Problems on time-varying port-controlled Hamiltonian systems: geometric structure and dissipative realization

doi:10.1016/j.automatica.2004.11.006

Automatica

Volume 41, Issue 4, April 2005, Pages 717-723

https://doi.org/10.1016/j.automatica.2004.11.006 Get rights and content

Abstract

To apply time-varying port-controlled Hamiltonian (PCH) systems to practical control designs, two basic problems should be dealt with: one is how to provide such time-varying systems a geometric structure to guarantee the completeness of representations in mathematics; and the other is how to express the practical system under consideration as a time-varying PCH system, which is called the dissipative Hamiltonian realization problem. The paper investigates the two basic problems. A suitable geometric structure for time-varying PCH systems is proposed first. Then the dissipative realization problem of time-varying nonlinear systems is investigated, and serval new methods and sufficient conditions are presented for the realization.

Introduction

In recent years, time-invariant port-controlled Hamiltonian (PCH) systems have been well investigated (see, e.g., van der Schaft, 1999, Nijmeijer and van der Schaft, 1990, Maschke et al., 2000, Ortega et al., 2002, Escobar et al., 1999). The Hamiltonian function in a PCH system is considered as the total energy, which is the sum of potential and kinetic energies in mechanical systems, and it can play the role of Lyapunov function for the system. Because of this, based on time-invariant PCH systems, various effective controllers have been designed for many control problems (see, e.g., Shen et al., 2000, Wang et al., 2003, Xi and Cheng, 2000). However, for some practical systems the time-invariant PCH structure does not easily apply and its time-varying form is really needed. Please see the following example.

Example 1

Consider a single-machine infinite-bus power system (Lu & Sun, 1993): $\begin{matrix} \dot{δ} = ω - ω_{0}, \\ \dot{ω} = \frac{ω_{0}}{M} P_{m} - \frac{D}{M} (ω - ω_{0}) - \frac{ω_{0} E_{q}^{'} V_{s}}{{Mx}_{d Σ}^{'}} \sin δ + w_{1}, \\ {\dot{E}}_{q}^{'} = - \frac{1}{T_{d}^{'}} E_{q}^{'} + \frac{1}{T_{do}} \frac{x_{d} - x_{d}^{'}}{x_{d Σ}^{'}} V_{s} \cos δ + \frac{1}{T_{d 0}} u_{f} + w_{2}, \end{matrix}$ where $w_{1}$ and $w_{2}$ are disturbances, $δ$ is the power angle, $ω$ the rotor speed, $E_{q}^{'}$ the q-axis internal transient voltage, $u_{f}$ the control input, and $V_{s}$ the infinite-bus voltage. As for other parameters, please refer to Lu and Sun (1993). In the case that all the parameters are constant, we can use the time-invariant PCH structure to design an effective controller to attenuate the disturbances $w_{1}$ and $w_{2}$ (Xi & Cheng, 2000). But as well known, in power systems there are always uncertainties caused by load-level variations, faults, or changes of network structure, etc. When a parameter of the above system is affected by a time-varying signal, say, $V_{s}$ is affected by a sine signal $\sin t$ , the time-invariant structure is no longer valid for the system. In this case, to design an effective energy-based controller, the time-varying PCH structure is really needed.

Therefore, it is necessary to develop the theory of time-varying PCH systems for some practical control problems. Recently, time-varying PCH systems have been studied by Fujimoto and Sugie, 2001a, Fujimoto and Sugie, 2001b, Fujimoto, Sakurama, and Sugie (2003) and Cheng (2002). It is worth noticing that Fujimoto et al. (2003) set up a very important way to the trajectory tracking control of time-varying PCH systems via generalized canonical transformations, whose key idea was to preserve the structure of PCH systems under both coordinate and feedback transformations. At present, in order to apply time-varying PCH systems to practical control designs, two basic problems should be dealt with: one is how to define a geometric structure on a manifold for such systems to guarantee the completeness of representations in mathematics; and the other is how to express the practical system under consideration into a time-varying PCH system. The latter is the so-called dissipative Hamiltonian realization problem.

This paper investigates the above-mentioned two problems. First, by defining a time-varying generalized Poisson bracket, we provide a geometric structure for time-varying PCH systems. Then, we deal with the dissipative Hamiltonian realization of time-varying nonlinear systems, and propose some new methods and sufficient conditions for the realization.

The rest of the paper is organized as follows. Section 2 briefly reviews the classical Poisson structure, and Section 3 provides the geometric structure for time-varying PCH systems. In Section 4, we deal with the dissipative Hamiltonian realization problem, which is followed by the conclusion in Section 5.

Section snippets

A brief review of Poisson structure

This section briefly reviews the classical Poisson structure with Lie algebraic properties, which will motivate the next section of the paper.

In order to define a Hamiltonian system on a manifold, one should equip the manifold with a suitable geometric structure first. Let $M$ be a smooth manifold and $C^{\infty} (M)$ be the set of smooth functions on $M$ . A Poisson bracket on $M$ , denoted by ${\cdot, \cdot}$ , is a map: $C^{\infty} (M) \times C^{\infty} (M) ⟼ C^{\infty} (M)$ , satisfying (Ortega and Planas-Bielsa, 2004, Olver, 1993):

(i)
Bilinearity: ${aF + bG, H} = a {F, H$

Geometric structure for time-varying PCH systems

This section is to provide a geometric structure for time-varying PCH systems. First, we give the concept of time-varying generalized Poisson brackets, and then, we present the geometric structure for time-varying PCH systems.

Definition 1

Let $M$ be an n-dimensional manifold and time $t \in R^{+} ≔ [0, \infty)$ . A time-varying generalized Poisson bracket (GPB), denoted by ${\cdot, \cdot}_{t}$ , is a map: $C^{\infty} (M \times R^{+}) \times C^{\infty} (M \times R^{+}) ⟼ C^{\infty} (M \times R^{+})$ , satisfying

(i)
Bilinearity: ${aF (x, t) + bG (x, t), H (x, t)}_{t} = a {F (x, t), H (x, t)}_{t} + b {G (x, t), H (x, t)}_{t}, {F (x, t), aG (x, t) + bH (x, t)}_{t} =$

Dissipative Hamiltonian realization

This section investigates the dissipative Hamiltonian realization of time-varying nonlinear systems, and proposes several new results. First, we give some concepts and properties.

Conclusion

Through defining a time-varying generalized Poisson bracket, we have provided a suitable geometric structure for time-varying PCH systems, which can guarantee the mathematical completeness of representations of time-varying PCH systems. In order to apply time-varying PCH systems to practical control problems, we have also investigated the dissipative Hamiltonian realization problem of time-varying nonlinear systems, and proposed serval new methods and sufficient conditions for the realization.

Yuzhen Wang was born in Shandong, China, in 1963. He 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. From 2001 to 2003, he worked as a Postdoctoral Fellow in Tsinghua University, Beijing, China. Now he is a professor with the School of Control Science and Engineering, Shandong University, Jinan, China. His research

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Daizhan Cheng received Ph.D. degree from Washington University, MO, in 1985. Since 1990, he has been a Professor with the Institute of Systems Science. Chinese Academy of Sciences. He was an Associate Editor of Math Sys., Est. Contr. (91–93), and Automatica (98–02). He is an Associate Editor of Asia Journal of Control, Deputy Chief Editor of Control and Decision, Journal of Control Theory and App. etc. He is Chairman of the Technical Committee on Control Theory, Chinese Automation Association. His research interests include nonlinear system and control, numerical method, etc.

Xiaoming Hu was born in Chengdu, China, in 1961. He received the B.S. degree from University of Science and Technology of China in 1983. He received the M.S. and Ph.D degrees from Arizona State University in 1986 and 1989, respectively. From 1989 to 1990 he was a Gustafson Postdoctoral Fellow at the Royal Institute of Technology, Stockholm, where he is currently an associate professor. His main research interests are in nonlinear feedback stabilization, nonlinear observer design, sensing and active perception, motion planning and control of mobile robots, and mobile manipulation.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in the revised form by Associate Editor H. Nijmeijer under the direction of the Editor H. K. Khalil. Supported by National Natural Science Foundation of China (G60474001).

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Brief paperProblems on time-varying port-controlled Hamiltonian systems: geometric structure and dissipative realization☆

Abstract

Introduction

Section snippets

A brief review of Poisson structure

Geometric structure for time-varying PCH systems

Dissipative Hamiltonian realization

Conclusion

A Hamiltonian viewpoint in the modelling of switching power converters

Automatica

Canonical transformations and stabilization of generalized Hamiltonian systems

Systems and Control Letters

Stabilization of Hamiltonian systems with nonholonomic constraints based on time-varying generalized canonical transformations

Systems and Control Letters

Trajectory tracking control of port-controlled Hamiltonian systems via generalized canonical transformations

Automatica

Symplectic geometry and analytic mechanics

Nonlinear control of power systems

Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation

IEEE Transactions on Automatic Control

Nonlinear dynamical control systems

Brief paper
Problems on time-varying port-controlled Hamiltonian systems: geometric structure and dissipative realization☆