Stability analysis and observer-based controllers design for T–S fuzzy positive systems

doi:10.1016/j.neucom.2017.09.087

Neurocomputing

Volume 275, 31 January 2018, Pages 1468-1477

https://doi.org/10.1016/j.neucom.2017.09.087 Get rights and content

Abstract

This paper investigates the stability analysis and observer-based controllers design for T–S fuzzy positive systems. A fuzzy copositive Lyapunov function is first proposed to analyze the stability of the T–S fuzzy positive systems via linear programming. In terms of the property of the fuzzy membership functions, the fuzzy copositive Lyapunov function is employed to derive the less conservative stability conditions. Then, the line-integral Lyapunov function is presented for the stability analysis. The time-derivatives of the membership functions do not appear in the stability analysis of the T–S fuzzy positive systems, therefore, the proposed stability conditions are more relaxed than those of the conventional Lyapunov function approaches. Based on the obtained stability conditions, observer-based control schemes are designed such that the resultant closed-loop systems are both stable and positive. Finally, two examples are provided to validate the effectiveness of the results proposed in this paper.

Introduction

Positive systems are dynamic systems whose state variables remain nonnegative whenever the trajectory is originated from the nonnegative orthant [1]. On account of the phenomenon that the positive quantitative value frequently occurs in nature and applications, the research of positive systems become even more significant. Many results for the common linear systems can also be applied to the positive linear systems. The positivity of the systems are more realistic in many practical applications compared with the common systems, but seriously give rise to difficulties when the systems are analyzed and the controllers are designed. Because of the positivity, variables elimination and linear transformation and other traditional approaches must be used circumspectly. In recent years, great progress has been made in the positive systems [2], [3], [4]. They can describe many branches of science such as electrical systems, chemical process systems [5], ecological systems [6] and physiological systems [7], [8].

The investigations on linear positive systems have received a lot of attention, however, in practical applications, many systems are inherently nonlinear, for example, [9], [10], [11], [12]. In this case, the existing results on positive linear systems can not be applied. Actually, the analysis and control problem are always difficult. T–S fuzzy model [12], [13], as an effective tool deal with the nonlinearities, can be regarded as an alternative way to overcome the difficulty. Then the analysis and synthesis of the nonlinear positive systems can be handled by the virtue of linear positive system theories. Because of the states of T–S fuzzy positive systems are nonnegative, a multitude of elegant stability conditions have been obtained. The sufficient stability conditions were derived in [14] by using the common Lyapunov $-$ Krasovskii function. A less conservative stability condition for switched positive T–S fuzzy systems is obtained in [15], which is formulated in terms of linear programming that is a powerful scheme to analyze the stability of positive systems. A sufficient condition of stability based on linear copositive Lyapunov function was derived which was not relevant to the magnitude of delays in [16]. The advantage is to make the results less conservative and the computation less demanding. The linear Lyapunov function was designed by [17] to reduce the conservatism for T–S fuzzy discrete system to determine whether the system was stable. The first motivation is to derive less conservative stability conditions by using the fuzzy copositive Lyapunov function and the property of the fuzzy membership functions. And then the line-integral Lyapunov function is proposed to further reduce the conservatism. The restriction of the time-derivatives of the membership functions do not appear in the stability analysis of the T–S fuzzy positive system.

States estimation and observation of nonlinear systems are important problems in modern control theory. The observers and the controllers design for positive systems are concerned by experts. A novel quadratic Lyapunov function was explored for observer design for a class of T–S fuzzy positive systems in [18] which gave less conservative conditions. An optimal l₁-induced state-feedback controller was designed for single-input multiple-output system in [19] which was easy to be computed. The authors of [20] studied the positive stabilization problem by dynamic output feedback controller through LMI formulations. A state-feedback controller was designed in [21] by using the linear Lyapunov function, under which the computation burden was released. In resent years, the authors in [22] provided observer-based controller design by the matrix structural decomposition and matrix transformation, but the positivity of the system was not be considered. In addition, there were some results which design either the positive observers or the state feedback controllers, however, there are no results concerning the observer-based controllers design for T–S fuzzy positive systems. In many practical problems, the states of the systems must be positive. However, the existing conclusions can not guarantee positivity when the observer-based controllers are designed for T–S fuzzy positive systems. The other motivation in this paper is to design the observer-based controllers for T–S fuzzy positive systems such that the stability and positivity of the closed-loop systems are guaranteed.

In this paper, the stability analysis and the observer-based state feedback controllers design for the T–S fuzzy positive systems are investigated. In the first place, a series of results are presented to analyze the stability of the T–S fuzzy positive system by using the fuzzy copositive Lyapunov function which is first used to analyze the stability of the T–S fuzzy positive system. And taking advantage of the property of the fuzzy membership functions, the conservatism of the stability conditions was reduced. Then line-integral Lyapunov functions are employed to make the time-derivatives of the membership functions do not appear in the stability analysis of the T–S fuzzy positive systems and the conservatism is further reduced. Afterwards, the observer-based controllers of T–S fuzzy positive systems are designed based on the stability conditions and the stability and the positivity of the closed-loop systems are guaranteed. Finally a numerical example and a practical example are provided to demonstrate the effectiveness of the results. The main contributions of this paper are from three aspects: (1) the stability of the system is analyzed by using the fuzzy copositive Lyapunov function which is first proposed in the T–S fuzzy positive system. And the less conservative stability conditions are obtained by employing the time derivatives of the fuzzy membership functions; (2) the stability conditions with less conservatism are derived by using the line-integral Lyapunov function when no information about the time-derivative of the membership functions is available; (3) improvements are made from the previous results of T–S fuzzy systems. The positive observer and the observer-based state feedback controller for the T–S fuzzy system are obtained simultaneously and the stability and the positivity of the closed-loop system are guaranteed.

The content of this paper is organized as follows. Section 2 gives the model of the system and presents some definitions and lemmas. Section 3 is devoted to stability analysis and the observer-based controllers design for the T–S fuzzy positive systems. Section 4 provides two illustrative examples to show the applicability of the results. The conclusion is given in Section 5.

In this paper, the matrix A ≥ 0 (respectively, A > 0) means that A is positive semi-definite(respectively, positive definite); the matrix A⪰0 and the vector x⪰0 (respectively, A≻0 and x≻0) mean that all elements of the matrix or the vector are non-negative (respectively, positive); Rⁿ and R^m × n denote the n dimensional Euclidean space and the set of all m × n real matrices; [A_ij]_n × n is a partitioned matrix where A_ij is the ith row and jth column entry of the matrix; $I_{m} \in R^{(n - m + 1) \times n}$ is a matrix composed of the rows m, $m + 1,$ ...,n of I_n, where I_n is an identity matrix; the matrix [a_ij]_n × n is called Metzler, if all its off-diagonal elements are positive, that is, ∀(i, j), i ≠ j, a_ij ≥ 0.

Section snippets

Problem formulation and preliminaries

Consider the following nonlinear system: $\begin{matrix} \begin{matrix} \dot{x} (t) & = f (x (t), u (t)), \\ y (t) & = g (x (t)), \end{matrix} \end{matrix}$ where f and g are nonlinear functions. x(t) ∈ Rⁿ is the state vector; y(t) ∈ R^q is the output vector; u(t) ∈ R^p is the input vector. We can represent (1) with the following fuzzy IF-THEN rules:

Model rule: IF θ₁(t) is $M_{1}^{i},$ θ₂(t) is $M_{2}^{i},$ ...,θ_r(t) is $M_{r}^{i}$ THEN $\begin{matrix} \dot{x} (t) & = A_{i} x (t) + B_{i} u (t), \\ y (t) & = C_{i} x (t), \end{matrix}$ where $M_{j}^{i}$ is the fuzzy set; r is the number of IF-THEN rules; A_i, B_i, C_i are real matrices with appropriate dimensions; $θ (t) = [θ_{1} (t), θ_{2} (t),$

Main results

Stability analysis is crucial to the systems. In this section, at first, three sufficient conditions are given for the stability of system (2)(when $u (t) = 0$ ).

Assumption 1

Let $\begin{matrix} | {\dot{h}}_{τ} (θ (t)) | \leq σ_{τ}, \end{matrix}$ where σ_τ are constants and σ_τ ≥ 0, $τ = 1, 2, \dots, r$ .

Theorem 1

The T–S fuzzy positive system (2) is stable if there exist vectors P_i≻0 and constants σ₁, σ₂, ..., σ_r such that $\begin{matrix} \sum_{τ = 1}^{r} σ_{τ} P_{τ}^{T} + P_{i}^{T} A_{j} ≺ 0, \end{matrix}$ where $i, j = 1, 2, \dots, r$ .

Proof

Consider the candidate of Lyapunov function $\begin{matrix} V (x (t)) = \sum_{i = 1}^{r} h_{i} (θ (t)) P_{i}^{T} x (t) . \end{matrix}$ Then $\begin{matrix} \dot{V} (x (t)) & = \sum_{τ = 1}^{r} {\dot{h}}_{τ} (θ (t)) P_{τ}^{T} x (t) + \sum_{i = 1}^{r} \sum_{j = 1}^{r} h_{i} (θ (t)) h \end{matrix}$

Examples

Example 1

Consider the following T–S fuzzy system: $\begin{matrix} \dot{x} (t) = \sum_{i = 1}^{2} h_{i} (θ (t)) (A_{i} x (t) + B_{i} u (t)), \\ y (t) = \sum_{i = 1}^{2} h_{i} (θ (t)) C_{i} x (t), \end{matrix}$ where $h_{1} (θ (t)) = s i n^{2} x_{1} (t),$ $h_{2} (θ (t)) = 1 - s i n^{2} x_{1} (t),$ $\begin{matrix} A_{1} & = & (\begin{matrix} - 4 & 3 \\ 3 & - 4 \end{matrix}), A_{2} = (\begin{matrix} - 3 & 1 \\ 2 & - 2 \end{matrix}), B_{1} = (\begin{matrix} 1 \\ 0 \end{matrix}), \\ B_{2} & = & (\begin{matrix} 0 \\ 1 \end{matrix}), C_{1} = C_{2} = (\begin{matrix} 1 & 1 \end{matrix}) . \end{matrix}$ The initial condition is $x_{0} = {(\begin{matrix} 5 & 5 \end{matrix})}^{T}$ .

In this case, it satisfies A_i are Metzler matrices, B_i⪰0 and C_i⪰0. It is easy to obtain that each subsystem of the T–S system is positive. Because of the membership functions are nonnegative and the T–S fuzzy system is the convex combination of the subsystems, the

Conclusions

In this paper, by choosing special Lyapunov functions, the results were derived which were more outstanding than the common ones, to demonstrate the stability of the positive systems. The stability conditions were derived without the upper bound of the time-derivatives of the membership functions. Then the observer-based controllers were designed and stability and positivity were guaranteed of the closed-loop systems. The effectiveness of the proposed results were shown through examples. In the

Acknowledgments

This work was supported by the Natural Science Foundation of China under Grant 61673099.

Bo Pang received the B. Sc. Degree in Mathematics from Liaoning Normal University, China, in 2014, the M. Sc. Degree in Operational Research and Cybernetics from Northeastern University, China, in 2016. She is now pursuing the Ph.D Degree in Control Theory and Control Engineering at Northeastern University, China. Her current research interests focus on T–S fuzzy system, positive system, and descriptor system.

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Qingling Zhang received B.S. and M.S. Degrees in Mathematics Department, and Ph.D Degree in Automatic Control Department from Northeastern University, Shenyang, China, in 1982, 1986 and 1995, respectively. He finished his two-year Postdoctoral work in Automatic Control Department of Northwestern Polytechnical University, Xian, China, in 1997. Since then he has been a Professor and serve College of Science at Northeastern University as dean from 1997 to 2006. He was also a member of the University Teaching Advisory Committee of National Ministry of Education, and now he is vice chairman of the Chinese Biomathematics Association, member of technical committee on control theory of the Chinese Association of Automation, member of the Chinese Association of Mathematics and Chairman of Mathematics Association of Liaoning Province. He has published 16 books and more than 600 papers about control theory and applications. Prof. Zhang received 14 prizes from central and local governments for his research. He has also received the Gelden Scholarship from Australia in 2000. During these periods, he visited Hong Kong University, Seoul University, Alberta University, Lakehead University, Sydney University, Western Australia University, Windsor University, Hongkong Polytechnic University and Kent University as a Research Associate, Research Fellow, Senior Research Fellow and Visiting Professor, respectively.

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Stability analysis and observer-based controllers design for T–S fuzzy positive systems

Abstract

Introduction

Section snippets

Problem formulation and preliminaries

Main results

Examples

Conclusions

Acknowledgments

Automatica

Syst. Control Lett.

Nonlinear Anal. Real World Appl.

Neurocomputing

ISA Trans.

Neurocomputing

Inf. Sci.

Neurocomputing

Automatica

Fuzzy Sets. Syst.

Automatica

Positive linear observer design via positive realization

Proceedings of the American Control Conference (ACC)

Positivity of continuous-time descriptor systems with time delays

IEEE Trans. Autom. Control

Positive solutions of a discrete-time descriptor system

Int. J. Syst. Sci.

A positive systems model of TCP-like congestion control: asymptotic results

IEEE/ACM Trans. Netw