On Exponential Stability of Integral Delay Systems - Semantic Scholar
2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013
On Exponential Stability of Integral Delay Systems Zhao-Yan Li1,4 , Bin Zhou2,4 and Zongli Lin3
infinite dimensional controllers can be safely implemented if and only if certain integral delay systems are asymptotically stable [15], [16]. Besides the two examples mentioned above, integral delay systems appear also in some other problems associated with delay systems, for example, delay approximations of the partial differential equations for describing the propagation phenomena in excitable media [17], and stability analysis of some difference operators in neutral type functional differential equations [7]. For more introduction on integral delay systems, the reader may refer to [12], [13], [16] and the references cited there. Because of their many applications, integral delay systems have received much attention in recent years. Lyapunov-Krasovskii theorems for integral delay systems have recently been introduced in [12] and [13]. With the help of this Lyapunov-Krasovskii theorem, some sufficient conditions in terms of linear matrix inequalities (LMIs) were recently established in [16] for the exponential stability of some classes of integral delay systems with analytic kernels, which include the integral delay systems encountered in the model transformation and predictor feedback for time-delay systems as special cases. In this paper, with the aid of the delay decomposition technique [4], we find that the Lyapunov-Krasovskii functional proposed in [16] can be written as an integral quadratic form of some generalized state vector obtained by fractionizing of the delay intervals of the state. This motivates us to propose a more general Lyapunov-Krasovskii functional in terms of this generalized state vector. Delay dependent LMI conditions guaranteeing the exponential stability of an integral delay system are then obtained by using this new Lyapunov-Krasovskii functional. It is proven that these conditions are always less conservative than those proposed in [16]. Numerical examples are worked out to illustrate the effectiveness of the proposed approaches. The remainder of this paper is organized as follows. The problem formulation and some preliminary results are presented in Section II. Section III contains the main results of this paper. Numerical examples are presented in Section IV to show the effectiveness of the proposed approach and Section V concludes the paper. Finally, some technical proofs are collected in the appendix. Notation. The notation used in this paper is fairly standard. For a matrix A ∈ Rn×n , we use AT , rank (A) , λ (A) , det (A) and He (A) to denote its transpose, rank, eigenvalue set, determinant, and the symmetric matrix A + AT . We use diag{A1 , A2 , · · · , Ap }, to denote a diagonal matrices whose diagonal elements are Ai , i = 1, 2, · · · , p and A ⊗ B to denote the Kronecker product of matrices A and B. For
Abstract— This paper deals with exponential stability of some classes of integral delay systems with a prescribed decay rate. By carefully exploring the literature on this topic, a delay decomposition approach is established to reduce the conservatism in the existing sufficient conditions by constructing new Lyapunov-Krasovskii functionals. It is proven that the proposed sufficient conditions are always less conservative than the existing ones. Numerical examples illustrate the effectiveness of the proposed approaches.
I. I NTRODUCTION Time delay arises frequently in many engineering systems such as nuclear reactors, long transmission lines in pneumatic systems, rolling mills, sampled-data control, manufacturing processes and networked control systems [3], [5], [7]. Because of their infinite dimensional nature, control problems, especially, the problems of asymptotic stability analysis and stabilization, for time-delay systems have been recognized to be very difficult. As a result, during the past several decades, control of time-delay systems has received considerable attentions from the researchers and a large number of results have been reported in the literature (see [2], [8], [9], [19], [20], [21], [22] and the references therein). Among the existing methods for carrying out asymptotic stability analysis and stabilization of time-delay systems, the Lyapunov-Krasovskii functional based methods are probably the most efficient ones. In these methods, one generally needs to start with certain system transformations that bring into the original delay system additional dynamics that can be described by the so-called integral delay system [6], [10]. The stability of such an integral delay system is thus important in the stability analysis of the original delay system. Another efficient approach to dealing with the stabilization of time-delay systems is the predictor feedback, which is especially effective for input delayed systems [1], [11], [14], [23]. However, it is now clear that the resulting This work was supported in part by the National Natural Science Foundation of China under grant numbers 61104124 and 61273028, by the Program for New Century Excellent Talents in University under Grant NCET-11-0815 and by the National Science Foundation of the United States under grant number CMMI-1129752. 1 Zhao-Yan Li is with the Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China. [email protected];
[email protected] 2 Bin Zhou is with the Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, 150001, China.
[email protected], [email protected] 3 Zongli Lin is with the Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, P.O. Box 400743, Charlottesville, VA 22904-4743, USA. [email protected] 4 On leave with the Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 229044743, USA.
978-1-4799-0176-0/$31.00 ©2013 AACC
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a semi-positive definite matrix P , we use λmin (P ) and λmax (P ) to denote respectively its minimal and maximal eigenvalues. For a positive scalar h and an integer m, let Cm,h = C ([−h, 0] , Rm ) denote the Banach space of continuous vector functions mapping the interval [−h, 0] into Rm with the topology of uniform convergence, and let xt ∈ Cm,h denote the restriction of x (t) to the interval [t − h, t] translated to [−h, 0] , that is, xt (θ) = x (t + θ) , θ ∈ [−h, 0]. For any ϕ ∈ Cm,h , the norm of ϕ is defined as kϕkh = supθ∈[−h,0] kϕ (θ)k .
Lemma 1: [12], [16] The integral delay system (1) is exponentially stable with a decay rate β ≥ 0 if there exists a functional V : Cm,h → R such that t → V (xt ) is differentiable on R+ and the following conditions hold: R0 2 1) α1 −h kx (t + θ)k dθ ≤ V (xt ) ≤ R0 2 α2 −h kx (t + θ)k dθ, for some 0 < α1 ≤ α2 ; R0 2 2) V˙ (xt ) + 2βV (xt ) ≤ −α3 −h kx (t + θ)k dθ, for some α3 > 0. In order to study the exponential stability of (1), in [16], the following Lyapunov-Krasovskii functional Z 0 xT (t + θ) B T (θ) e2βθ Vh (xt ) =
II. P ROBLEM F ORMULATION AND P RELIMINARIES In this paper we are interested in the following integral delay system Z 0 x (t) = GB (θ) x (t + θ) dθ, t ≥ 0, (1)
−h
× (P + (θ + h) Q) B (θ) x (t + θ) dθ,
was constructed, where P, Q ∈ Rn×n are two positive definite matrices. With this Lyapunov-Krasovskii functional (7), the following LMI based conditions for testing the exponential stability of (1) were obtained in [16]. Lemma 2: The integral delay system in (1) which satisfies (2) and (3) is exponentially stable with a decay rate β ≥ 0 if there exist two positive definite matrices P, Q ∈ Rn×n such that the following two LMIs are satisfied: Q + He M T (P + hQ) − hGT B T (P + hQ) BG > 0, (8)
−h
where h > 0 is a constant, G ∈ Rm×n , and B (θ) : [−h, 0] → Rn×m . Furthermore, we assume that B (θ) is continuously differentiable for all θ ∈ [−h, 0] and satisfies [16] B˙ (θ) = M B (θ) , ∀θ ∈ [−h, 0] , (2) with M ∈ Rn×n being a constant matrix. For notational simplicity, we denote B (0) = B. Moreover, according to [16], we should assume that there exists a γ > 0 such that min λmin B T (θ) B (θ) > γ. (3)
Q + M T P + P M − he2βh GT B T (P + hQ) BG > 0. (9) Finally, for easy reference, we recall from [13] and [16] the following result regarding algebraic conditions for testing the exponential stability of (1). Lemma 3: The integral delay system in (1) is exponentially stable if −1 h < hmax = max {kGB (s)k} . (10)
θ∈[−h,0]
Remark 1: Under the assumption in (2), the condition in (3) is equivalent to rank (B) = m.
(4)
In fact, (3) is satisfied if and only if rank (B (θ)) = m, ∀θ ∈ [−h, 0] , which, in view of the fact that (2) is equivalent to B (θ) = eM θ B (0) = eM θ B, ∀θ ∈ [−h, 0] ,
(5)
s∈[−h,0]
Mθ
is equivalent to (4) since e is nonsingular for all θ. Hence condition (4) instead of (3) is assumed to be satisfied hereafter. Remark 2: In [16], the matrix M is of dimension (N + 1) m for some integer N ≥ 0. Here in (1) the dimension of M is n, which is not necessary a multiple of m. Of course, to ensure that (4) is satisfied, we must have n ≥ m. In this paper we are interested in the stability analysis of the integral delay system (1). To this end, we first introduce the stability principle for this class of systems. Let ϕ (θ) ∈ Cm,h , θ ∈ [−h, 0], be an initial condition for system (1) and x (t) = x (t, ϕ) , t ≥ 0, be a solution of (1) satisfying x (t) = ϕ (t) , t ∈ [−h, 0] . Then we can recall the definition of exponential stability of the integral delay system (1). Definition 1: [16] The integral delay system (1) is said to be exponentially stable with a decay rate β ≥ 0 if there exists a µ > 0 such that every solution of (1) satisfies the inequality kx (t)k ≤ µ kϕkh e−βt , t ≥ 0.
(7)
III. M AIN R ESULTS A. Construction of the Lyapunov-Krasovskii Functional In order to reduce the conservatism of the LyapunovKrasovskii functional approach, motivated by the work in h i, i = 1, 2, · · · , N , of the [4], we consider the fractions N delay h, where N ≥ 1 is a given integer. For an s ∈ R, we define B1 (s) x1 (t + s) B2 (s) x2 (t + s) π (t + s) = (11) , .. . BN (s) xN (t + s) in which, for i = 1, 2, · · · , N , h (i − 1) , xi (s) = x s − N h Bi (s) = B s − (i − 1) . N
(6) 165
(12) (13)
h (i − 1) = s, i = 1, 2, · · · , N . Then, by some Let θ + N computation, it follows from (7) that Z 0 Vh (xt ) = π T (t + s) e2βs Ω (s) π (t + s) ds, (14)
For future use, for any integer N ≥ 1, by some computation, we can rewrite the integral delay system (1) as N Z − h (i−1) X N x (t) = GB (θ) x (t + θ) dθ
h −N
i=1
where Ω (s) = P + s +
h N
Q and we have denoted
P = diag {P1 , P2 , · · · , PN } ,
(15)
Q = diag {Q1 , Q2 , · · · , QN } ,
(16)
Z
h
Pi = e−2 N (i−1)β h
In
In
···
In
∈ Rn×N n .
(24)
In this subsection, we provide some sufficient conditions for testing the exponential stability of the integral delay system (1). Theorem 1: The integral delay system in (1) is exponentially stable with a decay rate β ≥ 0 if there exist an integer N ≥ 1, two positive definite matrices P ∈ RN n×N n and S ∈ Rn×n , and a symmetric matrix Q ∈ RN n×N n such that (21) and T In In Θ− S < 0, (25) 0N n×n 0N n×n h h MT P + Q + P + Q M N N h T T T G B SBGIN n > 0, (26) + Q − IN N n h h T T T He (PM) + Q − e2β N IN n G B SBGIN n > 0, N (27)
(18)
h −N
where P, Q ∈ RN n×N n are two given symmetric matrices. Before giving our main results, we present the following technical lemma whose proof is omitted due to space limitation. Lemma 4: Let P ∈ Rn×n and Q ∈ Rn×n be two symmetric matrices and R ∈ Rn×n be a semi-positive definite matrix. Let
are satisfied, where M = IN ⊗ M and Θ is defined as T h IN n IN n Θ= P+ Q 0n×N n 0n×N n N T h 0n×N n 0n×N n −2β N − P e . (28) IN n IN n
(20)
where s1 and s2 are two constants with s2 > s1 and f (s) is a concave function1 . Then Φ (s) > 0, ∀s ∈ [s1 , s2 ] if and only if Φ (s1 ) > 0 and Φ (s2 ) > 0. h Remark 3: Let Υ (s) = P + N + s Q. Then the Lyapunov-Krasovskii functional V (xt ) in (19) is positive definite only if Υ (s) is positive definite for all hif and s ∈ −N , 0 , which, according to Lemma 4, is the case h if and only if Υ − N = P > 0 and h Υ (0) = P + Q > 0. N
B. Some Sufficient Conditions
(17)
Notice that both P and Q are block diagonal matrices. Hence the conservatism of Lemma 2 can be further reduced by increasing the degrees of freedom of P and Q. More specifically, we can respectively use two symmetric matrices P and Q instead of the block diagonal matrices P and Q defined in (15)-(17). Namely, we consider the following Lyapunov-Krasovskii functional Z 0 V (xt ) = π T (t + s) e2βs Ω (s) π (t + s) ds, (19)
Φ (s) = P + sQ + f (s) R, ∀s ∈ [s1 , s2 ] ,
(23)
in which
Qi = e−2 N (i−1)β Q.
π (t + s) ds, h −N
IN n = h P + h− i Q , N
0
= GIN n
in which
h −N i
Proof: (Sketch) Consider the Lyapunov-Krasovskii functional in (19). Similarly to the derivation of (23), we can derive Z 0 π T (t + s) π (t + s) ds h −N
Z
(21)
0
=
xT (t + θ) B T (θ) B (θ) x (t + θ) dθ.
(29)
−h
For the same reason given in the above remark, the condition that Q > 0 in Lemma 2 can be weakened as P + hQ > 0.
With this identity, we can show that there exist two constants αi , i = 1, 2 such that Z 0 h 2 −2β N α1 e kx (t + s)k ds ≤ V (xt ) −h Z 0 2 ≤ α2 kx (t + s)k ds. (30)
(22)
For easy reference, we state the following corollary. Corollary 1: The integral delay system in (1) is exponentially stable with a decay rate β ≥ 0 if there exist a positive definite matrix P ∈ Rn×n and a symmetric matrix Q ∈ Rn×n such that (8)-(9) and (22) are satisfied.
−h
We next prove that
1A
V˙ (xt ) + 2βV (xt ) ≤ −α3
function f : D → R is said to be concave if, for any s1 , s2 ∈ D and θ ∈ [0, 1] , there holds f (θs1 + (1 − θ) s2 ) ≥ θf (s1 ) + (1 − θ) f (s2 ) .
Z
0
166
2
kx (t + s)k ds, −h
(31)
h −N Z 0
(32) = −α3
Z
which is just (31). The proof is completed. The following proposition shows that Theorem 1 is always less conservative than Lemma 2. The proof of this proposition is omitted due to space limitation. Proposition 1: For any given β ≥ 0, if there exist two positive definite matrices P, Q ∈ Rn×n such that (8) and (9) are satisfied, then, for any given integer N ≥ 1, there exist two positive definite matrices P ∈ RN n×N n and S ∈ Rn×n , and a symmetric matrix Q ∈ RN n×N n such that (21) and (25)-(27) are satisfied. In the particular case that M = 0, namely B (θ) = B is a constant matrix for all θ ∈ [−h, 0] , we can rewrite the integral delay system (1) as Z t x (t) = GBx (s) ds, (41)
t
π T (θ) e2β(θ−t) Ω (θ − t) π (θ) dθ. (33)
h t− N
In view of (2), we can compute d dt B (θ − t) x (θ) d h B θ − t − dt N x θ− π˙ (θ) = .. . N −1 d B θ − t − dt N h x θ−
h N N −1 N h
= −Mπ (θ) .
t−h
(34)
Define a generalized state vector B (0) x (t) π (t) η (t) = = . h B (−h) x (t − h) π t− N
where G ∈ Rm×n and B ∈ Rn×m are constant matrices. In this case, by noting that inequality (27) implies (26), we obtain the following corollary of Theorem 1. Corollary 2: The integral delay system (41) is exponentially stable with a delay rate β ≥ 0 if there exist an integer N ≥ 1 and matrices 0 < P ∈ RN n×N n , Q ∈ RN n×N n , and 0 ≤ S ∈ Rn×n such that (21), (25) and the following inequality are satisfied:
(35)
Then, the equation in (33) can be rewritten as V˙ (xt ) + 2βV (xt ) = η T (t) Θη (t) Z 0 π T (t + s) e2βs Γ (s) π (t + s) ds, −
(36)
Q−
h −N
where Θ is defined in (28) and h Γ (s) = MT P + +s Q N h + P+ + s Q M + Q. N
h 2β h T T T e N IN n G B SBGIN n > 0. N
(42)
In this particular case, we can add the following functional Z 0 Z t e2β(θ−t) π T (θ) X π (θ) dθdv, (43) W (xt ) = h −N
(37)
t+v
where 0 ≤ X ∈ RN n×N n is a semi-positive definite matrix, to the Lyapunov-Krasovskii function (19) to get new sufficient conditions guaranteeing stability. In fact, by letting Z t F (v) = e2β(θ−t) π T (θ) X π (θ) dθ ≥ 0, (44)
By applying the inequality in (25), Jensen inequality [5] and the system model (23), we can get Z 0 ˙ V (xt ) + 2βV (xt ) ≤ − e2βs π T (t + s) Φπ (t + s) ds, h −N
(40)
−h
π T (θ) e2β(θ−t) Qπ (θ) dθ
− 2β
2
kx (t + s)k ds,
≤ −α3 γ
h t− N t
h t− N
xT (t + s) B T (s) B (s) π (t + s) ds
−h Z 0
from which it follows that h V˙ (xt ) = π T (t) P + Q π (t) N h h h T −2β N −π Pπ t − t− e N N Z t π˙ T (θ) e2β(θ−t) Ω (θ − t) π (θ) dθ +2 −
π T (t + s) π (t + s) ds
≤ −α3
h t− N
Z
0
Z
for some α3 > 0. To do so, let t + s = θ. Then Z t V (xt ) = π T (θ) e2β(θ−t) Ω (θ − t) π (θ) dθ,
t+v
(38)
we can compute Z 0 W (xt ) ≤
where h T T T I G B SBGIN n . (39) N Nn Since −e−2βs is a concave function and h T T T I G B SBGI ≥ 0, it follows from Lemma N n N n N h 4 that Φ (s) > 0, ∀s ∈ − N , 0 if and only if Φ (0) > 0 h and Φ − N > 0, which are respectively equivalent to (26) and (27). Hence, there exists a sufficiently small number α3 such that Φ = Φ (s) = Γ (s) − e−2βs
max {F (v)} dv h − N
h − h ≤v≤0 −N N
h F N Z h 0 2βs T = e π (t + s) X π (t + s) ds N − Nh Z 0 h ≤ λmax (X ) π T (t + s) π (t + s) ds h N −N =
V˙ (xt ) + 2βV (xt ) 167
=
h λmax (X ) λmax B T B N
Z
0
larger N leads to a larger maximal delay value, which is reasonable since a larger N means more decision variables in the conditions. Example 2 (Example 12 in [16]): Consider an integral delay system in the form of (1) with G = B = I2 and 0 1 M= . (50) 0 0
2
kx (t + θ)k dθ, −h
(45) which implies that W (xt ) satisfies the conditions in Lemma 1 and V (xt ) = V (xt ) + W (xt ) , (46) is indeed a Lyapunov-Krasovskii functional candidate for the integral delay system (41). For this Lyapunov-Krasovskii functional, by applying a technique quite similar to the one used in the proof of Theorem 1, we can obtain the following result whose proof is provided in Appendix. Lemma 5: The integral delay system (41) is exponentially stable with a delay rate β ≥ 0 if there exist matrices 0 < P ∈ RN n×N n , Q ∈ RN n×N n , 0 ≤ X ∈ RN n×N n , and 0 ≤ S ∈ Rn×n such that (21), (25) and (42) are satisfied, where Q is replaced by Q + X in (25) and (42). However, the following proposition indicates that the introduction of W (xt ) is not helpful in the reduction of conservatism of Corollary 2. Proposition 2: For any given β ≥ 0 and N ≥ 1, the LMIs in Corollary 2 are feasible if and only if the LMIs in Lemma 5 are feasible. Proof: We need only to show that if the LMIs in Corollary 2 are feasible then the LMIs in Lemma 5 are also feasible. In fact, it is straightforward to verify that if (P∗ , Q∗ , X∗ , S∗ ) is a feasible solution to the LMIs in Corollary 2, then (P# , Q# , S# ) = (P∗ , Q∗ + X∗ , S∗ ) is a feasible solution to the LMIs in Lemma 5. The proof is completed.
Notice that in this case the matrix M is marginally unstable since λ (M ) = {0, 0}. Again, for different values of β, the corresponding maximal values of the delay h by applying different approaches are recorded in Table 2 shown in the next page. From this table we again find that Theorem 1 is always less conservative than Lemma 2 (Theorem 7 in [16]) for any β and N . Moreover, even though Corollary 8 in [16], where an argumentation has been utilized to increase the size of the decision variables, is better than Theorem 1 with N = 1, it is however more conservative than Theorem 1 with N = 5. V. C ONCLUSIONS This paper proposes some delay-dependent sufficient conditions for the exponentially stability of some classes of integral delay systems which are frequently encountered in studying stability problems of time-delay systems. The basic idea is to divide the delay interval into N small intervals so that more information of the delayed state can be utilized to construct the Lyapunov-Krasovskii functional, which in turns can reduce the conservatism of the resulting sufficient conditions expressed in linear matrix inequalities (LMIs). It is shown that the proposed LMIs conditions are always less conservative than the existing conditions for any number of divisions of the delay intervals (namely, the number N ). Numerical examples demonstrate the effectiveness of the proposed approaches.
IV. N UMERICAL E XAMPLES In this section, we present some examples to validate the effectiveness of the proposed approaches. Example 1 (Example 1 in [16]): Consider an integral delay system in the form of (1) with G = B = I2 and 0 1 M= . (47) −2 3
A PPENDIX : T HE P ROOF OF L EMMA 5 For easy reference, we write the two LMI conditions in Lemma 5 as follows T In In 0 Θ − S < 0, (51) 0N n×n 0N n×n h h T T T (52) Q + X − e2β N IN n G B SBGIN n > 0, N where Θ 0 is obtained by replaying Q with Q + X in (28), namely, T h IN n IN n 0 Θ = P+ (Q + X ) 0n×N n 0n×N n N T h 0n×N n 0n×N n − e−2β N P . (53) IN n IN n
For different values of β, by applying different approaches, the corresponding maximal values of the delay h are recorded in Table 1 shown in the next page. From this table we can observe the following facts: 1) Corollary 1 is less conservative than Lemma 2 (Theorem 7 in [16]), which indicates that the weakened condition (22) instead of Q > 0 can indeed reduce the conservatism. In fact, for β = 1 and h = 0.64, we obtain the following numerical values for matrices P and Q : 27.7386 −16.1260 P = , (48) −16.1260 10.6060 −18.9847 10.6050 Q= , (49) 10.6050 −6.5418
It is easy to show that V˙ (xt ) + 2βV (xt ) = η T (t) Θ 0 η (t) Z 0 − e2βs π T (t + s) (Q + X ) π (t + s) ds,
which means that Q is negative definite. 2) Theorem 1 is always less conservative than Lemma 2 (Theorem 7 in [16]) for any β and N. In particular, a
h −N
168
(54)
Table 1: Maximal delay value by using different approaches for Example 1 β 0 1 2 3 Lemma 3 (Lemma 6 in [16]) 0.988 − − − Lemma 2 (Theorem 7 in [16]) 1.990 0.601 0.426 0.349 Corollary 1 2.000 0.653 0.427 0.349 2.000 0.653 0.427 0.349 Theorem 1 with N = 1 Theorem 1 with N = 5 2.538 0.908 0.623 0.494 Theorem 1 with N = 10 2.775 0.952 0.656 0.520 2.950 0.967 0.668 0.529 Theorem 1 with N = 15
Table 2: Maximal delay value by using different approaches for Example 2 β 0 1 2 3 Lemma 3 (Lemma 6 in [16]) 0.707 − − − Lemma 2 (Theorem 7 in [16]) 0.999 0.567 0.426 0.349 Corollary 8 in [16] 0.999 0.628 0.471 0.384 0.999 0.567 0.426 0.349 Corollary 1 Theorem 1 with N = 1 0.999 0.567 0.426 0.349 Theorem 1 with N = 5 0.999 0.660 0.516 0.430 0.999 0.676 0.531 0.445 Theorem 1 with N = 10 Theorem 1 with N = 15 0.999 0.681 0.537 0.450 Theoretical Maximal Delay ( 0, s ∈ − N h z (0) > 0 and z − N > 0, which is equivalent to (52). By h noticing that z − N > 0 implies z (0) > 0, we complete the proof. R EFERENCES [1] Z. Artstein, Linear systems with delayed controls: A reduction, IEEE Transactions on Automatic Control, Vol. 27, pp. 869-879, 1982. [2] B. Du, J. Lam and Z. Shu, Stabilization for state/input delay systems via static and integral output feedback, Automatica, Vol. 46, No. 12, pp. 2000-2007, 2011. [3] E. Fridman, A refined input delay approach to sampled-data control, Automatica, Vol. 46, No. 2, pp. 421-427, 2010. [4] F. Gouaisbaut and D. Peaucelle, Delay-dependent stability analysis of linear time delay systems, IFAC Workshop on Time Delay System, Vol. 6, No. 1, pp. 54–59, 2006. [5] K. Gu, V. L. Kharitonov and J. Chen, Stability of Time-Delay Systems, Boston, MA: Birkh¨auser, 2003. [6] K. Gu and S.-I. Niculescu, Further remarks on additional dynamics in various model transformationsof linear delay systems, IEEE Transactions on Automatic Control, Vol. 46, pp. 497–500, Mar. 2001. [7] J. K. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, Vol. 3, Springer, New York, 1977. [8] Y. He, Q. G. Wang, C. Lin and M. Wu, Delay-range-dependent stability for systems with time-varying delay, Automatica, Vol. 43, No. 2, pp. 371-376, 2007.
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On Exponential Stability of Integral Delay Systems Zhao-Yan Li1,4 , Bin Zhou2,4 and Zongli Lin3
infinite dimensional controllers can be safely implemented if and only if certain integral delay systems are asymptotically stable [15], [16]. Besides the two examples mentioned above, integral delay systems appear also in some other problems associated with delay systems, for example, delay approximations of the partial differential equations for describing the propagation phenomena in excitable media [17], and stability analysis of some difference operators in neutral type functional differential equations [7]. For more introduction on integral delay systems, the reader may refer to [12], [13], [16] and the references cited there. Because of their many applications, integral delay systems have received much attention in recent years. Lyapunov-Krasovskii theorems for integral delay systems have recently been introduced in [12] and [13]. With the help of this Lyapunov-Krasovskii theorem, some sufficient conditions in terms of linear matrix inequalities (LMIs) were recently established in [16] for the exponential stability of some classes of integral delay systems with analytic kernels, which include the integral delay systems encountered in the model transformation and predictor feedback for time-delay systems as special cases. In this paper, with the aid of the delay decomposition technique [4], we find that the Lyapunov-Krasovskii functional proposed in [16] can be written as an integral quadratic form of some generalized state vector obtained by fractionizing of the delay intervals of the state. This motivates us to propose a more general Lyapunov-Krasovskii functional in terms of this generalized state vector. Delay dependent LMI conditions guaranteeing the exponential stability of an integral delay system are then obtained by using this new Lyapunov-Krasovskii functional. It is proven that these conditions are always less conservative than those proposed in [16]. Numerical examples are worked out to illustrate the effectiveness of the proposed approaches. The remainder of this paper is organized as follows. The problem formulation and some preliminary results are presented in Section II. Section III contains the main results of this paper. Numerical examples are presented in Section IV to show the effectiveness of the proposed approach and Section V concludes the paper. Finally, some technical proofs are collected in the appendix. Notation. The notation used in this paper is fairly standard. For a matrix A ∈ Rn×n , we use AT , rank (A) , λ (A) , det (A) and He (A) to denote its transpose, rank, eigenvalue set, determinant, and the symmetric matrix A + AT . We use diag{A1 , A2 , · · · , Ap }, to denote a diagonal matrices whose diagonal elements are Ai , i = 1, 2, · · · , p and A ⊗ B to denote the Kronecker product of matrices A and B. For
Abstract— This paper deals with exponential stability of some classes of integral delay systems with a prescribed decay rate. By carefully exploring the literature on this topic, a delay decomposition approach is established to reduce the conservatism in the existing sufficient conditions by constructing new Lyapunov-Krasovskii functionals. It is proven that the proposed sufficient conditions are always less conservative than the existing ones. Numerical examples illustrate the effectiveness of the proposed approaches.
I. I NTRODUCTION Time delay arises frequently in many engineering systems such as nuclear reactors, long transmission lines in pneumatic systems, rolling mills, sampled-data control, manufacturing processes and networked control systems [3], [5], [7]. Because of their infinite dimensional nature, control problems, especially, the problems of asymptotic stability analysis and stabilization, for time-delay systems have been recognized to be very difficult. As a result, during the past several decades, control of time-delay systems has received considerable attentions from the researchers and a large number of results have been reported in the literature (see [2], [8], [9], [19], [20], [21], [22] and the references therein). Among the existing methods for carrying out asymptotic stability analysis and stabilization of time-delay systems, the Lyapunov-Krasovskii functional based methods are probably the most efficient ones. In these methods, one generally needs to start with certain system transformations that bring into the original delay system additional dynamics that can be described by the so-called integral delay system [6], [10]. The stability of such an integral delay system is thus important in the stability analysis of the original delay system. Another efficient approach to dealing with the stabilization of time-delay systems is the predictor feedback, which is especially effective for input delayed systems [1], [11], [14], [23]. However, it is now clear that the resulting This work was supported in part by the National Natural Science Foundation of China under grant numbers 61104124 and 61273028, by the Program for New Century Excellent Talents in University under Grant NCET-11-0815 and by the National Science Foundation of the United States under grant number CMMI-1129752. 1 Zhao-Yan Li is with the Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China. [email protected];
[email protected] 2 Bin Zhou is with the Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, 150001, China.
[email protected], [email protected] 3 Zongli Lin is with the Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, P.O. Box 400743, Charlottesville, VA 22904-4743, USA. [email protected] 4 On leave with the Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 229044743, USA.
978-1-4799-0176-0/$31.00 ©2013 AACC
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a semi-positive definite matrix P , we use λmin (P ) and λmax (P ) to denote respectively its minimal and maximal eigenvalues. For a positive scalar h and an integer m, let Cm,h = C ([−h, 0] , Rm ) denote the Banach space of continuous vector functions mapping the interval [−h, 0] into Rm with the topology of uniform convergence, and let xt ∈ Cm,h denote the restriction of x (t) to the interval [t − h, t] translated to [−h, 0] , that is, xt (θ) = x (t + θ) , θ ∈ [−h, 0]. For any ϕ ∈ Cm,h , the norm of ϕ is defined as kϕkh = supθ∈[−h,0] kϕ (θ)k .
Lemma 1: [12], [16] The integral delay system (1) is exponentially stable with a decay rate β ≥ 0 if there exists a functional V : Cm,h → R such that t → V (xt ) is differentiable on R+ and the following conditions hold: R0 2 1) α1 −h kx (t + θ)k dθ ≤ V (xt ) ≤ R0 2 α2 −h kx (t + θ)k dθ, for some 0 < α1 ≤ α2 ; R0 2 2) V˙ (xt ) + 2βV (xt ) ≤ −α3 −h kx (t + θ)k dθ, for some α3 > 0. In order to study the exponential stability of (1), in [16], the following Lyapunov-Krasovskii functional Z 0 xT (t + θ) B T (θ) e2βθ Vh (xt ) =
II. P ROBLEM F ORMULATION AND P RELIMINARIES In this paper we are interested in the following integral delay system Z 0 x (t) = GB (θ) x (t + θ) dθ, t ≥ 0, (1)
−h
× (P + (θ + h) Q) B (θ) x (t + θ) dθ,
was constructed, where P, Q ∈ Rn×n are two positive definite matrices. With this Lyapunov-Krasovskii functional (7), the following LMI based conditions for testing the exponential stability of (1) were obtained in [16]. Lemma 2: The integral delay system in (1) which satisfies (2) and (3) is exponentially stable with a decay rate β ≥ 0 if there exist two positive definite matrices P, Q ∈ Rn×n such that the following two LMIs are satisfied: Q + He M T (P + hQ) − hGT B T (P + hQ) BG > 0, (8)
−h
where h > 0 is a constant, G ∈ Rm×n , and B (θ) : [−h, 0] → Rn×m . Furthermore, we assume that B (θ) is continuously differentiable for all θ ∈ [−h, 0] and satisfies [16] B˙ (θ) = M B (θ) , ∀θ ∈ [−h, 0] , (2) with M ∈ Rn×n being a constant matrix. For notational simplicity, we denote B (0) = B. Moreover, according to [16], we should assume that there exists a γ > 0 such that min λmin B T (θ) B (θ) > γ. (3)
Q + M T P + P M − he2βh GT B T (P + hQ) BG > 0. (9) Finally, for easy reference, we recall from [13] and [16] the following result regarding algebraic conditions for testing the exponential stability of (1). Lemma 3: The integral delay system in (1) is exponentially stable if −1 h < hmax = max {kGB (s)k} . (10)
θ∈[−h,0]
Remark 1: Under the assumption in (2), the condition in (3) is equivalent to rank (B) = m.
(4)
In fact, (3) is satisfied if and only if rank (B (θ)) = m, ∀θ ∈ [−h, 0] , which, in view of the fact that (2) is equivalent to B (θ) = eM θ B (0) = eM θ B, ∀θ ∈ [−h, 0] ,
(5)
s∈[−h,0]
Mθ
is equivalent to (4) since e is nonsingular for all θ. Hence condition (4) instead of (3) is assumed to be satisfied hereafter. Remark 2: In [16], the matrix M is of dimension (N + 1) m for some integer N ≥ 0. Here in (1) the dimension of M is n, which is not necessary a multiple of m. Of course, to ensure that (4) is satisfied, we must have n ≥ m. In this paper we are interested in the stability analysis of the integral delay system (1). To this end, we first introduce the stability principle for this class of systems. Let ϕ (θ) ∈ Cm,h , θ ∈ [−h, 0], be an initial condition for system (1) and x (t) = x (t, ϕ) , t ≥ 0, be a solution of (1) satisfying x (t) = ϕ (t) , t ∈ [−h, 0] . Then we can recall the definition of exponential stability of the integral delay system (1). Definition 1: [16] The integral delay system (1) is said to be exponentially stable with a decay rate β ≥ 0 if there exists a µ > 0 such that every solution of (1) satisfies the inequality kx (t)k ≤ µ kϕkh e−βt , t ≥ 0.
(7)
III. M AIN R ESULTS A. Construction of the Lyapunov-Krasovskii Functional In order to reduce the conservatism of the LyapunovKrasovskii functional approach, motivated by the work in h i, i = 1, 2, · · · , N , of the [4], we consider the fractions N delay h, where N ≥ 1 is a given integer. For an s ∈ R, we define B1 (s) x1 (t + s) B2 (s) x2 (t + s) π (t + s) = (11) , .. . BN (s) xN (t + s) in which, for i = 1, 2, · · · , N , h (i − 1) , xi (s) = x s − N h Bi (s) = B s − (i − 1) . N
(6) 165
(12) (13)
h (i − 1) = s, i = 1, 2, · · · , N . Then, by some Let θ + N computation, it follows from (7) that Z 0 Vh (xt ) = π T (t + s) e2βs Ω (s) π (t + s) ds, (14)
For future use, for any integer N ≥ 1, by some computation, we can rewrite the integral delay system (1) as N Z − h (i−1) X N x (t) = GB (θ) x (t + θ) dθ
h −N
i=1
where Ω (s) = P + s +
h N
Q and we have denoted
P = diag {P1 , P2 , · · · , PN } ,
(15)
Q = diag {Q1 , Q2 , · · · , QN } ,
(16)
Z
h
Pi = e−2 N (i−1)β h
In
In
···
In
∈ Rn×N n .
(24)
In this subsection, we provide some sufficient conditions for testing the exponential stability of the integral delay system (1). Theorem 1: The integral delay system in (1) is exponentially stable with a decay rate β ≥ 0 if there exist an integer N ≥ 1, two positive definite matrices P ∈ RN n×N n and S ∈ Rn×n , and a symmetric matrix Q ∈ RN n×N n such that (21) and T In In Θ− S < 0, (25) 0N n×n 0N n×n h h MT P + Q + P + Q M N N h T T T G B SBGIN n > 0, (26) + Q − IN N n h h T T T He (PM) + Q − e2β N IN n G B SBGIN n > 0, N (27)
(18)
h −N
where P, Q ∈ RN n×N n are two given symmetric matrices. Before giving our main results, we present the following technical lemma whose proof is omitted due to space limitation. Lemma 4: Let P ∈ Rn×n and Q ∈ Rn×n be two symmetric matrices and R ∈ Rn×n be a semi-positive definite matrix. Let
are satisfied, where M = IN ⊗ M and Θ is defined as T h IN n IN n Θ= P+ Q 0n×N n 0n×N n N T h 0n×N n 0n×N n −2β N − P e . (28) IN n IN n
(20)
where s1 and s2 are two constants with s2 > s1 and f (s) is a concave function1 . Then Φ (s) > 0, ∀s ∈ [s1 , s2 ] if and only if Φ (s1 ) > 0 and Φ (s2 ) > 0. h Remark 3: Let Υ (s) = P + N + s Q. Then the Lyapunov-Krasovskii functional V (xt ) in (19) is positive definite only if Υ (s) is positive definite for all hif and s ∈ −N , 0 , which, according to Lemma 4, is the case h if and only if Υ − N = P > 0 and h Υ (0) = P + Q > 0. N
B. Some Sufficient Conditions
(17)
Notice that both P and Q are block diagonal matrices. Hence the conservatism of Lemma 2 can be further reduced by increasing the degrees of freedom of P and Q. More specifically, we can respectively use two symmetric matrices P and Q instead of the block diagonal matrices P and Q defined in (15)-(17). Namely, we consider the following Lyapunov-Krasovskii functional Z 0 V (xt ) = π T (t + s) e2βs Ω (s) π (t + s) ds, (19)
Φ (s) = P + sQ + f (s) R, ∀s ∈ [s1 , s2 ] ,
(23)
in which
Qi = e−2 N (i−1)β Q.
π (t + s) ds, h −N
IN n = h P + h− i Q , N
0
= GIN n
in which
h −N i
Proof: (Sketch) Consider the Lyapunov-Krasovskii functional in (19). Similarly to the derivation of (23), we can derive Z 0 π T (t + s) π (t + s) ds h −N
Z
(21)
0
=
xT (t + θ) B T (θ) B (θ) x (t + θ) dθ.
(29)
−h
For the same reason given in the above remark, the condition that Q > 0 in Lemma 2 can be weakened as P + hQ > 0.
With this identity, we can show that there exist two constants αi , i = 1, 2 such that Z 0 h 2 −2β N α1 e kx (t + s)k ds ≤ V (xt ) −h Z 0 2 ≤ α2 kx (t + s)k ds. (30)
(22)
For easy reference, we state the following corollary. Corollary 1: The integral delay system in (1) is exponentially stable with a decay rate β ≥ 0 if there exist a positive definite matrix P ∈ Rn×n and a symmetric matrix Q ∈ Rn×n such that (8)-(9) and (22) are satisfied.
−h
We next prove that
1A
V˙ (xt ) + 2βV (xt ) ≤ −α3
function f : D → R is said to be concave if, for any s1 , s2 ∈ D and θ ∈ [0, 1] , there holds f (θs1 + (1 − θ) s2 ) ≥ θf (s1 ) + (1 − θ) f (s2 ) .
Z
0
166
2
kx (t + s)k ds, −h
(31)
h −N Z 0
(32) = −α3
Z
which is just (31). The proof is completed. The following proposition shows that Theorem 1 is always less conservative than Lemma 2. The proof of this proposition is omitted due to space limitation. Proposition 1: For any given β ≥ 0, if there exist two positive definite matrices P, Q ∈ Rn×n such that (8) and (9) are satisfied, then, for any given integer N ≥ 1, there exist two positive definite matrices P ∈ RN n×N n and S ∈ Rn×n , and a symmetric matrix Q ∈ RN n×N n such that (21) and (25)-(27) are satisfied. In the particular case that M = 0, namely B (θ) = B is a constant matrix for all θ ∈ [−h, 0] , we can rewrite the integral delay system (1) as Z t x (t) = GBx (s) ds, (41)
t
π T (θ) e2β(θ−t) Ω (θ − t) π (θ) dθ. (33)
h t− N
In view of (2), we can compute d dt B (θ − t) x (θ) d h B θ − t − dt N x θ− π˙ (θ) = .. . N −1 d B θ − t − dt N h x θ−
h N N −1 N h
= −Mπ (θ) .
t−h
(34)
Define a generalized state vector B (0) x (t) π (t) η (t) = = . h B (−h) x (t − h) π t− N
where G ∈ Rm×n and B ∈ Rn×m are constant matrices. In this case, by noting that inequality (27) implies (26), we obtain the following corollary of Theorem 1. Corollary 2: The integral delay system (41) is exponentially stable with a delay rate β ≥ 0 if there exist an integer N ≥ 1 and matrices 0 < P ∈ RN n×N n , Q ∈ RN n×N n , and 0 ≤ S ∈ Rn×n such that (21), (25) and the following inequality are satisfied:
(35)
Then, the equation in (33) can be rewritten as V˙ (xt ) + 2βV (xt ) = η T (t) Θη (t) Z 0 π T (t + s) e2βs Γ (s) π (t + s) ds, −
(36)
Q−
h −N
where Θ is defined in (28) and h Γ (s) = MT P + +s Q N h + P+ + s Q M + Q. N
h 2β h T T T e N IN n G B SBGIN n > 0. N
(42)
In this particular case, we can add the following functional Z 0 Z t e2β(θ−t) π T (θ) X π (θ) dθdv, (43) W (xt ) = h −N
(37)
t+v
where 0 ≤ X ∈ RN n×N n is a semi-positive definite matrix, to the Lyapunov-Krasovskii function (19) to get new sufficient conditions guaranteeing stability. In fact, by letting Z t F (v) = e2β(θ−t) π T (θ) X π (θ) dθ ≥ 0, (44)
By applying the inequality in (25), Jensen inequality [5] and the system model (23), we can get Z 0 ˙ V (xt ) + 2βV (xt ) ≤ − e2βs π T (t + s) Φπ (t + s) ds, h −N
(40)
−h
π T (θ) e2β(θ−t) Qπ (θ) dθ
− 2β
2
kx (t + s)k ds,
≤ −α3 γ
h t− N t
h t− N
xT (t + s) B T (s) B (s) π (t + s) ds
−h Z 0
from which it follows that h V˙ (xt ) = π T (t) P + Q π (t) N h h h T −2β N −π Pπ t − t− e N N Z t π˙ T (θ) e2β(θ−t) Ω (θ − t) π (θ) dθ +2 −
π T (t + s) π (t + s) ds
≤ −α3
h t− N
Z
0
Z
for some α3 > 0. To do so, let t + s = θ. Then Z t V (xt ) = π T (θ) e2β(θ−t) Ω (θ − t) π (θ) dθ,
t+v
(38)
we can compute Z 0 W (xt ) ≤
where h T T T I G B SBGIN n . (39) N Nn Since −e−2βs is a concave function and h T T T I G B SBGI ≥ 0, it follows from Lemma N n N n N h 4 that Φ (s) > 0, ∀s ∈ − N , 0 if and only if Φ (0) > 0 h and Φ − N > 0, which are respectively equivalent to (26) and (27). Hence, there exists a sufficiently small number α3 such that Φ = Φ (s) = Γ (s) − e−2βs
max {F (v)} dv h − N
h − h ≤v≤0 −N N
h F N Z h 0 2βs T = e π (t + s) X π (t + s) ds N − Nh Z 0 h ≤ λmax (X ) π T (t + s) π (t + s) ds h N −N =
V˙ (xt ) + 2βV (xt ) 167
=
h λmax (X ) λmax B T B N
Z
0
larger N leads to a larger maximal delay value, which is reasonable since a larger N means more decision variables in the conditions. Example 2 (Example 12 in [16]): Consider an integral delay system in the form of (1) with G = B = I2 and 0 1 M= . (50) 0 0
2
kx (t + θ)k dθ, −h
(45) which implies that W (xt ) satisfies the conditions in Lemma 1 and V (xt ) = V (xt ) + W (xt ) , (46) is indeed a Lyapunov-Krasovskii functional candidate for the integral delay system (41). For this Lyapunov-Krasovskii functional, by applying a technique quite similar to the one used in the proof of Theorem 1, we can obtain the following result whose proof is provided in Appendix. Lemma 5: The integral delay system (41) is exponentially stable with a delay rate β ≥ 0 if there exist matrices 0 < P ∈ RN n×N n , Q ∈ RN n×N n , 0 ≤ X ∈ RN n×N n , and 0 ≤ S ∈ Rn×n such that (21), (25) and (42) are satisfied, where Q is replaced by Q + X in (25) and (42). However, the following proposition indicates that the introduction of W (xt ) is not helpful in the reduction of conservatism of Corollary 2. Proposition 2: For any given β ≥ 0 and N ≥ 1, the LMIs in Corollary 2 are feasible if and only if the LMIs in Lemma 5 are feasible. Proof: We need only to show that if the LMIs in Corollary 2 are feasible then the LMIs in Lemma 5 are also feasible. In fact, it is straightforward to verify that if (P∗ , Q∗ , X∗ , S∗ ) is a feasible solution to the LMIs in Corollary 2, then (P# , Q# , S# ) = (P∗ , Q∗ + X∗ , S∗ ) is a feasible solution to the LMIs in Lemma 5. The proof is completed.
Notice that in this case the matrix M is marginally unstable since λ (M ) = {0, 0}. Again, for different values of β, the corresponding maximal values of the delay h by applying different approaches are recorded in Table 2 shown in the next page. From this table we again find that Theorem 1 is always less conservative than Lemma 2 (Theorem 7 in [16]) for any β and N . Moreover, even though Corollary 8 in [16], where an argumentation has been utilized to increase the size of the decision variables, is better than Theorem 1 with N = 1, it is however more conservative than Theorem 1 with N = 5. V. C ONCLUSIONS This paper proposes some delay-dependent sufficient conditions for the exponentially stability of some classes of integral delay systems which are frequently encountered in studying stability problems of time-delay systems. The basic idea is to divide the delay interval into N small intervals so that more information of the delayed state can be utilized to construct the Lyapunov-Krasovskii functional, which in turns can reduce the conservatism of the resulting sufficient conditions expressed in linear matrix inequalities (LMIs). It is shown that the proposed LMIs conditions are always less conservative than the existing conditions for any number of divisions of the delay intervals (namely, the number N ). Numerical examples demonstrate the effectiveness of the proposed approaches.
IV. N UMERICAL E XAMPLES In this section, we present some examples to validate the effectiveness of the proposed approaches. Example 1 (Example 1 in [16]): Consider an integral delay system in the form of (1) with G = B = I2 and 0 1 M= . (47) −2 3
A PPENDIX : T HE P ROOF OF L EMMA 5 For easy reference, we write the two LMI conditions in Lemma 5 as follows T In In 0 Θ − S < 0, (51) 0N n×n 0N n×n h h T T T (52) Q + X − e2β N IN n G B SBGIN n > 0, N where Θ 0 is obtained by replaying Q with Q + X in (28), namely, T h IN n IN n 0 Θ = P+ (Q + X ) 0n×N n 0n×N n N T h 0n×N n 0n×N n − e−2β N P . (53) IN n IN n
For different values of β, by applying different approaches, the corresponding maximal values of the delay h are recorded in Table 1 shown in the next page. From this table we can observe the following facts: 1) Corollary 1 is less conservative than Lemma 2 (Theorem 7 in [16]), which indicates that the weakened condition (22) instead of Q > 0 can indeed reduce the conservatism. In fact, for β = 1 and h = 0.64, we obtain the following numerical values for matrices P and Q : 27.7386 −16.1260 P = , (48) −16.1260 10.6060 −18.9847 10.6050 Q= , (49) 10.6050 −6.5418
It is easy to show that V˙ (xt ) + 2βV (xt ) = η T (t) Θ 0 η (t) Z 0 − e2βs π T (t + s) (Q + X ) π (t + s) ds,
which means that Q is negative definite. 2) Theorem 1 is always less conservative than Lemma 2 (Theorem 7 in [16]) for any β and N. In particular, a
h −N
168
(54)
Table 1: Maximal delay value by using different approaches for Example 1 β 0 1 2 3 Lemma 3 (Lemma 6 in [16]) 0.988 − − − Lemma 2 (Theorem 7 in [16]) 1.990 0.601 0.426 0.349 Corollary 1 2.000 0.653 0.427 0.349 2.000 0.653 0.427 0.349 Theorem 1 with N = 1 Theorem 1 with N = 5 2.538 0.908 0.623 0.494 Theorem 1 with N = 10 2.775 0.952 0.656 0.520 2.950 0.967 0.668 0.529 Theorem 1 with N = 15
Table 2: Maximal delay value by using different approaches for Example 2 β 0 1 2 3 Lemma 3 (Lemma 6 in [16]) 0.707 − − − Lemma 2 (Theorem 7 in [16]) 0.999 0.567 0.426 0.349 Corollary 8 in [16] 0.999 0.628 0.471 0.384 0.999 0.567 0.426 0.349 Corollary 1 Theorem 1 with N = 1 0.999 0.567 0.426 0.349 Theorem 1 with N = 5 0.999 0.660 0.516 0.430 0.999 0.676 0.531 0.445 Theorem 1 with N = 10 Theorem 1 with N = 15 0.999 0.681 0.537 0.450 Theoretical Maximal Delay ( 0, s ∈ − N h z (0) > 0 and z − N > 0, which is equivalent to (52). By h noticing that z − N > 0 implies z (0) > 0, we complete the proof. R EFERENCES [1] Z. Artstein, Linear systems with delayed controls: A reduction, IEEE Transactions on Automatic Control, Vol. 27, pp. 869-879, 1982. [2] B. Du, J. Lam and Z. Shu, Stabilization for state/input delay systems via static and integral output feedback, Automatica, Vol. 46, No. 12, pp. 2000-2007, 2011. [3] E. Fridman, A refined input delay approach to sampled-data control, Automatica, Vol. 46, No. 2, pp. 421-427, 2010. [4] F. Gouaisbaut and D. Peaucelle, Delay-dependent stability analysis of linear time delay systems, IFAC Workshop on Time Delay System, Vol. 6, No. 1, pp. 54–59, 2006. [5] K. Gu, V. L. Kharitonov and J. Chen, Stability of Time-Delay Systems, Boston, MA: Birkh¨auser, 2003. [6] K. Gu and S.-I. Niculescu, Further remarks on additional dynamics in various model transformationsof linear delay systems, IEEE Transactions on Automatic Control, Vol. 46, pp. 497–500, Mar. 2001. [7] J. K. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, Vol. 3, Springer, New York, 1977. [8] Y. He, Q. G. Wang, C. Lin and M. Wu, Delay-range-dependent stability for systems with time-varying delay, Automatica, Vol. 43, No. 2, pp. 371-376, 2007.
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