Parametric control to a type of quasi-linear high-order systems via output feedback

Abstract

This paper investigates the design problem of output feedback control to a type of quasi-linear high-order systems containing time-variant coefficient matrices which include time-variant parameters and system variables. Based on the solution to a class of high-order generalized Sylvester matrix equation, we can design the left and right closed-loop eigenvectors and obtain two groups of arbitrary parameters, further establish a general complete parametrized expression for a quasi-linear high-order output feedback controller. By the presented parametric approach, the closed-loop system can be transformed into a linear time-invariant one with the desired eigenstructure. Finally, a nonlinear feedback synchronization problem of Genesio-Tesi and Coullet chaotic systems is presented to prove the effectiveness of the proposed approach.

Introduction

In recent years, quasi-linear systems attract the attention of researchers because it is widely used in the fields of mechanical, civil engineering, robotics and control theory, etc, all of them mentioned can be modeled as quasi-linear systems. For example, the modeling of a single-link flexible-joint manipulator can be regarded as a fourth-order quasi-linear system [27]. The jerk system can be modeled by third-order quasi-linear system, which is the application of quasi-linear system in mechanical engineering [23]. Additionally, wind-induced vibrating comfort problem of high-rise steel structural buildings [25], a nonlinear feedback synchronization problem of Genesio-Tesi and Coullet chaotic systems [22], etc. can be modeled by quasi-linear high-order systems with time-varying parameters.

So far, there are many research achievements for quasi-linear systems. Hlupić et al. present a novel derivative-free algorithm to figure out quasi-linear systems, which greatly simplifies computation and realizes fast execution and numerical stability [19]. Dmitriev and Makarov concentrate on the nonlinear stable regulator designed for quasi-linear systems containing state-depend coefficients with small parameters, and build the suboptimal estimation of proposed method under the Bellman equations and Lyapunov functions [5]. Through static output feedback, Zaitsev investigates the problem of uniformly exponential stability (under a given index) of the zero-unperturbed solution to a quasi-linear n-order control system, and provide the sufficient conditions [31]. It is noted that all the above approaches are invalid in time-variant case. However, parametric method is an effective technique to deal with time-variant situation. For parametric control in quasi-linear systems with time-variant parameters, Duan utilizes state feedback [12], output feedback [13] and dynamical feedback [10] to solve one-order quasi-linear systems, and Gu proposes a class of parametrized output feedback control applying to second-order quasi-linear systems [16]. Unfortunately, they all aim at one-order and second-order quasi-linear systems to design control laws rather than high-order quasi-linear systems. Noteworthy, the main work of this study is to popularize parametric control method to high-order quasi-linear systems, which expands applied range of parametric method. Meanwhile, the significant advantages of proposed method are that the presented approach maintains the physical meaning of original system and avoid ill-conditioned matrix caused by transforming high-order system into one-order system, also simplifies complexity when designing control law.

This paper considers a parametric control method (under eigenstructure assignment [2], [6], [7], [24]) for output feedback control to a type of quasi-linear high-order systems based on the used parametric control methods for linear systems [8], [14], [29] and quasi-linear systems [7], [12], [13], [16]. The proposed approach is based on the solution of a class of generalized high-order Sylvester matrix equation [9], [11], [30], and aims for realizing a linear time-invariant closed-loop system. Specifically, the presented method gives a linear time-invariant closed-loop system with expected eigenstructure depending upon an arbitrary chosen matrix F including expected eigenvalues, and establishes a general complete parametrized expression for a quasi-linear high-order output feedback controller corresponding to the matrix F and two groups of parameters Z_b and Z_c, meanwhile it also obtains the general complete parametrized expressions of left and right closed-loop eigenvectors.

Noted that, on the one hand, compared with the commonly used feedback linearization for nonlinear systems [1], [4], [20], [21], the proposed approach possesses mainly distinct advantages in the following aspects. Firstly, feedback linearization mainly depends on cancelling nonlinearity term, therefore, it is sensitive to the modeling errors, especially under a poorly linearizing transformation. However, the proposed method absorbs the nonlinearity term via selecting a stabilization control law by choosing degrees of design freedom, and therefore is tolerant to exist certain degrees of system uncertainties. Secondly, after feedback linearization, the closed-loop system may still be nonlinear, its stability depends on the external input, initial value and many other affecting factors. With the proposed approach, the closed-loop system is transformed into a linear time-invariant one, its stability is determined by its eigenvalues. Thirdly, when the system is non-minimum phase, the closed-loop system formed via feedback linearization is unstable, but the proposed approach does not have such problem. On the other hand, compared with adaptive control for nonlinear systems [26], [32], [33], the proposed method also possesses the following benefits. Firstly, adaptive control can only handle constant parameters, while the presented approach can process not only constant parameters but also time-variant parameters, which illustrates that the proposed method has more widely applicative range. Secondly, the closed-loop system resulted in by adaptive control is generally a nonlinear one, although the closed-loop system can be proven to be stable by Lyapunov functions, one is not very familiar with the performance of this system. However, with the proposed method, the closed-loop system can be transformed into a linear constant one whose performance and stability are totally determined by its eigenvalues. Thirdly, when designing control law, adaptive control needs to consider the derivative of control law, while the proposed method has no such requirements, which simplifies computational complexity and reduces the difficulties of realizing controller.

The remaining part of this paper is respectively organized as follows. The problem formulation of quasi-linear high-order output control under eigenstructure assignment is presented, and some assumptions and lemmas are provided in Section 2. In Section 3, we propose the parametrized expressions of quasi-linear high-order output feedback controller, and give the general procedure of eigenstructure assignment. In Section 4, a numerical simulation is presented to prove that the proposed method is effective.