A Thesis by. Viral Shailesh Zaveri. Bachelor of Science, Wichita State University, 2009

Transcription

1 H CONTROL OF SINGULARLY PERTURBED AIRCRAFT SYSTEM A Thesis by Viral Shailesh Zaveri Bachelor of Science, Wichita State University, 9 Submitted to the Department of Electrical Engineering and Computer Science and the faculty of the Graduate School of Wichita State University in partial fulfillment of the requirements for the degree of Master of Science July

2 Copyright by Viral Shailesh Zaveri All Rights Reserved

3 H CONTROL OF SINGULARLY PERTURBED AIRCRAFT SYSTEM The following faculty members have examined the final copy of this thesis for form and content, and recommend that it be accepted in partial fulfillment of the requirement for the degree of Master of Science with a major in Electrical Engineering. Ravi Pendse, Committee Chair M. Edwin Sawan, Committee Member Linda K. Kliment, Committee Member Animesh Chakravarthy, Committee Member iii

4 DEDICATION To my parents, sister, and my dear friends iv

5 ACKNOWLEDGEMENTS First, I would like to thank my advisor, Dr. Ravi Pendse, for attempting to introduce me to the field of Computer Networking and providing a splendid opportunity to work with him. I express my sincere gratitude for guiding me through the tough transition from Bachelor's to Master's degree and explaining me the real world pros and cons. I also convey my deepest thanks to Dr. M. Edwin Sawan, my academic advisor, for his invaluable time, insightful advices, guidance and support through the course of this research. My further thanks to Dr. Linda K. Kliment and Dr. Animesh Chakravarthy for their valuable time and service rendered as committee members. Finally, I sincerely owe my accomplishments to my parents and sister for their encouragement, love and support without which this might have just been a dream. v

6 ABSTRACT The objective of this research is to develop an analytical approach to control two-timescale systems operating under certain noise parameters. This approach addresses two important design criteria: augmentation of large-scale system with disturbance model and its two-timescale representation, and order reduction of the large-scale systems for reduced controller design complexity. The problem of large-scale system with Gaussian noises is solved as the stochastic system implementing linear-quadratic Gaussian control. Order reduction method uses singular perturbation techniques for the simplicity of control algorithms. Control law design process for a singularly perturbed stochastic system includes implementation and comparative analysis of optimal, composite, and reduced controller techniques. Practical model, longitudinal dynamics of digital fly-by-wire F-8C fighter aircraft, illustrates the validation of the proposed concepts. vi

7 TABLE OF CONTENTS Chapter Page. INTRODUCTION.... Background and Motivation.... Historical Perspective and Past Efforts....3 Objectives of the Thesis....4 Scope of the Thesis.... TWO-TIME-SCALE SYSTEM AND SINGULAR PERTURBATION METHOD What is Singular Perturbation? Two-Time-Scale System Time Scale Analysis State Transformation Permutation Scaling Singular Perturbation Method STOCHASTIC CONTROL OF SINGULARLY PERTURBED SYSTEM Overview Steady-State Optimal LQG Controller Problem Statement Steady-State Regulator and Kalman Filter Controller Design Techniques Composite Control Reduced Control Singular Perturbation Composite Control of Singularly Perturbed System Reduced Control of Singularly Perturbed System Controller Comparison Criteria Optimal Cost Stochastic Cost H Norm LONGITUDINAL DYNAMICS OF F-8 AIRCRAFT Brief History Linearized Aircraft Equations of Motion Dynamics of the Wind Disturbance Model Aircraft Model Augmentation State Transformation... 7 vii

8 TABLE OF CONTENTS (Continued) Chapter Page 4.4. Permutation State Measurements Sensor Noise Intensities State-Space Model Time-Scale Modeling Standard Singular Perturbation From LINEAR-QUADRATIC GAUSSIAN CONTROL OF SINGULARLY PERTURBED AIRCRAFT MODEL Simulation Procedure Simulation Test Matrix Numerical Values for Simulation Open-Loop F-8 Aircraft Model Measurement Equation Controller Implementation Optimal LQG Control of Full-Order Model Composite Control Reduced Control Controller Results and Comparisons CONCLUSIONS Summary of Research and Results Recommendations for Future Work REFERENCES viii

9 LIST OF TABLES Table Page 4. Dimensional Stability Derivatives for the Longitudinal F-8 Aircraft Model Row Norms of the Longitudinal F-8 Aircraft Model Open-Loop Characteristics of the Longitudinal F-8 Aircraft Model Simulation Test Matrices Open-Loop Eigenvalues of Time-Scaled Singularly Perturbed F-8 Aircraft Model Sensor Noise Intensities Simulation Results for Optimal LQG Control of Full-Order Model Simulation Results for Composite Control of Lower-Order Slow and Fast Subsystems Simulation Results for Reduced Control of Reduced-Order Model Summary of Controller Comparison Criteria ix

10 LIST OF FIGURES Figure Page. Example open-loop response of a TTS aircraft longitudinal dynamics A typical stochastic linear dynamic system Block diagram of linear stochastic system with LQG controller Parallel computation of slow and fast Kalman filters for LQG control NASA F-8C digital fly-by-wire test aircraft view of digital fly-by-wire F-8C crusader Open-loop longitudinal F-8 aircraft initial condition response Closed-loop response using optimal LQG control (Case (a) and ε =.4) Closed-loop response using optimal LQG control (Case (a) and ε =.336) Closed-loop response using optimal LQG control (Case (b) and ε =.4) Closed-loop response using optimal LQG control (Case (b) and ε =.336) Closed-loop response using optimal LQG control (Case (c) and ε =.4) Closed-loop response using optimal LQG control (Case (c) and ε =.336) Closed-loop response using composite control (Case (a) and ε =.4) Closed-loop response using composite control (Case (a) and ε =.336) Closed-loop response using composite control (Case (b) and ε =.4) Closed-loop response using composite control (Case (b) and ε =.336) Closed-loop response using composite control (Case (c) and ε =.4) Closed-loop response using composite control (Case (c) and ε =.336) Closed-loop response using reduced control (Case (a) and ε =.4) Closed-loop response using reduced control (Case (a) and ε =.336) x

11 LIST OF FIGURES (continued) Figure Page 5.5 Closed-loop response using reduced control (Case (b) and ε =.4) Closed-loop response using reduced control (Case (b) and ε =.336) Closed-loop response using reduced control (Case (c) and ε =.4) Closed-loop response using reduced control (Case (c) and ε =.336) Maximum singular values of open-loop versus closed-loop system... 6 xi

12 LIST OF ABBREVIATIONS DFBW LQG LQR TTS Digital Fly-By-Wire Linear-Quadratic Gaussian Linear-Quadratic Regulator Two-Time-Scale xii

13 LIST OF SYMBOLS g Acceleration due to Gravity (ft/sec ) α δ R c δ e V τ ω H u Angle of Attack (rad) Control Vector Control Weighting Matrix for Optimal Control Elevator Position (rad) Equilibrium Velocity (ft/sec) First Order Low Pass Filter Time Constant (sec) Frequency (rad/sec) Hardy Space Incremental Velocity (ft/sec) O( ) Magnitude to the Order of v Normalized Incremental Velocity (nondimensional) J* Optimal Cost J θ q ζ w L ε T S R f Performance Index/Cost Function Pitch Attitude Angle (rad) Pitch Rate (rad/sec) Root Mean Square Value of Vertical Gust Velocity (ft/sec) Scale Length (of the turbulence) (ft) Singular Perturbation Parameter Settling Time State Measurement Spectral Density xiii

14 LIST OF SYMBOLS (continued) μ Q c J S ' δ T ξ w Φ g Q State Vector State Weighting Matrix for Optimal Control Stochastic Cost Time-Scaled Measurement White Noise Process Throttle Position (nondimensional) White Gaussian Noise Due to Wind Gust Wind Gust State (rad) Wind Gust Power Spectral Density Wind Disturbance Intensity Matrix xiv

15 CHAPTER INTRODUCTION. Background and Motivation The influence of stochastic processes on a deterministic system has been a topic of interest for many decades. So much so that many researchers in the field have dedicated a great deal of time and energy towards advancing the boundaries of working knowledge relative to this topic. Consequently, great strides have taken place in the past thirty years within the context of singular perturbation theory and state-space formulation of linear systems with stochastic processes. Realizing that performance of large-scale systems can suffer a great deal in the presence of external disturbances, nominal model of such systems must be solved and analyzed accounting for these noise parameters. Consequently, a step towards the optimal solution of stochastic control for the linear-quadratic Gaussian (LQG) or H problem becomes imminent.. Historical Perspective and Past Efforts Several approaches have been made for solving the stochastic control of the LQG problem. The problem of stochastic control for two-time-scale (TTS) systems has been extensively studied in the literature in the past thirty years. The main idea is to find statistical properties for the stochastic processes and incorporate them in the large-scale system, exhibiting TTS properties, for the LQG controller design. Michael Athans et al. [] first presented the method to augment the large-scale system with disturbance model and conducted stochastic control of the complete F-8C aircraft model using LQG design incorporating multiple model adaptive control method. A. H. Haddad and P. V. Kokotovic [] analyzed the stochastic control of the LQG problem for a TTS system and applied singular perturbation theory. They showed that the optimal control could be approximated by the combination of a slow and fast control

16 computed in separate time scales. Treating the F-8C aircraft model from reference [] as the TTS system, Petar V. Kokotovic, Hassan K. Khalil, and John O'Reily [3] applied singular perturbation technique to it and solved this LQG problem for optimal and composite control. John L. Vian and M. Edwin Sawan [4] extended the problem of TTS F-8C aircraft model to further investigate the H and H norms..3 Objectives of the Thesis The control of the aircraft dynamics under the influence of wind disturbance is a challenging task. Not only is the aircraft dynamics of very high order, but it may also have incomplete state information and have measurement noises that need to be addressed during the control design process. The objectives of this thesis are twofold: () To accurately augment the aircraft model with the wind dynamics and transform it into singularly perturbed form to represent the TTS system. () To systematically obtain lower-order models using singular perturbation methodology and use them to design the LQG/H controllers to control the fullorder system in the presence of wind turbulence and state measurement disturbances, and evaluate the effectiveness of these controllers. The achievement of these objectives should suggest the best approach for LQG/H controller design for the singularly perturbed aircraft system augmented with the wind dynamics in the presence of state measurement disturbances. To the author's knowledge, such a comparative study of various controller design techniques for a singularly perturbed aircraft has not been performed before and should make a significance contribution..4 Scope of the Thesis In this thesis, optimal linear-quadratic Gaussian or H control of singularly perturbed systems is considered. The remainder of this thesis is organized as follows: Chapter Two reviews

17 the theoretical backgrounds for necessary conditions for the application of singular perturbation methods, time-scale analysis, and state transformation techniques. Chapter Three presents the stochastic control of singularly perturbed systems with white Gaussian process and measurement noises. Kalman filter design process is explained for both slow and fast subsystems. Optimal control, composite control, and reduced control design techniques are implemented. Chapter Four provides all the details of the F-8 aircraft model including wind dynamics, model augmentation, state transformation, time-scale modeling, and measurement noises. Chapter Five essentially implements the procedural techniques discussed in Chapter Three to the real model presented in Chapter Four, and presents evaluation of simulation results and comparative analysis of different control techniques. Finally, the conclusions and scope for future work are summarized in Chapter Six. 3

18 CHAPTER TWO-TIME-SCALE SYSTEM AND SINGULAR PERTURBATION METHOD. What is Singular Perturbation? A fundamental dilemma in the control system theory is the mathematical modeling of a physical system. The realistic models of many systems require high-order dynamic equations that contain small parameters such as time constants, moments of inertia, masses, resistances, inductances, capacitances, and Reynolds number. These small parasitic parameters often increase the dynamic order of the model. From the control engineer's perspective, system modeling needs to be parsimonious because the model should not be more detailed than required by the specific task. However, enough details about the small parasitic parameters must be included to guarantee satisfactory performance of the system while attempting to keep the dynamic order of the model as low as possible to reduce controller complexity and avoid numerical ill-conditioning in the design process. Engineers sometimes want to ignore small parameters in an attempt to simplify the dynamic models. For that very purpose, singular perturbation techniques can legitimize these ad hoc simplifications of dynamic model that are corrected for the small parameters to within a known order of error without introducing additional numerical ill-conditioning. A high-order system whose dimension reduces by letting a small parameter, ε, approach zero is referred to as singularly perturbed system. Generally, a system of such type comprises two widely separated clusters of eigenvalues resulting in the system to exhibit TTS properties. A TTS system attributes simultaneous occurrence of "slow" and "fast" phenomena giving rise to stiffness in the problem that leads to complexity in controller design solutions. For instance, longitudinal dynamics of an airplane features phugoid and short period modes. These motions 4

19 occur simultaneously but their decay speeds are different. It is a computational burden to solve for such large-scale systems. Evidently, system order reduction is needed, and thus the implementation of singular perturbation method for a TTS system is encouraged because modern control procedures are numerically ill-conditioned to provide a solution for such problems. Considered as a boon to the control engineers, the application of singular perturbation methodology for TTS systems has become popular as it presents with remedial features like dimensional reduction and stiffness relief. Essentially, the singular perturbation method uses asymptotic expansions to separate the full-order model into two reduced-order models that are numerically well-conditioned as their eigenvalues are clustered in the same region [5]. The resulting two separated models are the reduced ('slow') model and the boundary layer ('fast') model. Typically, the solutions attained via singular perturbation method are more accurate and optimal to within a specific O(ε) compared to those that ignore the small parameters. The principal idea behind considering a large-scale system as the TTS system is to have a systematic classification of the state variables in order to decouple them as slow and fast modes, and be able to apply singular perturbation techniques to reduce the complexity of controller design process.. Two-Time-Scale System In general, a two-time-scale system possessing two widely separated clusters of eigenvalues is represented as x( t) A A x( t) B ut () z( t) A3 A 4 z( t) B (.) n m q where xt and zt represent the state vectors and ut is the control vector, and matrices A ij and B i are of appropriate dimensionality. Note that if we associate ε with zt, it 5

20 will represent the equation (.) in the singularly perturbed form. Thus, the singularly perturbed form is just another way to represent the general TTS system [6]... Time Scale Analysis The TTS system (.) is such that the n eigenvalues of the system are close to the origin (small) and the remaining m eigenvalues are far from the origin (large), thus, giving slow and fast responses respectively. The system (.) can also be said to possess n dominant modes and m non-dominant modes. For the TTS system in the form of equation (.), Figure. shows a typical open-loop response. Figure. Example open-loop response of a TTS aircraft longitudinal dynamics The eigenspectrum e(a) of system (.) is arranged in the increasing order of absolute values as follows: s,, s, f,, f e A (. a) n s s s m e A,, (. b) n f f f e A,, (. c) n 6

21 where λ denotes eigenvalues of the system, and (. d) s s sn f f fm The system (.) exhibits two-time-scale property [7] if the largest of the absolute eigenvalue of the slow eigenspectrum e(a s ) is much smaller than the smallest absolute eigenvalue of the fast eigenspectrum e(a f ), that yields. /. (.3) sn f where the small, positive singular perturbation parameter, ε, is a measure of separation of time scales. Thus, the following inequality for the TTS system (.) holds good: ( A).. ( A ) (.4) max s min f and by the norm properties of invertible matrices, (.4) can equivalently be written as A.. A f s (.5) Hence, as the inequality (.5) suggests, the system (.) must be decoupled into two lower-order models namely slow and fast subsystems..3 State Transformation To decouple a TTS physical system into two lower-order subsystems, the full-order system, however, first needs to be in the form of equation (.). In practice a real physical TTS system may have its state variables arranged in an arbitrary order and the units of the state variables may be out of scale. Thus, a system in such form may not satisfy the following inequalities required for it to exhibit the TTS property [6]: A4.. 3 A A L (.6) A.. A 4 (.7) where A A A L and L A A 4 3 7

22 This calls for the system to be transformed such that the absolute values of its eigenvalues are arranged in increasing order as represented by expressions (.), and the inequalities (.6) and (.7) are satisfied. A physical system can be made to exhibit TTS properties through the transformation techniques like permutation and scaling. Permutation re-arranges the state variables such that the first n states of the transformed state vector correspond to the slow states and the remaining m states correspond to the fast states. Scaling readjusts the units and reduces the norms of.3. Permutation A 4, A, A, and L as much as possible. Re-arranging the state variables of a given system is done by computing norms of all the rows [6]. The row with the lowest norm is assumed as the row corresponding to a slow state variable, and is the first state variable appearing in the transformed state vector. Separation between the magnitudes of the row norms generally gives a coherent idea as to how the slow and fast state variables can be classified. Continuing this technique for all the remaining state variables as explained in reference [6], a transformed state vector is obtained in which the first n states correspond to the slow states and the next m states correspond to the fast states. The permutation matrix required to re-index the state variables is defined as P e, e, e, e, e (.8) where e i is an elementary column vector whose i th entry is.the state transformation is achieved by the following equation: Atransform T P AP (.9).3. Scaling If any one of the matrices A 4, A, A, and L is ill-conditioned, the conditions (.6) and (.7) may not be satisfied even if the system possesses inherent TTS property [6]. Thus, diagonal 8

23 scaling technique is applied to the transformed system matrix, obtained in equation (.9), to readjust the units and reduce the norms of A 4, A, A, and L as low as possible. The diagonal elements of the scaling matrix are approximately the ratio of the highest to the lowest elements of the respective row of the system matrix and is constructed as S diag D, D (.) where D n and D m are diagonal matrices of dimensions n and m, respectively. The scaled system matrix is obtained as n m Ascaled SAS (.) Further transformation procedures such as time-scale modeling required for transforming the general TTS form (.) to standard singular perturbation form (.) are discussed more in detail under Sections Singular Perturbation Method As pointed out earlier, associating ε with zt to the system, given by equation (.), gives the deterministic linear time-invariant singularly perturbed continuous system as x t A x t A z t B u t (. a) z t A x t A z t B u t (. b) 3 4 Assuming that the equation (.) is in standard form, that is matrix; and < ε. A 4 is non-singular and a Hurwitz Aforementioned, the purpose of the singular perturbation method is to reduce the complexity of controller design process. In order to do so, the system given by equation (.) is decoupled into two lower-order subsystems where system response can be computed in two 9

24 separate time scales satisfying the inequality (.5). To obtain the reduced-order model, a slow subsystem with fast modes eliminated, we set ε = in (. b), and substitute (.3) into (. a) to get where z t A A x t B u t (.3) s 4 3 s s x t A x t B u t (.4) s s s A A A A A (.5) 4 3 B B A A B (.6) 4 This approach is termed as the quasi-steady state approximation [5], and (.4) is called the quasi-steady state model because z, whose velocity z g/ can be large when ε is small, rapidly decays to the solution of (.3), is the quasi-steady state from of (. b). For deriving the fast subsystem, slow variables are assumed to be constant during fast modes, that is, z and x s is constant. From equations (.) and (.3), we have 4 z t z t A z t z t B u t u t (.7) Let z t z t z t and u t ut u t f s f s s s, the fast subsystem is obtained as s z t A z t B u t (.8) f 4 f f Equations (.4) and (.8) clearly highlight the task of computing the response of the system (.) in two separate time scales, and thereby reducing the complexity of controller design process. The following chapter extends the results from this section for the stochastic form of system (.) for the LQG problem.

25 CHAPTER 3 STOCHASTIC CONTROL OF SINGULARLY PERTURBED SYSTEM 3. Overview Today's advanced and complicated mechanisms of the modern industry often features high-order dynamic systems operating in the presence of external disturbances that requires utmost attention towards the stability and performance of such systems. Moreover, in such dynamic systems lower and higher frequencies co-exist that make the controller design process complicated. Thus, this calls for the system's order degradation for the simplification of control laws. As reviewed in the previous chapter, a reduced-order model can simply be obtained by diminishing the effects of fast dynamics, and this chapter explains how this approach can easily be realized for the system operating under disturbances. This chapter demonstrates the method to apply singular perturbation theory to the stochastic control for LQG problem. Control law design process includes implementation and comparative analysis of optimal, composite, and reduced control techniques. It also defines controller comparison criteria based on which the above control techniques are analyzed. 3. Steady-State Optimal LQG Controller Standard LQG compensation is a combination of optimal observer via Kalman filter and state feedback control via linear-quadratic regulator (LQR). The separation principle allows for this independent computation primarily because the observer dynamics are sufficiently faster than the plant dynamics [8]. 3.. Problem Statement Consider that the stochastic linear time-invariant singularly perturbed continuous system with corresponding measurements is given below as

26 x( t) x( t) A. B. u( t) G. w( t) z( t) z( t) xt () y t C. v t zt () (3. a) (3. b) which is equivalent to A A B G A3 A4 B G where A,.. B,.. G,.. C C C x t A x t A z t B u t G w t (3. c) z t A x t A z t B u t G w t (3. d) 3 4 y t C x t C z t v t (3. e) n m q where xt and zt are the slow and fast states respectively, ut is the control p input, yt is the observed output, ε is a small parameter, wt r and r vt are system and measurement disturbances, respectively, assumed to be mutually uncorrelated, zeromean, stationary Gaussian white noise stochastic processes with intensities Q > and R f >. Covariance functions of wt and vt are given by T E w t w Q t (3.) T f E v t v R t (3.3) where Q is symmetric positive semi-definite and R f is symmetric positive definite. The steadystate linear-quadratic Gaussian control problem is to find a control law of the form that minimizes the performance index... u t f y t (3.4)

27 y where l c t f T T J yc t yc t u t Rcu tdt (3.5) t t is the controlled output to be regulated to zero, which is given by xt () yc t M. zt () (3.6 a) or equivalently where M M M c y t M x t M z t (3.6 b). Equations (3.) are visualized in the block diagram form in Figure 3.. Figure 3. A typical stochastic linear dynamic system 3.. Steady-State Regulator and Kalman Filter For the stochastic problem (3.), as described in the previous section, we now design the optimal feedback control via LQR and a state estimator via Kalman filter based on the approach suggested in reference [3]. As long as AB, and AG, are stabilizable as well as AC, and AM, are detectible, then a control law that minimizes the performance index, J, is given by ˆ ˆ u t KC x t K C z t (3.7) 3

28 where KC and KC are regulator gain matrices given by C c T T K R B P B P (3.8 a) C c T T 3 K R B P B P (3.8 b) where P, P, and P 3 comprise the solution of the algebraic Riccati equation as given below PA A P PBR B P M M (3.9) T T T c which is similar to standard LQR Riccati equation T T A P PA PBR B P Q (3.) c c with Qc T M M and P is the symmetric matrix. ˆx tand ẑt are the steady-state optimal estimates given by Kalman filter ˆ ˆ xˆ t A xˆ t A ˆ z t Bu t K f y t Cx t Cz t ˆ ˆ zˆ t A ˆ ˆ 3x t A4 z t Bu t K f y t C x t Cz t (3. a) (3. b) where K f and K f are Kalman filter gain matrices given by f T T K C C R (3. a) f T T T 3 f K. C C R (3. b) where Σ, Σ, and Σ 3 comprise the solution of the algebraic Riccati equation given below as f T T A A C R C Q (3.3) f f which is similar to the solution of Lyapunov equation with Qf GQG T and Σ is symmetric. Thus, the closed-loop LQG controller is a dynamic, output feedback, model based compensator composed of the regulator and filter equations xˆ t A B K ˆ ˆ C K f C x t A B KC K f C z t K f y t (3.4 a) 4

29 ˆ ˆ zˆ t A3 BKC K f C x t A4 BKC K f C z t K f y t (3.4 b) Figure 3. shows the complete block diagram of the stochastic system with the LQG controller. The poles of the regulator det si A BK C (3.5 a) and the poles of the filter det si A K f C (3.5 b) are guaranteed to be stable (asymptotically stable). It should be noted the poles of the compensator might not always be stable, however, the poles of the closed-loop system (3.6) is guaranteed to be stable. det si A BKC K fc (3.6) In fact, the closed-loop poles of the LQG system are simply the poles of the regulator and the poles of the filter, both of which are guaranteed to be stable by the virtue of separation principle. Figure 3. Block diagram of linear stochastic system with LQG controller 5

30 3.3 Controller Design Techniques In contrast to the optimal LQG control approach for full-order model, this section introduces two other control design methods namely composite control and reduced control implemented to the lower-order slow and fast models derived from the full-order stochastic system Composite Control u t u t u t (3.7) c s f where u t is the composite control composed of u t and u c s f t, the slow and fast control components respectively. The state equations now become 3.3. Reduced Control where s f x t A x t A z t B u t u t G w t (3.8 a) s f z t A x t A z t B u t u t G w t (3.8 b) 3 4 r s u t u t (3.9) u t is the reduced control consisting only the slow control component, u r equations now become s s t. The state x t A x t A z t B u t G w t (3. a) z t A x t A z t B u t G w t (3. b) 3 4 s 3.4 Singular Perturbation As Sections 3.3. and 3.3. suggests, we now apply the singular perturbation technique to the stochastic system given by equations (3.) to obtain the lower-order slow and fast subsystems to separately evaluate for the slow, u t, and fast, u s f t, control components. 6

31 3.4. Composite Control of Singularly Perturbed System As reviewed in Chapter Two, singular perturbation methodology entirely decouples the system into two separate subsystems. So it is appropriate to also consider the decomposition of the feedback controls such that us () t and u () t are separately designed for the slow and fast subsystems (.4) and (.8), respectively. f Using the technique presented in Section.4, we set ε = in equation (3.8 b) and solve for the resulting algebraic equation for zt s f z t A A x t A B u t u t A G w t (3.) The slow subsystem is obtained by replacing zt by its steady-state component. x t A x t B u t u t G w t (3.) s s s f where A ( A A A A )... B ( B A A B )... G ( G A A G ) And the output equation is obtained as y t C x t D u t u t S w t v t (3.3) s s f where C ( C C A A )... D ( C A B )... S ( C A G ) The fast variable zt is defined as z t z t z t. Thus, the fast subsystem is s f obtained by removing the slow bias from the zt is given by s zt and yt. Deducing from (3.), the slow bias of z t A A x t B u t (3.4) s 4 3 Thus, the fast components of defined by zt and yt, denoted by z t and y f f t respectively, are 7

32 Computing the derivative f f z t z t A A x t B u t (3.5) s 4 3 y t y t C x t C A A x t B u t s 4 3 y t C x t D u t C z t v t (3.6) z t with z f s s f t treated as constant, we obtain the fast subsystem as z t A z t B u t G w t (3.7) f 4 f f f y t C z t v t (3.8) f Similarly, the controlled output equation y t is obtained by substituting z t z t z t c in s f equation (3.6) and can be expressed as y t M x t N u t M z t (3.9) c s s f where M ( M M A A )... N ( M A B ) The controlled output equation decomposes as the sum of a slow component M x t N u t s s and a fast component M z f t. Thus, the corresponding performance indexes for the slow and fast subsystems respectively are given by t f T T J M x t N u t M x t N u t u t Ru t dt (3.3) s s s s s s s t t f T T T f f f f f t J z t M M z t u t Ru t dt (3.3) As the problem of stochastic system (3.) has been re-defined completely in terms of slow and fast subsystems, we now need to obtain decoupled slow and fast filter equations for each corresponding subsystem. The filter equations are solved separately in terms of slow and fast control components. For the slow control u t K xˆ t s Cs s 8

33 xˆ ˆ ˆ s t A xs t B us t u f t K fs y t Cxs t D us t u f t (3.3) and for the fast control u t K zˆ t f C f f zˆ ˆ ˆ f t A4 z f t Bu f t K f f y f t Cz f t (3.33) Considering the composite control u t u t u t K xˆ t K zˆ t, we have c s f Cs s C f f xˆ ˆ ˆ s t A xs t Buc t K fs y t Cxs t Duc t (3.34) Replacing with yt C x t D u t instead of y t in z ˆs s f ˆ f t equation ˆ ˆ ˆ zˆf t A4 BKC K f f C f z f t K f f y t Cxs t DK C x s s t (3.35) Figure 3.3 represent the equations (3.34) and (3.35) in the block diagram form. Figure 3.3 Parallel computation of slow and fast Kalman filters for LQG control 9

34 3.4. Reduced Control of Singularly Perturbed System Referring back to the primary objective of this research, which is to reduce the controller design complexity, the application of singular perturbation techniques helps to achieve this mark. In the previous Section 3.4., we could compute the response of the system in two separate time scales. However, this approach falls short of our target, as we still need to carry out the cumbersome computations for the fast subsystem that bears only the non-dominant modes of the system. The singular perturbation methodology also facilitates to solve only for the reducedorder (slow subsystem) model accounting for the fast dynamics while not explicitly solving for the fast control. Design process for reduced control is similar to the approach used for composite control. We set ε = in the equation (3. b) and solve for the steady-state model of zt. Thus, the reduced-order model and the output equation are obtained as x t A x t B u t G w t (3.36) r r r y t C x t D u t S w t v t (3.37) r r where A, B, G, C, D, and S are defined in Section Eliminating the fast bias from the z t z t z t and substituting in the controlled output s f equation (3.6) for yc t, we have y t M x t N u t (3.38) c r r Unlike the composite control, note that the controlled output equation (3.38) this time only consists of the slow component. Hence, the performance criterion of the reduced control is clearly dictated by the slow dynamics only.

35 t f T T J M x t N u t M x t N u t u t Ru t dt (3.39) r r r r r r r t As the problem of stochastic system (3.) has been re-defined completely in terms of slow control, we now need to obtain filter equations for the reduced-order model. For the reduced control u t K xˆ t the reduced-order filter equation is given as r Cr r xˆ ˆ ˆ r t A xr t Bur t K fr y t Cxr t Du r t (3.4) 3.5 Controller Comparison Criteria After devising the optimal, composite, and reduced control laws for the singularly perturbed stochastic system represented by equations (3.), these different controller techniques are subjected to comparative analysis to evaluate for their effectiveness toward the performance of the overall system Optimal Cost The optimal cost of using a controller in terms of initial state conditions is given by t x P t x (3.4) T J* The initial conditions of the system are known. Therefore, the equation (3.4) allows computation of the optimal cost before the control is actually applied to the system, or even before the optimal gain K(t) is computed. If the cost is too high, it allows the engineer to select different weighting matrices Q c, R c, and P(t f ) in the performance index and evaluate various designs Stochastic Cost For the stochastic problem, LQG approach has been implemented incorporating the Kalman filters. The key role of Kalman filter is to handle sensor noise and estimate the unknown

36 states. Thus, the main goal of Kalman filter is to reduce the mean-square error of the state estimates. For this reason, we use another controller comparison parameter namely stochastic cost function for the system with incomplete state information [9], which is defined below as t J trace. PGQ G dt trace K R K dt f f T T. (3.4) S C c C t t where P is the solution of the algebraic Riccati equation as given in equations (3.9) or (3.), and Σ is the error covariance solved by the equation (3.3) H Norm Kalman filter is also evaluated for minimizing the maximum singular value for different control techniques. In other words, we comapre the H norm of the closed-loop system for all three controller design techniques. The transfer function of the closed-loop system is given below as Thus, the H norm of T s, K is defined by [8] where t T( s, K) Ccl si Acl Bcl (3.43) * T s, K. trace T ( j, K) T ( j, K) d (3.44)... r.. i T ( j, K) i d (3.45) T * ( j, K) is the complex conjugate transpose of T( j, K), i denotes the i th singular value, and r is the rank of T( j, K).

37 CHAPTER 4 LONGITUDINAL DYNAMICS OF F-8 AIRCRAFT The practical model used in this research is the longitudinal model of F-8 aircraft, twotime-scale in nature, for the evaluation of various controller design techniques. 4. Brief History In 97, F-8C Crusader fighter aircraft as shown in Figures 4. and 4. (taken from reference []) served as the testbed for NASA's first digital fly-by-wire (DFBW) technology to validate the principal concepts of all-electric flight control systems. The DFBW project was conducted jointly by Dryden Flight Research Center and Langely Research Center []. Figure 4. NASA F-8C digital fly-by-wire test aircraft Figure 4. 3-view of digital fly-by-wire F-8C crusader 3

38 4. Linearized Aircraft Equations of Motion The longitudinal model of the F-8 aircraft in terms of incremental velocity, u (ft/s), and two control inputs, δ e and δ T, presented by Elliot [], is given below as q M q M u M M q e u X X X u X g d u e T e dt Zu Z Z T e (4.) where q, α, θ, δ e, and δ T are respectively incremental pitch rate (rad/s), angle of attack (rad), pitch angle (rad), elevator position (rad) and throttle position (nondimensional). M ( ), X ( ), Z ( ) denote the longitudinal dimensional stability derivatives. The linear model (4.) is the result of linearization of the full nonlinear equations [] about the trim flight conditions. The linearization of the longitudinal model of the F-8 aircraft in terms of straight, steady state flight with velocity V and one control input δ e [3] yields M q M q M vv M q e X g X e v X d v V V v V e dt ZvV Z Ze (4.) where v is the nondimensional, normalized incremental velocity (v = u/v ) and g is the acceleration due to gravity, 3. ft/s. Like references [3] and [4], the control vector in the equation (4.) neglects throttle position, δ T, as one of the control inputs because at any rate a pilot flying the aircraft would be able to control the speed of the aircraft himself. The reference [], an overview of the NASA-F8 control program, also provides the longitudinal stability derivatives (Table 4.) required for the formulation of the longitudinal model (4.). The flight conditions used for the analytical simulations, at also which the longitudinal stability derivatives are computed, are Flight Condition Number referring to the 4

39 Table in reference [] that corresponds to the altitude of, ft, Mach Number of.6 (airstream velocity V = 6 ft/s), and trim angle of attack α =.78 rad. TABLE 4. DIMENSIONAL STABILITY DERIVATIVES FOR THE LONGITUDINAL F-8 AIRCRAFT MODEL State Variables q v α θ δ e q M q = -.49 M v =.5 M α = M δe = -8.7 v - X v = -.5 X α = X δe = -. α - Z v = -.9 Z α = Z δe = -. θ The deterministic linear model (4.) is converted to stochastic form in the design process by augmenting it with the additional wind gust state and introducing disturbances in the state measurements. 4.3 Dynamics of the Wind Disturbance Model As stated in Section 4., a continuous-time wind gust state variable w(t) is included in the longitudinal dynamics to study the effects of wind turbulence during a steady-state flight. The turbulence spectrum provided in [] is an approximate model to that of von Kármán model, as given in [3] and extensively analyzed in [4] for turbulent conditions, and the Haines approximation []. The wind disturbance model, like the aircraft model, changes with different flight conditions; while, only the Flight Condition Number is used through the analyses. The vertical gust power spectral density used to derive the dynamics of the wind disturbance model [] to incorporate into a real-time simulation is given below as w L 4 g L V 4 V (4.3) 5

40 where Φ g is gust power spectral density and ω is the frequency (rad/s). L is the scale length (ft) and has the values of at sea level,,5 above,5 ft of altitude and linearly interpolated in between. ζ w is root mean square value of vertical gust velocity (ft/s) and has the values of 6, 5, 3 ft/s for nominal, cumulus cloud cover, and thunderstorm conditions, respectively. For the chosen flight conditions and assuming the intermediate case of cumulus cloud cover, L =,5 ft and ζ w = 5 ft/s. To obtain a state variable model for the wind gust, a normalized state variable w(t) (rad) is used for the longitudinal dynamics. The dynamics of the wind disturbance model [] are given below as V w t w t t L LV w (4.4) where the wind state w(t) is the result of the first-order system driven by continuous white noise ξ(t) with zero mean and unity covariance function as E t t (4.5) 4.3. Aircraft Model Augmentation To simulate the influence of wind disturbance on the aircraft during steady-state flight conditions, wind dynamics (4.4) are included into the longitudinal aircraft model (4.). The wind state w(t) affects the longitudinal dynamics in the same manner as the angle of attack []. Thus, the augmented longitudinal model of the aircraft is given below as M q M vv M M M q q e X g X X e v X v V V V v V d Z vv Z Z Z e t t dt e V w w L w LV (4.6) 6

41 4.4 State Transformation As previously discussed in Section., the longitudinal model (4.6) is required to transform to represent the TTS form (.). The row norms and the open-loop characteristics of augmented model (4.6) are given in the Tables 4. and 4.3. TABLE 4. ROW NORMS OF THE LONGITUDINAL F-8 AIRCRAFT MODEL State Variables q(t) v(t) α(t) θ(t) w(t) A i (i = matrix row) TABLE 4.3 OPEN-LOOP CHARACTERISTICS OF THE LONGITUDINAL F-8 AIRCRAFT MODEL State Variables Eigenvalues Damping (ζ) Undamped Frequencies(ω n rad/s) Short-Period Mode ± j Phugiod Mode -.69 ± j Wind Gust State Comparing the row norms (Table 4.), distinctly v and θ can be grouped as the slow variables, and α and q as the fast variables. However, it is difficult to form a judgment on the response of the wind state based solely on its norm as it may simply be influenced by the intensity of the vertical gust or turbulence. Alternatively, observation of the open-loop eigenvalues in Table 4.3 gives an apparent idea of the behavior of the wind state. Subsequently, the wind gust state w(t) is chosen to be a fast variable, but slower than α and q based on the row norms and decay of the eigenvalues. 7

42 4.4. Permutation Once the behavior of the state variables is identified, an appropriate permutation matrix is built as P e, e, e, e, e such that the transformed model (4.7) has its first n variables (v and θ) as slow and the remaining m (w, α and q) as fast. Following the approach mentioned in Section.3., the augmented model (4.6) can be rewritten to represent the TTS form (.) as g X X X e v X v V V V v V d w dt Z vv Z Z Z e q M vv M M M q M q e V w w e t LV t L (4.7) The open-loop response to a unity initial condition (Figure 4.3) of the TTS longitudinal model (4.7) distinctly characterizes the slow and fast variables Time (sec) v (nondim.) (rad) w (rad) (rad) q (rad/s) Time (sec) Figure 4.3 Open-loop longitudinal F-8 aircraft initial condition response 8

43 4.5 State Measurements Modern flight data recorders measure almost all the flight stability parameters that a flight controls engineer can think of. These flight records give details of the aircraft performance that are based on in-flight instruments and sensors accounting for noise. The control program carried out by Elliot [] layouts the procedure for sensor modeling used during the F-8 control law studies. Table III in reference [] lists out sensor noise observed in aircraft flight records, which are modeled through shaping white noise of proper spectral density with a first order lowpass filter. Elliot points out that these sensor noises are result of many correlated sources of disturbances like aeroelasticity, engine vibrations, instrument noise, etc. Thus, the values that Table III [] offers are conservative if an analysis accounts for these disturbances separately. The usage of these values in this research would mean putting the controller techniques to the test as in this case, like Elliot cautioned, the analyses accounts only for the influence of wind gust on the aircraft system. Suppose that all the states for the longitudinal model (4.), which are velocity v, pitch angle θ, angle of attack α, and pitch rate q, are measured. Then the measurement equation including intensity matrix with white noise processes, i, for the augmented model (4.7) can be written as v y w 3 4 q (4.8) 4.5. Sensor Noise Intensities Abovementioned, Table III in reference [] provides sensor noise parameters such as first order low-pass filter time constant, (sec), and intensity,, to model the white noise 9

44 processes. Each of the measurement noise processes i in equation (4.8) is reasonably modeled as white noise processes with spectral densities given as 4.5. State-Space Model i i [3]. The TTS system (4.7) and the output equation (4.8) together can be represented as the general state-space model as A. B. G. w (4.9 a) TTS TTS TTS y C v (4.9 b) TTS. where [ v w q] T is the state vector, T is the control vector, w () t is the disturbance and v is the measurement noise. Since the TTS system (4.9) is in the nonstandard form, it must be transform into the standard singularly perturbed form (.) so the controller design techniques developed in Chapter 3 can be applied. 4.6 Time-Scale Modeling Two well-known time-scale characteristics that are associated with the longitudinal motion of an airplane are slow "phugiod mode" and fast "short-period mode." To exhibit these characteristics, the TTS model (4.9) first needs to be scaled and then transformed into the standard singularly perturbed form (.) as per the Proposition 6. presented in reference [3]. Considering the magnitude of system matrix elements in equation (4.7), they can be e T grouped as either O() or O(ε). Elements Z, M, M q, V L are O() while the quantities X, g, X V ZV, MV.... This suggests that ε need to be introduced as the ratio of the V, largest of the small quantities to the smallest of the large quantities. For this purpose, the system matrix of equation (4.7) is rewritten as F( ) F F and partitioned into matrices as given below 3

45 F F F F F F F F F (4.) where Xv g X X V V V,... F F V L ZV v F,... F Z Z MV v M M M q due to which the non-zero elements of F, X, g V, and F, ZV, M V, are scaled to the order O() while the elements X V of F is O(ε) and ( V L) of F is O(). This results in the change of time scale from t to t' = tω and allows the singular perturbation parameter to be chosen as, where ω and ω are undamped natural frequencies of phugoid and short-period modes, respectively Standard Singular Perturbation From After the system matrix of equation (4.9) is correctly scaled, we now need to rightly transform it into the singularly perturbed form represented by the equations (.) to meet the objectives of this research. By following the procedure in Section.7 of [3], we form a transformation matrix T such that T T [ P.... Q] (4.) where the P and Q are chosen as P [ I... F F ] and Q [... I3] Using the transformation technique T, the system (4.8) is transform into A. B. G. w (4. a) y C. v (4. b) 3

46 Finally, letting [ x... z] T and u, we have the standard singularly perturbed form where x A x A z Bu Gw (4.3 a) z A3 x A4 z Bu Gw (4.3 b) y Cx Cz v (4.3 c) A F F F F... A A F F A F... A F F F F 3 4 (4.4) Similarly, the transformations for control matrix, B, and disturbance matrix, G, are carried out based on the same outline. Moreover, to account for the effect of changing the time scale on the white noise processes, and to match the problem statement (3.), the time-scaled white noise intensities are factored by the phugiod mode undamped natural frequency ω. Thus, the intensities i entering the noise intensity matrix will be as (4.5) i i i So far, the practical model, longitudinal dynamics of F-8 aircraft model, has been augmented with wind dynamics, accounted for the measurement sensor noises, and completely been transformed into the singularly perturbed form to represent a TTS system. Thus, the model is now ready on which the various control techniques presented in Chapter Three can be implemented, and based on the controller comparison criteria, their performance will be evaluated. 3

47 CHAPTER 5 LINEAR-QUADRATIC GAUSSIAN CONTROL OF SINGULARLY PERTURBED AIRCRAFT MODEL 5. Simulation Procedure In a typical large-scale system not all the states are generally measured, which calls for the complete LQG design incorporating Kalman filters to handle sensor errors and to reconstruct the state variables that are not available for measurements. Varying the number of states available for measurements and feedback, thus, essentially dictates the flight simulations in this research, and accordingly changes the measurement matrix, C, and the intensity matrix, R f. For the different cases listed in Table 5. controller comparison criteria outlines which controller techniques can be considered as the best approach for the system. TABLE 5. SIMULATION TEST MATRICES Singular Perturbation Parameter Case ε =.4 Availability of State Measurements ( a)... v q ( b)... v q () c q Case ε =.336 ( a)... v q ( b)... v q () c q In addition to the case of varying the number of available state measurements, the simulations also account for the two different cases of ε values. The simulations carried out in this research are for two different sets of test matrices which are defined in Table

48 5.. Simulation Test Matrix This section presents reasoning for the selection of the test matrix as seen in Table 5.. State Measurements: As discussed in section 4.5, Reference [] gives sensor noise parameters for all the longitudinal and lateral states of the aircraft estimated from the flight records. Thus, assuming that all the states are available for measurement and feedback, hence, case (a) [v θ α q] T is selected. During the stochastic study of the complete F-8 model [], Athans et al. devised LQG approach due to the fact that angle of attack and sideslip angle could not be measured, in addition to the wind state. Hence, the case (b) [v θ q] T is selected. Lastly, the case (c) [θ q] T is selected to compare the results of this research to the results obtained in reference [3]. Singular Perturbation Parameter - ε: As reviewed in Sections.. and 4.6, the singular perturbation parameter, ε, can be found in several different ways: (i) ratio of the largest of the absolute eigenvalue of the slow eigenspectrum e(a s ) and the smallest absolute eigenvalue of the fast eigenspectrum e(a f ), (ii) ratio of the largest of the small quantities to the smallest of the large quantities of a system matrix, and (iii) ratio of the undamped natural frequencies of phugoid and short-period modes, ω and ω, respectively. Cases (i) and (iii) yields the same value for ε, hence, only (iii) is considered for the simulation purpose. Thus, based on approach explained in (ii) and (iii), we have two different cases of ε values which are ε =.4 and ε = Numerical Values for Simulation The variants of the open-loop model such as augmented, transformed, and time-scaled models and the measurement matrices along with the sensor noise intensities are presented in this section. 34