Robust Stability Criteria for T-S Fuzzy Systems with Time-Varying Delays via Nonquadratic Lyapunov-Krasovskii Functional Approach

This paper tackles the issue of stability analysis for uncertain T-S fuzzy systems with interval time-varying delays, especially based on the nonquadratic Lyapunov-Krasovskii functional (NLKF). To this end, this paper first provides a less conservative relaxation technique and then derives a relaxed robust stability criterion that enhances the interactions among delayed fuzzy subsystems.The effectiveness of our method is verified by two examples.


Introduction
Over the past few decades, Takagi-Sugeno (T-S) fuzzy model has attracted great attention since it can systematically represent nonlinear systems via a kind of interpolation method that connects smoothly some local linear systems based on fuzzy weighting functions [1].In particular, the T-S fuzzy model has the advantage that it allows the well-established linear system theory to be applied to the analysis and synthesis of nonlinear systems.For this reason, the T-S fuzzy model has been a popular choice not only in consumer products but also in industrial processes (refer to [2] and references therein).
As well-known, time-delay phenomena are ubiquitous in practical engineering systems such as aircraft systems, biological systems, and chemical engineering system [3][4][5].Recently, thus, the research on nonlinear systems with state delays has been an important issue in the stability analysis of T-S fuzzy systems.In the literature, there are two major research trends to deal with such systems: one focuses on decreasing computational burdens required to solve a set of conditions from the Lyapunov-Krasovskii functional (LKF) approach, and the other focuses on improving the solvability of delay-dependent stability conditions despite significant computational efforts.Strictly speaking, the first trend is mainly based on Jensen's inequality approach [6][7][8][9][10][11] and the second one is based on the free-weighing matrix approach [12][13][14][15][16].
Recently, it is recognized that the common quadratic Lyapunov function approach leads to overconservative performance for a large number of fuzzy rules [17,18].For this reason, it is essential to tackle the issue of stability analysis in the light of the nonquadratic Lyapunov-Krasovskii functional (NLKF) [19][20][21][22][23].However, to our best knowledge, up to now, little progress has been made toward using NLKFs for the stability analysis.Motivated by the above concern, this paper proposes a relaxed stability criterion for uncertain T-S fuzzy systems with interval time-varying delays, especially obtained by the NLKF approach.To this end, this paper offers a proper relaxation method that can enhance the interactions among delayed fuzzy subsystems.Further, it is worth noticing that Jensen's inequality, given in [24], is applicable only to the case where the internal matrix is constant, that is, to the case where the common quadratic Lyapunov-Krasovskii functional (CQLKF) is employed.Thus, this paper focuses more on exploring the second trend in the direction of reducing the conservatism that stems from the CQLKF approach, without resorting to any delay-decomposition method.In this sense, this paper provides two examples numerically to show the effectiveness of our method.
Notation.Throughout this paper, standard notions will be adopted.The notations  ≥  and  >  mean that  −  is positive semidefinite and positive definite, respectively.In symmetric block matrices, ( * ) is used as an ellipsis for terms that are induced by symmetry.For a square matrix X, He(X) denotes X + X  , where X  is the transpose of X.The natation Conv(⋅) denotes the convex hull for any vector V  ; diag(A, B) denotes a diagonal matrix with diagonal entries A and B; and N +  = {1, 2, . . ., }.For any matrix S  or S  , All matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operation.
Lemma 2. Let Θ  ∈ S Θ be satisfied.Then, the following condition holds: if there are all decision variables such that where Lemma 3. Let Θ ∈ S Θ be satisfied.Then, the following condition holds: if there are all decision variables such that Proof.In view of Θ ∈ S Θ, we can get where coefficients  ℓ  are all positive and sum to one and N is a constant slack variable.Then, (9) leads to which holds if (10) holds because ∑ 2 ℓ  =1  ℓ  () ℓ  , (P  + N) ∈ Conv( ℓ  , (P  + N)), where ℓ  denotes the th element of ℓ ∈ L.

Θ 𝑡 -Dependent Stability Criterion
Based on a nonquadratic Lyapunov-Krasovskii functional (NLKF), this section provides a less conservative stability criterion.To this end, we first choose an NLKF of the following form: where (Θ  ),  1 (Θ  ),  2 (Θ  ),  1 (Θ  ), and  2 (Θ  ) are positive definite for all admissible grades.Then, the time derivative of each   () along the trajectories of ( 6) is given by which leads to where Remark 4. Indeed, it is hard to directly use Jensen's inequality approach to obtain the upper bounds of O 1 and O 2 because  1 (Θ  ) and  2 (Θ  ) are set to be dependent on Θ  , which motivates the present study.
Proof.The proof is omitted since it is analogous to the derivation of Lemma 5.

LMI-Based Stability Criterion
Based on Lemmas 2 and 3, to derive a finite number of solvable LMI conditions from (17), this paper simply sets all the decision variables to be of affine dependence on fuzzyweighting functions: Remark 7. As a way to improve the performance to be considered, we can increase the degree of polynomial dependence on fuzzy-weighting functions, as in [31][32][33] but this is outside of the intended scope of this paper.
Proof.Note that Θ ∈ S Θ.Thus, in view of Lemma 3, applying the Schur complement to ( 17) is given by where Further, from ( 26) and ( 27), ( 35) and ( 18) can be converted into where As a result, from the convexity of fuzzy-weighting functions, ( 17) and ( 18) can be assured by (30), Further, note that representing (38) in the form of ( 7) becomes where M ℓ,0 , M , , M  , and M  are defined in ( 31)- (33).Therefore, from Lemma 2, we can obtain (29) in the sequel without loss of generality.
The following corollary presents the LMI-based stability criterion for nominal T-S fuzzy systems with time-varying delays.

Numerical Examples
To verify the effectiveness of our methods, this paper provides two examples that make some comparisons with other results: one is related to the stability analysis for nominal T-S fuzzy systems and the other is related to the robust stability analysis for T-S fuzzy systems with uncertainties.
Example 1.Consider the following T-S fuzzy system, adopted in [25]: where  (45) The maximum allowable upper bound (MAUB) for each method is tabulated in Table 3.And, from Table 3, we can see that the proposed method (Theorem 8) achieves larger MAUBs than those of other methods [27][28][29][30].Hence, it can be concluded that the robust stability criterion in Theorem 8, established from the NLKF approach and Lemma 2, is less conservative than those of [27][28][29][30].

Concluding Remarks
This paper proposed an NLKF-based method of deriving a less conservative stability criterion for T-S fuzzy systems with time-varying delays.Of course, the proposed method may increase the burden of numerical computation.However, if the computational complexity is out of the practical problem, then our results can be significantly useful.

Table 2 :
Maximum allowable upper bound (MAUB) for each  1 , where  denotes the number of delay segments and ( − 1) denotes the degree of delay partitioning.