Robust Output Regulation of Singular Nonlinear Systems

Abstract

Abstract. Singular systems, defined as dynamical systems subject to algebraic constraints, arise in many engineering disciplines. The output regulation problem for singular nonlinear systems has been studied recently for the ideal case where the mathematical model is exactly known. This paper will consider the robust output regulation problem for a class of singular nonlinear systems which contain uncertain parameters. We will establish the conditions for the solvability of the problem, thus extending the existing results from normal nonlinear systems to singular nonlinear systems.1. Introduction. Singular systems arise in many areas of engineering including electrical networks, power system, aerospace engineering and chemical processing.Since the late 1970s singular systems have attracted the attention of many researchers. Several books and survey papers dealing with these systems have addressed the issues of solvability, controllability and observability, pole assignment, the elimination of impulsive behavior, and so on [6], [20], [8], [3]. This paper will consider the robust output regulation problem for a class of singular nonlinear systems to be described in Section 2. Briefly, the output regulation problem aims to design control laws for a plant so that the output of the closed-loop system is able to asymptotically track a class of reference inputs and reject a class of disturbances. Both the disturbance and reference are generated by an autonomous differential equation called exosystem. When the controller is also required to tolerate certain plant uncertainty, the problem is called a robust output regulation problem. For linear systems, this problem was thoroughly studied for the normal systems in the 1970s in [10], [11] among others. A salient outcome of these research activities is the internal model principle which is the extension of the well known PID control. The problem was also investigated for linear singular systems in 1980's [8]. Recently, a more clear-cut solution of this problem for linear singular systems was obtained in [21]. For nonlinear systems, the same problem was first treated for normal systems. The special case in which the exogenous signals are constant was studied in [11], [16]. The general case with time varying exogenous signals was studied in [19] without considering the parameter uncertainty. Subsequently, the robust version of the same problem was pursued in [13], [14], [12], [2]. More recently, the output regulation problem for singular nonlinear systems was formulated and solved in [18]. The objective of this paper is to extendthe research initiated in [18] by considering the plant uncertainty.

Global robust output regulation for a class of affine singular nonlinear systems

“…Remark 3 Assumption 3 is standard and necessary for the solvability of the robust output regulation problem of singular nonlinear systems [21,24], and the equations (18) are called singular regulator equations. Assumption 4 holds if the solution of the singular regulator equations in (18) is polynomial in v which is used in the literature on the robust output regulation problem for normal nonlinear systems.…”

“…Assumption 4 holds if the solution of the singular regulator equations in (18) is polynomial in v which is used in the literature on the robust output regulation problem for normal nonlinear systems. This additional condition is also required to solve the robust output regulation problem for singular nonlinear systems [21,24].…”

“…. , 0) = 0, such that for any initial conditions (η(0),σ i (0),ξ(0) ∈ n−ρ × i × ρ , the equilibrium col(η,σ,ξ) = 0 of the closed-loop system composed of (21) and (22) is global asymptotically stable for all v ∈ V and μ ∈ U . Consider the following associate control law γ(η, ξ)z (23) then we have the following result.…”

“…Moreover, [22] and [23] removed the normalizability condition in [20], and designed a normal output feedback control law for the output regulation problem of the singular nonlinear systems. The robust output regulation problem for singular nonlinear systems was studied in [21] and [24]. However, it should be noted that the existing results on the robust output regulation problem for singular nonlinear systems are limited to local solutions because the solvability conditions on the robust output regulation problem for singular nonlinear systems are generally obtained by the linearization model of the singular nonlinear systems.…”

High Gain Disturbance Observer-Based Control for Nonlinear Affine Systems

“…For example, the limit cycles frequently encountered in many engineering areas -such as aircraft wing fluttering, the hopping motion of a legged robot and that generated in mathematical expression by the well-known Van der Pol oscillator -cannot be produced by any linear system. Perhaps this is also the reason for why the extension from linear exo-systems to nonlinear exo-systems is being extensively studied under the nonlinear robust output regulation problem in recent years [16][17][18]. Unfortunately, and unlike linear exo-systems, when it comes to nonlinear exo-systems under the output regulation problem, it might be difficult to ascertain the availability of the solution as a set of nonlinear partial differential equations -called regulator equations -and, indeed, the conditions for finding such a solution are not clear at present [17,18].…”