for the model reference robust controller in Chapter 2 and Chapter 4 to achieve a nonlinear controloriented analysis with respect to thermal gradients on the HSV. By analyzing control error and input norms as well as transient and steadystate responses over a wide range of temperature profiles an optimal temperature profile range is suggested. 1.2 Outline and Contributions This dissertation focuses on designing a nonlinear controller for general disturbed LPV system. The controller is then modified for a specific airbreathing HSV. The dynamic inversion design is a technique that allows the multiplicative input matrices to be inverted, thus rendering the controller affine in the control. Previous results in [27] and [29] have examined full state and output feedback adaptive dynamic inversion controllers, but are limited because they contain constant uncertainties. The HSV system presents a new challenge because the uncertain state and input matrices are parameter varying. Specifically, the state and input matrices of the hypersonic vehicle vary as a function of temperature. This chapter provides some background and motivates the robust dynamic inversion control method subsequently developed. A brief outline of the following chapters follows. In Chapter 2 a tracking controller is presented that achieves exponential stability of a model reference system in the presence of uncertainties and disturbances. Specifically, the plant model contains timevarying parametric uncertainty with disturbances that are bounded and nonvanishing. The contribution of this result is that it represents the first ever development of an exponentially stable continuous robust model reference tracking controller for an LPV system with an unknown system matrix and uncertain input matrix with an additive unknown bounded disturbance. Lyapunov based methods are used to prove exponential stability of the system. Chapter 3 provides the nonlinear dynamics and temperature model of a HSV. The nonlinear and highly coupled dynamic equations are presented. The equations that define the aerodynamic and generalized moments and forces are provided explicitly in previous literature. This chapter is meant to serve as an overview of the dynamics of the HSV. In addition to the flight and structural dynamics, temperature profile modeling is provided. Temperature variations impact the HSV flight dynamics through changes in the structural dynamics which affect the mode shapes and natural frequencies of the vehicle. The presented model offers an approximate approach, whereby the natural frequencies of a continuous beam are described as a function of the mass distribution of a beam and its stiffness. Figures and tables are presented to emphasize the need to include such dynamics for control design. This chapter is designed to familiarize the reader with the HSV dynamic and temperature models, since these dynamics are used throughout this dissertation. This chapter is a precursor and introduction to Chapter 4 and Chapter 5. Using the controller developed in Chapter 2, the contribution in Chapter 4 is to illustrate an application to an airbreathing hypersonic vehicle system with additive bounded disturbances and aerothermoelastic effects, where the control input is multiplied by an uncertain, column deficient, parametervarying matrix. In addition to the stability proof, the control design is also validated through implementation in a full nonlinear dynamic simulation. The exogenous disturbances (e.g., wind gust, engine variations, etc.) and temperature profiles (aerodynamic driven thermal heating) are designed to examine the robustness of the developed controller. The results from the simulation illustrate the boundedness of the controller with favorable transient and steady state tracking errors and provide evidence that the control model used for development is valid. The contribution in Chapter 5 is to provide an analysis framework to examine the nonlinear control performance based on variations in the vehicle dynamics. Specifically, the changes occur in the structural dynamics via their response to different temperature profiles, and hence the observed vibration has different frequencies and shapes. Using an initial random search and evolving algorithms, approximate optimal gains are found for the controller for each temperature dependant plant model. Errors, control effort, transient and steadystate performance analysis is provided. The results from this analysis show that there is a temperature range for operation of the HSV that minimizes a given cost of performance versus control authority. Knowledge of a favorable range with regard to control performance provides designers an extra tool when developing the thermal protection system as well as the structural characteristics of the HSV. Chapter 6 summarizes the contributions of the dissertation and possible avenues for future work are provided. The brief contributions of the LPV controller, HSV example controller design application, and the HSV optimization procedure provide the base of this dissertation. After a brief summary, some of the drawbacks of the current control design are presented as directions for future research work. CHAPTER 2 LYAPUNOVBASED EXPONENTIAL TRACKING CONTROL OF LPV SYSTEMS WITH AN UNKNOWN SYSTEM MATRIX, UNCERTAIN INPUT MATRIX VIA DYNAMIC INVERSION 2.1 Introduction Linear parameter varying (LPV) systems have a wide range of practical engineering applications. Some examples include several missile autopilot designs as in [7, 39, 43], a turbofan engine [5], and active suspension design [18]. Traditionally, LPV systems have been developed using a gain scheduling control approach. Gain scheduling is a technique to develop controllers for nonlinear system using traditional linear control theory. Gain scheduling is a technique where the system is linearized about certain operating conditions. About these operating conditions, constant parameters are assumed and separate control schemes and gains are chosen. More than a decade ago, Shamma et. al. pointed out some of the potential hazards of gain scheduling in [42]. In particular, gain scheduling is a analytically noncontinuous method and stability is not guaranteed while switching from one region of linearization to another. In fact the two biggest downfalls of gain scheduling control design is the linearization of the plant models close to equilibrium or constant parameters states and the requirement that the parameters must change slowly. Because the linearization is required to be close to some operation condition or stability point, many different schedules have to be taken. And by requiring that parameters change slowly, the gain scheduling techniques are not appropriate for many quickly varying systems. Another approach to LPV problems is the use of linear matrix inequalities (LMIs). In a book on LMIs and their use in system and control theory in [11], Boyd et. al. states that LMIs are mathematically convex optimization problems with extensions to control theory. However in [11] it is pointed out that these typically require numerical solutions and there are only a few special cases where analytical solutions exist. These LPV solutions typically only provide norm based solutions. The most common of these is the L2norm because it allows for continuity with Ho, control when the systems become linear timeinvariant. For instance H, control is developed in [14] which uses LMIs to optimize the solution and in [3], the parameterization of LMIs was investigated in the context of control theory. H, control is developed in [14], which uses LMIs to optimize the solution and Saif et. al. in [48] shows that stabilization solutions exist for multiinputmultioutput (i\ li\ O) systems using LMIs. These designs allow for the continuous solution of LPV systems, however knowledge of the structure of the system must be known, and the parameters are assumed measurable online. In [25] minimax controllers are designed to handle only constant or small variations in the parameters, where the parameterized algebraic Riccati inequalities are converted into equivalent LMIs so that the convexity can be exploited and a controller developed. Continuous control design for uncertain LPV systems in [13] is designed using LMIs, however the procedure is limited to uncertainties in the state matrix, and does not cover uncertainties in the input matrix. Another approach uses linear fractional transformations LFTs in the context of LPV control design such as in [31] and are based on small gain theory. This approach cannot handle uncertain parameters. However, by extending the solution in [31] the design can include uncertain parameters which are not available to the controller. These solutions are /synthesis type controllers, however the solvability conditions are nonconvex and therefore a solution to the problem is not guaranteed even when a stable controller exists. Several examples of recursive /type solutions are given in [2, 22, 45]. More recently in [26], the /type solutions have been extended to a hypersonic aircraft example, but suffers the same nonconvexity problem as the formerly listed /type literature. Recent research on nonlinear inversion of the input dynamics based on Lyapunov stability theory provides a stepping stone to LPV dynamic inversion. In [27, 28], dynamic inversion techniques are used to design controllers that can adaptively and robustly stabilize a more general statespace system that has been considered in previous work with uncertain constant parameters and additive unknown bounded disturbances. However, this work is limited to timeinvarient parameters and therefore is not applicable to LPV systems. The work presented in this chapter is an extension of the work in [27, 28], and provides a continuous robust controller that is able to exponentially stabilize LPV systems with unknown bounded disturbances, when both the state, input matrices, timevarying parameters, and disturbances are unknown. 2.2 Linear Parameter Varying Model The dynamic model used for the subsequent control development is a combination of linearparametervarying (LPV) system with an added unmodeled disturbance as S=A(p(t))x+B(p(t))u+f(t) (21) y = Cx. (22) In (21) and (22), x (t) E R' is the state vector, A (p (t)) E IR"' denotes a linear parameter varying state matrix, B (p (t)) E R x denotes a linear parameter varying input matrix, C IRqxn denotes a known output matrix, u(t) E RP denotes control vector, p (t) represents the unknown timedependent parameters, f(t) E IR' represents a timedependent unknown, nonlinear disturbance, and y (t) E IRq represents the measured output vector. The subsequent control development is based on the assumption that p > q, meaning that at least one control input is available for each output state. When the system is overactuated in that there are more control inputs available than output states, then p > q and the resulting input dynamic inversion matrix will be row deficient. For this case, a right pseudoinverse can be used in conjunction with a singularity avoidance law. For instance, if a ]Rqxp then the pseudoinverse a+ = (T (TT)1 and satisfies (aT+ = Iqxq where Iqxq is an identity matrix of dimension q x q. The matrices A (p (t)) and B (p (t)) have the standard linear parametervarying form A(p,t) = Ao + E (p(t))A (23) i= B (p,t) =Bo + ((t))B, (24) i=1 where Ao e R'x" and Bo E represent known nominal matrices with unknown variations (p (t)) Ai and (p (t)) Bi for i = 1,2,..., s, where Ai E IRx" and Bi IRfxp are timeinvariant matrices, and (p (t)) ,' (p (t)) IR are parameterdependent weighting terms. Knowledge of the nominal matrix Bo will be exploited in the subsequent control design. To facilitate the subsequent control design, a reference model is given as xm = Ax, + B,6 (25) Ym = Cxm (26) where A, IR"x" and Bm IERxp denote the state and input matrices, respectively, where A, is Hurwitz, 6 (t) E RP is a vector of reference inputs, y, (t) E IRq are the reference outputs, and C was defined in (22). Assumption 1: The nonlinear disturbance f (t) and its first two time derivatives are assumed to exist and be bounded by known constants. Assumption 2: The dynamics in (21) are assumed to be controllable. Assumption 3: The matrices A (p (t)) and B (p (t)) and their time derivatives satisfy the following inequalities: IIA(p (t)) A(p (t)) < (Ad Bp(Pt)) < (Bd where (A, (B, (Ad, (Bd E R+ are known bounding constants, and I denotes the induced infinity norm of a matrix. As is typical in robust control methods, knowledge of the upper bounds in (27) are used to develop sufficient conditions on gains used in the subsequent control design. 2.3 Control Development 2.3.1 Control Objective The control objective is to ensure that the output y(t) tracks the timevarying output generated from the reference model in (25) and (26). To quantify the control objective, an output tracking error, denoted by e (t) E IR, is defined as e y m = C (x m). (28) To facilitate the subsequent analysis, a filtered tracking error denoted by r (t) E IR, is defined as r e + (29) where 7 IR2 is a positive definite diagonal, constant control gain matrix, and is se lected to place a relative weight on the error state verses its derivative. To facilitate the subsequent robust control development, the state vector x(t) is expressed as x (t) = x (t) + x (t) (210) where x (t) E IR' contains the p output states, and x, (t) E IR' contains the remaining n p states. Likewise, the reference states x,(t) can also be separated as in (210). Assumption 4: The states contained in x,(t) in (210) and the corresponding time derivatives can be further separated as X" (t) = Xp (t) + X(, (t) (211) Xu (t) = Xp (t) + MXu (t) where X, (t) xp, (t) x, (t) (t) E R" are upper bounded as I (t) _< c \1 1 1 x (t) II < P, (212) I (t)ll C< I I I \\I M (t)ll < I( 0 where z(t) E R2q is defined as T1 zA eT r rT (213) and cl, c, (C (u CE IR are known nonnegative bounding constants. The terms in (211) and (212) are used to develop sufficient gain conditions for the subsequent robust control design. 2.3.2 OpenLoop Error System The openloop tracking error dynamics can be developed by taking the time deriva tive of (29) and using the expressions in (21)(26) as r = e + 7e = C ( m) + e = C (Ax + A+. + B i + B + f (t) Amim BmK ) + = N + Nd + CBu + CBi e. (214) The auxiliary functions N (x, A, e, Xm, im, t) E iR and Nd (X ,m i, m t) e IR in (2 14) are defined as N A CA ( Im) + CA ( +,) + CAip,, + CAx, + + e (215) and Nd A Cf (t) + CAJI, + CAx(, + CAjm + CAx. CAmim CBm6. (216) Motivation for the selective grouping of the terms in (215) and (216) is derived from the fact that the following inequalities can be developed [38, 54] as N where po, (Nd E IR+ are known bounding constants. 2.3.3 ClosedLoop Error System Based on the expression in (214) and the subsequent stability analysis, the control input is designed as S= k, (CBo)1 [(ks + qxq) e (t) (k + qxq) e (0) + v (t)] (218) where v (t) E IRq is an implicit learning law with an update rule given by v (t) = k. u (t)  sgn (r (t)) + (k, + Iqxq) ye (t) + ksgn (r (t)) (219) and kr Rpxp, k, ks, k y E Rqxq denote positive definite, diagonal constant control gain matrices, Bo E P. 'is introduced in (24), sgn () denotes the standard signum function where the function is applied to each element of the vector argument, and Iqxq denotes a q x q identity matrix. After substituting the time derivative of (218) into (214), the error dynamics can be expressed as Ir = + Nd (k, u (t) sgn (r (t)) + CBu (220) Q (k, + Ipp) r (t) Qk sgn (r (t)) e where the auxiliary matrix Q (p (t)) E Rqxq is defined as ( A CBk (CBo)1 (221) where Q (p (t)) can be separated into diagonal (i.e., A (p (t)) E IRqxq) and offdiagonal (i.e., A (p (t)) E IRqxq) components as 0 = A+ A. (222) Assumption 5: The subsequent development is based on the assumption that the uncertain matrix Q (p (t)) is diagonally dominant in the sense that Amin (A) I A  > e (223) where e IR+ is a known constant. While this assumption cannot be validated for a generic system, the condition can be checked (within some certainty tolerances) for a specific system. Essentially, this condition indicates that the nominal value Bo must remain within some bounded region of B. In practice, bounds on the variation of B should be known, for a particular system under a set of operating conditions, and this bound can be used to check the sufficient conditions given in (223). Motivation for the structure of the controller in (218) and (219) comes from the desire to develop a closedloop error system to facilitate the subsequent Lyapunovbased stability analysis. In particular, since the control input is premultiplied by the uncertain matrix CB in (214), the term CBo1 is motivated to generate the relationship in (221) so that if the diagonal dominance assumption (Assumption 5) is satisfied, then the control can provide feedback to compensate for the disturbance terms. The bracketed terms in (218) include the state feedback, an initial condition term, and the implicit learning term. The implicit learning term v (t) is the generalized solution to (219). The structure of the update law in (219) is motivated by the need to reject the exogenous disturbance terms. Specifically, the update law is motivated by a sliding mode control strategy that can be used to eliminate additive bounded disturbances. Unlike sliding mode control (which is a discontinuous control method requiring infinite actuator bandwidth), the current continuous control approach includes the integral of the sgn(.) function. This implicit learning law is the key element that allows the controller to obtain an exponential stability result despite the additive nonvanishing exogenous disturbance. Other results in literature also have used the implicit learning structure include [33, 34, 35, 36, 37, 40]. Differential equations such as (224) and (225) have discontinuous righthand sides as S (t) = k Iu (t)  sgn (r (t)) + (k, + Ipxp) ye (t) + kysgn (r (t)) (224) r = N + Nd Qk, Iu (t)ll sgn (r (t)) + CBu (ks + Ipxp) r (t) .ksgn (r (t)) e. (225) Let ffij (y, t) E R2p denote the righthand side of (224) and (225). Since the subsequent analysis requires that a solution exist for y = ffi~ (y, t), it is important to show the existence of the generalized solution. The existence of Filippov's generalized solution [15] can be established for (224) and (225). First, note that ffil (y, t) is continuous except in the set {(y, t) r = 0}. Let F (y, t) be a compact, convex, upper semicontinuous setvalued map that embeds the differential equation y = ffil (x, t) into the differential inclusion y F (y, t). An absolute continuous solution exists to y = F (x, t) that is a generalized solution to ? = fe i (x, t). A common choice [15] for F (y, t) that satisfies the above conditions is the closed convex hull of ffi (y, t). A proof that this choice for F (y, t) is upper semicontinuous is given in [20]. 2.4 Stability Analysis Theorem: The controller given in (218) and (219) ensures exponential tracking in the sense that \e(t) < I (0)1 exp t) Vt [0, o), (226) where A, e R+, provided the control gains k,, k,, and ky introduced in (218) are selected according to the sufficient conditions ( Bd P ( Nd min (ku) > min (k) > r} Amin (k) > "" (227) E 4E min {7, E} e where po and (Nd are introduced in (217), e is introduced in (223), (Bd E R+ is a known positive constant, and Amin () denotes the minimum eigenvalue of the argument. The bounding constants are conservative upper bounds on the maximum expected values. The Lyapunov analysis indicates that the gains in (227) need to be selected sufficiently large based on the bounds. Therefore, if the constants are chosen to be conservative, then the sufficient gain conditions will be larger. Values for these gains could be determined through a physical understanding of the system (within some conservative % of uncertainty) and/or through numerical simulations. Proof: Let VL (z, t) : IR2q X [0, oc)  IR be a Lipschitz continuous, positive definite function defined as VL (z, t) ^ e + 2r r (228) 2 2 where e (t) and r (t) are defined in (28) and (29), respectively. After taking the time derivative of (228) and utilizing (29), (220), and (222), VL (z, t) can be expressed as VL (z, t)= yeTe + rTN + rTCBu rTA (k, + Ipxp) r rTA (k, + Ipxp) r (229) r A IIull I. 1.. (r) rTA IIUII 1,1. (r) rT Aksgn (r) rTAk sgn (r) + rTNd. By utilizing the bounding arguments in (217) and Assumptions 3 and 5, the upper bound of the expression in (229) can be explicitly determined. Specifically, based on (27) of Assumption 3, the term rTCBu in (229) can be upper bounded as rTCBu < Bd Ir IrJ u I (230) After utilizing inequality (223) of Assumption 5, the following inequalities can be developed: rTA (k, + Ipp) r rTA (k, + Ipxp<) r < (Amin (ks) + 1) Ir 2 rTA u (t) 1. .. (r) rTA 1u (t) 1. .,. (r) < Emin (k) r ll \u (231) rTAksgn (r) rTAkysgn (r) < Amiin (k,7) Ir . After using the inequalities in (230) and (231), the expression in (229) can be upper bounded as VL (, t) < 7 11e 2 + rT? + d Ij E (A (ks) + 1) Ir 2 (232) EAmin (ku) \r1\ IJu eAmin (ky) r1 +r TNd, where the fact that Irl > rl V r E IR was utilized. After utilizing the inequalities in (217) and rearranging the resulting expression, the upper bound for VL (z, t) can be expressed as S(z, t) < eI2 _ 2 _ EAmin (ks) IrI 2 +PO Ijrl I : (233) [eAmin (ku) (Bd] Ir \ IJu [eAmin (ky) (Nd] Ir If k, and ky satisfy the sufficient gain conditions in (227), the bracketed terms in (233) are positive, and VL (z, t) can be upper bounded using the squares of the components of z (t) as: (L (zt)< 7 e 12 Er12 [min (ks) r 2 _ P r  I ] (234) By completing the squares, the upper bound in (234) can be expressed in a more convenient form. To this end, the term 4Ei ( s) is added and subtracted to the right hand 4sAmn(kfs) side of (234) yielding 2 [o 2 20 VL(z t) < '7 11e 2 2 E Amin (ks) [ r  nin )l 4 Am inks (235) [_ 2EAnin (k,)JI 4EAin (k,) Since the square of the bracketed term in (235) is always positive, the upper bound can be expressed as VL (z, t) < zTdiag {7pp, Elpxp} + kP (236) da4 Amin (ks)' where z (t) is defined in (213). Hence, (236) can be used to rewrite the upper bound of L (z, t) as VL ( t) < min {7, E} ) (237) 4 Amin (ks) where the fact that zTdiag {Ylpxp, Elpxp} z > min {1 e I '. I was utilized. Provided the gain condition in (227) is satisfied, (228) and (237) can be used to show that VL (t) E L,; hence e (t) r (t) E L,. Given that e (t) ,r (t) E Lo, standard linear analysis methods can be used to prove that e (t) E L, from (29). Since e (t) e (t) e Lo, the assumption that the reference model outputs y, (t) ? (t) E Lo can be used along with (28) to prove that y (t) y (t) E Lo. Given that y (t) y (t) ,e (t) ,r (t) E Lo, the vector x (t) E Lo, the time derivative (t) E Lo, and (210)(212) can be used to show that x (t) (t) E Lo. Given that x (t) (t) E Lo, Assumptions 1, 2, and 3 can be utilized along with (21) to show that u (t) E Lo. The definition for VL (z, t) in (228) can be used along with inequality (237) to show that VL (z, t) can be upper bounded as VL(z,t)< AXVL (z,t) (238) provided the sufficient condition in (227) is satisfied. The differential inequality in (238) can be solved as VL (z, t) < VL (z (0) ,0) exp (A t) (239) Hence, (213), (228), and (239) can be used to conclude that e (t) 11 <  (0)  exp t) t E [0, oo). (240) 2.5 Conclusions A continuous exponentially stable controller was developed for LPV systems with an unknown state matrix, an uncertain input matrix, and an unknown additive disturbance. This work presents a new approach to LPV control by inverting the uncertain input dynamics and robustly compensating for other unknowns and disturbances. The controller is valid for LPV systems where there are at least as many control inputs as there are outputs. Using this technique it is possible control LPV systems where there is a high amount of uncertainty and nonlinearities that invalidate traditional LPV approaches. Robust dynamic inversion control is possible for a wide range of practical systems that are approximated as an LPV system with additive disturbances. Future work will focus on relaxing the assumptions while maintaining the stability and performance. CHAPTER 3 HYPERSONIC VEHICLE DYNAMICS AND TEMPERATURE MODEL 3.1 Introduction In this chapter the dynamics of the hypersonic vehicle (HSV) are introduced, in cluding both the standard flight dynamics and the structural vibration dynamics. After the dynamics are developed and the flight and structural components are explained, a temperature model is introduced. Because changes in temperature change the structural dynamics, coupled forcing terms change the the flight dynamics. Examples of linear tem perature profiles are provided, and some examples of the structural modes and frequencies are explained. 3.2 Rigid Body and Elastic Dynamics To incorporate structural dynamics and aerothermoelastic effects in the HSV dynamic model, an assumed modes model is considered for the longitudinal dynamics [53] as =Tco (a) D gsin (6 a) (31) h = Vsin (0 a) (32) L +T sin (a) g a = + Q + cos (0 a) (33) = mV V = Q (34) M Q = (35) Iyy ri = 2... 'i + N, i = 1, 2, 3. (36) In (31)(36), V (t) R denotes the forward velocity, h (t) E R denotes the altitude, a (t) E R denotes the angle of attack, 0 (t) R denotes the pitch angle, Q (t) E R is pitch rate, and ri (t) E R Vi = 1, 2, 3 denotes the ith generalized structural mode displacement. Also in (31)(36), m R denotes the vehicle mass, lyy E R is the moment of inertia, g e R is the acceleration due to gravity, (i (t) wi (t) E R are the damping factor and natural frequency of the ith flexible mode, respectively, T (x) E R denotes the thrust, D (x) E R denotes the drag, L (x) E R is the lift, M (x) E R is the pitching moment about the body yaxis, and Ni (x) E R Vi = 1, 2, 3 denotes the generalized elastic forces, where x (t) E R11 is composed of the 5 flight and 6 structural dynamic states as [ T x = V a Q h 0 rj9 r1 2 r2 r93 3j (37) The equations that define the aerodynamic and generalized moments and forces are highly coupled and are provided explicitly in previous work [10]. Specifically, the rigid body and elastic modes are coupled in the sense that T (x), D (x), L (x), are functions of qi (t) and that Ni (x) is a function of the other states. As the temperature profile changes, the modulus of elasticity of the vehicle changes and the damping factors and natural frequencies of the flexible modes will change. The subsequent development exploits an implicit learning control structure, designed based on an LPV approximation of the dynamics in (31)(36), to yield exponential tracking despite the uncertainty due to the unknown aerothermoelastic effects and additional unmodeled dynamics. 3.3 Temperature Profile Model Temperature variations impact the HSV flight dynamics through changes in the structural dynamics which affect the mode shapes and natural frequencies of the vehicle. The temperature model used assumes a freefree beam [10], which may not capture the actual aircraft dynamics properly. In reality, the internal structure will be made of a complex network of structural elements that will expand at different rates causing thermal stresses. Thermal stresses affect different modes in different manners, where it raises the frequencies of some modes and lowers others (compared to a uniform degradation with Young's modulus only). Therefore, the current model only offers an approximate approach. The natural frequencies of a continuous beam are a function of the mass distribution of the beam and the stiffness. In turn, the stiffness is a function of Young's Modulus (E) and admissible mode functions. Hence, by modeling Young's Modulus as a function of temperature, the effect of temperature on flight dynamics can be captured. Thermostructural dynamics are calculated under the material assumption that titanium is below the thermal protection system [9, 12]. Young's Modulus (E) and the natural dynamic frequencies for the first three modes of a titanium freefree beam are depicted in Figure 31 and Figure 32 respectively. 165 16 155  15 o 145 W 14 S135 0 H 13 125 12  115 0 100 200 300 400 500 600 700 800 900 Temperature (F) Figure 31: Modulus of elasticity for the first three dynamic modes of vibration for a free free beam of titanium. In Figure 31, the moduli for the three modes are nearly identical. The temperature range shown corresponds to the temperature range that will be used in the simulation section. Frequencies in Figure 32 correspond to a solid titanium beam, which will not correspond to the actual natural frequencies of the aircraft. The data shown in Figure 31 and Figure 32 are both from previous experimental work [47]. Using this data, different temperature gradients along the fuselage are introduced into the model and affect the structural properties of the HSV. The simulations in Chapter 4 and Chapter 5 use linearly decreasing gradients from the nose to the tail section. It's expected that the nose will be the hottest part of the structure due to aerodynamic heating behind the bow shock wave. Thermostructural dynamics are calculated under the assumption that there are nine constanttemperature sections in the aircraft [6] as shown in Figure 33. Since the aircraft is 100 feet long, the length of each of the nine sections is approximately 11.1 feet. 1st Dynamic Mode N 55 50 LL 45 0 100 200 300 400 500 600 700 800 900 2nd Dynamic Mode NT 160 140 L 120 0 100 200 300 400 500 600 700 800 900 3rd Dynamic Mode NT 300 S250 LL 200 0 100 200 300 400 500 600 700 800 900 Temperature (F) Figure 32: Frequencies of vibration for the first three dynamic modes of a freefree tita nium beam. Figure 33: Nine constant temperature sections of the HSV used for temperature profile modeling. Table 31: Natural frequencies for 5 linear temperature profiles (Nose/Tail) in degrees F. Percent difference is the difference between the maximum and minimum frequencies divided by the minimum frequency. Mode 900/500 800/400 700/300 600/200 500/100 % Difference 1 (Hz) 23.0 23.5 23.9 24.3 24.7 7.39 % 2 (Hz) 49.9 50.9 51.8 52.6 53.5 7.21 % 3 (Hz) 98.9 101.0 102.7 104.4 106.2 7.38 % The structural modes and frequencies are calculated using an assumed modes method with finite element discretization, including vehicle mass distribution and inertia effects. The result of this method is the generalized mode shapes and mode frequencies for the HSV. Because the beam is nonuniform in temperature, the modulus of elasticity is also nonuniform, which produces asymmetric mode shapes. An example of the asymmetric mode shapes is shown in Figure 35 and the asymmetry is due to variations in E resulting from the fact that each of the nine fuselage sections (see Figure 33) has a different temperature and hence different flexible dynamic properties. An example of some of the mode frequencies are provided in Table 1, which shows the variation in the natural frequencies for five decreasing linear temperature profiles shown in Figure 34. For all three natural modes, Table 31 shows that the natural frequency for the first temperature profile is almost 7% lower than that of the fifth temperature profile. The temperature profile in a HSV is a complex function of the state history, struc tural properties, thermal protection system, etc. For the simulations in Chapter 4 and Chapter 5, the temperature profile is assumed to be a linear function that decreases from the nose to the tail of the aircraft. The linear profiles are then varied to span a prese lected design space. Rather than attempting to model a physical temperature gradient for some vehicle design, the temperature profile in the simulations in Chapter 4 and Chap ter 5 is intended to provide an aggressive temperature dependent profile to examine the robustness of the controller to such fluctuations. 2 3 4 5 6 Fuselage section 7 8 9 Figure 34: Linear temperature profiles used to calculate values shown in Table 31. a) *. S0 a 0.1 0.2  1st 0.3 2nd ....... 3rd 20 40 60 Fuselage Position (ft) 80 100 Figure 35: Asymetric mode shapes for the hypersonic vehicle. The percent difference was calculated based on the maximum minus the minimum structural frequencies divided by the minimum. 800 700 600 g 500 a 400 300 200 3.4 Conclusion This chapter explains the overall flight and structural dynamics for a HSV, in the presence of different temperature profiles. These dynamics are important to understand because changes in the temperature profile modify the dynamics, hence can be modeled as additive parameter disturbances. In the following chapters, the HSV dynamics will be reduced to a LPV system with an additive disturbance, and the controller from Chapter 2 will be applied. The temperature profiles will act as the parameter variations. This chapter was meant to briefly introduce the overall system and explain the structural modes, shapes, and frequencies. Data was shown to motivate the fact that changes in temperature substantially affect the overall dynamics. CHAPTER 4 LYAPUNOVBASED EXPONENTIAL TRACKING CONTROL OF A HYPERSONIC AIRCRAFT WITH AEROTHERMOELASTIC EFFECTS 4.1 Introduction The design of guidance and control systems for airbreathing hypersonic vehicles (HSV) is challenging because the dynamics of the HSV are complex and highly coupled [10], and temperatureinduced stiffness variations impact the structural dynamics [21]. The structural dynamics, in turn, affect the aerodynamic properties. Vibration in the forward fuselage changes the apparent turn angle of the flow, which results in changes in the pressure distribution over the forebody of the aircraft. The resulting changes in the pressure distribution over the aircraft manifest themselves as thrust, lift, drag, and pitching moment perturbations [10]. To develop control laws for the longitudinal dynamics of a HSV capable of compensating for these structural and aerothermoelastic effects, structural temperature variations and structural dynamics must be considered. Aerothermoelasticity is the response of elastic structures to aerodynamic heating and loading. Aerothermoelastic effects cannot be ignored in hypersonic flight, because such effects can destabilize the HSV system [21]. A loss of stiffness induced by aerodynamic heating has been shown to potentially induce dynamic instability in supersonic/hypersonic flight speed regimes [1]. Yet active control can be used to expand the flutter boundary and convert unstable limit cycle oscillations (LCO) to stable LCO [1]. An active structural controller was developed [26], which accounts for variations in the HSV structural proper ties resulting from aerothermoelastic effects. The control design [26] models the structural dynamics using a LPV framework, and states the benefits to using the LPV framework are twofold: the dynamics can be represented as a single model, and controllers can be designed that have affine dependency on the operating parameters. Previous publications have examined the challenges associated with the control of HSVs. For example, HSV flight controllers are designed using genetic algorithms to search a design parameter space where the nonlinear longitudinal equations of motion contain uncertain parameters [4, 30, 49]. Some of these designs utilize Monte Carlo simulations to estimate system robustness at each search iteration. Another approach [4] is to use fuzzy logic to control the attitude of the HSV about a single low end flight condition. While such approaches [4, 30, 49] generate stabilizing controllers, the procedures are computationally demanding and require multiple evaluation simulations of the objective function and have large convergent times. An adaptive gainscheduled controller [55] was designed using estimates of the scheduled parameters, and a semioptimal controller is developed to adaptively attain Ho control performance. This controller yields uniformly bounded stability due to the effects of approximation errors and algorithmic errors in the neural networks. Feedback linearization techniques have been applied to a control oriented HSV model to design a nonlinear controller [32]. The model [32] is based on a previously developed [8] HSV longitudinal dynamic model. The control design [32] neglects variations in thrust lift parameters, altitude, and dynamic pressure. Linear output feedback tracking control methods have been developed [44], where sensor placement strategies can be used to increase observability, or reconstruct full state information for a statefeedback controller. A robust output feedback technique is also developed for the linear parameterizable HSV model, which does not rely on state observation. A robust setpoint regulation controller [17] is designed to yield asymptotic regulation in the presence of parametric and structural uncertainty in a linear parameterizable HSV system. An adaptive controller [19] was designed to handle (linear in the parameters) mod eling uncertainties, actuator failures, and nonminimum phase dynamics [17] for a HSV with elevator and fuel ratio inputs. Another adaptive approach [41] was recently devel oped with the addition of a guidance law that maintains the fuel ratio within its choking limits. While adaptive control and guidance control strategies for a HSV are investigated [17, 19, 41], neither addresses the case where dynamics include unknown and unmodeled disturbances. There remains a need for a continuous controller, which is capable of achiev ing exponential tracking for a HSV dynamic model containing aerothermoelastic effects and unmodeled disturbances (i.e., nonvanishing disturbances that do not satisfy the linear in the parameters assumption). In the context of the aforementioned literature, the contribution of the current ef fort (and the preliminary effort by the authors [52]) is the development of a controller that achieves exponential model reference output tracking despite an uncertain model of the HSV that includes nonvanishing exogenous disturbances. A nonlinear temperature dependent parametervarying statespace representation is used to capture the aerother moelastic effects and unmodeled uncertainties in a HSV. This model includes an unknown parametervarying state matrix, an uncertain parametervarying nonsquare (column deficient) input matrix, and a nonlinear additive bounded disturbance. To achieve an exponential tracking result in light of these disturbances, a robust, continuous Lyapunov based controller is developed that includes a novel implicit learning characteristic that compensates for the nonvanishing exogenous disturbance. That is, the use of the implicit learning method enables the first exponential tracking result by a continuous controller in the presence of the bounded nonvanishing exogenous disturbance. To illustrate the perfor mance of the developed controller during velocity, angle of attack, and pitch rate tracking, simulations for the full nonlinear model [10] are provided that include aerothermoelastic model uncertainties and nonlinear exogenous disturbances whose magnitude is based on airspeed fluctuations. 4.2 HSV Model The dynamic model used for the subsequent control design is based on a reduction of the dynamics in (31)(36) to the following combination of linearparametervarying (LPV) state matrices and additive disturbances arising from unmodeled effects as '= A(p(t)) + B (p(t))u + f (t) (41) y = Cx. (42) In (41) and (42), x (t) E R1 is the state vector, A (p (t)) E R11x11 denotes a linear parameter varying state matrix, B (p (t)) E R11x3 denotes a linear parameter varying input matrix, C E R3x11 denotes a known output matrix, u(t) E IR3 denotes a vector of 3 control inputs, p (t) represents the unknown timedependent parameters, f(t) E IR11 represents a timedependent unknown, nonlinear disturbance, and y (t) E IR3 represents the measured output vector of size 3. 4.3 Control Objective The control objective is to ensure that the output y(t) tracks the timevarying output generated from the reference model like stated in Chapter 2. To quantify the control objective, an output tracking error, denoted by e (t) E R3, is defined as e y ym = C (x m) (43) To facilitate the subsequent analysis, a filtered tracking error denoted by r (t) E IR3, is defined as r = + ye (44) where y R3 is a positive definite diagonal, constant control gain matrix, and is selected to place a relative weight on the error state verses its derivative. Based on the control design presented in Chapter 2 the control input is designed as S= kr (CBo)1 [(k, + I33) e t) (k, + I33) e (0) + v (t)] (45) where v (t) E IR3 is an implicit learning law with an update rule given by iv (t) = k  u (t) l sgn (r (t)) + (k, + I3x3) ye (t) + ksgn (r (t)) (46) and kr, ku, ks, k E IR3x3 denote positive definite, diagonal constant control gain matrices, Bo E R11x3 represents a known nominal input matrix, sgn (.) denotes the standard signum function where the function is applied to each element of the vector argument, and 13x3 denotes a 3 x 3 identity matrix. To illustrate the performance of the controller and practicality of the assumptions, a numerical simulation was performed on the full nonlinear longitudinal equations of motion [10] given in (31)(36). The control inputs were selected as u = [e (t) Oc (t) i (t as in previous research [41], where 6e (t) and c6 (t) denote the elevator and canard deflection angles, respectively, Of (t) is the fuel equivalence ratio. The diffuser area ratio is left at its operational trim condition without loss of generality (Ad (t) = 1). The reference outputs were selected as maneuver oriented outputs of velocity, angle of attack, and pitch rate as y = V (t) a (t) Q (t) where the output and state variables are introduced in (31)(35). In addition, the proposed controller could be used to control other output states such as altitude provided the following condition is valid. The auxiliary matrix 2 (p (t)) E Rqxq is defined as Sa CBkr (CBo)l (47) where 0 (p (t)) can be separated into diagonal (i.e., A (p (t)) E IRqx) and offdiagonal (i.e., A (p (t)) E RqIx) components as 0 = A + A. (48) The uncertain matrix 2 (p (t)) is diagonally dominant in the sense that Amin (A) IAo > e (49) where e R I+ is a known constant. While this assumption cannot be validated for a generic HSV, the condition can be checked (within some certainty tolerances) for a given aircraft. Essentially, this condition indicates that the nominal value Bo must remain within some bounded region of B. In practice, bands on the variation of B should be known, for a particular aircraft under a set of operating conditions, and this band could be used to check the sufficient conditions. For the specific HSV example this Chapter simulates, the assumtion in 49 is valid. 4.4 Simulation Results The HSV parameters used in the simulation are m = 75, 000 lbs Iyy = 86723 lbs ft2, and g = 32.174 ft/s2.as defined in (31)(36). The simulation was executed for 35 seconds to sufficiently cycle through the different temperature profiles. Other vehicle parameters in the simulation are functions of the temperature profile. Linear temperature profiles between the forebody (i.e., Tfb e [450, 900]) and aftbody (i.e., Tab e [100, 800]) were used to generate elastic mode shapes and frequencies by varying the linear gradients as {( T t) 450 + 350 cos (ft) if Tfb (t) > Tab (t) Tfb (t) = 675 + 225 cos t Tab (t) = (10/ Tfb (t) otherwise. (410) Figure 41 shows the temperature variation as a function of time. The irregularities seen in the aftbody temperatures occur because the temperature profiles were adjusted to ensure the tail of the aircraft was equal or cooler than the nose of the aircraft according to bow shockwave thermodynamics. While the shockwave thermodynamics motivated the need to only consider the case when the tail of the aircraft was equal or cooler than the nose of the aircraft, the shape of the temperature profile is not physically motivated. Specifically, the frequencies of oscillation in (410) were selected to aggressively span the available temperature ranges. These temperature profiles are not motivated by physical temperature gradients, but motivated by the desire to generate a temperature disturbance to illustrate the controller robustness to the temperature gradients. The simulation assumes the damping coefficient remains constant for the structural modes ((i = 0.02). In addition to thermoelasticity, a bounded nonlinear disturbance was added to the dynamics as f= [fv fa fQ 0 0 0 fii 0 fi2 0 fi3 (411) 1000 800 1 600 S400 z 200 0 0 5 10 15 20 25 30 35 Time (s) 800IIIIII S600 400 200 0 5 10 15 20 25 30 35 Time (s) Figure 41: Temperature variation for the forebody and aftbody of the hypersonic vehicle as a function of time. where f (t) E IR denotes a longitudinal acceleration disturbance, f (t) IR denotes a angle of attack rate of change disturbance, fQ(t) E IR denotes an angular acceleration distur bance, and fi (t), fi2 (t), fi3(t), E IR denote structural mode acceleration disturbances. The disturbances in (411) were generated as an arbitrary exogenous input (i.e., unmodeled nonvanishing disturbance that does not satisfy the linear in the parameters assumption) as depicted in Figure 42. However, the magnitudes of the disturbances were motivated by the scenario of a 300 ft/s change in airspeed. The disturbances are not designed to mimic the exact effects of a wind gust, but to demonstrate the proposed controller's robustness with respect to realistically scaled disturbances. Specifically, a relative force disturbance is determined by comparing the drag force D at Mach 8 at 85, 000 ft (i.e., 7355 ft/s) with the drag force after adding a 300 ft/s (e.g., a wind gust) disturbance. Using Newton's second law and dividing the drag force differential AD by the mass of the HSV m, a realistic upper bound for an acceleration disturbance fy (t) was determined. Similarly, the same procedure can be performed, to compare the change in pitching moment AM caused by a 300 ft/s head wind gust. By dividing the moment differential by the moment of x 10 1 0 5 10 15 20 25 30 35 10 D 0 S? 10cl nr S10 0 5 10 15 20 25 30 35 ~ 2 0 2 0 5 10 15 20 25 30 35 005 0 005 0 01 I  0 5 10 15 20 25 30 35 001 0 001 0 5 10 15 20 25 30 35 x 10 0 5 10 15 20 25 30 35 Time (s) Figure 42: In this figure, fi denotes the ith element in the disturbance vecor f. Distur bances from top to bottom: velocity fy, angle of attack fd, pitch rate fQ, the 1st elastic structural mode i1, the 2nd elastic structural mode i2, and the 3rd elastic structural mode 173, as described in (411). inertia of the HSV Iyy, a realistic upper bound for fQ (t) can be determined. To calculate a reasonable angle of attack disturbance magnitude, a vertical wind gust of 300 ft/s is considered. By taking the inverse tangent of the vertical wind gust divided by the forward velocity at Mach 8 and 85,000 ft, an upper bound for the angle of attack disturbance fa(t) can be determined. Disturbances for the structural modes fji(t) were placed on the acceleration terms with i((t), where each subsequent mode is reduced by a factor of 10 relative to the first mode, see Figure 42. The proposed controller is designed to follow the outputs of a well behaved reference model. To obtain these outputs, a reference model that exhibited favorable characteristics was designed from a static linearized dynamics model of the full nonlinear dynamics [10]. The reference model outputs are shown in Figure 43. The velocity reference output follows a 1000 ft/s smooth step input, while the pitch rate performs several 1 /s maneuvers. The angle of attack stays within 2 degrees. 8500 O 8000 > 7500 7000 0 2 E 0 ac 5 10 15 20 25 30 35 5 10 15 20 25 30 35 5 10 15 20 25 30 35 Time (s) Figure 43: Reference model ouputs yi, which are the desired trajectories for top: velocity Vm (t), middle: angle of attack a, (t), and bottom: pitch rate Qm (t). The control gains for (43)(44) and (45)(46) are selected as y= diag {10, 10} k, = diag {5,1, 300} k = diag {0.1, 0.01, 0.1} k, = diag {0.01, 0.001, 0.01} kr = diag {1, 0.5, 1}. The control gains in (412) were obtained using the same method as in Chapter 5. In contrast to this suboptimal approach used, the control gains could have been adjusted using more methodical approaches as described in various survey papers on the topic [24, 46]. The C matrix and knowledge of some nominal Bo matrix must be known. The C matrix is given by: 1 0 0 0 0 0 0 0 0 0 0 o1000000000 C= 0 1 0 0 0 0 0 0 0 0 0 (412) (413) 8400 8200 8000 S7800 > 7600 7400 7200 0 5 10 15 20 25 30 35 02 S0 4  LU 0 6 08 1 0 5 10 15 20 25 30 35 Time (s) Figure 44: Top: velocity V (t), bottom: velocity tracking error ev (t). for the output vector of (42), and the Bo matrix is selected as T 32.69 0.017 9.07 0 0 0 2367 0 1132 0 316 Bo = 25.72 0.0111 9.39 0 0 0 3189 0 2519 0 2067 (414) 42.84 0.0016 0.0527 0 0 0 42 13 0 92.12 0 80.0 based on a linearized plant model about some nominal conditions. The HSV has an initial velocity of Mach 7.5 at an altitude of 85, 000 ft. The velocity, and velocity tracking errors are shown in Figure 44. The angle of attack and angle of attack tracking error is shown in Figure 45. The pitch rate and pitch tracking error is shown in Figure 46. The control effort required to achieve these results is shown in Figure 47. In addition to the output states, other states such as altitude and pitch angle are shown in Figure 48. The structural modes are shown in Figure 49. 4.5 Conclusion This result represents the first ever application of a continuous, robust model refer ence control strategy for a hypersonic vehicle system with additive bounded disturbances 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Time (s) Figure 45: Top: angle of attack a (t), bottom: angle of attack tracking error e (t). 0i 15 05 10 15 20 25 30 35 D 005 0 S 01 f n a r GC " "'" ~ Figure 46: Top: pitch rate Q (t), bottom: pitch rate tracking error eQ (t). 0 5 10 15 20 25 30 35 Time (s) 15 u_ 2o 15 10 10 15 20 25 30 35 o0 5 10 15 20 25 30 35 10 0 10 0 5 10 15 Time (s) 20 25 30 35 fuel equivalence ratio of. Middle: elevator deflection 6,. Bottom: Canard x 10 85 84 S83 82 81 10 15 20 25 30 35 10 15 20 25 30 35 Time (s) Figure 48: Top: altitude h (t), bottom: pitch angle 0 (t). Figure 47: Top: deflection cc. 5 10 15 20 25 30 35 S10 5 10 0 5 10 15 20 25 30 35 5i  0 5 10 15 20 25 30 35 Time (s) Figure 49: Top: 1st structural elastic mode rll. Middle: 2nd structural elastic mode l2. Bottom: 3'd structural elastic mode r3. and aerothermoelastic effects, where the control input is multiplied by an uncertain, col umn deficient, parametervarying matrix. A potential drawback of the result is that the control structure requires that the product of the output matrix with the nominal control matrix be invertible. For the output matrix and nominal matrix, the elevator and canard deflection angles and the fuel equivalence ratio can be used for tracking outputs such as the velocity, angle of attack, and pitch rate or velocity and the flight path angle, or veloc ity, flight path angle and pitch rate. Yet, these controls can not be applied to solve the altitude tracking problem because the altitude is not directly controllable and the product of the output matrix with the nominal control matrix is singular. However, the integrator backstepping approach that has been examined in other recent results for the hypersonic vehicle could potentially be incorporated in the control approach to address such objec tives. A Lyapunovbased stability analysis is provided to verify the exponential tracking result. Although the controller was developed using a linear parameter varying model of the hypersonic vehicle, simulation results for the full nonlinear model with temperature variations and exogenous disturbances illustrate the boundedness of the controller with favorable transient and steady state tracking errors. These results indicate that the LPV model with exogenous disturbances is a reasonable approximation of the dynamics for the control development. CHAPTER 5 CONTROL PERFORMANCE VARIATION DUE TO NONLINEAR AEROTHERMOELASTICITY IN A HYPERSONIC VEHICLE: INSIGHTS FOR STRUCTURAL DESIGN 5.1 Introduction Typically, controllers are developed to achieve some performance metrics for a given HSV model. However, improved performance and robustness to thermal gradients could result if the structural design and control design were optimized in unison. Along this line of reasoning in [16, 23], the advantage of correctly placing the sensors is discussed, representing a move towards implementing a control friendly design. A previous control oriented design analysis in [6] states that simultaneously optimizing both the structural dynamics and control is an intractable problem, but that controloriented design may be performed by considering the closedloop performance of an optimal controller on a series of different openloop design models. The best performing design model is then said to have the optimal dynamics in the sense of controllability. Knowledge of the better performing thermal gradients can provide design engineers insight to properly weight the HSV's thermal protection system for both steadystate and transient flight. The preliminary work in [6] provides a controloriented design architecture by investigating control performance variations due to thermal gradients using an too con troller. Chapter 5 seeks to extend the control oriented design concept to examine control performance variations for HSV models that include nonlinear aerothermoelastic distur bances. Given these disturbances, Chapter 5 focuses on examining control performance variations for our previous model reference robust controller in [52] and previous chapters to achieve a nonlinear controloriented analysis with respect to thermal gradients. By analyzing the control error and input norms over a wide range of temperature profiles an optimal temperature profile range is suggested. Based on preliminary work done in [50], a number of linear temperature profile models are examined for insight into the structural design. Specifically, the full set of nonlinear flight dynamics will be used and control effort, errors, and transients such as steadystate time and peak to peak error will be examined across the design space. 5.2 Dynamics and Controller The HSV dynamics used in this chapter are the same is in Chapter 3 and equations (31)(36). Similarly as in the results in Chapter 4, the dynamics in (31)(36) are reduced to the linear parameter model used in (21) and (22) with p = q = 2. For the controloriented design analysis, a number of different linear profiles are chosen [6, 50] with varying nose and tail temperatures as illustrated in Figure 51. This set of profiles define the space from which the controloriented analysis will be performed. As seen in Figure 51, the temperature profiles are linear and decreasing towards the tail. These profiles are realistic based on shock formation at the front of the vehicle and that the temperatures are within the expected range for hypersonic flight. Based on previous IIII I 21 II I  41111 1 2 46 1 2 3 4 5 6 Fuselage Station Figure 51: HSV surface temperature profiles. Tnose [100F, 800F]. 7 8 9 [450F, 900F], and Ttil control development in [52] and in the previous Chapters, the control input is designed as S= kr (CBo)1 [(ks + I33) e (t) (k + I33) e (0) + v (t)] (51) where v (t) E IR2 is an implicit learning law with an update rule given by S (t) = k, Iu (a)  sgn (r (a)) + (k, + 13x3) ye (a) + kysgn (r (a)) (52) where kr, k,, k, k, E R 2x2 denote positive definite, diagonal constant control gain matrices, Bo0 R112 represents a known nominal input matrix, sgn () denotes the standard signum function where the function is applied to each element of the vector argument, and I2x2 denotes a 2 x 2 identity matrix. 5.3 Optimization via Random Search and Evolving Algorithms For each of the individual temperature profiles examined, the control gains kr, k,, k,, k~, and 7 in (51)(52) were optimized for the specific plant model using a combination of random search and evolving algorithms. Since both the plant model simulation dynamics and the control scheme itself are nonlinear, traditional methods for linear gain tuning optimization could not be used. The selected method is a combination of a control gain random search space, combined with an evolving algorithm scheme which allows the search to find a nearest set of optimal control gains for each individual plant. This method allows one nearoptimal controller/plant to be compared to the other nearoptimal controller/plants and provides a more accurate way of comparing cases. The first step in the control gain optimization search is a random initialization. For this numerical study, 1000 randomly selected sets of control gains are used for a given plant model. A 1000 initial random set was chosen to provide sufficient sampling to insure global convergence. The following section has a specific example case for one of the temperature profiles. After the 1000 control gain sets are selected, all the sets are simulated on the given plant model and the controller in (51) and (52) is applied to track a certain trajectory as well as reject disturbances. The trajectory and disturbances were chosen the same throughout the entire study so that the only variations will be due to the plant model and control gains. The example case section explicitly shows both the desired trajectory and the disturbances injected. After the 1000 initial random control gain search is performed, the top five perform ing sets of control gains are chosen as the seeds for the evolving algorithm process. This process is repeated for four generations, each with the best five performing sets of control gains at each step. All evolving algorithms have some or all of the following characteris tics: elitism, crossover, and random mutation. This particular numerical study uses all three as follows. The best five performing sets in each subsequent generation, are chosen as elite and move onto the next iteration step. From those five, each set of control gains is averaged with all other permutations of control gains in the elite set. For instance, if parent #1 is averaged with #2 to form an offspring set of control gains. Parent #1 is also averaged with parent #3 for a separate set of offspring control gains. In this way, all combinations of crossover are performed. The permutations of the five elite parents yield a total of 10 offspring. The next generation contains the five elite parents from the generation before, as well as the 10 crossover offspring, for a total of 15. Each of these 15 sets of control gains is then mutated by a certain percentage. Based on preliminary numerical studies performed on this specific example, the random mutations were chosen to be 20% for the first two generations and 5% for the final two generations. This produced both global search in the beginning, and refinement at the end of the optimization procedure. The set of 15 remains, with the addition of 20 mutated sets for each of the 15. This gives a total control gain set for the next generation of search of 315. As stated, there are four evolving generations after the first 1000 random control sets. The combined number of simulations with different control gains performed for a single temperature profile case is 2260. These particular numbers were chosen based on preliminary trial optimization cases, with the goal to provide sufficient search to achieve convergence of a minimum for the cost function. The following section illustrates the entire procedure for a single temperature profile case. The cost function is designed such that the errors and control inputs are the same order of magnitudes, so that they can more easily be added and interpreted. This is important because for example, the desired velocity is high (in the thousands of ft/s) and the desired pitch rate is small (fraction of radians). Explicitly, the cost function is taken as the sum of the control and error norms and is scaled as Gerr= 100e 1000 eo (53) 2 and co = j 10 (54) L 2 where ev (t) Q (t) E IR are the velocity and pitch rate errors, respectively, and 6, (t), Of (t) E IR are the elevator and fuel ratio control inputs, respectively, and 2  denotes the standard 2norm. The combined cost function is the sum of the individual components and can be explicitly written as Ltot = Qerr + Lcon (55) where Qtot is the cost value associated with all subsequent optimal gain selection. 5.4 Example Case The HSV parameters used in the simulation are m = 75, 000 lbs Iyy = 86723 lbs ft2, and g = 32.174 ft/s2.as defined in (31)(36). To illustrate how the random search and evolving optimization algorithms work, this section is provided as a detailed example. First the output tracking signal and disturbances are provided, followed by the optimization and convergence procedure. The goal of this section is to demonstrate that the specific number of elites, offspring, mutations, and generations listed in the previous section are justified in that the cost function shows asymptotic convergence to a minimum. The desired trajectory is shown in Figure 52 and the disturbance is depicted in Figure 53, where the magnitudes are chosen based on previous analysis performed in [52]. The example case is based on a temperature profile with Tnose = 3500F and Trtil = 2000F. For 6) 0.5 . 0 n 2 0.5 1 2 4 6 8 0 2 4 6 8 1( 7900 7850 7800 Time (s) Figure 52: Desired trajectories: pitch rate Q (top) and velocity V (bottom). x 10 1 gE 0 _ 0 1 2 3 4 5 6 7 8 9 10 5 5 0 1 2 3 4 5 6 7 8 9 10 0 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 Time (s) 6 7 8 9 10 Figure 53: Disturbances for velocity V (top), angle of attack a (second from top), pitch rate Q (second from bottom) and the 1st structural mode (bottom). 0 0 05 0 ou 005 0 c? S0.02 ) IG 0.02 0 2 4 6 8 10 0.5 0 S0.5 1 1.5 0 2 4 6 8 10 Time (s) Figure 54: Tracking errors for the pitch rate Q in degrees/sec (top) and the velocity V in ft/sec (bottom). this particular case, Figure 54 and Figure 55 show the tracking errors and control inputs, respectively, for the control gains 11.17 0 14.55 0 25.99 0 y ks k  0 39.61 0 224.0 0 0.618 20.7 0 0.915 0 k = kr= (56) 0 0.369 0 0.898 The cost functions have values as seen in Figure 56. In Figure 56 the control input cost remains approximately the same, but as the control gains evolve, the error cost and hence total cost decrease asymptotically. The 1st five iterations correspond to the top five performers in the first 1000 random sample, and each subsequent five correspond to the top five for the subsequent evolution generations. To limit the optimization search design space, all simulations are performed with two inputs and two outputs. As indicated in the cost functions listed in (53)(55), the inputs include the elevator deflection 6, (t) and the fuel ratio 0f (t), and the outputs are the velocity V (t) and the pitch rate Q (t). Time (s) Figure 55: Control inputs for the elevator 6, in degrees (top) and the fuel ratio of (bot tom). x 105 18 0 16  1 0 4 12 0 5 10 15 20 25 x 10 9596 g 9 5955 2 9595 9 5945 9 594 0 x 10 7 6 S5 U 4 Iteration # Figure 56: Cost function values for the total cost Qtot and the error cost err (bottom). (top), the input cost Qco, (middle) 5.5 Results The results of this section cover all the temperature profiles shown in Figure 51. The data presented includes the cost functions as well as other steadystate and transient data. Included in this analysis are the control cost function, the error cost function, the peak topeak transient response, the time to steadystate, and the steadystate peaktopeak, for both control and error signals. Because the data contains noise, a smoothed version of each plot is also provided. The smoothed plots use a standard 2dimensional filtering, where each point is averaged with its neighbors. For instance for some variable cv, the averaged data is generated as (4'...' ; + i,+lj + il,j + .' ;+1 + '.' 1)57 = i(57) 8 The averaging formula shown in (57) is used for filtering of all subsequent data. Also, note that the lower right triangle formation is due to the design space only containing temperature profiles where the nose is hotter than the tail. This is due to the assumption that because of aerodynamic heating from the extreme speeds of the HSV, that this will always be the case. These temperature profiles relate to the underlying structural temperature, not necessarily the skin surface temperature. Figure 57 and Figure 58 show the control cost function value ,co,. Note that there is a global minimum, however also note for all of the control norms the total values are approximately the same. This data indicates that while other performance metrics varied widely as a function of temperature profile, the overall input cost remains approximately the same. In Figure 59 and Figure 510, the error cost is shown. Note that there is variability, but that there seems to be a region of smaller errors in the cooler section of the design space. Namely, where Tnose E [200, 600] F and Trail e [100, 250] F. Combining the control cost function with the error cost function yields the total cost function (and its filtered counterpart) depicted in Figure 511 (and Figure 512, respectively). The importance of this plot is that the total cost function was the criteria for which the control gains were optimized. In this 900 300 200 100 100 200 300 400 500 600 Tail Temp (F) 700 800 900 Figure 57: Control cost function co,, data as a function of tail and nose temperature profiles. 400 500 600 Tail Temp (F) x 104 9.5956 9.5954 9 5952 9.595 9.5948 9 .5944 9.5944 9.5942 Figure 58: Control cost function co,, data (filtered) as a function of tail and nose tem perature profiles. x104 9.5956 9.5954 9.5952 9.595 9.5948 9.5946 9.5944 9.5942 9.594 9.5938 200 100 100 200 300 400 500 600 700 800 900 Tail Temp (F) Figure 59: Error cost function ,,er data as a function of tail and nose temperature pro files. 300 400 500 600 Tail Temp (F) Figure 510: Error cost function 2,er data (filtered) as a function of tail and nose tempera ture profiles. x 105 1.4 E  1.35 300 200 1.3 100 100 200 300 400 500 600 700 800 900 Tail Temp (F) Figure 511: Total cost function Qt,t data as a function of tail and nose temperature pro files. sense, the total cost plots represent where the temperature parameters are best suited for control based on the given cost function. Since the cost of the control input is relatively constant, the total cost largely shows the same pattern as the error cost. In addition to the region between T,,os E [200, 600] F and Trail e [100, 250] F, there also seems to be a region between Tose = 9000F and Trail e [600, 900] F, where the performance is also improved. The control cost, error cost, and total cost were important in the optimization of the control gains and were used as the criteria for selecting which gain combination was considered near optimal. However, there are potentially other performance metrics of value. In addition to the optimization costs, the peaktopeak transient errors, time to steadystate, and steadystate peaktopeak errors were examined for further investigation. The peaktopeak transient error is produced by taking the difference from the maximum and minimum transient tracking errors. The peaktopeak error for the pitch rate Q (t) is plotted in Figure 513 and Figure 514, and the peaktopeak for the velocity V (t) is x 105 900 1.5 800 700 1.45 0 1.4 11,I,,1.35 100 200 300 400 500 600 700 800 900 Tail Temp (F) Figure 512: Total cost function Qtot data (filtered) as a function of tail and nose tempera ture profiles. plotted in Figure 515 and Figure 516. The pitch rate peaktopeak errors do not have a large variation for the different plants, other than a noticeable poor performing region around Tnose = 5500F and Ttail = 4500F. The velocity peaktopeak has a minimum around the similar Tnose E [200, 600] F and Trail e [100, 250] F. The velocity peakto peak has minimums when the pitch rate has maximums, indicating a degree of trade off between better velocity performance, but worse pitch rate performance, and vice versa. An examination of the time to steadystate plots for pitch rate and velocity shown in Figures 517520 indicates relatively similar transient times, with a few outliers. Having little variation means that all the plant models are similar in the transient times with this particular control design. The time to steadystate is calculated by looking at the transient performance and extracting the time it takes for the error signals to decay below the steadystate peaktopeak error value. 700 " 600 E . 500 0 Z .n" 100 200 300 400 500 600 700 800 900 Tail Temp (F) Figure 513: Peaktopeak transient error for the pitch rate Q (t) tracking error in deg./sec.. 900 800 0.4 700 0.35 600 S0.3 500 0 z 400 0.25 300 0.2 200 0.15 100 100 200 300 400 500 600 700 800 900 Tail Temp (F) Figure 514: Peaktopeak transient error (filtered) for the pitch rate Q (t) tracking error in deg./sec.. 1UU 2UU 3UU 4UU bUU 6UU 7UU BUU 9UU Tail Temp (F) Figure 515: Peaktopeak transient error for the velocity V (t) tracking error in ft/sec.. 1.65 1.6 1.55 1.5 I (11C1 100 200 300 400 500 600 700 800 900 Tail Temp (F) Figure 516: Peaktopeak transient error (filtered) for the velocity V (t) tracking error in ft./sec.. S600 E . 500 0 Z Anr 100 200 300 400 500 600 Tail Temp (F) 700 800 900 Figure 517: Time to steadystate for the pitch rate Q (t) tracking error in seconds. 200 300 400 500 600 Tail Temp (F) 700 800 900 Figure 518: Time to steadystate (filtered) for the pitch rate Q (t) tracking error in sec onds. ~ 600 E 4 500 0 Z Anr 100 100 11.5 4UU bUU bUU Tail Temp (F) Figure 519: Time to steadystate for the velocity V (t) tracking error in seconds. 400 500 600 Tail Temp (F) Figure 520: Time to steadystate (filtered) for the velocity V (t) tracking error in seconds. 900 0.022 0.02 0.018 0.016 ,.,,.,0.014 0.012 0.01 0.008 "' 0.006 0.004 0.002 100 100 200 300 400 500 600 700 800 900 Tail Temp (F) Figure 521: Steadystate peaktopeak error for the pitch rate Q (t) in deg./sec.. Finally, the steadystate peaktopeak error values can be examined for both output signals. The steadystate peaktopeak errors are calculated by waiting until the error signal falls to within some nonvanishing steadystate bound after the initial transients have died down, and then measuring the maximum peaktopeak error within that bound. The plots for steadystate peaktopeak error for the pitch rate and velocity are shown in Figures 521 524. The steadystate peaktopeak errors show a minimum in the similar region as seen for other performance metrics, i.e. Tnose E [200, 600] F and Ttai e [100, 250] F. By normalizing all of the previous data about the minimum of each set of data, and then adding the plots together, a combined plot is obtained. This plot assumes that the designer weights each of the plots equally, but the method could be modified if certain aspects were deemed more important than others. Explicitly, data from each metric was combined as according to I1 A ,j (A) (58) A =l min (i,j (A)) 900 800 700 600 0 1 2 100 100 700 800 900 Figure 522: Steadystate peaktopeak error (filtered) for the pitch rate Q (t) in deg./sec.. ZUU JUU 4UU oUU oUU Tail Temp (F) 700 800 900 Figure 523: Steadystate peaktopeak error for the velocity V (t) in ft./sec.. 200 300 400 500 600 Tail Temp (F) x 103 12 10 8 6 4 2 300 100 100 x 103 10 IIII8 i.. .6 0 4 100 200 300 400 500 600 700 800 900 Tail Temp (F) Figure 524: Steadystate peaktopeak error (filtered) for the velocity V (t) in ft./sec. where ip is the new combined and normalized temperature profile data, A is the number of data sets being combined, and i, j are the location coordinates of the temperature data. Figure 525 shows this combination of control cost, error cost, peaktopeak error, time to steadystate, and steadystate peaktopeak error for both pitch rate and velocity tracking errors. By examining this cost function, an optimal region between Tnose E [200, 600] F and Trail e [100, 250] F is determined. In addition, optimal regions for the control gains can be examined. The control gains used for this problem are shown in (51) and (52) having the form 7= 0s k, 0 k, O 0 72 0 k,2 O k2 0 kY2 0 kFr By examining the control gains the maximum, minimum, mean, and standard deviation can be computed for all sets of control gains found to be near optimal. Table 51 SUU 7 6 4 Z 4C0C0 100 200 300 400 500 600 700 800 900 Tail Temp (F) Figure 525: Combined optimization p chart of the control and error costs, transient and steadystate values. Table 51: Optimization Control Gain Search Statistics ^71 '72 kC1 ks 2 k kU1 kk2 k kY1 kY2 kr1 kr2 Mean 25.35 36.60 16.07 265.3 28.38 9.65 27.43 14.12 0.972 0.8958 Std. 7.72 7.64 7.05 85.6 13.1 7.98 13.5 10.6 0.1565 0.133 Max 44.6 55.3 53.6 423.5 57.3 36.4 62.1 39.1 1.318 1.201 Min 7.14 3.58 6.30 9.762 0.360 0.050 0.392 0.110 0.658 0.6640 shows the control gain statistics. This data is useful in describing the optimal range for which control gains were selected. By knowing the region of near optimal attraction for the control gains, a future search could be confined to that region. The standard deviation also says something about the sensitivity of the control/aircraft dynamics, where larger standard deviations mean that particular gain has less effect on the overall system and vice a versa. 5.6 Conclusion A controloriented analysis of thermal gradients for a hypersonic vehicle (HSV) is presented. By incorporating nonlinear disturbances into the HSV model, a more representative controloriented analysis can be performed. Using the nonlinear controller developed in Chapter 2 and Chapter 4, performance metrics were calculated for a number of different HSV temperature profiles based on the design process initially developed in [6, 50]. Results from this analysis show that there is a range of temperature profiles that maximizes the controller effectiveness. For this particular study, the range was Tnose E [200,600] F and Ttail e [100, 250] F. In addition, this research has shown the range of control gains, useful for future design and numerical studies. This control oriented analysis data is useful for HSV structural designs and thermal protection systems. Knowledge of a desirable temperature profile and control gains will allow engineers and designers to build a HSV with the proper thermal protection that will keep the vehicle within a desired operating range based on control performance. In addition, this numerical study provides information that can be further used in more elaborate analysis processes and demonstrates one possible method for obtaining performance data for a given controller on the complete nonlinear HSV model. CHAPTER 6 CONCLUSIONS AND FUTURE WORK 6.1 Conclusions A new type on controller is developed for LPV systems that robustly compensates for the unknown state matrix, disturbances, and compensates for the uncertainty in the input dynamic inversion. In comparison with previous results, this work presents a novel approach in control design that stands out from the classical gain scheduling techniques such as standard scheduling, the use of LMIs, and the more recent development of LFTs, including their nonconvex /type optimization methods. Classical problems such as gain scheduling suffer from stability issues and the requirement that parameters only change slowly, limiting their use to quasilinear cases. LMIs use convex optimization, but typically require the use of numerical optimization schemes and are analytically intractable except in rare cases. LFTs further the control design for LPV systems by using small gain the ory, however they cannot deal explicitly with uncertain parameters. To handle uncertain parameters, the LFT problem is converted into a numerical optimization problem such as /type optimization. /type optimization is nonconvex and therefore solutions may not be found even when they exist. The robust dynamic inversion control developed for uncertain LPV systems alleviates these problems. As long as some knowledge of the input matrix is known and certain invertability requirements are met then a stabilizing con troller always exists. Proofs provided show that the controller is robust to disturbances, state dynamics, and uncertain parameters by using a new robust controller technique with exponential stability. Common applications for LPV systems are flight controllers. This is because his torically flight trajectories vary slowly with time and are well suited to the previously mentioned LPV control schemes such as gain scheduling. Recent advances in technology and aircraft design as well as more dynamic and demanding flight profiles have increased the demand on the controllers. In these demanding dynamic environments, parameters no longer change slowly and may be unknown or uncertain. This renders previous con trol designs limiting. Motivated by this fact and specifically using the dynamics of an airbreathing HSV, the dynamics are shown to be modeled as an LPV system with un certainties and disturbances. This work motivates the design and testing of the robust dynamic inversion controller on a temperature varying HSV. Using unknown temperature profiles, while simultaneously tracking an output trajectory, the robust controller is shown to compensate for unknown timevarying parameters in the presence of disturbances for the HSV. Using one set of control gains it was shown that stable control was maintained over the entire design space while performing maneuvers. Even though the control was de veloped for LPV systems, the simulation results are performed on the full nonlinear HSV flight and structural dynamics, hence validated the controloriented modeling assumptions. Finally, a numerical optimization scheme was performed on the same HSV model, using a combination of random search and evolving algorithms to produce dynamic optimization data for the combined vehicle and controller. Regions of optimality were shown to provide feedback to design engineers on the best suitable temperature profile parameter space. To remove ambiguity, the controller for each individual temperature profile case was optimally tuned and the tracking trajectory and disturbances were kept the same. Analytical methods do not exist for optimal gain tuning nonlinear controllers on nonlinear systems Hence, a numerical optimizing scheme was developed. By strategically searching the control gain space values were obtained, and the performance metrics at that point were compared across the vehicle design space. This work may be useful for future design problems for HSVs where the structural and dynamic design are performed in conjunction with the control design. 6.2 Contributions A new robust dynamic inversion controller was developed for general perturbed LPV systems. The control design requires knowledge of a best guess input matrix and at least as many inputs as tracked outputs. In the presence an unknown state matrix, parameters, and disturbance, and with an uncertain input matrix, the developed control design provides exponential tracking provided certain assumptions are met. The developed control method takes a different approach to traditional LPV design and provides a framework for future control design. * Because the assumptions required of the controller are met by the HSV, a numerical simulation was performed. After reducing the HSV nonlinear dynamics to that of an LPV system motivation was provided to implement the controller designed. A simulation is provided where the full nonlinear HSV dynamics are used. The simulation demonstrates the efficacy of the proposed control design on this particular HSV application. A wide range of temperature variations were used and tracking control was implemented to demonstrate the performance of the controller. * Further performance evaluation was conducted by designing an optimization proce dure to analyze the interplay between the HSV dynamics, temperature parameters, and controller performance. A number of different temperature plant models for HSV were near optimally tuned using a combination of a random search and evolv ing algorithms. Next, the control performance was evaluated and compared to the other HSV temperature models. Comparative analysis is provided that suggests regions where the temperature profiles of the HSV in conjunction with the proposed control design achieve improved performance results. These results may provide insight to structural systems designers for HSVs as well as provide scaffolding for future numerical design optimization and control tuning. 6.3 Future Work * The robust dynamic inversion control design in this dissertation requires knowledge of the sign of the error signal derivative terms. While these measurements may be available for specific applications, this underlying necessity reduces the generality of the controller. Future work could focus on removing this restriction, and producing an output feedback only robust dynamic inversion control. * Another requirement of the control design is the requirement of the diagonal dominance of the best guess feed forward input matrix. While this requirement is not unreasonable because it only requires that the guess be within the vicinity of the actual value, future work could focus on relaxing that requirement. Alleviating this restriction could potentially be done by using partial adaptation laws while simultaneously using robust algorithms to counter the parameter variations. * It was shown that the controller developed is able to track innerloop states for the HSV, however it would be beneficial to adapt this inner loop control design to an outer loop flight planning controller. In this way, more practical planned trajectories can be tracked (e.g., altitude) by using the inner loop of pitch rate and pitch angle control. Additionally, this same result can be attained by using backstepping techniques. By backstepping through other state dynamics (e.g., altitude) and into the control dynamics (e.g., pitch rate), a combined controller could be developed. * The temperature and control gain optimization provides a good framework for finding HSV designs with increased performance. It would be interesting in future work to reanalyze the optimal control gain space, and see if it could be converged to a smaller set. If the optimal set could be further converged, then through numerous iterations a very precise and narrow range may be found. Finding a more optimal design space may aid in future structural optimization searches. * It would also be beneficial for the optimization work to have more accurate nonlinear models. Obtaining better models will require working in collusion with HSV designers. Getting high quality feedback on the design constraints and flight trajectory constraints would further aid the search for optimality in regards to control gains and temperature profiles. In addition, the dynamics could be modeled and simulated with higher certainty if more details were known. Combining extra data on the dynamics into the control design would help further the development of actual flight worthy vehicles. APPENDIX A OPTIMIZATION DATA The data presented in the following tables is the raw data from the images presented in Chapter 5. The rows contains all of the Tos, in F and the columns contain the Trtil in F. Empty spaces are places where the tail temperature is higher than the nose temperature, and are outside the design space of this work and committed. Table Al: Total cost function, used to generate Figure 511 and 512 (Part 1) Ttail F 250 300 350 112 ','7 134531 140t233 144110 140079 140945 1 11,'"1 138181 140t633 140904 149182 141681 146708 143591 14, 1., 147113 147159 151291 1 1.;'','I , 1440t27 143730 129812 14. 1'", 142439 1 1I; 17 140439 143303 1 1. .'. I. 145086 11'27 145212 140633 400 450 500 140353 l1 11110 145178 139847 140785 144025 1 11,10 1401110. 142817 144027 149182 142468 139083 139202 145853 149182 139965 144790 140848 11'27 141932 1 1.:.:ns 144040 1117 I  129812 146527 1; 110 140940 140110. Table A2: Total cost function, used to generate Figure 511 and 512 (Part 2) TtailF Tnose 550 oF 600 650 750 800 850 550 144322 600 144420 111"7 650 145109 141262 127435 700 140633 144027 111. 27 14011i0. 750 140 11.. 11.:.:'", 139825 140233 146708 800 144948 143418 145297 135394 142384 140069 850 111',.: 141883 148014 136336 1.:1',1 145803 145941 900 11.:';s 147566 129349 138888 131875 11'11", 1;: ',1 13403 144526 143210 141588 141588 1l1. I> 1;;.;17 143490 129673 14011h 143730 143730 1 1.; 1 146708 11 11.10 140) "1; 141959 1 1.;*' ,'I , 145071 140254 140254 140577 145807 1 1. :. i, 141 . 1; ;' i11,1[ 144033 145599 137784 142439 149182 146015 138801 145086 143199 129636 139825 141577 141863 137552 1;;" 1.; l 145621 1 Iii.;'. 129812 139499 1 I:.:I 1 11110t Table A3: Control input cost function, used to generate Figure 57 and 58 (Part 1) TtailF T nose 100 150 200 250 300 350 400 450 500 100 95951 150 95949 95953 200 95948 95949 95952 250 95948 95949 95952 95952 300 95949 95950 95953 95953 95949 350 95952 95952 95957 95957 95953 95953 400 95948 95953 95957 95953 95954 95952 95953 450 95951 ',L.' 95952 95950 'i,.' 95953 95952 95953 500 95950 95950 95954 95948 95953 95949 95952 95948 95953 550 95952 95950 95957 95949 95954 95952 95948 95937 95949 600 95952 95949 95950 95952 95951 95952 95948 95937 95952 650 95953 95952 95953 95953 95953 95954 95949 95953 95950 700 95954 95952 95953 '.'il1 95953 95953 95952 95953 95952 750 95952 95953 95952 95953 95953 95954 95950 95950 95954 800 95953 95953 95953 95949 95953 95952 95953 95952 95957 850 95952 95953 95953 95953 95953 95957 95953 95949 95953 900 95953 95953 95952 95953 95952 95953 95953 95954 95950 Table A4: Control input cost function, used to generate Figure 57 and 58 (Part 2) TtailOF T ose 550 600 650 700 750 800 850 900 OF 550 95952 600 95952 95952 650 95949 95952 95952 700 95953 95953 95954 95950 750 95950 95953 95957 95953 95954 800 95953 95949 95953 95949 95953 95949 850 95953 95952 95953 95952 95953 95952 95953 900 95948 95953 95951 95940 95949 95953 r.'l1i 95950 Table A5: Error cost function, used to generate Figure 59 and 510 (Part 1) T nose 100 150 200 250 300 350 400 450 500 oF 100 1.74 150 17_', I 49118 200 1.. 11.:1 1. 4 47444 250 1. 11..;' I l.: 14 47444 li.,.i . 300 47136 11i_'. 47245 47942 I,117m. 350 37525 1',, 33679 38574 45727 47, 1 400 17. 11 4711.: 1.;',.7 44280 50754 11, 17 44400 450 33721 45337 l,_, 40418 7ID. 1 ., .S *..:' 500 44516 43114 1'"11s 48162 1l,1' 44490 1' 11,19 45979 550 47777 lN_' 41594 44129 51159 17.:.1 1.' 43146 47358 600 IT7 ,7 1i11l' 42479 11'i', 51208 47673 11 7 .'2.1 48088 650 47930 41831 1'i,.7 'li, 55337 .:, 1 48075 49900 I .:1 700 50754 1. 17 44400 122'., 48002 49133 '..S ',.._22s 33860 750 l.s ,.:' 33860 446.S0 48074 50572 44516 44015 50572 800 11'2 50062 1.. 1'. 44954 47776 33474 50911 1i.,7 ::'l" ' 850 46007 42848 46978 49969 1i'.:7. l'i. 1i.. 44898 11' 1. 900 48002 49133 48658 ..:_S 33860 44G680 48074 50572 44516 Table A6: Error cost function, used to generate Figure 59 and 510 (Part 2) TtailOF T Those 550 600 650 700 750 800 850 900 OF 550 137 600 1 1i. 48905 650 49160 45310 311"I 700 44S680 48074 50572 44516 750 44516 4711.: 1.;:,,7 44280 50754 800 48995 47469 1'. 1.: ;'i 1. l'i. 1.11 44120 850 1" ": 45931 52060 I.;'" 1 I 'I lI7';. 1'""7 900 47880 51613 33397 42947 ::.'. 11 ..42 39514 38653 Table A7: Pitch rate, peaktopeak error, used to generate Figure 513 and 514 (Part 1) TtailF Tnose F 0.1951 0.1678 0.2057 0.2057 0.1450 0.1374 0.1399 0.1535 0.2175 0 2..;" 0.1738 0.1560 0.1336 0.1434 0.1510 0.1939 0.1539 150 200 250 300 350 400 450 500 0.1377 0.1722 0.1722 (I 2",. 0.1712 0.1530 0.2478 0.2197 0.1624 (i I I "., 0.1327 0.1448 0.11.:, 0.1331 0.1468 0.1415 0.1421 0.1421 0.1365 0.1427 0.1500 0.1278 0.1505 0.1491 0.1 11'.. 0.1857 0.1849 0.1530 0.1502 0.1532 0.1434 0.1842 0.1803 0.1372 0.1835 0.2214 0.1728 0.1430 0.1548 0.1553 0.2928 0.1573 0.1573 0.1532 0.11.:, 0.1669 0.1536 0.1336 0.1590 0.131.: 0.2071 0.1406 0.1539 0.1692 0.1595 0.1992 0.1530 0.1601 0.1448 0.1421 0.1672 0.1867 0.1394 0.1400 0.1415 0.1655 0.1916 0.1 11 . 0.1573 0.1849 0.1434 0.1481 0.1848 0.1799 0.1374 0.1434 0.2175 0.1655 0.1432 0.1688 0.1 1 .:1 0.2174 0.4458 0.4561 0.1665 0.11.':. 0.2200 0.1832 ( 0.6, [ 0.1655 0.1292 0.2287 0.1471 0.1530 0.1530 0.1655 0.1473 0.1409 0.2175 Table A8: Pitch rate, peaktopeak error, used to generate Figure 513 and 514 (Part 2) TtailF Tnose 5 550 600 650 700 750 800 850 900 0.1787 0.1960 0.1309 0.1719 0.1947 0.1353 0.1573 0.1692 0.1655 0.2175 0.2912 0.1530 0.1612 0.1835 0.1939 0.1471 0.2673 0.1356 0.1:.1 0.1658 0.1833 0.1641 0.1323 0.1398 0.1507 0.1 1.. 0.1733 0.1395 0.1491 0.1 '.; 0.1499 0.3929 0.2276 0.1822 0.2941 0.2615 Table A9: Pitch rate, steadystate peaktopeak error, used to generate Figure 521 and 522 (Part 1) TtailF Tnose 0F 150 200 250 300 350 400 450 500 0.0170 0.0163 0.0179 0.0179 0.0176 0.0027 0.0232 0.0012 0.0053 0.0196 0.0173 0.0222 0.0186 0.0207 0.0165 0.0179 0.0182 0.0170 0.0156 0.0156 0.0166 0.0233 0.0169 0.0200 0.0048 0.0169 0.0186 0.0027 0.0154 0.0211 0.0213 0.0030 0.0202 0.0178 0.0178 0.0173 0.0016 0.0048 0.0173 0.0070 0.0036 0.0045 0.0164 0.0144 0.0008 0.0146 0.0161 0.0221 0.0163 0.0184 0.0028 0.0149 0.0031 0.0177 0.0166 0.0163 0.0152 0.0039 0.0154 0.0151 0.0170 0.0221 0.0167 0.0150 0.0186 0.0192 0.0166 0.0173 0.0178 0.0178 0.0172 0.0202 0.0163 0.0049 0.0008 0.0173 0.0154 0.0183 0.0166 0.0193 0.0183 0.0210 0.0191 0.0171 0.0029 0.0174 0.0160 0.0144 0.0221 0.0167 0.0159 0.0154 0.0173 0.0207 0.0053 0.0214 0.0150 0.0204 0.0221 0.0184 0.0032 0.0034 0.0202 0.0211 0.0056 0.0166 0.0176 0.0169 0.0160 0.0185 0.0185 0.0171 0.0008 0.0171 0.0050 0.0074 0.0053 Table A10: Pitch rate, steadystate peaktopeak error, used to generate Figure 521 and 522 (Part 2) Ttail F 600 650 700 750 800 850 900 0.0210 0.0185 0.0202 0.0169 0.0187 0.0198 0.0179 0.0019 0.0171 0.001 0.0163 0.0199 0.0009 0.0053 0.0149 0.0026 0.0034 0.0058 0.0180 0.0170 0.0163 0.0015 0.0193 0.0204 0.0183 0.0173 0.0041 0.0021 Tnose OF 0.0188 0.0180 0.0193 0.0154 0.0048 0.0171 0.0171 0.0226 Table A11: Pitch rate, time to steadystate, used to generate Figure 517 and 518 (Part 1) TtailF T 1ose 100 150 200 250 300 350 400 450 500 oF 100 0.439 150 0.429 0.433 200 0.412 0.338 0.451 250 0.471 0.287 0.472 0.541 300 0.450 0.381 0.518 0.518 0.515 350 0.394 0.499 0.431 0.540 0.512 0.472 400 0.471 0..12 0.412 0.494 0.402 0.511 0.474 450 0.556 0.407 0.405 0.475 0.473 0.444 0. 1I 0.519 500 0.580 0.613 0..12 0.450 0.424 0.473 0.404 0.1'", 0.473 550 0. 1.:'. 0.358 0.444 0.461 0.442 0.468 0.450 0.618 0.427 600 0.447 0. 1' 0.518 0.475 0.457 0.513 0.506 0.593 0.473 650 0.518 0.570 0.489 0.457 0.601 0.475 0.495 0.533 0.408 700 0.1. 0.471 0.449 0.677 0.1',7 0.464 0.449 0.1'",1 0.450 750 0. 1'7 0. 1'11 2.143 0.474 0.453 0.445 0.564 0.692 0.470 800 0.442 0.491 0.450 0.471 0.494 0. 1l' 0.473 0.495 0.587 850 0.432 0.576 0.470 0.527 0.593 0.464 0.477 0.517 0.572 900 0.494 0.1'12 0.1'7 0.427 0. 11. 0.1' 0.11.7 0.455 0.537 Table A12: Pitch rate, time to steadystate, used to generate Figure 517 and 518 (Part 2) TtailOF Tns ose 550 600 650 700 750 800 850 900 oF 550 0.450 600 0.421 0.476 650 0.453 0. I 1. 0.423 700 0.472 0.471 0.445 0.564 750 0.548 0.473 0.430 0.518 0.479 800 0.451 0.495 0.460 0.449 0.474 0.522 850 0.503 0.537 0.558 0.404 0.478 0.1,', 0.469 900 0.421 0.491 0.559 0.592 0.818 0.449 0.521 0.692 Table A13: Velocity, peaktopeak error, used to generate Figure 515 and 516 (Part 1) TtailF T nose 100 150 200 250 300 350 400 450 500 100 1.5670 150 1.1. l'. 1 .1 17 200 1.611. 1.6669 1.5972 250 1.6445 1.6668 1.5973 1.5986 300 1.6839 1.6663 1.6344 1.6055 1.6081 350 1.5596 1.5904 1.5401 1.5366 1.5580 1.6235 400 1.5836 1.6022 1.4917 1.5910 1.6735 1.5357 1.5910 450 1.5634 1.7.'1 1..'11 1.4420 1.7098 1.1lIII 1.5872 1.5456 500 1.111,' 1.11 .1 1.5321 1.5966 1.6579 1.1..;:' 1.5408 1.6075 1.5710 550 1.5893 1.7447 1.1.' 1.6417 1.6516 1.6166 1..'7 1.4219 1.5953 600 1.6537 1.5934 1.5359 1.5993 1.7076 1.6038 1.5962 1.4238 1.5834 650 1.5961 1.6176 1.6366 1.612' 1.4089 1.6170 1.7221 1.5990 1.6525 700 1.6735 1.5357 1.5910 1.4344 1.5949 1.5890 1.5876 1.5456 1.5294 750 1.5876 1.5456 1.5294 1.5980 1.6078 1.6965 1. 1i' 1.406t 1.6965 800 1.5948 1.6270 1.5828 1.6800 1.6248 1.5124 1.6033 1.6058 1.5128 850 1.5855 1.5205 1.'". I 1.6675 1.6205 1..1.:.; 1.5966 1.1~,.:1 1.6128 900 1.5940 1.5890 1.5872 1.5456 1.5294 1.5980 1.6078 1.6965 1.111i, Table A14: Velocity, peaktopeak error, used to generate Figure 515 and 516 (Part 2) TtailF Tns ose 550 600 650 700 750 800 850 900 OF 550 1.5814 600 1.'" 1., 1.'.7".. 650 1.6916 1.5845 1."7. I 700 1.5980 1.6078 1.6965 1.111'"1 750 1.4662 1.6022 1...1 1.5910 .1.61.:' 800 1.5951 1.6737 1.5572 1.1 '2 1.5873 1.6127 850 1.5830 1.5817 1.4735 1.5305 1.6027 1.5670 1.6405 900 1.6025 1.5693 1...12 1.4585 1.4581 1.6060 1.4721 1. i. Table A15: Velocity, steadystate peaktopeak, used to generate Figure 523 and 524 (Part 1) Ttail F 150 200 250 0.0037 0.0088 0.0050 0.0018 0.0036 0.001' 0.0019 0.0038 0.0047 0.0039 0.0037 0.0030 0.0034 0.0035 0.0016 0.0033 0.0010 0.0026 0.0035 0.0028 0.0013 0.0029 0.0002 0.0017 0.0027 0.0004 0.0021 0.0017 0.0015 0.0027 0.0069 0.0032 0.0012 0.0037 0.0038 0.0041 0.0022 0.0033 0.0035 0.0010 0.0070 0.0045 0.0059 0.0021 0.0031 0.0024 0.0014 0.0040 0.0008 0.0029 0.0035 0.0094 0.0038 0.0034 0.0028 0.0003 0.0075 0.0068 0.0023 0.0031 0.0031 0.0008 300 350 400 450 500 0.0131 0.0015 0.0066 0.0059 0.0021 0.0031 0.0105 0.0032 0.0014 0.0031 0.0037 0.0069 0.0022 0.0014 0.0027 0.0103 0.0022 0.001'. 0.0026 0.0035 0.0108 0.0018 0.0038 0.0027 0.0035 0.0041 0.0051 0.0040 0.0084 0.0066 0.0028 0.0039 0.0014 0.0040 0.0008 0.0037 0.0055 0.0021 0.0022 0.0055 0.0118 0.0009 0.0126 0.0033 0.0008 0.0006 0.0045 0.0024 0.0023 0.0016 0.0031 0.0040 0.0099 0.0021 0.0027 Table A16: Velocity, steadystate peaktopeak, used to generate Figure 523 and 524 (Part 2) TtailF Tnose 5 550 600 650 700 750 800 850 900 0.0022 0.0040 0.0027 0.0034 0.0036 0.0008 0.0029 0.0037 0.0055 0.0021 0.0016 0.0028 0.0013 0.0029 0.0041 0.0035 0.0054 0.0032 0.0015 0.0030 0.0028 0.0027 0.0007 0.0013 0.0018 0.0101 0.0041 0.0057 0.0027 0.0033 0.0002 0.0006 0.0005 0.0107 0.0005 0.0009 Table A17: Velocity, time to steadystate, used to generate Figure 519 and 520 (Part 1) TtailF T 1ose 100 150 200 250 300 350 400 450 500 oF 100 2.012 150 1.119 0.915 200 0.539 0 2'i. 0. 1'1. 250 0.498 0 1 0.528 0.506 300 0.' 1.: 0.314 0.586 0. 1'' 1.201 350 0.383 0.520 0.474 0.472 0.522 0.701 400 0.501 0.515 0.403 0.1',' 0.355 1.94 0.516 450 0.747 0.521 0.491 0.821 0..1.' 0.491 0.513 0.637 500 0.494 0.472 0.484 0.983 0.378 0.502 0.481 0. 12 0.656 550 0. 1'' 0.339 0.385 0.,11 0.568 1.208 0. 1'i. 0.841 1.201 600 0.562 0.500 0. 1". 0.563 0.578 1.01.: 0.1'1' 0.708 0.712 650 0.562 1.681 0.627 0.400 0.705 0.521 1.396 1.760 0.932 700 0.383 0. 1"r 0.704 0.808 0.836 0.491 0.504 0.516 3.330 750 0.459 0.587 3.300 1.347 0.539 0.679 0.473 0. 1',, 0.680 800 0.678 0.817 0.459 0.538 1.114 0.309 0.929 0.675 0.403 850 0.702 1.356 0.522 0.776 0.514 0.430 0.541 0.511 0.363 900 0.798 0.480 0.519 0.632 2.844 0.822 0.'1.' 0.607 0.518 Table A18: Velocity, time to steadystate, used to generate Figure 519 and 520 (Part 2) TtailOF Tns ose 550 600 650 700 750 800 850 900 oF 550 0.473 600 0.398 0.568 650 0.515 0.516 0.519 700 0.821 0.520 0.679 0.473 750 0.518 0.541 0.473 0.541 0.671 800 0.467 0.705 0.702 0.337 0.802 0.550 850 0..12 0.564 0.518 0.300 0.688 0.818 0.607 900 0.474 0.545 2.293 0.564 1.095 1.393 0.i.12 0.559 REFERENCES [1] L. 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MacKunis, Z. D. Wilcox, K. Kaiser, and W. E. Dixon, "Global adaptive output feedback MRAC," in Proc. IEEE Conf. Decis. Control Chin. Control Conf., 2009. [30] C. I. Marrison and R. F. Stengel, "Design of robust control systems for a hypersonic aircraft," J. Guid. Contr. Dynam., vol. 21, no. 1, pp. 5863, 1998. [31] A. Packard, "Gain scheduling via linear fractional transformations," Systems & Control Letters, vol. 22, no. 2, pp. 7992, 1994. [32] J. T. Parker, A. Serrani, S. Yurkovich, M. A. Bolender, and D. B. Doman, "Control oriented modeling of an airbreathing hypersonic vehicle," J. Guid. Contr. Dynam., vol. 30, no. 3, pp. 856869, 2007. [33] P. Patre, W. Mackunis, M. Johnson, and W. Dixon, "Composite adaptive control for EulerLagrange systems with additive disturbances," Automatica, vol. 46, no. 1, pp. 140147, 2010. [34] P. M. Patre, K. Dupree, W. MacKunis, and W. E. Dixon, "A new class of modular adaptive controllers, part II: Neural network extension for nonLP systems," in Proc. IEEE Am. Control Conf., 2008, pp. 12141219. [35] P. M. Patre, W. MacKunis, K. Dupree, and W. E. Dixon, "A new class of modular adaptive controllers, part I: Systems with linearintheparameters uncertainty," in Proc. IEEE Am. Control Conf., 2008, pp. 12081213. [36] RISEBased Robust and Adaptive Control of Nonlinear Systems. Boston: Birkhauser, 2009, under contract. [37] P. M. Patre, W. MacKunis, K. Kaiser, and W. E. Dixon, "Asymptotic tracking for uncertain dynamic systems via a multilayer neural network feedforward and RISE feedback control structure," IEEE Trans. Autom. Control, vol. 53, no. 9, pp. 21802185, 2008. [38] P. M. Patre, W. Mackunis, C. Makkar, and W. E. Dixon, "Asymptotic tracking for systems with structured and unstructured uncertainties," in IEEE Transactions on Control Systems Technology, vol. 16, No. 2, 2008, pp. 373379. [39] P. Pellanda, P. Apkarian, and H. D. Tuan, "Missile autopilot design via a multi channel Ift/lpv control method," Int. J. 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Dixon, "Control performance variation due to aerothermoelasticity in a hypersonic vehicle: Insights for structural design," in Proc. AIAA Guid. Navig. Control Conf., August 2009. [51] Z. D. Wilcox, W. MacKunis, S. Bhat, R. Lind, and W. E. Dixon, "Robust nonlinear control of a hypersonic aircraft in the presence of aerothermoelastic effects," in Proc. IEEE Am. Control Conf., St. Louis, MO, June 2009, pp. 25332538. [52] "Lyapunovbased exponential tracking control of a hypersonic aircraft with aerothermoelastic effects," J. Guid. Contr. Dynam., vol. 33, no. 4, Jul./Aug 2010. [53] T. Williams, M. A. Bolender, D. B. Doman, and O. Morataya, "An aerothermal flexible mode analysis of a hypersonic vehicle," in AIAA Paper 20066647, Aug. 2006. [54] B. Xian, D. M. Dawson, M. S. de Queiroz, and J. Chen, "A continuous asymptotic tracking control strategy for uncertain nonlinear systems," IEEE Trans. Autom. Control, vol. 49, no. 7, pp. 12061211, Jul. 2004. [55] M. Yoshihiko, "Adaptive gainscheduled Hinfinity control of linear parametervarying systems with nonlinear components," in Proc. IEEE Am. Control Conf., Denver, CO, June 2003, pp. 208213. BIOGRAPHICAL SKETCH Zach Wilcox grew up in Yarrow Point, a city just outside of Seattle, Washington, and lived there until moving to Florida to attend college in 2001. He received dual Bachelor of Science degrees from the University of Florida's Aerospace and Mechanical Engineering department in the spring of 2006. During his undergraduate work, Zach participated as a diver on UF's Men's Swimming Diving Team. In addition, he did research work for UF's Micro Air Vehicle (MAV) group and participated in International MAV competitions. He recieved his Masters of Science in Aerospace Engineering from University of Florida in the spring of 2008. His Doctoral studies were in the Nonlinear Controls and Robotics Group in the Department of Mechanical and Aerospace Engineering under the advisement of Dr. Dixon. He received his Ph.D. in Aerospace Engineering in August 2010. PAGE 1 NONLINEARCONTROLOFLINEARPARAMETERVARYINGSYSTEMSWITH APPLICATIONSTOHYPERSONICVEHICLES By ZACHARYDONALDWILCOX ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2010 1 PAGE 2 c 2010ZacharyDonaldWilcox 2 PAGE 3 Thisworkisdedicatedtomyparents,family,friends,andadvisor,whohaveprovidedme withsupportduringthechallengingmomentsinthisdissertationwork. 3 PAGE 4 ACKNOWLEDGMENTS Iwouldliketoexpresssinceregratitudetomyadvisor,Dr.WarrenE.Dixon,who isapersonwithremarkablea ability.Asanadvisor,heprovidedthenecessaryguidance andallowedmetodevelopmyownideas.Asamentor,hehelpedmeunderstandthe intricaciesofworkinginaprofessionalenvironmentandhelpeddevelopmyprofessional skills.Ifeelfortunateingettingtheopportunitytoworkwithhim. 4 PAGE 5 TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................4 LISTOFTABLES........................... ..........7 LISTOFFIGURES...... .................... ..........8 ABSTRACT........................................10 CHAPTER 1INTRODUCTION.... .................... ..........12 1 .1MotivationandProblemStatement......................12 1.2OutlineandContributions...........................16 2LYAPUNOVBASEDEXPONENTIALTRACKINGCONTROLOFLPVSYSTEMSWITHANUNKNOWNSYSTEMMATRIX,UNCERTAININPUTMATRIXVIADYNAMICINVERSION........................19 2.1Introduction ...................................19 2.2LinearParameterVaryingModel.......................21 2.3ControlDevelopm ent..............................23 2.3.1ControlObjective............................23 2.3.2OpenLoopErrorSystem........................24 2.3.3ClosedLoopErrorSystem.......................25 2.4StabilityAnalysi s................. ...............27 2.5Conclusions..... .................... ..........30 3HYPERSONICVEHICLEDYNAMICSANDTEMPERATUREMODEL...32 3.1Introduction ...................................32 3.2RigidBodyandElasticDynamics.......................32 3.3TemperaturePro leModel...........................33 3.4Conclusion...... .................... ..........38 4LYAPUNOVBASEDEXPONENTIALTRACKINGCONTROLOFAHYPERSONICAIRCRAFTWITHAEROTHERMOELASTICEFFECTS.....39 4.1Introduction ...................................39 4.2HSVModel..... .................... ..........41 4.3ControlObjectiv e................. ...............42 4.4SimulationResult s...............................44 4.5Conclusion...... .................... ..........48 5 PAGE 6 5CONTROLPERFORMANCEVARIATIONDUETONONLINEARAEROTHERMOELASTICITYINAHYPERSONICVEHICLE:INSIGHTSFORSTRUCTURALDESIGN..... .................... ..........53 5.1Introduction ...................................53 5.2DynamicsandController............................54 5.3OptimizationviaRandomSearchandEvolvingAlgorithms.........55 5.4ExampleCase.... .................... ..........57 5.5Results............................ ..........61 5.6Conclusion...... .................... ..........73 6CONCLUSIONSANDFUTUREWORK......................75 6.1Conclusions..... .................... ..........75 6.2Contributions ..................................76 6.3FutureWork..... .................... ..........77 APPENDIX AOPTIMIZATIONDAT A...............................79 REFERENCES......... .................... ..........89 BIOGRAPHICALSKETCH ................. ...............94 6 PAGE 7 LISTOFTABLES Table page 31Naturalfrequenciesfor5lineartemperaturepro les(Nose/Tail)indegreesF. Percentdi erenceisthedi erencebetweenthemaximumandminimumfrequenciesdividedbytheminimumfrequency.....................36 51OptimizationControlGainSearchStatistics....................73 A1Totalcostfunction,usedtogenerateFigure511and512(Part1).......80 A2Totalcostfunction,usedtogenerateFigure511and512(Part2).......80 A3Controlinputcostfunction,usedtogenerateFigure57and58(Part1)....81 A4Controlinputcostfunction,usedtogenerateFigure57and58(Part2)....81 A5Errorcostfunction,usedtogenerateFigure59and510(Part1)........82 A6Errorcostfunction,usedtogenerateFigure59and510(Part2)........82 A7Pitchrate,peaktopeakerror,usedto generateFigure513and514(Part1).83 A8Pitchrate,peaktopeakerror,usedto generateFigure513and514(Part2).83 A9Pitchrate,steadystatepeaktopeakerror,usedtogenerateFigure521and522(Part1)............................ ..........84 A10Pitchrate,steadystatepeaktopeakerror,usedtogenerateFigure521and522(Part2)............................ ..........84 A11Pitchrate,timetosteadystate,usedtogenerateFigure517and518(Part1)85 A12Pitchrate,timetosteadystate,usedtogenerateFigure517and518(Part2)85 A13Velocity,peaktopeakerror,usedtogenerateFigure515and516(Part1)..86 A14Velocity,peaktopeakerror,usedtogenerateFigure515and516(Part2)..86 A15Velocity,steadystatepeaktopeak,usedtogenerateFigure523and524(Part 1)....................... ....................87 A16Velocity,steadystatepeaktopeak,usedtogenerateFigure523and524(Part 2)....................... ....................87 A17Velocity,timetosteadystate,usedtogenerateFigure519and520(Part1)..88 A18Velocity,timetosteadystate,usedtogenerateFigure519and520(Part2)..88 7 PAGE 8 LISTOFFIGURES Figure page 31Modulusofelasticityforthe rstthreedynamicmodesofvibrationforafreefreebeamoftitanium .................. ...............34 32Frequenciesofvibrationforthe rstthreedynamicmodesofafreefreetitaniumbeam............................. ..........35 33NineconstanttemperaturesectionsoftheHSVusedfortemperaturepro le modeling.......... .................... ..........35 34Lineartemperaturepro lesusedtocalculatevaluesshowninTable31......37 35Asymetricmodeshapesforthehypersonicvehicle.Thepercentdi erencewas calculatedbasedonthemaximumminustheminimumstructuralfrequencies dividedbytheminimu m................. ...............37 41Temperaturevariationfortheforebodyandaftbodyofthehypersonicvehicle asafunctionoftime. .................................45 42Inthis gure, denotesthe elementinthedisturbancevecor .Disturbances fromtoptobottom:velocity ,angleofattack ,pitchrate ,the 1elasticstructuralmode 1,the 2elasticstructuralmode 2,andthe 3elastic structuralmode 3,asdescribedin(411)......................46 43Referencemodelouputs ,whicharethedesiredtrajectoriesfortop:velocity ( ) ,middle:angleofattack ( ) ,andbottom:pitchrate ( ) .......47 44Top:velocity ( ) ,bottom:velocitytrackingerror ( ) .............48 45Top:angleofattack ( ) ,bottom:angleofattacktrackingerror ( ) ......49 46Top:pitchrate ( ) ,bottom:pitchratetrackingerror ( ) ..........49 47Top:fuelequivalenceratio .Middle:elevatorde ection .Bottom:Canard de ection ................... ....................50 48Top:altitude ( ) ,bottom:pitchangle ( ) ...................50 49Top: 1structuralelasticmode 1.Middle:2structuralelasticmode 2.Bottom: 3structuralelasticmode 3..........................51 51HSVsurfacetemperaturepro les. [450 900 ] ,and [100 800 ] .54 52Desiredtrajectories:pitchrate (top)andvelocity (bottom).........58 53Disturbancesforvelocity (top),angleofattack (secondfromtop),pitch rate (secondfrombottom)andthe 1structuralmode(bottom)........58 8 PAGE 9 54Trackingerrorsforthepitchrate indegrees/sec(top)andthevelocity in ft/sec(bottom). ....................................59 55Controlinputsfortheelevator indegrees(top)andthefuelratio (bottom).60 56Costfunctionvaluesforthetotalcost (top),theinputcost (middle) andtheerrorcost (bottom)...........................60 57Controlcostfunction dataasafunctionoftailandnosetemperatureproles.................. .........................62 58Controlcostfunction data( ltered)asafunctionoftailandnosetemperaturepro les.................. ....................62 59Errorcostfunction dataasafunctionoftailandnosetemperaturepro les.63 510Errorcostfunction data( ltered)asafunctionoftailandnosetemperaturepro les.......................................63 511Totalcostfunction dataasafunctionoftailandnosetemperaturepro les.64 512Totalcostfunction data( ltered)asafunctionoftailandnosetemperaturepro les.......................................65 513Peaktopeaktransienterrorforthepitchrate ( ) trackingerrorindeg./sec...66 514Peaktopeaktransienterror( ltered)forthepitchrate ( ) trackingerrorin deg./sec........... .................... ..........66 515Peaktopeaktransienterrorforthevelocity ( ) trackingerrorinft/sec.....67 516Peaktopeaktransienterror( ltered)forthevelocity ( ) trackingerrorin ft./sec............ .................... ..........67 517Timetosteadystateforthepitchrate ( ) trackingerrorinseconds.. .....68 518Timetosteadystate( ltered)forthepitchrate ( ) trackingerrorinseconds.68 519Timetosteadystateforthevelocity ( ) trackingerrorinseconds........69 520Timetosteadystate( ltered)forthevelocity ( ) trackingerrorinseconds..69 521Steadystatepeaktopeakerrorforthepitchrate ( ) indeg./sec.........70 522Steadystatepeaktopeakerror( ltered)forthepitchrate ( ) indeg./sec....71 523Steadystatepeaktopeakerrorforthevelocity ( ) inft./sec...........71 524Steadystatepeaktopeakerror( ltered)forthevelocity ( ) inft./sec.....72 525Combinedoptimization chartofthecontrolanderrorcosts,transientand steadystatevalues. .................................. 7 3 9 PAGE 10 AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFul llmentofthe RequirementsfortheDegreeofDoctorofPhilosophy NONLINEARCONTROLOFLINEARPARAMETERVARYINGSYSTEMSWITH APPLICATIONSTOHYPERSONICVEHICLES By ZacharyDonaldWilcox August2010 Chair:WarrenE.Dixon Major:AerospaceEngineering Thefocusofthisdissertationistodesignacontrollerforlinearparametervarying (LPV)systems,applyitspeci callytoairbreathinghypersonicvehicles,andexaminethe interplaybetweencontrolperformanceandthestructuraldynamicsdesign.Speci callya Lyapunovbasedcontinuousrobustcontrollerisdevelopedthatyieldsexponentialtracking ofareferencemodel,despitethepresenceofbounded,nonvanishingdisturbances.The hypersonicvehiclehastimevaryingparameters,speci callytemperaturepro les,andits dynamicscanbereducedtoanLPVsystemwithadditivedisturbances.SincetheHSV canbemodeledasanLPVsystemtheproposedcontroldesignisdirectlyapplicable.The controlperformanceisdirectlyexaminedthroughsimulations. Awidevarietyofapplicationsexistthatcanbee ectivelymodeledasLPVsystems. Inparticular, ightsystemshavehistoricallybeenmodeledasLPVsystemsandassociated controltoolshavebeenappliedsuchasgainscheduling,linearmatrixinequalities(LMIs), linearfractionaltransformations(LFT),and types.However,asthetypeof ight environmentsandtrajectoriesbecomemoredemanding,thetraditionalLPVcontrollers maynolongerbesu cient.Inparticular,hypersonic ightvehicles(HSVs)presentan inherentlydi cultproblembecauseofthenonlinearaerothermoelasticcouplinge ectsin thedynamics.HSV ightconditionsproducetemperaturevariationsthatcanalterboth thestructuraldynamicsand ightdynamics.Startingwiththefullnonlineardynamics, theaerothermoelastice ectsaremodeledbyatemperaturedependent,parametervarying 10 PAGE 11 statespacerepresentationwithaddeddist urbances.Themodelincludesanuncertain parametervaryingstatematrix,anuncert ainparametervaryingnonsquare(column de cient)inputmatrix,andanadditiveboundeddisturbance.Inthisdissertation,a robustdynamiccontrollerisformulatedforauncertainanddisturbedLPVsystem.The developedcontrolleristhenappliedtoaHSVmodel,andaLyapunovanalysisisusedto proveglobalexponentialreferencemodeltrackinginthepresenceofuncertaintyinthe stateandinputmatricesandexogenousdisturbances.Simulationswithaspectrumof gainsandtemperaturepro lesonthefullnonlineardynamicmodeloftheHSVisusedto illustratetheperformanceandrobustnessofthedevelopedcontroller. Inaddition,thisworkconsidershowtheperformanceofthedevelopedcontroller variesoverawidevarietyofcontrolgainsandtemperaturepro lesandareoptimized withrespecttodi erentperformancemetrics.Speci cally,varioustemperaturepro le modelsandrelatednonlineartemperaturedependentdisturbancesareusedtocharacterize therelativecontrolperformanceande ortforeachmodel.Examiningsuchmetricsas afunctionoftemperatureprovidesapotentialinroadtoexaminetheinterplaybetween structural/thermalprotectiondesignandcontroldevelopmentandhasapplicationfor futureHSVdesignandcontrolimplementation. 11 PAGE 12 CHAPTER1 INTRODUCTION 1.1MotivationandProblemStatement RecentresearchonnonlinearinversionoftheinputdynamicsbasedonLyapunov stabilitytheoryprovidesasteppingstonetoLPVdynamicinversion.In[27,28],dynamic inversiontechniquesareusedtodesigncontrollersthatcanadaptivelyandrobustly stabilizestatespacesystemswithuncertainconstantparametersandadditiveunknown boundeddisturbances.However,thisworkislimitedtotimeinvarientparametersand thereforeisnotapplicabletoLPVsystems.Theworkpresentedinthischapterisan extensionoftheworkin[27,28],andprovidesacontinuousrobustcontrollerthatisable tostabilizegeneralperturbedLPVsystemswithdisturbances,whenboththestate,input matrices,timevaryingparameters,anddisturbancesareunknown. ThedesignofguidanceandcontrolsystemsforairbreathingHSVischallengingbecausethedynamicsoftheHSVarecomplexandhighlycoupledasin[10],and temperatureinducedsti nessvariationsimpactthestructuraldynamicssuchasin[21]. Muchofthisdi cultyarisesfromtheaerodynamic,thermodynamic,andelasticcoupling (aerothermoelasticity)inherentinHSVsystems.BecauseHSVtravelatsuchhighvelocities(inexcessofMach5)therearelargeamountsofaerothermalheating.Aerothermal heatingisnonuniform,generallyproducin gmuchhighertemperatu resatthestagnation pointofair ownearthefrontofthevehicle.Coupledwithadditionalheatingdueto theengine,HSVshavelargethermalgradientsbetweenthenoseandtail.Thestructural dynamics,inturn,a ecttheaerodynamicproperties.Vibrationintheforwardfuselage changestheapparentturnangleofthe ow,whichresultsinchangesinthepressure distributionovertheforebodyoftheaircraft.Theresultingchangesinthepressuredistributionovertheaircraftmanifestthemselvesasthrust,lift,drag,andpitchingmoment perturbationsasin[10].TodevelopcontrollawsforthelongitudinaldynamicsofaHSV 12 PAGE 13 capableofcompensatingforthesestructuralandaerothermoelastice ects,structural temperaturevariationsandstructu raldynamicsmustbeconsidered. Aerothermoelasticityistheresponseofelasticstructurestoaerodynamicheatingand loading.Aerothermoelastice ectscannotbeignoredinhypersonic ight,becausesucheffectscandestabilizetheHSVsystemasin[21].Alossofsti nessinducedbyaerodynamic heatinghasbeenshowntopotentiallyinducedynamicinstabilityinsupersonic/hypersonic ightspeedregimesasin[1].Yetactivecontrolcanbeusedtoexpandthe utterboundaryandconvertunstablelimitcycleoscillations(LCO)tostableLCOasshownin[1].An activestructuralcontrollerwasdevelopedin[26],whichaccountsforvariationsintheHSV structuralpropertiesresultingfromaerothermoelastice ects.Thecontroldesignin[26] modelsthestructuraldynamicsusingaLPVframework,andstatesthebene tstousing theLPVframeworkaretwofold:thedynamicscanberepresentedasasinglemodel,and controllerscanbedesignedthathavea nedependencyontheoperatingparameters. Previouspublicationshaveexaminedthechallengesassociatedwiththecontrol ofHSVs.Forexample,HSV ightcontrollersaredesignedusinggeneticalgorithmsto searchadesignparameterspacewherethenonlinearlongitudinalequationsofmotion containuncertainparametersasin[4,30,49 ].SomeofthesedesignsutilizeMonteCarlo simulationstoestimatesystemrobustnessateachsearchiteration.Anotherapproach [4]istousefuzzylogictocontroltheattitudeoftheHSVaboutasinglelowend ight condition.Whilesuchapproachesasin[4,30,49]generatestabilizingcontrollers,the proceduresarecomputationallydemandingandrequiremultipleevaluationsimulations oftheobjectivefunctionandhavelargeconvergenttimes.Anadaptivegainscheduled controllerin[55]wasdesignedusingestimatesofthescheduledparameters,andasemioptimalcontrollerisdevelopedtoadaptivelyattain controlperformance.This controlleryieldsuniformlyboundedstabilityduetothee ectsofapproximationerrors andalgorithmicerrorsintheneuralnetworks.Feedbacklinearizationtechniqueshave beenappliedtoacontrolorientedHSVmodeltodesignanonlinearcontrollerasin[32]. 13 PAGE 14 Themodelin[32]isbasedonapreviouslydevelopedHSVlongitudinaldynamicmodel in[8].Thecontroldesignin[32]neglectsvariationsinthrustliftparameters,altitude, anddynamicpressure.Linearoutputfeedbacktrackingcontrolmethodshavebeen developedin[44],wheresensorplacementstrategiescanbeusedtoincreaseobservability, orreconstructfullstateinformationforas tatefeedbackcontroller.Arobustoutput feedbacktechniqueisalsodevelopedforthelinearparameterizableHSVmodel,which doesnotrelyonstateobservation.Arobustsetpointregulationcontrollerin[17]is designedtoyieldasymptoticregulationinthepresenceofparametricandstructural uncertaintyinalinearpar ameterizableHSVsystem. Anadaptivecontrollerin[19]wasdesignedtohandle(linearintheparameters) modelinguncertainties,actuatorfailures,andnonminimumphasedynamicsasin[17] foraHSVwithelevatorandfuelratioinputs.Anotheradaptiveapproachin[41]was recentlydevelopedwiththeadditionofaguidancelawthatmaintainsthefuelratio withinitschokinglimits.Whileadaptivecontrolandguidancecontrolstrategiesfora HSVareinvestigatedin[17,19,41],neitheraddressesthecasewheredynamicsinclude unknownandunmodeleddisturbances.Thereremainsaneedforacontinuouscontroller, whichiscapableofachievingexponentialtrackingforaHSVdynamicmodelcontaining aerothermoelastice ectsandunmodeleddisturbances(i.e.,nonvanishingdisturbancesthat donotsatisfythelinearintheparametersassumption). Inthecontextoftheaforementionedliterature,acontributionofthisdissertation (andinthepublicationsin[51]and[52])isthedevelopmentofacontrollerthatachieves exponentialmodelreferenceoutputtrackingdespiteanuncertainmodeloftheHSV thatincludesnonvanishingexogenousdisturbances.Anonlineartemperaturedependent parametervaryingstatespacerepresentationisusedtocapturetheaerothermoelasticeffectsandunmodeleduncertaintiesinaHSV.Thismodelincludesanunknownparametervaryingstatematrix,anuncertainparametervaryingnonsquare(columnde cient)input matrix,andanonlinearadditiveboundeddisturbance.Toachieveanexponentialtracking 14 PAGE 15 resultinlightofthesedisturbances,arobust,continuousLyapunovbasedcontrolleris developedthatincludesanovelimplicitlearningcharacteristicthatcompensatesforthe nonvanishingexogenousdisturbance.Thatis,theuseoftheimplicitlearningmethod enablesthe rstexponentialtrackingresultbyacontinuouscontrollerinthepresenceof theboundednonvanishingexogenousdisturbance.Toillustratetheperformanceofthe developedcontroller,simulationsareperformedonthefullnonlinearmodelgivenin[10] thatincludesaerothermoelasticmodeluncertaintiesandnonlinearexogenousdisturbances whosemagnitudeisbasedonairspeed uctuations. Inadditiontothecontroldevelopment,thereexiststheneedtounderstandthe interplayofacontroldesignwithrespecttothevehicledynamics.Apreviouscontrol orienteddesignanalysisin[6]statesthatsimultaneouslyoptimizingboththestructural dynamicsandcontrolisanintractableproblem,butthatcontrolorienteddesignmay beperformedbyconsideringtheclosedloopperformanceofanoptimalcontrollerona seriesofdi erentopenloopdesignmodels.Thebestperformingdesignmodelisthensaid tohavetheoptimaldynamicsinthesenseofcontrollability.Knowledgeoftheoptimal thermalgradientswillprovideinsighttoengineersonhowtoproperlyweighttheHSVs thermalprotectionsystemforbothsteadystateandtransient ight.Thepreliminary workbyauthorsin[6]providesacontrolorienteddesignarchitecturebyinvestigating controlperformancevariationsduetothermalgradientsusingan Hcontroller.Chapter 5seekstoextendthecontrolorienteddesignconcepttoexaminecontrolperformance variationsforHSVmodelsthatincludenonlinearaerothermoelasticdisturbances.Given thesedisturbances,Chapter5focusesonexaminingcontrolperformancevariationsfor themodelreferencerobustcontrollerinChapter2andChapter4toachieveanonlinear controlorientedanalysiswithrespecttothermalgradientsontheHSV.Byanalyzing controlerrorandinputnormsaswellastransientandsteadystateresponsesoverawide rangeoftemperaturepro lesanoptimaltemperaturepro lerangeissuggested. 15 PAGE 16 1.2OutlineandContributions Thisdissertationfocusesondesigninganonlinearcontrollerforgeneraldisturbed LPVsystem.Thecontrolleristhenmodi edforaspeci cairbreathingHSV.The dynamicinversiondesignisatechniquethatallowsthemultiplicativeinputmatricesto beinverted,thusrenderingthecontrollera neinthecontrol.Previousresultsin[27]and [29]haveexaminedfullstateandoutputfeedbackadaptivedynamicinversioncontrollers, butarelimitedbecausetheycontainconstantuncertainties.TheHSVsystempresents anewchallengebecausetheuncertainstate andinputmatricesareparametervarying. Speci cally,thestateandinputmatricesofthehypersonicvehiclevaryasafunctionof temperature.Thischapterprovidessomebackgroundandmotivatestherobustdynamic inversioncontrolmethodsubsequentlydeveloped.Abriefoutlineofthefollowingchapters follows. InChapter2atrackingcontrollerispresentedthatachievesexponentialstabilityof amodelreferencesysteminthepresenceofuncertaintiesanddisturbances.Speci cally, theplantmodelcontainstimevaryingparametricuncertaintywithdisturbancesthatare boundedandnonvanishing.Thecontributionofthisresultisthatitrepresentsthe rst everdevelopmentofanexponentiallystablecontinuousrobustmodelreferencetracking controllerforanLPVsystemwithanunknownsystemmatrixanduncertaininputmatrix withanadditiveunknownboundeddisturbance.Lyapunovbasedmethodsareusedto proveexponentialstabilityofthesystem. Chapter3providesthenonlineardynamicsandtemperaturemodelofaHSV.The nonlinearandhighlycoupleddynamicequationsarepresented.Theequationsthat de netheaerodynamicandgeneralizedmomentsandforcesareprovidedexplicitlyin previousliterature.Thischapterismeanttoserveasanoverviewofthedynamicsofthe HSV.Inadditiontothe ightandstructuraldynamics,temperaturepro lemodelingis provided.TemperaturevariationsimpacttheHSV ightdynamicsthroughchangesinthe structuraldynamicswhicha ectthemodeshapesandnaturalfrequenciesofthevehicle. 16 PAGE 17 Thepresentedmodelo ersanapproximateapproach,wherebythenaturalfrequencies ofacontinuousbeamaredescribedasafunctionofthemassdistributionofabeam anditssti ness.Figuresandtablesarepresentedtoemphasizetheneedtoincludesuch dynamicsforcontroldesign.Thischapterisdesignedtofamiliarizethereaderwiththe HSVdynamicandtemperaturemodels,sincethesedynamicsareusedthroughoutthis dissertation.ThischapterisaprecursorandintroductiontoChapter4andChapter5. UsingthecontrollerdevelopedinChapter2,thecontributioninChapter4isto illustrateanapplicationtoanairbreathinghypersonicvehiclesystemwithadditive boundeddisturbancesandaerothermoelastice ects,wherethecontrolinputismultiplied byanuncertain,columnde cient,parametervaryingmatrix.Inadditiontothestability proof,thecontroldesignisalsovalidatedthroughimplementationinafullnonlinear dynamicsimulation.Theexogenousdisturbances(e.g.,windgust,enginevariations,etc.) andtemperaturepro les(aerodynamicdriventhermalheating)aredesignedtoexamine therobustnessofthedevelopedcontroller.Theresultsfromthesimulationillustratethe boundednessofthecontrollerwithfavorabletransientandsteadystatetrackingerrorsand provideevidencethatthecontrolmodelusedfordevelopmentisvalid. ThecontributioninChapter5istoprovideananalysisframeworktoexaminethe nonlinearcontrolperformancebasedonvariationsinthevehicledynamics.Speci cally, thechangesoccurinthestructuraldynamicsviatheirresponsetodi erenttemperature pro les,andhencetheobservedvibrationhasdi erentfrequenciesandshapes.Using aninitialrandomsearchandevolvingalgorithms,approximateoptimalgainsarefound forthecontrollerforeachtemperaturedependantplantmodel.Errors,controle ort, transientandsteadystateperformanceanalysisisprovided.Theresultsfromthisanalysis showthatthereisatemperaturerangeforoperationoftheHSVthatminimizesagiven costofperformanceversuscontrolauthorit y.Knowledgeofafavorablerangewithregard tocontrolperformanceprovidesdesignersanextratoolwhendevelopingthethermal protectionsystemaswellasthestructuralcharacteristicsoftheHSV. 17 PAGE 18 Chapter6summarizesthecontributionsofthedissertationandpossibleavenuesfor futureworkareprovided.Thebriefcontri butionsoftheLPVcontroller,HSVexample controllerdesignapplication,andtheHSVoptimizationprocedureprovidethebaseofthis dissertation.Afterabriefsummary,someofthedrawbacksofthecurrentcontroldesign arepresentedasdirectionsforfutureresearchwork. 18 PAGE 19 CHAPTER2 LYAPUNOVBASEDEXPONENTIALTRACKINGCONTROLOFLPVSYSTEMS WITHANUNKNOWNSYSTEMMATRIX,UNCERTAININPUTMATRIXVIA DYNAMICINVERSION 2.1Introduction Linearparametervarying(LPV)systemsha veawiderangeofpracticalengineering applications.Someexamplesincludeseveralmissileautopilotdesignsasin[7,39,43], aturbofanengine[5],andactivesuspensiondesign[18].Traditionally,LPVsystems havebeendevelopedusingagainschedulingcontrolapproach.Gainschedulingisa techniquetodevelopcontrollersfornonlinearsystemusingtraditionallinearcontrol theory.Gainschedulingisatechniquewherethesystemislinearizedaboutcertain operatingconditions.Abouttheseoperatingconditions,constantparametersareassumed andseparatecontrolschemesandgainsarechosen.Morethanadecadeago,Shammaet. al.pointedoutsomeofthepotentialhazardsofgainschedulingin[42].Inparticular,gain schedulingisaanalyticallynoncontinuousmethodandstabilityisnotguaranteedwhile switchingfromoneregionoflinearizationtoanother.Infactthetwobiggestdownfallsof gainschedulingcontroldesignisthelinearizationoftheplantmodelsclosetoequilibrium orconstantparametersstatesandtherequirementthattheparametersmustchange slowly.Becausethelinearizationisrequiredtobeclosetosomeoperationcondition orstabilitypoint,manydi erentscheduleshavetobetaken.Andbyrequiringthat parameterschangeslowly,thegainschedulingtechniquesarenotappropriateformany quicklyvaryingsystems. AnotherapproachtoLPVproblemsistheuseoflinearmatrixinequalities(LMIs). InabookonLMIsandtheiruseinsystemandcontroltheoryin[11],Boydet.al.states thatLMIsaremathematicallyconvexoptimizationproblemswithextensionstocontrol theory.Howeverin[11]itispointedoutthatthesetypicallyrequirenumericalsolutions andthereareonlyafewspecialcaseswhereanalyticalsolutionsexist.TheseLPV solutionstypicallyonlyprovidenormbasedsolutions.Themostcommonoftheseisthe 19 PAGE 20 2normbecauseitallowsforcontinuitywith controlwhenthesystemsbecomelinear timeinvariant.Forinstance controlisdevelopedin[14]whichusesLMIstooptimize thesolutionandin[3],theparameterizationofLMIswasinvestigatedinthecontextof controltheory. controlisdevelopedin[14],whichusesLMIstooptimizethesolution andSaifet.al.in[48]showsthatstabilizationsolutionsexistformultiinputmultioutput (MIMO)systemsusingLMIs.ThesedesignsallowforthecontinuoussolutionofLPV systems,howeverknowledgeofthestructureofthesystemmustbeknown,andthe parametersareassumedmeasurableonline.In[25]minimaxcontrollersaredesignedto handleonlyconstantorsmallvariationsintheparameters,wheretheparameterized algebraicRiccatiinequalitiesareconvertedintoequivalentLMIssothattheconvexity canbeexploitedandacontrollerdeveloped.ContinuouscontroldesignforuncertainLPV systemsin[13]isdesignedusingLMIs,howevertheprocedureislimitedtouncertaintiesin thestatematrix,anddoesnotcoveruncertaintiesintheinputmatrix. AnotherapproachuseslinearfractionaltransformationsLFTsinthecontextofLPV controldesignsuchasin[31]andarebasedonsmallgaintheory.Thisapproachcannot handleuncertainparameters.However,byextendingthesolutionin[31]thedesigncan includeuncertainparameterswhicharenotavailabletothecontroller.Thesesolutions are synthesistypecontrollers,howeverthesolvabilityconditionsarenonconvexand thereforeasolutiontotheproblemisnotguaranteedevenwhenastablecontrollerexists. Severalexamplesofrecursive typesolutionsaregivenin[2,22,45].Morerecentlyin [26],the typesolutionshavebeenextendedtoahypersonicaircraftexample,butsu ers thesamenonconvexityproblemastheformerlylisted typeliterature. RecentresearchonnonlinearinversionoftheinputdynamicsbasedonLyapunov stabilitytheoryprovidesasteppingstonetoLPVdynamicinversion.In[27,28],dynamic inversiontechniquesareusedtodesigncontrollersthatcanadaptivelyandrobustly stabilizeamoregeneralstatespacesystemthathasbeenconsideredinpreviousworkwith uncertainconstantparametersandadditiveunknownboundeddisturbances.However, 20 PAGE 21 thisworkislimitedtotimeinvarientparametersandthereforeisnotapplicabletoLPV systems.Theworkpresentedinthischapterisanextensionoftheworkin[27,28],and providesacontinuousrobustcontrollerthatisabletoexponentiallystabilizeLPVsystems withunknownboundeddisturbances,whenboththestate,inputmatrices,timevarying parameters,anddisturbancesareunknown. 2.2LinearParameterVaryingModel Thedynamicmodelusedforthesubsequentcontroldevelopmentisacombinationof linearparametervarying(LPV)systemwithanaddedunmodeleddisturbanceas = ( ( )) + ( ( )) + ( ) (21) = (22) In(2)and(22), ( ) Risthestatevector, ( ( )) R denotesalinear parametervaryingstatematrix, ( ( )) denotesalinearparametervarying inputmatrix, R denotesaknownoutputmatrix, ( ) Rdenotescontrol vector, ( ) representstheunknowntimedependentparameters, ( ) Rrepresentsa timedependentunknown,nonlineardisturbance,and ( ) Rrepresentsthemeasured outputvector.Thesubsequentcontroldevelopmentisbasedontheassumptionthat ,meaningthatatleastonecontrolinputisavailableforeachoutputstate.Whenthe systemisoveractuatedinthattherearemorecontrolinputsavailablethanoutputstates, then andtheresultinginputdynamicinversionmatrixwillberowde cient.For thiscase,arightpseudoinversecanbeusedinconjunctionwithasingularityavoidance law.Forinstance,if R thenthepseudoinverse += 1andsatis es += where isanidentitymatrixofdimension Thematrices ( ( )) and ( ( )) havethestandardlinearparametervaryingform ( )= 0+P =1( ( )) (23) ( )= 0+P =1( ( )) (24) 21 PAGE 22 where 0 R and 0 R representknownnominalmatriceswithunknown variations ( ( )) and ( ( )) for =1 2 ,where R and R aretimeinvariantmatrices,and ( ( )) ( ( )) R areparameterdependent weightingterms.Knowledgeofthenominalmatrix 0willbeexploitedinthesubsequent controldesign. Tofacilitatethesubsequentcontroldesign,areferencemodelisgivenas = + (25) = (26) where R and R denotethestateandinputmatrices,respectively,where isHurwitz, ( ) Risavectorofreferenceinputs, ( ) Rarethereference outputs,and wasde nedin(22). Assumption1:Thenonlineardisturbance ( ) andits rsttwotimederivativesare assumedtoexistandbeboundedbyknownconstants. Assumption2:Thedynamicsin(2)areassumedtobecontrollable. Assumption3:Thematrices ( ( )) and ( ( )) andtheirtimederivativessatisfy thefollowinginequalities: k ( ( )) k k ( ( ))k (27) ( ( )) ( ( )) where R+areknownboundingconstants,and kkdenotestheinduced in nitynormofamatrix.Asistypicalinrobustcontrolmethods,knowledgeoftheupper boundsin(27)areusedtodevelopsu cientconditionsongainsusedinthesubsequent controldesign. 22 PAGE 23 2.3ControlDevelopment 2.3.1ControlObjective Thecontrolobjectiveistoensurethattheoutput ( ) tracksthetimevaryingoutput generatedfromthereferencemodelin(2)and(2).Toquantifythecontrolobjective, anoutputtrackingerror,denotedby ( ) R,isde nedas = ( ) (28) Tofacilitatethesubsequentanalysis,a lteredtrackingerrordenotedby ( ) R,is de nedas + (29) where R2isapositivede nitediagonal,constantcontrolgainmatrix,andisselectedtoplacearelativeweightontheerrorst ateversesitsderivative.Tofacilitatethe subsequentrobustcontroldevelopment,thestatevector ( ) isexpressedas ( )= ( )+ ( ) (2) where ( ) Rcontainsthe outputstates,and ( ) Rcontainstheremaining states.Likewise,thereferencestates ( ) canalsobeseparatedasin(2). Assumption4:Thestatescontainedin ( ) in(2)andthecorrespondingtime derivativescanbefurtherseparatedas ( )= ( )+ ( ) (2) ( )= ( )+ ( ) where ( ) ( ) ( ) ( ) Rareupperboundedas k ( ) k 1k kk ( ) k (2) k ( ) k 2k kk ( ) k 23 PAGE 24 where ( ) R2 isde nedas (2) and 12 R areknownnonnegativeboundingconstants.Thetermsin(2) and(2)areusedtodevelopsu cientgainconditionsforthesubsequentrobustcontrol design. 2.3.2OpenLoopErrorSystem Theopenlooptrackingerrordynamicscanbedevelopedbytakingthetimederivativeof(29)andusingtheexpressionsin(21)(2)as = + = ( )+ = + + + + ( ) + = + + + (2) Theauxiliaryfunctions ( ) Rand Rin(2) arede nedas ( )+ ( )+ + + + (2) and ( )+ + + + (2) Motivationfortheselectivegroupingofthetermsin(2)and(2)isderivedfromthe factthatthefollowinginequalitiescanbedeveloped[38,54]as 0k kk k (2) where 0 R+areknownboundingconstants. 24 PAGE 25 2.3.3ClosedLoopErrorSystem Basedontheexpressionin(24)andthesubsequentstabilityanalysis,thecontrol inputisdesignedas = ( 0) 1[( + ) ( ) ( + ) (0)+ ( )] (2) where ( ) Risanimplicitlearninglawwithanupdaterulegivenby ( )= k ( ) k ( ( ))+( + ) ( )+ ( ( )) (2) and R R denotepositivede nite,diagonalconstantcontrolgain matrices, 0 R isintroducedin(24), ( ) denotesthestandardsignumfunction wherethefunctionisappliedtoeachelementofthevectorargument,and denotesa identitymatrix. Aftersubstitutingthetimederivativeof(2)into(2),theerrordynamicscanbe expressedas = + k ( ) k ( ( ))+ (2) ( + ) ( ) ( ( )) wheretheauxiliarymatrix ( ( )) R isde nedas ( 0) 1(2) where ( ( )) canbeseparatedintodiagonal(i.e., ( ( )) R )ando diagonal(i.e., ( ( )) R )componentsas = + (2) Assumption5:Thesubsequentdevelopmentisbasedontheassumptionthatthe uncertainmatrix ( ( )) isdiagonallydominantinthesensethat min( ) k k (2) 25 PAGE 26 where R+isaknownconstant.Whilethisassumptioncannotbevalidatedfora genericsystem,theconditioncanbechecked(withinsomecertaintytolerances)fora speci csystem.Essentially,thisconditionindicatesthatthenominalvalue 0must remainwithinsomeboundedregionof .Inpractice,boundsonthevariationof should beknown,foraparticularsystemunderasetofoperatingconditions,andthisboundcan beusedtocheckthesu cientconditionsgivenin(2). Motivationforthestructureofthecontrollerin(2)and(2)comesfromthe desiretodevelopaclosedlooperrorsystemtofacilitatethesubsequentLyapunovbased stabilityanalysis.Inparticular,sincethecontrolinputispremultipliedbytheuncertain matrix in(2),theterm 1 0ismotivatedtogeneratetherelationshipin(21) sothatifthediagonaldominanceassumption(Assumption5)issatis ed,thenthecontrol canprovidefeedbacktocompensateforthedisturbanceterms.Thebracketedtermsin (28)includethestatefeedback,aninitialconditionterm,andtheimplicitlearningterm. Theimplicitlearningterm ( ) isthegeneralizedsolutionto(29).Thestructureofthe updatelawin(219)ismotivatedbytheneedtorejecttheexogenousdisturbanceterms. Speci cally,theupdatelawismotivatedbyaslidingmodecontrolstrategythatcanbe usedtoeliminateadditiveboundeddisturbances.Unlikeslidingmodecontrol(which isadiscontinuouscontrolmethodrequiringin niteactuatorbandwidth),thecurrent continuouscontrolapproachincludestheintegralofthe ( ) function.Thisimplicit learninglawisthekeyelementthatallowsthecontrollertoobtainanexponentialstability resultdespitetheadditivenonvanishingexog enousdisturbance.Otherresultsinliterature alsohaveusedtheimplicitlearningstructureinclude[33,34,35,36,37,40]. 26 PAGE 27 Di erentialequationssuchas(224)and(225)havediscontinuousrighthandsides as ( )= k ( ) k ( ( ))+(+ ) ( )+ ( ( )) (2) = + k ( ) k ( ( ))+ ( + ) ( ) ( ( )) (2) Let ( ) R2 denotetherighthands ideof(224)and(225).Sincethesubsequent analysisrequiresthatasolutionexistfor = ( ) ,itisimportanttoshowthe existenceofthegeneralizedsolution.TheexistenceofFilippovsgeneralizedsolution [15]canbeestablishedfor(2)and(2).First,notethat ( ) iscontinuous exceptintheset { ( )  =0 } .Let ( ) beacompact,convex,uppersemicontinuous setvaluedmapthatembedsthedi erentialequation = ( ) intothedi erential inclusion ( ) .Anabsolutecontinuoussolutionexiststo = ( ) thatisa generalizedsolutionto = ( ) .Acommonchoice[15]for ( ) thatsatis esthe aboveconditionsistheclosedconvexhullof ( ) .Aproofthatthischoicefor ( ) isuppersemicontinuousisgivenin[20]. 2.4StabilityAnalysis Theorem: Thecontrollergivenin(2)and(2)ensuresexponentialtrackingin thesensethat k ( ) k k (0) k exp 1 2 [0 ) (2) where 1 R+,providedthecontrolgains ,and introducedin(28)areselected accordingtothesu cientconditions min( ) min( ) 2 0 4 min { } min( ) (2) where 0and areintroducedin(2), isintroducedin(223), R+isa knownpositiveconstant,and min( ) denotestheminimumeigenvalueoftheargument. Theboundingconstantsareconservativeupperboundsonthemaximumexpected 27 PAGE 28 values.TheLyapunovanalysisindicatesthatthegainsin(227)needtobeselected su cientlylargebasedonthebounds.Therefore,iftheconstantsarechosentobe conservative,thenthesu cientgainconditionswillbelarger.Valuesforthesegainscould bedeterminedthroughaphysicalunderstandingofthesystem(withinsomeconservative %ofuncertainty)and/orthroughnumericalsimulations. Proof :Let ( ): R2 [0 ) R beaLipschitzcontinuous,positivede nite functionde nedas ( ) 1 2 + 1 2 (2) where ( ) and ( ) arede nedin(2)and(2),respectively.Aftertakingthetime derivativeof(2)andutilizing(2),(2),and(2), ( ) canbeexpressedas ( )= + + ( + ) ( + ) (2) k k ( ) k k ( ) ( ) ( )+ Byutilizingtheboundingargumentsin(217)andAssumptions3and5,theupperbound oftheexpressionin(229)canbeexplicitlydetermined.Speci cally,basedon(27)of Assumption3,theterm in(2)canbeupperboundedas k kk k (2) Afterutilizinginequality(23)ofAssumption5,thefollowinginequalitiescanbe developed: ( + ) ( + ) ( min( )+1) k k2 k ( ) k ( ) k ( ) k ( ) min( )  k k (2) ( ) ( ) min( )   28 PAGE 29 Afterusingtheinequalitiesin(230)and(231),theexpressionin(229)canbeupper boundedas ( ) k k2+ + k kk k ( min( )+1) k k2(2) min( ) k kk k min( ) k k + wherethefactthat   k k Rwasutilized.Afterutilizingtheinequalitiesin (27)andrearrangingtheresultingexpression,theupperboundfor ( ) canbe expressedas ( ) k k2 k k2 min( ) k k2+ 0k kk k (2) [ min( ) ] k kk k [ min( ) ] k k If and satisfythesu cientgainconditionsin(227),thebracketedtermsin(23) arepositive,and ( ) canbeupperboundedusingthesquaresofthecomponentsof ( ) as: ( ) k k2 k k2 min( ) k k2 0k kk k (2) Bycompletingthesquares,theupperboundin(234)canbeexpressedinamore convenientform.Tothisend,theterm2 0k k2 4 min( )isaddedandsubtractedtotherighthand sideof(2)yielding ( ) k k2 k k2 min( ) k k 0k k 2 min( ) 2+ 2 0k k2 4 min( ) (2) Sincethesquareofthebracketedtermin(25)isalwayspositive,theupperboundcan beexpressedas ( ) { } + 2 0k k2 4 min( ) (2) where ( ) isde nedin(2).Hence,(2)canbeusedtorewritetheupperboundof ( ) as ( ) min { } 2 0 4 min( ) k k2 (2) 29 PAGE 30 wherethefactthat { } min { }k k2wasutilized.Provided thegainconditionin(2)issatis ed,(228)and(237)canbeusedtoshowthat ( ) ;hence ( ) ( ) .Giventhat ( ) ( ) ,standardlinearanalysis methodscanbeusedtoprovethat ( ) from(2).Since ( ) ( ) ,the assumptionthatthereferencemodeloutputs ( ) ( ) canbeusedalongwith (2)toprovethat ( ) ( ) .Giventhat ( ) ( ) ( ) ( ) ,thevector ( ) ,thetimederivative ( ) ,and(2)(2)canbeusedtoshowthat ( ) ( ) .Giventhat ( ) ( ) ,Assumptions1,2,and3canbeutilized alongwith(2)toshowthat ( ) Thede nitionfor ( ) in(2)canbeusedalongwithinequality(2)toshow that ( ) canbeupperboundedas ( ) 1( ) (2) providedthesu cientconditionin(27)issatis ed.Thedi erentialinequalityin(238) canbesolvedas ( ) ( (0) 0)exp( 1 ) (2) Hence,(2),(2),and(2)canbeusedtoconcludethat k ( ) k k (0) k exp 1 2 [0 ) (2) 2.5Conclusions AcontinuousexponentiallystablecontrollerwasdevelopedforLPVsystemswithan unknownstatematrix,anuncertaininputmatrix,andanunknownadditivedisturbance. ThisworkpresentsanewapproachtoLPVcontrolbyinvertingtheuncertaininput dynamicsandrobustlycompensatingforotherunknownsanddisturbances.Thecontroller isvalidforLPVsystemswherethereareatleastasmanycontrolinputsasthereare outputs.UsingthistechniqueitispossiblecontrolLPVsystemswherethereisahigh amountofuncertaintyandnonlinearitiesthatinvalidatetraditionalLPVapproaches. 30 PAGE 31 Robustdynamicinversioncontrolispossibleforawiderangeofpracticalsystemsthatare approximatedasanLPVsystemwithadditivedisturbances.Futureworkwillfocuson relaxingtheassumptionswhilemaintainingthestabilityandperformance. 31 PAGE 32 CHAPTER3 HYPERSONICVEHICLEDYNAMICSANDTEMPERATUREMODEL 3.1Introduction Inthischapterthedynamicsofthehypersonicvehicle(HSV)areintroduced,includingboththestandard ightdynamicsandthestructuralvibrationdynamics.After thedynamicsaredevelopedandthe ightandstructuralcomponentsareexplained,a temperaturemodelisintroduced.Becausechangesintemperaturechangethestructural dynamics,coupledforcingtermschangethethe ightdynamics.Examplesoflineartemperaturepro lesareprovided,andsomeexamplesofthestructuralmodesandfrequencies areexplained. 3.2RigidBodyandElasticDynamics Toincorporatestructuraldynamicsandaerothermoelastice ectsintheHSVdynamic model,anassumedmodesmodelisconsideredforthelongitudinaldynamics[53]as = cos( ) sin( ) (31) = sin( ) (32) = + sin( ) + + cos( ) (33) = (34) = (35) = 2 2 + =1 2 3 (36) In(3)(3), ( ) denotestheforwardvelocity, ( ) denotesthealtitude, ( ) denotestheangleofattack, ( ) denotesthepitchangle, ( ) ispitch rate,and ( ) =1 2 3 denotesthe generalizedstructuralmodedisplacement. Alsoin(31)(3), denotesthevehiclemass, isthemomentofinertia, istheaccelerationduetogravity, ( ) ( ) arethedampingfactorand naturalfrequencyofthe exiblemode,respectively, ( ) denotesthethrust, 32 PAGE 33 ( ) denotesthedrag, ( ) isthelift, ( ) isthepitchingmomentabout thebody axis,and ( ) =1 2 3 denotesthegeneralizedelasticforces,where ( ) 11iscomposedofthe 5 ightand 6 structuraldynamicstatesas = 1 12 23 3 (37) Theequationsthatde netheaerodynamicandgeneralizedmomentsandforcesare highlycoupledandareprovidedexplicitlyinpreviouswork[10].Speci cally,therigid bodyandelasticmodesarecoupledinthesensethat ( ) ( ) ( ) ,arefunctions of ( ) andthat ( ) isafunctionoftheotherstates.Asthetemperaturepro le changes,themodulusofelasticityofthevehiclechangesandthedampingfactorsand naturalfrequenciesofthe exiblemodeswillchange.Thesubsequentdevelopmentexploits animplicitlearningcontrolstructure,designedbasedonanLPVapproximationofthe dynamicsin(31)(36),toyieldexponentialtrackingdespitetheuncertaintyduetothe unknownaerothermoelastice ectsandadditionalunmodeleddynamics. 3.3TemperaturePro leModel TemperaturevariationsimpacttheHSV ightdynamicsthroughchangesinthe structuraldynamicswhicha ectthemodeshapesandnaturalfrequenciesofthevehicle. Thetemperaturemodelusedassumesafreefreebeam[10],whichmaynotcapturethe actualaircraftdynamicsproperly.Inreali ty,theinternalstructurewillbemadeofa complexnetworkofstructuralelementsthatwillexpandatdi erentratescausingthermal stresses.Thermalstressesa ectdi erentmodesindi erentmanners,whereitraises thefrequenciesofsomemodesandlowersothers(comparedtoauniformdegradation withYoungsmodulusonly).Therefore,thecurrentmodelonlyo ersanapproximate approach.Thenaturalfrequenciesofacontinuousbeamareafunctionofthemass distributionofthebeamandthesti ness.Inturn,thesti nessisafunctionofYoungs Modulus(E)andadmissiblemodefunctions.Hence,bymodelingYoungsModulusasa functionoftemperature,thee ectoftemperatureon ightdynamicscanbecaptured. 33 PAGE 34 Thermostructuraldynamicsarecalculatedunderthematerialassumptionthattitanium isbelowthethermalprotectionsystem[9,12].YoungsModulus(E)andthenatural dynamicfrequenciesforthe rstthreemodesofatitaniumfreefreebeamaredepictedin Figure31andFigure32respectively. 0 100 200 300 400 500 600 700 800 900 11.5 12 12.5 13 13.5 14 14.5 15 15.5 16 16.5 Temperature (F)E (Modulus of Elasticity in psi)Figure31:Modulusofelasticityforthe rstthreedynamicmodesofvibrationforafreefreebeamoftitanium. InFigure31,themoduliforthethreemodesarenearlyidentical.Thetemperature rangeshowncorrespondstothetemperaturerangethatwillbeusedinthesimulation section.FrequenciesinFigure32correspondtoasolidtitaniumbeam,whichwillnot correspondtotheactualnaturalfrequenciesoftheaircraft.ThedatashowninFigure31 andFigure32arebothfrompreviousexperimentalwork[47].Usingthisdata,di erent temperaturegradientsalongthefuselageareintroducedintothemodelanda ectthe structuralpropertiesoftheHSV.ThesimulationsinChapter4andChapter5uselinearly decreasinggradientsfromthenosetothetailsection.Itsexpectedthatthenosewill bethehottestpartofthestructureduetoaerodynamicheatingbehindthebowshock wave.Thermostructuraldynamicsarecalculatedundertheassumptionthattherearenine constanttemperaturesectionsintheaircraft[6]asshowninFigure33.Sincetheaircraft is100feetlong,thelengthofeachoftheninesectionsisapproximately11.1feet. 34 PAGE 35 0 100 200 300 400 500 600 700 800 900 45 50 55 1st Dynamic ModeFrequency (Hz) 0 100 200 300 400 500 600 700 800 900 120 140 160 2nd Dynamic ModeFrequency (Hz) 0 100 200 300 400 500 600 700 800 900 200 250 300 3rd Dynamic Mode Temperature (F)Frequency (Hz)Figure32:Frequenciesofvibrationforthe rstthreedynamicmodesofafreefreetitaniumbeam. Figure33:NineconstanttemperaturesectionsoftheHSVusedfortemperaturepro le modeling. 35 PAGE 36 Table31:Naturalfrequenciesfor5lineartemperaturepro les(Nose/Tail)indegrees F.Percentdi erenceisthedi erencebetweenthemaximumandminimumfrequencies dividedbytheminimumfrequency. Mode 900 500800 400700 300600 200500 100 %Di erence 1(Hz)23.023.523.924.324.77.39 % 2(Hz)49.950.951.852.653.57.21 % 3(Hz)98.9101.0102.7104.4106.27.38 % Thestructuralmodesandfrequenciesarecalculatedusinganassumedmodesmethod with niteelementdiscretization,includingvehiclemassdistributionandinertiae ects. Theresultofthismethodisthegeneralizedmodeshapesandmodefrequenciesforthe HSV.Becausethebeamisnonuniformintemperature,themodulusofelasticityisalso nonuniform,whichproducesasymmetricmodeshapes.Anexampleoftheasymmetric modeshapesisshowninFigure35andtheasymmetryisduetovariationsin resulting fromthefactthateachoftheninefuselagesections(seeFigure33)hasadi erent temperatureandhencedi erent exibledynamicproperties.Anexampleofsomeof themodefrequenciesareprovidedinTable 1 ,whichshowsthevariationinthenatural frequenciesfor vedecreasinglineartemperaturepro lesshowninFigure34.Forall threenaturalmodes,Table31showsthatthenaturalfrequencyforthe rsttemperature pro leisalmost 7% lowerthanthatofthe fthtemperaturepro le. Thetemperaturepro leinaHSVisacomplexfunctionofthestatehistory,structuralproperties,thermalprotectionsystem,etc.ForthesimulationsinChapter4and Chapter5,thetemperaturepro leisassumedtobealinearfunctionthatdecreasesfrom thenosetothetailoftheaircraft.Thelinearpro lesarethenvariedtospanapreselecteddesignspace.Ratherthanattemptingtomodelaphysicaltemperaturegradientfor somevehicledesign,thetemperaturepro leinthesimulationsinChapter4andChapter5isintendedtoprovideanaggressivetemperaturedependentpro letoexaminethe robustnessofthecontrollertosuch uctuations. 36 PAGE 37 1 2 3 4 5 6 7 8 9 100 200 300 400 500 600 700 800 900 Fuselage sectionTemperature (F)Figure34:Lineartemperaturepro lesusedtocalculatevaluesshowninTable31. 0 20 40 60 80 100 .4 .3 .2 .1 0 0.1 0.2 0.3 Fuselage Position (ft)Displacement 1st 2nd 3rd Figure35:Asymetricmodeshapesforthehypersonicvehicle.Thepercentdi erencewas calculatedbasedonthemaximumminustheminimumstructuralfrequenciesdividedby theminimum. 37 PAGE 38 3.4Conclusion Thischapterexplainstheoverall ightandstructuraldynamicsforaHSV,inthe presenceofdi erenttemperaturepro les.Thesedynamicsareimportanttounderstand becausechangesinthetemperaturepro lemodifythedynamics,hencecanbemodeled asadditiveparameterdisturbances.Inthef ollowingchapters,theHSVdynamicswillbe reducedtoaLPVsystemwithanadditivedisturbance,andthecontrollerfromChapter 2willbeapplied.Thetemperaturepro leswillactastheparametervariations.This chapterwasmeanttobrie yintroducetheoverallsystemandexplainthestructural modes,shapes,andfrequencies.Datawasshowntomotivatethefactthatchangesin temperaturesubstantiallya ecttheoveralldynamics. 38 PAGE 39 CHAPTER4 LYAPUNOVBASEDEXPONENTIALTRACKINGCONTROLOFAHYPERSONIC AIRCRAFTWITHAEROTHERMOELASTICEFFECTS 4.1Introduction Thedesignofguidanceandcontrolsystemsforairbreathinghypersonicvehicles (HSV)ischallengingbecausethedynamicsoftheHSVarecomplexandhighlycoupled [10],andtemperatureinducedsti nessvariationsimpactthestructuraldynamics[21]. Thestructuraldynamics,inturn,a ecttheaerodynamicproperties.Vibrationinthe forwardfuselagechangestheapparentturnangleofthe ow,whichresultsinchanges inthepressuredistributionovertheforebodyoftheaircraft.Theresultingchangesin thepressuredistributionovertheaircraftmanifestthemselvesasthrust,lift,drag,and pitchingmomentperturbations[10].Todevelopcontrollawsforthelongitudinaldynamics ofaHSVcapableofcompensatingforthesestructuralandaerothermoelastice ects, structuraltemperaturevariationsandstructuraldynamicsmustbeconsidered. Aerothermoelasticityistheresponseofelasticstructurestoaerodynamicheatingand loading.Aerothermoelastice ectscannotbeignoredinhypersonic ight,becausesuch e ectscandestabilizetheHSVsystem[21].Alossofsti nessinducedbyaerodynamic heatinghasbeenshowntopotentiallyinducedynamicinstabilityinsupersonic/hypersonic ightspeedregimes[1].Yetactivecontrolcanbeusedtoexpandthe utterboundary andconvertunstablelimitcycleoscillations(LCO)tostableLCO[1].Anactivestructural controllerwasdeveloped[26],whichaccountsforvariationsintheHSVstructuralpropertiesresultingfromaerothermoelastice ects.Thecontroldesign[26]modelsthestructural dynamicsusingaLPVframework,andstatesthebene tstousingtheLPVframework aretwofold:thedynamicscanberepresentedasasinglemodel,andcontrollerscanbe designedthathavea nedependencyontheoperatingparameters. Previouspublicationshaveexaminedthechallengesassociatedwiththecontrolof HSVs.Forexample,HSV ightcontrollersaredesignedusinggeneticalgorithmstosearch adesignparameterspacewherethenonlinearlongitudinalequationsofmotioncontain 39 PAGE 40 uncertainparameters[4,30,49].SomeofthesedesignsutilizeMonteCarlosimulations toestimatesystemrobustnessateachsearchiteration.Anotherapproach[4]istouse fuzzylogictocontroltheattitudeoftheHSVaboutasinglelowend ightcondition. Whilesuchapproaches[4,30,49]generatestabilizingcontrollers,theproceduresare computationallydemandingandrequiremultipleevaluationsimulationsoftheobjective functionandhavelargeconvergenttimes.An adaptivegainscheduledcontroller[55]was designedusingestimatesofthescheduledparameters,andasemioptimalcontrolleris developedtoadaptivelyattain controlperformance.Thiscontrolleryieldsuniformly boundedstabilityduetothee ectsofapproximationerrorsandalgorithmicerrorsin theneuralnetworks.FeedbacklinearizationtechniqueshavebeenappliedtoacontrolorientedHSVmodeltodesignanonlinearcontroller[32].Themodel[32]isbasedon apreviouslydeveloped[8]HSVlongitudinaldynamicmodel.Thecontroldesign[32] neglectsvariationsinthrustliftparameters,altitude,anddynamicpressure.Linearoutput feedbacktrackingcontrolmethodshavebeendeveloped[44],wheresensorplacement strategiescanbeusedtoincreaseobservability,orreconstructfullstateinformation forastatefeedbackcontroller.Arobustoutputfeedbacktechniqueisalsodeveloped forthelinearparameterizableHSVmodel,whichdoesnotrelyonstateobservation.A robustsetpointregulationcontroller[17]isdesignedtoyieldasymptoticregulationinthe presenceofparametricandstructuraluncertaintyinalinearparameterizableHSVsystem. Anadaptivecontroller[19]wasdesignedto handle(linearintheparameters)modelinguncertainties,actuatorfailures,andnonminimumphasedynamics[17]foraHSV withelevatorandfuelratioinputs.Anotheradaptiveapproach[41]wasrecentlydevelopedwiththeadditionofaguidancelawthatmaintainsthefuelratiowithinitschoking limits.WhileadaptivecontrolandguidancecontrolstrategiesforaHSVareinvestigated [17,19,41],neitheraddressesthecasewheredynamicsincludeunknownandunmodeled disturbances.Thereremainsaneedforacontinuouscontroller,whichiscapableofachievingexponentialtrackingforaHSVdynamicmodelcontainingaerothermoelastice ects 40 PAGE 41 andunmodeleddisturbances(i.e .,nonvanishingdisturbances thatdonotsatisfythelinear intheparametersassumption). Inthecontextoftheaforementionedliterature,thecontributionofthecurrenteffort(andthepreliminarye ortbytheauthors[52])isthedevelopmentofacontroller thatachievesexponentialmodelreferenceoutputtrackingdespiteanuncertainmodelof theHSVthatincludesnonvanishingexogenousdisturbances.Anonlineartemperaturedependentparametervaryingstatespacerepresentationisusedtocapturetheaerothermoelastice ectsandunmodeleduncertaintiesinaHSV.Thismodelincludesanunknown parametervaryingstatematrix,anuncertainparametervaryingnonsquare(column de cient)inputmatrix,andanonlinearaddit iveboundeddisturbance.Toachievean exponentialtrackingresultinlightofthesedisturbances,arobust,continuousLyapunovbasedcontrollerisdevelopedthatincludesanovelimplicitlearningcharacteristicthat compensatesforthenonvanishingexogenous disturbance.Thatis,theuseoftheimplicit learningmethodenablesthe rstexponentialtrackingresultbyacontinuouscontrollerin thepresenceoftheboundednonvanishingexogenousdisturbance.Toillustratetheperformanceofthedevelopedcontrollerduringvelocity,angleofattack,andpitchratetracking, simulationsforthefullnonlinearmodel[10]areprovidedthatincludeaerothermoelastic modeluncertaintiesandnonlinearexogenousdisturbanceswhosemagnitudeisbasedon airspeed uctuations. 4.2HSVModel Thedynamicmodelusedforthesubsequentcontroldesignisbasedonareduction ofthedynamicsin(3)(3)tothefollowing combinationoflinearparametervarying (LPV)statematricesandadditivedisturbancesarisingfromunmodelede ectsas = ( ( )) + ( ( )) + ( ) (41) = (42) 41 PAGE 42 In(4)and(42), ( ) R11isthestatevector, ( ( )) R11 11denotesalinear parametervaryingstatematrix, ( ( )) 11 3denotesalinearparametervaryinginput matrix, R3 11denotesaknownoutputmatrix, ( ) R3denotesavectorof 3 control inputs, ( ) representstheunknowntimedependentparameters, ( ) R11representsa timedependentunknown,nonlineardisturbance,and ( ) R3representsthemeasured outputvectorofsize 3 4.3ControlObjective Thecontrolobjectiveistoensurethattheoutput ( ) tracksthetimevaryingoutput generatedfromthereferencemodellikesta tedinChapter2.Toquantifythecontrol objective,anoutputtrackingerror,denotedby ( ) R3,isde nedas = ( ) (43) Tofacilitatethesubsequentanalysis,a lteredtrackingerrordenotedby ( ) R3,is de nedas + (44) where R3isapositivede nitediagonal,constantcontrolgainmatrix,andisselected toplacearelativeweightontheerrorstateversesitsderivative.Basedonthecontrol designpresentedinChapter2thecontrolinputisdesignedas = ( 0) 1[( + 3 3) ( ) ( + 3 3) (0)+ ( )] (45) where ( ) R3isanimplicitlearninglawwithanupdaterulegivenby ( )= k ( ) k ( ( ))+(+ 3 3) ( )+ ( ( )) (46) and R3 3denotepositivede nite,diagonalconstantcontrolgainmatrices, 0 R11 3representsaknownnominalinputmatrix, ( ) denotesthestandard signumfunctionwherethefunctionisappliedtoeachelementofthevectorargument, and 3 3denotesa 3 3 identitymatrix.Toillustratetheperformanceofthecontroller 42 PAGE 43 andpracticalityoftheassumptions,anumericalsimulationwasperformedonthefull nonlinearlongitudinalequationsofmotion[10]givenin(3)(36).Thecontrolinputs wereselectedas = ( ) ( ) ( ) asinpreviousresearch[41],where ( ) and ( ) denotetheelevatorandcanardde ectionangles,respectively, ( ) isthefuel equivalenceratio.Thedi userarearatioisleftatitsoperationaltrimconditionwithout lossofgenerality( ( )=1 ).Thereferenceoutputswereselectedasmaneuveroriented outputsofvelocity,angleofattack,andpitchrateas = ( ) ( ) ( ) where theoutputandstatevariablesareintroducedin(3)(3).Inaddition,theproposed controllercouldbeusedtocontrolotheroutputstatessuchasaltitudeprovidedthe followingconditionisvalid.Theauxiliarymatrix ( ( )) R isde nedas ( 0) 1(47) where ( ( )) canbeseparatedintodiagonal(i.e., ( ( )) R )ando diagonal(i.e., ( ( )) R )componentsas = + (48) Theuncertainmatrix ( ( )) isdiagonallydominantinthesensethat min( ) k k (49) where R+isaknownconstant.Whilethisassumptioncannotbevalidatedfora genericHSV,theconditioncanbechecked(withinsomecertaintytolerances)foragiven aircraft.Essentially,thisconditionindicatesthatthenominalvalue 0mustremain withinsomeboundedregionof .Inpractice,bandsonthevariationof shouldbe known,foraparticularaircraftunderasetofoperatingconditions,andthisbandcould beusedtocheckthesu cientconditions.Forthespeci cHSVexamplethisChapter simulates,theassumtionin49isvalid. 43 PAGE 44 4.4SimulationResults TheHSVparametersusedinthesimulationare =75 000 = 86723 2,and =32 174 2 asde nedin(3)(3).Thesimulationwasexecutedfor 35 secondstosu cientlycyclethroughthedi erenttemperaturepro les.Othervehicle parametersinthesimulationarefunctionsofthetemperaturepro le.Lineartemperature pro lesbetweentheforebody(i.e., [450 900] )andaftbody(i.e., [100 800] ) wereusedtogenerateelasticmodeshapesandfrequenciesbyvaryingthelineargradients as ( )=675+225cos 10 ( )= 450+350cos 3 if ( ) ( ) ( ) otherwise. (4) Figure41showsthetemperaturevariationa safunctionoftime.Theirregularitiesseen intheaftbodytemperaturesoccurbecausethetemperaturepro leswereadjustedto ensurethetailoftheaircraftwasequalorco olerthanthenoseoftheaircraftaccording tobowshockwavethermodynamics.Whiletheshockwavethermodynamicsmotivated theneedtoonlyconsiderthecasewhenthetailoftheaircraftwasequalorcoolerthan thenoseoftheaircraft,theshapeofthetemperaturepro leisnotphysicallymotivated. Speci cally,thefrequenciesofoscillationin(410)wereselectedtoaggressivelyspanthe availabletemperatureranges.Thesetemperaturepro lesarenotmotivatedbyphysical temperaturegradients,butmotivatedbythedesiretogenerateatemperaturedisturbance toillustratethecontrollerrobustnesstothetemperaturegradients.Thesimulation assumesthedampingcoe cientremainsconstantforthestructuralmodes ( =0 02) Inadditiontothermoelasticity,aboundednonlineardisturbancewasaddedtothe dynamicsas = 000 10 20 3 (4) 44 PAGE 45 0 5 10 15 20 25 30 3 5 0 200 400 600 800 1000 Nose Temperature (F)Time (s) 0 5 10 15 20 25 30 3 5 0 200 400 600 800 Time (s)Tail Temperature (F)Figure41:Temperaturevariationfortheforebodyandaftbodyofthehypersonicvehicle asafunctionoftime. where ( ) R denotesalongitudinalaccelerationdisturbance, ( ) R denotesaangle ofattackrateofchangedisturbance, ( ) R denotesanangularaccelerationdisturbance,and 1( ) 2( ) 3( ) R denotestructuralmodeaccelerationdisturbances.The disturbancesin(4)weregeneratedasana rbitraryexogenousinput(i.e.,unmodeled nonvanishingdisturbancethatdoesnotsatis fythelinearintheparametersassumption) asdepictedinFigure42.However,themagni tudesofthedisturbancesweremotivatedby thescenarioofa 300 changeinairspeed.Thedisturbancesarenotdesignedtomimic theexacte ectsofawindgust,buttodemonstratetheproposedcontrollersrobustness withrespecttorealisticallyscaleddisturbances.Speci cally,arelativeforcedisturbanceis determinedbycomparingthedragforce atMach 8 at 85 000 (i.e., 7355 )with thedragforceafteraddinga 300 (e.g.,awindgust)disturbance.UsingNewtons secondlawanddividingthedragforcedi erential bythemassoftheHSV ,a realisticupperboundforan accelerationdisturbance ( ) wasdetermined.Similarly,the sameprocedurecanbeperformed,tocomparethechangeinpitchingmoment caused bya 300 headwindgust.Bydividingthemomentdi erentialbythemomentof 45 PAGE 46 0 5 10 15 20 25 30 3 5 0 1 x 10 f1 (ft/s2) 0 5 10 15 20 25 30 3 5 0 10 f2 (deg/s) 0 5 10 15 20 25 30 3 5 0 2 f3 (deg/s2) 0 5 10 15 20 25 30 3 5 .05 0 0.05 f7 (1/s2) 0 5 10 15 20 25 30 3 5 .01 0 0.01 f9 (1/s2) 0 5 10 15 20 25 30 3 5 0 1 x 10 f11 (1/s2)Time (s)Figure42:Inthis gure, denotesthe elementinthedisturbancevecor .Disturbancesfromtoptobottom:velocity ,angleofattack ,pitchrate ,the 1elastic structuralmode 1,the 2elasticstructuralmode 2,andthe 3elasticstructuralmode 3,asdescribedin(411). inertiaoftheHSV ,arealisticupperboundfor ( ) canbedetermined.Tocalculate areasonableangleofattackdisturbancemagnitude,averticalwindgustof 300 is considered.Bytakingtheinversetangentoftheverticalwindgustdividedbytheforward velocityatMach 8 and 85 000 ,anupperboundfortheangleofattackdisturbance ( ) canbedetermined.Disturbancesforthestructuralmodes ( ) wereplacedonthe accelerationtermswith ( ) ,whereeachsubsequentmodeisreducedbyafactorof10 relativetothe rstmode,seeFigure42. Theproposedcontrollerisdesignedtofollowtheoutputsofawellbehavedreference model.Toobtaintheseoutputs,areferencemodelthatexhibitedfavorablecharacteristics wasdesignedfromastaticlinearizeddynamicsmodelofthefullnonlineardynamics [10].Thereferencemodeloutputsareshowni nFigure43.Thevelocityreferenceoutput followsa 1000 smoothstepinput,whilethepitchrateperformsseveral 1 maneuvers.Theangleofattackstayswithin 2 degrees. 46 PAGE 47 0 5 10 15 20 25 30 3 5 7000 7500 8000 8500 Vm (ft/s) 0 5 10 15 20 25 30 3 5 0 2 m (deg) 0 5 10 15 20 25 30 3 5 0 1 2 Qm (deg/s)Time (s)Figure43:Referencemodelouputs ,whicharethedesiredtrajectoriesfortop:velocity ( ) ,middle:angleofattack ( ) ,andbottom:pitchrate ( ) Thecontrolgainsfor(43)(4)and(45)(46)areselectedas = { 10 10 } = { 5 1 300 } = { 0 01 0 001 0 01 } = { 0 1 0 01 0 1 } = { 1 0 5 1 } (4) Thecontrolgainsin(4)wereobtainedusingthesamemethodasinChapter5.In contrasttothissuboptimalapproachused,thecontrolgainscouldhavebeenadjusted usingmoremethodicalapproachesasdescribedinvarioussurveypapersonthetopic [24,46]. The matrixandknowledgeofsomenominal 0matrixmustbeknown.The matrixisgivenby: = 10000000000 01000000000 00100000000 (4) 47 PAGE 48 0 5 10 15 20 25 30 3 5 7200 7400 7600 7800 8000 8200 8400 Velocity (ft/s) 0 5 10 15 20 25 30 3 5 .2 .8 .6 .4 .2 0 0.2 Velocity Error (ft/s)Time (s)Figure44:Top:velocity ( ) ,bottom:velocitytrackingerror ( ) fortheoutputvectorof(4),andthe 0matrixisselectedas 0= 32 69 0 017 9 0700023670 11320 316 25 72 0 01119 3900031890251902067 42 84 0 00160 052700042 13092 120 80 0 (4) basedonalinearizedplantmodelaboutsomenominalconditions. TheHSVhasaninitialvelocityofMach 7 5 atanaltitudeof 85 000 .Thevelocity, andvelocitytrackingerrorsareshowninFigure44.Theangleofattackandangleof attacktrackingerrorisshowninFigure45.Thepitchrateandpitchtrackingerror isshowninFigure46.Thecontrole ortrequiredtoachievetheseresultsisshownin Figure47.Inadditiontotheoutputstates,o therstatessuchasaltitudeandpitchangle areshowninFigure48.ThestructuralmodesareshowninFigure49. 4.5Conclusion Thisresultrepresentsthe rsteverapplicationofacontinuous,robustmodelreferencecontrolstrategyforahypersonicvehiclesystemwithadditiveboundeddisturbances 48 PAGE 49 0 5 10 15 20 25 30 3 5 0 1 2 AoA (deg) 0 5 10 15 20 25 30 3 5 .01 0 0.01 0.02 0.03 0.04 0.05 0.06 AoA Error (deg)Time (s)Figure45:Top:angleofattack ( ) ,bottom:angleofattacktrackingerror ( ) 0 5 10 15 20 25 30 35 .5 .5 0 0.5 1 1.5 Pitch Rate (deg/s) 0 5 10 15 20 25 30 35 .2 .15 .1 .05 0 0.05 0.1 0.15 Pitch Rate Error (deg/s)Time (s) Figure46:Top:pitchrate ( ) ,bottom:pitchratetrackingerror ( ) 49 PAGE 50 0 5 10 15 20 25 30 3 5 0 0.5 1 1.5 Fuel Ratio f 0 5 10 15 20 25 30 3 5 10 15 20 25 Elevator (deg) 0 5 10 15 20 25 30 3 5 0 10 20 Canard (deg)Time (s)Figure47:Top:fuelequivalenceratio .Middle:elevatorde ection .Bottom:Canard de ection 0 5 10 15 20 25 30 35 8 8.1 8.2 8.3 8.4 8.5 x 104 Altitude (ft) 0 5 10 15 20 25 30 35 0 1 2 3 Pitch Angle (deg)Time (s) Figure48:Top:altitude ( ) ,bottom:pitchangle ( ) 50 PAGE 51 0 5 10 15 20 25 30 3 5 0 20 40 1 0 5 10 15 20 25 30 3 5 0 5 10 2 0 5 10 15 20 25 30 3 5 0 5 3Time (s)Figure49:Top: 1structuralelasticmode 1.Middle: 2structuralelasticmode 2. Bottom: 3structuralelasticmode 3. andaerothermoelastice ects,wherethecontrolinputismultipliedbyanuncertain,columnde cient,parametervaryingmatrix.Apot entialdrawbackoftheresultisthatthe controlstructurerequiresthattheproduct oftheoutputmatrixwiththenominalcontrol matrixbeinvertible.Fortheoutputmatrixandnominalmatrix,theelevatorandcanard de ectionanglesandthefuelequivalenceratiocanbeusedfortrackingoutputssuchas thevelocity,angleofattack,andpitchrateorvelocityandthe ightpathangle,orvelocity, ightpathangleandpitchrate.Yet,thesecontrolscannotbeappliedtosolvethe altitudetrackingproblembecausethealtitudeisnotdirectlycontrollableandtheproduct oftheoutputmatrixwiththenominalcontrolmatrixissingular.However,theintegrator backsteppingapproachthathasbeenexaminedinotherrecentresultsforthehypersonic vehiclecouldpotentiallybeincorporatedinthecontrolapproachtoaddresssuchobjectives.ALyapunovbasedstabilityanalysisisprovidedtoverifytheexponentialtracking result.Althoughthecontrollerwasdevelopedusingalinearparametervaryingmodelof thehypersonicvehicle,simulationresultsforthefullnonlinearmodelwithtemperature variationsandexogenousdisturbancesillustratetheboundednessofthecontrollerwith 51 PAGE 52 favorabletransientandsteadystatetrackingerrors.TheseresultsindicatethattheLPV modelwithexogenousdisturbancesisareasonableapproximationofthedynamicsforthe controldevelopment. 52 PAGE 53 CHAPTER5 CONTROLPERFORMANCEVARIATIONDUETONONLINEAR AEROTHERMOELASTICITYINAHYPERSONICVEHICLE:INSIGHTSFOR STRUCTURALDESIGN 5.1Introduction Typically,controllersaredevelopedtoachievesomeperformancemetricsforagiven HSVmodel.However,improvedperformanceandrobustnesstothermalgradientscould resultifthestructuraldesignandcontroldesignwereoptimizedinunison.Alongthis lineofreasoningin[16,23],theadvantageofcorrectlyplacingthesensorsisdiscussed, representingamovetowardsimplementingacontrolfriendlydesign.Apreviouscontrol orienteddesignanalysisin[6]statesthatsimultaneouslyoptimizingboththestructural dynamicsandcontrolisanintractableproblem,butthatcontrolorienteddesignmaybe performedbyconsideringtheclosedloopperformanceofanoptimalcontrolleronaseries ofdi erentopenloopdesignmodels.Thebestperformingdesignmodelisthensaidto havetheoptimaldynamicsinthesenseofcontrollability. Knowledgeofthebetterperformingthermalgradientscanprovidedesignengineers insighttoproperlyweighttheHSVsthermal protectionsystemforbothsteadystateand transient ight.Thepreliminaryworkin[6]providesacontrolorienteddesignarchitecture byinvestigatingcontrolperformancevariationsduetothermalgradientsusingan Hcontroller.Chapter5seekstoextendthecontrolorienteddesignconcepttoexaminecontrol performancevariationsforHSVmodelsthatincludenonlinearaerothermoelasticdisturbances.Giventhesedisturbances,Chapter5focusesonexaminingcontrolperformance variationsforourpreviousmodelreferencerobustcontrollerin[52]andpreviouschapters toachieveanonlinearcontrolorientedanalysiswithrespecttothermalgradients.By analyzingthecontrolerrorandinputnormsoverawiderangeoftemperaturepro lesan optimaltemperaturepro lerangeissuggested.Basedonpreliminaryworkdonein[50],a numberoflineartemperaturepro lemodelsareexaminedforinsightintothestructural design.Speci cally,thefullsetofnonlinear ightdynamicswillbeusedandcontrole ort, 53 PAGE 54 errors,andtransientssuchassteadystatetimeandpeaktopeakerrorwillbeexamined acrossthedesignspace. 5.2DynamicsandController TheHSVdynamicsusedinthischapterarethesameisinChapter3andequations (3)(36).SimilarlyasintheresultsinC hapter4,thedynamicsin(3)(3)are reducedtothelinearparametermodelusedin(2)and(22)with = =2 .Forthe controlorienteddesignanalysis,anumberofdi erentlinearpro lesarechosen[6,50] withvaryingnoseandtailtemperaturesasillustratedinFigure51.Thissetofpro les de nethespacefromwhichthecontrolorientedanalysiswillbeperformed.Asseenin Figure51,thetemperaturepro lesarelinearanddecreasingtowardsthetail.These pro lesarerealisticbasedonshockformationatthefrontofthevehicleandthatthe temperaturesarewithintheexpectedrangeforhypersonic ight.Basedonprevious 1 2 3 4 5 6 7 8 9 100 200 300 400 500 600 700 800 900 Fuselage StationTemperature (F)Figure51:HSVsurfacetemperaturepro les. [450 900 ] ,and [100 800 ] controldevelopmentin[52]andinthepreviousChapters,thecontrolinputisdesignedas = ( 0) 1[( + 3 3) ( ) ( + 3 3) (0)+ ( )] (51) 54 PAGE 55 where ( ) R2isanimplicitlearninglawwithanupdaterulegivenby ( )= k ( ) k ( ( ))+(+ 3 3) ( )+ ( ( )) (52) where R2 2denotepositivede nite,diagonalconstantcontrolgain matrices, 0 R11 2representsaknownnominalinputmatrix, ( ) denotesthe standardsignumfunctionwherethefunctionisappliedtoeachelementofthevector argument,and 2 2denotesa 2 2 identitymatrix. 5.3OptimizationviaRandomSearchandEvolvingAlgorithms Foreachoftheindividualtemperaturepro lesexamined,thecontrolgains and in(51)(5)wereoptimizedforthespeci cplantmodelusinga combinationofrandomsearchandevolvingalgorithms.Sinceboththeplantmodel simulationdynamicsandthecontrolschemeitselfarenonlinear,traditionalmethodsfor lineargaintuningoptimizationcouldnotbeused.Theselectedmethodisacombination ofacontrolgainrandomsearchspace,combinedwithanevolvingalgorithmscheme whichallowsthesearchto ndanearestsetofoptimalcontrolgainsforeachindividual plant.Thismethodallowsonenearoptimalcontroller/planttobecomparedtotheother nearoptimalcontroller/plantsandprovi desamoreaccuratewayofcomparingcases. The rststepinthecontrolgainoptimizationsearchisarandominitialization.For thisnumericalstudy, 1000 randomlyselectedsetsofcontrolgainsareusedforagiven plantmodel.A 1000 initialrandomsetwaschosentoprovidesu cientsamplingto insureglobalconvergence.Thefollowingsectionhasaspeci cexamplecaseforoneof thetemperaturepro les.Afterthe 1000 controlgainsetsareselected,allthesetsare simulatedonthegivenplantmodelandthecontrollerin(5)and(52)isappliedto trackacertaintrajectoryaswellasrejectdisturbances.Thetrajectoryanddisturbances werechosenthesamethroughouttheentirestudysothattheonlyvariationswillbedue totheplantmodelandcontrolgains.Theexamplecasesectionexplicitlyshowsboththe desiredtrajectoryandthedisturbancesinjected. 55 PAGE 56 Afterthe 1000 initialrandomcontrolgainsearchisperformed,thetop veperformingsetsofcontrolgainsarechosenastheseedsfortheevolvingalgorithmprocess.This processisrepeatedforfourgenerations,eachwiththebest veperformingsetsofcontrol gainsateachstep.Allevolvingalgorithmshavesomeorallofthefollowingcharacteristics:elitism,crossover,andrandommutation.Thisparticularnumericalstudyusesall threeasfollows.Thebest veperformingsetsineachsubsequentgeneration,arechosen aseliteandmoveontothenextiterationstep.Fromthose ve,eachsetofcontrolgains isaveragedwithallotherpermutationsofcontrolgainsintheeliteset.Forinstance,if parent #1 isaveragedwith #2 toformano springsetofcontrolgains.Parent #1 is alsoaveragedwithparent #3 foraseparatesetofo springcontrolgains.Inthisway,all combinationsofcrossoverareperformed.Thepermutationsofthe veeliteparentsyielda totalof 10 o spring. Thenextgenerationcontainsthe veeliteparentsfromthegenerationbefore, aswellasthe 10 crossovero spring,foratotalof 15 .Eachofthese 15 setsofcontrol gainsisthenmutatedbyacertainpercentage.Basedonpreliminarynumericalstudies performedonthisspeci cexample,therandommutationswerechosentobe 20% forthe rsttwogenerationsand 5% forthe naltwogenerations.Thisproducedbothglobal searchinthebeginning,andre nementattheendoftheoptimizationprocedure.The setof 15 remains,withtheadditionof 20 mutatedsetsforeachofthe 15 .Thisgivesa totalcontrolgainsetforthenextgenerationofsearchof 315 .Asstated,therearefour evolvinggenerationsafterthe rst 1000 randomcontrolsets.Thecombinednumberof simulationswithdi erentcontrolgainsperformedforasingletemperaturepro lecaseis 2260 .Theseparticularnumberswerechosenbasedonpreliminarytrialoptimizationcases, withthegoaltoprovidesu cientsearchtoachieveconvergenceofaminimumforthecost function.Thefollowingsectionillustratestheentireprocedureforasingletemperature pro lecase. 56 PAGE 57 Thecostfunctionisdesignedsuchthattheerrorsandcontrolinputsarethesame orderofmagnitudes,sothattheycanmoreeasilybeaddedandinterpreted.Thisis importantbecauseforexample,thedesiredvelocityishigh(inthethousandsofft/s)and thedesiredpitchrateissmall(fractionofradians).Explicitly,thecostfunctionistakenas thesumofthecontrolanderrornormsandisscaledas = 100 1000180 2(53) and = 180 10 2(54) where ( ) ( ) R arethevelocityandpitchrateerrors,respectively,and ( ) ( ) R aretheelevatorandfuelratiocontrolinputs,respectively,and kk2denotesthestandard 2 norm.Thecombinedcostfunctionisthesumoftheindividual componentsandcanbeexplicitlywrittenas = + (55) where isthecostvalueassociatedwithallsubsequentoptimalgainselection. 5.4ExampleCase TheHSVparametersusedinthesimulationare =75 000 = 86723 2,and =32 174 2 asde nedin(31)(36).Toillustratehowtherandom searchandevolvingoptimizationalgorithmswork,thissectionisprovidedasadetailed example.Firsttheoutputtrackingsignalanddisturbancesareprovided,followedbythe optimizationandconvergenceprocedure.Thegoalofthissectionistodemonstratethat thespeci cnumberofelites,o spring,mutations,andgenerationslistedintheprevious sectionarejusti edinthatthecostfunctionshowsasymptoticconvergencetoaminimum. ThedesiredtrajectoryisshowninFigure52andthedisturbanceisdepictedinFigure 53,wherethemagnitudesarechosenbasedonpreviousanalysisperformedin[52].The examplecaseisbasedonatemperaturepro lewith =350 and =200 .For 57 PAGE 58 0 2 4 6 8 10 .5 0 0.5 1 Pitch Rate (deg./s) 0 2 4 6 8 10 7800 7850 7900 7950 Time (s)Velocity ft/s Figure52:Desiredtrajectories:pitchrate (top)andvelocity (bottom). 0 1 2 3 4 5 6 7 8 9 10 0 1 x 10 fVdot (ft/s2) 0 1 2 3 4 5 6 7 8 9 10 0 5 f dot (Deg./s) 0 1 2 3 4 5 6 7 8 9 10 .5 0 0.5 fQdot (Deg./s2) 0 1 2 3 4 5 6 7 8 9 10 .05 0 0.05 fetadot (1/s2)Time (s) Figure53:Disturbancesforvelocity (top),angleofattack (secondfromtop),pitch rate (secondfrombottom)andthe 1structuralmode(bottom). 58 PAGE 59 0 2 4 6 8 10 .02 0 0.02 0.04 eQ (deg./s) 0 2 4 6 8 10 .5 .5 0 0.5 Time (s)eV (ft/s) Figure54:Trackingerrorsforthepitchrate indegrees/sec(top)andthevelocity in ft/sec(bottom). thisparticularcase,Figure54andFigure55s howthetrackingerrorsandcontrolinputs, respectively,forthecontrolgains = 11 170 039 61 = 14 550 0224 0 = 25 990 00 618 = 20 70 00 369 = 0 9150 00 898 (56) ThecostfunctionshavevaluesasseeninFigure56.InFigure56thecontrolinput costremainsapproximatelythesame,butasthecontrolgainsevolve,theerrorcostand hencetotalcostdecreaseasymptotically.The 1 veiterationscorrespondtothetop ve performersinthe rst 1000 randomsample,andeachsubsequent vecorrespondtothe top veforthesubsequentevolutiongenerations.Tolimittheoptimizationsearchdesign space,allsimulationsareperformedwithtwoinputsandtwooutputs.Asindicatedinthe costfunctionslistedin(53)(55),theinputsincludetheelevatorde ection ( ) andthe fuelratio ( ) ,andtheoutputsarethevelocity ( ) andthepitchrate ( ) 59 PAGE 60 0 2 4 6 8 10 9 9.5 10 10.5 11 e (deg.) 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 Time (s)f Figure55:Controlinputsfortheelevator indegrees(top)andthefuelratio (bottom). 0 5 10 15 20 25 1.2 1.4 1.6 1.8 x 105 Total Cost 0 5 10 15 20 25 9.594 9.5945 9.595 9.5955 9.596 x 104 Control Cost 0 5 10 15 20 25 3 4 5 6 7 x 104 Error CostIteration # Figure56:Costfunctionvaluesforthetotalcost (top),theinputcost (middle) andtheerrorcost (bottom). 60 PAGE 61 5.5Results Theresultsofthissectioncoverallthetemperaturepro lesshowninFigure51.The datapresentedincludesthecostfunctionsaswellasothersteadystateandtransientdata. Includedinthisanalysisarethecontrolcostfunction,theerrorcostfunction,thepeaktopeaktransientresponse,thetimetosteady state,andthesteadystatepeaktopeak, forbothcontrolanderrorsignals.Becausethedatacontainsnoise,asmoothedversion ofeachplotisalsoprovided.Thesmoothedplotsuseastandard2dimensional ltering, whereeachpointisaveragedwithitsneighbors.Forinstanceforsomevariable ,the averageddataisgeneratedas = (4+ +1 + 1 + +1+ 1) 8 (57) Theaveragingformulashownin(5)isusedfor lteringofallsubsequentdata.Also, notethatthelowerrighttriangleformationisduetothedesignspaceonlycontaining temperaturepro leswherethenoseishotterthanthetail.Thisisduetotheassumption thatbecauseofaerodynamicheatingfromtheextremespeedsoftheHSV,thatthis willalwaysbethecase.Thesetemperaturepro lesrelatetotheunderlyingstructural temperature,notnecessarilytheskinsurfacetemperature.Figure57andFigure58show thecontrolcostfunctionvalue .Notethatthereisaglobalminimum,howeveralso noteforallofthecontrolnormsthetotalvaluesareapproximatelythesame.Thisdata indicatesthatwhileotherperformancemetricsvariedwidelyasafunctionoftemperature pro le,theoverallinputcostremainsapproximatelythesame.InFigure59andFigure 510,theerrorcostisshown.Notethatthe reisvariability,butthatthereseemsto bearegionofsmallererrorsinthecoolersectionofthedesignspace.Namely,where [200 600] and [100 250] .Combiningthecontrolcostfunctionwith theerrorcostfunctionyieldsthetotalcostfunction(andits lteredcounterpart)depicted inFigure511(andFigure512,respectively).Theimportanceofthisplotisthatthe totalcostfunctionwasthecriteriaforwhichthecontrolgainswereoptimized.Inthis 61 PAGE 62 Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 9.5938 9.594 9.5942 9.5944 9.5946 9.5948 9.595 9.5952 9.5954 9.5956 x 104 Figure57:Controlcostfunction dataasafunctionoftailandnosetemperature pro les. Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 9.5942 9.5944 9.5946 9.5948 9.595 9.5952 9.5954 9.5956 x 104 Figure58:Controlcostfunction data( ltered)asafunctionoftailandnosetemperaturepro les. 62 PAGE 63 Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 3.5 4 4.5 5 x 104 Figure59:Errorcostfunction dataasafunctionoftailandnosetemperatureproles. Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 4 4.2 4.4 4.6 4.8 5 5.2 5.4 x 104 Figure510:Errorcostfunction data( ltered)asafunctionoftailandnosetemperaturepro les. 63 PAGE 64 Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 1.3 1.35 1.4 1.45 1.5 x 105 Figure511:Totalcostfunction dataasafunctionoftailandnosetemperatureproles. sense,thetotalcostplotsrepresentwherethetemperatureparametersarebestsuitedfor controlbasedonthegivencostfunction.Sincethecostofthecontrolinputisrelatively constant,thetotalcostlargelyshowsthesamepatternastheerrorcost.Inadditionto theregionbetween [200 600] and [100 250] ,therealsoseemstobe aregionbetween =900 and [600 900] ,wheretheperformanceisalso improved. Thecontrolcost,errorcost,andtotalcos twereimportantintheoptimizationof thecontrolgainsandwereusedasthecriteriaforselectingwhichgaincombinationwas considerednearoptimal.However,therearepotentiallyotherperformancemetricsof value.Inadditiontotheoptimizationcosts,thepeaktopeaktransienterrors,timeto steadystate,andsteadystatepeaktopeakerrorswereexaminedforfurtherinvestigation. Thepeaktopeaktransienterrorisproducedbytakingthedi erencefromthemaximum andminimumtransienttrackingerrors.T hepeaktopeakerrorforthepitchrate ( ) isplottedinFigure513andFigure514,andthepeaktopeakforthevelocity ( ) is 64 PAGE 65 Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 1.35 1.4 1.45 1.5 x 105 Figure512:Totalcostfunction data( ltered)asafunctionoftailandnosetemperaturepro les. plottedinFigure515andFigure516.Thepitchratepeaktopeakerrorsdonothave alargevariationforthedi erentplants,otherthananoticeablepoorperformingregion around =550 and =450 .Thevelocitypeaktopeakhasaminimum aroundthesimilar [200 600] and [100 250] .Thevelocitypeaktopeakhasminimumswhenthepitchratehasmaximums,indicatingadegreeoftradeo betweenbettervelocityperformance,butworsepitchrateperformance,andviceversa. Anexaminationofthetimetosteadystateplotsforpitchrateandvelocityshownin Figures517520indicatesrelativelysimilartransienttimes,withafewoutliers.Having littlevariationmeansthatalltheplantmodelsaresimilarinthetransienttimeswith thisparticularcontroldesign.Thetimetosteadystateiscalculatedbylookingatthe transientperformanceandextractingthetimeittakesfortheerrorsignalstodecaybelow thesteadystatepeaktopeakerrorvalue. 65 PAGE 66 Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 0.15 0.2 0.25 0.3 0.35 0.4 Figure513:Peaktopeaktransienterrorforthepitchrate ( ) trackingerrorin deg./sec.. Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 0.15 0.2 0.25 0.3 0.35 0.4 Figure514:Peaktopeaktransienterror( ltered)forthepitchrate ( ) trackingerror indeg./sec.. 66 PAGE 67 Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 1.45 1.5 1.55 1.6 1.65 1.7 Figure515:Peaktopeaktra nsienterrorforthevelocity ( ) trackingerrorinft/sec.. Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 1.5 1.55 1.6 1.65 1.7 Figure516:Peaktopeaktransienterror( ltered)forthevelocity ( ) trackingerrorin ft./sec.. 67 PAGE 68 Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Figure517:Timetosteadystateforthepitchrate ( ) trackingerrorinseconds. Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Figure518:Timetosteadystate( ltered)forthepitchrate ( ) trackingerrorinseconds. 68 PAGE 69 Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 0.5 1 1.5 2 2.5 3 Figure519:Timetosteadystateforthevelocity ( ) trackingerrorinseconds. Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 0.5 1 1.5 2 2.5 3 Figure520:Timetosteadystate( ltered)forthevelocity ( ) trackingerrorinseconds. 69 PAGE 70 Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 Figure521:Steadystatepeaktopeakerrorforthepitchrate ( ) indeg./sec.. Finally,thesteadystatepeaktopeakerrorvaluescanbeexaminedforbothoutput signals.Thesteadystatepeaktopeakerrorsarecalculatedbywaitinguntiltheerror signalfallstowithinsomenonvanishingsteadystateboundaftertheinitialtransients havedieddown,andthenmeasuringthemaximumpeaktopeakerrorwithinthat bound.Theplotsforsteadystatepeaktopeakerrorforthepitchrateandvelocityare showninFigures521524.Thesteadystatepeaktopeakerrorsshowaminimumin thesimilarregionasseenforotherperformancemetrics,i.e. [200 600] and [100 250] Bynormalizingallofthepreviousdataabouttheminimumofeachsetofdata,and thenaddingtheplotstogether,acombinedplotisobtained.Thisplotassumesthatthe designerweightseachoftheplotse qually,butthemethodcouldbemodi edifcertain aspectsweredeemedmoreimportantthanothers.Explicitly,datafromeachmetricwas combinedasaccordingto = 1 P1( ) min( ( )) (58) 70 PAGE 71 Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 Figure522:Steadyst atepeaktopeakerror( ltered)forthepitchrate ( ) indeg./sec.. Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 2 4 6 8 10 12 x 10 Figure523:Steadystatepeaktopeakerrorforthevelocity ( ) inft./sec.. 71 PAGE 72 Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 2 4 6 8 10 12 x 10 Figure524:Steadystatepeaktopeakerror( ltered)forthevelocity ( ) inft./sec. where isthenewcombinedandnormalizedtemperaturepro ledata, isthenumberof datasetsbeingcombined,and arethelocationcoordinatesofthetemperaturedata. Figure525showsthiscombinationofcontrolcost,errorcost,peaktopeakerror,timeto steadystate,andsteadystatepeaktopeakerrorforbothpitchrateandvelocitytracking errors.Byexaminingthiscostfunction,anoptimalregionbetween [200 600] and [100 250] isdetermined. Inaddition,optimalregionsforthecontr olgainscanbeexamined.Thecontrolgains usedforthisproblemareshownin(5)and(5)havingtheform = 10 0 2 = 10 0 2 = 10 0 2 = 10 0 2 = 10 0 2 (59) Byexaminingthecontrolgainsthemaximum,minimum,mean,andstandard deviationcanbecomputedforallsetsofcontrolgainsfoundtobenearoptimal.Table51 72 PAGE 73 Tail Temp (F)Nose Temp (F) 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 2 3 4 5 6 7 Figure525:Combinedoptimization chartofthecontrolanderrorcosts,transientand steadystatevalues. Table51:OptimizationControlGainSearchStatistics 1212121212 Mean25.3536.6016.07265.328. 389.6527.4314.120.9720.8958 Std.7.727.647.0585.613.17.9813.510.60.15650.133 Max44.655.353.6423.557.336.462.139.11.3181.201 Min7.143.586.309.7620.3600.0500.3920.1100.6580.6640 showsthecontrolgainstatistics.Thisdataisusefulindescribingtheoptimalrangefor whichcontrolgainswereselected.Byknowingtheregionofnearoptimalattractionfor thecontrolgains,afuturesearchcouldbecon nedtothatregion.Thestandarddeviation alsosayssomethingaboutthesensitivityofthecontrol/aircraftdynamics,wherelarger standarddeviationsmeanthatparticulargainhaslesse ectontheoverallsystemand viceaversa. 5.6Conclusion Acontrolorientedanalysisofthermalgradientsforahypersonicvehicle(HSV) ispresented.ByincorporatingnonlineardisturbancesintotheHSVmodel,amore representativecontrolorientedanalysiscanbeperformed.Usingthenonlinearcontroller developedinChapter2andChapter4,performancemetricswerecalculatedforanumber 73 PAGE 74 ofdi erentHSVtemperaturepro lesbasedonthedesignprocessinitiallydeveloped in[6,50].Resultsfromthisanalysisshowthatthereisarangeoftemperaturepro les thatmaximizesthecontrollere ectiveness.Forthisparticularstudy,therangewas [200 600] and [100 250] Inaddition,thisresearchhasshown therangeofcontrolgains,usefulforfuturedesignandnumericalstudies.ThiscontrolorientedanalysisdataisusefulforHSVstructuraldesignsandthermalprotectionsystems. Knowledgeofadesirabletemperaturepro leandcontrolgainswillallowengineers anddesignerstobuildaHSVwiththeproperthermalprotectionthatwillkeepthe vehiclewithinadesiredoperatingrangebasedoncontrolperformance.Inaddition,this numericalstudyprovidesinformationthatcanbefurtherusedinmoreelaborateanalysis processesanddemonstratesonepossiblemethodforobtainingperformancedatafora givencontrolleronthecompletenonlinearHSVmodel. 74 PAGE 75 CHAPTER6 CONCLUSIONSANDFUTUREWORK 6.1Conclusions AnewtypeoncontrollerisdevelopedforLPVsystemsthatrobustlycompensates fortheunknownstatematrix,disturbances,andcompensatesfortheuncertaintyinthe inputdynamicinversion.Incomparisonwithpreviousresults,thisworkpresentsanovel approachincontroldesignthatstandsoutfromtheclassicalgainschedulingtechniques suchasstandardscheduling,theuseofLMIs,andthemorerecentdevelopmentofLFTs, includingtheirnonconvex typeoptimizationmethods.Classicalproblemssuchasgain schedulingsu erfromstabilityissuesandtherequirementthatparametersonlychange slowly,limitingtheirusetoquasilinearcases.LMIsuseconvexoptimization,buttypically requiretheuseofnumericaloptimizationschemesandareanalyticallyintractableexcept inrarecases.LFTsfurtherthecontroldesignforLPVsystemsbyusingsmallgaintheory,howevertheycannotdealexplicitlywithuncertainparameters.Tohandleuncertain parameters,theLFTproblemisconvertedi ntoanumericaloptimizationproblemsuch as typeoptimization. typeoptimizationisnonconvexandthereforesolutionsmay notbefoundevenwhentheyexist.Therobustdynamicinversioncontroldevelopedfor uncertainLPVsystemsalleviatestheseproblems.Aslongassomeknowledgeoftheinput matrixisknownandcertaininvertabilityrequirementsaremetthenastabilizingcontrolleralwaysexists.Proofsprovidedshowthatthecontrollerisrobusttodisturbances, statedynamics,anduncertainparametersbyusinganewrobustcontrollertechniquewith exponentialstability. CommonapplicationsforLPVsystemsare ightcontrollers.Thisisbecausehistorically ighttrajectoriesvaryslowlywithtimeandarewellsuitedtothepreviously mentionedLPVcontrolschemessuchasgainscheduling.Recentadvancesintechnology andaircraftdesignaswellasmoredynamicanddemanding ightpro leshaveincreased thedemandonthecontrollers.Inthesedemandingdynamicenvironments,parameters 75 PAGE 76 nolongerchangeslowlyandmaybeunknownoruncertain.Thisrenderspreviouscontroldesignslimiting.Motivatedbythisfactandspeci callyusingthedynamicsofan airbreathingHSV,thedynamicsareshowntobemodeledasanLPVsystemwithuncertaintiesanddisturbances.Thisworkmotivatesthedesignandtestingoftherobust dynamicinversioncontrolleronatemperaturevaryingHSV.Usingunknowntemperature pro les,whilesimultaneouslytrackinganoutputtrajectory,therobustcontrollerisshown tocompensateforunknowntimevaryingpara metersinthepresenceofdisturbancesfor theHSV.Usingonesetofcontrolgainsitwasshownthatstablecontrolwasmaintained overtheentiredesignspacewhileperformingmaneuvers.EventhoughthecontrolwasdevelopedforLPVsystems,thesimulationresultsareperformedonthefullnonlinearHSV ightandstructuraldynamics,hencevalidatedthecontrolorientedmodelingassumptions. Finally,anumericaloptimizationschemewasperformedonthesameHSVmodel, usingacombinationofrandomsearchandevolvingalgorithmstoproducedynamic optimizationdataforthecombinedvehicleandcontroller.Regionsofoptimalitywere showntoprovidefeedbacktodesignengineersonthebestsuitabletemperaturepro le parameterspace.Toremoveambiguity,thecontrollerforeachindividualtemperature pro lecasewasoptimallytunedandthetrackin gtrajectoryanddisturbanceswerekept thesame.Analyticalmethodsdonotexistforoptimalgaintuningnonlinearcontrollerson nonlinearsystemsHence,anumericaloptimizingschemewasdeveloped.Bystrategically searchingthecontrolgainspacevalueswereobtained,andtheperformancemetricsat thatpointwerecomparedacrossthevehicledesignspace.Thisworkmaybeusefulfor futuredesignproblemsforHSVswherethestructuralanddynamicdesignareperformed inconjunctionwiththecontroldesign. 6.2Contributions AnewrobustdynamicinversioncontrollerwasdevelopedforgeneralperturbedLPV systems.Thecontroldesignrequiresknowledgeofabestguessinputmatrixandat leastasmanyinputsastrackedoutputs.Inthepresenceanunknownstatematrix, 76 PAGE 77 parameters,anddisturbance,andwithanuncertaininputmatrix,thedeveloped controldesignprovidesexponentialtrackingprovidedcertainassumptionsaremet. Thedevelopedcontrolmethodtakesadi erentapproachtotraditionalLPVdesign andprovidesaframeworkforfuturecontroldesign. BecausetheassumptionsrequiredofthecontrolleraremetbytheHSV,anumerical simulationwasperformed.AfterreducingtheHSVnonlineardynamicstothat ofanLPVsystemmotivationwasprovidedtoimplementthecontrollerdesigned. AsimulationisprovidedwherethefullnonlinearHSVdynamicsareused.The simulationdemonstratesthee cacyoftheproposedcontroldesignonthisparticular HSVapplication.Awiderangeoftemperaturevariationswereusedandtracking controlwasimplementedtodemonstratetheperformanceofthecontroller. FurtherperformanceevaluationwasconductedbydesigninganoptimizationproceduretoanalyzetheinterplaybetweentheHSVdynamics,temperatureparameters, andcontrollerperformance.Anumberofdi erenttemperatureplantmodelsfor HSVwerenearoptimallytunedusingacombinationofarandomsearchandevolvingalgorithms.Next,thecontrolperformancewasevaluatedandcomparedtothe otherHSVtemperaturemodels.Comparativeanalysisisprovidedthatsuggests regionswherethetemperaturepro lesoftheHSVinconjunctionwiththeproposed controldesignachieveimprovedperformanceresults.Theseresultsmayprovide insighttostructuralsystemsdesignersforHSVsaswellasprovidesca oldingfor futurenumericaldesignoptimizationandcontroltuning. 6.3FutureWork Therobustdynamicinversioncontroldesigninthisdissertationrequiresknowledge ofthesignoftheerrorsignalderivativeterms.Whilethesemeasurementsmaybe availableforspeci capplications,thisunderlyingnecessityreducesthegeneralityof thecontroller.Futureworkcouldfocuson removingthisrestriction,andproducing anoutputfeedbackonlyrobustdynamicinversioncontrol. 77 PAGE 78 Anotherrequirementofthecontroldesignistherequirementofthediagonal dominanceofthebestguessfeedforwardinputmatrix.Whilethisrequirementis notunreasonablebecauseitonlyrequiresthattheguessbewithinthevicinityof theactualvalue,futureworkcouldfocusonrelaxingthatrequirement.Alleviating thisrestrictioncouldpotentiallybedonebyusingpartialadaptationlawswhile simultaneouslyusingrobustalgorithmstocountertheparametervariations. Itwasshownthatthecontrollerdevelopedisabletotrackinnerloopstatesforthe HSV,howeveritwouldbebene cialtoadaptthisinnerloopcontroldesigntoan outerloop ightplanningcontroller.Inthisway,morepracticalplannedtrajectories canbetracked(e.g.,altitude)byusingtheinnerloopofpitchrateandpitchangle control.Additionally,thissameresultcanbeattainedbyusingbackstepping techniques.Bybacksteppingthroughotherstatedynamics(e.g.,altitude)andinto thecontroldynamics(e.g.,pitchrate),acombinedcontrollercouldbedeveloped. Thetemperatureandcontrolgainoptimizationprovidesagoodframeworkfor ndingHSVdesignswithincreasedperformance.Itwouldbeinterestinginfuture worktoreanalyzetheoptimalcontrolgainspace,andseeifitcouldbeconvergedto asmallerset.Iftheoptimalsetcouldbefurtherconverged,thenthroughnumerous iterationsaverypreciseandnarrowrangemaybefound.Findingamoreoptimal designspacemayaidinfuturestructuraloptimizationsearches. Itwouldalsobebene cialfortheoptimizationworktohavemoreaccuratenonlinear models.ObtainingbettermodelswillrequireworkingincollusionwithHSV designers.Gettinghighqualityfeedbackonthedesignconstraintsand ight trajectoryconstraintswouldfurtheraidthesearchforoptimalityinregardsto controlgainsandtemperaturepro les.Inaddition,thedynamicscouldbemodeled andsimulatedwithhighercertaintyifmoredetailswereknown.Combiningextra dataonthedynamicsintothecontroldesignwouldhelpfurtherthedevelopmentof actual ightworthyvehicles. 78 PAGE 79 APPENDIXA OPTIMIZATIONDATA Thedatapresentedinthefollowingtablesistherawdatafromtheimagespresented inChapter5.Therowscontainsallofthe in andthecolumnscontainthe in .Emptyspacesareplaceswherethetailtemperatureishigherthanthenose temperature,andareoutsidethedesignspaceofthisworkandommitted. 79 PAGE 80 TableA1:Totalcostfunction,usedtogenerateFigure511and512(Part1) 100150200250300350400450500 100144526 150143210145071 200141588140254143397 250141588140254143397142557 300143086140577143199143895142656 350133478145807129636134531141681143496 400143490143396139825140233146708142439140353 450129673141283141577136368143591144789144610149182 500140466139064141863144110145435140439145178142468141932 550143730144033137552140079147113143303139847139083143308 600143730145599138430140945147159143625140785139202144040 650143884137784145621144958151291148236144025145853144782 700146708142439140353138181143955145086144610149182129812 750144610149182129812140633144027146527140466139965146527 800140845146015139499140904143730129426146864144790135440 850141959138801142931145923138328145212142817140848140940 900143955145086144610149182129812140633144027146527140466 TableA2:Totalcostfunction,usedtogenerateFigure511and512(Part2) 550600650700750800850900 550144322 600144420144857 650145109141262127435 700140633144027146527140466 750140466143396139825140233146708 800144948143418145297135394142384140069 850144253141883148014136336143641145803145941 900143828147566129349138888131875142296135461134603 80 PAGE 81 TableA3:Controlinputcostfunction,usedtogenerateFigure57and58(Part1) 100150200250300350400450500 10095951 1509594995953 200959489594995952 25095948959499595295952 3009594995950959539595395949 350959529595295957959579595395953 40095948959539595795953959549595295953 4509595195946959529595095946959539595295953 500959509595095954959489595395949959529594895953 550959529595095957959499595495952959489593795949 600959529594995950959529595195952959489593795952 650959539595295953959539595395954959499595395950 700959549595295953959469595395953959529595395952 750959529595395952959539595395954959509595095954 800959539595395953959499595395952959539595295957 850959529595395953959539595395957959539594995953 900959539595395952959539595295953959539595495950 TableA4:Controlinputcostfunction,usedtogenerateFigure57and58(Part2) 550600650700750800850900 55095952 6009595295952 650959499595295952 70095953959539595495950 7509595095953959579595395954 800959539594995953959499595395949 85095953959529595395952959539595295953 9009594895953959519594095949959539594695950 81 PAGE 82 TableA5:Errorcostfunction,usedtogenerateFigure59and510(Part1) 100150200250300350400450500 10048574 1504726049118 200456394430447444 25045639443044744446605 3004713644626472454794246706 350375254985533679385744572747542 40047541474434386744280507544648744400 4503372145337456254041847644488354865853228 500445164311445908481624948244490492254651945979 550477774808241594441295115947350438984314647358 600478574964942479449925120847673448374326448088 650479304183149667490055533752281480754990048831 700507544648744400422354800249133486585322833860 750486585322833860446804807450572445164401550572 800448925006243546449544777633474509114883739482 850460074284846978499694237549254468644489844986 900480024913348658532283386044680480745057244516 TableA6:Errorcostfunction,usedtogenerateFigure59and510(Part2) 550600650700750800850900 55048370 6004846748905 650491604531031482 70044680480745057244516 7504451647443438674428050754 800489954746949343394384643044120 85048299459315206040384476884985049987 9004788051613333974294735925463423951438653 82 PAGE 83 TableA7:Pitchrate,peaktopeakerror,usedtogenerateFigure513and514(Part1) 100150200250300350400450500 1000.1951 1500.16780.1377 2000.20570.17220.1421 2500.20570.17220.14210.1842 3000.14500.25880.13650.18030.1669 3500.13740.17120.14270.13720.15360.1601 4000.13990.15300.15000.18350.13360.14480.1849 4500.15350.24780.12780.22140.28390.14210.14340.1436 5000.21750.21970.15050.17280.15900.16720.14810.21740.1292 5500.23380.16240.14910.14300.13430.18670.18480.44580.2287 6000.17380.20850.14650.15480.20710.13940.17990.45610.1471 6500.15600.13270.18570.15530.14060.14000.13740.16650.1530 7000.13360.14480.18490.29280.15390.14150.14340.14360.1530 7500.14340.14360.15300.15730.16920.16550.21750.22000.1655 8000.15100.13310.15020.15730.15950.19160.16550.18320.1473 8500.19390.14680.15320.15320.19920.14640.14320.20640.1409 9000.15390.14150.14340.14360.15300.15730.16880.16550.2175 TableA8:Pitchrate,peaktopeakerror,usedtogenerateFigure513and514(Part2) 550600650700750800850900 5500.1787 6000.19600.1309 6500.17190.19470.1353 7000.15730.16920.16550.2175 7500.29120.15300.16120.18350.1939 8000.14710.26730.13560.13540.16580.1833 8500.16410.13230.13980.15070.14380.17330.1395 9000.14910.14930.14990.39290.22760.18220.29410.2615 83 PAGE 84 TableA9:Pitchrate,steadystatepeaktopeakerror,usedtogenerateFigure521and 522(Part1) 100150200250300350400450500 1000.0170 1500.01630.0170 2000.01790.01560.0178 2500.01790.01560.01780.0163 3000.01760.01660.01730.01840.0167 3500.00270.02330.00160.00280.01500.0173 4000.02320.01690.00480.01490.01860.01540.0144 4500.00120.02000.01730.00310.01920.01830.02210.0221 5000.00530.00480.00700.01770.01660.01660.01670.01840.0160 5500.01960.01690.00360.01660.01730.01930.01590.00320.0185 6000.01730.01860.00450.01630.01780.01830.01540.00340.0185 6500.02220.00270.01640.01520.01780.02100.01730.02020.0171 7000.01860.01540.01440.00390.01720.01910.02070.02110.0008 7500.02070.02110.00080.01540.02020.01710.00530.00560.0171 8000.01650.02130.01460.01510.01630.00290.02140.01660.0050 8500.01790.00300.01610.01700.00490.01740.01500.01760.0074 9000.01820.02020.02210.02210.00080.01600.02040.01690.0053 TableA10:Pitchrate,steadystatepeaktopeakerror,usedtogenerateFigure521and 522(Part2) 550600650700750800850900 5500.0188 6000.01800.0210 6500.01930.01850.0019 7000.01540.02020.01710.0053 7500.00480.01690.00460.01490.0180 8000.01710.01870.01630.00260.01700.0193 8500.01710.01980.01990.00340.01630.02040.0173 9000.02260.01790.00090.00580.00150.01830.00410.0021 84 PAGE 85 TableA11:Pitchrate,timetosteadystate,usedtogenerateFigure517and518(Part 1) 100150200250300350400450500 1000.439 1500.4290.433 2000.4120.3380.451 2500.4710.2870.4720.541 3000.4500.3810.5180.5180.515 3500.3940.4990.4310.5400.5120.472 4000.4710.5420.4120.4940.4020.5110.474 4500.5560.4070.4050.4750.4730.4440.4820.519 5000.5800.6130.5420.4500.4240.4730.4040.4960.473 5500.4360.3580.4440.4610.4420.4680.4500.6180.427 6000.4470.4930.5180.4750.4570.5130.5060.5930.473 6500.5180.5700.4890.4570.6010.4750.4950.5330.408 7000.4250.4710.4490.6770.4970.4640.4490.4960.450 7500.4970.4962.1430.4740.4530.4450.5640.6920.470 8000.4420.4910.4500.4710.4940.4260.4730.4950.587 8500.4320.5760.4700.5270.5930.4640.4770.5170.572 9000.4940.4920.4970.4270.4860.4960.4670.4550.537 TableA12:Pitchrate,timetosteadystate,usedtogenerateFigure517and518(Part 2) 550600650700750800850900 5500.450 6000.4210.476 6500.4530.4030.423 7000.4720.4710.4450.564 7500.5480.4730.4300.5180.479 8000.4510.4950.4600.4490.4740.522 8500.5030.5370.5580.4040.4780.4950.469 9000.4210.4910.5590.5920.8180.4490.5210.692 85 PAGE 86 TableA13:Velocity,peaktopeakerror,usedtogenerateFigure515and516(Part1) 100150200250300350400450500 1001.5670 1501.66491.6847 2001.64461.66691.5972 2501.64451.66681.59731.5986 3001.68391.66631.63441.60551.6081 3501.55961.59041.54011.53661.55801.6235 4001.58361.60221.49171.59101.67351.53571.5910 4501.56341.72541.59461.44201.70981.60641.58721.5456 5001.46861.46511.53211.59661.65791.63291.54081.60751.5710 5501.58931.74471.48591.64171.65161.61661.56271.42191.5953 6001.65371.59341.53591.59931.70761.60381.59621.42381.5834 6501.59611.61761.63661.64261.40891.61701.72211.59901.6525 7001.67351.53571.59101.43441.59491.58901.58761.54561.5294 7501.58761.54561.52941.59801.60781.69651.46861.46061.6965 8001.59481.62701.58281.68001.62481.51241.60331.60581.5128 8501.58551.52051.59841.66751.62051.54331.59661.68341.6128 9001.59401.58901.58721.54561.52941.59801.60781.69651.4686 TableA14:Velocity,peaktopeakerror,usedtogenerateFigure515and516(Part2) 550600650700750800850900 5501.5814 6001.58431.5756 6501.69161.58451.5754 7001.59801.60781.69651.4686 7501.46621.60221.52541.59101.6436 8001.59511.67371.55721.48521.58731.6127 8501.58301.58171.47351.53051.60271.56701.6405 9001.60251.56931.55421.45851.45811.60601.47211.4935 86 PAGE 87 TableA15:Velocity,steadystatepeaktopeak,usedtogenerateFigure523and524 (Part1) 100150200250300350400450500 1000.0037 1500.00880.0050 2000.00180.00360.0046 2500.00190.00380.00470.0039 3000.00370.00300.00340.00350.0131 3500.00160.00330.00100.00260.00150.0066 4000.00350.00280.00130.00290.00590.00210.0031 4500.00020.00170.00270.00040.01050.00320.00140.0031 5000.00210.00170.00150.00270.00370.00690.00220.00140.0027 5500.00690.00320.00120.00370.01030.00220.00460.00260.0035 6000.00380.00410.00220.00330.01080.00180.00380.00270.0035 6500.00350.00100.00700.00450.00410.00510.00400.00840.0066 7000.00590.00210.00310.00240.00280.00390.00140.00400.0008 7500.00140.00400.00080.00290.00370.00550.00210.00220.0055 8000.00350.00940.00380.00340.01180.00090.01260.00330.0008 8500.00280.00030.00750.00680.00060.00450.00240.00230.0016 9000.00230.00310.00310.00080.00310.00400.00990.00210.0027 TableA16:Velocity,steadystatepeaktopeak,usedtogenerateFigure523and524 (Part2) 550600650700750800850900 5500.0022 6000.00400.0027 6500.00340.00360.0008 7000.00290.00370.00550.0021 7500.00160.00280.00130.00290.0041 8000.00350.00540.00320.00150.00300.0028 8500.00270.00070.00130.00180.01010.00410.0057 9000.00270.00330.00020.00060.00050.01070.00050.0009 87 PAGE 88 TableA17:Velocity,timetosteadystate,usedtogenerateFigure519and520(Part1) 100150200250300350400450500 1002.012 1501.1190.915 2000.5390.2680.496 2500.4980.2840.5280.506 3000.5430.3140.5860.4921.201 3500.3830.5200.4740.4720.5220.701 4000.5010.5150.4030.4920.3551.940.516 4500.7470.5210.4910.8210.5430.4910.5130.637 5000.4940.4720.4840.9830.3780.5020.4810.5420.656 5500.4920.3390.3850.5460.5681.2080.4960.8411.201 6000.5620.5000.4930.5630.5781.0430.4920.7080.712 6500.5621.6810.6270.4000.7050.5211.3961.7600.932 7000.3830.4980.7040.8080.8360.4910.5040.5163.330 7500.4590.5873.3001.3470.5390.6790.4730.4960.680 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[55]M.Yoshihiko,AdaptivegainscheduledHin nitycontroloflinearparametervarying systemswithnonlinearcomponents,in Proc.IEEEAm.ControlConf. ,Denver,CO, June2003,pp.208. 93 PAGE 94 BIOGRAPHICALSKETCH ZachWilcoxgrewupinYarrowPoint,acityjustoutsideofSeattle,Washington,and livedthereuntilmovingtoFloridatoattendcollegein2001.HereceiveddualBachelor ofSciencedegreesfromtheUniversityofFloridasAerospaceandMechanicalEngineering departmentinthespringof2006.Duringhisundergraduatework,Zachparticipatedasa diveronUFsMensSwimmingDivingTeam.Inaddition,hedidresearchworkforUFs MicroAirVehicle(MAV)groupandparticipatedinInternationalMAVcompetitions.He recievedhisMastersofScienceinAerospaceEngineeringfromUniversityofFloridainthe springof2008.HisDoctoralstudieswereintheNonlinearControlsandRoboticsGroup intheDepartmentofMechanicalandAerospaceEngineeringundertheadvisementofDr. Dixon.HereceivedhisPh.D.inAerospaceEngineeringinAugust2010. 94 