[ProQuest: [...] denotes non US-ASCII text; see PDF]
Academic Editor:Olfa Boubaker
School of Electrical Engineering, University of Ulsan, Daehak-ro 93, Nam-Gu, Ulsan 680-749, Republic of Korea
Received 17 May 2016; Revised 13 September 2016; Accepted 5 October 2016
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. 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 [3-5]. 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 [6-11] and the second one is based on the free-weighing matrix approach [12-16].
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) [19-23]. 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.
The rest of the paper is organized as follows. Section 2 gives a mathematical description of the system considered here and presents a useful lemma. Section 3 presents the main result of this paper. Furthermore, through numerical examples, Section 4 shows the verification of our results. Finally, Section 5 makes the concluding remarks.
Notation . Throughout this paper, standard notions will be adopted. The notations X≥Y and X>Y mean that X-Y is positive semidefinite and positive definite, respectively. In symmetric block matrices, ([low *]) is used as an ellipsis for terms that are induced by symmetry. For a square matrix X, He(X) denotes X+XT , where XT is the transpose of X. The natation Conv(·) denotes the convex hull; col(v1 ,v2 ,...,vn )=(v1Tv2T [...]vnT )T for any vector vi ; diag(A,B) denotes a diagonal matrix with diagonal entries A and B; and Nr+ ={1,2,...,r}. For any matrix Si or Sij , [figure omitted; refer to PDF] All matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operation.
2. System Description and Preliminaries
Consider the following uncertain T-S fuzzy system, which represents a class of nonlinear systems: for i∈Nr+ ,
Plant Rule i . IF η1 (t) is Fi1 and [...] ηs (t) is Fis , THEN [figure omitted; refer to PDF] where x(t)∈Rnx and x(t-d(t))∈Rnx denote the state and the delayed state, respectively; the initial condition ψ(t) is a continuously differentiable vector-valued function; Fij denotes a fuzzy set; ηi (t) denotes the ith premise variable; and r denotes the number of IF-THEN rules. In (2), p(t)∈Rnp and q(t)∈Rnq are used to describe the structured feedback uncertainty such that p(t)=Δ(t)q(t) and ΔT (t)Δ(t)<=I∈Rnq ×nq . Further, the state delay d(t) is assumed to be unknown and time-varying with known bounds as follows: d1 <=d(t)<=d2 , where d1 and d2 are constant. Then, the overall T-S fuzzy model is inferred as follows: [figure omitted; refer to PDF] where A(Θt )=∑i=1r [...]θiAi , Ad (Θt )=∑i=1r [...]θiAd,i , E(Θt )=∑i=1r [...]θiEi , G(Θt )=∑i=1r [...]θiGi , and Gd (Θt )=∑i=1r [...]θiGd,i in which θi (=θi (η(t)) denotes the normalized fuzzy weighting function for the ith rule; η(t)=col(η1 (x(t)),...,ηs (x(t))) denotes the premise variable vector; and Θt =col(θ1 ,...,θr ) belongs to [figure omitted; refer to PDF]
Assumption 1.
The fuzzy weighting functions θi are differentiable and Θ t =col(θ 1 ,...,θ r ) belongs to [figure omitted; refer to PDF]
To simplify the notations, we use θid1 =θi (η(t-d1 )) and θid2 =θi (η(t-d2 )). And, for later convenience, we define x¯(t)=col(x(t),x(t-d1 ),x(t-d(t)),x(t-d2 )), η(t)=col(x¯(t),p(t))∈Rnη , and nη =4nx +np . And we use some block entry matrices ei (i=1,2,...,5) such that x(t)=e1 η(t), x(t-d1 )=e2 η(t), x(t-d(t))=e3 η(t), x(t-d2 )=e4 η(t), and p(t)=e5 η(t), which implies x¯(t)=e14 η(t) by defining e14T =(e1T [...]e4T ). Then, (3) becomes [figure omitted; refer to PDF] where Φt =A(Θt )e1 +Ad (Θt )e3 +E(Θt )e5 and Ψt =G(Θt )e1 +Gd (Θt )e3 .
Lemma 2.
Let Θt ∈SΘ be satisfied. Then, the following condition holds: [figure omitted; refer to PDF] if there are all decision variables such that [figure omitted; refer to PDF] where L0 =M0 +He(S0 -∑i=1r [...]αiβiXi ), Li =Mi +Si -S0 +(αi +βi )Xi , Lii =Mii +He(-Si -Xi ), and Lij =Mij -Si -Sj .
Lemma 3.
Let Θ t ∈SΘ be satisfied. Then, the following condition holds: [figure omitted; refer to PDF] if there are all decision variables such that [figure omitted; refer to PDF]
Proof.
In view of Θ t ∈SΘ , we can get [figure omitted; refer to PDF] where coefficients λli are all positive and sum to one and N is a constant slack variable. Then, (9) leads to [figure omitted; refer to PDF] which holds if (10) holds because ∑li =12 [...]λli (t)[varrho]li ,i (Pi +N)∈Conv([varrho]li ,i (Pi +N)), where li denotes the ith element of l∈L.
3. Θt -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: [figure omitted; refer to PDF] where P(Θt ), Q1 (Θα ), Q2 (Θα ), R1 (Θβ ), and R2 (Θβ ) are positive definite for all admissible grades. Then, the time derivative of each Vi (t) along the trajectories of (6) is given by [figure omitted; refer to PDF] which leads to [figure omitted; refer to PDF] where [figure omitted; refer to PDF]
Remark 4.
Indeed, it is hard to directly use Jensen's inequality approach to obtain the upper bounds of O1 and O2 because R1 (Θα ) and R2 (Θα ) are set to be dependent on Θα , which motivates the present study.
Lemma 5.
Suppose that there exist matrices U0 (Θt ), U1 (Θt ), and U2 (Θt )∈R4nx ×nx and symmetric matrices 0<P(Θt ),P (Θt ), 0<Q1 (Θt-d1 ), 0<Q2 (Θt-d2 ), 0<Q1 (Θt ), 0<Q2 (Θt ), 0<R1 (Θα ), 0<R2 (Θαp ), 0<R1 (Θt ), 0<R2 (Θt )∈Rnx ×nx , M0 (Θt ), M1 (Θt ), and M2 (Θt )∈R4nx ×4nx such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Then, (6) is robustly asymptotically stable for d1 <=d(t)<=d2 .
Proof.
First of all, by incorporating the following equalities into (15), [figure omitted; refer to PDF] we can get [figure omitted; refer to PDF] where [figure omitted; refer to PDF] in which λ1 (t)=(d2 -d(t))/(d2 -d1 ) and λ2 (t)=(d(t)-d1 )/(d2 -d1 ). Next, the structured feedback uncertainty, given as 0<=qT (t)q(t)-pT (t)p(t), can be converted into 0<=ηT (t)(ΨtTΨt -e5Te5 )η(t), which yields V (t)<=ηT (t)(Π1 +Π2 +Γ¯p )η(t)+O¯1 +O¯2 . That is, the robust stability for (6) is assured by 0>ηT (t)(Π1 +Π2 +Γ¯p )η(t)+O¯1 +O¯2 . Therefore, if (18) holds, then O¯1 +O¯2 <=0, and hence the robust stability criterion is given by (17) because ∑p=12 [...]λp (t)Mp (t)∈Conv(Mp (Θt )).
In the absence of uncertainties, the T-S fuzzy system becomes x (t)=Φt x¯(t), where Φt =A(Θt )e1 +Ad (Θt )e3 . The following corollary presents the stability criterion for nominal T-S fuzzy systems with time-varying delays.
Corollary 6.
Suppose that there exist matrices U0 (Θt ), U1 (Θt ), and U2 (Θt )∈R4nx ×nx and symmetric matrices 0<P(Θt ), P (Θt ), 0<Q1 (Θt-d1 ), 0<Q2 (Θt-d2 ), 0<Q1 (Θt ), 0<Q2 (Θt ), 0<R1 (Θα ), 0<R2 (Θαp ), 0<R1 (Θt ), 0<R2 (Θt )∈Rnx ×nx , M0 (Θt ),M1 (Θt ), and M2 (Θt )∈R4nx ×4nx such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Then, (6) without uncertainties is asymptotically stable for d1 <=d(t)<=d2 .
Proof.
The proof is omitted since it is analogous to the derivation of Lemma 5.
4. 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 fuzzy-weighting functions: [figure omitted; refer to PDF]
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 [31-33] but this is outside of the intended scope of this paper.
Theorem 8.
Let Θ t ∈SΘ be satisfied. Suppose that there exist matrices U0,i ,U1,i ,U2,i ∈R4nx ×nx and S0 ,Si ,Xi ∈Rnc ×nc (nc =6nx +np +nq ), for i∈Nr+ , symmetric matrices N,0<Pi , 0<Q1,i , 0<Q2,i , 0<R1,i , and 0<R2,i ∈Rnx ×nx , for i∈Nr+ , and M0,i ,M1,i ,M2,i ∈R4nx ×4nx such that, for all q,[varphi]∈Nr+ , p∈N2+ , and l∈L, [figure omitted; refer to PDF] where Llq[varphi],0 =Mlq[varphi],0 +He(S0 -∑i=1r [...]αiβiXi ), Lp,i =Mp,i +Si -S0 +(αi +βi )Xi , Lii =Mii +He(-Si -Xi ), and Lij =Mij -Si -Sj in which [figure omitted; refer to PDF] Then, the system in (6) is robustly asymptotically stable for d1 <=d(t)<=d2 .
Proof.
Note that Θ t ∈SΘ . Thus, in view of Lemma 3, applying the Schur complement to (17) is given by [figure omitted; refer to PDF] where Ωl =He(e1T P(Θt )Φt )-e2TQ1 (Θt-d1 )e2 -e4TQ2 (Θt-d2 )e4 -e5Te5 +e1T (∑i=1r [...][varrho]li ,i (Pi +N)+Q1 (Θt )+Q2 (Θt ))e1 . Further, from (26) and (27), (35) and (18) can be converted into [figure omitted; refer to PDF] where [figure omitted; refer to PDF] As a result, from the convexity of fuzzy-weighting functions, (17) and (18) can be assured by (30), [figure omitted; refer to PDF] Further, note that representing (38) in the form of (7) becomes [figure omitted; refer to PDF] where Mlq[varphi],0 , Mp,i , Mii , and Mij 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.
Corollary 9.
Let Θ t ∈SΘ be satisfied. Suppose that there exist matrices U0,i ,U1,i ,U2,i ∈R4nx ×nx and S0 ,Si ,Xi ∈Rnc ×nc (nc =6nx ), for i∈Nr+ , symmetric matrices N,0<Pi , 0<Q1,i , 0<Q2,i , 0<R1,i , and 0<R2,i ∈Rnx ×nx , for i∈Nr+ , and M0,i ,M1,i ,M2,i ∈R4nx ×4nx such that, for all q,[varphi]∈Nr+ , p∈N2+ , and l∈L, [figure omitted; refer to PDF] where Llp[varphi],0 =Mlp[varphi],0 +He(S0 -∑i=1r [...]αiβiXi ), Lp,i =Mp,i +Si -S0 +(αi +βi )Xi , Lii =Mii +He(-Si -Xi ), and Lij =Mij -Si -Sj in which [figure omitted; refer to PDF] Then, (6) without uncertainties is asymptotically stable for d1 <=d(t)<=d2 .
Proof.
The proof is omitted since it is analogous to the derivation of Theorem 8.
Remark 10.
The number of scalar variables involved in Theorem 8 and Corollary 9 is given as follows: #=nc2 (2r+1)+0.5nx (nx +1)+nx (38.5nx +8.5)r. Table 1 shows the number for each case of (nx ,r). Since the use of slack variables requires more computation cost compared with other methods, there may be the need to balance the tradeoffs between the computational cost and the performance enhancement.
Table 1: # involved in Corollary 9 and Theorem 8 (np =1,nq =1).
( n x , r ) | (2, 2) | (2, 3) | (2, 4) | (3, 2) | (3, 3) | (3, 4) | (4, 2) | (4, 3) | (4, 4) |
Corollary 9 | 1065 | 1524 | 1983 | 2370 | 3390 | 4410 | 4190 | 5992 | 7794 |
Theorem 8 | 1325 | 1888 | 2451 | 2750 | 3922 | 5094 | 4690 | 6692 | 8694 |
5. 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]: [figure omitted; refer to PDF] where θ1 =1/(1+exp[...](-2x1 (t))) and θ2 =1-θ1 . Table 2 shows the maximum allowable upper bound (MAUB) for each d1 ∈{0.0,0.4,0.8,1.0,1.2}, where m denotes the number of delay segments and (m-1) denotes the degree of delay partitioning. From Table 2, we can see that our method (Corollary 9) provides larger MAUBs in comparison with those of [25, 26]. Hence it can be concluded that the stability criterion in Corollary 9, obtained based on the NLKF, is less conservative than other results. In particular, for d1 =1.2 and d2 =1.531, Corollary 9 offers the following solutions: [figure omitted; refer to PDF]
Table 2: Maximum allowable upper bound (MAUB) for each d1 , where m denotes the number of delay segments and (m-1) denotes the degree of delay partitioning.
d 1 | 0.0 | 0.4 | 0.8 | 1.0 | 1.2 | (m-1) |
[25] | 0.982 | 1.038 | 1.158 | 1.252 | 1.359 | 0 |
[26] | 1.221 | 1.277 | 1.311 | 1.358 | 1.419 | 1 |
[26] | 1.278 | 1.303 | 1.316 | 1.361 | 1.425 | 2 |
Corollary 9 | 1.302 | 1.380 | 1.413 | 1.462 | 1.531 | 0 |
Example 2.
Consider the following T-S fuzzy system: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] 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 [27-30]. 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 [27-30].
Table 3: Maximum allowable upper bound (MAUB) for d1 =0.
Methods | [27] | [28] | [29] | [30] | Theorem 8 |
d 2 | -- | 0.443 | 0.499 | 1.081 | 1.132 |
6. 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.
Acknowledgments
This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2015R1A1A1A05001131).
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Abstract
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.
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Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer