Finite-time robust passive control for a class of uncertain Lipschitz nonlinear systems with time-delays

doi:10.1016/j.neucom.2015.01.038

Neurocomputing

Volume 159, 2 July 2015, Pages 275-281

https://doi.org/10.1016/j.neucom.2015.01.038 Get rights and content

Abstract

The finite-time passive control for a class of nonlinear uncertain systems with time-delays and uncertainties is studied. The nonlinear parameters are satisfied Lipschitz conditions. An optimal robust passive controller with respect to the finite-time interval is designed while the exogenous disturbances are unknown but energy bounded. Based on passive control theory, the sufficient condition for the existence of finite-time robust passive controller is given. This condition such that the resulting closed-loop system is finite-time boundedness (FTB) for all admissible uncertainties and satisfies the given passive control index. By using the constructed Lyapunov function, and applying linear matrix inequalities techniques (LMIs), the design method of the finite-time optimal passive controller is derived and can be obtained. Simulation results demonstrate the validity of the proposed approach.

Introduction

Over the past decades, the nonlinear systems with the nonlinearities satisfied Lipschitz conditions have been extensively studied by many researchers (see [1], [2], [3], [4], [37], [38]). Time delays encountered in practical situations often result in unsatisfactory performance, and much work has been done for the analysis and synthesis of these time-delayed systems [5], [6], [26], [27]. In recent years, the Lipschitz nonlinear systems with time-delay have received great attention, and a lot of results have also been obtained. For more results on this topic, we refer readers to [7], [18], [22] and the references therein.

It necessary to point out that the results of aforementioned papers are mostly based on Lyapunov theory. As we all known, Lyapunov theory pays more attention to the asymptotic pattern of systems over an infinite-time interval. But in some practical process, the main attention may be related to the behavior of the dynamical systems over fixed finite time interval, for instance, large values of the states cannot be accepted in the presence of saturations. To deal with such situations, Dorato [8] first presented the concept of finite-time stability in 1961. Since then, a lot of attempts on finite-time stability have been made in [9], [10], [28], [33], [34]. It should be noticed that the studies about finite-time boundedness (FTB, [32]) of the systems with time-delays have received more and more attention, see such as [15], [29], [30], [31]. However, to the best knowledge of authors, the problems of finite-time robust passive control for a class of Lipschitz nonlinear systems with time-delays are not studied. This motivates our research.

In fact, passivity has played an important role in analysis and control design of linear and nonlinear systems [11], [12], [19], and it provides a competitive tool for studying stability of uncertain or nonlinear systems. In many engineering problems, stability issues are often linked to the theory of dissipative systems which postulate the energy dissipated inside a dynamic system is less than the one supplied from external source. The main idea of passivity theory is that the passive properties of systems can keep the systems internal stable [13], [15]. In this paper, we deal with the problem of finite-time robust passive control for a class of Lipschitz nonlinear systems with time-delays and uncertainties, and the nonlinear function is assumed to satisfy Lipschitz conditions. The uncertain parameters are assumed to be time-varying and norm-bounded. The purpose is to construct a memory-less state feedback controller such that the resulting closed-loop system is finite-time bounded and satisfies the given passive index. The desired state feedback controlled can be constructed by solving the presented given linear matrix inequalities (LMIs) with optimization techniques. Finally, a simulation example illustrates the effectiveness of the developed techniques.

This paper is divided into six sections. Section 2 formulates the problem to be solved in this paper. Section 3 gives a sufficient condition for the finite-time boundedness of the resulting closed-loop system. Based on Section 3, a sufficient condition of finite-time robust passive controller is given in Section 4. Section 5 gives a simulation example. Conclusions follow in Section 6.

Notations. In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions. The notation $M > (\geq, <, \leq)$ 0 is used to denote a symmetric positive-definite (positive-semidefinite, negative, negative-semidefinite) matrix. $λ_{\min} (\cdot)$ , $λ_{\max} (\cdot)$ denote the minimum and the maximum eigenvalue of the corresponding matrix, respectively. $‖ ⁎ ‖$ denotes the Euclidean norm for vectors or the spectral norm of matrices. $L_{2}^{n} [\begin{matrix} 0 & N \end{matrix}]$ is the space of n-dimensional square integrable function vector over $[\begin{matrix} 0 & N \end{matrix}]$ . In symmetric block matrices, we use “ $⁎$ ” represents the elements below the main diagonal of a symmetric block matrix. The superscript “T” represents the transpose.

Section snippets

Problem formulation

Consider a class of Lipschitz nonlinear systems with time-delays and parameter uncertainties described by ${\begin{array}{l} \dot{x} (t) = [A + Δ A (t)] x (t) + [A_{d} + Δ A_{d} (t)] x (t - d) \\ + [B + Δ B (t)] u (t) + G w (t) + F f [x (t), x (t - d)] \\ y (t) = [C + Δ C (t)] x (t) + [C_{d} + Δ C_{d} (t)] x (t - d) \\ + [E + Δ E (t)] u (t) + D w (t) \\ x (t) = σ (t), \forall t \in [\begin{matrix} - d & 0 \end{matrix}] \end{array}$ where $x (t) \in ℜ^{n}$ is the state; $x (t - d) \in ℜ^{n}$ is the time-delayed state; $w (t) \in L_{2}^{q} [\begin{matrix} 0 & + \infty \end{matrix})$ is the exogenous disturbances input, which can be either an exogenous disturbance input or a reference signal, and $w (0) = w_{0}$ is unknown; $u (t) \in ℜ^{p}$ is the controlled input;

Finite-time boundedness

In this section, we present a sufficient condition for the finite-time boundedness of the closed-loop system (8). First, we introduce the following lemma which will be used in the development of our main results.

Lemma 1

Ref. [17] let X and Y are real matrices of appropriate dimensions. For a given scalar $ε > 0$ and vectors $x, y \in ℜ^{n}$ , then

2 x^{T} X Y y \leq ε^{- 1} x^{T} X^{T} X x + ε y^{T} Y^{T} Y y .

Theorem 1

For given positive constants $c_{1}$ , δ, T, ε and α, two weighting matrices $η_{1}$ and $η_{2}$ , and a symmetric matrix $R > 0$ , the closed-loop system (8) is FTB with

Finite-time passive control

Theorem 1

We give the finite-time boundedness sufficient condition of the closed-loop system (8). In this section, we will derive a sufficient condition for finite-time robust passive control of the uncertain time-delayed Lipschitz nonlinear systems.

Theorem 2

For some given positive constants $c_{1}$ , δ, T, ε and α, two weighting matrices $η_{1}$ and $η_{2}$ , and a symmetric matrix $R > 0$ , the closed-loop system (8) is FTB with respect to $(\begin{matrix} c_{1} & c_{2} & δ & T & R \end{matrix})$ and satisfies the cost function (10) for all admissible $w (t)$ with the constraint

Simulation example

In this section, we shall present an example to demonstrate the effectiveness and applicability of the proposed method. Subject to a time-delay Chua’s circuit [23] described by ${\begin{array}{l} {\dot{x}}_{1} (t) = a [x_{2} (t) - m x_{1} (t)] + f_{1} [x_{1} (t)] - c x_{1} (t - τ) \\ {\dot{x}}_{2} (t) = x_{1} (t) - x_{2} (t) + x_{3} (t) - c x_{1} (t - τ) \\ {\dot{x}}_{3} (t) = - b x_{2} (t) + c [2 x_{1} (t - τ) - x_{3} (t - τ)] \\ y (t) = 5 x_{1} (t) + 2 x_{2} (t) + 3 x_{3} (t) \end{array}$ where the nonlinear characteristics is $f_{1} [x_{1} (t)] = (1 / 2) (| x_{1} (t) + 1 | - | x_{1} (t) - 1 |)$ , the parameters $m_{0} = - (1 / 7)$ , $m_{1} = 2 / 7$ , $a = 9$ , $b = 14.28$ , $c = 0.1$ and the system time-delay $τ = 0.04$ . Furthermore, we rewrite

Conclusions

In this paper, we have investigated to design a finite-time robust passive controller for a class of Lipschitz nonlinear systems with time-varying norm-bounded parameter uncertainties, energy- bounded exogenous disturbance input, and time-delays. By using passive control theory and the Lyapunov function approach, a sufficient condition is derived such that the closed-loop controlled system satisfies finite-time boundedness and a condition of passivity. The controller design can be completed in

Acknowledgment

This work was supported in part by the Joint Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20123401120010), the National Natural Science Foundation of China (Grant no. 61203051) and the Key Program of Natural Science Foundation of Education Department of Anhui Province (Grant no. KJ2012A014).

Jun Song was born in 1989. He received his B.Eng. in Electronic Science and Technology from Anhui University. Currently, he is a Master candidate in School of Electrical Engineering and Automation of Anhui University. His current research includes adaptive control, stochastic control, and optimal control, etc. E-mail: [email protected].

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Shuping He was born in 1983. He received his B.Eng. and Ph.D. from Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Automation, Jiangnan University, Wuxi, China. From 2011 to 2013, he was successively a Lecturer in Anhui University. Since 2013, he has been a Professor with School of Electrical Engineering and Automation, Anhui University, Hefei, China. From 2010 to 2011, he was a Visiting Scholar with the Control Systems Centre, School of Electrical and Electronic Engineering, The University of Manchester, UK. His current research focuses on control theory and applications, includes robust control of stochastic systems, nonlinear and optimal control of nonlinear systems and complex system modeling, control, filtering, fault detection and their applications. E-mail: [email protected].

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Brief PapersFinite-time robust passive control for a class of uncertain Lipschitz nonlinear systems with time-delays

Abstract

Introduction

Section snippets

Problem formulation

Finite-time boundedness

Finite-time passive control

Simulation example

Conclusions

Acknowledgment

Nonlinear Anal. Real World Appl.

Syst. Control Lett.

Neurocomputing

Nonlinear Anal. Real World Appl.

Automatica

Neurocomputing

J. Franklin Inst.

J. Franklin Inst.

Neurocomputing

Signal Proc.

Nonlinear Anal. Hybrid Syst

Automatica

Comput. Electron. Agric.

Automatica

Automatica

Inf. Sci.

Syst. Control Lett.

Automatica

Observers for Lipschitz nonlinear systems

IEEE Trans. Autom. Control

Brief Papers
Finite-time robust passive control for a class of uncertain Lipschitz nonlinear systems with time-delays