# Delay-Dependent Finite-TimeH∞ Controller Design for a Kind of Nonlinear Descriptor Systems via a T-S Fuzzy Model

Authors: Baoyan Zhu; Jingjun Zhang; Daqing Zhang; Aibin Liu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Delay-dependent finite-timeH∞ controller design problems are investigated for a kind of nonlinear descriptor system via a T-S fuzzy model in this paper. The solvable conditions of finite-time H∞ controller are given to guarantee that the loop-closed system is impulse-free and finite-time bounded and holds the H∞ performance to a prescribed disturbance attenuation level γ. The method given is the ability to eliminate the impulsive behavior caused by descriptor systems in a finite-time interval, which confirms the existence and uniqueness of solutions in the interval. By constructing a nonsingular matrix, we overcome the difficulty that results in an infeasible linear matrix inequality (LMI). Using the FEASP solver and GEVP solver of the LMI toolbox, we perform simulations to validate the proposed methods for a nonlinear descriptor system via the T-S fuzzy model, which shows the application of the T-S fuzzy method in studying the finite-time control problem of a nonlinear system. Meanwhile the method was also applied to the biological economy system to eliminate impulsive behavior at the bifurcation value, stabilize the loop-closed system in a finite-time interval, and achieve a H∞ performance level.


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## Body

## 1. Introduction

The Lyapunov stability theory can be regarded as a relatively mature field, which focuses on the asymptotic stability of a system state on an infinite interval. However, it is noteworthy that the behavior of the system over a fixed finite-time interval needs to be considered in many practical problems, and this just leads to the finite-time or short time stability issues. In short, a system is said to be finite-time stable, if its state keeps within the prescribed bounds on a fixed time interval for given some initial conditions [1, 2]. It has been shown that the finite-time stability issues play an important role in the theory and practical application; for instance, the trajectory control of space vehicles, the disease control, the population quantity control, and so on will lead to a discussion on the finite-time stability or boundedness. This fact also motivates many scholars to study the finite-time stability or boundedness and robust control problems [1–6].As is known to all, time-delays exist inevitably in many practical engineering systems, such as nuclear reactors, electronics, chemical processes, and hydraulic and biological systems. They can frequently lead to instability and poor performance. So there has been an increasing interest in the stability analysis and stabilization for time-delay systems in the last decades [4, 5, 7–15]. The results and discussions may be classified as delay-independent or -dependent. Delay-dependent discussions which take into account the size of delays are of less conservative, especially in the case of small time-delays [7]. The main issue of delay-dependent is that the presence of an integral term normally does not found in LMIs. There may be several ways to deal with the integral term, such as Jesson’s inequality [10, 11], Moon et al.’s inequality [12], Park’s inequality [13], Wirtinger’s inequality [14], and the free-weighting-matrix approach [15]. However, each method has its own benefits and shortcomings in the conservative viewpoint. According to the characteristics of our research issue, we will investigate the delay-dependent finite-time H∞ control problems of T-S descriptor system in this paper by using Jesson’s inequality approach to overcoming the conservativeness.The stability or robust stabilization of a nonlinear descriptor system has been an acknowledged difficult problem [16], especially with regard to the finite-time robust stabilization issues. Fortunately, Taniguchi et al. established a T-S fuzzy descriptor system model and proposed an innovative and simple method to solve control problems of a kind of nonlinear descriptor systems [17]. Many nonlinear dynamic systems are represented as T-S fuzzy systems, which is a universal fuzzy approximator [18–26]. The descriptor systems have been one of the major research fields of control theory, which is due to the fact that a descriptor system may exhibit impulsive or noncausal behavior along with the derivatives of these impulses. It just results in the fact that the discussion of descriptor systems is much more complex and more challenging than ordinary systems [7]. However, it is noteworthy that singular systems not only describe actual physical systems better but also are also more widely utilized than ordinary ones. So in the past decades, descriptor systems have attracted much attention [7, 17, 19, 25–30].On the other hand, the problem ofH∞ control has been a topic of growing interestover the past decades. A great number of results on H∞ control for the state-space and descriptor systems have been reported in literatures [29, 31–35]. However, the related research on finite-time H∞ control of fuzzy descriptor systems is relatively less.In view of the above-mentioned facts, this paper investigates the delay-dependent finite-timeH∞ controller design problems for a kind of nonlinear descriptor system via a T-S fuzzy model. The solvable conditions of finite-time H∞ controller are given to guarantee that the loop-closed system is impulse-free and finite-time bounded and holds the H∞ performance to a prescribed disturbance attenuation level γ. The method given is the ability to eliminate the impulsive behavior caused by descriptor systems in a finite-time interval, which confirms the existence and uniqueness of solutions in the interval. By constructing a nonsingular matrix, we overcome the difficulty that results in an infeasible linear matrix inequality (LMI). By using the FEASP solver and GEVP solver of the LMI toolbox together with SIMULINK simulation technology, we perform simulations to validate the proposed methods for a nonlinear descriptor system via the T-S fuzzy model, which shows the application of the T-S fuzzy method in studying the finite-time control problem for a nonlinear system. Meanwhile the method was also applied to the biological economy system to eliminate impulsive behavior at the bifurcation value, stabilize the loop-closed system in a finite-time interval, and achieve a H∞ performance level.This paper is organized as follows. Section2 provides preliminaries and the formulation. The finite-time H∞ control design scheme is proposed in Section 3. In Section 4, two design examples are given to show the advantage of developed results. Finally, concluding remark is made in Section 5.


## 2. System Formulation and Preliminaries

In the following, we consider the delay-depended finite-timeH∞ control problems for a kind of nonlinear descriptor systems via T-S fuzzy models. The nonlinear dynamical descriptor system can be described by the following fuzzy IF-THEN rules: (1)Model rulei: IFξ1tisMi1andξ2tisMi2⋯andξptisMip,THENEx˙t=Aixt+Aidxt-d+Biut+D1iwt,zt=Cixt+D2iwt,i=1,2,…,r,xt=φt,t∈-d,0,where r is the number of IF-THEN rules and Mij(j=1,2,…,p) are fuzzy sets; x(t)∈Rn is the state; u(t)∈Rm is control input; z(t)∈Rq is the controlled output; w(t)∈Rp is the exogenous disturbance and satisfies wT(t)w(t)≤cw, where cw is a positive number; the matrix E∈Rn×n may be singular; and we let rank⁡E=r≤n. The matrices Ai, Aid, Bi, Ci, D1i, and D2i are known real constant matrices with appropriate dimensions. The delay d is a positive constant. φ(t) is a continuous initial function on [-d,0]. ξ1t,ξ2t,…,ξp(t) are premise variables and write ξt=[ξ1t,ξ2t,…,ξpt]T.By using the fuzzy inference method of singleton fuzzifier and weighted average defuzzifier, the overall fuzzy model can be inferred as follows:(2)Ex˙t=A~hxt+A~hdxt-d+B~hut+D~1hwt,zt=C~hxt+D~2hwt,xt=φt,t∈-d,0,where(3)A~h=∑i=1rhiξtAi,A~hd=∑i=1rhiξtAid,B~h=∑i=1rhiξtBi,D~1h=∑i=1rhiξtD1i,C~h=∑i=1rhiξtCi,D~2h=∑i=1rhiξtD2i,as follows: Mij(ξj(t)) is the grade of membership of ξj(t) in Mij. It can be seen that βiξt=∏j=1pMijξjt≥0, i=1,2,…,r. Thus, hi(ξ(t))=βiξt/∑i=1rβiξt≥0, ∑i=1rhiξt=1, t∈[0,T]. In this paper, let ∑i=1rβiξt>0, t∈[0,T].Note thatrank⁡E=r<n. Without loss of generality, decompose matrices in (2) as follows:(4)E=Ir000,A~h=A~h11A~h12A~h21A~h22,A~hd=A~hd11A~hd12A~hd21A~hd22,where A~h11,A~hd11∈Rr×r, A~h22∈R(n-r)×(n-r), and A~hd22∈R(n-r)×(n-r).Now we consider the systems(5)Ex˙t=A~hxt,(6)Ex˙t=A~hxt+A~hdxt-d,xt=φt,t∈-d,0.Definition 1 (see [7]).
System (5) is said to be regular if det⁡(sE-A~h) is not identically 0 for t∈[0,T]. The system is said to be impulse-free if degsdet⁡(sE-A~h)=rank⁡E, for t∈[0,T].Lemma 2 (see [7]).
System (6) is regular and impulse-free, if system (5) is regular and impulse-free. The descriptor system (5) is impulse-free if and only if A~h22 for t∈[0,T] is nonsingular.Lemma 3 (Jensen’s inequality [10]).
For any positive symmetric constant matrixZ∈Rn×n, scalars a, b satisfying a<b, a vector function x in [a,b]→Rn such that the integrations concerned, are well defined, and then(7)∫abxTtZxtdt≥1b-a∫abxTtdtZ∫abxtdt.Lemma 4 (see [35]).
Given any real matricesX, Y, and W with appropriate dimensions such that Y>0 and is symmetric, then one has(8)XTYX+XTW+WTX+WTY-1W≥0.Definition 5 (see [4, 5] (finite-time stability)).
The fuzzy descriptor time-delay system (6) without impulse is said to be finite-time stable with respect to c1,c2,T, 0<c1<c2, if supt∈[-d,0]φT(t)φ(t)≤c1 implies xT(t)ETEx(t)<c2, ∀t∈[0,T].Remark 6.
In Definition5, the finite-time stability of dynamic state is defined for the time-delay descriptor system. In some actual systems, we should focus on the dynamic state and the static state stability or focus only on the dynamic state stability and so forth. For example, in power systems and economic systems, there exists the question on the dynamic state stability analysis to be considered. When we need to consider the finite-time stability of dynamic and static state, we should redefine it (see the following Definition 7). In this paper, we discuss only the finite-time stability analysis and H∞ control problems for the dynamic state of a fuzzy descriptor system with time-delay.Definition 7 (finite-time stability).
The fuzzy descriptor system (6) without impulse is said to be finite-time bounded with respect to c1,c2,T,R, 0<c1<c2, R>0, if supt∈[-d,0]⁡φT(t)Rφ(t)≤c1 implies xT(t)Rx(t)<c2, ∀t∈[0,T].Amato et al. pointed out in [1] that the Lyapunov asymptotic stability and the finite-time stability are independent concepts: a system which is finite-time stable may be not Lyapunov asymptotic stable, while a Lyapunov asymptotic stable system may be not finite-time stable if its state exceeds the prescribed bounds during the transients.


## 3. Finite-TimeH∞ Control

For the fuzzy model (2), we consider the following fuzzy controller via the PDC:(9)ut=∑i=1rhiξtKixt.We apply this controller u(t) to system (2) which will result in the following closed-loop systems:(10)Ex˙t=∑i=1r∑j=1rhiξthjξtAi+BiKjxt+Aidxt-d+D1iwt≔A~hkxt+A~hdxt-d+D~1hwt,xt=φt,t∈-d,0,zt=∑i=1rhiξtCixt+D2iwt≔C~hxt+D~2hwt,t∈0,T.The so-called finite-timeH∞ fuzzy control is to design a state feedback controller (9) such that the closed-loop system (10) is impulse-free and finite-time bounded and holds the H∞ performance to the prescribed disturbance attenuation level γ. And the controller is called a finite-time H∞ fuzzy controller. We also say that a finite-time H∞ fuzzy control problem is solvable.Definition 8 (finite-time boundedness).
The time-delay descriptor system without impulse (10) is said to be finite-time boundedness with respect to c1,c2,T,cw where 0≤c1<c2 and cw≥0, if supt∈[-d,0]⁡φT(t)φ(t)≤c1 implies xT(t)ETEx(t)<c2, ∀t∈[0,T], ∀w(t):wT(t)w(t)≤cw.Theorem 9.
Given positive numbersd0 and γ, then for any delay 0<d≤d0, system (10) is impulse-free and finite-time bounded with respect to c1,c2,T,cw and holds the H∞ performance level to the prescribed disturbance attenuation level γ, if there exist matrices P=P10P2P3  (P1∈Rr×r,P3∈Rn-r×n-r,P1>0,P3≠0), P^=P100P^3(P^3∈R(n-r)×(n-r),P^3≠0), Ki, positive matrices Q=Q1Q2Q2TQ3, Z=Z1Z2Z2TZ3, and a positive scalar ρ such that the following set of matrix inequalities hold:(11)Ψii<0,i=1,2,…,r,(12)Ψij+Ψji<0,j<i=1,2,…,r,(13)λmax⁡P1c1+dλmax⁡Qc1+d2λmax⁡Zc0+γ2e-ρTcwT<c2e-ρTλmin⁡P^,where(14)
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sup
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x
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.Proof.
We prove first that the systemEx˙(t)=A~hkx(t) is impulse-free. Thus the system Ex˙(t)=A~hkx(t)+A~hdx(t-d) is impulse-free by Lemma 2. This also implies that it is regular [7]. To achieve this goal, we set Ψhl=∑i=1r∑j=1rhi(ξ(t))hj(ξ(t))Ψij. Then it can be derived by inequalities (11) and (12) along with Ψhl=∑i=1rhi(ξ(t))hi(ξ(t))Ψii+∑j<i∑hi(ξ(t))hj(ξ(t))Ψij+Ψji that(15)
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.This implies A~hkTP+PTA~hk+Q-1/dETZE-ρETP<0, ∀t∈[0,T]. We decompose the matrix A~hk into A~hk11A~hk12A~hk21A~hk22 which is compatible with that of A~h in (4). Then it turns out with some classical manipulations that ⊗⊗⊗A~hk22TP3+P3TA~hk22+Q3<0, ∀t∈[0,T], in which ⊗ stands for the matrix that is not relevant in the discussion. It follows that A~hk22TP3+P3TA~hk22<0, ∀t∈[0,T], which implies that A~hk22 is nonsingular for ∀t∈[0,T]. So the system Ex˙(t)=A~hkx(t) is impulse-free from Lemma 2.
Then we proof that system (10) is finite-time bounded. To this end, we consider the following Lyapunov-Krasovskii functional candidate:(17)Vxt=xTtETPxt+∫t-dtxTsQxsds+∫-d0∫t+θtx˙TsETZEx˙sdsdθ.
It can be obtained thatETP=PTE≥0 from the structures of the matrices E and P. Then it is shown by classical computations that the derivative of the functional (17) along the trajectories of system (10) satisfies(18)V˙xt=x˙TtETPxt+xTtPTEx˙t+xTtQxt-xTt-dQxt-d+dx˙TtETZEx˙t-∫-d0x˙Tt+θETZEx˙t+θdθ=A~hkxt+A~hdxt-d+D~1hwtTPxt+xTtPTA~hkxt+A~hdxt-d+D~1hwt+xTtQxt-xTt-dQxt-d+dA~hkxt+A~hdxt-d+D~1hwtTZA~hkxt+A~hdxt-d+D~1hwt-∫-d0x˙Tt+θETZEx˙t+θdθ.By Jensen’s inequality in Lemma 3 and considering (18) together with inequality (16), then it can be reduced that(19)
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.Therefore, the following inequalities can be derived:(20)V˙xt-ρVxt<γ2e-ρTwTtwt.Before and after multiplying (20) by e-ρt, then integrating the inequality form 0 to t leads to(21)Vxt<eρtVx0+γ2e-ρTcwT,∀t∈0,T.It is easy to see that ETP=ETP^E=P1000 from the structures of the matrices E, P, and P^. Then we can derive that(22)Vx0=xT0ETPx0+∫-d0xTsQxsds+∫-d0∫θ0x˙TsETZEx˙sdsdθ=xT0ETP^Ex0+∫-d0xTsQxsds+∫-d0∫θ0x˙TsETZEx˙sdsdθ≤λmax⁡P1c1+dλmax⁡Qc1+d2λmax⁡Zc0,λmin⁡P^xTtETExt≤xTtETP^Ext=xTtETPxt<Vxt.Thus from inequalities (22) and (13), we can obtain that(23)xTtETExt<eρT1λmin⁡P^λmax⁡P1c1+dλmax⁡Qc1+d2λmax⁡Zc0+γ2e-ρTcwT<c2,∀t∈0,T.By Definition 8, it is easy to see that system (10) is finite-time bounded.
Finally we prove that system (10) also achieves H∞ performance to the prescribed disturbance attenuation level γ.
Noting inequality (19) and by some algebraic manipulations, then it is derived that(24)
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t
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t
-
d
w
t
-
γ
2
e
-
ρ
T
w
T
t
w
t
.By inequalities (15) and (24), it is obtained that(25)V˙xt-ρVxt<-zTtzt+γ2e-ρTwTtwt.Before and after multiplying (25) by e-ρt, then integrating the inequality from 0 to T leads to (26)0≤e-ρTVxT=e-ρTVxT-Vx0<∫0T-e-ρtzTtzt+γ2e-ρTe-ρtwTtwtdt.Thus, we can obtain that (27)e-ρT∫0TzTtztdt≤∫0Te-ρtzTtztdt≤e-ρT∫0Te-ρtγ2wTtwtdt≤e-ρTγ2∫0TwTtwtdt.That is, ∫0TzT(t)z(t)dt≤γ2∫0TwT(t)w(t)dt. Thus system (10) holds the H∞ performance for the prescribed disturbance attenuation level γ. This completes the proof.Remark 10.
In the proceeding of the finite-timeH∞ controller design, one of the difficulties is a nonpositive matrix produced from a descriptor system, which will result in an infeasible linear matrix inequality. For overcoming the difficulty, we construct a nonsingular matrix P^=P100P^3 satisfying ETP=ETP^E, which can provide the feasibility conditions to solve the finite-time boundedness problem of a descriptor system via LMIs. And the proof is simpler and easier to understand than one of the existing literatures on the descriptor systems.Remark 11.
In the following, we give the design method of the finite-timeH∞ fuzzy controller design via LMIs. In order to simplify the process of controller design, we select the positive matrix Z of Theorem 9 as Z100Z3. The main issue of the delay-dependent robust control is that the presence of inverse matrices of Q, Z normally does not found LMIs. This problem was successfully solved By Lemma 4 and its deformation.Theorem 12.
Given positive numbersd0, ρ, and γ, then for any delay 0<d≤d0, system (10) is impulse-free and finite-time bounded with respect to c1,c2,T,cw and holds H∞ performance for the prescribed disturbance attenuation level γ, if there exist a common matrix X=P-T=P1-1-P1-1P2TP3-T0P3-T≔X1X20X3, matrices Ni=KiP-1, nonsingular matrix X^=P^-1=P1-100P^3-1=P1-100P^3-1≔X100X^3, and positive matrices Q=Q1Q2Q2TQ3, Z-=Z-1=Z1-100Z3-1≔Z-100Z-3, such that the following set of LMIs hold:(28)Ψ-ii<0,i=1,2,…,r,(29)Ψ-ij+Ψ-ji<0,j<i=1,2,…,r,(30)X1IIλ1I>0,(31)Q<λ2In,(32)λ3IIIZ->0(33)X100X^3<I,(34)λ1c1+dλ2c1+d2λ3c0+γ2e-ρTcwTeρT<c2,where(35)
Ψ
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i
j
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N
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ρ
T
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D
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0
0
A
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B
i
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j
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i
d
D
1
i
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1
d
Z
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T
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D
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I
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Z
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<
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.Proof.
Before and after multiplying inequality (11) by diag⁡P-T,I,I,I,I and its transpose respectively, then it is obtained that(36)
P
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P
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ρ
P
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P
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P
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P
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1
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Q
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1
d
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E
0
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γ
2
e
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T
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d
Z
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.By Lemma 4, it can be seen that -1/dP-TETZEP-1≤1/dETZ-1E+1/dETEP-1+1/dP-TETE. Therefore, inequality (36) gives(37)
P
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P
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.Now, set X=P-T, Ni=KiP-1. Then, by inequality (37), it can be obtained that(38)
X
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.From Lemma 4, it can be seen that(39)1dXET0000Z0E000+1d0ET000ZEXT0000≤1dXET0000Z·Z-1ZEXT0000+1d0ET000Z0E000.Thus, inequalities (38) and (39) give that(40)
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.Then, by Schur complement, inequality (40) is equivalent to (41)
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.By Lemma 4, we have -Q-1≤Q-2I. Let Z-=Z-1. Then, inequality (41) implies Ψ-ii<0, i=1,2,…,r.
Similarly, we can prove thatΨij+Ψji<0 only if Ψ-ij+Ψ-ji<0. The proof is omitted to save space.
Inequality (30) is equivalent to X1>1/λ1I, which implies λmax(X1-1)=λmax(P1)<λ1. Inequality (31) gives λmax(Q)<λ2. Inequality (32) is equivalent to 0<λ3I-Z--1, which results in λmax(Z)<λ3. Inequality (33) is equivalent to P^-1<I, which can derive λmin(P^)>1. Thus, from the above inequalities and (34), it can be derived that(42)λmax⁡P1c1+dλmax⁡Qc1+d2λmax⁡Zc0+γ2e-ρTcwTeρT≤λ1c1+dλ2c1+d2λ3c0+γ2e-ρTcwTeρT<c2<c2λmin⁡P^.That is, condition (13) of Theorem 9 holds only if all conditions of inequalities (30), (31), (32), (33), and (34) in Theorem 12 are met. This completes the proof.By using Theorem12, we can also obtain the following finite-time H∞ controller design method for system (10) without time-delay.Corollary 13.
Given positive numbersρ and γ, then system (10) without time-delay is impulse-free and finite-time bounded with respect to c1,c2,T,cw and holds H∞ performance for the prescribed disturbance attenuation level γ, if there exist a common matrix X=P-T=P1-1-P1-1P2TP3-T0P3-T≔X1X20X3, matrices Ni=KiP-1, and nonsingular matrices X^=P^-1=P1-100P^3-1=P1-100P^3-1≔X100X^3, such that the following set of LMIs hold:(43)Ψ^ii<0,i=1,2,…,r,Ψ^ij+Ψ^ji<0,j<i=1,2,…,r,X1IIλ1I>0,X100X^3<I,λ1c1+γ2e-ρTcwTeρT<c2,where(44)Ψ^ij=XAiT+NjTBiT+AiXT+BiNj-ρXETD1iXCiTD1iT-γ2e-ρTID2iTCiXTD2i-I.


## 4. Design Examples

Example 1.
We know that it is more convenient to express practical models as the descriptor systems than ordinary ones such as a biological economic model with differential and algebraic equation established in [27]. The following economic model is expressed by a descriptor system instead of the ordinary one as a result of an algebraic equation:(45)ς˙1t=-αβr2-ηcpς1t+ας2t-cpς3t-ης12t-ς1tς3t+b11wt,ς˙2t=βς1t-r2ς2t,0=pαβr2-r1-β-ηcpς1t+pς1tς3t-m,where m=0 is a bifurcation value. Equation (45) describes a biological economic model obtained by the transformation. It is a differential algebraic equation, where the meanings of parameters can be found in [27]. When m transitions 0 from negative to positive, an impulse phenomenon is formed. Carry outing a control for system (45), then the following system is given:(46)ς˙1t=-αβr2-ηcpς1t+ας2t-cpς3t-ης12t-ς1tς3t+b11wt,ς˙2t=βς1t-r2ς2t,0=pαβr2-r1-β-ηcpς1t+pς1tς3t+ut,zt=ς1t.Equation (46) can be represented by the following fuzzy descriptor system:(47)Ex˙t=∑i=12λiς1tAixt+D1iwt+Biut,zt=∑i=12λiς1tCixt,where (48)E=100010000,A1=-αβr2-ηcp+ηlα-cp+lβ-r20pαβr2-r1-β-ηcp0-pl,A2=-αβr2-ηcp-ηlα-cp-lβ-r20pαβr2-r1-β-ηcp0pl,D11=D12=b1100,B1=B2=001,D21=D22=0,C1=C2=100,λ1ς1t=121-ς1tl,λ2ς1t=121+ς1tl,ς1t<l,l>0,xt=ς1t,ς2t,ς3tT.Select the coefficients of the matrices α=0.15, β=0.5, r1=0.2, r2=0.1, η=0.001, p=1, c=40, b11=0.1, and l=10 in [27]. Let T=2, c1=0.2, cw=1, ρ=0.02, and λ1=0.01. By using Corollary 13 and the FEASP solver and GEVP solver of the LMI toolbox, we can obtain min⁡c2=0.2100. For c2=22, we can obtain the feasible solutions of a finite-time fuzzy controller with the norm H∞ bound γ=0.01 as follows:(49)X=103·0.0006-0.00020.0143-0.00020.00070.000000-1.6642,X^3=-1.6617·103,N1=104·-4.97840.0000-1.7224,N2=104·-8.3353-0.00001.6061,ut=121-ς1t10·104·-8.7569-2.11060.0010xt+121+ς1t10·104·-1.4594-0.3518-0.0001xt.Let w(t)=e-0.1tsin⁡t. Then the trajectory of xT(t)ETEx(t) for the nonlinear system approximated by a fuzzy system is shown in Figure 1.Figure 1
The trajectory ofxT(t)ETEx(t) for the nonlinear system approximated by the fuzzy system.Example 2.
Consider the following nonlinear descriptor system with time-delay:(50)ς˙1t=-αβr2-ηcpς1t+ας2t-cpς3t-ης12t-ς1tς3t+d11wt+τ11ς1t-d,ς˙2t=βς1t-r2ς2t,0=pαβr2-r1-β-ηcpς1t+pς1tς3t+b11ut,zt=c11ς1t.Defining x(t)=[ς1(t),ς2(t),ς3(t)]T, ςi(t)∈R1(i=1,2,3). The above equation can be rewritten as(51)100010000x˙t=-αβr2-ηcp-ης1tα-cp-ς1tβ-r20pαβr2-r1-β-ηcp0pς1txt+τ1100000000xt-d+00b11ut+d1100wt,zt=c1100xt.Choose, respectively, the membership functions of the fuzzy sets M1, M2 as λ1(ς1(t))=1/21-ς1(t)/l, λ2(ς1(t))=1/21+ς1(t)/l, and ς1(t)<l, l>0. The following T-S fuzzy model exactly represents the dynamics of nonlinear descriptor system (50) or (51):(52)Model Rule1: Ifς1tisM1,ThenEx˙t=A1xt+A1dxt-d+B1ut+D11wt,zt=C1xt+D21wt,xt=φt,t∈-d,0,Model Rule2: Ifς1tisM2,ThenEx˙t=A2xt+A2dxt-d+B2ut+D12wt,zt=C2xt+D22wt,xt=φt,t∈-d,0,where(53)E=100010000,A1=-αβr2-ηcp+ηlα-cp+lβ-r20pαβr2-r1-β-ηcp0-pl,A2=-αβr2-ηcp-ηlα-cp-lβ-r20pαβr2-r1-β-ηcp0pl,A1d=A2d=τ1100000000,D11=D12=d1100,B1=B2=00b11,D21=D22=0,C1=C2=c1100.Equation (50) can be represented by the following fuzzy descriptor system:(54)Ex˙t=∑i=12λiς1tAixt+Aidxt-d+Biut+D1iwt,zt=∑i=12λiς1tCixt+D2iut.Select the coefficients of the matrices α=0.15, β=0.5, r1=0.2, r2=0.1, η=0.001, p=1, c=220, τ11=2.0034·10-5, d11=-4.2344·10-5, b11=-0.0725, c11=0.0317, and l=220. Let d=0.0001, T=2, c0=0c1=0.02, cw=2, and ρ=0.00002. By using Theorem 12 and the FEASP solver and GEVP solver of the LMI toolbox, we can obtain min⁡c2=20. For c2=32, we can obtain the feasible solutions of a finite-time fuzzy controller with the H∞ norm bound γ=0.0001 as follows:(55)X1=10-7·0.2887-0.0000-0.00000.2900,X3=X^3=-0.0122,N1=103·0.00000.00004.3297,N2=103·-8.3353-0.00004.3320,Q=0.9973-0.00000-0.00000.99730000.9973,Z=107·0.0000-0.00000-0.00000.00000002.6151.The finite-time H∞ fuzzy controller is as follows:(56)ut=121-ς1t220105·0.02450.0000-3.5489xt+121+ς1t220105·0.28280.0000-3.5508xt.Let w(t)=e-tsin⁡t. Then the trajectories of xT(t)ETEx(t) for the approximated nonlinear system are shown in Figure 2.Figure 2
The trajectory ofxT(t)ETEx(t) for the nonlinear system approximated by the fuzzy system.


## 5. Conclusions

This paper investigates the delay-dependent finite-timeH∞ controller design problems for a kind of nonlinear descriptor system via a T-S fuzzy model. The solvable conditions of finite-time H∞ controller are given to guarantee that the loop-closed system is impulse-free and finite-time bounded and holds the H∞ performance to a prescribed disturbance attenuation level γ. The method given is the ability to eliminate the impulsive behavior caused by descriptor systems in a finite-time interval, which confirms the existence and uniqueness of solutions in the interval. We perform simulations to validate the proposed methods for a nonlinear descriptor system via the T-S fuzzy model, which shows the application of the T-S fuzzy method in studying the finite-time control problem of a nonlinear system. Meanwhile the method was also applied to the biological economy system to eliminate impulsive behavior at the bifurcation value, stabilize the loop-closed system in a finite-time interval, and achieve a H∞ performance level.



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*Source: 546835-2015-10-28.xml*