Chun-Ning Yang, Ya-Zhou Yue, and Hui Li

1. Introduction

Comparing with traditional control methods in nonlinear systems, the fuzzy controller has shown better performance that could model uncertainty or nonlinear characteristic[1]. But it is difficult to describe the stability of this nonlinear fuzzy control system. Takagi and Sugeno presented an analytical model of fuzzy system in 1985,called T-S fuzzy model[2]. They analyzed the stability condition and designed the fuzzy controller using the Lyapunov direct method. This T-S model becomes the mostly adopted method in stability analysis of fuzzy control system. The focus of stability analysis about T-S fuzzy system is how to relax the stability conditions and how to find a better controller in an enlarged solution range for the closed-loop system. Tanaka proved a sufficient condition of fuzzy system by seeking a common positive definite matrix P, and the system was asymptotically stable if there existed such matrix[3]. Tanaka also proposed several new relaxed stability conditions and controller design procedures in[3]-[6]. Cao found it was difficult to get a common positive definite matrix and tried to find a set of positive definite matrixpi(i=1, 2,…,r) instead of seeking the single common positive definite matrix P[7]. Yanget al. derived the relaxed stability of the fuzzy control system with uncertain grades of membership which represented the parameter uncertainties of the fuzzy systems[8]. Xieet al. discussed the relaxed stability conditions of continuous T-S fuzzy systems via non-quadratic Lyapunov function technique[9],[10]. They solved the problems (equal to configure the eigenvalues of the closed-loop system) by the stability condition mentioned before, which are all the singly feasible solution. However, sometimes it is not sufficient to all the subsystems, so Chadliet al. studied the stability farther but the conclusion is still not perfect for all subsystems[11]. This paper studies more deeply on the stability research works of [2], [10], and [11]; a different stability condition is proved through Lyapunov stability theory, and a simulation result of flight control system is given comparing with the basic stability conditions in [2].

The paper is organized as follows. Section 2 outlines the T-S fuzzy system model and the basic stability condition by a different analysis method and proves a new stability condition. Section 3 mainly presents the simulation effectiveness of flight control law with the proposed method. Conclusions are given in Section 4.

2. T-S Fuzzy Model and the Different Relaxed Stability Results

2.1 Description of Fuzzy Model

The ordinary T-S continuous fuzzy system is described as

Ifz1(t) isMi1,z2(t) isMi2, …, andzg(t) isMig, then whereMi,j(j=1, 2, …,g) is the fuzzy sets, x(t) is the state vector, u(t) is the input vector, and y(t) is output vector,x(t)∈Rn×1,y(t)∈Rk×1, u(t)∈Rm×1, Ai∈Rn×n, Bi∈Rn×m,Ci∈Rk×n, and Fi∈Rm×n,ris the number of “IF-THEN”rules,nis the number of state variables,mis the number of input variables,kis the number of output variables,andz1(t),z2(t), …,zg(t) are the premise variables.

Given a pair of (x(t), u(t)), if we design fuzzy local state feedback controller based on each fuzzy sub-system(such as parallel distributed compensation[6]) and use fuzzy reasoning method and fuzzification process, the finally output of the fuzzy system (1) is given by

According to the design method of linear system theory,if (Ai, Bi),i=1, 2,…,ris a pair of controllable matrices, the fuzzy system is local controllable, then we can assign the eigenvalues of (Ai, Bi) at will.

The open-loop system (u(t)=0) of (2) is:

Every function described by Aix(t) is called a sub-system,where

and z(t)=[z1(t),z2(t), …,zg(t)],Mi,j(zj(t)) is the grade of the membership function of premise variablezj(t) in setMi,j.

2.2 Stability Condition of T-S Fuzzy System

Takaki and Sugeno have analyzed the stability via Lyapunov’s direct method, proved the stability condition of a fuzzy system, and gave a controller design method[2].Firstly, the system model is described by a T-S fuzzy model,then a designed controller seeks a common symmetric matrix P to satisfy the Lyapunov function; if there exists such matrix P, the fuzzy system is asymptotically stable.

Theorem 1[2]. If there exists a common positive definite matrix Pto all the fuzzy subsystems that satisfies

then the fuzzy system (3) is asymptotically stable in the large to all the subsystems.

The mainly relaxed stability result of continuous fuzzy system was given by Tanaka[7].

Theorem 2[2],[3]. Assume that the number of rules that fire for alltis less than or equal tos, where 1≤s≤r. The equilibrium of the continuous fuzzy control system (2) is asymptotically stable in the large if there exists a common positive definite matrix P and a common positive semidefinite matrix Q such that

for alliandjexcepting the pairs (i,j).

The stability condition of Theorem 1 relies on each subsystem Ai, Biand Fi,i=1, 2, …,r, but the grades of the membership function is not taken into account.

In the work below, we will propose an extension of [8],with a different strategy from [7] and derive a different stability result. Due to

wherehi(z(t))≥0.

whenm≥r,mis an integer variable. Then

Assume that the number of fuzzy rules that fire for alltisr0, 1≤r0≤r, whenm≥r0, (10b) is also equivalent. Because

Substitute (11) to (10) and

Rewrite (2), we get

and then obtain the following theorem.

Theorem 3. Assume that the number of rules that fire for alltisr0, 1≤r0≤r, and is less than or equal tomwhere 1＜m≤r. If there exists the common matrix P＞0 and Q≥0 such that

the equilibrium of the continuous fuzzy control system (13)is asymptotically stable in the large.

Proof.Select the Lyapunov function as bellow:

we get

From the conditions and (8)-(12), we get the following results:

Note that: in fact, the obtained stability condition has a different definition from Theorem 5 in [7], though it is similar in form. Because the solving procedures based on linear matrix inequalities always get a single feasible solution, the proposed design method could give many solution through tuning the value ofm(we can definema positive number indeed, as shown in the following example). So it is helpful to select a favorable result and avoid the non-anticipant eigenvalues of closed-loop fuzzy system.

3. Robust Fuzzy Controller Design in Flight Control System

A flight control system is a typical nonlinear,uncertainty parameter modeled and time varying system. It is not easy to keep stable and robust when designing controller in flight. We select the simple airplane as same as that in [9] and design the fuzzy controller with the proposed method.

The known conditions are: the height of flighth=3 km,and the mach numberMa=0.5, 0.8, and 0.9. The purpose is to design a feedback controller that could stabilize an airplane in the range ofMa∈[1.4, 1.0]. The design procedure of fuzzy control law is shown as below.

We definite three fuzzy subsetsM11=0.5,M21=0.8,M31=0.9, considering the height of flighthis a constant. Fig.1 shows the grade of the fuzzy membership function of the mach number.

Fig.1. Grade of membership function about mach number.

Fuzzy rules according to mach number are taken as the premise variables.

Theith fuzzy rulei: ifMaisMi1, then

The global controller is

where u is the control input of tuning angle of elevator. And

The following results show the solution of Theorem 3(m=1.5):

Fig.2 to Fig.4 show the simulation results under the initial condition x=[0 2/57.3 0 0 0]T. We can see that the results from Theorem 3 are better in performance evaluation, such as the state value, adjustment time and overshoot, and also a better robustness to reject the parameter perturbation. A different damping constant could be obtained through tuning the value of this variable, so it helps to achieve better control effectiveness.

Fig.2. Response of angle of attack α, pitch rate ωZ, and angle of pitch θ.

Fig.3. Response of altitude.

Fig.4. Response of velocity.

4. Conclusions

This paper uses a different strategy to discuss the relaxed stability condition via Lyapunov’s direct method.The derived result is different from the previous papers,which is flexible to gain a better control performance by adjusting a variable, and can be utilized to search a better control rule in large range varied models. The application of the new strategy in the flight control system shows that the obtained stability condition is useful to design a stable controller of a nonlinear system and the controller also has good robustness to reject the parameter perturbation of the plane.

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