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

Abstract

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.

1. 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].

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.

2. System Description and Preliminaries

Consider an uncertain T-S fuzzy continuous model for a nonlinear system as follows: IF𝑧₁(𝑡)is𝑀_𝑖1,…,𝑧_𝑝(𝑡)is𝑀_𝑖𝑝,THEN𝑥̇(𝑡)=𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑡)+𝐵_𝑖+Δ𝐵_𝑖𝑢(𝑡)𝑖=1,…,𝑟, (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)∈ℜ^𝑛 is the state vector, 𝑢(𝑡)∈ℜ^𝑚 is the input vector, 𝐴_𝑖∈ℜ^𝑛×𝑛, 𝐵_𝑖∈ℜ^𝑛×𝑚 are constant real matrices, and 𝑧₁(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: 𝑥̇(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡))𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑡)+𝐵_𝑖+Δ𝐵_𝑖𝑢(𝑡), (2.2) where 𝑧(𝑡)=𝑧₁(𝑡)𝑧₂(𝑡)⋯𝑧_𝑝(𝑡,ℎ_𝑖)(𝑧(𝑡=𝑤))^𝑖(𝑧(𝑡))∑^𝑟_𝑖=1𝑤_𝑖(𝑧(𝑡,𝑤))_𝑖(𝑧(𝑡))=^𝑝_𝑗=1𝑀_𝑖𝑗𝑧_𝑗(𝑡)∀𝑡. (2.3)

The term 𝑀_𝑖1(𝑧_𝑗(𝑡)) is the grade of membership of 𝑧_𝑗(𝑡) in 𝑀_𝑖1Since^𝑟_𝑖∑₌₁𝑤_𝑖(𝑧(𝑡𝑤))≻0_𝑖(𝑧(𝑡))≥0,𝑖=1,2,…,𝑟wehave^𝑟_𝑖∑₌₁ℎ_𝑖(𝑧(𝑡ℎ))=1,_𝑖(𝑧(𝑡))≥0,𝑖=1,2,…,𝑟 (2.4) for all 𝑡.

We have the following property: ℎ̇^𝑟_𝑘=1𝑘(𝑧(𝑡))=0. (2.5) 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 𝑢(𝑡)=−^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡))𝐹_𝑖𝑥(𝑡)−^𝑟_𝜌=1ℎ̇_𝜌(𝑧(𝑡))𝐾_𝜌+𝑅𝑥(𝑡). (2.6) 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 𝑥̇(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡))𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑡). (2.7) By substituting (2.6) into (2.2), the closed-loop fuzzy system can be represented as: 𝑥̇(𝑡)=^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝐴_Δ𝑖−𝐵_Δ𝑖𝐹_𝑗−^𝑟_𝜌=1ℎ̇_𝜌(𝑧(𝑡))𝐵_Δ𝑖𝐾_𝜌+𝑅𝑥(𝑡), (2.8) where 𝐴_Δ𝑖=𝐴_𝑖+Δ𝐴_𝑖 and 𝐵_Δ𝑖=𝐵_𝑖+Δ𝐵_𝑖.

Assumption 2.1. The time derivative of the premises membership function is upper bound such that |ℎ̇_𝑘|≤𝜙_𝑘, for 𝑘=1,…,𝑟, where, 𝜙_𝑘,𝑘=1,…,𝑟 are given positive constants.

Assumption 2.2. The matrices denote the uncertainties in the system and take the form of Δ𝐴_𝑖=𝐷_𝑎𝑖𝐹_𝑎𝑖(𝑡)𝐸_𝑎𝑖,Δ𝐵_𝑖=𝐷_𝑏𝑖𝐹_𝑏𝑖(𝑡)𝐸_𝑏𝑖, (2.9) where 𝐷_𝑎𝑖,𝐷_𝑏𝑖,𝐸_𝑎𝑖, and 𝐸_𝑏𝑖 are known constant matrices and 𝐹_𝑎𝑖(𝑡) and 𝐹_𝑏𝑖(𝑡) are unknown matrix functions satisfying: 𝐹^𝑇_𝑎𝑖(𝑡)𝐹_𝑎𝑖(𝑡)𝐹≤𝐼,∀𝑡,_𝑇𝑏𝑖(𝑡)𝐹_𝑏𝑖(𝑡)≤𝐼,∀𝑡, (2.10) where 𝐼 is an appropriately dimensioned identity matrix.

Lemma 2.3 (Boyd et al. Schur complement [12]). Given constant matrices Ω₁,Ω₂, and Ω₃ with appropriate dimensions, where Ω₁=Ω^𝑇₁ and Ω₂=Ω^𝑇₂, then Ω₁+Ω^𝑇₃Ω₂⁻¹Ω₃≺0 (2.11) if and only if Ω₁Ω^𝑇₃−Ω₂≻0𝑜𝑟−Ω₂Ω₃Ω₁≻0. (2.12)

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 𝑄+𝐻𝐹(𝑡)𝐸+𝐸^𝑇𝐹^𝑇(𝑡)𝐻^𝑇≺0 (2.13) is true, if and only if the following inequality holds for any 𝜆≻0𝑄+𝜆⁻¹𝐻𝐻^𝑇+𝜆𝐸^𝑇𝐸≺0. (2.14)

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.

3. Main Results

Consider the closed-loop system without uncertainties 𝑥̇(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡))𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡). (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: 2𝑥^𝑇(𝑡)𝑀+𝑥̇^𝑇(𝑡)𝜇𝑀×𝑥̇(𝑡)−^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡))𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡)=0. (3.2)

3.1. A PDC Controller with Derivative Membership Function

By substituting (2.6) into (3.1), the closed-loop fuzzy system can be represented as: 𝑥̇(𝑡)=^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝐴_𝑖−𝐵_𝑖𝐹_𝑗−^𝑟_𝑚=1ℎ̇_𝑚(𝑧(𝑡))𝐵_𝑖𝐾_𝑚+𝑅𝑥(𝑡). (3.3) 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_𝑖𝐻^−𝑇, 𝐾_𝜌=𝑉^𝑇_𝜌𝐻^−𝑇, and 𝑅=𝑉^𝑇𝐻^−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. 𝑇_𝑖𝑇≻0,_𝑖(),Λ+𝑌≻0𝑖=1,2,…,𝑟_𝑖𝑖≺0,Λ_𝑖𝑗≺0,Λ_𝑖𝑗=Λ_𝑖𝑗+Λ_𝑗𝑖, (3.4)(3.5) where Λ_𝑖𝑗=𝑇_𝜙−𝐴_𝑖𝐻^𝑇−𝐻𝐴^𝑇_𝑖+𝐵_𝑖𝑆^𝑇_𝑗+𝑆_𝑗𝐵^𝑇_𝑖+𝐵_𝑖𝑉^𝑇_𝜙+𝑉_𝜙𝐵^𝑇_𝑖∗𝑇_𝑖−𝜇𝐴_𝑖𝐻^𝑇−𝐵_𝑖𝑆^𝑇_𝑗−𝐵_𝑖𝑉^𝑇_𝜙+𝐻𝜇𝐻^𝑇,+𝐻𝑉_𝜙=^𝑟_𝜌=1𝜙_𝜌𝑉^𝑇_𝜌+𝑉^𝑇,𝑇_𝜙=^𝑟_𝑘=1𝜙_𝑘𝑇_𝑘+𝑌. (3.6)

Proof of Theorem 3.1. Let’s consider the fuzzy weighting-dependent Lyapunov-Krasovskii functional as: 𝑉(𝑥(𝑡))=^𝑟_𝑘=1ℎ_𝑘(𝑧(𝑡))⋅𝑉_𝑘(𝑥(𝑡)), (3.7) with 𝑉_𝑘(𝑥(𝑡))=𝑥^𝑇(𝑡)(𝑃_𝑘+𝜀𝑋)𝑥(𝑡), 𝑘=1,2,…,𝑟, where 𝑃_𝑘=𝑃^𝑇_𝑘, 𝑋=𝑋^𝑇, 𝜀≥0, and (𝑃_𝑘+𝜀𝑋)≥0.
This candidate Lyapunov function satisfies (i)𝑉(𝑥(𝑡)) is 𝐶¹,(ii)𝑉(0)=0 and 𝑉(𝑥(𝑡))≥0 for 𝑥(𝑡)≠0,(iii)‖𝑥(𝑡)‖→∞⇒𝑉(𝑥(𝑡))→∞.
The time derivative of 𝑉(𝑥(𝑡)) is given by: 𝑉̇(𝑥(𝑡))=^𝑟_𝑘=1ℎ̇_𝑘(𝑧(𝑡))𝑉_𝑘(𝑥(𝑡))+^𝑟_𝑘=1ℎ_𝑘(𝑧(𝑡))𝑉̇_𝑘(𝑥(𝑡)). (3.8)
Adding the null product, then 𝑉̇(𝑥(𝑡))=^𝑟_𝑘=1ℎ̇_𝑘(𝑧(𝑡))𝑉_𝑘(𝑥(𝑡))+^𝑟_𝑘=1ℎ_𝑘(𝑧(𝑡))𝑉̇_𝑘(𝑥(𝑡))+2𝑥^𝑇(𝑡)𝑀+𝑥̇^𝑇(𝑡)×𝑥̇(𝑡)−𝜇𝑀_{𝑟𝑟𝑖=1𝑗=1}ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝐴_𝑖−𝐵_𝑖𝐹_𝑗−^𝑟_𝜌=1ℎ̇_𝜌(𝑧(𝑡))𝐵_𝑖𝐾_𝜌+𝑅𝑥(𝑡). (3.9)
Equation (3.9) can be rewritten as, 𝑉̇(𝑥(𝑡))=Υ₁(𝑥,𝑧)+Υ₂(𝑥,𝑧), (3.10) where Υ₁(𝑥,𝑧)=𝑥^𝑇(𝑡)^𝑟_𝑘=1ℎ̇_𝑘(𝑧(𝑡))⋅𝑃_𝑘+𝜀𝑋𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑀+𝑥̇^𝑇(𝑡)×𝜇𝑀_{𝑟𝑟𝑖=1𝜌=1}ℎ_𝑖(𝑧(𝑡))ℎ̇_𝜌(𝑧(𝑡))𝐵_𝑖𝐾_𝜌𝑥(𝑡),Υ+𝑅₂(𝑥,𝑧)=^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡))2𝑥^𝑇(𝑡)𝑃_𝑖+𝜀𝑋𝑥̇(𝑡)+2𝑥^𝑇(𝑡)𝑀+𝑥̇^𝑇(𝑡)×𝑥̇(𝑡)−𝜇𝑀_{𝑟𝑟𝑖=1𝑗=1}ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝐴_𝑖−𝐵_𝑖𝐹_𝑗𝑥(𝑡). (3.11)
Then, based on Assumption 2.1, an upper bound of Υ₁(𝑥,𝑧) is obtained as: Υ₁(𝑥,𝑧)≤^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡))^𝑟_𝑘=1𝜙_𝑘⋅𝑥(𝑡)^𝑇𝑃_𝑘+𝜀𝑋𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑀+𝑥̇^𝑇(𝑡)×𝜇𝑀_𝑟𝜌=1𝜙_𝜌𝐵_𝑖𝐾_𝜌+𝑅𝑥(𝑡). (3.12)
Based on (2.5), it follows that ∑^𝑟_𝑘=1ℎ̇_𝑘(𝑧(𝑡))(1−𝜀)𝑋=0 where 𝑋 is any symmetric matrix of proper dimension.
Suppose that 𝑋=∑^𝑟_𝑘=1ℎ̇_𝑘(𝑧(𝑡))(1−𝜀)𝑋 and adding 𝑋 to (3.12), then Υ₁(𝑥,𝑧)≤^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡))^𝑟_𝑘=1𝜙_𝑘⋅𝑥(𝑡)^𝑇𝑃_𝑘+𝑋𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑀+𝑥̇^𝑇(𝑡)×𝜇𝑀_𝑟𝜌=1𝜙_𝜌𝐵_𝑖𝐾_𝜌𝑥(𝑡)=+𝑅_𝑟𝑖=1ℎ_𝑖(𝑧(𝑡))𝑥(𝑡)^𝑇𝑃_𝜙𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑀⋅𝐵_𝑖𝐾_𝜙𝑥(𝑡)+2𝑥̇^𝑇(𝑡)𝜇𝑀⋅𝐵_𝑖𝐾_𝜙𝑥(𝑡), (3.13) where 𝑃_𝜙=^𝑟_𝑘=1𝜙_𝑘⋅𝑃_𝑘+𝑋,𝐾_𝜙=^𝑟_𝜌=1𝜙_𝜌𝐾_𝜌+𝑅. (3.14)
Then, 𝑉̇(𝑥(𝑡))≤^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡×𝑥(𝑡)))_𝑇𝑃_𝜙𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑀⋅𝐵_𝑖𝐾_𝜙𝑥(𝑡)+2𝑥̇^𝑇(𝑡)𝜇𝑀⋅𝐵_𝑖𝐾_𝜙𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃_𝑖+𝜀𝑅𝑥̇(𝑡)+2𝑥^𝑇(𝑡)𝑀𝑥̇(𝑡)+2𝑥̇^𝑇(𝑡)𝜇𝑀𝑥̇(𝑡)−2𝑥^𝑇(𝑡)𝑀𝐴_𝑖−𝐵_𝑖𝐹_𝑗𝑥(𝑡)−2𝑥̇^𝑇(𝑡)𝜇𝑀𝐴_𝑖−𝐵_𝑖𝐹_𝑗𝑥(𝑡). (3.15)
Using vector 𝜂^𝑇=[𝑥^𝑇(𝑡)𝑥̇^𝑇(𝑡)]^𝑇, (3.15) can be rewritten as 𝑉̇(𝑥(𝑡))≤^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝜂^𝑇Ξ_𝑖𝑗𝜂=^𝑟_𝑖=1ℎ²_𝑖(𝑧(𝑡))𝜂^𝑇Ξ_𝑖𝑖𝜂+^𝑟_{𝑖=1𝑖≺𝑗}ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝜂^𝑇Ξ_𝑖𝑗+Ξ_𝑗𝑖𝜂, (3.16) where Ξ_𝑖𝑗=𝑃_𝜙−𝑀𝐴_𝑖−𝐵_𝑖𝐹_𝑗−𝐵_𝑖𝐾_𝜙−𝐴_𝑖−𝐵_𝑖𝐹_𝑗−𝐵_𝑖𝐾_𝜙^𝑇𝑀^𝑇∗𝑃_𝑖+𝜀𝑋−𝜇𝑀𝐴_𝑖−𝐵_𝑖𝐹_𝑗−𝐵_𝑖𝐾_𝜙+𝑀^𝑇𝜇𝑀+𝑀^𝑇. (3.17)
If Ξ_𝑖𝑖≺0 and (Ξ_𝑖𝑗+Ξ_𝑗𝑖)≺0, then 𝑉̇(𝑥(𝑡))≺0 and (3.3) is stable. Pre- and postmultiplying Ξ_𝑖𝑖≺0 and (Ξ_𝑖𝑗+Ξ_𝑗𝑖)≺0 by nonsingular matrices diag(𝑀⁻¹,𝑀⁻¹) and diag(𝑀^−𝑇,𝑀^−𝑇), respectively, and pre- and postmultiplying by 𝑀⁻¹ and 𝑀^−𝑇, respectively, then we obtain 𝑀⁻¹𝑃_𝑖+𝜀𝑋𝑀^−𝑇(),𝑀≻0𝑖=1,2,…,𝑟₋₁𝑃_𝑖+𝑋𝑀^−𝑇(),Λ≻0𝑖=1,2,…,𝑟_𝑖𝑖≺0(𝑖=1,2,…,𝑟),Λ_𝑖𝑗≺0(𝑖≺=1,2,…,𝑟), (3.18) where Λ_𝑖𝑗=Λ_𝑖𝑗+Λ_𝑗𝑖, and Λ_𝑖𝑗=𝑀⁻¹𝑃_𝜙𝑀^−𝑇−𝐴_𝑖−𝐵_𝑖𝐹_𝑗−𝐵_𝑖𝐾_𝜙𝑀^−𝑇−𝑀⁻¹𝐴_𝑖−𝐵_𝑖𝐹_𝑗−𝐵_𝑖𝐾_𝜙^𝑇∗𝑀⁻¹𝑃_𝑖+𝜀𝑋𝑀^−𝑇−𝜇𝐴_𝑖−𝐵_𝑖𝐹_𝑗−𝐵_𝑖𝐾_𝜙𝑀^−𝑇+𝑀⁻¹𝜇𝑀^−𝑇+𝑀⁻¹ (3.19) for the following variables definition: 𝐻=𝑀⁻¹,𝑇_𝑖=𝐻𝑃_𝑖+𝜀𝑋𝐻^𝑇,𝑇_𝜙=𝐻𝑃_𝜙𝐻^𝑇,𝑆_𝑗=𝐻𝐹^𝑇_𝑗,𝑉_𝑗=𝐻𝐾^𝑇_𝜌,𝑉=𝐻𝑅^𝑇,𝑌=𝐻𝑋𝐻^𝑇. (3.20) If LMI in holds then the closed-loop continuous fuzzy system (3.3) is asymptotically stable. The control gains are given by 𝐹_𝑖=𝑆^𝑇_𝑖𝐻^−𝑇, 𝐾_𝜌=𝑉^𝑇_𝜌𝐻^−𝑇, and 𝑅=𝑉^𝑇𝐻^−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.

3.2. 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: 𝑥(𝑡)−𝑥(𝑡)→0as𝑡→∞, (3.21) 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: 𝑥̇(𝑡)=^𝑟_𝑗=1ℎ_𝑖(𝑧(𝑡))𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡)+𝐿_𝑗𝐶_𝑖𝑥(𝑡)(𝑡,𝑦(𝑡)−𝑦)=−^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡))𝐶_𝑖𝑥(𝑡), (3.22) where 𝑦(𝑡) denotes the output vector.

We consider the proposed PDC controller given by (2.6): 𝑢(𝑡)=−^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡))𝐹_𝑖𝑥(𝑡)−^𝑟_𝜌=1ℎ̇_𝜌(𝑧(𝑡))𝐾_𝜌+𝑅𝑥(𝑡). (3.23)

Replacing the fuzzy controller (2.6) in fuzzy observer we obtain the closed-loop fuzzy system as: 𝑥̇(𝑡)=^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝐴_𝑖−𝐵_𝑖𝐹_𝑗−^𝑟_𝜌=1ℎ̇_𝜌(𝑧(𝑡))𝐾_𝜌𝑥(𝑡)++𝑅_{𝑟𝑟𝑖=1𝑗=1}ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝐵_𝑖𝐹_𝑗+^𝑟_𝜌=1ℎ̇_𝜌(𝑧(𝑡))𝐾_𝜌𝑒(𝑡)𝑒̇(𝑡)=+𝑅_{𝑟𝑟𝑖=1𝑗=1}ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝐴_𝑖−𝐿_𝑖𝐶_𝑗𝑒(𝑡). (3.24)

The augmented system is represented as follows: 𝑥̇_𝑎(𝑡)=^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝐺_𝑖𝑗𝑥_𝑎(𝑡)=^𝑟_𝑗=1ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝐺_𝑖𝑖𝑥_𝑎(𝑡)+2^𝑟𝑟_{𝑖=1𝑖≺𝑗}ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡𝐺))^𝑖𝑗+𝐺_𝑗𝑖2𝑥_𝑎(𝑡), (3.25) where 𝑥_𝑎(𝑡)=𝑥(𝑡)𝑒(𝑡),𝐺_𝑖𝑗=𝐴_𝑖−𝐵_𝑖𝐹_𝑗−∑^𝑟_𝜌=1ℎ̇_𝜌𝐵_𝑖𝐾_𝜌+𝑅𝐵_𝑖𝐹_𝑗+∑^𝑟_𝜌=1ℎ̇_𝜌𝐵_𝑖𝐾_𝜌+𝑅0𝐴_𝑖−𝐿_𝑖𝐶_𝑗. (3.26)

By applying Theorem 1 in [16] in the augmented system (3.25) we derive the following Theorem.

Theorem 3.4. Under Assumptions 2.1 and 2.2, and for 0≤𝜇≤1, the Takagi-Sugeno fuzzy system (3.1) is stable if there exist positive definite symmetric matrices 𝑃_𝑘,𝑘=1,2,…,𝑟, and 𝑅, matrices 𝐹₁,…,𝐹_𝑟 such that the following LMIs holds. 𝑃_𝑘{},𝑃+𝑅≻0,𝑘∈1,…,𝑟_𝑗𝑃+𝜇𝑅≥0,𝑗=1,2,…,𝑟,_𝜙+𝐺^𝑇_𝑖𝑖𝑃_𝑘+𝜇𝑅+𝑃_𝑘+𝜇𝑅𝐺_𝑖𝑖{},𝐺≺0,𝑖,𝑘∈1,…,𝑟_𝑖𝑗+𝐺_𝑗𝑖2^𝑇𝑃_𝑘+𝜇𝑅+𝑃_𝑘𝐺+𝜇𝑅^𝑖𝑗+𝐺_𝑗𝑖2≺0,𝑓𝑜𝑟𝑖,𝑗,𝑘=1,2,…,𝑟𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑖≺𝑗, (3.27) where 𝐺_𝑖𝑗=𝐴_𝑖−𝐵_𝑖𝐹_𝑗−∑^𝑟_𝜌=1ℎ̇_𝜌𝐵_𝑖𝐾_𝜌+𝑅𝐵_𝑖𝐹_𝑗+∑^𝑟_𝜌=1ℎ̇_𝜌𝐵_𝑖𝐾_𝜌+𝑅0𝐴_𝑖−𝐿_𝑖𝐶_𝑗,𝑃_𝜙=^𝑟_𝑘=1𝜙_𝑘𝑃_𝑘+𝑅. (3.28)

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

4. 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 𝐹_𝑖=𝑆^𝑇_𝑖𝐻^−𝑇, 𝐾_𝜌=𝑉^𝑇_𝜌𝐻^−𝑇, and 𝑅=𝑉^𝑇𝐻^−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. 𝑇_𝑖𝑇≻0,_𝑖+𝑌≻0(𝑖=1,2,…,𝑟_𝑖𝑖),≺0,_𝑖𝑗≺0,_𝑖𝑗=_𝑖𝑗+_𝑗𝑖,𝑇_𝜙=^𝑟_𝑘=1𝜙_𝑘𝑇_𝑘+𝑌, (4.1)(4.2) where Σ_𝑖𝑗=Φ₁₁Φ₁₂𝑀𝐷_𝑎𝑖𝑀𝐷_𝑏𝑖𝐸^𝑇_𝑎𝑖−𝐸_𝑏𝑖𝐹_𝑖+𝐾_𝜙^𝑇Φ₂₂𝜇𝑀𝐷_𝑎𝑖𝜇𝑀𝐷_𝑏𝑖00∗−𝜆𝐼000∗∗−𝜆𝐼00∗∗∗−𝜆⁻¹𝐼0∗∗∗∗−𝜆⁻¹𝐼, (4.3) with Φ₁₁=𝑃_𝜙+𝑀⋅𝐵_𝑖𝐾_𝜙+𝐾^𝑇_𝜙𝐵^𝑇_𝑖𝑀−𝑀𝐺_𝑖𝑖+𝐺^𝑇_𝑖𝑖𝑀^𝑇,Φ₁₂=𝑃_𝑖+𝜀𝑋−𝜇𝐺_𝑖𝑖−𝐵_𝑖𝐾_𝜙^𝑇𝑀^𝑇Φ+𝑀,₂₂=𝜇𝑀+𝑀^𝑇, (4.4) where 𝐺_𝑖𝑗=𝐴_𝑖−𝐵_𝑖𝐹_𝑗, 𝐺_𝑖𝑖=𝐴_𝑖−𝐵_𝑖𝐹_𝑖, and 𝑃_𝜙=∑^𝑟_𝑘=1𝜙_𝑘(𝑃_𝑘+𝑅).

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_𝑎𝑖𝐹(𝑡)𝐸_𝑎𝑖 and 𝐵_𝑖 with 𝐵_𝑖+𝐷_𝑏𝑖𝐹(𝑡)𝐸_𝑏𝑖, we obtain the following inequality: 𝑉̇(𝑥(𝑡))≤^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡×𝑥(𝑡)))_𝑇𝑃_𝜙𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑀⋅𝐵_𝑖𝐾_𝜙𝑥(𝑡)+2𝑥̇^𝑇(𝑡)𝜇𝑀⋅𝐵_𝑖𝐾_𝜙𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃_𝑖+𝜀𝑅𝑥̇(𝑡)+2𝑥^𝑇(𝑡)𝑀𝑥̇(𝑡)+2𝑥̇^𝑇(𝑡)𝜇𝑀𝑥̇(𝑡)−2𝑥^𝑇(𝑡)𝑀𝐴_𝑖−𝐵_𝑖𝐹_𝑗𝑥(𝑡)−2𝑥^𝑇(𝑡)𝑀×𝐷_𝑎𝑖𝐷_𝑏𝑖𝐹_𝑎𝑖(𝑡)00𝐹_𝑏𝑖(𝑡)𝐸_𝑎𝑖−𝐸_𝑏𝑖𝐹_𝑗+𝐾_𝜙𝑥(𝑡)−2𝑥̇^𝑇(𝑡)𝜇𝑀𝐴_𝑖−𝐵_𝑖𝐹_𝑗𝑥(𝑡)−2𝑥̇^𝑇(𝑡)×𝐷𝜇𝑀_𝑎𝑖𝐷_𝑏𝑖𝐹_𝑎𝑖(𝑡)00𝐹_𝑏𝑖(𝑡)𝐸_𝑎𝑖−𝐸_𝑏𝑖𝐹_𝑗+𝐾_𝜙𝑥(𝑡). (4.5)
Using vector 𝜂^𝑇=[𝑥^𝑇(𝑡)𝑥̇^𝑇(𝑡)]^𝑇, (4.5) can be rewritten as 𝑉̇(𝑥(𝑡))≤^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝜂^𝑇Ξ_𝑖𝑗𝜂=^𝑟_𝑖=1ℎ²_𝑖(𝑧(𝑡))𝜂^𝑇Ξ_𝑖𝑖𝜂+^𝑟_{𝑖=1𝑖≺𝑗}ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝜂^𝑇Ξ_𝑖𝑗+Ξ_𝑗𝑖𝜂, (4.6) where Ξ_𝑖𝑗=Ξ_𝑖𝑗+𝑀𝐷_𝑎𝑖𝐷_𝑏𝑖𝐹_𝑎𝑖(𝑡)00𝐹_𝑏𝑖(𝑡)𝐸_𝑎𝑖−𝐸_𝑏𝑖𝐹_𝑗+𝐾_𝜙+𝐷_𝑎𝑖𝐷_𝑏𝑖𝐹_𝑎𝑖(𝑡)00𝐹_𝑏𝑖(𝑡)𝐸_𝑎𝑖−𝐸_𝑏𝑖𝐹_𝑗+𝐾_𝜙^𝑇𝑀^𝑇∗𝜇𝑀𝐷_𝑎𝑖𝐷_𝑏𝑖𝐹_𝑎𝑖(𝑡)00𝐹_𝑏𝑖(𝑡)𝐸_𝑎𝑖−𝐸_𝑏𝑖𝐹_𝑗+𝐾_𝜙0,Ξ_𝑖𝑗=Ξ_𝑖𝑗+𝑀𝐷_𝑎𝑖𝐷_𝑏𝑖𝜇𝑀𝐷_𝑎𝑖𝐷_𝑏𝑖𝐹_𝑎𝑖(𝑡)00𝐹_𝑏𝑖(𝑡)𝐸_𝑎𝑖0−𝐸_𝑏𝑖𝐹_𝑗+𝐾_𝜙0+𝐸^𝑇_𝑎𝑖−𝐸_𝑏𝑖𝐹_𝑗+𝐾_𝜙^𝑇𝐹00^𝑎𝑖(𝑡)00𝐹_𝑏𝑖(𝑡)^𝑇𝐷^𝑇_𝑎𝑖𝑀^𝑇𝜇𝐷^𝑇_𝑎𝑖𝑀^𝑇𝐷^𝑇_𝑏𝑖𝑀^𝑇𝜇𝐷^𝑇_𝑏𝑖𝑀^𝑇. (4.7)
Then, based on Lemma 2.4, an upper bound of Ξ_𝑖𝑗 obtained as: Ξ_𝑖𝑗=Ξ_𝑖𝑗+𝜆⁻¹𝑀𝐷_𝑎𝑖𝐷_𝑏𝑖𝜇𝐷_𝑎𝑖𝜇𝐷_𝑏𝑖𝐷^𝑇_𝑎𝑖𝜇𝐷^𝑇_𝑎𝑖𝐷^𝑇_𝑏𝑖𝜇𝐷^𝑇_𝑏𝑖𝑀^𝑇𝐸+𝜆^𝑇𝑎𝑖−𝐸_𝑏𝑖𝐹_𝑗+𝐾_𝜙^𝑇𝐸00^𝑎𝑖0−𝐸_𝑏𝑖𝐹_𝑗+𝐾_𝜙0≺0 (4.8) by Schur complement, we obtain, Φ₁₁Φ₁₂𝑀𝐷_𝑎𝑖𝑀𝐷_𝑏𝑖𝐸^𝑇_𝑎𝑖−𝐸_𝑏𝑖𝐹_𝑗+𝐾_𝜙^𝑇Φ₂₂𝜇𝑀𝐷_𝑎𝑖𝜇𝑀𝐷_𝑏𝑖00∗−𝜆𝐼000∗∗−𝜆𝐼00∗∗∗−𝜆⁻¹𝐼0∗∗∗∗−𝜆⁻¹𝐼≺0, (4.9) with Φ₁₁=𝑃_𝜙+𝑀⋅𝐵_𝑖𝐾_𝜙+𝐾^𝑇_𝜙𝐵^𝑇_𝑖𝑀−𝑀𝐴_𝑖−𝐵_𝑖𝐹_𝑗+𝐴_𝑖−𝐵_𝑖𝐹_𝑗^𝑇𝑀^𝑇,Φ₁₂=𝑃_𝑖+𝜀𝑋−𝜇𝐴_𝑖−𝐵_𝑖𝐹_𝑗−𝐵_𝑖𝐾_𝜙^𝑇𝑀^𝑇Φ+𝑀,₂₂=𝜇𝑀+𝑀^𝑇. (4.10) If the LMI 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_𝑖𝐻^−𝑇, 𝐾_𝜌=𝑉^𝑇_𝜌𝐻^−𝑇, and 𝑅=𝑉^𝑇𝐻^−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 𝑇_𝑖𝑇≻0,_𝑖+(),Λ𝑌≻0𝑖=1,2,…,𝑟_𝑖𝑖≺0,Λ_𝑖𝑗≺0,Λ_𝑖𝑗=Λ_𝑖𝑗+Λ_𝑗𝑖,𝑇_𝜙=^𝑟_𝑘=1𝜙_𝑘𝑇_𝑘+𝑌,𝑉_𝜙=^𝑟_𝜌=1𝜙_𝜌𝑉^𝑇_𝜌+𝑉^𝑇, (4.11)(4.12) where Λ_𝑖𝑗=Φ₁₁Φ₁₂𝐷_𝑎𝑖𝐷_𝑏𝑖𝐻𝐸^𝑇_𝑎𝑖−𝑆_𝑗𝐸^𝑇_𝑏𝑖−𝑉_𝜙𝐸^𝑇_𝑏𝑖Φ₂₂𝜇𝐷_𝑎𝑖𝜇𝐷_𝑏𝑖00∗−𝜆𝐼000∗∗−𝜆𝐼00∗∗∗−𝜆⁻¹𝐼0∗∗∗∗−𝜆⁻¹𝐼, (4.13) with Φ₁₁=𝑇_𝜙−𝐴_𝑖𝐻^𝑇−𝐻𝐴^𝑇_𝑖+𝐵_𝑖𝑆^𝑇_𝑗+𝑆_𝑗𝐵^𝑇_𝑖+𝐵_𝑖𝑉^𝑇_𝜙+𝑉_𝜙𝐵^𝑇_𝑖,Φ₁₂=𝑇_𝑖−𝜇𝐴_𝑖𝐻^𝑇−𝐵_𝑖𝑆^𝑇_𝑗−𝐵_𝑖𝑉^𝑇_𝜙+Φ𝐻,₂₂=𝜇𝐻^𝑇+𝐻. (4.14)

Proof of Theorem 4.2. Consider (4.8) and pre- and postmultiplying by 𝑀⁻¹ and 𝑀^−𝑇, respectively, then we obtain Λ_𝑖𝑗=Λ_𝑖𝑗+𝜆⁻¹𝐷_𝑎𝑖𝐷_𝑏𝑖𝜇𝐷_𝑎𝑖𝜇𝐷_𝑏𝑖𝐷^𝑇_𝑎𝑖𝜇𝐷^𝑇_𝑎𝑖𝐷^𝑇_𝑏𝑖𝜇𝐷^𝑇_𝑏𝑖+𝜆𝑀⁻¹𝐸^𝑇_𝑎𝑖−𝐸_𝑏𝑖𝐹_𝑗+𝐾_𝜙^𝑇𝐸00^𝑎𝑖0−𝐸_𝑏𝑖𝐹_𝑗+𝐾_𝜙0𝑀^−𝑇≺0, (4.15) with Λ_𝑖𝑗 defined by (3.5).
By Schur complement, we obtain, Φ₁₁Φ₁₂𝐷_𝑎𝑖𝐷_𝑏𝑖𝑀⁻¹𝐸^𝑇_𝑎𝑖−𝑀⁻¹𝐸_𝑏𝑖𝐹_𝑗+𝐾_𝜙^𝑇Φ₂₂𝜇𝐷_𝑎𝑖𝜇𝐷_𝑏𝑖00∗−𝜆𝐼000∗∗−𝜆𝐼00∗∗∗−𝜆⁻¹𝐼0∗∗∗∗−𝜆⁻¹𝐼≺0, (4.16) with Φ₁₁=𝑀⁻¹𝑃_𝜙𝑀^−𝑇−𝐴_𝑖−𝐵_𝑖𝐹_𝑗−𝐵_𝑖𝐾_𝜙𝑀^−𝑇−𝑀⁻¹𝐴_𝑖−𝐵_𝑖𝐹_𝑗−𝐵_𝑖𝐾_𝜙^𝑇,Φ₁₂=𝑀⁻¹𝑃_𝑖+𝜀𝑋𝑀^−𝑇−𝜇𝐴_𝑖−𝐵_𝑖𝐹_𝑗−𝐵_𝑖𝐾_𝜙𝑀^−𝑇+𝑀⁻¹,Φ₂₂=𝜇𝑀^−𝑇+𝑀⁻¹ (4.17) for the following variables definition: 𝐻=𝑀⁻¹,𝑇_𝑖=𝐻𝑃_𝑖+𝜀𝑋𝐻^𝑇,𝑇_𝜙=𝐻𝑃_𝜙𝐻^𝑇,𝑆_𝑗=𝐻𝐹^𝑇_𝑗,𝑉_𝑗=𝐻𝐾^𝑇_𝜌,𝑉=𝐻𝑅^𝑇,𝑌=𝐻𝑋𝐻^𝑇. (4.18)
If LMI in holds then the closed-loop continuous fuzzy system (2.8) is asymptotically stable.
The control gains are given by 𝐹_𝑖=𝑆^𝑇_𝑖𝐻^−𝑇, 𝐾_𝜌=𝑉^𝑇_𝜌𝐻^−𝑇, and 𝑅=𝑉^𝑇𝐻^−𝑇. This completes the proof.

5. 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.

Example 5.1. Consider the following continuous T-S fuzzy system: 𝑥̇(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡))𝐴_𝑖𝑥(𝑡), (5.1) with 𝑟=2;𝐴₁=3.6−1.66.2−4.3,𝐴₂=,𝐵−𝑎−1.66.2−4.3₁=−0.45−3,𝐵₂=,−𝑏−3 (5.2) where 𝑎∈[0,25], 𝑏∈[0,2], considering 𝜇=0.04 and 𝜙_1,2=1.
The proposed approach (Theorem 3.4) gives less conservative stabilization conditions (Figure 1) than some recent results provided by [14, 17].

Figure 1 

Feasible area provided by [17] (+), [14] (◯) and Theorem 3.1 (•).

Example 5.2. Consider the uncertain continuous T-S fuzzy system given by (2.8) with 𝑟=2;𝐴₁=3.6−1.66.2−4.3,𝐴₂=,𝐵−𝑎−1.66.2−4.3₁=−0.45−3,𝐵₂=,𝐷−𝑏−3_𝑎1=𝐸_𝑎1=0.3000,𝐷_𝑏1=𝐸_𝑏1=0,𝐷0.3_𝑎2=𝐸_𝑎2=−0.2000.3,𝐷_𝑏2=𝐸_𝑏2=−0.20.3 (5.3) for 𝑎=1, 𝑏=0.5, considering 𝜇=0.04, 𝜆=0.1 and 𝜙_1,2=1, we find the following gains values: 𝐹₁=[−5.03401.5230],𝐹₂=[],𝐾3.7868−0.8082₁=[−180.233465.7330],𝐾₂=[−180.949465.9073],𝑅=[201.9216−71.7345]. (5.4)

6. 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.