Research Article Convex Polyhedron Method to Stability of Continuous Systems with Two Additive Time-Varying Delay Components

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1 Applied Mathematics Volume 202, Article ID , 3 pages doi:0.55/202/ Research Article Convex Polyhedron Method to Stability of Continuous Systems with Two Additive Time-Varying Delay Components Tiejun Li, 2, 2, 3 and Junkang Tian State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 60500, China 2 School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 60500, China 3 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 673, China Correspondence should be addressed to Junkang Tian, junkangtian@63.com Received 27 November 20; Revised 26 December 20; Accepted 27 December 20 Academic Editor: Chong Lin Copyright q 202 T. Li and J. Tian. 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. This paper is concerned with delay-dependent stability for continuous systems with two additive time-varying delay components. By constructing a new class of Lyapunov functional and using a new convex polyhedron method, a new delay-dependent stability criterion is derived in terms of linear matrix inequalities. The obtained stability criterion is less conservative than some existing ones. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.. Introduction Robust stability of dynamic interval systems covering interval matrices and interval polynomials has attracted considerable attention over last decades. Reference presents some necessary and sufficient conditions for the quadratic stability and stabilization of dynamic interval systems. It is well known that time delay frequently occurs in many industrial and engineering systems, such as manufacturing systems, telecommunication, and economic systems, and is a major cause of instability and poor performance. Over the past decades, much efforts have been invested in the stability analysis of time-delay systems 2 6. Reference 2 deals with the problem of quadratic stability analysis and quadratic stabilization for uncertain linear discrete time systems with state delay. Reference 3 deals with the quadratic stability and linear state-feedback and output-feedback stabilization of switched delayed

2 2 Applied Mathematics linear dynamic systems. However, almost all the reported results mentioned above on timedelay systems are based on the following basic mathematical model: ẋ t Ax t A d x t d t,. where d t is a time delay in the state x t, which is often assumed to be either constant or time-varying satisfying certain conditions, for example, 0 d t d<, d t τ<..2 Sometimes in practical situations, however, signals transmitted from one point to another may experience a few segments of networks, which can possibly induce successive delays with different properties due to the variable network transmission conditions. Thus, in recent papers 5, 6, a new model for time-delay systems with multiple additive timevarying delay components has been proposed: ( ) ẋ t Ax t A d x t n d i t,.3 0 d i t d i <, i d i t τ i <..4 To make the stability analysis simpler, we proceed by considering the system.3 with two additive delay components as follows: ẋ t Ax t A d x t d t d 2 t, x t φ t, t d, 0..5 Here, x t R n is the state vector; d t and d 2 t represent the two delay components in the state, and we denote d t d t d 2 t ; A, A d are system matrices with appropriate dimensions. It is assumed that 0 d t d <, d t τ <, 0 d 2 t d 2 <, d 2 t τ 2 <,.6 and d d d 2, τ τ τ 2. φ t is the initial condition on the segment d, 0. The purpose of our paper is to derive new stability conditions under which system.5 is asymptotically stable for all delays d t and d 2 t satisfying.6. One possible approach to check the stability of this system is to simply combine d t and d 2 t into one delay d t with 0 d t d d 2 <, d t τ τ 2 <..7

3 Applied Mathematics 3 Then, the system.5 becomes ẋ t Ax t A d x t d t, x t φ t, t d, 0..8 By using some existing stability conditions, the stability of system.8 can be readily checked. As discussed in 5, 6, however, since this approach does not make full use of the information on d t and d 2 t, it would be inevitably conservative for some situations. Recently, some new delay-dependent stability criteria have been obtained for system.5 in 5, 6, by making full use of the information on d t and d 2 t. However, the stability result is conservative because of overly bounding some integrals appearing in the derivative of the Lyapunov functional. On the one hand, the integral ẋ T s Z ẋ s ds in 5 was enlarged as t ẋt s Z ẋ s ds, with d t ẋ T s Z ẋ s ds discarded. On the other hand, some integrals were estimated conservatively. Take t ẋt s Z ẋ s ds as an example, by introducing [ 0 2ζ T S x t x t d t t ] ẋ s ds.9 with an appropriate vector ζ t and a matrix S, respectively, it was estimated as 2ζ T t S x t x t d t ζ T t d SZ ST ζ t.0 with d t SZ ST enlarged as d SZ ST. The problem of delay-dependent stability criterion for continuous systems with two additive time-varying delay components has been considered in this paper. By constructing a new class of Lyapunov functional and using a new convex polyhedron method, a new stability criterion is derived in terms of linear matrix inequalities. The obtained stability criterion is less conservative than some existing ones. Finally, numerical examples are given to indicate less conservatism of the stability results. Definition.. Let Φ, Φ 2,..., Φ N : R m R n be a given finite number of functions such that they have positive values in an open subset D of R m. Then, a reciprocally convex combination of these functions over D is a function of the form α Φ α 2 Φ 2 α N Φ N : D R n,. where the real numbers α i satisfy α i > 0and i α i. The following Lemma.2 suggests a lower bound for a reciprocally convex combination of scalar positive functions Φ i f i.

4 4 Applied Mathematics Lemma.2 See 0. Let f,f 2,...,f N : R m R have positive values in an open subset D of R m. Then, the reciprocally convex combination of f i over D satisfies min {α i α i >0, f i t i α i } α i i i f i t max g i,j t g i,j t i / j.2 subject to { [ ] } fi t g g i,j : R m i,j t R, g j,i t Δ g i,j t, 0..3 g i,j t f j t In the following, we present a new stability criterion by a convex polyhedron method and Lemma Main Results Theorem 2.. System.5 with delays d t and d 2 t satisfying.6 is asymptotically stable if there exist symmetric positive definite matrices P, Q, Q 2, Q 3, Q 4, Q 5, Q 6, Z, Z, Z 2 and any matrices S 2, N, M, L, S, P, P 2 with appropriate dimensions, such that the following LMIs hold: [ ] Z S2 0, Z 2. E d N d 2 L E 3 d Z 0 < 0, d 2 Z E d N d 2 S E 4 d Z 0 < 0, d 2 Z E d M d 2 L E 23 d Z 0 < 0, d 2 Z E d M d 2 S E 24 d Z 0 < 0, d 2 Z 2 2.5

5 Applied Mathematics 5 where E E 2 S T E 8 E 22 E A T d P T 2 E E E [ N L 0 0 M N M S L S 0 ] Q E Q 6 0 E 88 [ N L 0 0 M N M S L S 0 ] T, E Q Q 2 Q 3 Q 4 Q 5 Q 6 Z P A A T P T, E 2 S T 2 Z P A d, E 8 P P A T P T 2, E 22 τ Q 2Z S 2 S T 2, E 23 S T 2 Z, E 33 Q 2 Z, E 44 τ Q 3, E 66 τ 2 Q 5, E 88 d 2 Z d Z d 2 Z 2 P 2 P T Proof. Construct a new Lyapunov functional candidate as V x t V x t V 2 x t V 3 x t V 4 x t, V x t x T t Px t, V 2 x t x T s Q x s ds x T s Q 2 x s ds x T s Q 3 x s ds t d t t d t x T s Q 4 x s ds x T s Q 5 x s ds x T s Q 6 x s ds, t d 2 t d 2 t 0 V 3 x t d ẋ T s Zẋ s ds dθ, d t θ 0 V 4 x t ẋ T s Z ẋ s ds dθ d t θ 0 d 2 t θ ẋ T s Z 2 ẋ s ds dθ. 2.7 Remark 2.2. Our paper fully uses the information about d t, d t, andd 2 t, but 5, 6 only use the information about d t and d 2 t, when constructing the Lyapunov functional V x t. So the Lyapunov functional in our paper is more general than that in 5, 6, and the stability criteria in our paper may be more applicable.

6 6 Applied Mathematics The time derivative of V x t along the trajectory of system.5 is given by V x t 2x T t Pẋ t, 2.8 V 2 x t x T t Q Q 2 Q 3 Q 4 Q 5 Q 6 x t τ x T t d t Q x t d t x T t d Q 2 x t d τ x T t d t Q 3 x t d t x T t d Q 4 x t d τ 2 x T t d 2 t Q 5 x t d 2 t 2.9 x T t d 2 Q 6 x t d 2, d t V 3 x t d 2 ẋ T t Zẋ t d ẋ T s Zẋ s ds d ẋ T s Zẋ s ds, t d t d t d t V 4 x t ẋ T t d Z d 2 Z 2 ẋ t ẋ T s Z ẋ s ds ẋ T s Z ẋ s ds t d2 t ẋ T s Z 2 ẋ s ds ẋ T s Z 2 ẋ s ds. t d 2 t t d The V 3 x t is upper bounded by V 3 x t d 2 ẋ T t Zẋ t d d d t ζt t e 2 e 3 Z e 2 e 3 T ζ t d d t ζt t e e 2 Z e e 2 T ζ t e T d 2 ẋ T t Zẋ t ζ T 2 T et 3 Z S 2 e T 2 et 3 t e T et 2 Z e T ζ t, et 2 S T where the inequality in 2.2 comes from the Jensen inequality lemma, and that of 2.3 from Lemma.2 as T β ζ T t α e 2 e 3 T β Z S 2 α β e e 2 T S T α e 2 e 3 T 2 Z α β e e 2 T ζ t 0, 2.4

7 Applied Mathematics 7 where ζ T t [ x T t x T t d t x T t d x T t d t x T t d x T t d 2 t x T t d 2 ẋ T t ], e ( I ) T, e2 ( 0 I ) T, e3 ( 0 0 I ) T, 2.5 α d d t /d, β d t /d. Note that when d t d or d t 0, one can obtain ζ T t e 2 e 3 0orζ T t e e 2 0, respectively. So the relation 2.3 also holds. By the Jensen inequality lemma, it is easy to obtain ẋ T s Z ẋ s ds d t U T Z U, t d t ẋ T s Z ẋ s ds d d t U T 2 Z U 2, ẋ T s Z 2 ẋ s ds d 2 t U T 3 Z 2U 3, t d 2 t 2.6 d2 t ẋ T s Z 2 ẋ s ds d 2 d 2 t U T 4 Z 2U 4, t d 2 where U d t t ẋ s ds, U 3 d 2 t t d 2 t ẋ s ds, U 2 U 4 d d t d 2 d 2 t d t d2 t t d 2 ẋ s ds, ẋ s ds, 2.7 lim d t 0 d t t ẋ s ds ẋ t, lim d t d d d t lim d 2 t 0 d 2 t t d 2 t d t ẋ s ds ẋ t d, ẋ s ds ẋ t, 2.8 lim d 2 t d 2 d 2 d 2 t d2 t t d 2 ẋ s ds ẋ t d 2.

8 8 Applied Mathematics From the Leibniz-Newton formula, the following equations are true for any matrices N, M, L, S, P, P 2 with appropriate dimensions 2ζ T t N x t x t d t d t U 0, 2ζ T t M x t d t x t d d d t U 2 0, 2ζ T t L x t x t d 2 t d 2 t U 3 0, 2.9 2ζ T t S x t d 2 t x t d 2 d 2 d 2 t U 4 0, ] 2 [x T t P ẋ T t P 2 ẋ t Ax t A d x t d t 0. Hence, according to , we can obtain V x t ξ T t Eξ t, 2.20 where ξ T t [ ζ T t U T U T 2 U T 3 U T ] 4, E d t N d d t M d 2 t L d 2 d 2 t S d t Z E d d t Z 0 0. d 2 t Z 2 0 d 2 d 2 t Z If E<0, then there exists a scalar ε>0, such that V x t ξ T t Eξ t εξ T t ξ t εx T t x t < 0, x t / The E<0leadsford t d to E < 0 and leads for d t 0toE 2 < 0, where E d N d 2 t L d 2 d 2 t S d Z 0 0 E < 0, 2.23 d 2 t Z 2 0 d 2 d 2 t Z 2 E d M d 2 t L d 2 d 2 t S d Z 0 0 E 2 < d 2 t Z 2 0 d 2 d 2 t Z 2

9 Applied Mathematics 9 It is easy to see that E results from E d t d, where we deleted the zero row and the zero column. Define ξ T t [ ζ T t U T U T 3 U T ] 4, ξ T 2 t [ ζ T t U T 2 U T 3 U T ] , The LMI 2.23 and 2.24 imply 2.22 because d t ζ T d t E ζ t d d t ζ T 2 d t E 2ζ 2 t ξ T t Eξ t εx T t x t 2.26 and E is convex in d t 0,d. LMI 2.23 leads for d 2 t d 2 to LMI 2.2 and for d 2 t 0toLMI 2.3. Itiseasy to see that E 3 results from E d2 t d 2, where we deleted the zero row and the zero column. The LMI 2.2 and 2.3 imply 2.23 because d 2 t ζ T 3 d t E 3ζ 3 t d 2 d 2 t ζ T 4 2 d t E 4ζ 4 t ξ T t E ξ t < and E is convex in d 2 t 0,d 2, where ξ T 3 t [ ζ T t U T U T ] 3, ξ T 4 t [ ζ T t U T U T ] E 3 and E 4 are defined in Theorem 2.. Similarly, the LMI 2.4 and 2.5 imply 2.24 because d 2 t ζ T 23 d t E 23ζ 23 t d 2 d 2 t ζ T 24 2 d t E 24ζ 24 t ξ T 2 t E 2ξ 2 t < and E 2 is convex in d 2 t 0,d 2, where ξ T 23 t [ ζ T t U T 2 U T ] 3, ξ T 24 t [ ζ T t U T 2 U T ] E 23 and E 24 are defined in Theorem 2.. According to the above analysis, one can conclude that the system.5 with delays d t and d 2 t satisfying.6 is asymptotically stable if the LMIs hold. From the proof of Theorem 2., one can obtain that E is negative definite in the rectangle 0 d t d,0 d 2 t d 2, only if it is negative definite at all vertices. We call this method as the convex polyhedron method.

10 0 Applied Mathematics Remark 2.3. To avoid the emergence of the reciprocally convex combination in 2.2, similar to 9, the integral terms in 2.0 can be upper bounded by d ẋ T s Zẋ s ds x t τ t x t d T Z x t τ t x t d t d x t x t τ t T Z x t z t τ t ( γ ) x t τ t x t d T Z x t τ t x t d 2.3 γ x t x t τ t T Z x t x t τ t which results in a convex combination on γ. However, Theorem 2. directly handles the inversely weighted convex combination of quadratic terms of integral quantities by utilizing the result of Lemma.2, which achieves performance behavior identical to the approaches based on the integral inequality lemma but with much less decision variables, comparable to those based on the Jensen inequality lemma. Remark 2.4. Compared to some existing ones, the estimation of V x t in the proof of Theorem 2. is less conservative due to the convex polyhedron method is employed. More specifically, d t ẋ T s Z ẋ s ds is retained, while ẋ T s Z ẋ s ds is divided into t ẋt s Z ẋ s ds and d t ẋ T s Z ẋ s ds. When the two integrals together with others are handled by using free weighting matrix method, instead of enlarging some term d t SZ ST as d SZ ST. The convex polyhedron method is employed to verify the negative definiteness of V x t. Therefore, Theorem 2. is expected to be less conservative than some results in the literature. Remark 2.5. The case in which only two additive time-varying delay components appear in the state has been considered, and the idea in this paper can be easily extended to the system.3 with multiple additive delay components satisfying.4. Choose the Lyapunov functional as V x t V x t V 2 x t V 3 x t V 4 x t, V x t x T t Px t, V 2 x t t d t n i x T s Q x s ds x T s Q 2 x s ds t d t d i x T s Q 4i x s ds, n i t d i t x T s Q 3i x s ds 0 V 3 x t d V 4 x t n i d t θ 0 d i ẋ T s Zẋ s ds dθ, t θ ẋ T s Z i ẋ s ds dθ. 2.32

11 Applied Mathematics Table : Calculated delay bounds for different cases. Stability conditions Delay bound d 2 for given d Delay bound d for given d 2 d d.2 d.5 d d d , 2, Infeasible Theorem Then, the corresponding stability result can be easily derived similar to the proof of Theorem 2.. The result is omitted due to complicated notation. Remark 2.6. The stability condition presented in Theorem 2. is for the nominal system. However, it is easy to further extend Theorem 2. to uncertain systems, where the system matrices A and A d contain parameter uncertainties either in norm-bounded or polytopic uncertain forms. The reason why we consider the simplest case is to make our idea more lucid and to avoid complicated notations. 3. Illustrative Example Example 3.. Consider system.5 with the following parameters: [ ] [ ] A, A d 0 0.9, assuming d t 0., d 2 t Our purpose is to calculate the upper bound d of delay d t, ord 2 of delay d 2 t, when the other is known, below which the system is asymptotically stable. By combining the two delay components together, some existing stability results can be applied to this system. The calculation results obtained by Theorem 2., in this paper, Theorem in 6, 2, 5, 6, 4, Theorem 2 for different cases are listed in Table. It can be seen from the Table that Theorem 2., in this paper, yields the least conservative stability test than other results. Example 3.2. Consider system.5 with the following parameters: [ ] [ ] A, A d We assume condition : d t 0.2, d 2 t 0.5; condition 2: d t 0.2, d 2 t 0.3, and under the two cases above, respectively. Table 2 lists the corresponding upper bounds of d 2 for given d. This numerical illustrates the effectiveness of the derived results.

12 2 Applied Mathematics Table 2: Allowable upper bound of d 2 for various d. d Condition d Condition 2 d Conclusions This paper has investigated the stability problem for continuous systems with two additive time-varying delay components. By constructing a new class of Lyapunov functional and using a new convex polyhedron method, a new delay-dependent stability criterion is derived in terms of linear matrix inequalities. The obtained stability criterion is less conservative than some existing ones. Finally, numerical examples are given to illustrate the effectiveness of the proposed method. Acknowledgments The authors would like to thank the editors and the reviewers for their valuable suggestions and comments which have led to a much improved paper. This work was supported by research on the model and method of parameter identification in reservoir simulation under Grant PLN2. References W.-J. Mao and J. Chu, Quadratic stability and stabilization of dynamic interval systems, IEEE Transactions on Automatic Control, vol. 48, no. 6, pp , M. da la Sen, Quadratic stability and stabilization of switched dynamic systems with uncommensurate internal point delays, Applied Mathematics and Computation, vol. 85, no., pp , S. Xu, J. Lam, and C. Yang, Quadratic stability and stabilization of uncertain linear discrete-time systems with state delay, Systems & Control Letters, vol. 43, no. 2, pp , Y. He, Q.-G. Wang, C. Lin, and M. Wu, Delay-range-dependent stability for systems with timevarying delay, Automatica, vol. 43, no. 2, pp , X. Jiang and Q.-L. Han, On H control for linear systems with interval time-varying delay, Automatica, vol. 4, no. 2, pp , X.-J. Jing, D.-L. Tan, and Y.-C. Wang, An LMI approach to stability of systems with severe timedelay, IEEE Transactions on Automatic Control, vol. 49, no. 7, pp , E. Fridman, U. Shaked, and K. Liu, New conditions for delay-derivative-dependent stability, Automatica, vol. 45, no., pp , P. Park and J. W. Ko, Stability and robust stability for systems with a time-varying delay, Automatica, vol. 43, no. 0, pp , H. Shao, New delay-dependent stability criteria for systems with interval delay, Automatica, vol. 45, no. 3, pp , P. Park, J. W. Ko, and C. K. Jeong, Reciprocally convex approach to stability of systems with timevarying delays, Automatica, vol. 47, no., pp , 20. P. Park, A delay-dependent stability criterion for systems with uncertain time-invariant delays, IEEE Transactions on Automatic Control, vol. 44, no. 4, pp , E. Fridman and U. Shaked, Delay-dependent stability and H control: constant and time-varying delays, International Control, vol. 76, no., pp , J.-H. Kim, Delay and its time-derivative dependent robust stability of time-delayed linear systems with uncertainty, IEEE Transactions on Automatic Control, vol. 46, no. 5, pp , M. Wu, Y. He, J.-H. She, and G.-P. Liu, Delay-dependent criteria for robust stability of time-varying delay systems, Automatica, vol. 40, no. 8, pp , 2004.

13 Applied Mathematics 3 5 J. Lam, H. Gao, and C. Wang, Stability analysis for continuous systems with two additive timevarying delay components, Systems & Control Letters, vol. 56, no., pp. 6 24, H. Gao, T. Chen, and J. Lam, A new delay system approach to network-based control, Automatica, vol. 44, no., pp , 2008.

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