New Approach to Stability Analysis and Stabilization of Discrete-Time T-S Fuzzy Descriptor Systems

，

（Space Control and Inertial Technology Research Center，Harbin Institute of Technology，Harbin 150080，China）

New Approach to Stability Analysis and Stabilization of Discrete-Time T-S Fuzzy Descriptor Systems

Mao Wang?，Jia Shi and Kan Liu

（Space Control and Inertial Technology Research Center，Harbin Institute of Technology，Harbin 150080，China）

The problems of stability and state feedback control for a class of discrete-time T-S fuzzy descriptor systems are investigated in this paper.Based on fuzzy Lyapunov function，a set of slack variables is introduced to remove the basic semi-definite matrix inequality condition to check the regularity，causality and stability of discrete-time T-S fuzzy descriptor systems；a new sufficient condition for the discrete-time T-S fuzzy descriptor systems to be admissible is proposed in terms of strict linear matrix inequalities（LMIs）.And a sufficient condition is proposed for the existence of state feedback controller in terms of a set of coupled strict LMIs. Finally，an illustrative example is presented to demonstrate the effectiveness of the proposed approach.

T-S fuzzy descriptor systems；parameter uncertainties；stability；state feedback control；linear matrix inequality（LMI）

1 Introduction

Descriptor systems are also known as singular systems，generalized state-space systems，implicit systems or differential-algebraic systems，which are well-known to be able to describe a wider class of systems，including physical models and non-dynamic constraints，and thus their stability issues are more complicated［1］.Taniguchi and Tanaka et al.［2-3］investigated the fuzzy descriptor systems firstly.A variety of results concerning on the analysis and synthesis for the nonlinear descriptor systems described in T-S model have been proposed［4-8］.In recent years，more and more attention has been paid to derive the strict LMI condition for stability analysis and controller design.

Nevertheless，to date and to the best of our knowledge，there are few research results concerning on discrete-time T-S fuzzy descriptor systems.And main approaches to admissibility analysis for fuzzy descriptor systems have been based on a single Lyapunov function. These methods are basically reduced to the problem of finding a common Lyapunov function for a set of admissible conditions，which may lead to significant conservativeness［9］.On the other hand，the most frequently used admissible condition for discrete-time descriptor systems is derived using a non-strict LMI and has the drawback that it cannot be directly solved using classical numerical tools.

In this paper，it tries to remove the semi-definite matrix inequality in admissible condition.The objective in this paper is to remove such inequalities and establish new strict matrix inequality conditions.Based on the fuzzy Lyapunov function，the admissible condition for the discrete-time T-S fuzzy descriptor systems will be investigated.The fuzzy Lyapunov function shares the same membership function with the T-S fuzzy descriptor model of a descriptor system. Then，based on this result，this paper derives a sufficient condition for state feedback control of the discrete-time T-S fuzzy descriptor systems.We believe our results will find further applications as a fundamental tool for the discrete-time T-S fuzzy descriptor systems.

The notations used throughout the paper are standard.Rndenotes then-dimensional Euclidean space，andRm×nrefers to the set of all real matrices withmrows andncolumns.ATrepresents the transpose of the matrixA，andA-1denotes the inverse ofA．For real symmetric matricesXandY，the notationX≥Y（respectively，X＞Y）means that the matrixX-Yis positive-semi definite（respectively，positive definite）. The notation?is used to indicate the terms which can be induced by symmetry.Iis the identity matrix with appropriate dimensions.

2 Problem Formulation and Preliminaries

In this paper，we consider a class of uncertain discrete-time T-S fuzzy descriptor systems described by，

Rulei：IFTHEN

Then，the final output of the uncertain discrete TS fuzzy descriptor system is inferred as follows：

Considering the following unforced discrete fuzzy descriptor system：

Based on system（4），we will introduce the following definitions.

Definition 1［5］The system（4）is said to be regular if is not identically zero.

Definition 2［5］The system（4）is said to be causal if

Definition 3［5］The system（4）is said to be stable if it is regular and the system trajectory satisfies

under arbitrary switching signals.

Definition 4［5］The system（4）is said to be admissible if it is regular，causal and stable.

Following lemmas are given for developing the main results.We end this section by recalling the following lemmas which will be used to derive our main results in this paper.

3 Main Results

In this section，we focus on the problems of stability and stabilization for the discrete-time T-S fuzzy descriptor system（3）based on the fuzzy Lyapunov function.And the results will be described by strict linear matrix inequalities.Firstly，we present the following theorem，which gives the condition to satisfy that the unforced system is admissible.

Theorem 1System（4）is admissible，if there exist symmetric positive definite matricesand invertible symmetric matricessatisfying the following LMI：

ProofTo facilitate the following discussion，we define

Sincerank（E）＝r，there exist two nonsingular matricessuch that［10］

According to formulas（6）-（10），it follows that

It is obviously that

Then，we will prove that system（4）is stable.

Let

And from formulas（6）and（7）-（9），it is obtained that

Pre-and post-multiply inequality（5）byandrespectively，together with formulas（6），（7）-（9）and（12）-（13），it follows that

Obviously

From formulas（6）and（7）-（9），system（4）is equally transformed into

From system（15），we can see that ifx1（k）→0，ask→∞，thenx2（k）→0，k→∞．So，we only need to prove that the first system of system（15）is stable.

A fuzzy Lyapunov function is constructed as

Then，it is obtained that

From formula（14），we can see thatΔV（k）＜0．Then，it follows that the first system of system（15）is stable.This completes the proof.

Remark 1Theorem 1 provides a sufficient condition for the discrete-time T-S fuzzy descriptor system（4）to be admissible.It is noted that the condition in formula（5）is strict LMI，which is in contrast to that in Refs.［1］and［5］where a non-strict LMI was included.

However，Theorem 1 cannot be used directly to design controllers or observers.In order to design a feedback controller in strict LMI setting，we need a new formulation to Theorem 1.The following lemma is needed to derive the result of controller design.

Theorem 2Letεandδbe given scalars.System（4）is admissible if there exist a nonsingular matrixGand symmetric positive-definite matricesYi，Ψisuch that the following LMI holds

ProofSinceis of full row rank，then

From formula（17），it is obtained that system（4）is admissible if there exist symmetric positivedefinite matricesPi，symmetric nonsingular matricesUiand matrixFsatisfying the following LMI：

Applying the Schur complement lemma，formula（19）is equivalent to

Note that，for any scalarsεandδ，and let0，the following inequalities hold

Then，the above inequalities together with formula（20）results in

The proof is completed.

Remark 2In Theorem 2，scalarsεandδare introduced to reduce the conservativeness.It is easy to see that scalarsεandδmay be positive or negative. Given values of parametersεandδhave an important effect on the feasible solution of the coupled LMIs（16）.If the values of parametersεandδare not chosen properly，the coupled LMIs（16）may have no feasible solutions.

Next，we construct a state-feedback controller which shares the same fuzzy sets in the premise parts with the fuzzy model（1）and has local linear controller in the consequent parts as follows：

Using the feedback control law（21）to system（3）withwe can obtain the closed-loop system

Apply Theorem 2 to the closed-loop system（22）and letHi＝KiG，then we have the following theorem.

Theorem 3Letεandδbe given scalars. Considering the discrete-time T-S fuzzy descriptor system（3）withTi（k）＝0，there exists a state feedback controller of the form（21）such that the closed-loop system（22）is admissible if there exist a non-singular matrixG，matricesHj，and symmetric positive-definite matricesYi，Ψisuch that

and the local gains of the stabilizing state feedback controller are given by：

ProofConsider the closed-loop system（22），according Theorem 2，and letwe have the following result.

Letεandδbe given scalars.Considering system（4），if there exist a nonsingular matrixG，matricesHiand symmetric positive-definite matricesYi，Ψisuch that the following LMI holds

and then system（22）is admissible via the state feedback controllerand the local gains are given by：

This completes the proof［15］.

In the following，we focus on the design of a state feedback control for a class of uncertain discrete-time TS fuzzy descriptor systems such that the resulting closedloop system is admissible for all parameter uncertainties satisfying formula（2）and

Using the feedback control law（21）to system（3），we can obtain the closed-loop system

To facilitate the discussion，we give the following lemma.

Lemma 1Given real matricesD，HandF（k）with appropriate dimensions，andF（k）satisfyingFT（k）F（k）≤I．Then，for any scalarρ＞0，we have

DF（k）H+HTFT（k）DT≤ρ-1DDT+ρHTH

Based on Theorem 3 and Lemma 1，we can get the corresponding robust condition as follows.

Theorem 4Letεandδ＜0 be given scalars. Considering the uncertain discrete-time T-S fuzzy descriptor system（3），there exists a state feedback controller of the form（21）such that the closed-loop system（24）is admissible if there exist positive scalarsρ1ij＞0，ρ2ij＞0，a non-singular matrixG，matricesHj，and symmetric positive-definite matricesYi，Ψisuch that

and the local gains of the stabilizing state feedback controller are given by：Ki＝HiG-1．

4 Numerical Example

In this section，we present an illustrative example to demonstrate the applicability and effectiveness of the proposed approach.

Considering uncertain discrete-time T-S fuzzy descriptor system（3）with parameters as

The aim is to design a state feedback controller（21）to stabilize the system（3）to ensure correspond closed-loop system is robust admissible.

The open system is not admissible.In particular，let the initial conditionx（0）＝［1-0.5 0.5］T，then the state responses for the open-loop are shown in Fig.1.

Fig.1 State responses of the open-loop system

In Ref.［4］，the matrices of the system need to do some transformation before we try to find feedback gain matrices.And the conditions of the given theorems are not given in stric LMIs，which cannot be solved by LMI control toolbox in Matlab.Similarly，theorems in Ref.［5］are also given in non-stric LMIs，which also causes difficulties in finding feedback gain matrices.

The state responses for the closed-loop are shown in Fig.2.It can be clearly observed from the simulation curves that the designed state feedback controller stabilizes the above system，which illustrates the effectiveness of the proposed design approach.

Fig.2 State responses of the closed-loop system

5 Conclusions

The problems of stability，state feedback control for discrete-time T-S fuzzy descriptor systems have been studied.Based on the fuzzy Lyapunov approach，a new sufficient condition for a discrete-time T-S fuzzy descriptor system to be admissible has been proposed in terms of strict LMIs.An explicit construction of a desired state feedback control law has also been given. A numerical example has been used to illustrate the main results.

［1］Dai L.Singular Control Systems.Berlin：Springer-Verlag，Inc.，1989.

［2］Taniguchi T，Tanaka K，Yamafuji K，et al.Fuzzy descriptor systems：stability analysis and design via LMIs. American Control Conference，1999.Proceedings of the 1999.Piscataway：IEEE，1999，3：1827-1831.

［3］Taniguchi T，Tanaka K，Wang H O.Fuzzy descriptor systems and nonlinear model following control.IEEE Transactions on Fuzzy Systems，2000，8（4）：442-452.

［4］Xu S，Song B，Lu J，et al.Robust stability of uncertain discrete-time singular fuzzy systems.Fuzzy Sets and Systems，2007，158（20）：2306-2316.

［5］Zhu B Y，Zhang Q L，Tong S C.Stability criteria for a class of Takagi-Sugeno fuzzy discrete descriptor system. Kongzhi Lilun yu Yingyong／Control Theory＆Applications，2007，24（1）：113-116.

［6］Huang C P.Stability analysis of discrete singular fuzzy systems.Fuzzy Sets and Systems，2005，151（1）：155-165.

［7］Wang C，Huang T，Qiu J.RobustH∞control of discretetime singular T-S fuzzy systems.7th World Congress on Intelligent Control and Automation，2008.WCICA 2008. Piscataway：IEEE，2008.2159-2163.

［8］Huang C P.Stability analysis and controller synthesis for fuzzy descriptor systems.International Journal of Systems Science，2013，44（1）：23-33.

［9］Yuan Y，Zhang Q，Zhang D Q，et al.Admissible conditions of fuzzy descriptor systems based on fuzzy Lyapunov function approach.International Journal of Information and Systems Sciences，2008，4（2）：219-232.

［10］Zhang G，Xia Y，Shi P.New bounded real lemma for discrete-time singular systems.Automatica，2008，44（3）：886-890.

［11］Ma S，Boukas E K.Stability and stabilization for a class of discrete-time piecewise affine singular systems.Proceedings of the 48th IEEE Conference on Decision and Control，2009 held jointly with the 2009 28th Chinese Control Conference.CDC／CCC 2009.Piscataway：IEEE，2009：6395-6400.

［12］Xia Y，Zhang J，Boukas E K.Control for discrete singular hybrid systems.Automatica，2008，44（10）：2635-2641.

［13］Yang C，Zhang Q，Zhou L.Stability Analysis and Design for Nonlinear Singular Systems.Berlin：Springer，2013.

［14］Chadli M，Darouach M.Novel bounded real lemma for discrete-time descriptor systems：Application toH∞control design.Automatica，2012，48（2）：449-453.

［15］Lee H J，Kau S W，Lee C H，et al.H∞control for discrete-time fuzzy descriptor systems.IEEE International Conference on Systems，Man and Cybernetics，2006.SMC＇06.Piscataway：IEEE，2006，6：5059-5064.

TP273

：

：1005-9113（2015）05-0055-06

10.11916／j.issn.1005-9113.2015.05.009

2014-07-14.

Sponsored by the National Natural Science Foundation of China（Grant No.61004038）.

?Corresponging author.E-mail：wangmao0451＠sina.com.