Local H ∞ Control for Continuous-Time T-S Fuzzy Systems via Generalized Non-Quadratic Lyapunov Functions

: This paper further develops a relaxed method to reduce conservatism in H ∞ feedback control for continuous-time T-S fuzzy systems via a generalized non-quadratic Lyapunov function. Different from the results of some exisiting works, the generalized H ∞ state feedback controller is designed. The relaxed stabilization conditions are obtained by applying Finsler’s lemma with the homogenous polynomial multipliers, and the H ∞ performance is acquired by solving an optimization problem. In addition, the proposed method could be expanded to handle other control problems for fuzzy systems. Two examples are given to show the validity of the proposed results.


Introduction
Owing to its better approximation properties, the Takagi-Sugeno (T-S) system [1] has attracted much attention from different communities. The system comprises a set of linear models and normalized membership functions (MFs).
The analysis and synthesis of fuzzy systems have been widely studied, such as stability analysis [2][3][4][5], observer design [6,7], filter [8][9][10], etc. Due to some limitations of the proposed methods in the existing results, researchers are seeking new methods to obtain better results, such as larger domain of attraction or stability region, and better H∞ or H 2 performance. To reduce conservatism, many methods usually focus on the form of Lyapunov functions (LFs), the structure of slack variables and analysis of MFs and its derivatives.
Considering the drawbacks of common quadratic LFs, the complex LFs such as the fuzzy Lyapunov functions (FLFs) [11], the non-quadratic Lyapunov functions (NQLFs) [12], the line-integral FLFs [13], and homogeneous polynomial Lyapunov functions (HPLFs) [14], the homogeneous polynomial non-quadratic Lyapunov functions (HPNQLFs) [15] were proposed successively to further reduce the conservatism of stability conditions. For instance, Ref. [12] proposed new relaxed linear matrix inequality (LMI) conditions based on NQLFs. Ref. [14] first provided local asymptotic stability conditions and obtained different estimations of the attraction domain by using HPLFs. Ref. [15] gave the asymptotically necessary and sufficient stability conditions for discrete-time fuzzy systems via non-parallel distributed compensation law (NPDCL) with HPNQLFs. In addition, Ref. [12] designed an NPDCL to outperform previous results. By using the multi-indexed matrix approach, a homogeneous polynomial nonparallel distribution compensator (HPNQDC) was designed in [16], but the Lyapunov matrix used is linearly dependent on MFs. Ref. [17] further generalized previous HPNQDC and enlarged the stabilization region. Based on congruence transformation and Polya's theorem, inner and outer slack variables were introduced in [18][19][20][21] to obtain a less (1) The generalized NQLFs and NPCL depending on multi-index MFs are designed, including that found in [27] and double-fuzzy-summation in [20] as a special case, and more variables are introduced to reduce the conservatism. (2) The new LMIs conditions, which reduce the adjusted parameters to be calculated [7] and avoid redundant restrictions such as LFs matrices, slack variables in [23,28], are obtained to bound the time derivatives of MFs with disturbance. (3) The extended stabilization conditions for H∞ performance are obtained by polynomial technology. As q increases, conservatism of obtained conditions will reduce, and the proposed method can be generalized to handle other cases, such as output feedback controller design [5], finite-time annular domain stability [29],mean-square strong stability [30].

The T-S Fuzzy System
The T-S fuzzy system (1) in C x (C x = {x : |x i | ≤ θ xi , i = 1, 2, · · · , n}) can be obtained by applying the local approximation method.
where the state x(t) ∈ R n×1 , the measured output y(t) ∈ R n y ×1 , the external disturbance ω(t) ∈ R n w ×1 , the control input u(t) ∈ R n u ×1 and the controlled output z(t) ∈ R n z ×1 . A i ∈ R n×n , B 1i ∈ R n×n ω , B 2i ∈ R n×n u , C 1i ∈ R n z ×n , C 2i ∈ R n y ×n , D 1i ∈ R n z ×n ω , D 2i ∈ R n z ×n u and D 3i ∈ R n y ×n ω . h i (x) is the normalised MF satisfying ∑ r i=1 h i (x(t)) = 1.
The objective of this paper is to design a controller u(t) such that the system (1) with ω = 0 is locally asymptotically stable, and guarantees t 0 z(t) T z(t)dt ≤ γ 2 t 0 ω(t) T ω(t)dt (t > 0) under zero initial conditions.

Property 3 ([19]).
Let Ω i q , i q ∈ I + q be the matrices of proper dimensions. Then, Ω h q < 0 holds if

Main Results
, then, the closed-loop system (1) in Ω x (:= {x : x T P −1 h q x ≤ ρ}) is locally asymptotically stable with a disturbance attenuation level γ under the controller (9) Proof. Designing the generalized NQLF candidate as and where On one hand, Γ ≤ 0 is guaranteed by the following formula: Multiplying left and right by diag{P h q , I, I} and using the relationṖ −1 Starting from the characteristics where ∑ r j=1ḣ j = 0, thus ∑ r j=1ḣ j M h + q+1 = 0 for any sym- with suitable size, adding ∑ r j=1ḣ j M h + q+1 to (13) and using Property 2, we get where ).
For (17), one has where Utilizing Schur complement to (19) and multiplying left and right by diag{P h q , I}, one has  Therefore, if (7) and (8) hold, we have Γ < 0, which means ∞ 0 z T zdt ≤ γ 2 ∞ 0 ω T ωdt. The proof is completed. [20,23] were bounded without disturbance and some assumed conditions such as matrix P (P i > λ 2

Remark 1. TDMFs in
x +λ 2 k 2β k I ) and the free variable S (S h = P h ) in [28,33] were limited. This paper eliminates these restrictions or assumptions.

Remark 2.
Due to ( ∂h j ∂x ) T ∂h j ∂x ≤ σ 2 j being restricted, Theorem 1 means a local result. The Lyapunov level Ω x (x T P −1 h q x ≤ ρ) is an estimated region for H ∞ performance, which must be contained in Θ x . Applying the Lagrange multiplier method, we get Thus, according to Property 3, conditions (22) guarantee (21).
Remark 3. There are two parameters µ j and ρ in Theorem 1 to be searched. By solving the following optimization problem, one can get H∞ performance with given µ j and ρ.
Proof. The proof is similar to that of Theorem 1, and is thus omitted here.

Simulation Example
All the experiments were performed on a computer with an Intel(R) Core(TM)i5-7200U CPU @ 2.50 GHz 2.70 GHz, 12 GB(RAM), using Matlab2017a. Example 1. Considering the following nonlinear system form [33].
The above system is expressed as a T-S fuzzy system with two rules in the C x = {x : |x i | ≤ π 2 , i = 1, 2}. One gets the system matrices: Since ∂h 1 ∂x = 2sin(x 1 )cos(x 1 ) and ∂h 2 ∂x = −2sin(x 1 )cos(x 1 ), thus . The parameters required by Theorem 1 in the region Ω x (x T P −1 h q x ≤ ρ = 1) are as follows: All results are solved by function minx in Matlab Toolbox. The minimal γ obtained are shown in Table 1 under different methods. Notice, a line indicates that SeDuMi solver is unable to converge to a solution. This clearly shows that the Theorem 1 proposed in this paper is less conservative than other methods. Moreover, as q increases, better results are obtained.
On the other hand, when systems include unknown parameters, such as B 22 = −2.06 + λ −1 , we compare the minimal γ with the different λ shown in Table 2. Although the minimal γ increases with the parameter λ, Theorem 1's results are better than other ones under the same parameter λ. Choosing the initial four points four trajectory curves in the domain of attraction Ω x are shown in Figure 1, which are asymptotically driven to the origin under the controller. Therefore, this shows that the designed controller is effective. Example 2. The state equation of motion for the inverted pendulum controlled by a separately excited direct current (DC) motor from [35].
where x 1 ∈ [−π π] denotes the angle, x 2 denotes the angular velocity, x 3 represents the current of the DC motor, u is the control input voltage. The parameter values of the system are m = 1 kg, l = 1 m, g = 9.8 m/s 2 , N = 10, K 2 = 0.1 Vs/rad, K 1 = 0.1 Nm/A and 0.6 Ω ≤R ≤ 3.5 Ω.
Equation (29) can be converted to We consider z = x 1 + x 2 . Applying local approximation method, the T-S fuzzy model (1) is given with Here, we consider that the derivative of MFs (µ 1 = µ 2 ) in Theorem 1 affects the conservativeness of the different method. The results are shown in Table 3. It can be seen that [23,27,33] cannot converge to a solution when µ = 1, but Theorem 1 can. Moreover, as µ i increases, γ decreases, and Theorem 1's results with q = 2 are better than other ones under the same parameter µ. Choosing q = 2, µ = 100, we get γ = 6.3759 × 10 −6 and control gain matrixes as follow: From the starting point x 0 = [3, −3] T , the response trajectories of the states, Lyapunov function and control input are shown in Figures 2-4, respectively. Therefore, the closed-loop system (30) is locally asymptotically stable with disturbance attenuation level γ under the controller (9).

Conclusions
This paper has presented the local H∞ control for continuous-time T-S fuzzy systems via generalized nonquadratic Lyapunov functions. Using polynomial technology, relaxed LMIs conditions satisfying H∞ performance are obtained, which are easily solved by the optimization problem. The simulation examples provided show the validity of the proposed method.
Furthermore, the method proposed in this paper could be generalized to handle problems regarding output feedback controllers, filters or observers. Now, our group is considering the output feedback controller design for nonlinear systems, finite-time boundedness, finite-time annular domain stability and mean-square strong stability for stochastic systems using our method.