Robust Stabilization for Continuous Takagi-Sugeno Fuzzy System Based on Observer Design

This paper investigates the influence of a new parallel distributed controller PDC on the stabilization region of continuous Takagi-Sugeno T-S fuzzy models. Using a nonquadratic Lyapunov function, a new sufficient stabilization criterion is established in terms of linear matrix inequality. The criterion examines the derivative membership function; an approach to determine state variables is given based on observer design. In addition, a stabilization condition for uncertain system is given. Finally, numeric simulation is given to validate the developed approach.


Introduction
Fuzzy control systems have experienced a big growth of industrial applications in the recent decades, because of their reliability and effectiveness.Many researches have investigated the Takagi-Sugeno models 1-3 during the last decades.Two classes of Lyapunov functions are used to analyze these systems: quadratic Lyapunov functions and nonquadratic Lyapunov ones which are less conservative than the first class.Many researches have investigated with nonquadratic Lyapunov functions 4-10 .
As the information about the time derivatives of membership function is considered by the PDC fuzzy controller, it allows the introduction of slack matrices to facilitate the stability analysis.The relationship between the membership function of the fuzzy model and the fuzzy controllers is used to introduce some slack matrix variables.The boundary information of the membership functions is brought to the stability condition and thus offers some relaxed stability conditions 6 .In order to determine the state variables many approaches of observer design are given 9-11 .

Mathematical Problems in Engineering
In this paper, new stabilization conditions for Takagi-Sugeno uncertain fuzzy models based on the use of fuzzy Lyapunov function are presented.This criterion is expressed in terms of linear matrix inequalities LMIs which can be efficiently solved by using various convex optimization algorithms 12, 13 .The presented method is less conservative than existing results.
The organization of the paper is as follows.In Section 2, we present the system description and problem formulation and we give some preliminaries which are needed to derive results.Section 3 will be concerned with stabilization analysis for continuous T-S fuzzy systems by the use of new PDC controller based on derivative membership functions.An observer approach design is derived to estimate state variables.In Section 4, a new stabilization condition for uncertain system is given.Next, a new robust PDC controller design approach is presented.Illustrative examples are given in Section 5 for a comparison of previous results to demonstrate the advantage of the proposed method.Finally Section 6 makes the conclusion.
Notation.Throughout this paper, a real symmetric matrix S > 0 denotes S being a positive definite matrix.The superscript "T" is used for the transpose of a matrix.

System Description and Preliminaries
Consider an uncertain T-S fuzzy continuous model for a nonlinear system as follows:

2.1
where M ij i 1, 2, . . ., r, j 1, 2, . . ., p is the fuzzy set and r is the number of model rules, x t ∈ n is the state vector, u t ∈ m is the input vector, A i ∈ n×n , B i ∈ n×m are constant real matrices, and z 1 t , . . ., z p t are known premise variables.ΔA i , and ΔB i are time-varying matrices representing parametric uncertainties in the plant model.These uncertainties are admissibly norm-bound and structured.The final outputs of the fuzzy systems are: where M ij z j t ∀t.

2.3
The term M i1 z j t is the grade of membership of we have for all t.
We have the following property: This study investigates the PDC controller influence on the closed-loop stability region and gives robustness analysis of uncertain Takagi-Sugeno fuzzy system.Thus, we consider a PDC fuzzy controller which examines the derivative membership function and it is given by The fuzzy controller design consists to determine the local feedback gains F i , K ρ , and R in the consequent parts.The state variables are determined by an observer and are detailed in the next section.The open-loop system is given by By substituting 2.6 into 2.2 , the closed-loop fuzzy system can be represented as: if and only if is true, if and only if the following inequality holds for any λ 0 The aim of the next section is to find conditions for the stabilization of the closed-loop T-S fuzzy system by using the Lyapunov theory.

Main Results
Consider the closed-loop system without uncertainties 3.1 In order to give stability conditions, the slack matrix variables and the membership function boundary Mozelli et al. 14 are used.Consider the following null product that will serve stability analysis purposes:

A PDC Controller with Derivative Membership Function
By substituting 2.6 into 3.1 , the closed-loop fuzzy system can be represented as: The next Theorem gives sufficient conditions to guarantee stability of system 3.3 .
Theorem 3.1.Under Assumptions 2.1 and 2.2, and for μ 0, ε ≥ 0, the Takagi-Sugeno fuzzy system 3.1 is stabilizable with the PDC controller 2.6 , with gains given by F i S T i H −T , K ρ V T ρ H −T , and R V T H −T , if there exist positive definite symmetric matrices T k , k 1, 2, . . ., r, Y , and any matrices H, S i , V ρ , and V with appropriate dimensions such that the following LMIs hold. where

3.6
Proof of Theorem 3.1.Let's consider the fuzzy weighting-dependent Lyapunov-Krasovskii functional as: The time derivative of V x t is given by: Adding the null product, then 3.9 Equation 3.9 can be rewritten as, where

3.11
Then, based on Assumption 2.1, an upper bound of Υ 1 x, z is obtained as:

3.12
Based on 2.5 , it follows that r k 1 ḣk z t 1 − ε X 0 where X is any symmetric matrix of proper dimension.
Suppose that X r k 1 ḣk z t 1 − ε X and adding X to 3.12 , then

3.15
Using vector η T x T t ẋT t T , 3.15 can be rewritten as where

3.18
where Λ ij Λ ij Λ ji , and for the following variables definition: T i H P i εX H T , T φ HP φ H T , S j HF T j , V j HK T ρ , V HR T , Y HXH T .

3.20
If LMI in 3.4 holds then the closed-loop continuous fuzzy system 3.3 is asymptotically stable.The control gains are given by F i S T i H −T , K ρ V T ρ H −T , and R V T H −T .This completes the proof.
Remark 3.2.The selection of φ k given in Assumption 2.1 is performed by using a simple procedure given in 15 .
Remark 3.3.The major contribution of the Theorem 3.1 is represented by the proposed PDC controller given by 2.6 .The contribution appears in the gains K ρ R introduced in the controller term based on derivative membership functions.The stabilization condition proposed is less conservative than some of those in the literature, as is shown in the example below.

Observer Design
In order to determine state variables of the system, this section gives a solution by the means of fuzzy observer design.The following condition is to be satisfied by the observer: where x t denotes the state vector estimated by a fuzzy observer.This condition guarantees that the steady-state error between x t and x t converge to 0 and we denote this error by e t x t − x t .A stabilizing observer-based controller can be formulated as follow:

3.22
where y t denotes the output vector.We consider the proposed PDC controller given by 2.6 :

3.23
Replacing the fuzzy controller 2.6 in fuzzy observer 3.22 we obtain the closed-loop fuzzy system as:

3.24
The augmented system is represented as follows:

3.26
By applying Theorem 1 in 16 in the augmented system 3.25 we derive the following Theorem.

3.28
Proof of Theorem 3.4.The result follows immediately from the proof of Theorem 1 in 16 .

Robust Stability Condition with PDC Controller
Consider the uncertain closed-loop system 2.8 .A sufficient robust stability condition is given as follows.
Theorem 4.1.Under Assumptions 2.1 and 2.2, and for μ 0, ε ≥ 0, the Takagi-Sugeno fuzzy system 2.2 is stabilizable with the PDC controller 2.6 , with gains given by and R V T H −T , if there exist positive definite symmetric matrices T k , k 1, 2, . . ., r, Y , and any matrices H, S i , V ρ , and V with appropriate dimensions such that the following LMIs hold. where with

Mathematical Problems in Engineering
Proof.Proof of Theorem 4.1 The result follows immediately from the proof of Theorem 3.1 by replacing in the matrix inequality A i with A i D ai F t E ai and B i with B i D bi F t E bi , we obtain the following inequality:

4.5
Using vector η T x T t ẋT t T , 4.5 can be rewritten as where

Mathematical Problems in Engineering 13
Then, based on Lemma 2.4, an upper bound of Ξ ij obtained as: by Schur complement, we obtain,

4.10
If the LMI 4.1 holds then the system 2.8 is stable.This completes the proof.
The following theorem gives sufficient conditions for robust PDC controller design.
Theorem 4.2.Under Assumptions 2.1 and 2.2, and for μ 0, ε ≥ 0, the Takagi-Sugeno robust fuzzy system 2.2 is stabilizable with the PDC controller 2.6 , with gains given by F i S T i H −T , Mathematical Problems in Engineering K ρ V T ρ H −T , and R V T H −T , if there exist positive definite symmetric matrices T k , k 1, 2, . . ., r, Y , and any matrices H, S i , V ρ , and V with appropriate dimensions such that the following LMIs hold.

4.12
where with

4.14
Proof of Theorem 4.2.Consider 4.8 and pre-and postmultiplying by M −1 and M −T , respectively, then we obtain with Λ ij defined by 3.5 .

Mathematical Problems in Engineering 15
By Schur complement, we obtain, with for the following variables definition: S j HF T j , V j HK T ρ , V HR T , Y HX H T .

4.18
If LMI in 4.11 holds then the closed-loop continuous fuzzy system 2.8 is asymptotically stable.
The control gains are given by F i S T i H −T , K ρ V T ρ H −T , and R V T H −T .This completes the proof.

Numerical Examples
In order to show the improvements of the proposed approaches over some existing results, in this section, we present a numerical example in which we present the feasible area for a T-S fuzzy system.Indeed, we compare the proposed fuzzy Lyapunov approaches Theorem 3.4 with result provided by 17 , and in 14, Theorem 6 .A second example is given to improve the given gains of robust PDC controller.The proposed approach Theorem 3.4 gives less conservative stabilization conditions Figure 1 than some recent results provided by 14

Conclusion
This paper provided new conditions for the stabilization with a class of PDC controller of Takagi-Sugeno fuzzy systems in terms of a combination of the LMI approach and the use of nonquadratic Lyapunov function as fuzzy Lyapunov function.In addition, the time derivative of membership function is considered by the PDC fuzzy controller and the slack matrix variables are introduced in order to facilitate the stability analysis.An approach to design an observer is derived in order to estimate variable states.In addition, a new condition of the stabilization of uncertain system is given in this paper.
The stabilization condition proposed in this paper is less conservative than some of those in the literature, which has been illustrated via examples.
Δi A i ΔA i and B Δi B i ΔB i .

12 Lemma 2.4 Peterson and
Hollot 2 .Let Q Q T , H, E, and F t satisfying F T t F t ≤ I are appropriately dimensional matrices then the following inequality , 17 .Example 5.2.Consider the uncertain continuous T-S fuzzy system given by 2.8 with , b 0.5, considering μ 0.04, λ 0.1 and φ 1,2 1, we find the following gains values: