Elsevier

Automatica

Volume 39, Issue 8, August 2003, Pages 1437-1443
Automatica

Brief Paper
Generalized Hamiltonian realization of time-invariant nonlinear systems

https://doi.org/10.1016/S0005-1098(03)00132-8Get rights and content

Abstract

A key step in applying the Hamiltonian function method is to express the system under consideration into a generalized Hamiltonian system with dissipation, which yields the so-called generalized Hamiltonian realization (GHR). In this paper, we investigate the problem of GHR. Several new methods and the corresponding sufficient conditions are presented. A major result is that if the Jacobian matrix of a time-invariant nonlinear system is nonsingular, the system has a GHR whose structure matrix and Hamiltonian function are given in simple forms. Then the orthogonal decomposition method and a sufficient condition for the feedback dissipative realization are proposed.

Introduction

In recent years, port-controlled Hamiltonian (PCH) systems, proposed by Maschke and van der Schaft (1992) and van der Schaft and Maschke (1995), have been investigated in detail by van der Schaft (1999), Maschke, Ortega, and van der Schaft (2000), Escobar, van der Schaft, and Ortega (1999), Ortega, Lorı́a, Nicklasson, and Sira-Ramı́rez (1998), Ortega, van der Schaft, Maschke, and Escobar (2002), Fujimoto and Sugie (2001), and Cheng, Spurgeon, and Xiang (2000). Indeed, the Hamiltonian function in the PCH system is considered as the total energy and can play the role of Lyapunov function for the system. Because of this, the Hamiltonian function method (the energy-based Lyapunov function method) has been developed (Maschke et al., 2000) and applied to many practical control problems (Cheng, Xi, Hong, & Qin, 1999; Shen, Ortega, Lu, Mei, & Tamura, (2000); Wang, Chen, & Hong, 2001; Xi, Cheng, Lu, & Mei, 2002). It has been shown in Cheng et al. (1999), Wang et al. (2001) and Xi et al. (2002) that the Hamiltonian function method has some advantages.

The key point in applying the Hamiltonian function method is to express the system concerned into a Hamiltonian system with dissipation, which is called the dissipative Hamiltonian realization. In general, to complete the dissipative Hamiltonian realization, we first express the system into a generalized Hamiltonian system, i.e., obtain the generalized Hamiltonian realization (GHR), and then eliminate the non-dissipative part of the GHR by a state feedback to get a Hamiltonian system with dissipation. We recall some related concepts first:

Definition 1 Cheng et al., 2000

(1) A dynamic systemẋ=f(x),x∈Rnis said to have a generalized Hamiltonian realization (GHR) if there exists a suitable coordinate chart and a Hamiltonian function H such that (1) can be expressed asẋ=T(x)H,where T(x) is an n×n matrix called the structure matrix and ∇H=∂H/∂x. If the structure matrix can be expressed as T(x)=J(x)−R(x), with skew-symmetric J(x) and symmetric positive semi-definite R(x), then system (2) is called a dissipative Hamiltonian realization. Furthermore, if R(x)>0, (2) is called a strict dissipative Hamiltonian realization.

(2) A controlled dynamic systemẋ=f(x)+g(x)uis said to have a state feedback Hamiltonian realization if there exists a suitable state feedback u=α(x)+v such that the closed-loop system can be expressed asẋ=T(x)H+g(x)v.If T(x) can be expressed as T(x)=J(x)−R(x), J(x) is skew-symmetric and R(x)⩾0(>0), then (4) is called a feedback (strict) dissipative Hamiltonian realization.

The GHR problem has been studied in some recent works (Hebertt Sira-Ramı́rez, 1998; Cheng et al., 2000). There are, however, no effective methods to handle it yet. In this paper, we investigate the GHR problem of nonlinear systems. We propose several new methods to handle the problem and give some new sufficient conditions for the realization.

Section snippets

Generalized Hamiltonian realization

In this section, we investigate the GHR problem of system (1) and propose several new methods and sufficient conditions for the GHR. Let Jf denote the Jacobian matrix ∂f/∂x.

In system (1), setAi=∂f∂xiT=∂f1∂xi,…,∂fn∂xi,ai=∂xi,i=1,2,…,n.Construct two equations as follows:A2−A1A3−A1An−A1A3−A2An−A2An−An−1X1(x)X2(x)Xn(x)=0anda2−a1a3−a1an−a1a3−a2an−a2an−an−1⊗InX1(x)X2(x)Xn(x)=0,where Xi(x)(i=1,2,…,n) are n-dimensional column vector fields, ⊗ is the Kronecker product and In is the n×n

Feedback dissipative Hamiltonian realization

In this section, we investigate the feedback dissipative Hamiltonian realization. First, we propose a new approach to the GHR called the orthogonal decomposition method. Then we use the method to study the feedback dissipative realization.

Let us introduce the following concept. A function V(x) is called a regular positive definite function if V(x)>0 (x≠0), V(0)=0, ∂V/∂x|x=0=0 and ∂V/∂x|x≠0≠0. For example, H(x)=12i=1nxi2 is a regular positive definite function on Rn.

Consider systemẋ=f(x),

Conclusion

We have investigated the Hamiltonian realization problem of time-invariant nonlinear systems, proposed several operable and comparatively systematic methods and given some sufficient conditions for the realization. The main results are as follows.

(1) If the Jacobian matrix of a system is nonsingular, the system has a GHR whose structure matrix and Hamiltonian function are given in very simple forms.

(2) If the Jacobian matrix of the system is singular and has a nonsingular main diagonal block,

Yuzhen Wang graduated from Tai'an Teachers College in 1986, received his M.S. degree from Shandong University of Science & Technology in 1995 and his Ph.D. degree from the Institute of Systems Science, Chinese Academy of Sciences in 2001. He is now the Postdoctoral Fellow in Tsinghua University, Beijing, China. His research interests include nonlinear control systems, Hamiltonian systems and robust control. Dr. Wang received the Prize of Guan Zhaozhi in 2002, and the Prize of Huawei from the

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Yuzhen Wang graduated from Tai'an Teachers College in 1986, received his M.S. degree from Shandong University of Science & Technology in 1995 and his Ph.D. degree from the Institute of Systems Science, Chinese Academy of Sciences in 2001. He is now the Postdoctoral Fellow in Tsinghua University, Beijing, China. His research interests include nonlinear control systems, Hamiltonian systems and robust control. Dr. Wang received the Prize of Guan Zhaozhi in 2002, and the Prize of Huawei from the Chinese Academy of Sciences in 2001.

Chunwen Li received his B.S. degree and Ph.D. degree from Department of Automation, Tsinghua University in 1982 and 1989, respectively. Since 1994, he has been a Professor with Department of Automation, Tsinghua University. His research interests include nonlinear control systems, inverse systems, CAD and simulation of nonlinear systems, and robust control. Prof. Li received the National Youth Prize in 1991 and the Prize of Chinese Outstanding Ph.D. Degree Receiver in 1992.

Daizhan Cheng graduated from Tsinghua University in 1970, and received the M.S. degree from Graduate School, the Chinese Academy of Sciences and the Ph.D. degree from Washington University, St. Louis, MO, in 1981 and 1985, respectively. Since 1990, he has been a Professor with the Institute of Systems Science, Chinese Academy of Sciences. He was an Associate Editor of Mathematical Systems, Estimator and Control (91-93), and Automatica (98-02). He is an Associate Editor of Asia J. Control. Deputy Chief Editor of Control and decision, and a member of the editorial board of Systems Science and Complex, Systems Science and Mathematics, Control Theory and Applications. He is the Chairman of the Control Theoretical Committee, Chinese Automation Association. His research interests include nonlinear system and control, Hamiltonian systems and numerical method in system and control.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Henk Nijmeijer under the direction of Editor Hassan Khalil. Supported by Project 973 of China (G1998020307, G1998020308) and China Postdoctoral Science Foundation.

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