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Über dieses Buch

This book focuses on methods that relate, in one form or another, to the “small-gain theorem”. It is aimed at readers who are interested in learning methods for the design of feedback laws for linear and nonlinear multivariable systems in the presence of model uncertainties. With worked examples throughout, it includes both introductory material and more advanced topics.

Divided into two parts, the first covers relevant aspects of linear-systems theory, the second, nonlinear theory. In order to deepen readers’ understanding, simpler single-input–single-output systems generally precede treatment of more complex multi-input–multi-output (MIMO) systems and linear systems precede nonlinear systems. This approach is used throughout, including in the final chapters, which explain the latest advanced ideas governing the stabilization, regulation, and tracking of nonlinear MIMO systems. Two major design problems are considered, both in the presence of model uncertainties: asymptotic stabilization with a “guaranteed region of attraction” of a given equilibrium point and asymptotic rejection of the effect of exogenous (disturbance) inputs on selected regulated outputs.

Much of the introductory instructional material in this book has been developed for teaching students, while the final coverage of nonlinear MIMO systems offers readers a first coordinated treatment of completely novel results. The worked examples presented provide the instructor with ready-to-use material to help students to understand the mathematical theory.

Readers should be familiar with the fundamentals of linear-systems and control theory. This book is a valuable resource for students following postgraduate programs in systems and control, as well as engineers working on the control of robotic, mechatronic and power systems.



Chapter 1. An Overview

This is a book intended for readers, familiar with the fundamentals of linear system and control theory, who are interested in learning methods for the design of feedback laws for (linear and nonlinear) multivariable systems, in the presence of model uncertainties. One of the main purposes of the book is to offer a parallel presentation for linear and nonlinear systems. Linear systems are dealt with in Chaps. 2–5, while nonlinear systems are dealt with in Chaps. 6–12. Among the various design options in the problem of handling model uncertainties, the focus of the book is on methods that appeal—in one form or in another—to the so-called “Small-Gain Theorem.” In this respect, it should be stressed that, while some of such methods may require, for their practical implementation, a “high-gain feedback” on selected measured variables, their effectiveness is proven anyway with the aid of the Small-Gain Theorem. Methods of this kind lend themselves to a presentation that is pretty similar for linear and nonlinear systems and this is the viewpoint adopted in the book. While the target of the book are multi-input multi-output (MIMO) systems, for pedagogical reasons in some cases (notably for nonlinear systems) the case of single-input single-output (SISO) systems is handled first in detail. Two major design problems are addressed (both in the presence of model uncertainties): asymptotic stabilization “in the large” (that is, with a “guaranteed region of attraction”) of a given equilibrium point and asymptotic rejection of the effect of exogenous (disturbance and/or commands) inputs on selected regulated outputs. This second problem, via the design of an “internal model” of the exogenous inputs, is reduced to the problem of asymptotic stabilization of a closed invariant set.

Alberto Isidori

Linear Systems


Chapter 2. Stabilization of Minimum-Phase Linear Systems

It is well-known from the elementary theory of servomechanisms that a single-input single-output linear system, whose transfer function has all zeros in the left-half complex plane can be stabilized via output feedback. If the transfer function of the system has n poles and m zeros, the feedback in question is a dynamical system of dimension $$n-m-1$$n-m-1, whose eigenvalues (in case $$n-m>1$$n-m>1) are far away in the left-half complex plane. In this chapter, this result is reviewed using a state-space approach. This makes it possible to systematically handle the case of systems whose coefficients depend on uncertain parameters and serves as a preparation to a similar set of results that will be presented in Chap. 6 for nonlinear systems.

Alberto Isidori

Chapter 3. The Small-Gain Theorem for Linear Systems and Its Applications to Robust Stability

In a system consisting of the interconnection of several component subsystems, some of which could be only poorly modeled, stability analysis and feedback design might not be easy tasks. Thus, methods allowing to understand the influence of interconnections on stability and asymptotic behavior are important. The methods in question are based on the use of a concept of gain, which can take alternative forms and can be evaluated by means of a number of alternative methods. This chapter describes the various alternative forms of such concept of gain, and shows why this is useful in the analysis of stability of interconnected systems. A major consequence is the development of a systematic method for stabilization in the presence of (general class of) model uncertainties.

Alberto Isidori

Chapter 4. Regulation and Tracking in Linear Systems

In this chapter, we will study in some generality the problem of designing a feedback law to the purpose of making a controlled plant stable, and securing exact asymptotic tracking of external commands (respectively, exact asymptotic rejection of external disturbances) which belong to a fixed family of functions. The problem in question can be seen as a (broad) generalization of the classical set point control problem in the elementary theory of servomechanisms.

Alberto Isidori

Chapter 5. Coordination and Consensus of Linear Systems

In this chapter, we will see how the theory of asymptotic tracking can be fruitfully extended to address problems in which a large set of systems is controlled in such a way that certain variables of interest asymptotically coincide. The specific challenge addressed in this chapter resides in the fact that there is a limited exchange of information between individual systems, each one of which has access only to measurements of the outputs of limited number of neighbors.

Alberto Isidori

Nonlinear Systems


Chapter 6. Stabilization of Nonlinear Systems via State Feedback

This chapter considers single-input single-output nonlinear systems that can be brought, by change of coordinates, to a special form having essentially the same structure as the normal form of a linear system. For such systems, it is also possible to define concepts and properties which identify a class of systems that can be seen as a nonlinear analogue of the class of linear system having all zeros with negative real part. This makes it possible to systematically develop stabilization methods that, in various forms, extend to nonlinear systems the stabilization methods presented in Chap. 2.

Alberto Isidori

Chapter 7. Nonlinear Observers and Separation Principle

This chapter considers the design of asymptotic state observers for a single-output nonlinear system. A fundamental property that makes the design of such observers possible is the existence of change of coordinates by means of which the system is brought to a special form in which a property of observability, uniform with respect to the input, is highlighted. For such systems, it is possible to design global asymptotic state observers. Then, a nonlinear equivalent of the so-called separation principle of linear system theory is developed. It is shown how to combine a state feedback stabilizer with a nonlinear observer, so as to obtain a dynamic output feedback by means of which asymptotic stability with guaranteed domain of attraction is obtained.

Alberto Isidori

Chapter 8. The Small-Gain Theorem for Nonlinear Systems and Its Applications to Robust Stability

As it is the case for linear systems, understanding the influence of interconnections on stability and asymptotic behavior is of paramount importance. In the case of nonlinear systems, a powerful concept in the analysis of interconnections is the notion of gain function of an input-to-state stable system. Using this concept, it is possible to develop a nonlinear version of the small-gain theorem, which is useful in the analysis as well as in the design of feedback laws. This chapter describes this theorem and how it can be used in the design of stabilizing feedback laws for nonlinear systems.

Alberto Isidori

Chapter 9. The Structure of Multivariable Nonlinear Systems

The purpose of this chapter is to analyze, in a multi-input multi-output nonlinear system (having the same number of input and output components), the notion of invertibility. A major consequence of such property is the existence of a change of variables that plays, for a multivariable system, a role equivalent to the change of variable leading to the normal form of a single-input single-output system. A class of special relevance consists of those invertible systems in which it is possible to force, by means of state feedback, a linear input–output behavior. A further subclass is that of those systems for which a (multivariable version of the concept of) relative degree can be defined.

Alberto Isidori

Chapter 10. Stabilization of Multivariable Nonlinear Systems: Part I

This chapter addresses the problem of asymptotically stabilizing a multivariable system having vector relative degree. To this end, a multivariable version of the property of being strongly minimum phase is introduced and it is shown that, if a system has this property, global asymptotic stability via state feedback can be achieved. The resulting control, though, requires exact cancellation of nonlinear terms and access to all state variables. Such control, therefore, is unsuitable if robust stability in the presence of modeling errors is sought. In the second part of the chapter an alternative approach is pursued, based on the design of a robust observer, by means of which all state variables and nonlinear terms whose knowledge is needed for stabilization are approximately estimated. In this way, a dynamic output feedback is designed, by means of which asymptotic stability with guaranteed domain of attraction can be obtained.

Alberto Isidori

Chapter 11. Stabilization of Multivariable Nonlinear Systems: Part II

This chapter addresses the problem of asymptotically stabilizing invertible multivariable systems, in the more general case in which the system does not have a vector relative degree. The case of input–output linearizable systems is addressed in the first part of the chapter, where it is shown how the robust stabilization method presented in Chap. 10 can be extended. Then, in the second part of the paper, a more general class of invertible systems is considered.

Alberto Isidori

Chapter 12. Regulation and Tracking in Nonlinear Systems

In this chapter, the problem of asymptotic tracking/rejection of exogenous commands/disturbances for nonlinear systems is discussed. Results that extend those developed earlier in Chap. 4 for linear systems are presented. The discussion follows very closely the analysis of necessary conditions presented in Sect. 4.3 and the construction of a regulator presented in the second part of Sect. 4.6. The construction of an internal model, though, requires a different and more elaborate analysis, for which two alternatives are offered. The chapter is complemented with a discussion of a simple problem of inducing consensus in a network of nonlinear agents.

Alberto Isidori


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