Robust Gradient-Like Property and Controller Design for Uncertain Pendulum-Like Systems
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GRADIENT-LIKE PROPERTY AND CONTROLLER DESIGN FOR UNCERTAIN PENDULUM-LIKE SYSTEMS JIN-ZHI WANG, ZHI-SHENG DUAN, and LIN HUANG Abstract. In this paper the gradient-like behavior for the pendulumlike systems with additive, multiplicative, and H∞ uncertainties is analyzed. Then some conditions ensuring existence of controllers for uncertain pendulum-like systems are given. The corresponding controller design problem is transformed into H∞ control problem. Two examples are given to illustrate the effectiveness of the method.
1. Introduction The pendulum-like systems form a special class of nonlinear systems with periodic nonlinearity and multiple equilibria. The typical motions in such systems are well known in mechanics, i.e., damped oscillatory motions and rotatory motions under external forces. In addition, they cover a wide class of systems in engineering and power systems such as phase-locked loops and synchronous machines. There appear new types of stability problems for such systems which are different from the systems with a single stationary point. The Lagrange stability, dichotomy, and gradient-like behavior of pendulum-like systems are important global properties. The frequencydomain methods established in the 1950–60s were firstly applied and developed in the frames of the absolute stability theory for investigation of global stability of a single stationary point [1, 5, 8, 11]. In the 1970–80s, the frequency-domain methods were successfully applied to the investigation of the global properties of solutions for the pendulum-like systems with multiple equilibria [6, 7, 9]. The frequency-domain inequality conditions ensuring the systems with the corresponding global properties were provided [6, 7]. We note that most results above on the pendulum-like systems are focused on analysis of global properties for certain systems. Recently, a Lagrange 2000 Mathematics Subject Classification. 93C10. Key words and phrases. Uncertainty, nonlinear pendulum-like systems, gradient-like property, controller design. This work was partially supported by National Science Foundation of China under Grant 60334030, 10472001, 10272001, 60204007.
229 c 2006 Springer Science+Business Media, Inc. 1079-2724/06/0400-0229/0
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JIN-ZHI WANG, ZHI-SHENG DUAN, and LIN HUANG
stabilization problem for the pendulum-like systems was proposed and output feedback controller was constructed by feasible solutions of a certain set of linear matrix inequalities [12]. The problem of controller design for a class of certain pendulum-like systems guaranteeing the dichotomy and gradient-like property of the closed-loop systems was investigated [4]. The corresponding problem was transformed into the extended strictly positive real control problem. As we know, many practical pendulum-like systems are uncertain, where the uncertainties can be caused by parameter changes or by neglected dynamics, or by a host of other unspecified effects. Obviously, the following important problems for these systems are needed to deal with: to what domain the uncertainties
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