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About this book

This textbook provides readers with a good working knowledge of adaptive control theory through applications. It is intended for students beginning masters or doctoral courses, and control practitioners wishing to get up to speed in the subject expeditiously.

Readers are taught a wide variety of adaptive control techniques starting with simple methods and extending step-by-step to more complex ones. Stability proofs are provided for all adaptive control techniques without obfuscating reader understanding with excessive mathematics.

The book begins with standard model-reference adaptive control (MRAC) for first-order, second-order, and multi-input, multi-output systems. Treatment of least-squares parameter estimation and its extension to MRAC follow, helping readers to gain a different perspective on MRAC. Function approximation with orthogonal polynomials and neural networks, and MRAC using neural networks are also covered.

Robustness issues connected with MRAC are discussed, helping the student to appreciate potential pitfalls of the technique. This appreciation is encouraged by drawing parallels between various aspects of robustness and linear time-invariant systems wherever relevant.

Following on from the robustness problems is material covering robust adaptive control including standard methods and detailed exposition of recent advances, in particular, the author’s work on optimal control modification. Interesting properties of the new method are illustrated in the design of adaptive systems to meet stability margins. This method has been successfully flight-tested on research aircraft, one of various flight-control applications detailed towards the end of the book along with a hybrid adaptive flight control architecture that combines direct MRAC with least-squares indirect adaptive control. In addition to the applications, understanding is encouraged by the use of end-of-chapter exercises and associated MATLAB® files.

Readers will need no more than the standard mathematics for basic control theory such as differential equations and matrix algebra; the book covers the foundations of MRAC and the necessary mathematical preliminaries.

Table of Contents


Chapter 1. Introduction

This chapter provides a brief summary of the recent advancements made in model-reference adaptive control theory. Adaptive control is a promising technology that can improve the performance of control systems in the presence of uncertainty due to a variety of factors such as degradation and modeling uncertainty. During the past decade, the adaptive control research community has produced several advancements in adaptive control theory along with many novel adaptive control methods as a result of the increased government research funding. Many of these new adaptive control methods have added new capabilities in terms of improved performance and robustness that further increase the viability of model-reference adaptive control as a future technology. Flight test validation of adaptive control on full-scale aircraft and unmanned arial vehicles has increased the confidence in model-reference adaptive control as a possible new flight control technology for aerospace vehicles in the near future. In spite of the five decades of research in adaptive control, the fact still remains that currently no adaptive control system has ever been deployed on any safety-critical or human-rated production systems. Many technical problems remain unresolved. As a nonlinear control method, the lack of well-accepted metrics for adaptive control system design presents a major hurdle for certification. The development of certifiable adaptive control systems is viewed as a major technical challenge for the adaptive control research community to address in the current research.
Nhan T. Nguyen

Chapter 2. Nonlinear Systems

A brief overview of nonlinear systems is presented in this chapter. Nonlinear systems are inherently more complex to study than linear systems. Nonlinear systems possess many complex behaviors that are not observed in linear systems. Multiple equilibrium points, limit cycle, finite escape time, and chaos are illustrative of some of the complex behaviors of nonlinear systems. Global stability of a nonlinear system over its entire solution domain is difficult to analyze. Linearization can provide information on the local stability of a region about an equilibrium point. The phase plane analysis of a nonlinear system is related to that of its linearized systems because the local behaviors of the nonlinear system can be approximated by the behaviors of its linearized systems in the vicinity of the equilibrium points. Because nonlinear systems can have multiple equilibrium points, one important fact to note is that the trajectories of a nonlinear system can exhibit unpredictable behaviors.
Nhan T. Nguyen

Chapter 3. Mathematical Preliminaries

This chapter presents some basic mathematical fundamentals for adaptive control theory. Vector and matrix norms are defined. The existence and uniqueness of a solution of a nonlinear differential equation are stated by the Cauchy theorem and the Lipschitz condition. Positive-valued functions are an important class of functions in adaptive control theory. Positive definiteness of a real-valued function is defined. The properties of a positive-definite matrix are given.
Nhan T. Nguyen

Chapter 4. Lyapunov Stability Theory

Stability of nonlinear systems are discussed in this chapter. Lyapunov stability, asymptotic stability, and exponential stability of an equilibrium point of a nonlinear system are defined. The Lyapunov’s direct method is introduced as an indispensable tool for analyzing stability of nonlinear systems. The Barbashin–Krasovskii theorem provides a method for global stability analysis. The LaSalle’s invariant set theorem provides a method for analyzing autonomous systems with invariant sets. Stability of non-autonomous systems involves the concepts of uniform stability, uniform boundedness, and uniform ultimate boundedness. The Barbalat’s lemma is an important mathematical tool for analyzing asymptotic stability of adaptive control systems in connection with the concept of uniform continuity of a real-valued function.
Nhan T. Nguyen

Chapter 5. Model-Reference Adaptive Control

This chapter presents the fundamental theory of model-reference adaptive control. Various types of uncertainty are defined. The composition of a model-reference adaptive control system is presented. Adaptive control theory for first-order single-input single-output (SISO) systems, second-order SISO systems, and multiple-input multiple-output (MIMO) systems is presented. Both direct and indirect adaptive control methods are discussed. The direct adaptive control methods adjust the control gains online directly, whereas the indirect adaptive control methods estimate unknown system parameters for use in the update of the control gains. Asymptotic tracking is the fundamental property of model-reference adaptive control which guarantees that the tracking error tends to zero in the limit. On the other hand, adaptive parameters are only bounded in the model-reference adaptive control setting.
Nhan T. Nguyen

Chapter 6. Least-Squares Parameter Identification

This chapter presents the fundamental theory of least-squares parameter identification. Least-squares methods are central to function approximation theory and data regression analysis. Least-squares methods can also be used in adaptive control as indirect adaptive control methods to estimate unknown system parameters to provide the information for adjusting the control gains. The batch least-squares method is often used for data regression analysis. Least-squares gradient and recursive least-squares methods are well-suited for on-line time series analysis and adaptive control. The concept of persistent excitation is introduced as a fundamental requirement for exponential parameter convergence of least-squares methods. Indirect least-squares adaptive control theory is introduced. The adaptation signal is based on the plant modeling error in contrast to the tracking error for model-reference adaptive control. An important notion to recognize is that the plant modeling error is the source of the tracking error and not vice versa. The combined least-squares model-reference adaptive control uses both the plant modeling error and tracking error for adaptation. As a result, the adaptation mechanism is highly effective. Both the least-squares gradient and recursive least-squares methods can also be used separately in adaptive control without combining with model-reference adaptive control. A fundamental difference with the least-squares adaptive control methods from model-reference adaptive control is that a parameter convergence to true system parameters is guaranteed in the presence of a persistently exciting input signal.
Nhan T. Nguyen

Chapter 7. Function Approximation and Adaptive Control with Unstructured Uncertainty

This chapter presents the fundamental theories of least-squares function approximation and least-squares adaptive control of systems with unstructured uncertainty. The function approximation theory based on polynomials, in particular the Chebyshev orthogonal polynomials, and neural networks is presented. The Chebyshev orthogonal polynomials are generally considered to be optimal for function approximation of real-valued functions. The Chebyshev polynomial function approximation is therefore more accurate than function approximation with regular polynomials. The neural network function approximation theory for a two-layer neural network is presented for two types of activation functions: sigmoidal function and radial basis function. Model-reference adaptive control of systems with unstructured uncertainty is developed in connection with the function approximation theory. Least-squares direct adaptive control methods with polynomial approximation and neural network approximation are presented. Because the least-squares methods can guarantee the parameter convergence, the least-squares adaptive control methods are shown to achieve uniform ultimate boundedness of control signals in the presence of unstructured uncertainty. The standard model-reference adaptive control can be used for systems with unstructured uncertainty using polynomial or neural network approximation. Unlike the least-squares adaptive control methods, boundedness of tracking error is guaranteed but boundedness of adaptive parameters cannot be mathematically guaranteed. This can lead to robustness issues with model-reference adaptive control, such as the well-known parameter drift problem. In general, least-squares adaptive control achieves better performance and robustness than model-reference adaptive control.
Nhan T. Nguyen

Chapter 8. Robustness Issues with Adaptive Control

This chapter discusses limitations and weaknesses of model-reference adaptive control. Parameter drift is the result of the lack of a mathematical guarantee of boundedness of adaptive parameters. Systems with bounded external disturbances under feedback control actions using model-reference adaptive control can experience a signal growth of a control gain or an adaptive parameter even though both the state and control signals remain bounded. This signal growth associated with the parameter drift can cause instability of adaptive systems. Model-reference adaptive control for non-minimum phase systems presents a major challenge. Non-minimum phase systems have unstable zeros in the right half plane. Such systems cannot tolerate large control gain signals. Model-reference adaptive control attempts to seek the ideal property of asymptotic tracking. In so doing, an unstable pole-zero cancelation occurs that leads to instability. For non-minimum phase systems, adaptive control designers generally have to be aware of the limiting values of adaptive parameters in order to prevent instability. Time-delay systems are another source of challenge for model-reference adaptive control. Many real systems have latency which results in a time delay at the control input. Time delay is caused by a variety of sources such as communication bus latency, computational latency, transport delay, etc. Time-delay systems are a special class of non-minimum phase systems. Model-reference adaptive control of time-delay systems is sensitive to the amplitude of the time delay. As the time delay increases, robustness of model-reference adaptive control decreases. As a consequence, instability can occur. Model-reference adaptive control is generally sensitive to unmodeled dynamics. In a control system design, high-order dynamics of internal states of the system sometimes are neglected in the control design. The neglected internal dynamics, or unmodeled dynamics, can result in loss of robustness of adaptive control systems. The mechanism of instability for a first-order SISO system with a second-order unmodeled actuator dynamics is presented. The instability mechanism can be due to the frequency of a reference command signal or an initial condition of an adaptive parameter that coincides with the zero phase margin condition. Fast adaptation is referred to the use of a large adaptation rate to achieve the improved tracking performance. An analogy of an integral control action of a linear time-invariant system is presented. As the integral control gain increases, the cross-over frequency of the closed-loop system increases. As a consequence, the phase margin or time-delay margin of the system decreases. Fast adaptation of model-reference adaptive control is analogous to the integral control of a linear control system whereby the adaptation rate plays the equivalent role as the integral control gain. As the adaptation rate increases, the time-delay margin of an adaptive control system decreases. In the limit, the time-delay margin tends to zero as the adaptation rate tends to infinity. Thus, the adaptation rate has a strong influence on the closed-loop stability of an adaptive control system.
Nhan T. Nguyen

Chapter 9. Robust Adaptive Control

This chapter presents several techniques for improved robustness of model-reference adaptive control. These techniques, called robust modification, achieve increased robustness through two general principles: (1) limiting adaptive parameters and (2) adding damping mechanisms in the adaptive laws to bound adaptive parameters. The dead-zone method and the projection method are two common robust modification schemes based on the principle of limiting adaptive parameters. The dead-zone method prevents the adaptation when the tracking error norm falls below a certain threshold. This method prevents adaptive systems from adapting to noise which can lead to a parameter drift. The projection method is widely used in practical adaptive control applications. The method requires the knowledge of a priority bounds on system parameters. Once the bounds are given, a convex set is established. The projection method then permits the normal adaptation mechanism of model-reference adaptive control as long as the adaptive parameters remain inside the convex set. If the adaptive parameters reach the boundary of the convex set, the projection method changes the adaptation mechanism to bring the adaptive parameters back into the set. The \(\sigma \) modification and e modification are two well-known robust modification techniques based on the principle of adding damping mechanisms to bound adaptive parameters. These two methods are discussed, and the Lyapunov stability proofs are provided. The optimal control modification and the adaptive loop recovery modification are two recent robust modification methods that also add damping mechanisms to model-reference adaptive control. The optimal control modification is developed from the optimal control theory. The principle of the optimal control modification is to explicitly seek a bounded tracking as opposed to the asymptotic tracking with model-reference adaptive control. The bounded tracking is formulated as a minimization of the tracking error norm bounded from an unknown lower bound. A trade-off between bounded tracking and stability robustness can therefore be achieved. The damping term in the optimal control modification is related to the persistent excitation condition. The optimal control modification exhibits a linear asymptotic property under fast adaptation. For linear uncertain systems, the optimal control modification causes the closed-loop systems to tend to linear systems in the limit. This property can be leveraged for the design and analysis of adaptive control systems using many existing well-known linear control techniques. The adaptive loop recovery modification is designed to minimize the nonlinearity in a closed-loop plant so that the stability margin of a linear reference model could be preserved. This results in a damping term proportional to the square of the derivative of the input function. As the damping term increases, in theory the nonlinearity of a closed-loop system decreases so that the closed-loop plant can follow a linear reference model which possesses all the required stability margin properties. The \(\mathscr {L}_{1}\) adaptive control has gained a lot of attention in the recent years due to its ability to achieve robustness with fast adaptation for a given a priori bound on the uncertainty. The underlying principle of the \(\mathscr {L}_{1}\) adaptive control is the use of fast adaptation for improved transient or tracking performance coupled with a low-pass filter to suppress high-frequency responses for improved robustness. As a result, the \(\mathscr {L}_{1}\) adaptive control can be designed to achieve stability margins under fast adaptation for a given a priori bound on the uncertainty. The basic working concept of the \(\mathscr {L}_{1}\) adaptive control is presented. The bi-objective optimal control modification is an extension of the optimal control modification designed to achieved improved performance and robustness of systems with input uncertainty. The adaptation mechanism relies on two sources of errors: the normal tracking error and the predictor error. A predictor model of a plant is constructed to estimate the open-loop response of the plant. The predictor error is formed as the difference between the plant and the predictor model. This error signal is then added to the optimal control modification adaptive law to enable the input uncertainty to be estimated. Model-reference adaptive control of singularly perturbed systems is presented to address slow actuator dynamics. The singular perturbation method is used to decouple the slow and fast dynamics of a plant and its actuator. The asymptotic outer solution of the singularly perturbed system is then used in the design of model-reference adaptive control. This modification effectively modifies an adaptive control signal to account for slow actuator dynamics by scaling the adaptive law to achieve tracking. Adaptive control of linear uncertain systems using the linear asymptotic property of the optimal control modification method is presented for non-strictly positive real (SPR) systems and non-minimum phase systems. The non-SPR plant is modeled as a first-order SISO system with a second-order unmodeled actuator dynamics. The plant has relative degree 3 while the first-order reference model is SPR with relative degree 1. By invoking the linear asymptotic property, the optimal control modification can be designed to guarantee a specified phase margin of the asymptotic linear closed-loop system. For non-minimum phase systems, the standard model-reference adaptive control is known to be unstable due to the unstable pole-zero cancellation as a result of the ideal property of asymptotic tracking. The optimal control modification is applied as an output feedback adaptive control design that prevents the unstable pole-zero cancellation by achieving bounded tracking. The resulting output feedback adaptive control design, while preventing instability, can produce poor tracking performance. A Luenberger observer state feedback adaptive control method is developed to improve the tracking performance. The standard model-reference adaptive control still suffers the same issue with the lack of robustness if the non-minimum phase plant is required to track a minimum phase reference model. On the other hand, the optimal control modification can produce good tracking performance for both minimum phase and non-minimum phase reference models.
Nhan T. Nguyen

Chapter 10. Aerospace Applications

This chapter presents several adaptive control applications with a particular focus on aerospace flight control applications. Two relatively simple pendulum applications are used to illustrate a nonlinear dynamic inversion adaptive control design method for tracking a linear reference model. The rest of the chapter presents several adaptive flight control applications for rigid aircraft and flexible aircraft. The chapter concludes with an application of the optimal control modification to a F-18 aircraft model. The flight control applications for rigid aircraft include the \(\sigma \) modification, e modification, bi-objective optimal control modification, least-squares adaptive control, neural network adaptive control, and hybrid adaptive control which combines model-reference adaptive control with a recursive least-squares parameter identification method. An adaptive control design for a flexible aircraft is presented using a combined adaptive law with the optimal control modification and adaptive loop recovery modification to suppress the dynamics of the aircraft flexible modes. An adaptive linear quadratic gaussian (LQG) design based on the optimal control modification for flutter suppression is presented. By taking advantage of the linear asymptotic property, the adaptive flutter suppression can be designed to achieve a closed-loop stability with output measurements.
Nhan T. Nguyen


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