Delay-Dependent Stability Criteria for Linear Systems with Multiple Time Delays
|
|
- Tyrone Wilkerson
- 5 years ago
- Views:
Transcription
1 Delay-Dependent Stability Criteria for Linear Systems with Multiple Time Delays Yong He, Min Wu, Jin-Hua She Abstract This paper deals with the problem of the delay-dependent stability of linear systems with multiple time delays. A new method is first presented for a system with two time delays, in which free weighting matrices are used to express the relationships among the terms of the Leibniz-Newton formula. Next, this method is used to show the equivalence between a system with two identical time delays and a system with a single time delay. Then, a numerical example verifies that the criterion given in this paper is effective and is a significant improvement over existing ones. Finally, the basic idea is extended to a system with multiple time delays. Key words: delay-dependent criteria, stability, linear matrix inequality (LMI), multiple delays, free weighting matrix. 1 Introduction The problem of deriving delay-dependent stability criteria for linear delay systems has attracted the attention of many researchers in the last decade (e.g. [1 22]). They have mainly employed a Lyapunov functional to derive delay-dependent criteria; and the This work was supported in part by the National Science Foundation of China under Grant Nos and ; the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of the Ministry of Education, P. R. China; and the Doctor Subject Foundation of China under Grant No Y. He and M. Wu are with the School of Information Science and Engineering, Central South University, Changsha , China. Y. He is aslo with the School of Mathematical Science and Computing Technology, Central South University, Changsha , China. Corresponding author: min@mail.csu.edu.cn J.-H. She is with the School of Bionics, Tokyo University of Technology, Tokyo, , Japan. 1
2 resulting methods are generally classified into two categories: the discretized-lyapunovfunctional method and methods based on the Leibniz-Newton formula. It is difficult to extend the former type (for instance, [6, 7, 10]) to solve the synthesis problem for a control system, and the latter type generally requires a system transformation. Four basic fixed model transformations have been presented (e.g., see [5]); but they all entail a certain degree of conservativeness, which leaves room for further investigation. Recently, He et al. [14, 15] and Wu et al. [21, 22] devised a new method that employed free weighting matrices to express the relationships between the terms of the Leibniz- Newton formula. This overcomes the conservativeness of methods involving a fixed model transformation. On the other hand, it is well known that, if a linear system with a single time delay, h, is not stable for a delay of any length but is stable for h = 0, then there must exist a positive number h for which the system is stable for 0 h h. Many researchers have simply extended this idea to a system with multiple time delays. However, this simple extension may lead to conservatism. For example, [4, 5] dealt with a linear system containing two time delays: ẋ(t) = A 0 x(t) + A 1 x(t h 1 ) + A 2 x(t h 2 ). (1) The upper bounds h 1 and h 2 on h 1 and h 2, respectively, are selected such that (1) is stable for 0 h 1 h 1 and 0 h 2 h 2. However, the ranges of h 1 and h 2 that ensure the stability of the system (1) are conservative because they are guaranteed from zero to the upper bound, even though it may not be necessary for them to start from zero. One reason for this is that the relationship between h 1 and h 2 was not taken into account in the procedure for finding the upper bounds. Specifically, the criterion for a linear system with a single delay ẋ(t) = A 0 x(t) + (A 1 + A 2 )x(t h), (2) should be equivalent to that for a system with two delays (1) in which h 1 = h 2 ; but this equivalence cannot be demonstrated by the methods in [4, 5]. In this paper, new delay-dependent criteria are proposed based on a new method. Concretely, a criterion is first established for a linear system with two time delays. Not only the relationships between x(t h 1 ) and x(t) x(t) ẋ(s)ds, but also that between x(t h 2 ) and x(t h 1 ) ẋ(s)ds, and x(t h 2 ) and h1 ẋ(s)ds are taken into account. Note that the last relationship is precisely that between h 1 and h 2. All these relationships are expressed in terms of weighting matrices and are determined 2
3 from the solutions of linear matrix inequalities (LMIs). Furthermore, the equivalence between systems (1) and (2) for h 1 = h 2 is demonstrated. A numerical example is used to show that the criteria given in this paper are effective and are a significant improvement over existing ones. Finally, the basic idea is extended to a system with multiple time delays. 2 Delay-Dependent Stability Consider the following linear system, Σ, with multiple time delays: ẋ(t) = A i x(t h i ), t > 0, Σ : i=0 x(t) = φ(t), t [ h, 0], (3) where x(t) R n is the state vector; h 0 = 0 and h i 0 (i = 1, 2,, m) are constant time delays; h := max{h 1, h 2,, h m }; and A i R n n (i = 0, 1,, m) are constant matrices. The initial condition, φ(t), denotes a continuous vector-valued initial function of t [ h, 0]. 2.1 Two Time Delays First, let s consider the case m = 2. The following delay-dependent stability criterion, which takes the relationship between h 1 and h 2 into account, is established for Σ. Theorem 1 For given scalars h i 0 (i = 1, 2), the system Σ is asymptotically stable if there exist symmetric positive definite matrices P = P T > 0 and Q i = Q T i > 0 (i = 1, 2), symmetric semi-positive definite matrices W i = W T i 0, X ii = X T ii 0, Y ii = Y T ii 0 and Z ii = Z T ii 0 (i = 1, 2, 3), and any matrices N i, S i, T i (i = 1, 2, 3) and X ij, Y ij, Z ij (1 i < j 3) such that the following LMIs hold: Φ 11 Φ 12 Φ 13 Φ = Φ T 12 Φ 22 Φ 23 < 0, (4) Φ T 13 Φ T 23 Φ 33 Ψ 1 = X 11 X 12 X 13 N 1 X12 T X 22 X 23 N 2 X13 T X23 T X 33 N 3 N1 T N2 T N3 T W 1 0, (5) 3
4 Y 11 Y 12 Y 13 S 1 Y12 T Y 22 Y 23 S 2 Ψ 2 = Y13 T Y23 T Y 33 S 3 S1 T S2 T S3 T W 2 0, (6) Ψ 3 = Z 11 Z 12 Z 13 kt 1 Z12 T Z 22 Z 23 kt 2 Z13 T Z23 T Z 33 kt 3 kt1 T kt2 T kt3 T W 3 0, (7) where k = { 1; if h1 h 2, 1; if h 1 < h 2, and Φ 11 = P A 0 + A T 0 P + Q 1 + Q 2 + N 1 + N T 1 + S 1 + S T 1 + AT 0 HA 0 + h 1 X 11 + h 2 Y 11 + h 1 h 2 Z 11, Φ 12 = P A 1 N 1 + N T 2 + ST 2 T 1 + A T 0 HA 1 + h 1 X 12 + h 2 Y 12 + h 1 h 2 Z 12, Φ 13 = P A 2 + N T 3 + ST 3 S 1 + T 1 + A T 0 HA 2 + h 1 X 13 + h 2 Y 13 + h 1 h 2 Z 13, Φ 22 = Q 1 N 2 N T 2 T 2 T T 2 + AT 1 HA 1 + h 1 X 22 + h 2 Y 22 + h 1 h 2 Z 22, Φ 23 = N T 3 S 2 + T 2 T T 3 + AT 1 HA 2 + h 1 X 23 + h 2 Y 23 + h 1 h 2 Z 23, Φ 33 = Q 2 S 3 S T 3 + T 3 + T T 3 + AT 2 HA 2 + h 1 X 33 + h 2 Y 33 + h 1 h 2 Z 33, H = h 1 W 1 + h 2 W 2 + h 1 h 2 W 3. Proof. First, let s consider the case h 1 h 2. Choose the following candidate Lyapunov-Krasovskii functional: V 2 (x t ) := x T (t)p x(t) + x T (s)q 1 x(s)ds h 1 t+θ h2 h 1 t+θ ẋ T (s)w 1 ẋ(s)dsdθ + ẋ T (s)w 3 ẋ(s)dsdθ, 0 h 2 x T (s)q 2 x(s)ds t+θ ẋ T (s)w 2 ẋ(s)dsdθ (8) 4
5 where P = P T > 0, Q i = Q T i > 0 (i = 1, 2), and W i = Wi T 0 (i = 1, 2, 3) are to be determined. Calculating the derivative of V 2 (x t ) along the solutions of Σ yields V 2 (x t ) = 2x T (t)p [A 0 x(t) + A 1 x(t h 1 ) + A 2 x(t h 2 )] +x T (t)q 1 x(t) x T (t h 1 )Q 1 x(t h 1 ) +x T (t)q 2 x(t) x T (t h 2 )Q 2 x(t h 2 ) +h 1 ẋ T (t)w 1 ẋ(t) +h 2 ẋ T (t)w 2 ẋ(t) t +(h 1 h 2 )ẋ T (t)w 3 ẋ(t) ẋ T (s)w 1 ẋ(s)ds ẋ T (s)w 2 ẋ(s)ds h2 ẋ T (s)w 3 ẋ(s)ds. (9) Using the Leibniz-Newton formula, it is clear that, for any matrices N i, S i and T i (i = 1, 2, 3), the following are true. ] 2 [x T (t)n 1 + x T (t h 1 )N 2 + x T (t h 2 )N 3 [ ] x(t) x(t h 1 ) ẋ(s)ds = 0, ] 2 [x T (t)s 1 + x T (t h 1 )S 2 + x T (t h 2 )S 3 [ x(t) x(t h 2 ) ] 2 [x T (t)t 1 + x T (t h 1 )T 2 + x T (t h 2 )T 3 [ x(t h 2 ) x(t h 1 ) h2 ] ẋ(s)ds = 0, ẋ(s)ds On the other hand, for any appropriately dimensioned matrices X ii = X T ii 0, Y ii = Y T ii ] = 0. 0, Z ii = Z T ii 0 (i = 1, 2, 3), and X ij, Y ij, Z ij (1 i < j 3), the following equation holds: where (10) (11) (12) T x(t) Λ 11 Λ 12 Λ 13 x(t) x(t h 1 ) Λ T 12 Λ 22 Λ 23 x(t h 1 ) = 0, (13) x(t h 2 ) Λ T 13 Λ T 23 Λ 33 x(t h 2 ) Λ ij = h 1 (X ij X ij ) + h 2 (Y ij Y ij ) + (h 1 h 2 )(Z ij Z ij ), 1 i j 3. If we add the terms on the left in (10)-(13) to V 2 (x t ), then based on the fact that, for any r 0 and any function f(t) t r f(t)ds = rf(t), 5
6 we can write V 2 (x t ) as V 2 (x t ) = ξ1 T (t)φξ 1(t) ξ2 T (t, s)ψ 1 ξ 2 (t, s)ds h 1 t h2 ξ2 T (t, s)ψ 2 ξ 2 (t, s)ds ξ2 T (t, s)ψ 3 ξ 2 (t, s)ds, (14) where ξ 1 (t) = [ T [ T x T (t) x T (t h 1 ) x T (t h 2 )], ξ2 (t, s) = ξ1 T (t) ẋ (s)] T, and Φ and Ψ i (i = 1, 2, 3) (k = 1 in Ψ 3 ) are defined in (4)-(7). If Φ < 0 and Ψ i 0 (i = 1, 2, 3), then V 2 (x t ) < 0 for any ξ 1 (t) 0. So, Σ is asymptotically stable if LMIs (4)-(7) hold. is On the other hand, when h 1 < h 2, one candidate Lyapunov-Krasovskii functional V 2 (x t ) := x T (t)p x(t) + x T (s)q 1 x(s)ds h 1 t+θ h1 h 2 t+θ and Eq. (12) can be rewritten as ẋ T (s)w 1 ẋ(s)dsdθ + ẋ T (s)w 3 ẋ(s)dsdθ, ] 2 [x T (t)t 1 + x T (t h 1 )T 2 + x T (t h 2 )T 3 [ 0 h 2 x T (s)q 2 x(s)ds t+θ x(t h 2 ) x(t h 1 ) + ẋ T (s)w 2 ẋ(s)dsdθ h1 ẋ(s)ds ] = 0. (15) (16) Then, following the procedure for the case h 1 h 2 yields a similar result; but note that, in this case, k = 1 in (7). Remark 1: The main modification of the candidate Lyapunov-Krasovskii functional is that we added the last term, which contains an integral of the state with respect to h 1 and h 2. This term plays an important role. Without it, the stability is guaranteed from 0 to the upper bounds on h 1 and h 2 ; but with it, the stability range is from some lower (possibly non-zero) bound to the upper bound for each h i (i = 1, 2). This enlarges the stability range and, as a result, reduces the conservatism. This paper employs Eqs (12) and (16) for the first time in the calculation of the derivative of the Lyapunov-Krasovskii functional. This improvement reduces the conservatism of previous treatments arising from the replacement of the term x(t h) with x(t) in some places but not in others. 6 t h ẋ(s)ds
7 x(t) h1 Remark 2: In Theorem 1, the relationships between the terms x(t h 1 ) and ẋ(s)ds, x(t h 2 ) and x(t) ẋ(s)ds, and x(t h 2 ) and x(t h 1 ) ẋ(s)ds have been taken into account through the free weighting matrices N i, S i and T i (i = 1, 2, 3), respectively. They are determined by solving the LMIs in Theorem 1. The criterion for the case h 1 = h 2 should be equivalent to a criterion for a single delay. However, this cannot be demonstrated by previous methods. In contrast, since the relationship between h 1 and h 2 is taken into account in the above theorem, the equivalence between Theorem 1 for two identical delays and a criterion for a single delay can easily be shown, as explained below. Let s begin with a criterion for a single delay derived directly from Theorem 1. Corollary 1 For m = 1 and a given scalar h 1 0, the linear system Σ is asymptotically stable if there exist symmetric positive definite matrices P = P T > 0 and Q = Q T > 0, symmetric semi-positive definite matrices W = W T 0 and X ii = X ii T 0 (i = 1, 2, and any matrices X 12 and N i (i = 1, 2) such that the following LMIs hold. Φ 11 Φ12 < 0, (17) Φ T 12 Φ 22 X 11 X12 N1 X 12 T X 22 N2 0, (18) N 1 T N 2 T W where Φ 11 = P A 0 + A T P 0 + Q + N 1 + N 1 T + AT HA h 1 X11, Φ 12 = P A 1 N 1 + N 2 T + AT HA h 1 X12, Φ 22 = Q N 2 N 2 T + AT HA h 1 X22, H = h 1 W. We now show that Corollary 1 is equivalent to Theorem 1 for h 1 = h 2 when A 1 is replaced with A 1 + A 2 in Φ 12 and Φ 22. If the third row and column of (4) are inserted into the second row and column, respectively, then (4) is equivalent to the following inequality: Φ 11 Π 12 Φ 13 Π = Π 12 Π 22 Π 23 < 0, (19) Φ T 13 Π T 23 Φ 33 7
8 where Π 12 = P A 1 + P A 2 + N T 2 + N T 3 N 1 + S T 2 + ST 3 S 1 + A T 0 H(A 1 + A 2 ) + h 1 (X 12 + X 13 ) + h 2 (Y 12 + Y 13 ) + h 1 h 2 (Z 12 + Z 13 ), Π 22 = (Q 1 +Q 2 ) N 3 N T 3 S 3 S T 3 N 2 N T 2 S 2 S T 2 +(A 1+A 2 ) T H(A 1 +A 2 )+ h 1 (X 22 +X 23 +X T 23 +X 33)+h 2 (Y 22 +Y 23 +Y T 23 +Y 33)+ h 1 h 2 (Z 22 +Z 23 +Z T 23 +Z 33), Π 23 = Q 2 S 3 S T 3 + T 3 N T 3 S 2 + T 2 + (A 1 + A 2 ) T HA 2 + h 1 (X 23 + X 33 ) + h 2 (Y 23 + Y 33 ) + h 1 h 2 (Z 23 + Z 33 ), and Φ 11, Φ 13, Φ 33, and H are defined in (4). First, if LMIs (17) and (18) in Corollary 1 hold (where A 1 is replaced with A 1 +A 2 ), the solutions can be expressed as appropriate forms of the feasible solutions of LMIs (19), (5), (6), and (7). In fact, for the feasible solutions of LMIs (17) and (18) in Corollary 1, we can set P = P, S i = 0 (i = 1, 2, 3), N 1 = N 1, N 2 = N 2, N 3 = 0, 0 < Q 2 < Q, Q 1 = Q Q 2, T 1 = P A 2 A T 0 HA 2, T 2 = Q 2 (A 1 + A 2 ) T HA2, T 3 = 0, W 1 = W, W 2 = 0, X 11 = X 11, X 12 = X 12, X 13 = 0, X 22 = X 22, X 23 = 0, X 33 = 0 and Y ij = 0 (1 i j 3). Then, Z ij (1 i j 3) and W 3 are the feasible solutions of the following LMI: Z 11 Z 12 Z 13 T 1 Z T 12 Z 22 Z 23 T 2 Z13 T Z23 T 0. (20) Z 33 0 T1 T T2 T 0 W 3 The above matrices must be the feasible solutions of LMIs (19), (5), (6), and (7). Consequently, Theorem 1 for h 1 = h 2 contains Corollary 1. On the other hand, for the feasible solutions of LMIs (19), (5), (6), and (7), setting P = P, Q = Q 1 +Q 2, W = W 1 + W 2, N 1 = N 1 +S 1, N 2 = N 2 +N 3 +S 2 +S 3, X 11 = X 11 + Y 11, X12 = X 12 +Y 12 +X 13 +Y 13, and X 22 = X 22 +Y 22 +X 23 +Y 23 +X T 23 +Y T 23 +X 33+Y 33 yields the feasible solutions of LMIs (17) and (18) in Corollary 1. That is, Corollary 1 contains Theorem 1 for h 1 = h 2. Thus, Corollary 1 and Theorem 1 are equivalent for the case h 1 = h Numerical Example Consider the stability of the system Σ with m = 2 and A 0 = , A 1 =, A 2 = (21) 8
9 If h 1 = h 2, the system is equivalent to Σ with m = 1 and A 0 = , A 1 = (22) [4, 5] and Corollary 1 show that the system (22) is asymptotically stable only for 0 h However, [4, 5] also showed that the system (21) was asymptotically stable for 0 h 1 = h This result is conservative for multiple delays. The main reason for the conservatism is that they did not take the relationship between h 1 and h 2 into account. In contrast, Theorem 1 in this paper shows that (21) is asymptotically stable for 0 h 1 = h Clearly, the upper bound on h 1 = h 2 is much larger than that in [4,5]; and it is the same as that obtained for a single delay. Regarding the calculated range of h 2 that ensures that (21) is asymptotically stable for a given h 1, a detailed comparison of the results of our method and the method in [4, 5] is shown in Table 1; and the results are also illustrated in Figure 2.2. It is clear that our method significantly enlarges the stability domains of h 1 and h 2. The stable range for a single delay is generally from 0 to an upper bound; so, we usually just need to find the upper bound. Since Fridman and Shaked simply extended the method for a single delay to two delays in [4, 5], they were only able to provide a stable upper bound, but not an appropriate (possibly non-zero) lower bound, for h 2. In the numerical example, their method yielded h 1 < 2.25; and it was impossible to find the stable range of h 2 for h In contrast, our method employs a cross term for h 1 and h 2 (the last term of (8) and (15)) to construct a new type of Lyapunov- Krasovskii functional. Unlike existing methods, this is not a simple extension of the treatment for a single delay; and the relationship between the two delays is taken into account. Consequently, our method yields a stable range for h 2 rather than a simple upper bound. More specifically, in the numerical example, the stable range of h 2 is much larger than that given by the method in [4, 5] when h 1 < 2.25; and we can even obtain the stable range of h 2 when h Note that in this case, the stable range of h 2 no longer starts from 0. Remark 3: Unlike the discretized-lyapunov-functional approach, our method can easily be extended to the synthesis of a control system, as explained in [4, 5, 19]. 9
10 Table 1: Range of h 2 ensuring asymptotic stability of system (21) for a given h 1. h h 2 (Theorem 1) [0, + ] [0, 3.36] [0, 3.35] [0, 3.34] [0, 3.33] h 2 (Method in [4, 5]) [0, + ] [0, 1.84] [0, 1.81] [0, 1.78] [0, 1.71] h h 2 (Theorem 1) [0, 3.33] [0, 3.33] [0, 3.36] [0, 3.39] [0, 3.43] h 2 (Method in [4, 5]) [0, 1.64] [0, 1.57] [0, 1.42] [0, 1.22] [0, 0.88] h h 2 (Theorem 1) [0, 3.47] [0, 3.52] [0, 3.55] [0.08, 3.57] [0.22, 3.61] h 2 (Method in [4, 5]) [0, 0.40] [0, 0.07] [0, 0] h h 2 (Theorem 1) [0.35, 3.65] [1.04, 3.77] [1.88, 3.90] [3.59, 4.18] [4.47, 4.47] h 2 (Method in [4, 5]) 2.3 Multiple Time Delays In this subsection, Theorem 1 is extended to a system with multiple time delays. For convenience, it is assumed that in (3) 0 = h 0 h 1 h 2 h m. (23) Theorem 2 For given scalars h i 0 (i = 1, 2,, m) that satisfy (23), the system (3) with multiple time delays is asymptotically stable if there exist symmetric positive definite matrices P = P T > 0 and Q i = Q T i > 0 (i = 1, 2,, m), symmetric semipositive definite matrices X (ij) [X (ij) X (ij) 00 X (ij) 01 X (ij) 0m = 01 ]T X (ij) 11 X (ij) 1m (0 i <. [X (ij) 0m ]T [X (ij) 1m ]T X mm (ij) j m) and W (ij) = [W (ij) ] T 0 (0 i < j m), and any matrices N (ij) l (l = 0, 1, 2,, m, 0 i < j m) such that the following LMIs hold: 10
11 4.5 Area 1: stable area using the method in [4,5] Area 1 & Area 2: stable area using our method h Area 1 Area h 1 Figure 1: Comparison of stability ranges obtained with the method in [4, 5] and our method. where Γ (ij) = Ξ 00 Ξ 01 Ξ 0m Ξ T Ξ = 01 Ξ 11 Ξ 1m..... < 0, (24). Ξ T 0m ΞT 1m Ξ mm X (ij) 00 X (ij) 01 X (ij) 0m [X (ij) 01 ]T X (ij) 11 X (ij) 1m [X (ij) 0m ]T [X (ij) 1m ]T X (ij) mm N (ij) 0 N (ij) 1. N (ij) m [N (ij) 0 ] T [N (ij) 1 ] T [N (ij) m ] T W (ij) Ξ 00 = ( P A 0 + A T 0 P + Q i + N (0j) 0 + [N (0j) 0 ] T ) i=1 j=1 +A T 0 GA 0 + (h j h i )X (ij) 00, Ξ 0k = k 1 P A k N (ik) 0 + [N (0i) k ] T + N (kj) 0 i=0 i=1 11 j=k+1 0 (0 i < j m), (25)
12 +A T 0 GA k + k 1 Ξ kk = Q k i=0 +A T k GA k + k 1 Ξ lk = G = i=0 N (ik) l +A T l GA k + ( N (ik) k + [N (ik) l 1 i=0 (h j h i )X (ij) k ] T ) + 0k (h j h i )X (ij) [N (il) k ] T + (h j h i )W (ij). kk j=k+1 (k = 1, 2,, m), j=k+1 ( N (kj) k + [N (kj) k ] T ) (k = 1, 2,, m), N (kj) l + j=l+1 [N (lj) k ] T (h j h i )X (ij) (l = 1, 2,, m, l < k m), Proof. Choose the candidate Lyapunov-Krasovskii functional to be lk V m (x t ) := x T (t) T P x(t) + m 1 + i=1 hi h j t h i x T (s)q i x(s)ds t+θ ẋ T (s)w (ij) ẋ(s)dsdθ, (26) where P = P T > 0, Q i = Q T i > 0 (i = 1, 2,, m), and W (ij) = [W (ij) ] T 0 (0 i < j m) are free matrices that need to be determined. According to the Leibniz-Newton formula, for 0 i < j m and for any matrices N (ij) l (l = 0, 1, 2,, m), the following equation holds: [ m ] [ ] 2 x T (t h l )N (ij) t hi l x(t h i ) x(t h j ) ẋ(s)ds = 0. (27) t h j l=0 On the other hand, for 0 i < j m and for any matrices X (ij) 0, m 1 (h j h i )ζ T 1 (t)[x (ij) X (ij) ]ζ 1 (t) = 0, (28) where ζ 1 (t) = [x T (t) x T (t h 1 ) x T (t h 2 ) x T (t h m )] T. So, the derivative of V m (x t ) along the solutions of Σ can be written as where m 1 V m (x t ) = ζ1 T (t)ξζ 1 (t) hi t h j ζ T 2 (t, s)γ (ij) ζ 2 (t, s)ds, (29) 12
13 ζ 2 (t, s) = [ζ T 1 (t) ẋ(s)]t, and Ξ and Γ (ij) (0 i < j m) are defined in (24) and (25). asymptotically stable if LMIs (24) and (25) hold. From (23), Σ is Remark 5: If i [1,, m 1] such that h i = h i+1, then the system can be transformed into a system with m 1 time delays. Following the explanation in Theorem 1 and Corollary 1, it is easy to see that the delay-dependent condition is equivalent to that for a system with m 1 time delays. 3 Conclusion This paper presented some new delay-dependent stability criteria for linear systems with multiple delays. The free-weighting-matrix method was used in the derivation. Since the method does not employ a system transformation, does not use an inequality to estimate the upper bound on a cross term, and uses free matrices to take the relationships between h i (i = 0, 1, 2,, m) into account, these criteria overcome the conservatism of previous methods. Free weighting matrices that express the reciprocal influences of the terms in the Leibniz-Newton formula can easily be calculated and are determined by LMIs. Unlike existing methods, the stability domain consists of ranges for the time delays rather than just upper bounds. A numerical example demonstrated that the method described in this paper is a significant improvement over previous methods. References [1] Cao, Y. Y., Sun, Y. X., Cheng, C. W.: Delay-dependent robust stabilization of uncertain systems with multiple state delays, IEEE Trans. Automat. Contr. 1998, 43, pp [2] de Souza, C. E., Li, X.: Delay-dependent robust H control of uncertain linear state-delayed systems, Automatica, 1999, 35, pp [3] Fridman, E., Shaked, U.: A descriptor system approach to H control of linear time-delay systems, IEEE Trans. Automat. Contr., 2002, 47, pp [4] Fridman, E., Shaked, U.: An improved Stabilization method for linear time-delay systems, IEEE Trans. Automat. Contr. 2002, 47, pp
14 [5] Fridman, E., Shaked, U.: Delay-dependent stability and H control: constant and time-varying delays, Int. J. Control, 2003, 76, pp [6] Gu, K.: Discretized LMI set in the stability problem of linear uncertain time-delay systems, Int. J. Control, 1997, 68, pp [7] Gu, K.: A generalized discretization scheme of Lyapunov functional in the stability problem of linear uncertain time-delay systems, Int. J. Robust & Nonlinear Contr., 1999, 9, pp [8] Gu, K., Niculescu, S. I.: Additional dynamics in transformed time delay systems, IEEE Trans. Automat. Contr., 2000, 45, pp [9] Gu, K.: A further refinement of discretized Lyapunov functional method for the stability of time-delay systems, Int. J. Control, 2001, 74, pp [10] Gu, K., Niculescu, S. I.: Further remarks on additional dynamics in various model transformations of linear delay systems, IEEE Trans. Automat. Contr., 2001, 46, pp [11] Han, Q. L.: New results for delay-dependent stability of linear systems with timevarying delay, Int. J. Sys. Sci., 2002, 33, pp [12] Han, Q. L.: On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty, Automatica, 2004, 40, pp [13] Han, Q. L.: A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays, Automatica, 2004, 40, pp [14] He, Y., Wu, M., She, J.-H., and Liu, G. P.: Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays, Systems & Control Letters, 2004, 51, pp [15] He, Y., Wu, M., She, J.-H., and Liu, G. P.: Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic type uncertainties, IEEE Trans. Automat. Contr., 2004, 49, pp [16] Kim, J. H.: Delay and its time-derivative dependent robust stability of timedelayed linear systems with uncertainty, IEEE Trans. Automat. Contr., 2001, 46, pp [17] Li, X., de Souza, C. E.: Criteria for robust stability and stabilization of uncertain linear systems with state delay, Automatica, 1997, 33, pp
15 [18] Li, X., de Souza, C. E.: Delay-dependent robust stability and stabilization of uncertain linear delay systems: A Linear Matrix Inequality approacmh, IEEE Trans. Automat. Contr., 1997, 42, pp [19] Moon, Y. S., Park, P., Kwon W. H., and Lee Y.S.: Delay-dependent robust stabilization of uncertain state-delayed systems, Int. J. Control, 2001, 74, pp [20] Park, P.: A delay-dependent stability criterion for systems with uncertain timeinvariant delays, IEEE Trans. Automat. Contr., 1999, 44, pp [21] Wu, M., He, Y., She, J.-H., and Liu, G.P.: Delay-dependent criteria for robust stability of time-varying delay systems, Automatica, 2004, 40, pp [22] Wu, M., He, Y., and She, J.-H.: New delay-dependent stability criteria and stabilizing method for neutral systems, IEEE Trans. Automat. Contr., 2004, 49, pp
Delay-Dependent Exponential Stability of Linear Systems with Fast Time-Varying Delay
International Mathematical Forum, 4, 2009, no. 39, 1939-1947 Delay-Dependent Exponential Stability of Linear Systems with Fast Time-Varying Delay Le Van Hien Department of Mathematics Hanoi National University
More informationResearch Article Convex Polyhedron Method to Stability of Continuous Systems with Two Additive Time-Varying Delay Components
Applied Mathematics Volume 202, Article ID 689820, 3 pages doi:0.55/202/689820 Research Article Convex Polyhedron Method to Stability of Continuous Systems with Two Additive Time-Varying Delay Components
More informationA new robust delay-dependent stability criterion for a class of uncertain systems with delay
A new robust delay-dependent stability criterion for a class of uncertain systems with delay Fei Hao Long Wang and Tianguang Chu Abstract A new robust delay-dependent stability criterion for a class of
More informationDelay-dependent stability and stabilization of neutral time-delay systems
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust Nonlinear Control 2009; 19:1364 1375 Published online 6 October 2008 in Wiley InterScience (www.interscience.wiley.com)..1384 Delay-dependent
More informationSTABILITY ANALYSIS FOR SYSTEMS WITH LARGE DELAY PERIOD: A SWITCHING METHOD. Received March 2011; revised July 2011
International Journal of Innovative Computing, Information and Control ICIC International c 2012 ISSN 1349-4198 Volume 8, Number 6, June 2012 pp. 4235 4247 STABILITY ANALYSIS FOR SYSTEMS WITH LARGE DELAY
More informationDelay-dependent Stability Analysis for Markovian Jump Systems with Interval Time-varying-delays
International Journal of Automation and Computing 7(2), May 2010, 224-229 DOI: 10.1007/s11633-010-0224-2 Delay-dependent Stability Analysis for Markovian Jump Systems with Interval Time-varying-delays
More informationStability Analysis and H Synthesis for Linear Systems With Time-Varying Delays
Stability Analysis and H Synthesis for Linear Systems With Time-Varying Delays Anke Xue Yong-Yan Cao and Daoying Pi Abstract This paper is devoted to stability analysis and synthesis of the linear systems
More informationResults on stability of linear systems with time varying delay
IET Control Theory & Applications Brief Paper Results on stability of linear systems with time varying delay ISSN 75-8644 Received on 8th June 206 Revised st September 206 Accepted on 20th September 206
More informationNew Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems
Systems & Control Letters 43 (21 39 319 www.elsevier.com/locate/sysconle New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems E. Fridman Department of Electrical
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution
More informationOn Computing the Worst-case Performance of Lur'e Systems with Uncertain Time-invariant Delays
Article On Computing the Worst-case Performance of Lur'e Systems with Uncertain Time-invariant Delays Thapana Nampradit and David Banjerdpongchai* Department of Electrical Engineering, Faculty of Engineering,
More informationImproved delay-dependent globally asymptotic stability of delayed uncertain recurrent neural networks with Markovian jumping parameters
Improved delay-dependent globally asymptotic stability of delayed uncertain recurrent neural networks with Markovian jumping parameters Ji Yan( 籍艳 ) and Cui Bao-Tong( 崔宝同 ) School of Communication and
More informationMin Wu Yong He Jin-Hua She. Stability Analysis and Robust Control of Time-Delay Systems
Min Wu Yong He Jin-Hua She Stability Analysis and Robust Control of Time-Delay Systems Min Wu Yong He Jin-Hua She Stability Analysis and Robust Control of Time-Delay Systems With 12 figures Authors Min
More informationImproved Stability Criteria for Lurie Type Systems with Time-varying Delay
Vol. 37, No. 5 ACTA ATOMATICA SINICA May, 011 Improved Stability Criteria for Lurie Type Systems with Time-varying Delay RAMAKRISHNAN Krishnan 1 RAY Goshaidas 1 Abstract In this technical note, we present
More informationOn Dwell Time Minimization for Switched Delay Systems: Free-Weighting Matrices Method
On Dwell Time Minimization for Switched Delay Systems: Free-Weighting Matrices Method Ahmet Taha Koru Akın Delibaşı and Hitay Özbay Abstract In this paper we present a quasi-convex minimization method
More informationNew Stability Criteria for Recurrent Neural Networks with a Time-varying Delay
International Journal of Automation and Computing 8(1), February 2011, 128-133 DOI: 10.1007/s11633-010-0564-y New Stability Criteria for Recurrent Neural Networks with a Time-varying Delay Hong-Bing Zeng
More informationSTABILIZATION FOR A CLASS OF UNCERTAIN MULTI-TIME DELAYS SYSTEM USING SLIDING MODE CONTROLLER. Received April 2010; revised August 2010
International Journal of Innovative Computing, Information and Control ICIC International c 2011 ISSN 1349-4198 Volume 7, Number 7(B), July 2011 pp. 4195 4205 STABILIZATION FOR A CLASS OF UNCERTAIN MULTI-TIME
More informationDelay and Its Time-derivative Dependent Robust Stability of Uncertain Neutral Systems with Saturating Actuators
International Journal of Automation and Computing 74, November 200, 455-462 DOI: 0.007/s633-00-0527-3 Delay and Its ime-derivative Dependent Robust Stability of Uncertain Neutral Systems with Saturating
More informationOn backwards and forwards reachable sets bounding for perturbed time-delay systems
*Manuscript Click here to download Manuscript: 0HieuNamPubuduLeAMC2015revision 2.pdf Click here to view linked References On backwards and forwards reachable sets bounding for perturbed time-delay systems
More informationEects of small delays on stability of singularly perturbed systems
Automatica 38 (2002) 897 902 www.elsevier.com/locate/automatica Technical Communique Eects of small delays on stability of singularly perturbed systems Emilia Fridman Department of Electrical Engineering
More informationOn Delay-Dependent Robust H Control of Uncertain Continuous- and Discrete-Time Linear Systems with Lumped Delays
On Delay-Dependent Robust H Control of Uncertain Continuous- and Discrete-Time Linear Systems with Lumped Delays R. M. Palhares, C. D. Campos, M. C. R. Leles DELT/UFMG Av. Antônio Carlos 6627 3127-1, Belo
More informationResearch Article Delay-Range-Dependent Stability Criteria for Takagi-Sugeno Fuzzy Systems with Fast Time-Varying Delays
Journal of Applied Mathematics Volume 2012rticle ID 475728, 20 pages doi:10.1155/2012/475728 Research Article Delay-Range-Dependent Stability Criteria for Takagi-Sugeno Fuzzy Systems with Fast Time-Varying
More informationFilter Design for Linear Time Delay Systems
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 11, NOVEMBER 2001 2839 ANewH Filter Design for Linear Time Delay Systems E. Fridman Uri Shaked, Fellow, IEEE Abstract A new delay-dependent filtering
More informationTHE phenomena of time delays are often encountered in
0 American Control Conference on O'Farrell Street San Francisco CA USA June 9 - July 0 0 Robust stability criteria for uncertain systems with delay and its derivative varying within intervals Luis Felipe
More informationRobust sampled-data stabilization of linear systems - An input delay approach
Robust sampled-data stabilization of linear systems - An input delay approach Emilia Fridman, Alexandre Seuret, Jean-Pierre Richard To cite this version: Emilia Fridman, Alexandre Seuret, Jean-Pierre Richard.
More informationA Delay-dependent Condition for the Exponential Stability of Switched Linear Systems with Time-varying Delay
A Delay-dependent Condition for the Exponential Stability of Switched Linear Systems with Time-varying Delay Kreangkri Ratchagit Department of Mathematics Faculty of Science Maejo University Chiang Mai
More informationEXPONENTIAL STABILITY OF SWITCHED LINEAR SYSTEMS WITH TIME-VARYING DELAY
Electronic Journal of Differential Equations, Vol. 2007(2007), No. 159, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXPONENTIAL
More informationON THE ROBUST STABILITY OF NEUTRAL SYSTEMS WITH TIME-VARYING DELAYS
ON THE ROBUST STABILITY OF NEUTRAL SYSTEMS WITH TIME-VARYING DELAYS V. J. S. Leite P. L. D. Peres E. B. Castelan S. Tarbouriech UnED Divinópolis CEFET-MG R. Monte Santo, 319 35502-036, Divinópolis - MG
More informationDecentralized Disturbance Attenuation for Large-Scale Nonlinear Systems with Delayed State Interconnections
Decentralized Disturbance Attenuation for Large-Scale Nonlinear Systems with Delayed State Interconnections Yi Guo Abstract The problem of decentralized disturbance attenuation is considered for a new
More informationObserver design for systems with non small and unknown time-varying delay
Observer design for systems with non small and unknown time-varying delay Alexandre Seuret, Thierry Floquet, Jean-Pierre Richard, Sarah Spurgeon To cite this version: Alexandre Seuret, Thierry Floquet,
More informationDelay-Dependent H 1 Control of Uncertain Discrete Delay Systems
European Journal of Control ():9 # EUCA Delay-Dependent H Control of Uncertain Discrete Delay Systems E. Fridman and U. Shaked Department of Electrical Engineering-Systems Tel-Aviv University Tel-Aviv
More informationSTABILIZATION OF LINEAR SYSTEMS VIA DELAYED STATE FEEDBACK CONTROLLER. El-Kébir Boukas. N. K. M Sirdi. Received December 2007; accepted February 2008
ICIC Express Letters ICIC International c 28 ISSN 1881-83X Volume 2, Number 1, March 28 pp. 1 6 STABILIZATION OF LINEAR SYSTEMS VIA DELAYED STATE FEEDBACK CONTROLLER El-Kébir Boukas Department of Mechanical
More informationMemory State Feedback Control for Singular Systems with Multiple Internal Incommensurate Constant Point Delays
Vol. 35, No. 2 ACTA AUTOMATICA SINICA February, 2009 Memory State Feedback Control for Singular Systems with Multiple Internal Incommensurate Constant Point Delays JIANG Zhao-Hui 1 GUI Wei-Hua 1 XIE Yong-Fang
More informationInput/output delay approach to robust sampled-data H control
Systems & Control Letters 54 (5) 71 8 www.elsevier.com/locate/sysconle Input/output delay approach to robust sampled-data H control E. Fridman, U. Shaked, V. Suplin Department of Electrical Engineering-Systems,
More informationAsymptotic stability of solutions of a class of neutral differential equations with multiple deviating arguments
Bull. Math. Soc. Sci. Math. Roumanie Tome 57(15) No. 1, 14, 11 13 Asymptotic stability of solutions of a class of neutral differential equations with multiple deviating arguments by Cemil Tunç Abstract
More informationCircuits, Systems, And Signal Processing, 2012, v. 31 n. 1, p The original publication is available at
Title Stability analysis of markovian jump systems with multiple delay components and polytopic uncertainties Author(s Wang, Q; Du, B; Lam, J; Chen, MZQ Citation Circuits, Systems, And Signal Processing,
More informationROBUST STABILITY TEST FOR UNCERTAIN DISCRETE-TIME SYSTEMS: A DESCRIPTOR SYSTEM APPROACH
Latin American Applied Research 41: 359-364(211) ROBUS SABILIY ES FOR UNCERAIN DISCREE-IME SYSEMS: A DESCRIPOR SYSEM APPROACH W. ZHANG,, H. SU, Y. LIANG, and Z. HAN Engineering raining Center, Shanghai
More informationDelay-Dependent α-stable Linear Systems with Multiple Time Delays
Contemporary Engineering Sciences, Vol 4, 2011, no 4, 165-176 Delay-Dependent α-stable Linear Systems with Multiple Time Delays E Taghizadeh, Y Ordokhani 1 and D Behmardi Department of Mathematics, Alzahra
More informationObserver-based sampled-data controller of linear system for the wave energy converter
International Journal of Fuzzy Logic and Intelligent Systems, vol. 11, no. 4, December 211, pp. 275-279 http://dx.doi.org/1.5391/ijfis.211.11.4.275 Observer-based sampled-data controller of linear system
More informationResearch Article Delay-Dependent Exponential Stability for Discrete-Time BAM Neural Networks with Time-Varying Delays
Discrete Dynamics in Nature and Society Volume 2008, Article ID 421614, 14 pages doi:10.1155/2008/421614 Research Article Delay-Dependent Exponential Stability for Discrete-Time BAM Neural Networks with
More informationA DELAY-DEPENDENT APPROACH TO DESIGN STATE ESTIMATOR FOR DISCRETE STOCHASTIC RECURRENT NEURAL NETWORK WITH INTERVAL TIME-VARYING DELAYS
ICIC Express Letters ICIC International c 2009 ISSN 1881-80X Volume, Number (A), September 2009 pp. 5 70 A DELAY-DEPENDENT APPROACH TO DESIGN STATE ESTIMATOR FOR DISCRETE STOCHASTIC RECURRENT NEURAL NETWORK
More informationDelay-dependent robust stabilisation of discrete-time systems with time-varying delay
Delay-dependent robust stabilisation of discrete-time systems with time-varying delay X.G. Liu, R.R. Martin, M. Wu and M.L. Tang Abstract: The stability of discrete systems with time-varying delay is considered.
More informationCOMPUTATION OF ROBUST H CONTROLLERS FOR TIME-DELAY SYSTEMS USING GENETIC ALGORITHMS
Control and Intelligent Systems, Vol. 35, No. 4, 2007 COMPUTATION OF ROBUST H CONTROLLERS FOR TIME-DELAY SYSTEMS USING GENETIC ALGORITHMS H. Du, N. Zhang, and J. Lam Abstract This paper presents an evolutionary
More informationPAijpam.eu DELAY-RANGE-DEPENDENT MEAN SQUARE STABILITY OF STOCHASTIC SYSTEMS WITH INTERVAL TIME-VARYING DELAYS
International Journal of Pure and Applied Mathematics Volume 94 No. 4 2014, 489-499 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v94i4.4
More informationStability of Linear Distributed Parameter Systems with Time-Delays
Stability of Linear Distributed Parameter Systems with Time-Delays Emilia FRIDMAN* *Electrical Engineering, Tel Aviv University, Israel joint with Yury Orlov (CICESE Research Center, Ensenada, Mexico)
More informationSynchronization criteria for coupled Hopfield neural networks with time-varying delays
Synchronization criteria for coupled Hopfield neural networks with time-varying delays M.J. Park a), O.M. Kwon a), Ju H. Park b), S.M. Lee c), and E.J. Cha d) a) School of Electrical Engineering, Chungbuk
More informationRobust sampled-data stabilization of linear systems: an input delay approach
Automatica 40 (004) 44 446 www.elsevier.com/locate/automatica Technical Communique Robust sampled-data stabilization of linear systems: an input delay approach Emilia Fridman a;, Alexandre Seuret b, Jean-Pierre
More informationA new method to obtain ultimate bounds and convergence rates for perturbed time-delay systems
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR ONTROL Int. J. Robust. Nonlinear ontrol 212; 22:187 188 ublished online 21 September 211 in Wiley Online Library (wileyonlinelibrary.com)..179 SHORT OMMUNIATION
More informationStatic Output Feedback Controller for Nonlinear Interconnected Systems: Fuzzy Logic Approach
International Conference on Control, Automation and Systems 7 Oct. 7-,7 in COEX, Seoul, Korea Static Output Feedback Controller for Nonlinear Interconnected Systems: Fuzzy Logic Approach Geun Bum Koo l,
More informationLinear matrix inequality approach for synchronization control of fuzzy cellular neural networks with mixed time delays
Chin. Phys. B Vol. 21, No. 4 (212 4842 Linear matrix inequality approach for synchronization control of fuzzy cellular neural networks with mixed time delays P. Balasubramaniam a, M. Kalpana a, and R.
More informationControl under Quantization, Saturation and Delay: An LMI Approach
Proceedings of the 7th World Congress The International Federation of Automatic Control Control under Quantization, Saturation and Delay: An LMI Approach Emilia Fridman Michel Dambrine Department of Electrical
More informationRobust Stability Analysis of Teleoperation by Delay-Dependent Neutral LMI Techniques
Applied Mathematical Sciences, Vol. 8, 2014, no. 54, 2687-2700 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.42139 Robust Stability Analysis of Teleoperation by Delay-Dependent Neutral
More informationNEW SUPERVISORY CONTROL USING CONTROL-RELEVANT SWITCHING
NEW SUPERVISORY CONTROL USING CONTROL-RELEVANT SWITCHING Tae-Woong Yoon, Jung-Su Kim Dept. of Electrical Engineering. Korea University, Anam-dong 5-ga Seongbuk-gu 36-73, Seoul, Korea, twy@korea.ac.kr,
More informationResearch Article Stabilization Analysis and Synthesis of Discrete-Time Descriptor Markov Jump Systems with Partially Unknown Transition Probabilities
Research Journal of Applied Sciences, Engineering and Technology 7(4): 728-734, 214 DOI:1.1926/rjaset.7.39 ISSN: 24-7459; e-issn: 24-7467 214 Maxwell Scientific Publication Corp. Submitted: February 25,
More informationLyapunov Stability of Linear Predictor Feedback for Distributed Input Delays
IEEE TRANSACTIONS ON AUTOMATIC CONTROL VOL. 56 NO. 3 MARCH 2011 655 Lyapunov Stability of Linear Predictor Feedback for Distributed Input Delays Nikolaos Bekiaris-Liberis Miroslav Krstic In this case system
More informationReducing the Computational Cost of the Sum-of-Squares Stability Test for Time-Delayed Systems
Reducing the Computational Cost of the Sum-of-Squares Stability Test for Time-Delayed Systems Yashun Zhang Matthew Peet and Keqin Gu Abstract This paper considers the problem of reducing the computational
More informationLinear matrix inequality approach for robust stability analysis for stochastic neural networks with time-varying delay
Linear matrix inequality approach for robust stability analysis for stochastic neural networks with time-varying delay S. Lakshmanan and P. Balasubramaniam Department of Mathematics, Gandhigram Rural University,
More informationOn Absolute Stability of Lur e Control Systems. with Multiple Nonlinearities
On Absolute Stability of Lur e Control Systems with Multiple Nonlinearities Min Wu a, Yong He a, Guo-Ping Liu b, Jin-Hua She c a School of Information Science and Engineering, Central South University,
More informationMultiple-mode switched observer-based unknown input estimation for a class of switched systems
Multiple-mode switched observer-based unknown input estimation for a class of switched systems Yantao Chen 1, Junqi Yang 1 *, Donglei Xie 1, Wei Zhang 2 1. College of Electrical Engineering and Automation,
More informationResearch Article Delay-Dependent H Filtering for Singular Time-Delay Systems
Discrete Dynamics in Nature and Society Volume 211, Article ID 76878, 2 pages doi:1.1155/211/76878 Research Article Delay-Dependent H Filtering for Singular Time-Delay Systems Zhenbo Li 1, 2 and Shuqian
More informationStability and Stabilizability of Linear Parameter Dependent System with Time Delay
Proceedings of te International MultiConference of Engineers and Computer Scientists 2008 Vol II IMECS 2008, 19-21 Marc, 2008, Hong Kong Stability and Stabilizability of Linear Parameter Dependent System
More informationResearch Article Robust Tracking Control for Switched Fuzzy Systems with Fast Switching Controller
Mathematical Problems in Engineering Volume 212, Article ID 872826, 21 pages doi:1.1155/212/872826 Research Article Robust Tracking Control for Switched Fuzzy Systems with Fast Switching Controller Hong
More informationLINEAR QUADRATIC OPTIMAL CONTROL BASED ON DYNAMIC COMPENSATION. Received October 2010; revised March 2011
International Journal of Innovative Computing, Information and Control ICIC International c 22 ISSN 349-498 Volume 8, Number 5(B), May 22 pp. 3743 3754 LINEAR QUADRATIC OPTIMAL CONTROL BASED ON DYNAMIC
More informationFloor Control (kn) Time (sec) Floor 5. Displacement (mm) Time (sec) Floor 5.
DECENTRALIZED ROBUST H CONTROL OF MECHANICAL STRUCTURES. Introduction L. Bakule and J. Böhm Institute of Information Theory and Automation Academy of Sciences of the Czech Republic The results contributed
More informationTakagi Sugeno Fuzzy Sliding Mode Controller Design for a Class of Nonlinear System
Australian Journal of Basic and Applied Sciences, 7(7): 395-400, 2013 ISSN 1991-8178 Takagi Sugeno Fuzzy Sliding Mode Controller Design for a Class of Nonlinear System 1 Budiman Azzali Basir, 2 Mohammad
More informationRobust absolute stability criteria for a class of uncertain Lur e systems of neutral type
Proceedings of the 6th World Congress on Intelligent Control and Automation, June 21-23, 2006, Dalian, China Robust absolute stability criteria for a class of uncertain Lur e systems of neutral type Qing-Long
More informationRobust H control of Takagi Sugeno fuzzy systems with state and input time delays
Fuzzy Sets and Systems 6 (29 43 422 www.elsevier.com/locate/fss Robust H control of Takagi Sugeno fuzzy systems with state and input time delays Bing Chen a,, Xiaoping Liu b, Chong Lin a,kefuliu b a Institute
More informationConstructing Lyapunov-Krasovskii Functionals For Linear Time Delay Systems
Constructing Lyapunov-Krasovskii Functionals For Linear Time Delay Systems Antonis Papachristodoulou, Matthew Peet and Sanjay Lall Abstract We present an algorithmic methodology for constructing Lyapunov-Krasovskii
More informationH State-Feedback Controller Design for Discrete-Time Fuzzy Systems Using Fuzzy Weighting-Dependent Lyapunov Functions
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL 11, NO 2, APRIL 2003 271 H State-Feedback Controller Design for Discrete-Time Fuzzy Systems Using Fuzzy Weighting-Dependent Lyapunov Functions Doo Jin Choi and PooGyeon
More informationStability of Delay Impulsive Systems with Application to Networked Control Systems
Stability of Delay Impulsive Systems with Application to Networked Control Systems Payam Naghshtabrizi, João P. Hespanha, and Andrew R. Teel Abstract We establish asymptotic and exponential stability theorems
More informationNew summation inequalities and their
New summation inequalities and their applications to discrete-time delay systems Le Van Hien and Hieu Trinh arxiv:505.06344v [math.oc] 3 May 05 Abstract This paper provides new summation inequalities in
More informationROBUST PASSIVE OBSERVER-BASED CONTROL FOR A CLASS OF SINGULAR SYSTEMS
INTERNATIONAL JOURNAL OF INFORMATON AND SYSTEMS SCIENCES Volume 5 Number 3-4 Pages 480 487 c 2009 Institute for Scientific Computing and Information ROBUST PASSIVE OBSERVER-BASED CONTROL FOR A CLASS OF
More informationResearch Article On Exponential Stability Conditions of Descriptor Systems with Time-Varying Delay
Applied Mathematics Volume 2012, Article ID 532912, 12 pages doi:10.1155/2012/532912 Research Article On Exponential Stability Conditions of Descriptor Systems with Time-Varying Delay S. Cong and Z.-B.
More informationState estimation of uncertain multiple model with unknown inputs
State estimation of uncertain multiple model with unknown inputs Abdelkader Akhenak, Mohammed Chadli, Didier Maquin and José Ragot Centre de Recherche en Automatique de Nancy, CNRS UMR 79 Institut National
More informationH Synchronization of Chaotic Systems via Delayed Feedback Control
International Journal of Automation and Computing 7(2), May 21, 23-235 DOI: 1.17/s11633-1-23-4 H Synchronization of Chaotic Systems via Delayed Feedback Control Li Sheng 1, 2 Hui-Zhong Yang 1 1 Institute
More informationConvex Optimization Approach to Dynamic Output Feedback Control for Delay Differential Systems of Neutral Type 1,2
journal of optimization theory and applications: Vol. 127 No. 2 pp. 411 423 November 2005 ( 2005) DOI: 10.1007/s10957-005-6552-7 Convex Optimization Approach to Dynamic Output Feedback Control for Delay
More informationAn LMI Approach to Robust Controller Designs of Takagi-Sugeno fuzzy Systems with Parametric Uncertainties
An LMI Approach to Robust Controller Designs of akagi-sugeno fuzzy Systems with Parametric Uncertainties Li Qi and Jun-You Yang School of Electrical Engineering Shenyang University of echnolog Shenyang,
More informationCorrespondence should be addressed to Chien-Yu Lu,
Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2009, Article ID 43015, 14 pages doi:10.1155/2009/43015 Research Article Delay-Range-Dependent Global Robust Passivity Analysis
More informationPartial-State-Feedback Controller Design for Takagi-Sugeno Fuzzy Systems Using Homotopy Method
Partial-State-Feedback Controller Design for Takagi-Sugeno Fuzzy Systems Using Homotopy Method Huaping Liu, Fuchun Sun, Zengqi Sun and Chunwen Li Department of Computer Science and Technology, Tsinghua
More informationStability Analysis of Linear Systems with Time-Varying Delays: Delay Uncertainty and Quenching
Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 2-4, 2007 Stability Analysis of Linear Systems with Time-Varying Delays: Delay Uncertainty and Quenching Antonis
More informationTrajectory Tracking Control of Bimodal Piecewise Affine Systems
25 American Control Conference June 8-1, 25. Portland, OR, USA ThB17.4 Trajectory Tracking Control of Bimodal Piecewise Affine Systems Kazunori Sakurama, Toshiharu Sugie and Kazushi Nakano Abstract This
More informationOn Design of Reduced-Order H Filters for Discrete-Time Systems from Incomplete Measurements
Proceedings of the 47th IEEE Conference on Decision and Control Cancun, Mexico, Dec. 9-11, 2008 On Design of Reduced-Order H Filters for Discrete-Time Systems from Incomplete Measurements Shaosheng Zhou
More informationFIR Filters for Stationary State Space Signal Models
Proceedings of the 17th World Congress The International Federation of Automatic Control FIR Filters for Stationary State Space Signal Models Jung Hun Park Wook Hyun Kwon School of Electrical Engineering
More informationA novel control scheme for teleoperation with guaranteed performance under time-varying delays
A novel control scheme for teleoperation with guaranteed performance under time-varying delays Bo Zhang, Alexandre Kruszewski, Jean-Pierre Richard To cite this version: Bo Zhang, Alexandre Kruszewski,
More informationStability and stabilization of 2D continuous state-delayed systems
20 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, December 2-5, 20 Stability and stabilization of 2D continuous state-delayed systems Mariem Ghamgui,
More informationStability of linear time-varying systems through quadratically parameter-dependent Lyapunov functions
Stability of linear time-varying systems through quadratically parameter-dependent Lyapunov functions Vinícius F. Montagner Department of Telematics Pedro L. D. Peres School of Electrical and Computer
More informationFinite-Dimensional H Filter Design for Linear Systems with Measurement Delay
Proceedings of the 17th World Congress The International Federation of Automatic Control Finite-Dimensional H Filter Design for Linear Systems with Measurement Delay Michael Basin Peng Shi Dario Calderon-Alvarez
More informationDynamic Integral Sliding Mode Control of Nonlinear SISO Systems with States Dependent Matched and Mismatched Uncertainties
Milano (Italy) August 28 - September 2, 2 Dynamic Integral Sliding Mode Control of Nonlinear SISO Systems with States Dependent Matched and Mismatched Uncertainties Qudrat Khan*, Aamer Iqbal Bhatti,* Qadeer
More informationStability of Hybrid Control Systems Based on Time-State Control Forms
Stability of Hybrid Control Systems Based on Time-State Control Forms Yoshikatsu HOSHI, Mitsuji SAMPEI, Shigeki NAKAURA Department of Mechanical and Control Engineering Tokyo Institute of Technology 2
More informationInput-output framework for robust stability of time-varying delay systems
Input-output framework for robust stability of time-varying delay systems Yassine Ariba, Frédéric Gouaisbaut To cite this version: Yassine Ariba, Frédéric Gouaisbaut. Input-output framework for robust
More informationSOS Methods for Delay-Dependent Stability of Neutral Differential Systems
SOS Methods for Delay-Dependent Stability of Neutral Differential Systems Matthew M. Peet, Catherine Bonnet Özbay and Hitay 1 Introduction In this paper, we consider the problem of verification of certain
More informationResearch Article Input and Output Passivity of Complex Dynamical Networks with General Topology
Advances in Decision Sciences Volume 1, Article ID 89481, 19 pages doi:1.1155/1/89481 Research Article Input and Output Passivity of Complex Dynamical Networks with General Topology Jinliang Wang Chongqing
More informationTime-delay feedback control in a delayed dynamical chaos system and its applications
Time-delay feedback control in a delayed dynamical chaos system and its applications Ye Zhi-Yong( ), Yang Guang( ), and Deng Cun-Bing( ) School of Mathematics and Physics, Chongqing University of Technology,
More informationRobust stabilization of linear uncertain plants with polynomially time varying parameters
Preprints of the 19th World Congress The International Federation of Automatic Control Robust stabilization of linear uncertain plants with polynomially time varying parameters L. Jetto V. Orsini R. Romagnoli
More informationFINITE HORIZON ROBUST MODEL PREDICTIVE CONTROL USING LINEAR MATRIX INEQUALITIES. Danlei Chu, Tongwen Chen, Horacio J. Marquez
FINITE HORIZON ROBUST MODEL PREDICTIVE CONTROL USING LINEAR MATRIX INEQUALITIES Danlei Chu Tongwen Chen Horacio J Marquez Department of Electrical and Computer Engineering University of Alberta Edmonton
More informationStability Analysis of Linear Systems with Time-varying State and Measurement Delays
Proceeding of the th World Congress on Intelligent Control and Automation Shenyang, China, June 29 - July 4 24 Stability Analysis of Linear Systems with ime-varying State and Measurement Delays Liang Lu
More informationRobust Stabilization of the Uncertain Linear Systems. Based on Descriptor Form Representation t. Toru ASAI* and Shinji HARA**
Robust Stabilization of the Uncertain Linear Systems Based on Descriptor Form Representation t Toru ASAI* and Shinji HARA** This paper proposes a necessary and sufficient condition for the quadratic stabilization
More informationCHATTERING-FREE SMC WITH UNIDIRECTIONAL AUXILIARY SURFACES FOR NONLINEAR SYSTEM WITH STATE CONSTRAINTS. Jian Fu, Qing-Xian Wu and Ze-Hui Mao
International Journal of Innovative Computing, Information and Control ICIC International c 2013 ISSN 1349-4198 Volume 9, Number 12, December 2013 pp. 4793 4809 CHATTERING-FREE SMC WITH UNIDIRECTIONAL
More informationDELAY-DEPENDENT STABILITY OF DISCRETE-TIME SYSTEMS WITH MULTIPLE DELAYS AND NONLINEARITIES. Siva Kumar Tadepalli and Venkata Krishna Rao Kandanvli
International Journal of Innovative Computing, Information and Control ICIC International c 2017 ISSN 1349-4198 Volume 13, Number 3, June 2017 pp. 891 904 DELAY-DEPENDENT STABILITY OF DISCRETE-TIME SYSTEMS
More informationH State Feedback Control of Discrete-time Markov Jump Linear Systems through Linear Matrix Inequalities
H State Feedback Control of Discrete-time Markov Jump Linear Systems through Linear Matrix Inequalities A. P. C. Gonçalves, A. R. Fioravanti, M. A. Al-Radhawi, J. C. Geromel Univ. Estadual Paulista - UNESP.
More information