Kevin A. Wise

Robust and Adaptive Control

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Robust Adaptive Control

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Output Feedback Control

This chapter is devoted to the design of adaptive controllers for dynamical systems that operate in the presence of parametric uncertainties and bounded noise. Four MRAC design modifications for robustness are discussed: (1) the dead zone, (2) the \( \sigma \)-modification, (3) the \( e \)-modification, and (4) the Projection Operator. We argue that out of the four modifications, the dead zone and the Projection Operator are essential for any MRAC system designed to predictably operate in a realistic environment.

Lyapunov Stability of Motion

Output feedback design methods are needed when the states are not available for feedback. There are many output feedback control design approaches. This chapter presents three design methods that have proven to be useful in developing output feedback flight control designs in aerospace applications. The first method is called projective control. This method is used to replicate the eigenstructure of a state feedback controller using static and/or dynamic output feedback. By selecting the dominant eigenvalues and associated eigenvectors from the state feedback design, the projective control retains those performance and robustness properties exhibited by that eigenstructure. For static output feedback, a partial eigenstructure can be retained equal to the number of feedback variables. For dynamic output feedback, a low-order compensator can be built that retains the entire state feedback eigenstructure. The second and third methods are based upon linear quadratic Gaussian with Loop Transfer Recovery (LQG/LTR). Both these methods use an optimal control state feedback control implemented with a full-order observer called a Kalman filter to estimate the states needed in the control law. These two variants of LQG/LTR have very different asymptotic properties for recovering frequency domain loop properties.

Optimal Control and the Linear Quadratic Regulator

The main intent of this chapter is to introduce the essential mathematical tools for stability analysis of continuous finite-dimensional dynamical systems. We begin with an overview of sufficient conditions to guarantee existence and uniqueness of the system solutions, followed by a collection of Lyapunov-based methods for studying stability of the system equilibriums and trajectories. The beginning of what is known today as the Lyapunov stability theory can be traced back to the original publication of Alexander Mikhailovich Lyapunov’s doctoral thesis on “the general problem of the stability of motion,” which he defended at the University of Moscow in 1892. Our interest and emphasis on the Lyapunov’s stability methods stem from the fact that these methodologies lay out the much needed theoretical framework and the foundation for performing design and analysis of adaptive controllers. In this chapter, we review selected (but not inclusive) methods due to Lyapunov. This selection is primarily driven by our interest in justifying the design of stable model reference adaptive controllers, with predictable and guaranteed closed-loop performance, for a wide class of nonlinear nonautonomous multi-input multi-output dynamical systems.

Command Tracking and the Robust Servomechanism

In this chapter, we introduce optimal control theory and the linear quadratic regulator. In the introduction, we briefly discuss and compare classical control, modern control, and optimal control, and why optimal control designs have emerged as a popular design method of control in aerospace problems. We then begin by introducing optimal control problems and the resulting Hamilton–Jacobi–Bellman partial differential equation. Then, for linear systems with a quadratic performance index, we develop the linear quadratic regulator. We will cover both finite-time and infinite-time problems and will explore some very important stability and robustness properties of these systems. Central to the design of optimal control laws is the selection of the penalty matrices in the performance index. In the last section, we discuss some asymptotic properties with regard to the penalty matrices that will set the stage for detailed design of these controllers in later chapters.

State Feedback H∞ Optimal Control

In this chapter, we discuss requirements and control system architectures that provide command tracking. This is a very important attribute in aerospace, automotive, and other industrial control problems. We begin by reviewing classical control terminology on system types and then extend this to the servomechanism problem. We then use the servomechanism problem formulation within an optimal control setting to design optimal command tracking controllers, optimal in the sense of the numerical weights used in the performance index. We shall spend considerable time discussing examples on how to select these weights using aerospace control example problems.

Frequency Domain Analysis

This chapter presents full information state feedback H∞ optimal control. This control synthesis method uses state space methods to achieve stability, performance, and robustness and allows for the direct loop shaping in the frequency domain. This chapter begins with an introduction of various norms used in control system design and analysis, followed by methods of specifying stability and performance specifications in the frequency domain. This logically leads into loop shaping using frequency-dependent weights. The state feedback control law is then synthesized using an algebraic Riccati equation approach called γ-iteration. This method is applied to a UAV design example. This control synthesis method is an excellent approach that teaches design engineers important properties in both the time domain and frequency domain and more importantly how to achieve these properties in a closed-loop design.

Direct Model Reference Adaptive Control: Motivation and Introduction

This chapter presents frequency domain analysis methods for both single-input single-output and multi-input multi-output control systems. Transfer function matrices, Nyquist theory for multivariable system, and singular value frequency response methods are discussed in detail. Modeling techniques for robust stability analysis are covered in which both complex and real parametric uncertainties are covered. Theory and examples for the structured singular value μ and the real stability margin are presented. Flight control systems designed using classical and optimal control theories are analyzed to determine their robust stability.

State Feedback Direct Model Reference Adaptive Control

This chapter presents essential concepts for the now-classical model reference adaptive control. We begin with motivational examples from aerospace applications, followed by basic definitions, and a brief description of control-theoretic tools for the design and analysis of state feedback adaptive controllers that are applicable to an aircraft-like general class of multi-input multi-output systems. Our primary goal here is to motivate, introduce, and outline the material that will be discussed in the remainder of the book.