Volume 2012, Article ID 475728,20pages doi:10.1155/2012/475728
Research Article
Delay-Range-Dependent Stability
Criteria for Takagi-Sugeno Fuzzy Systems with
Fast Time-Varying Delays
Pin-Lin Liu
Department of Automation Engineering, Institute of Mechatronoptic System, Chienkuo Technology University, Changhua 500, Taiwan
Correspondence should be addressed to Pin-Lin Liu,[email protected] Received 23 April 2012; Revised 16 June 2012; Accepted 23 June 2012 Academic Editor: Reinaldo Martinez Palhares
Copyrightq 2012 Pin-Lin Liu. 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.
The problem of delay-range-dependent stability for T-S fuzzy system with interval time-varying delay is investigated. The constraint on the derivative of the time-varying delay is not required, which allows the time delay to be a fast time-varying function. By developing delay decomposition approach, integral inequalities approachIIA, and Leibniz-Newton formula, the information of the delayed plant states can be taken into full consideration, and new delay-dependent sufficient stability criteria are obtained in terms of linear matrix inequalities LMIs which can be easily solved by various optimization algorithms. Simulation examples show resulting criteria outperform all existing ones in the literature. It is worth pointing out that our criteria are carried out more efficiently for computation and less conservatism of the proposed criteria.
1. Introduction
It is well known that time delay often appears in the dynamic systems, which is an important source of instability and degradation in the control performance. Fuzzy system in the form of Takagi-Sugeno T-S model has been paid considerable attention in the past two decades 1, 2. It has been shown that the T-S model method gives an effective way to
represent complex nonlinear systems by some simple local linear dynamic systems, and some analysis methods in the linear systems can be effectively extended to the T-S fuzzy systems. However, all the aforementioned criteria aim at time delay free T-S fuzzy systems. In practice, time delay, one of the instability sources in dynamical systems, is a common and complex phenomenon in many industrial and engineering systems such as chemical process, metallurgical processes, biological systems, rolling mill systems, and communication
networks. As a result, stability analysis for T-S fuzzy systems with time delay is of great significance both in theory and in practice. Some approaches developed for general delay systems have been borrowed to deal with fuzzy systems with time delay. In recent years, the problems of stability and stabilization of the T-S fuzzy systems with time delay have attracted rapidly growing interests3–21. Among these references, great efforts have been focused
on effective reduction of the conservation of the delayed T-S fuzzy model. Many effective methods, such as new bounding technique for cross-terms 7, 9, augmented Lyapunov
functional method7, and free-weighing matrix method 6,10,17–20, have been proposed.
We can see that the free-weighting matrix approach is used as a main tool to make the criteria less conservative in the literature, and only the lower and upper bounds of delay function ht are considered. When stability analysis of delayed systems is concerned, a very effective strategy is to apply the Gu’s Lyapunov-Krasovskii functional discretization technique22. However, this discretization technique has been developed for linear systems
subject to constant time delay. Besides, it is very hard to extend the stability analysis conditions obtained via this technique to control design since several products between decision variables will be generated, leading to nonconvex formulations. Therefore, less-conservative conditions for stability and control of T-S fuzzy systems subject to uncertain time delay are proposed based on a fuzzy weighting-dependent Lyapunov function, the Gu discretization technique 22, and extra strategies to introduce slack matrix variables
by13,23. However, these results have conservatism to some extent, which exist room for
further improvement.
The delay varying in an interval has strong application background, which commonly exists in many practical systems. The investigation for the systems with interval time-varying delay has caused considerable attention, see13,18,24–26 and the references therein. In 15,
Lyapunov-Krasovskii function and augmented Lyapunov-Krasovskii function to construct uncorrelated augmented matrixUAM and to deal with cross terms in the UAM through improved Jessen’s inequality. An improved delay-dependent criterion is derived in 18
by constructing a new Lyapunov functional and using free-weighting matrices. In 24, a
weighting delay method is used to deal with the stability of system with time varying delay. In25,26, by developing a delay decomposition approach, the integral interval t − h, t is
decomposed intot − h, t − αh and t − αh, t. Since a tuning parameter α is introduced, the information about xt − αh can be taken into full consideration; thus, the upper bound of t
t−h ˙xTsR ˙xsds can be estimated more exactly no matter the delay derivative exists or not.
However, it has been realized that too many free variables introduced in the free-weighting matrix method will complicate the system synthesis and consequently lead to a significant computational demand7. The problem of improving system performance while reducing
the computational demand will be addressed in this paper.
The main contributions of this paper are highlighted as follows.1 delay-dependent stability criteria are developed, which are an improvement over the latest results available from the open literature 3, 5–7, 9, 10, 12, 13, 15, 17–21, 23, 27; 2 theoretical proof is
provided to show that the results in 6 are a special case of the results derived in this
paper. The approach developed in this work uses the least number of unknown variables and consequently is the least mathematically complex and most computationally efficient. This implies that some redundant variables in the existing stability criteria can be removed while maintaining the efficiency of the stability conditions. With the present stability conditions, the computational burden is largely reduced; 3 since the delay decomposition approach is introduced in delay interval, it is clear that the stability results are based on the delay decomposition approach. When the positions of delay decomposition are varied, the stability
results of the proposed criteria are also different. In order to obtain the optimal delay decomposition sequence, we proposed an implementation based on optimization methods.
Motivated by the above discussions, we propose new stability criteria for T-S fuzzy system with interval time-varying delay. By developing a delay decomposition approach, the information of the delayed plant states can be taken into full consideration, and new delay-dependent sufficient stability criteria are obtained in terms of linear matrix inequalities LMIs which can be easily solved by various optimization algorithms. Since the delay terms are concerned more exactly, less conservative results are presented. Moreover, the restriction on the change rate of time-varying delays is relaxed in the proposed criteria. The proposed stability conditions are much less conservative and are more general than some existing results. Numerical examples are given to illustrate the effectiveness of our theoretical results.
2. Stability Analysis
Consider a T-S fuzzy system with a time-varying delay, which is represented by a T-S fuzzy model, composed of a set of fuzzy implications, and each implication is expressed by a linear system model. The ith rule of this T-S fuzzy model is of the following form:
Plant rule i : if z1t is Mi1, and . . ., and zpt is Mipthen
˙xt Aixt Adixt − ht, 2.1a
xt φt, t ∈ −h, 0, i 1, 2, ..., r, 2.1b
where z1t, z2t, ..., zpt are the premise variables; Mij, i 1, 2, ..., r, j 1, 2, ..., p are the
fuzzy sets; xt ∈ Rn is the state; φt is a vector-valued initial condition; A
i and Adi are constant real matrices with appropriate dimensions; the scalars ris the number of if-then rules; time delay, ht, is a varying delay. We will consider the following two cases for the time-varying delay.
Case 1. ht is a differentiable function satisfying
0≤ h1≤ ht ≤ h2, ˙ht ≤ hd, ∀t ≥ 0. 2.2
Case 2. ht is a differentiable function satisfying
0≤ h1 ≤ ht ≤ h2, 2.3
where h1 and h2 are the lower and upper delay bounds, respectively; h1, h2, and hd are
constants. Here h1 the lower bound of delay may not be equal to 0, and when hd 0 we
have h1 h2. Both Cases1and2have considered the upper and nonzero lower delay bounds of the interval time-varying delay. Case1is a special case of Case2. If the time-varying delay is differentiable and hd < 1, one can obtain a less conservative result using Case1than that
By fuzzy blending, the overall fuzzy model is inferred as follows: ˙xt r i1wiztA ixt Adixt − ht r i1wizt r i1 θiztAixt Adixt − ht Axt Adxt − ht xt r i1 θiztφit, t ∈ −h, 0, 2.4
where z z1, z2, ..., zp; wi : Rp → 0, 1, i 1, 2, ..., r, is the membership function
of the system with respect to the plant rule i; θizt wizt/
r
i1wizt; A
r
i1θiztAi, Ad
r
i1θiztAdi, and It is assumed that wizt ≥ 0, i
1, 2,...,r,ri1wizt ≥ 0 for all t, so we have θizt ≥ 0,ri1θizt 1.
In the following, we will develop some practically computable stability criteria for the system described2.1a. The following lemmas are useful in deriving the criteria. First, we
introduce the following technical Lemma2.1of integral inequality approachIIA.
Lemma 2.1 see 11. For any positive semidefinite matrices
X ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ X11 X12 X13 X12T X22 X23 XT 13 XT23 X33 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦≥ 0. 2.5
Then, one obtains
− t t−h ˙xTsX33˙xsds ≤ t t−h xTt xTt − h ˙xTs ⎡ ⎣XX11T X12 X13 12 X22 X23 XT 13 X23T 0 ⎤ ⎦ ⎡ ⎣xt − hxt ˙xs ⎤ ⎦ds. 2.6
Lemma 2.2 see 28. The following matrix inequality
Qx Sx STx Rx
< 0, 2.7
where Qx QTx, Rx RTx and Sx depend on affine on x is equivalent to
Rx < 0, Qx < 0,
Qx − SxR−1xSTx < 0.
In this paper, a new Lyapunov functional is constructed, which contains the informa-tion of the lower bound of delay h1and upper bound h2.The following Theorem2.3presents a delay-range-dependent result in terms of LMIs and expresses the relationships between the terms of the Leibniz-Newton formula.
Theorem 2.3. Under Case1, for given scalars h1, h2, hd, and α0 < α < 1, System 2.4 subject to
2.2 is asymptotically stable if there exist symmetry positive-definite matrices P PT > 0, Q
1
QT1 > 0, Q2 QT2 > 0, Q3 Q3T > 0, R1 RT1 > 0, R2 RT2 > 0, and positive semidefinite matrices
X ⎡ ⎣XX11T X12 X13 12 X22 X23 XT13 X23T X33 ⎤ ⎦ ≥ 0, Y ⎡ ⎣YY11T Y12 Y13 12 Y22 Y23 Y13T Y23T Y33 ⎤ ⎦ ≥ 0, Z ⎡ ⎣ZZ11T Z12 Z13 12 Z22 Z23 ZT13 ZT23 Z33 ⎤ ⎦ ≥ 0 2.9
such that the following LMIs hold:
Ξi ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Ξ11 Ξ12 0 Ξ14 Ξ15 Ξ16 ΞT 12 Ξ22 Ξ23 Ξ24 Ξ25 Ξ26 0 ΞT 23 Ξ33 0 0 0 ΞT 14 ΞT24 0 Ξ44 0 0 ΞT 15 ΞT25 0 0 Ξ55 0 ΞT 16 ΞT26 0 0 0 Ξ66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ < 0, 2.10a R1− X33≥ 0, 2.10b R2− Y33≥ 0, 2.10c R2− Z33≥ 0, 2.10d where Ξ11 ATiP P Ai Q1 Q2 Q3 αδX11 X13 X13T, Ξ12 PAdi, Ξ14 αδX12− X13 X23T, Ξ15 αδATiR1, Ξ16 h2− αδATiR2, Ξ22 −1 − hdQ2 h2− αδY22− Y23− Y23T h2− αδZ11 Z13 Z13T, Ξ23 h2− αδZ12− Z13 ZT23, Ξ24 h2− αδY12T − Y13T Y23, Ξ25 αδATdiR1, Ξ26 h2− αδATdiR2, Ξ33 −Q3 h2− αδZ22− Z23− Z23T, Ξ44 −Q1 h2− αδY11 Y13 Y13T αδX22− X23− XT23, Ξ55 −δR1, Ξ66 −h2− αδR2. 2.11
Proof. If we can proof that Theorem 2.3holds for two cases, that is, αδ ≤ ht ≤ h2 and
Case 1. When αδ ≤ ht ≤ h2.
Construct a Lyapunov-Krasovskii functional candidate as
Vxt xTtPxt t t−αδ xTsQ1xsds t t−htx TsQ 2xsds t t−h2 xTsQ3xsds 0 −αδ t t θ ˙xTsR1˙xsds dθ −αδ −h2 t t θ ˙xTsR2˙xsds dθ. 2.12 Calculating the derivative of2.12 with respect to t > 0 along the trajectories of 2.1a
and2.1b leads to ˙ Vxt xTt P A ATPxt xTtPAdxt − ht xTt − htATdP xt xTtQ 1 Q2 Q3xt − xTt − αδQ1xt − αδ − xTt − ht1− ˙htQ 2xt − ht − xTt − h2Q3xt − h2 ˙xTtαδR1˙xt ˙xTth 2− αδR2˙xt − t t−δ ˙xTsR1˙xsds − t−δ t−h2 ˙xTsR2˙xsds ≤ xTtP A ATPxt xTtPA dxt − ht xTt − htATdP xt xTtQ 1 Q2 Q3xt − xTt − αδQ1xt − αδ − xTt − ht1 − h dQ2xt − ht − xTt − h2Q3xt − h2 ˙xTtαδR 1 h2− αδR2 ˙xt − t t−αδ ˙xTsR1˙xsds − t−αδ t−h2 ˙xTsR2˙xsds, 2.13 with the operator for the term ˙xTtαδR
1 h2− αδR2 ˙xt as follows: ˙xTtαδR1 h2− αδR2 ˙xt Axt Adxt − htTαδR1 h2− αδR2Axt Adxt − ht xTtATαδR 1 h2− αδR2Axt xTtATαδR1 h2− αδR2Adxt − ht xTt − htAT dαδR1 h2− αδR2Axt xTt − htAT dαδR1 h2− αδR2Adxt − ht. 2.14
Alternatively, the following equations are true: − t t−αδ ˙xTsR1˙xsds − t−αδ t−h2 ˙xTsR2˙xsds − t t−αδ ˙xTsR 1˙xsds − t−αδ t−ht ˙x TsR 2˙xsds − t−ht t−h2 ˙xTsR 2˙xsds − t t−αδ ˙xTsR1− X33 ˙xsds − t−αδ t−ht ˙x TsR 2− Y33 ˙xsds − t−ht t−h2 ˙xTsR 2− Z33 ˙xsds − t t−αδ ˙xTsX 33˙xsds − t−αδ t−ht ˙x TsY 33˙xsds − t−ht t−h2 ˙xTsZ33 ˙xsds. 2.15
By utilizing Lemma2.1and the Leibniz-Newton formula, we have
− t t−αδ ˙xTsX33˙xsds ≤ t t−αδ xTt xTt − αδ ˙xTs ⎡ ⎣XX11T X12 X13 12 X22 X23 X13T XT23 0 ⎤ ⎦ ⎡ ⎣xt − αδxt ˙xs ⎤ ⎦ds ≤ xTtαδX 11xt xTtαδX12xt − αδ xTt − αδαδXT12xt xTt − αδαδX 22xt − αδ xTtX13T t t−αδ ˙xsds xTt − αδXT 23 t t−αδ ˙xsds t t−αδ ˙xTsdsX13xt t t−αδ ˙xTsdsX23xt − αδ xTtαδX 11 X13T X13 xt xTtαδX12− X13 X23T xt − αδ xTt − αδδXT 12 X23− X13T xt xTt − αδαδX22− X23− XT23 xt − αδ. 2.16
Similarly, we obtain − t−αδ t−ht ˙x TsY 33˙xsds ≤ xTt − αδh 2− αδY11 Y13T Y13 xt − αδ xTt − αδh 2− αδY12− Y13 Y23T xt − ht xTt − hth 2− αδY12T − Y13T Y23 xt − αδ xTt − hth 2− αδY22− Y23− Y23T xt − ht, 2.17 − t−ht t−h2 ˙xTsZ 33˙xsds ≤ xTt − hth 2− αδZ11 Z13 ZT13 xt − ht xTt − hth 2− αδZ12− Z13 ZT23 xt − h2 xTt − h 2 h2− αδZ12T − ZT13 Z23 xt − ht xTt − h 2 h2− αδZ22− Z23− ZT23 xt − h2. 2.18
Substituting the above equations2.14–2.18 into 2.13, we obtain
˙ Vxt ≤ ξTtΩξt − t t−αδ ˙xTsR1− X33 ˙xsds − t−αδ t−ht ˙x TsR 2− Y33 ˙xsds − t−ht t−h2 ˙xTsR2− Z33 ˙xsds, 2.19 where ξTt xTt xTt − ht xTt − h 2 xTt − αδ and Ω ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Ω11 Ω12 0 Ω14 ΩT 12 Ω22 Ω23 Ω24 0 ΩT 23 Ω33 0 ΩT 14 ΩT24 0 Ω44 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , 2.20
where Ω11 ATP P A Q1 Q2 Q3 αδX11 X13 XT13 ATαδR1 h2− αδR2A, Ω12 PAd ATαδR1 h2− αδR2Ad, Ω14 αδX12− X13 X23T, Ω22 −1 − hdQ2 h2− αδZ11 Z13 ZT13 h2− αδY22− Y23− Y23T AT dαδR1 h2− αδR2Ad, Ω23 h2− αδZ12− Z13 Z23T, Ω24 h2− αδY12T − Y13T Y23, Ω33 −Q3 h2− αδZ22− Z23− Z23T, Ω44 −Q1 h2− αδY11 Y13 Y13T αδX22− X23− X23T. 2.21
IfΩ < 0, R1− X33 ≥ 0, R2− Y33 ≥ 0, and R2− Z33 ≥ 0, then ˙V xt < −εxt2 for a
sufficiently small ε > 0. By the Schur complement of Lemma2.2,Ω < 0 is equivalent to the following inequality and is true:
Ψ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Ψ11 Ψ12 0 Ψ14 Ψ15 Ψ16 ΨT 12 Ψ22 Ψ23 Ψ24 Ψ25 Ψ26 0 ΨT 23 Ψ33 0 0 0 ΨT 14 ΨT24 0 Ψ44 0 0 ΨT 15 ΨT25 0 0 Ψ55 0 ΨT 16 ΨT26 0 0 0 Ψ66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ < 0, 2.22 where Ψ11 ATP P A Q1 Q2 Q3 αδX11 X13 XT13, Ψ12 PAd, Ψ14 αδX12− X13 XT23, Ψ15 αδATR1, Ψ16 h2− αδATR2, Ψ22 −1 − hdQ2 h2− αδY22− Y23− Y23T h2− αδZ11 Z13 ZT13, Ψ23 h2− αδZ12− Z13 ZT23, Ψ24 h2− αδY12T − Y13T Y23, Ψ25 αδATdR1, Ψ26 h2− αδATdR2, Ψ33 −Q3 h2− αδZ22− Z23− ZT23, Ψ44 −Q1 h2− αδY11 Y13 Y13T αδX22− X23− X23T, Ψ55 −αδR1, Ψ66 −h2− αδR2. 2.23 That is to say, ifΨ < 0, R1−X33≥ 0, R2−Y33≥ 0, and R2−Z33 ≥ 0, then ˙V t < −εxt2 for a sufficiently small ε > 0. Furthermore, 2.10a implies ri1θiztΩi < 0, which is
equivalent to2.22. Therefore, if LMIs 2.10a are feasible, the system 2.4 is asymptotically
stable.
Case 2. When h1 ≤ ht ≤ αδ.
For this case, the Lyapunov-Krasovskii functional candidate is chosen
Vxt xTtPxt t t−αδ xTsQ1xsds t t−htx TsQ 2xsds t t−h1 xTsQ3xsds 0 −αδ t t θ ˙xTsR1˙xsds dθ −h1 −αδ t t θ ˙xTsR2˙xsds dθ, 2.24 where P > 0, R1> 0, R2> 0, Qi > 0 i 1, 2, 3. Choosing ξTt xTt xTt − ht xTt − h
1 xTt − αδ and then using a proof process similar to that for Case 1, we derive the same condition2.10a, 2.10b, 2.10c, and
2.10d as that for Case 1. This completes the proof.
When the information of the time derivative of delay is unknown, by eliminating Q2we have the following result from Theorem2.3.
Corollary 2.4. For given scalars h1, h2, and α0 < α < 1, the system 2.4 is asymptotically stable
if there exist positive-definite matrices P PT > 0, Q
1 Q1T > 0, Q3 QT3 > 0, R1 RT1 > 0,
R2 RT2 > 0 and positive semidefinite matrices
X ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ X11 X12 X13 X12T X22 X23 XT 13 XT23 X33 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦≥ 0, Y ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ Y11 Y12 Y13 Y12T Y22 Y23 YT 13 Y23T Y33 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦≥ 0, Z ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ Z11 Z12 Z13 ZT12 Z22 Z23 ZT 13 ZT23 Z33 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦≥ 0, 2.25
such that the following LMIs hold:
Ξi ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Ξ11 Ξ12 0 Ξ14 Ξ15 Ξ16 ΞT 12 Ξ22 Ξ23 Ξ24 Ξ25 Ξ26 0 ΞT 23 Ξ33 0 0 0 ΞT 14 ΞT24 0 Ξ44 0 0 ΞT 15 ΞT25 0 0 Ξ55 0 ΞT 16 ΞT26 0 0 0 Ξ66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ < 0, 2.26a R1− X33≥ 0, 2.26b
R2− Y33≥ 0, 2.26c
R2− Z33≥ 0, 2.26d
whereΞij i, j 1, 2, ..., 6 are defined in (8) and
Ξ11 ATP P A Q1 Q3 αδX11 X13 X13T,
Ξ22 h2− αδY22− Y23− Y23T h2− αδZ11 Z13 ZT13.
2.27
Proof. If the matrix Q2 0 is selected in 2.10a, 2.10b, 2.10c, and 2.10d, this proof can be completed in a similar formulation to Theorem2.3.
When h1 0, Theorem2.3reduces to the following Corollary2.5.
Corollary 2.5. For given scalars h, hd, and α 0 < α < 1, the system 2.4 is asymptotically stable if
there exist symmetry positive-definite matrices P PT> 0, Q
1 QT1 > 0, Q2 QT2 > 0,R RT> 0,
and positive semidefinite matrices
X ⎡ ⎣XX11T X12 X13 12 X22 X23 X13T XT23 X33 ⎤ ⎦ ≥ 0, Y ⎡ ⎣YY11T Y12 Y13 12 Y22 Y23 Y13T Y23T Y33 ⎤ ⎦ ≥ 0, 2.28
such that the following LMIs hold for i 1, 2, ..., r,
Σi ⎡ ⎢ ⎢ ⎣ Σ11 Σ12 0 Σ14 ΣT 12 Σ22 Σ23 Σ24 0 ΣT 23 Σ33 0 ΣT 14 ΣT24 0 Σ44 ⎤ ⎥ ⎥ ⎦ < 0, R − X33≥ 0, R − Y33≥ 0, 2.29 where Σ11 ATiP P Ai Q1 Q2 αhX11 X13 X13T, Σ12 PAdi αhX12− X13 X23T, Σ14 αhATiR, Σ22 −1 − hdQ2 αhX22− X23− XT23 hY11 Y13 Y13T, Σ23 αhY12− Y13 Y23T, Σ24 αhATdiR, Σ33 −Q1 αhY22− Y23− Y23T, Ω44 −αhR. 2.30
Proof. Choose the following fuzzy Lyapunov-Krasovskii functional candidate to be
where V1t xTtPxt V2t t t−αh xTsQ1xsds V3t t t−htx TsQ 2xsds V4t 0 −αh t t θ ˙xTsR ˙xsds dθ. 2.32
Then, taking the time derivative of V t with respect to t along the system 2.4 yield
˙
Vt ˙V1t ˙V2t ˙V3t ˙V4t. 2.33 Then the proof follows a linear similar to the proof of Theorem2.3and thus is omitted here.
Remark 2.6. In our Theorems2.3, hdcan be any value or unknown due toΞ22 −1 − hdQ2 h2−αδY22−Y23−Y23T h2−αδZ11 Z13 ZT13. Therefore, Theorem2.3is applicable to both cases of fast and slow time varying delay. We will show the other characteristic of Theorem2.3. When the distance between h1and h2is sufficiently small, the upper bound h2 of delay for unknown hd will be very close to the upper bound for hd 0. This characteristic is not
included in previous Lyapunov functional based work where the upper bound of delay for
hd/ 0 is always less than that for hd 0.
Remark 2.7. It is seen from the proof of Theorem 2.3 and Corollary 2.4 that the main characteristics of the method developed in this paper can be generalized as the following two steps.i Construct a Lyapunov function to integrated both lower and upper delay bounds, for example, αδ h2 h1/2 in 2.12 and 2.24. ii Employ Lemma 2.1 to deal with cross-product terms, for example, those in2.15–2.18. It is also seen from the proof that
neither model transformation nor free-weighting matrices have been employed to deal with the cross-product terms. Therefore, the stability criteria derived in this paper are expected to be less conservative. This will be demonstrated later through numerical examples. It is noted that although it has been observed that using αδ h2 h1/2 in the constructed Lyapunov function can improve stability performance for many examples, theoretical evidence has not been found so far to explain the observations.
Based on that, a convex optimization problem is formulated to find the bound on the allowable delay time 0≤ h1≤ ht ≤ h2which maintains the delay-dependent stability of the time delay system2.4.
Remark 2.8. It is interesting to note that h1, h2 appears linearly in2.10a and 2.26a. Thus, a generalized eigenvalue problemGEVP as defined in Boyd et al. 28 can be formulated
to solve the minimum acceptable 1/h1 or 1/h2 and, therefore, the maximum h1 or h2 to maintain robust stability as judged by these conditions.
In this way, our optimization problem becomes a standard generalized eigenvalue problem, which can be then solved using GEVP technique. From this discussion, we have the following Remark2.9.
Remark 2.9. Theorem2.3provides delay-dependent asymptotic stability criteria for the T-S fuzzy systems with an interval time-varying delay2.4 in terms of solvability of LMIs 28.
Based on them, we can obtain the maximum allowable delay boundMADB 0 ≤ h1 ≤ ht ≤
h2such that2.4 is stable by solving the following convex optimization problem:
Maximize h2
Subject to Theorem 2.3Corollary 2.4 2.34
Inequality2.34 is a convex optimization problem and can be obtained efficiently
using the MATLAB LMI Toolbox.
About how to seek an appropriate α satisfying 0 < α < 1, such that the upper bound h of delay ht subjecting to 2.29 is maximal, we give an algorithm as follows.
Algorithm 2.10 Maximizing h > 0. Step 1: For given hd, choose an upper bound on h
satisfying2.29, and then select this upper bound as the initial value h0of h.
Step 2: Set appropriate step lengths, hstepand αstep, for h and α, respectively. Set k as a counter, and choose k 1. Meanwhile, let h h0 hstepand the initial value α0of α equals to
αstep.
Step 3: Let α kαstep, if inequality2.29 is feasible, go to Step 4; otherwise, go to Step 5.
Step 4: Let h0 h, α0 α, k 1 and h h0 hstep, go to step 3. Step 5: Let k k 1. If kαstep< 1, then go to step 3; otherwise, stop.
Remark 2.11. For Algorithm2.10, the final h0is the desired maximum of the upper bound of delay ht satisfying 2.29 and α0is the corresponding value of α.
3. Illustrative Examples
To illustrate the usefulness of our results, this section will provide numerical examples. It will be shown that the proposed results can provide less conservative results that recent ones have given3,5–7,9,10,12,13,15,17–21,23,27. It is worth pointing out that our criteria
carried out more efficiently for computation.
Example 3.1. Consider a time-delayed fuzzy system without controlling input. The T-S fuzzy
model of this fuzzy system is of the following form. Plant rules:
Rule 1: If x1t is M1, then
Rule 2: If x1t is M2, then
˙xt A2xt Ad2xt − ht, 3.2
and the membership functions for rule 1 and rule 2 are
M1x1t 1− 1 1 e−5x1t−π/6 1 1 e−5x1t−π/6, M2x1t 1 − M1x1t, 3.3 where A1 −2 0 0 −0.9 , Ad1 −1 0 −1 −1 , A2 −1.5 1 0 −0.75 , Ad2 −1 0 1 −0.85 . 3.4
Solution 3.2. For system3.1 and 3.2, by taking the parameter hd 0 and α 0.6, we get
the Corollary2.5which remains feasible for any delay time h ≤ 2.2459. In case of maximum allowable delay boundMADB h 2.2459, solving Corollary2.5yields the following set of feasible solutions: P 33.2342 7.9992 7.9992 28.4879 , Q1 0.0039 −0.0007 −0.0007 0.0029 , Q2 78.8437 −2.5471 −2.5471 22.8291 , R 1.6280 4.5093 4.5093 13.7942 , X11 1.8933 1.5877 1.5877 8.4013 , X12 0.1663 −3.3981 −3.4381 −6.7700 , X13 −1.2051 −3.3469 −3.3454 −10.2327 , X22 2.0259 1.5108 1.5108 8.4337 , X23 1.2030 3.3488 3.3473 10.2310 , X33 1.6246 4.5097 4.5097 13.7915 , Y11 5.1299 0.4116 0.4116 5.9858 , Y12 −0.1102 −0.1624 −0.1588 −0.5302 , Y13 0.1971 0.5872 0.5915 1.7945 , Y22 0.2962 0.6522 0.6522 2.0458 , Y23 0.3456 0.8055 0.8058 2.5092 , Y33 0.8093 2.1141 2.1141 6.4977 . 3.5
Applying the criteria in9, 10,12,17 and in this paper, the maximum values of h
for the stability of system under considerations are listed in Table 1. It is easy to see that the stability criterion in this paper gives a much less conservative result than the ones in 9,10,12,17.
Furthermore, by taking the various h1 hd 0.5, and from Theorem2.3, we obtain the
maximum allowable delay boundMADB h2as shown in Table2. From the above results of Table2, if the h1increases, the delay time length increases.
Table 1: Computation MADB h for varying hdin Example3.1. hd 0.2 0.4 0.6 0.8 0.9 ≥1 Li et al.9 1.0078 0.8164 0.6042 0.3356 0.1509 Fail Liu12 1.0655 0.9247 0.8040 0.7306 0.7232 0.7232 Lien10 1.2052 1.0766 0.9589 0.8468 0.7968 0.7843 Tian et al.17 1.2682 1.2071 1.1565 1.1415 1.1415 1.1415 Corollary2.5α 0.6 2.0087 1.7944 1.5981 1.4114 1.3280 1.3071
Table 2: MADB h2hd 0.5 for various h1in Example3.1.
h1 0.1 0.3 0.5 0.7 0.9 1.0 2.0
Theorem2.3α 0.6 1.3451 1.4016 1.4598 1.5208 1.5903 1.6357 2.1182
Remark 3.3. Similar to Algorithm2.10, an algorithm for seeking an appropriate α such that the upper bound of delay 0≤ h1≤ ht ≤ h2, subjecting to2.10a, 2.10b, 2.10c, and 2.10d is maximal can be easily obtained.
Example 3.4. Consider a T-S fuzzy system with time-varying delay. The T-S fuzzy model of
this fuzzy system is of the following form: Rule 1: If x1t is M1, then
˙xt A1xt Ad1xt − ht. 3.6
Rule 2: If x1t is M2, then
˙xt A2xt Ad2xt − ht, 3.7
and the membership functions for rule 1 and rule 2 are
M1zt 1 1 exp−2x1t, M2x1t 1 − M1zt, 3.8 where A1 −3.2 0.6 0 −2.1 , Ad1 1 0.9 0 2 , A2 −1 0 1 −3 , Ad2 0.9 0 1 1.6 . 3.9
Solution 3.5. By taking the parameter hd 0, using Corollary2.5, the maximum value of delay
time for the System3.6 and 3.7 to be asymptotically stable is h ≤ 1.0245. By the criteria
in9,16, the system 3.6 and 3.7 is asymptotically stable for h ≤ 0.58 and h ≤ 0.6148,
respectively. Hence, for this example, the criteria proposed here significantly improve the estimate of the stability limit compared with the result of9,16. Employing the LMIs in 18–
21 and those in Corollary2.5yields upper bounds on h2that guarantee the stability of system 3.6 and 3.7 for various hd, which are listed in Table3, in which “–” means that the results
Table 3: MADB h for various hdin Example3.4.
hd 0.1 0.5 0.9 ≥1
Tian and Peng18 0.3950 0.3950 0.3950 —
Zuo and Wang21 0.4808 0.4746 0.4454 —
Wu and Li19 0.4809 0.472 0.4455 —
Z. Yang and Y. P. Yang20 0.5030 0.4995 0.4988 —
Kwon et al.8 0.7236 0.7154 0.7014 —
Corollary2.5α 0.6 0.8014 0.7908 0.7418 0.7402
are not applicable to the corresponding cases. It can be seen from Table3that Corollary2.5
in this paper yields the least conservative stability test than other approaches, showing the advantage of the stability result in this paper.
Example 3.6. Consider a T-S fuzzy system with time-varying delay is of the following form:
Plant rules.
Rule 1: If x1t is M1, then
˙xt A1xt Ad1xt − ht. 3.10
Rule 2: If x1t is M2, then
˙xt A2xt Ad2xt − ht, 3.11
and the membership functions for rule 1 and rule 2 are
M1zt 1 1 exp−2x1t, M2x1t 1 − M1zt, 3.12 where A1 0.1 1 0 0.2 , Ad1 −0.3 2 0 −0.5 , A2 0.1 2 0 −0.2 , Ad2 −0.34 2 0.01 −0.4 . 3.13
Solution 3.7. By using Corollary 2.4, the maximum allowable delay bound MADB can be calculated as h2 2.3725. The results for stability conditions in different methods are compared in Table 4. It can be seen that the delay-dependent stability condition in this paper is less conservative than earlier reported ones in the literature 5,7,10,15, 18,27.
Compared with Guan and chen7 who used 5 LMI variables, Yoneyama 27 employed 20
LMI variables to get better stability results. To obtain improved stability results than those in 15,18,27, we need 8 variables in Corollary2.4the same as5,10. It is also seen from Table4
that the larger r is the more unknown LMI variables are required in18,27. However, the
unknown number of LMI variables is independent of r in the results of this paper. It can be shown that the delay-dependent stability condition in this paper is the best performance.
Table 4: MADB h2for unknown hd h1 0 in Example3.6for different methods Number of variables: Nv; Number of fuzzy rules: r.
Method h2 Nv
Guan and Chen7 1,2214 5
Yoneyama27 1.8294 8 6r Tian andPeng18 1.9187 8 8r Lien10 1.9187 8 Chen et al.5 1.9187 8 Peng et al.15 2.2812 15 Corollary2.4α 0.6 2.3725 8
Remark 3.8. Similar to Algorithm2.10, we can also find an appropriate scalar α, such that the upper bound of delay 0 ≤ h1 ≤ ht ≤ h2, subjecting to2.26a, 2.26b, 2.26c, and 2.26d reaches the maximum.
Example 3.9. Consider a nominal time-delay fuzzy system. The T-S fuzzy model of this fuzzy
system is of the following form. Rule 1: If x1t is M1, then
˙xt A1xt Ad1xt − ht. 3.14a
Rule 2: If x1t is M2, then
˙xt A2xt Ad2xt − ht, 3.14b
and the membership function for rules 1 and 2 are
M1zt 1 1 exp−2x1t, M2x1t 1 − M1zt, 3.15 where A1 −2.1 0.1 −0.2 −0.9 , Ad1 −1.1 0.1 −0.8 −0.9 , A2 −1.9 0 −1.1 −1.1 , Ad2 −0.9 0 −1.1 −1.2 . 3.16
Solution 3.10. The objective is to determine the maximum value of constant time-delay h h2h1 0for which the system is stable. Table5compares works based on common quadratic functionals7,18–20,23 with the fuzzy functional of Corollary2.4. It is clear by inspecting Table5that Corollary2.4provides the largest time delays. For comparison, Table6also lists the maximum allowable delay boundMADB h obtained from the criterion 3. It is clear
that Corollary2.5gives much better results than those obtained by3. It is illustrated that the
proposed stability criteria are effective in comparison to earlier and newly published results existing in the literature.
Table 5: MADB h h2h1 0 for different methods in Example3.9.
Method 7 18 19 20 23 Corollary2.4α 0.6
h 1.25 1.85 3.37 3.85 4.61 5.4873
Table 6: MADB h for different hdfor Example3.9.
hd 0.1 0.3 0.5 0.7 1.1
An and Wen3 3.036 2.240 1.747 1.523 1.497
Corollary2.5α 0.6 4.4274 3.2378 2.5016 1.9494 1.5926
Example 3.11. Consider a nominal time-delay fuzzy system. The T-S fuzzy model of this fuzzy
system is of the following form. Rule 1: If x1t is M1, then
˙xt A1xt Ad1xt − ht. 3.17
Rule 2: If x1t is M2, then
˙xt A2xt Ad2xt − ht, 3.18
and the membership functions for rule 1 and rule 2 are
M1zt sin2x1t, M2x1t cos2x1t, 3.19 where A1 0 1 −8 −6 , A2 0 1 −8 0 , Ad1 Ad2 0 0 1 −2 . 3.20
Solution 3.12. Considering a constant time delay, by using Corollary 2.4, the maximum allowable delay bound MADB can be calculated as h2 0.4130h1 0, α 0.5. By the criteria in13,23, the systems 3.17 and 3.18 is asymptotically stable for any h that satisfies
h ≤ 0.322 and h ≤ 0.4060, respectively. Hence, for this example, the criteria proposed here
significantly improve the estimate of the stability limit compared with the result of13,23.
4. Conclusion
In this paper, we have dealt with the stability problem for T-S fuzzy systems with interval time-varying delay. By constructing a Lyapunov-Krasovskii functional, the supplementary requirement that the time derivative of time-varying delays must be smaller than one is released in the proposed delay-range-dependent stability criterion. By developing a delay decomposition approach, the information of the delayed plant states can be taken into full consideration, and new delay-range-dependent sufficient stability criteria are obtained in
terms of linear matrix inequalitiesLMIs which can be easily solved by various optimization algorithms. Since the delay term is concerned more exactly, it is less conservative and more computationally efficient than those obtained from existing methods. Thus, the present method could largely reduce the computational burden in solving LMIs. Both theoretical and numerical comparisons have been provided to show the effectiveness and efficiency of the present method. Numerical examples are given to illustrate the effectiveness of our theoretical results.
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