Nonlinear and Adaptive Control Systems
An adaptive system for linear systems with unknown parameters is a nonlinear system. The analysis of such adaptive systems requires similar techniques to analyse nonlinear systems. Therefore it is natural to treat adaptive control as a part of nonlinear control systems. Nonlinear and Adaptive Control Systems treats nonlinear control and adaptive control in a unified framework, presenting the major results at a moderate mathematical level, suitable for MSc students and engineers with undergraduate degrees. Topics covered include introduction to nonlinear systems; state space models; describing functions for common nonlinear components; stability theory; feedback linearization; adaptive control; nonlinear observer design; backstepping design; disturbance rejection and output regulation; and control applications, including harmonic estimation and rejection in power distribution systems, observer and control design for circadian rhythms, and discretetime implementation of continuoustime nonlinear control laws.
Inspec keywords: control system synthesis; adaptive control; nonlinear control systems; feedback; linearisation techniques; stability; linear systems; statespace methods; observers
Other keywords: disturbance rejection; state space models; nonlinear observer design; describing functions; advanced stability theory; linear systems; nonlinear control system; adaptive control systems; output regulation; feedback linearisation; control applications; backstepping design
Subjects: General and management topics; Selfadjusting control systems; Stability in control theory; Control system analysis and synthesis methods; Nonlinear control systems
 Book DOI: 10.1049/PBCE084E
 Chapter DOI: 10.1049/PBCE084E
 ISBN: 9781849195744
 eISBN: 9781849195751
 Page count: 287
 Format: PDF

Front Matter
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1 Introduction to nonlinear and adaptive systems
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This chapter discusses typical nonlinearities and nonlinear behaviours, and introduces some basic concepts for nonlinear system analysis and control.

2 State space models
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The nonlinear systems under consideration in this paper are described by differential equations. In the same way as for linear systems, it has system state variables, inputs and outputs. The paper provides basic definitions for state space models of nonlinear systems, and tools for preliminary analysis, including linearisation around operating points. Typical nonlinear behaviours such as limit cycles and chaos are also discussed with examples.

3 Describing functions
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This chapter presents basic concept of describing functions, calculation of describing functions of common nonlinear elements and how to use describing functions to predict the existence of limit cycles.

4 Stability theory
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For control systems, design, one important objective is to ensure the stability of the closedloop system. For a linear system, the stability can be evaluated in time domain or frequency domain, by checking the eigenvalues of the system matrix or the poles of the transfer function. For nonlinear systems, the dynamics of the system cannot be described by equations in linear state space or transfer functions in general. We need more general definitions about the stability of nonlinear systems. In this chapter, we will introduce basic concepts of stability theorems based on Lyapunov functions.

5 Advanced stability theory
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Lyapunov direct method provides a tool to check the stability of a nonlinear system if a Lyapunov function can be found. For linear systems, a Lyapunov function can always be constructed if the system is asymptotically stable. In many nonlinear systems, a part of the system may be linear, such as linear systems with memoryless nonlinear components and linear systems with adaptive control laws. For such a system, a Lyapunov function for the linear part may be very useful in the construction for the Lyapunov function for the entire nonlinear system. In this chapter, we will introduce one specific class of linear systems, strict positive real systems, for which, an important result, KalmanYakubovich lemma, is often used to guarantee a choice of the Lyapunov function for stability analysis of several types of nonlinear systems. The application of KalmanYakubovich lemma to analysis of adaptive control systems will be shown in later chapters, while in this chapter, this lemma is used for stability analysis of systems containing memoryless nonlinear components and the related circle criterion. In Section 5.3 of this chapter, inputtostate stability (ISS) is briefly introduced.

6 Feedback linearisation
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Nonlinear systems can be linearised around operating points and the behaviours in the neighbourhoods of the operating points are then approximated by their linearised models. The domain for a locally linearised model can be fairly small, and this may result in that a number of linearised models are needed to cover an operating range of a system. In this chapter, we will introduce another method to obtain a linear model for nonlinear systems via feedback control design. The aim is to convert a nonlinear system to a linear one by state transformation and redefining the control input. The resultant linear model describes the system dynamics globally. Of course, there are certain conditions for the nonlinear systems to satisfy so that this feedback linearisation method can be applied.

7 Adaptive control of linear systems
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The principle of feedback control is to maintain a consistent performance when there are uncertainties in the system or changes in the setpoints through a feedback controller using the measurements of the system performance, mainly the outputs. Many controllers are with fixed controller parameters, such as the controllers designed by normal state feedback control, and H_{∞} control methods. The basic aim of adaptive control also is to maintain a consistent performance of a system in the presence of uncertainty or unknown variation in plant parameters, but with changes in the controller parameters, adapting to the changes in the performance of the control system. Hence, there is an adaptation in the controller setting subject to the performance of the closedloop system. How the controller parameters change is decided by the adaptive laws, which are often designed based on the stability analysis of the adaptive control system. A number of design methods have been developed for adaptive control. Model Reference Adaptive Control (MRAC) consists of a reference model which produces the desired output, and the difference between the plant output and the reference output is then used to adjust the control parameters and the control input directly. MRAC is often in continuoustime domain, and for deterministic plants. SelfTuning Control (STC) estimates system parameters and then computes the control input from the estimated parameters. STC is often in discretetime and for stochastic plants. Furthermore, STC often has a separate identification procedure for estimation of the system parameters, and is referred to as indirect adaptive control, while MRAC adapts to the changes in the controller parameters, and is referred to as direct adaptive control. In general, the stability analysis of direct adaptive control is less involved than that of indirect adaptive control, and can often be carried out using Lyapunov functions. In this chapter, we focus on the basic design method of MRAC.

8 Nonlinear observer design
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Observers are needed to estimate unmeasured state variables of dynamic systems. They are often used for output feedback control design when only the outputs are available for the control design. Observers can also be used for other estimation purposes such as fault detection and diagnostics. There are many results on nonlinear observer design in literature, and in this chapter, we can introduce only a number of results. Observer design for linear systems is briefly reviewed before the introduction of observers with linear error dynamics. We then introduce another observer design method based on Lyapunov's auxiliary theorem, before the observer design for systems with Lipschitz nonlinearities. At the end of this chapter, adaptive observers are briefly described.

9 Backstepping design
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For a nonlinear system, the stability around an equilibrium point can be established if one can find a Lyapunov function. Nonlinear control design can be carried out by exploring the possibility of making a Lyapunov function candidate as a Lyapunov function through control design. In this chapter, we start with the fundamental form of adding an integrator, and then introduce the method for iterative backstepping with state feedback. We also introduce backstepping using output feedback, and adaptive backstepping for certain nonlinear systems with unknown parameters.

10 Disturbance rejection and output regulation
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Disturbance rejection and output regulation are big topics in control system design. In this chapter, rejection of deterministic disturbances in a class of nonlinear output feedback systems is discussed. The chapter starts from sinusoidal disturbances with unknown frequencies, then disturbances generated from nonlinear exosystems and then general periodical disturbances, etc. Adaptive control techniques are used to deal with the unknown disturbance frequencies and unknown parameters in the system. The presented design concepts can be applied to other classes of nonlinear systems.

11 Control applications
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In this chapter, we will address a few issues about control applications. Several methods of disturbance rejection are presented in Chapter 10, including rejection of general periodic disturbances. A potential application can be the estimation and rejection of undesirable harmonics in power systems. Harmonics, often referred to highorder harmonics in power systems, are caused by nonlinearities in power systems, and the successful rejection depends on accurate estimation of amplitudes and phase of harmonics. We will show an iterative estimation method based on a new observer design method.

Back Matter
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