Furtherstudiesonrelaxedstabilizationconditionsfordiscrete-timetwo-dimensionTakagi-Sugenofuzzysystems资源-CSDN文库

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Further studies on relaxed stabilization conditions for discrete-time two-dimension Takagi-Sugeno fuzzy systems 本文研究的是离散时间二维Takagi-Sugeno模糊系统的稳定性问题。Takagi-Sugeno（T-S）模糊模型自20世纪80年代中期以来，由于其能有效逼近非线性系统，一直受到工业界和学术研究者的极大关注。在介绍二维T-S模糊系统稳定性的放松条件的研究中，本文提出了一种新的非并行分布补偿（non-PDC）控制策略和一个新型的非二次Lyapunov函数，从而发展了更为宽松的稳定性条件。在研究中，非二次Lyapunov函数采用了一种基于隶属度函数的均匀多项式参数依赖形式。随着Lyapunov函数的阶数增加，可以逐渐减少所得稳定性条件的保守性。通过利用隶属度函数的代数性质，稳定性条件可以在精确性上收敛，即逐渐达到精确度。与现有的方法相比，本文所提出的控制合成方法中没有引入多余的松弛变量，因此可以在较低的计算成本下获得相同或更为宽松的结果。文章还通过一个数值例子展示了所提方法的有效性。在此文中，研究者利用了线性矩阵不等式（LMIs）来表达稳定性条件，这种方法在控制理论中是一个常用工具。在数学上，线性矩阵不等式通常用于表达线性矩阵的各种不等式约束，这在设计控制器时非常有用，特别是在保证系统的稳定性和性能时。该研究的关键点在于放宽了稳定性条件，这在控制系统设计中是十分重要的，因为它直接影响到控制器设计的可行性和系统的性能。过于保守的设计可能会导致系统性能不佳，例如，对扰动过于敏感或者响应时间过长，而放松的稳定性条件可以在保证系统稳定性的同时，尽可能地提高系统性能。此外，研究者提出的方法不仅提高了稳定条件的精确度，还降低了控制器设计的复杂性。这在实际应用中是非常宝贵的，因为简化的设计过程意味着更易于实现，并且可能减少实施成本。对于T-S模糊系统的研究，目前是控制理论和应用领域内的一个热点。T-S模糊模型是一种基于模糊逻辑的系统建模方法，它可以处理不确定性，并且适用于非线性系统建模。二维T-S模糊系统则是在一个更加复杂的框架下进行研究，因为它涉及到两个独立的变量。在模糊逻辑控制中，隶属度函数扮演了至关重要的角色，它决定了模糊集合中元素的归属程度。在本文的研究中，隶属度函数不仅用于建立模糊模型，还被用于构造Lyapunov函数，这是分析系统稳定性的一个重要工具。通过隶属度函数的不同表达形式，可以调整Lyapunov函数的结构，从而得到不同程度的稳定性分析结果。本文的研究为离散时间二维T-S模糊系统的稳定性分析提供了一种新方法，并通过理论推导和数值示例证明了该方法的有效性。这一成果对控制系统设计以及模糊逻辑控制领域具有重要的理论和应用价值。

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Further studies on relaxed stabilization conditions for discrete-time

two-dimension Takagi–Sugeno fuzzy systems

Da-Wei Ding

a,b,

⇑

, Xiaoli Li

a,b

, Yixin Yin

a,b

, Xiang-Peng Xie

School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, PR China

Key Laboratory of Advanced Control of Iron and Steel Process (Ministry of Education), University of Science and Technology Beijing, Beijing 100083, PR China

WISDRI Engineering & Research Incorporation Limited, MCC Group of China, Wuhan 420223, PR China

article info

Article history:

Received 18 June 2010

Received in revised form 10 September 2011

Accepted 15 November 2011

Available online 25 November 2011

Keywords:

2-D systems

Takagi–Sugeno fuzzy model

Homogeneous polynomially parameter-

dependent Lyapunov function

Linear matrix inequalities (LMIs)

abstract

This paper investigates the stabilization problem of discrete-time two-dimension (2-D)

Takagi–Sugeno (T–S) fuzzy systems. Based on a novel non-parallel distributed compensa-

tion (non-PDC) control scheme combined with a new non-quadratic Lyapunov function,

less conservative stabilization conditions are developed. The proposed non-quadratic

Lyapunov function is homogeneous polynomially parameter-dependent on membership

functions. As the degree of the Lyapunov function increases, the conservatism of the

obtained stabilization conditions is gradually reduced. By exploiting the algebraic property

of membership functions, the stabilization conditions approach to exactness in the sense of

convergence. Compared with the existing methods, no slack variables are introduced in

control synthesis, and hence the same or less conservative results can be obtained with

a lower computational cost. A numerical example is given to illustrate the effectiveness

of the proposed method.

1. Introduction

Since the middle of the 1980s, Takagi–Sugeno (T–S) fuzzy models [29] have attracted a great deal of attention from indus-

trial practitioners and academic researchers, especially because they can effectively approximate a wide class of nonlinear

systems. The issue of quadratic stabilization of T–S fuzzy systems [8,18] has been widely investigated based on a common

quadratic Lyapunov function and parallel distributed compensation (PDC) [31]. However, these results might be quite

conservative due to the common Lyapunov function, and then various approaches [1–7,9,11,12,14–17,19–21,25,30,36,37]

have been proposed. Among these approaches, non-quadratic Lyapunov functions and non-PDC control schemes

[1,11,12,14,19,21,36] have attracted particular attention. It is worth noticing that the PDC control scheme [31] is linearly

dependent on membership functions, and the non-PDC control scheme [11] is quadratically dependent on membership func-

tions. In addition, fuzzy Lyapuonv functions [11,16] are also linearly dependent on membership functions.

On the other hand, two-dimensional (2-D) systems [10,24] have drawn great attention due to their extensive applications

in practice [23,27,28], such as image data processing and transmission, thermal process, signal ﬁltering, etc. H

control for

2-D discrete state delay systems described by the second Fornasini–Marchesini (FM) state-space model has been investi-

gated in [35]. H

ﬁltering for 2-D Markovian jump systems has been studied in [32]. Stability analysis of 2-D discrete

systems described by the FM second model with state saturation has been considered in [26]. However, the aforementioned

doi:10.1016/j.ins.2011.11.031

⇑

Corresponding author at: School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, PR China.

Tel.: +86 15011248835; fax: +86 010 62332926.

E-mail addresses: ddaweiauto@163.com (D.-W. Ding), lixiaoli@hotmail.com (X. Li), yyx@ies.ustb.edu.cn (Y. Yin), xiexiangpeng1953@sina.com (X.-P. Xie).

Information Sciences 189 (2012) 143–154

Contents lists available at SciVerse ScienceDirect

Information Sciences

journal homepage: www.elsevier.com/locate/ins

results are only applicable to linear 2-D systems. As it is well known, most practical 2-D systems are nonlinear and hence the

results do not work in this case [33]. More recently, based on the T–S fuzzy modeling approach, non-quadratic stabilization

conditions for a class of Roesser-type nonlinear 2-D systems have been proposed in [33], where the non-quadratic control

scheme [16] and the right-hand-side slack variable approach [18,31] are extended to the 2-D T–S fuzzy setting. To further

reduce the conservatism, relaxed stabilization conditions for discrete-time 2-D T–S fuzzy systems have been developed

via improved homogeneous polynomial techniques in [34]. In the existing results [33,34], the so-called right-hand-side slack

variable technique plays an important role in reducing the conservatism. However, the introduction of slack variables leads

to a high computational cost as a trade-off. Now, it seems that the challenge is to improve the quality and efﬁciency of relax-

ations, i.e., to obtain the same results with lower computational complexity, or to extend a given stability condition to con-

trol synthesis. As stated in [33,34], the conservatism can be further reduced if we change ‘‘something’’, such as the control

law, the form of Lyapunov functions, or exploit the algebraic property of membership functions. This motivates us to carry

out the present work.

This paper considers the stabilization problem of a class of discrete-time 2-D T–S fuzzy systems. Based on a novel non-

PDC scheme and a new homogeneous polynomially parameter-dependent (HPPD) Lyapunov function, new relaxed stabiliza-

tion conditions are proposed. As the degree of HPPD Lyapunov function increases, the conservatism of the obtained stabil-

ization conditions is gradually reduced due to the introduction of extra degrees of freedom. Moreover, by exploiting the

algebraic property of membership functions, the stabilization conditions approach to exactness in the sense of convergence.

In other words, the stabilization conditions are asymptotically necessary and sufﬁcient. Compared with the existing methods

[33,34], no slack variables are introduced in control synthesis, and hence the same or less conservative results can be ob-

tained with a lower computational cost. The effectiveness of the proposed method is illustrated by a numerical example.

The rest of the paper is organized as follows. Problem statement and preliminaries are given in Section 2. New relaxed

non-quadratic stabilization conditions are presented in Section 3. In Section 4, an example is given to illustrate the effective-

ness of the proposed method. Finally, some conclusions are drawn in Section 5.

The notations used in the paper are fairly standard. For example, X > 0 (or X P 0) means the matrix X is symmetric and

positive deﬁnite (or symmetric and positive semideﬁnite). For a square matrix E, He(E) is deﬁned as E + E

. X

denotes the

transpose of X. The symbol I represents the identity matrix with appropriate dimension. A star ⁄ in a symmetric matrix de-

notes the transposed element in the symmetric position. For a matrix P, min(P) (respectively, max(P)) means the smallest

(respectively, largest) eigenvalue of P. Z

denotes the set of negative integers {0,1,2,...} and M! represents factorial. C

de-

notes the space of continuously differentiable functions.

2. Problem formulation and preliminaries

2.1. Discrete-time 2-D T–S fuzzy model

Consider a Roesser-type discrete-time nonlinear 2-D system described by

ðs; lÞ¼Zðxðs; lÞÞþSðxðs; lÞÞuðs; lÞ; ð1Þ

ð0; lÞ¼f ðlÞ; x

ðs; 0Þ¼gðsÞ ð2Þ

with

xðs; lÞ¼

ðs; lÞ

; x

ðs; lÞ¼

ðs þ 1; lÞ

ðs; l þ 1Þ

;

where x

ðÞ 2 R

is the horizonal state, x

ðÞ 2 R

is the vertical state, u() 2 R

is the control input. ZðÞ and SðÞ are general

nonlinear functions satisfying Z; S2C

. s, l are two integers in Z

. f(l) and g(s) are the corresponding boundary conditions

along two independent directions.

By extending the 1-D T–S fuzzy modeling method [29] to the 2-D case, we can obtain a discrete-time 2-D T–S fuzzy model

described by the following rules

IF z

ðs; lÞ is M

; and ...; and z

ðs; lÞ is M

; Then;

ðs; lÞ¼A

xðs; lÞþB

uðs; lÞ; i ¼ 1; ...; r;

ð0; lÞ¼f ðlÞ; x

ðs; 0Þ¼gðsÞ

ð3Þ

with

; B

;

where z

(s,l), for p =1,...,L are the premise variables, M

is the fuzzy set, r is the number of IF-THEN rules. A

2 R

n

;

2 R

n

; A

2 R

n

; A

2 R

n

; B

2 R

m

; B

2 R

m

, respectively.

144 D.-W. Ding et al. / Information Sciences 189 (2012) 143–154

剩余11页未读，继续阅读

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