Analysis and Design of Nonlinear Control Systems

Frontmatter

Chapter 1. Introduction

Abstract

In this chapter we give an introduction to control theory and nonlinear control systems. In Section 1.1 we briefly review some basic concepts and results for linear control systems. Section 1.2 describes some basic characteristics of nonlinear dynamics. A few typical nonlinear control systems are presented in Section 1.3.

Daizhan Cheng, Xiaoming Hu, Tielong Shen

Chapter 2. Topological Space

Abstract

The purpose of this chapter is to present some basic topological concepts of point sets. What we discuss here is very elementary, thus should not be considered as a comprehensive introduction to topology. But it suffices for our goal-providing a foundation for further discussion, particularly for the introduction of differential manifold and the geometrical framework for nonlinear control systems. Many standard text books such as [1, 2, 3] can serve as further references.

Daizhan Cheng, Xiaoming Hu, Tielong Shen

Chapter 3. Differentiable Manifold

Abstract

This chapter provides an outline of Differential Geometry. First we describe the fundamental structure of a differentiable manifold and some related basic concepts, including mappings between manifolds, smooth functions, sub-manifolds. The concept of fiber bundle is also introduced. Then vector fields, their integral curves, Lie derivatives, distributions are discussed intensively. The dual concepts, namely, covector fields, their Lie derivatives with respect to a vector field, co-distributions and the relations with the prime ones are also discussed. Finally, some important theorems and formulas, such as Frobenius’ theorem, Lie series expansions and Chow’s theorem etc. are presented. This chapter provides a fundamental tool for the analysis of nonlinear control systems.

Daizhan Cheng, Xiaoming Hu, Tielong Shen

Chapter 4. Algebra, Lie Group and Lie Algebra

Abstract

Geometry, algebra, and analysis are usually called the three main branches of mathematics. This chapter introduces some fundamental results in algebra that are mostly useful in systems and control. In section 4.1 some basic concepts of group and three homomorphism theorems are discussed. Ring and algebra are introduced briefly in section 4.2. As a tool, homotopy is investigated in section 4.3. Sections 4.4 and 4.5 contain some primary knowledge about algebraic topology, such as fundamental group, covering space etc. In sections 4.6 and 4.7, Lie group and its Lie algebra are discussed. Section 4.8 considers the structure of Lie algebra.

Daizhan Cheng, Xiaoming Hu, Tielong Shen

Chapter 5. Controllability and Observability

Abstract

In section 5.1 we consider controllability of nonlinear systems. The materials are mainly based on [11, 10]. We also refer to [5, 9] for related results, to [4] for later developments. Section 5.2 is about observability of nonlinear systems. The observability of nonlinear control systems is closely related to their controllability [1, 3, 8]. The Kalman decomposition of nonlinear systems is investigated in section 5.3. This section is based on [2]. We refer to [6, 7] for decomposition of nonlinear control systems.

Daizhan Cheng, Xiaoming Hu, Tielong Shen

Chapter 6. Global Controllability of Affine Control Systems

Abstract

This chapter investigates global controllability of general affine control systems, including linear and nonlinear, switched and non-switched ones. The main purpose is to establish criteria for the controllability of nonlinear systems, which is similar to those for linear systems. Since controllability is a property of a set of vector fields, there is no essential difference between switched and non-switched systems as far as controllability is concerned. That is the reason we treat all types of systems together.

Daizhan Cheng, Xiaoming Hu, Tielong Shen

Chapter 7. Stability and Stabilization

Abstract

Stability is a fundamental property for dynamic systems. In most engineering projects unstable systems are useless. Therefore in system analysis and control design the stability and stabilization become the first priority to be consider. This chapter considers the stability of dynamic systems and the stabilization and stabilizer design of nonlinear control systems. In Section 7.1 the concepts about stability of dynamic systems are presented. Section 7.2 considers the stability of nonlinear systems via its linear approximation. The Lapunov direct method is discussed in Section 7.3. Section 7.4 presents the LaSalle’s invariance principle. The converse theory of Lyapunov stability is introduced in Section 7.5. Section 7.6 is about the invariant set. The input-output stability of control systems is discussed in Section 7.7. In section 7.8 the semi-tensor product is used to find the region of attraction. Many results in this chapter are classical, hence the proofs are omitted.

Daizhan Cheng, Xiaoming Hu, Tielong Shen

Chapter 8. Decoupling

Abstract

Section 8.1 considers the (f, g)-invariant distribution of a nonlinear system. Quaker lemma is proved, which assures the equivalence between two kinds of (f, g)- invariances. Quaker lemma is the foundation of the feedback decoupling of nonlinear systems [2]. The local disturbance decoupling problem is discussed in Section 8.2. In Section 8.3, the controlled invariant distribution is introduced. The problem of decomposition of the state equations is discussed in Section 8.4 and Section 8.5. In Section 8.4 only a coordinate change is used, while in Section 8.5 a state feedback control is also used. We refer to [3, 5] for feedback decomposition. More details can be found in their books [2, 4].

Daizhan Cheng, Xiaoming Hu, Tielong Shen

Chapter 9. Input-Output Structure

Abstract

This chapter considers the structure of an affine nonlinear systems from a viewpoint of the relation between inputs and outputs. Section 9.1 considers the relative degree and the decoupling matrix. Section 9.2 considers the Morgan’s problem, that is, the input-output decoupling problem. The invertibility of the input-output mapping is discussed in Section 9.3. Section 9.4 provides a dynamic solution to Morgan’s problem. In Section 9.5 the Byrnes-Isidori normal form of nonlinear control systems is introduced. In Section 9.6 a generalization of Byrnes-Isidori normal form is provided. Section 9.7 presents the Fliess functional expansion, which describes the input-output behavior of nonlinear control systems. In Section 9.8 the Fliess functional expansion has been used to missile guide control.

Daizhan Cheng, Xiaoming Hu, Tielong Shen

Chapter 10. Linearization of Nonlinear Systems

Abstract

Linearization is one of the most powerful tools for dealing with nonlinear systems. Some person says that in fact, what the mathematicians can really deal with is linear problems. Believe it or not, the control theory can treat linear systems perfectly. Hence linearization is an ideal method to deal with nonlinear systems.

Daizhan Cheng, Xiaoming Hu, Tielong Shen

Chapter 11. Design of Center Manifold

Abstract

This chapter provides a systematic technique for designing center manifold of closed loop of nonlinear systems to stabilize the system. The method was firstly presented in [6]. Section 11.1 introduces some fundamental concepts and results about center manifold theory. Section 11.2 considers the case when the zero dynamics has minimum phase. A powerful tool, called the Lyapunov function with homogeneous derivative, is developed in Section 11.3. Section 11.4 and Section 11.5 consider the stabilization of systems with zero center and oscillatory center respectively. The application to generalized normal form is considered in Section 11.6. Section 11.7 is for the stabilization of general control systems.

Daizhan Cheng, Xiaoming Hu, Tielong Shen

Chapter 12. Output Regulation

Abstract

Output regulation is the problem of finding a control law by which the output of the concerning plant can asymptotically track a prescribed trajectories and/or asymptotically reject undesired disturbances. Meanwhile, the unforced closed-loop system is required to be asymptotically stable. For linear systems the internal model principle is a powerful tool to solve this problem. It is a central problem in control theory.

Daizhan Cheng, Xiaoming Hu, Tielong Shen

Chapter 13. Dissipative Systems

Abstract

This chapter discusses the dissipative systems. First, the definition and the properties of two kinds of dissipative systems, mainly passive system and γ-dissipative system, are introduced in Section 13.1. Then, the conditions for verifying passivity and dissipativity are presented in Section 13.2. Based on these conditions, the controller design problem is investigated in Section 13.3. Finally, two classes of main dissipative systems, mainly Lagrange systems and Hamiltonian systems are studied in Section 13.4 and Section 13.5 respectively.

Daizhan Cheng, Xiaoming Hu, Tielong Shen

Chapter 14. L 2 -Gain Synthesis

Abstract

This chapter investigates L₂-gain synthesis problem. As L₂-induced norm, L₂-gain is an extension of H _∞ norm of linear systems. Hence, L₂-gain synthesis problem is usually called the nonlinear H _∞ control problem. Section 14.1 discusses the H _∞ norm and L₂-gain. Then, the H _∞ design problem is formulated in Section 14.2 and the design principle for linear systems is introduced. Section 14.3 focuses on the L ₂- gain synthesis problem for nonlinear systems. In Section 14.4, a constructive design approach is presented. Finally, some application examples are presented to illustrate the design techniques in Section 14.5.

Daizhan Cheng, Xiaoming Hu, Tielong Shen

Chapter 15. Switched Systems

Abstract

As the simplest hybrid system a switched system has many industrial backgrounds and engineering applications. Theoretically, it is also challenging: Switching adds complexity, and at the same time provides more freedom for control design. This chapter considers switched affine (control) systems. Section 15.1 investigates the problem of common quadratic Lyapunov function. It provides a tool for stability analysis and stabilization of switched linear systems. Section 15.2 gives a necessary and sufficient condition for quadratic stabilization of planar switched linear systems. Controllability of switched linear and bilinear control systems are studied in Sections 15.3 and 15.4 respectively. As an application, Section 15.5 considers the consensus of multi-agent systems.

Daizhan Cheng, Xiaoming Hu, Tielong Shen

Chapter 16. Discontinuous Dynamical Systems

Abstract

This chapter discusses analysis and design problems of discontinuous dynamic systems represented by differential equations with discontinuous right-hand side. After a brief review of discontinuous dynamic systems, which is summarized in Section 16.1, Filippov framework including solution and some analyzing tools for control system design is explained in Section 16.2. Furthermore, a feedback design problem is investigated in Section 16.3 for a class of nonlinear systems which have cascaded structure. Section 16.4 demonstrates an application example of stability analysis for PD controlled mechanical systems.

Daizhan Cheng, Xiaoming Hu, Tielong Shen

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