Robust H ∞ Control of Uncertain TS Fuzzy Time-Delay System : A Delay Decomposition Approach

This paper is concerned with the problem of robust H ∞ control for a class of uncertain time-delay fuzzy systems with normbounded parameter uncertainties. By utilizing the instrumental idea of delay decomposition, the decomposed Lyapunov-Krasovskii functional is introduced to uncertain T-S fuzzy system, and some delay-dependent conditions for the existence of robust controller are formulated in the form of linearmatrix inequalities (LMIs).When these LMIs are feasible, a controller is presented. A numerical example is given to demonstrate the effectiveness of the proposed method.


Introduction
It is well known that time delay is built-in features in various nonlinear systems such as tandem mills, remote control systems, long transmission lines in pneumatic systems, and chemical system.The time delay is recognized to be a source of instability and performance deterioration of control systems.Therefore, stability analysis and controller synthesis for time-delay system have been one of the most hot research area in the control community over the past years [1][2][3][4][5][6][7][8][9][10][11][12][13][14].
Fuzzy systems in the form of the Takagi-Sugeno (T-S) model have attracted rapidly growing interest in recent years.It has been shown that the T-S model method is a simple and effective way to represent complex nonlinear systems by a set of simple local linear dynamic systems with their linguistic description [12,[15][16][17][18][19].Over the past few years, most work has been devoted to analysis and synthesis of T-S fuzzy control systems.See the survey papers [16,17] and the reference citied therein for the most recent advances on this topic.The appeal and superiority of T-S fuzzy models is that the analysis and synthesis of the overall fuzzy systems can be carried out in the Lyapunov-function-based framework.To mention a few, by using LMI, Cao and Frank presented controller design for a class of fuzzy dynamic systems with time delay in both continuous and discrete cases in [20,21].Wu et al. studied the model approximation problem and  2 - ∞ control problem for nonlinear time-delay systems in [22,23].Moreover, great attention from researchers has been drawn to the study of stability analysis and controller design for T-S fuzzy systems with time delays [24][25][26][27][28]. On the other hand, type-2 fuzzy mode are considered in [29,30].
Recently, many scholars studied the stability problem based on the piecewise Lyapunov-Krasovskii functional [31][32][33].Reference [31] investigated the linear continuous/discrete systems with time-varying delay and divided the variation interval of the time delay into several subintervals.based on this method, [32]addressed the problem of the robust  ∞ filtering for singular linear parameter varying (LPV).Reference [33] researched the stability of linear time-invariant systems and divided the delay interval into  subintervals.The simulations show these methods can lead to much less conservative results than those in the existing references.
Motivated by the above observations, in this paper, we will investigate the problem of robust  ∞ control of uncertain T-S fuzzy systems with constant delay.Attention is focused on the design of robust  ∞ controllers via the parallel distributed compensation scheme such that the closed-loop fuzzy time-delay system is asymptotically stable and the  ∞ disturbance attenuation is below a prescribed level.

Mathematical Problems in Engineering
Based on delay decomposition approach [33], the decomposed Lyapunov-Krasovskii functional is introduced, and some delay-dependent conditions have been obtained.These conditions are formulated in the form of LMIs, and the controller design is cast into a convex optimization problem subject to LMI constraints, which can be readily solved via standard numerical software.Finally, a numerical example is provided to show the effectiveness and less conservatism of the proposed results.
The rest of this paper is organized as follows.In Section 2, the model description and problem are first formulated.The main results for delay-dependent robust  ∞ controller are presented in Section 3. Illustrative examples are given in Section 4, and the paper is concluded in Section 5.
Notations.The notations used throughout this paper are fairly standard.The superscript "" stands for matrix transpose, and the notation  > 0 ( ≥ 0) means that matrix  is real symmetric and positive (or being positive semidefinite). and 0 are used to denote appropriate dimensions identity matrix and zero matrix, respectively.The notation * in a symmetric always denotes the symmetric block in the matrix.The parameter diag{⋅ ⋅ ⋅} denotes a block-diagonal matrix.Matrices, if not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

System Descriptions and Preliminaries
Consider the uncertain nonlinear system with state delay that is described by the following T-S model with uncertain parameter matrices.
where  1 (),  2 (), . ..,   () are the premise variables that are measurable and each   ( = 1, 2, . . ., ) is fuzzy set.() ∈   is the state vector and () ∈   is the control input vector.() is the output vector.() ∈   is the disturbance input vector belongs to  2 [0, ∞). is the number of IF-THEN rules,  is the constant delay in the state.() is a vector-valued initial continuous function.
The matrices Δ  , Δ  , and Δ 1 denote the parameters uncertainties, which are assumed of the form where ,   ,   , and  1 are known constant matrices and () is an unknown time-varying matrix function satisfying   ()() ≤ .
Then, the overall output of the controller rules is given by Substituting ( 6) into (4), the closed-loop system can be given as Mathematical Problems in Engineering 3 with its compact form where Before ending this section, we introduce the following definitions and lemmas, which will be used in the derivation of our main results.
Definition 1 ( ∞ performance).Given a scalar  > 0 and under zero initial condition, the system (1) is said to be asymptotically stable with -disturbance attenuation if the system (4) is asymptotically stable and the output () satisfies that is, for all nonzero () ∈  2 [0, ∞).

Main Results
In this section, some delay-dependent sufficient conditions on the existence of robust  ∞ controller for T-S fuzzy system (7) will be presented.A Lyapunov-Krasovskii functional, based on the idea of delay decomposition approach, will be introduced, which can potentially reduce the conservatism of the results.
To this end, we first consider the following nominal closed-loop system: Firstly, the sufficient condition of  ∞ performance analysis for the unforced case of system ( 16) is established in Proposition 4.

Mathematical Problems in Engineering
In the following, based on Proposition 4, we design robust state feedback  ∞ controller for the system (7).
According to Proposition 4, it is easy to know that the  ∞ performance requirement of the nominal case of closed-loop system (7) implies where Σ is a matrix derived from ( 18) by changing the term   to   +  1   .
Replace   , A  , and   with   + Δ  , A  + ΔA  , and  1 + Δ 1 , respectively; then we obtain from ( 2) and ( 38) that By Lemma 3, we can know (39) holds, if and only if the following inequality holds: where  is a positive scalar.Applying Schur complement to (40), we have that Therefore, condition (33) can guarantee that condition (41) holds.This completes the proof.
It should be noted that the obtained conditions in Theorem 5 are not strict LMI conditions due to the existence of nonlinear term  1  in (33).It cannot be directly solved by standard LMI Toolbox.In the following, we present an approach to solving the condition in Theorem 5.
Introduce additional matrix variable  > 0 such that By Schur complement, it follows from (43) that Then, we readily obtain the following theorem.

Numerical Example
In this section, we use an example to show the applicability of the results proposed in this paper.
Example 8. Consider the truck trailer system borrowed from [38], which can be represented by the following uncertain time-delay T-S fuzzy model.
For this example, the prescribed  ∞ performance level is chosen as  = 0.5.In order to design a robust  ∞ state feedback controller for the given T-S fuzzy model, choose  = 1,  = 4.According to Theorem 5, the gain matrix of controller is given as  According to [38], take the membership function as   Figure 1 shows the controlled output () and the disturbance input ().According to Figure 1, the resulting output energy of the robust  ∞ controller is ∫ 100 0  2 () = 0.832, while the input energy is ∫ 100 0  2 () = 5.Simulation result for the ratio of the output energy to the disturbance energy is 0.1664, and the  2 -norm is √ 0.1664 = 0.41 <  = 0.5 (due to the fact that the state has been stable for a long time, we can regard the value 0.41 as the  2 -norm).
State response of the closed-loop system and controller input are shown in Figure 2.
The simulation results show that the algorithm proposed in this paper is effective for robust  ∞ control problem of the truck trailer system with time delay.

Conclusion
The problem of robust  ∞ controller design has been addressed for a class of T-S fuzzy-model-based systems with constant delay and norm-bounded parameter uncertainty.Based on the Lyapunov-Krasovskii functional approach, a sufficient condition for the existence of the robust  ∞ controller, which robustly stabilizes the T-S fuzzy-model-based uncertain systems and guarantees a prescribed level on disturbance attenuation, has been obtained in an LMI form.The given numerical example has shown the effectiveness of the proposed method.In addition, the filtering problems of T-S fuzzy delayed systems by using the delay decomposition approach are also challenging, which could be our further work.

Figure 2 :
Figure 2: Response of the closed-loop system and controller input.