About this Journal Submit a Manuscript Table of Contents
Journal of Control Science and Engineering
Volume 2011 (2011), Article ID 579871, 7 pages
http://dx.doi.org/10.1155/2011/579871
Research Article

Hybrid Feedback Stabilization of Fuzzy Nonlinear Systems

1College of Science, Henan University of Technology, Zhengzhou 450001, China
2Department of Mathematics, Zhengzhou University, Zhengzhou 450001, China

Received 24 September 2010; Accepted 8 January 2011

Academic Editor: G. Yin

Copyright © 2011 Yingqi Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with the problem of stabilizing one family of fuzzy nonlinear systems by means of fuzzy quantized feedback. The hybrid control strategy originating in earlier work by Brockett and Liberzon (2000) and Liberzon (2003) relies on the possibility of making discrete online adjustments of quantizer parameters. We explore this method here for one class of fuzzy nonlinear systems with fuzzy quantizers affecting the state of the system. New results on the stabilization of the family of fuzzy nonlinear systems are obtained by choosing appropriately quantized strategies. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed method.

1. Introduction

In recent years, there has been increasing interest in stability analysis and controller design for hybrid and switched systems see, for example, [1, 2]. In the presence of quantization, the state space of the system is divided into a finite number of quantization regions, each corresponding to a fixed value of the quantizer. At the time of passage from one quantization region to another, the dynamics of the closed-loop system change abruptly. Therefore, systems with quantization can be naturally viewed as hybrid systems. Thus, considerable efforts have been devoted to the study of quantized control, for instance, see [37] and the references therein. Among these results, mainly two approaches for studying control problems with quantized feedback are chosen, which are called static quantization policies (e.g., [810]) and dynamic quantization policies (e.g., [5, 11]).

Liberzon [5] gave the conditions of hybrid feedback stabilization of systems with quantized signal under the assumption of the systems being stabilized by a feedback law. De Persis [12] extended Liberzon's [5] results to the problem of stabilizing a nonlinear system by means of quantized output feedback. Gao and Chen [13] presented a new approach to quantized feedback control systems which provided stability and 𝐻 performance analysis as well as controller synthesis for discrete-time state-feedback control systems with logarithmic quantizers. The most significant feature is the utilization of a quantization dependent Lyapunov function. Ceragioli and De Persis [14] discussed discontinuous stabilization of nonlinear systems with quantized and switching controls, that is, considering the classical problem of stabilizing nonlinear systems in the case of the control laws which take values in a discrete set.

The well-known Takagi-Sugeno (T-S) fuzzy model (e.g., [15]) has been recognized as a popular and powerful tool in approximating and describing complex nonlinear systems. Thus, over the past ten years, the study of T-S systems has been attracting increasing attention, for instance, see [1623]. However, so far, the study of fuzzy systems with quantized feedback was rare, for instance, [24]. In this paper, we concentrate on the problem of stabilizing fuzzy nonlinear systems via fuzzy quantized feedback. We extend the results (see, [5]) to a class of T-S fuzzy nonlinear systems with general types of quantizers affecting the state of the system. New results on the stabilization of fuzzy nonlinear systems are obtained by choosing appropriately quantized strategies and applying the Lyapunov function approach.

The paper is organized as follows. Section 2 gives the concept of quantizer and the description of fuzzy systems. New results on the stabilization of fuzzy nonlinear systems with fuzzy quantized feedback are presented in Section 3. In Section 4, an example is given to show the effectiveness of the proposed method. Conclusions are presented in Section 5.

2. Problem Statement

In this section, some notations and definition of quantizer are introduced, and the problem statement is given.

As in [5], a quantizer with general form is defined as follows.

Let 𝑧𝑙 be the variable being quantized. A quantizer is defined as a piecewise constant function 𝑞𝑙𝐷, where 𝐷 is a finite subset of 𝑙. This leads to a partition of 𝑙 into a finite number of quantization regions of the form {𝑧𝑙𝑞(𝑧)=𝑖}, 𝑖𝐷. These quantization regions are not assumed to have any particular shapes. We assume that there exist positive real numbers 𝑀 and Δ such that the following conditions hold:|||||𝑧|𝑀𝑞(𝑧)𝑧Δ,(1)|||||𝑧|>𝑀𝑞(𝑧)>𝑀Δ.(2)

Throughout this paper, we denote by || the standard Euclidean norm in the n-dimensional vector space 𝑛 and denote by the corresponding induced matrix norm in 𝑛×𝑛. Condition (1) gives a bound on the quantization error when the quantizer does not saturate. Condition (2) provides a way to detect the possibility of saturation. We will refer to 𝑀 and Δ as the range of 𝑞 and the quantization error, respectively. We also assume that {𝑥𝑞(𝑥)=0} for 𝑥 in some neighborhood of the origin which is needed to preserve the origin as an equilibrium.

In the control strategy to be developed below, we will use quantized measurements of same the form as in [3, 4]𝑞𝜇𝑧(𝑧)=𝜇𝑞𝜇,(3) where 𝜇>0 is an adjustable parameter, called the “zoom” variable, that is updated at discrete instants of time.

To be convenient, we denoted that 𝑟𝑖,𝑗=1=𝑟𝑖=1𝑟𝑗=1, 𝑖=𝑖(𝑥(𝑡)), 𝑞𝜇𝑖(𝑥)=𝑖(𝑞𝜇(𝑥(𝑡))), and 𝑤𝑖=𝑤𝑖(𝑥(𝑡)).

The T-S fuzzy system, suggested by Takagi and Sugeno [15] can represent a general class of nonlinear systems. It is based on “fuzzy partition” of input space and it can be viewed as the expansion of piecewise linear partition. Considering a nonlinear dynamic multi-input-multi-output system modeled by the T-S fuzzy system, it can be represented by the following forms.(i)If-then form:

𝑅𝑖: IF 𝑥1(𝑡) is 𝑀𝑖1, 𝑥2(𝑡) is 𝑀𝑖2 and 𝑥𝑛(𝑡) is 𝑀𝑖𝑛

theṅ𝑥(𝑡)=𝐴𝑖𝑥(𝑡)+𝐵𝑖𝑢(𝑡).(4)(ii)Input-output form:̇𝑥(𝑡)=𝑟𝑖=1𝑤𝑖𝐴𝑖𝑥(𝑡)+𝐵𝑖𝑢(𝑡)𝑟𝑖=1𝑤𝑖=𝑟𝑖=1𝑖𝐴(𝑥(𝑡))𝑖𝑥(𝑡)+𝐵𝑖,𝑤𝑢(𝑡)𝑖=𝑛𝑗=1𝑀𝑖𝑗𝑥𝑗,(𝑡)𝑟𝑖=1𝑤𝑖>0,𝑤𝑖0,𝑖=𝑤𝑖𝑟𝑖=1𝑤𝑖,𝑟𝑖=1𝑖=1,𝑖0,(5) where 𝑥(𝑡)=[𝑥1(𝑡),𝑥2(𝑡),,𝑥𝑛(𝑡)]𝑇 is the state, 𝑢(𝑡)𝑚 is the control input, 𝑅𝑖  (𝑖=1,2,,𝑟) is the 𝑖th fuzzy rule, 𝑟 is the number of rule, 𝑀𝑖1,𝑀𝑖2,,𝑀𝑖𝑛 are fuzzy variable, and 𝑖 is fuzzy basis function.

For the nonlinear plant represented by (4) or (5), we consider the fuzzy controller as follows.(iii)If-then form:

𝑅𝑖: if 𝑥1(𝑡) is 𝑀𝑖1, 𝑥2(𝑡) is 𝑀𝑖2 and 𝑥𝑛(𝑡) is 𝑀𝑖𝑛

then𝑢(𝑡)=𝐿𝑖𝑥(𝑡),(6) or𝑢(𝑡)=𝐿𝑖𝑞𝜇(𝑥).(7)(iv)Input-output form:𝑢(𝑡)𝑟𝑖=1=𝑖𝐿𝑖𝑥(𝑡),(8) or𝑢(𝑡)=𝑟𝑖=1𝑞𝜇𝑖(𝑥)𝐿𝑖𝑞𝜇(𝑥).(9)

The system (5) with (8) or the system (5) with (9) can, respectively, be written in the form of the T-S fuzzy control system as follows:̇𝑥(𝑡)=𝑟𝑖,𝑗=1𝑖𝑗𝐴𝑖+𝐵𝑖𝐿𝑗=𝑥(𝑡)𝑟𝑖,𝑗=1𝑖𝑗𝐻𝑖𝑗𝑥(𝑡),(10) or=̇𝑥(𝑡)𝑟𝑖,𝑗=1𝑖𝑞𝜇𝑗(𝑥)𝐴𝑖+𝐵𝑖𝐿𝑗𝑥(𝑡)+𝐵𝑖𝐿𝑗𝜇𝑞𝑥𝜇𝑥𝜇=𝑟𝑖,𝑗=1𝑖𝑞𝜇𝑗(𝑥)𝐻𝑖𝑗𝑥(𝑡)+𝐵𝑖𝐿𝑗𝜇𝑞𝑥𝜇𝑥𝜇,(11) where 𝐻𝑖𝑗 denotes 𝐻𝑖𝑗=𝐴𝑖+𝐵𝑖𝐿𝑗.

3. Fuzzy Hybrid Feedback Stabilization

In this section, in order to find some sufficient conditions which stabilize the fuzzy nonlinear systems (11) by choosing appropriately quantized strategies, we require the following assumption 1 and an important lemma is given as in Lemma 1 as follows.

Assumption. Assume that there exists a sequence of matrices {𝐿𝑖}𝑟𝑖=1 and a common positive definite matrix 𝑃 and a sequence of positive matrices {𝑄𝑖𝑗}𝑟𝑖,𝑗=1 such that 𝑄𝑖𝑗𝐴=𝑖+𝐵𝑖𝐿𝑗𝑇𝐴𝑃+𝑃𝑖+𝐵𝑖𝐿𝑗=𝐻𝑇𝑖𝑗𝑃+𝑃𝐻𝑖𝑗.(12) Moreover, both 𝐵𝑖 and 𝐿𝑖 for all 𝑖,𝑗{1,2,,𝑟} are nonzero matrices, which cause no loss of generality because the case of interest is when 𝐴𝑖 is not a stable matrix for all 𝑖{1,2,,𝑟}.

Remark 1. If Assumption 1 holds, the system (5) with fuzzy control law (8) or the T-S fuzzy system (10) is asymptotically stable by using Lyapunov approach (e.g., see [16, 17]).

Remark 2. As in [5], it is necessary to suppose that systems are stabilizable. To be convenient, we suppose that Assumption 1 holds so that the system (5) is stabilizable.

Lemma 1. Assume that Assumption 1 holds. an arbitrary 𝜎>0, and 𝑀 is large enough compared to Δ such that 𝜆min(𝑃)𝜆max(𝑃)𝑀>Θ𝑥Δ(1+𝜎),(13) where Θ𝑥𝜆𝑄=2𝜃/𝜆,𝜆=min𝑖𝑗,𝑖,𝑗=1,2,,𝑟𝜃=max𝑃𝐵𝑖𝐿𝑗.𝑖,𝑗=1,2,,𝑟(14) Let 1(𝜇)=𝑥𝑥𝑇𝑃𝑥𝜆min(𝑃)𝑀2𝜇2,(15)2(𝜇)=𝑥𝑥𝑇𝑃𝑥𝜆max(𝑃)Θ2𝑥Δ2(1+𝜎)2𝜇2.(16) Then all solutions of (11) that start in the ellipsoid 1(𝜇) enter the smaller ellipsoid 2(𝜇) in finite time.

Proof. We consider the Lyapunov function candidate 𝑉(𝑥)=𝑥𝑇𝑃𝑥 for the closed-loop system (11) the derivative of 𝑉(𝑥) along solutions of (11) is computed as ̇𝑥𝑉(𝑥)=𝑇𝑃𝑥=𝑟𝑖,𝑗=1𝑖𝑞𝜇𝑗(𝑥)𝑥𝑇𝐻𝑇𝑖𝑗𝑃+𝑃𝐻𝑖𝑗𝑥(𝑡)+2𝑥𝑇𝑃𝐵𝑖𝐿𝑗𝜇×𝑞𝑥𝜇𝑥𝜇=𝑟𝑖,𝑗=1𝑖𝑞𝜇𝑗(𝑥)𝑥𝑇𝑄𝑖𝑗𝑥(𝑡)+2𝑥𝑇𝑃𝐵𝑖𝐿𝑗𝜇×𝑞𝑥𝜇𝑥𝜇𝑟𝑖,𝑗=1𝑖𝑞𝜇𝑗(𝑥)𝜆|𝑥|2+2𝜃|𝑥|𝜇Δ𝑟𝑖,𝑗=1𝑖𝑞𝜇𝑗(𝑥)|𝜆|𝑥|𝑥|Θ𝑥𝜇Δ=𝜆|𝑥||𝑥|Θ𝑥.𝜇Δ(17) According to (13), for any nonzero 𝑥, we can find a positive scalar 𝜇 such that Θ𝑥𝜇Δ(1+𝜎)|𝑥|𝑀𝜇.(18)
This is also true in the case of 𝑥=0, where we set 𝜇=0 as an extreme case and consider the output of the quantizer as zero.
When Θ𝑥𝜇Δ(1+𝜎)|𝑥|𝑀𝜇 holds, we have𝑥𝑇𝑃𝑥|𝑥|𝜆Θ𝑥𝜇Δ𝜎.(19)

Claim 1. Both 1(𝜇) and 2(𝜇) are invariant sets of the system (11).

Proof. we only prove that 1(𝜇) is an invariant set of the system (11). Assuming 𝑥(𝑡0)1(𝜇), we denote 𝜏=sup𝑡𝑥(𝑡)1𝑡(𝜇),𝑡0,𝜏,(20) where 𝑥(𝑡) is a solution of the system (11) with the initial condition 𝑥(𝑡0). If 𝜏<+, then there exists a positive constant 𝜏 such that 𝑥(𝜏)𝑇𝑃𝑥(𝜏)=𝜆min(𝑃)𝑀2𝜇2.(21) By the virtue of condition (13), we have 𝜆max(𝑃)Θ2𝑥Δ2(1+𝜎)2𝜇2<𝜆min(𝑃)𝑀2𝜇2=𝑥(𝜏)𝑇𝑃𝑥(𝜏)𝜆max||||(𝑃)𝑥(𝜏)2.(22) Hence, we obtain Θ𝑥||𝑥||𝜇Δ(1+𝜎)<(𝑡)𝑡=𝜏.(23) Using (13), we have 2(𝜇)1(𝜇) and ̇𝑉(𝑥(𝑡))𝑡=𝜏||𝑥||||𝑥||𝜆(𝑡)(𝑡)𝜇ΔΘ𝑥𝑡=𝜏||||<𝜆𝑥(𝑡)𝑡=𝜏Θ𝑥𝜇Δ𝜎0.(24) By the continuity of 𝑉(𝑥(𝑡)), there exists a positive constant 𝜖 such that for all 𝑡[0,𝜖] satisfying 𝜆min||||(𝑃)𝑥(𝑡+𝜏)2𝑥(𝑡+𝜏)𝑇𝑃𝑥(𝑡+𝜏)=𝑉(𝑥(𝑡+𝜏))𝑉(𝑥(𝜏))=𝜆min(𝑃)𝑀2𝜇2.(25) Hence, we have 𝑉(𝑥(𝑡+𝜏))𝜆min(𝑃)𝑀2𝜇2 for all 𝑡[0,𝜖] that is to say that 𝑥(𝑡+𝜏)1(𝜇) holds for all 𝑡[0,𝜖] this is a contradiction with the definition 𝜏. Thus, 𝜏=+. we complete the proof of Claim 1.

Fixed an arbitrary 𝑥(𝑡0)1(𝜇), and for all 𝑥, we can find a positive scalar 𝜇 satisfying (18). Then, integrating (19) from 𝑡0 to 𝑡0+𝑇, we have𝑥𝑇𝑡0𝑡+𝑇𝑃𝑥0+𝑇𝑥𝑇𝑡0𝑡𝑃𝑥0=𝑡0𝑡+𝑇0𝑥(𝑠)𝑇𝑃𝑥(𝑠)𝑑𝑠𝑡0𝑡+𝑇0||||𝑥(𝑠)𝜆Θ𝑥𝜇Δ𝜎𝑑𝑠𝑡0𝑡+𝑇0Θ𝑥𝜇Δ(1+𝜎)𝜆Θ𝑥𝜇Δ𝜎𝑑𝑠𝑇Θ2𝑥𝜇2Δ2𝜎(1+𝜎)𝜆.(26) Hence, we obtain𝑥𝑇𝑡0𝑡+𝑇𝑃𝑥0+𝑇𝑥𝑇𝑡0𝑡𝑃𝑥0𝑇Θ2𝑥𝜇2Δ2𝜎(1+𝜎)𝜆𝜆min(𝑃)𝑀2𝜇2𝑇Θ2𝑥𝜇2Δ2𝜎(1+𝜎)𝜆.(27) If we choose𝑇=𝑇𝑥=𝜆min(𝑃)𝑀2𝜆max(𝑃)Θ2𝑥Δ2(1+𝜎)2Θ2𝑥Δ2𝜎(1+𝜎)𝜆,(28) we have 𝑥(𝑡0+T𝑥)2(𝜇).

Using Lemma 1 and assuming that the fuzzy control law (9) of the system (5) satisfies𝑢(𝑡)=0,0𝑡<𝑡0,𝑟𝑖=1𝑞𝜇𝑖(𝑥)𝐿𝑖𝑞𝜇(𝑥),𝑡𝑡0,(29) We have the following theorem 1.

Theorem 1. Assume that Assumption 1 holds. Assume that 𝑀 is large enough compared to Δ such that 𝜆min(𝑃)𝜆max(𝑃)𝑀>2Δmax1,Θ𝑥(30) holds, where Θ𝑥 is the same as in Lemma 1. Then there exists a fuzzy quantized feedback control strategy such that the system (5) with fuzzy quantized control law (9) or the closed fuzzy nonlinear system (11) is globally asymptotically stable.

Proof. The “zooming-out” stage. Let 𝑢=0. In this case, we rewrite the system (11) for ̇𝑥(𝑡)=𝑟𝑖,𝑗=1𝑖𝑞𝜇𝑗(𝑥)𝐻𝑖𝑗𝑥(𝑡)+𝐵𝑖𝐿𝑗𝜇𝑞𝑥𝜇𝑥𝜇=𝑟𝑖=1𝑖𝐴𝑖𝑥(𝑡).(31) Let 𝐴=argmax𝐴𝑖𝑗𝑖,𝑗{1,2,,𝑟}𝐴𝑖𝑗(32)
Let 𝜇0=𝜇(0)=1, and then increase 𝜇 in a piecewise constant fashion, fast enough to dominate the rate of 𝑒𝑟𝐴𝑡. Then, there is a time 𝑡0 such that||||𝑥(𝑡)||||𝜇(𝑡)𝜆min(𝑃)𝜆max(𝑃)𝑀2Δ.(33) By condition (1) in Section 2, it is implied ||||𝑞𝑥(𝑡)||||𝜇(𝑡)𝜆min(𝑃)𝜆max(𝑃)𝑀Δ.(34) We can pick a 𝑡0 such that (34) holds with 𝑡=𝑡0. Again, applying conditions (1) and (2) of Section 2, we obtain ||||𝑥𝑡0𝜇𝑡0||||𝜆min(𝑃)𝜆max(𝑃)𝑀.(35) Hence, we have 𝑥(𝑡0)1(𝜇(𝑡0)) given by (15).

The “zooming-in” stage. Define the sequence of times {𝑡𝑗}𝑗 satisfying𝑥𝑡01𝜇𝑡0,𝑡𝑗+1=𝑡𝑗+𝑇𝑥,(36) and the sequence of positive real numbers𝜇0𝑡=𝜇0𝜇=1,𝑗𝑡=𝜇𝑗𝑡=Ω𝜇𝑗1=Ω𝑗𝜇0=Ω𝑗.(37) where Ω denotes 𝜆max(𝑃)/𝜆min(𝑃)(Θ𝑥𝜇Δ(1+𝜎)/𝑀) and 𝑇𝑥 is the same as in (28).

Define also the control law𝑢(𝑡)=𝑟𝑗=1𝑞𝜇𝑗𝑗(𝑥)𝐿𝑗𝑞𝜇𝑗𝑡(𝑥(𝑡)),𝑡𝑗,𝑡𝑗+1,𝑗.(38)

By (30) and Lemma 1, we have Ω<1 and 2(𝜇(𝑡𝑗+1))=1(𝜇(𝑡𝑗)). Hence, 𝜇𝑗=Ω𝑗𝜇0=Ω𝑗0 as 𝑡+, and the above analysis implies 𝑥(𝑡)0 as 𝑡+.

In order to prove the stability of the equilibrium 𝑥=0 of system (11), take an arbitrary 𝜖>0 and notice that 𝑢(𝑡)=0 as 0𝑡𝑡0 firstly, finding a positive integer 𝐾=ln(𝜖/𝑀)/Ω+1, 𝑡[(𝐾1)𝑇𝑥,𝐾𝑇𝑥), we have ||𝑥||𝑡(𝑡)𝑀𝜇𝐾1=𝑀Ω𝐾1𝜇𝑡0=𝑀Ω𝐾1𝜖.(39) This implies 1(Ω𝐾1){𝑥|𝑥|<𝜖}.

By the virtue of 𝑞(𝑥), there exists a positive constant 𝜖0 such that 𝑞(𝑥)=0 holds for all 𝑥{𝑥|𝑥|<𝜖0}. With no loss of generality, we assume 𝜖0𝜖. We define 𝜖𝛿=min0𝑒𝑟𝐴𝑗𝑇𝑥𝑗=1,2,,𝐾=𝜖0𝑒𝑟𝐴𝐾𝑇𝑥.(40) Then for all |𝑥(0)|<𝛿 and for all 𝑗=1,2,,𝐾, we have ||||𝑥(𝑡)𝜇𝑡𝑗||||||𝑥||𝑒(0)𝑟𝐴𝑗𝑇𝑥||𝑥||𝑒(0)𝑟𝐴𝐾𝑇𝑥𝜖0,(41) Hence there exists a positive constant 𝛿=𝜖0𝑒𝑟𝐴𝐾𝑇𝑥, and the solutions of ̇𝑥=𝑟𝑖=1𝑖𝐴𝑖𝑥 with |𝑥(0)|<𝛿 stay in the intersection of this 𝜖0 with the region {𝑥𝑞(𝑥)=𝑞(𝑥/Ω)=𝑞(𝑥/Ω2)==𝑞(𝑥/Ω𝐾1)}=0 for all 𝑡[0,𝐾𝑇𝑥]. Therefore, these solutions satisfy |𝑥(𝑡)|𝜖 for all 𝑡0.

4. Numerical Example

In this section, we consider the following nonlinear system: ̇𝑥1=𝑎𝑥1(𝑡)+𝑏𝑥2(𝑡),̇𝑥2=𝑐𝑥2𝑥(𝑡)+𝑑1𝑥(𝑡)2(𝑡)+𝑢(𝑡),(42) where 𝑎,𝑏,𝑐,𝑑 are constants, 𝑢(𝑡) is control input, and𝑥1=(𝑡)sin𝑥1(𝑡)𝑥1(𝑡),𝑥1(𝑡)0,1,𝑥1(𝑡)=0.(43) It follows that the nonlinear system can be represented by the following T-S fuzzy model.(i)If-then rule: if 𝑥1(𝑡) is 𝐹1, then ̇𝑥=𝐴1𝑥(𝑡)+𝐵1𝑢(𝑡); IF 𝑥1(𝑡) is 𝐹2, then ̇𝑥=𝐴2𝑥(𝑡)+𝐵2𝑢(𝑡), where𝑥𝑥(𝑡)=1𝑥(𝑡)2(𝑡),𝐴1=,𝐴𝑎𝑏0𝑐+𝑑2=𝑎𝑏0𝑐,𝐵1=𝐵2=01.(44) Moreover, the 𝐹1 and 𝐹2 are fuzzy sets defined as 𝐹1(𝑥(𝑡))=(𝑥1(𝑡)) and 𝐹2(𝑥(𝑡))=1(𝑥1(𝑡)).

For the simplicity of simulation, the quantizer is chosen to be logarithmic, which satisfies general quantizer (9), see [5, 9, 10]. That is to say, we choose the quantization level to be described as𝑈𝜌=±𝑢𝑖,𝑢𝑖=𝜌𝑖𝑢0𝜌,𝑖=1,2,𝑖𝑢0{0},(45) and the associated quantizer 𝑞() is defined as follows:𝑢𝑞(𝑧)=𝑖,1+𝜌2𝑢𝑖<𝑧1+𝜌𝑢2𝜌𝑖,𝑧>0,0,𝑧=0,𝑞(𝑧),𝑧<0.(46) Thus, the corresponding fuzzy quantized controller can be chosen aŝ𝑢=𝑞(𝑧)=𝑞2𝑖=1𝑖𝐿(𝑥(𝑡))𝑖.𝑥(𝑡)(47) Now define the quantization error by𝑒(𝑧)=𝑞(𝑧)𝑧=Φ𝑧.(48) Therefore, ̂𝑢(𝑡) can be expressed aŝ𝑢=𝑞(𝑧)=(1+Φ)𝑞2𝑖=1𝑖𝐿(𝑥(𝑡))i,𝑥(𝑡)(49) where Φ[𝛿,𝛿]. Thus the above closed-loop system with quantized control law can be written as followṡ𝑥(𝑡)=2𝑖,𝑗=1𝑖𝑗𝐴𝑖+(1+Φ)𝐵𝑖𝐿𝑗𝑥(𝑡).(50)

In this paper, the system parameters are 𝑎=10, 𝑏=2, 𝑐=0.2, 𝑑=0.1, and quantized parameters are 𝛿=0.4. It can be easily seen that both matrices 𝐴1 and 𝐴2 are unstable and the corresponding feedback gain matrix and Lyapunov function matrix of the fuzzy system with quantized controller (49) in Lemma 1 are obtained, respectively:𝐿1=𝐿2=0.98652.0583𝑇.,𝑃=0.09900.12160.12161.0403(51) Moreover, for the quantized control of system (42), we can obtain 𝑀>43.8764Δ from Theorem 1. Then the response of state and control law with quantized control law (49) is showed in Figures 1 and 2, respectively, where the initial condition is 𝑥0=[1.2,0.85]𝑇.

579871.fig.001
Figure 1: System state.
579871.fig.002
Figure 2: System control law.

5. Conclusions

In this paper, we extend the results (see, [5]) to a class of T-S fuzzy nonlinear systems and obtain the conditions of stabilizing a fuzzy nonlinear system via fuzzy quantized feedback. We present new results on the stabilization of fuzzy nonlinear systems by choosing appropriately quantized strategies and applying the Lyapunov function approach. An example has been given to illustrate the effectiveness of the proposed method.

Acknowledgments

The authors are very grateful to all the anonymous reviewers and the editors for their helpful comments and suggestions. This paper was supported by the National Natural Science Foundation of P. R. China under Grant 60874006, Doctoral Foundation of Henan University of Technology under Grant 2009BS048, by the Natural Science Foundation of Henan Province of China under Grant 102300410118, Foundation of Henan Educational Committee under Grant 2011A120003, and Foundation of Henan University of Technology under Grant 09XJC011.

References

  1. D. Liberzon and A. S. Morse, “Basic problems in stability and design of switched systems,” IEEE Control Systems Magazine, vol. 19, no. 5, pp. 59–70, 1999. View at Publisher · View at Google Scholar · View at Scopus
  2. D. Liberzon, Switching in Systems and Control, Birkhäuser, Boston, Mass, USA, 2003.
  3. R. W. Brockett and D. Liberzon, “Quantized feedback stabilization of linear systems,” IEEE Transactions on Automatic Control, vol. 45, no. 7, pp. 1279–1289, 2000. View at Publisher · View at Google Scholar · View at Scopus
  4. H. Ishii and B. A. Francis, “Stabilizing a linear system by switching control with dwell time,” IEEE Transactions on Automatic Control, vol. 47, no. 12, pp. 1962–1973, 2002. View at Publisher · View at Google Scholar · View at Scopus
  5. D. Liberzon, “Hybrid feedback stabilization of systems with quantized signals,” Automatica, vol. 39, no. 9, pp. 1543–1554, 2003. View at Publisher · View at Google Scholar · View at Scopus
  6. H. Ishii and B. A. Francis, “Quadratic stabilization of sampled-data systems with quantization,” Automatica, vol. 39, no. 10, pp. 1793–1800, 2003. View at Publisher · View at Google Scholar · View at Scopus
  7. J. Liu and N. Elia, “Quantized feedback stabilization of non-linear affine systems,” International Journal of Control, vol. 77, no. 3, pp. 239–249, 2004. View at Publisher · View at Google Scholar · View at Scopus
  8. D. F. Delchamps, “Stabilizing a linear system with quantized state feedback,” IEEE Transactions on Automatic Control, vol. 35, no. 8, pp. 916–924, 1990. View at Publisher · View at Google Scholar · View at Scopus
  9. N. Elia and S. K. Mitter, “Stabilization of linear systems with limited information,” IEEE Transactions on Automatic Control, vol. 46, no. 9, pp. 1384–1400, 2001. View at Publisher · View at Google Scholar · View at Scopus
  10. M. Fu and L. Xie, “The sector bound approach to quantized feedback control,” IEEE Transactions on Automatic Control, vol. 50, no. 11, pp. 1698–1711, 2005. View at Publisher · View at Google Scholar · View at Scopus
  11. S. Tatikonda and S. Mitter, “Control under communication constraints,” IEEE Transactions on Automatic Control, vol. 49, no. 7, pp. 1056–1068, 2004. View at Publisher · View at Google Scholar · View at Scopus
  12. C. De Persis, “On feedback stabilization of nonlinear systems under quantization,” in Proceedings of the 44th IEEE Conference on Decision and Control (CDC '05), pp. 7698–7703, December 2005. View at Publisher · View at Google Scholar · View at Scopus
  13. H. Gao and T. Chen, “A new approach to quantized feedback control systems,” Automatica, vol. 44, no. 2, pp. 534–542, 2008. View at Publisher · View at Google Scholar · View at Scopus
  14. F. Ceragioli and C. De Persis, “Discontinuous stabilization of nonlinear systems: quantized and switching controls,” Systems and Control Letters, vol. 56, no. 7-8, pp. 461–473, 2007. View at Publisher · View at Google Scholar · View at Scopus
  15. T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Transactions on Systems, Man and Cybernetics, vol. 15, no. 1, pp. 116–132, 1985. View at Scopus
  16. K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Sets and Systems, vol. 45, no. 2, pp. 135–156, 1992. View at Scopus
  17. K. Tanaka, T. Ikeda, and H. O. Wang, “Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs,” IEEE Transactions on Fuzzy Systems, vol. 6, no. 2, pp. 250–265, 1998. View at Scopus
  18. H. O. Wang, K. Tanaka, and M. F. Griffin, “An approach to fuzzy control of nonlinear systems: stability and design issues,” IEEE Transactions on Fuzzy Systems, vol. 4, no. 1, pp. 14–23, 1996. View at Scopus
  19. H. D. Tuan, P. Apkarian, T. Narikiyo, and Y. Yamamoto, “Parameterized linear matrix inequality techniques in fuzzy control system design,” IEEE Transactions on Fuzzy Systems, vol. 9, no. 2, pp. 324–332, 2001. View at Publisher · View at Google Scholar · View at Scopus
  20. C. S. Tseng, B. S. Chen, and H. J. Uang, “Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model,” IEEE Transactions on Fuzzy Systems, vol. 9, no. 3, pp. 381–392, 2001. View at Publisher · View at Google Scholar · View at Scopus
  21. M. C. M. Teixeira, E. Assunção, and R. G. Avellar, “On relaxed LMI-based designs for fuzzy regulators and fuzzy observers,” IEEE Transactions on Fuzzy Systems, vol. 11, no. 5, pp. 613–623, 2003. View at Publisher · View at Google Scholar · View at Scopus
  22. R. J. Wang, “Observer-based fuzzy control of fuzzy time-delay systems with parametric uncertainties,” International Journal of Systems Science, vol. 35, no. 12, pp. 671–683, 2004. View at Publisher · View at Google Scholar · View at Scopus
  23. C. Lin, Q. G. Wang, T. H. Lee, and Y. He, “Design of observer-based H control for fuzzy time-delay systems,” IEEE Transactions on Fuzzy Systems, vol. 16, no. 2, pp. 534–543, 2008. View at Publisher · View at Google Scholar · View at Scopus
  24. L. Hou, A. N. Michel, and H. Ye, “Some qualitative properties of sampled-data control systems,” IEEE Transactions on Automatic Control, vol. 42, no. 12, pp. 1721–1725, 1997. View at Scopus