Generalized Jensen Inequalities with Application to Stability Analysis
arXiv:1602.05281v1 [cs.SY] 17 Feb 2016
Generalized Jensen Inequalities with Application to Stability Analysis of Systems with Distributed Delays over Infinite Time-Horizons ⋆ Kun Liu a Emilia Fridman b Karl Henrik Johansson c Yuanqing Xia a a
School of Automation, Beijing Institute of Technology, 100081 Beijing, China (e-mails: kunliubit, xia [email protected]). b
c
School of Electrical Engineering, Tel Aviv University, 69978 Tel Aviv, Israel (e-mail: [email protected]).
ACCESS Linnaeus Centre and School of Electrical Engineering, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden (e-mail: [email protected]).
Abstract The Jensen inequality has been recognized as a powerful tool to deal with the stability of time-delay systems. Recently, a new inequality that encompasses the Jensen inequality was proposed for the stability analysis of systems with finite delays. In this paper, we first present a generalized integral inequality and its double integral extension. It is shown how these inequalities can be applied to improve the stability result for linear continuous-time systems with gamma-distributed delays. Then, for the discrete-time counterpart we provide an extended Jensen summation inequality with infinite sequences, which leads to less conservative stability conditions for linear discrete-time systems with poisson-distributed delays. The improvements obtained thanks to the introduced generalized inequalities are demonstrated by examples. Key words: new integral and summation inequalities, gamma-distributed delays, poisson-distributed delays, Lyapunov method.
1
Introduction
Time-delay often appears in many control systems either in the state, the control input, or the measurements. During the last two decades, the stability of time-delay systems has received considerable attention (e.g., [3], [8], [15], [17] and references therein). One of the most popular approaches is the use of Lyapunov-Krasovskii functionals (LKF) to derive stability conditions (e.g., [1], [5], [9], [26]). The choice of the Lyapunov functional and the method of bounding an integral term in the derivative of the LKF are important ways to reduce conservativeness of the stability results. The Jensen inequality [8], has been widely used as an efficient bounding technique, although at a price of an unavoidable conservativeness [7], [12]. The Jensen inequality claims that for any continuous function ω : [a, b] → Rn and n × n positive definite ⋆ This work was partially supported by the National Natural Science Foundation of China (grant no. 61503026, 61440058), the Knut and Alice Wallenberg Foundation, the Swedish Research Council, and the Israel Science Foundation (grant no. 754/10 and 1128/14).
Preprint submitted to *****
matrix R, Rb a
ω T (s)Rω(s)ds ≥
1 b−a
Rb a
ω T (s)dsR
Rb a
ω(s)ds
holds. There is a discrete counterpart, which involves sums instead of integrals [3], [4]. Some recent efforts have been made to overcome the conservativeness induced by the Jensen inequality when applied to the stability analysis of time-delay systems. The bound on the gap of the Jensen inequality was analyzed in [2] by using the Gr¨ uss inequality. Based on the Wirtinger inequality [11], Seuret and Gouaisbaut [19] derived an extended integral inequality, which encompasses Jensen inequality as a particular case. Recently, the inequality they proposed was further refined in [20]. By combining the newly developed integral inequality and an augmented Lyapunov functional, a remarkable result was obtained for systems with constant discrete and distributed delays. Let us recall the inequality provided in [20] (see [21] for the discrete counterpart): for any continuous function ω : [a, b] → Rn and n × n posi-
18 February 2016
Question 1 Is it possible to derive more accurate lower bounds to reduce the conservativeness of integral inequalities (4) and (5)? If so, how much improvements can we obtain by applying the generalized inequalities to the stability analysis of continuous-time systems with gammadistributed delays?
tive definite matrix R, the inequality Rb a
ω T (s)Rω(s)ds ≥
Rb T Rb 1 b−a a ω (s)dsR a 3 + b−a ΩT RΩ
ω(s)ds
(1)
holds, where Ω=
Rb a
ω(s)ds −
2 b−a
RbRs a
a
ω(r)drds.
(2)
We further analyze the discrete-time case. Poissondistribution is widespread in queuing theory [6]. In [18], the experimental data on the arrivals of pulses in indoor environments revealed that each cluster’s timedelay is poisson-distributed (see also [10]). Therefore, we study the stability of linear discrete-time systems with poisson-distributed delays via appropriate Lyapunov functionals. The Lyapunov-based analysis uses the discrete counterpart of integral inequalities (4) and (5), i.e., Jensen inequalities with infinite sequences [13], [25]. The following question corresponds to Question 1 in the discrete case:
To prove (1), a function f (u), u ∈ [a, b], was introduced in [20] as follows: f (u) = z(u) ˙ = ω(u) −
1 b−a
Rb a
ω(s)ds −
a+b−2u (b−a)2 Θ,
(3)
where Θ ∈ R Rn is a constant be defined R b vector to(b−u)(u−a) u and z(u) = a ω(s)ds − u−a ω(s)ds − Θ, b−a a (b−a)2 Rb u ∈ [a, b]. It is noted that a (a + b − 2u)du = 0 plays an important role in the utilization of (3). Since Θ is a constant vector, it is obvious that in (3), a + b − 2u could be replaced by c(a + b − 2u), c ∈ R\{0}, because Rb c(a + b − 2u)du = 0. By using a more general auxa Rb iliary function g(u) with a g(u)du = 0, an extended integral inequality, which included the one proposed in [20] as a particular case, was provided in [16].
Question 2 Is it possible to generalize Jensen summation inequalities with infinite sequences? If so, how much improvements can be achieved by applying the generalized inequalities to the stability analysis of discrete-time systems with poisson-distributed delays? The central aim of the present paper is to answer the above questions. First, we present generalized Jensen integral inequality and its double integral extension, which are over infinite intervals of integration. We show how they can be applied to improve the stability result for linear continuous-time systems with gammadistributed delays. Then, for the discrete counterpart we provide extended Jensen summation inequality with infinite sequences, which leads to less conservative stability conditions for linear discrete-time systems with poisson-distributed delays. In both the continuous-time and discrete-time cases, the considered infinite distributed delays are shown to have stabilizing effects. Following [22], we derive the results via augmented Lyapunov functionals.
Recently, the stability analysis of systems with gammadistributed delays was studied [22]. The Lyapunov-based analysis was based on two kinds of integral inequalities with infinite intervals of integration: given an n × n positive definite matrix R, a scalar h ≥ 0, a vector function ω : [0, +∞) → Rn and a scalar function K : [0, +∞) → R+ such that the integrations concerned are well defined, the following inequalities R +∞
K(s)ω T (s)Rω(s)ds R +∞ R +∞ ≥ K0−1 0 K(s)ω T (s)dsR 0 K(s)ω(s)ds 0
and
R +∞ R t
T t−θ−h K(θ)ω (s)Rω(s)dsdθ R R −1 +∞ t ≥ K1h K(θ)ω T (s)dsdθR R +∞0 R t t−θ−h × 0 t−θ−h K(θ)ω(s)dsdθ, R +∞ R +∞ hold, where K0 = 0 K(s)ds and K1h = 0
(4)
0
(5)
The structure of this paper is as follows. In Section 2 we derive generalized Jensen integral inequalities. Section 3 presents stability results for linear continuous-time systems with gamma-distributed delays to illustrate the efficiency of the proposed inequalities. Sections 4 and 5 discuss the corresponding extended Jensen summation inequality with infinite sequences and its application to the stability analysis of linear discrete-time systems with poisson-distributed delays, respectively. The conclusions and the future work will be stated in Section 6.
K(s)(s+ h)ds. The inequalities (4) and (5) were used in [23] to the stability and passivity analysis for diffusion partial differential equations with infinite distributed delays. To obtain more accurate lower bounds of integral inequalities (4) and (5) over infinite intervals of integration, the method developed in [20] for the integral inequality over finite intervals of integration seems not to be applicable, since the function f of (3) is directly dependent on both the lower limit a and the upper limit b. Therefore, an interesting question arises:
Notations: The notations used throughout the paper are standard. The superscript ‘T ’ stands for matrix transposition, Rn denotes the n dimensional Euclidean space with vector norm | · |, Rn×m is the set of all n × m
2
real matrices, and the notation P ≻ 0, for P ∈ Rn×n means that P is symmetric and positive definite. The symmetric term in a symmetric matrix is denoted by ∗. The symbols R, R+ , Z+ and N denote the set of real numbers, non-negative real numbers, non-negative integers and positive integers, respectively. 2
Rewriting the last two terms as sum of squares yields Rb T Rb 1 ω (s)dsR a ω(s)ds ω T (s)Rω(s)ds ≥ b−a a hR i−1 R Rb b b T + a g 2 (s)ds a g(s)ω (s)dsR a g(s)ω(s)ds hR i b − a g 2 (s)ds [Θ − Υ]T R[Θ − Υ], (8) where Rb a
Extended Jensen integral inequalities
The objective of this section is to provide extended Jensen integral inequalities over infinite intervals. To do so, we first prove the generalized Jensen integral inequality introduced in [16] over finite intervals in a simpler way. Then we extend the method to prove the inequality over infinite intervals. 2.1
Υ=
hZ
b
g 2 (s)ds
a
i−1 Z
b
g(s)ω(s)ds.
a
Since (8) holds independently of the choice of Θ, we may choose Θ = Υ, which leads to the maximum of the righthand side of (8) and thus, (6) holds. This concludes the proof.
Extended Jensen integral inequality over finite intervals
Remark 1 In [16], the proof was more complicated as the corresponding construction of (7) relied on an auxiliary function g¯(u), where g¯(u) satisfies g(u) = g¯(u) − Rb 1 ¯(s)ds. b−a a g
By changing a + b − 2u of (3) to a more general scalar Rb function g(u) with a g(u)du = 0 and g(u) not identically zero, we first present the extended Jensen inequality over finite intervals of integration. Lemma 1 [16] If there exist an n × n matrix R ≻ 0, a scalar function g : [a, b] → R and a vector function ω : [a, b] → Rn such that the integrations concerned are well Rb defined and a g(s)ds = 0, where g(s) is not identically zero, then the following inequality holds:
2.2
Generalized Jensen integral inequalities over infinite intervals
We extend the method used for proving Lemma 1 from finite intervals of integration to infinite ones in the following result.
Rb
Rb T Rb 1 ω T (s)Rω(s)ds ≥ b−a ω (s)dsR a ω(s)ds a hR i−1 R Rb b b T + a g 2 (s)ds a g(s)ω (s)dsR a g(s)ω(s)ds. (6)
a
Theorem 1 For a given n×n matrix R ≻ 0, scalar functions g : [0, +∞) → R, K : [0, +∞) → R+ and a vector function ω : [0, +∞) → Rn , assume that the integrations R +∞ concerned are well defined and 0 K(s)g(s)ds = 0 with g(s) not identically zero. Then the following inequality holds:
Proof: Define a function f (u) for all u ∈ [a, b] by f (u) = ω(u) −
1 b−a
Rb a
ω(s)ds − g(u)Θ,
(7)
R +∞
K(s)ω T (s)Rω(s)ds R +∞ R +∞ ≥ K0−1 0 K(s)ω T (s)dsR 0 K(s)ω(s)ds hR i−1 +∞ ¯ T RΩ, ¯ + 0 K(s)g 2 (s)ds Ω 0
where Θ ∈ Rn is a constant vector to be defined. Then, since R ≻ 0 it follows that 0≤ =
Rb
f T (s)Rf (s)ds a h iT h i Rb Rb Rb 1 1 ω(s)− ω(θ)dθ ω(θ)dθ ds R ω(s)− b−a a b−a a ah iR Rb T b 2 + b−a a ω (s)dsRΘ a g(s)ds i hR Rb b + a g 2 (s)ds ΘT RΘ − 2ΘT R a g(s)ω(s)ds.
Noting that
Rb a
where
R +∞ K0 = 0 K(s)ds, R ¯ = +∞ K(s)g(s)ω(s)ds. Ω
(10)
0
Proof:
g(s)ds = 0, we obtain
Define a function f¯(u) for all u ∈ [0, +∞) by
Rb T Rb 1 ω (s)dsR a ω(s)ds ω T (s)Rω(s)ds ≥ b−a a a h i Rb Rb − a g 2 (s)ds ΘT RΘ + 2ΘT R a g(s)ω(s)ds.
Rb
(9)
f¯(u) =
3
i h p R +∞ ¯ , K(u) ω(u)−K0−1 0 K(s)ω(s)ds− g(u)Θ
¯ ∈ Rn is a constant vector to be defined. Because where Θ R ≻ 0 we have
Corollary 1 For a given n × n matrix R ≻ 0, a scalar function K : [0, +∞) → R+ and a vector function ω : [0, +∞) → Rn , assume that the integrations concerned are well defined. Then the following inequality holds:
R +∞ 0 ≤ 0 f¯T (s)Rf¯(s)ds iT p R +∞hp R +∞ = 0 K(s)ω(s)−K0−1 K(s) 0 K(θ)ω(θ)dθ R i hp p R +∞ × K(s)ω(s)−K0−1 K(s) 0 K(θ)ω(θ)dθ ds h iR R ¯ T R +∞ K(s)ω(s)ds +∞ K(s)g(s)ds + 2K0−1 Θ 0 0 i hR +∞ ¯ T RΘ ¯ + 0 K(s)g 2 (s)ds Θ R ¯ T R +∞ K(s)g(s)ω(s)ds. −2Θ 0
R +∞
(16) where K0 , K1 and K2 are given by (10), (12) and (15), respectively, and
RepresentingR the last two terms as sum of squares to+∞ gether with 0 K(s)g(s)ds = 0 yields R +∞
K(s)ω T (s)Rω(s)ds R +∞ R +∞ ≥ K0−1 0 K(s)ω T (s)dsR 0 K(s)ω(s)ds i−1 hR +∞ ¯ T RΩ ¯ Ω + 0 K(s)g 2 (s)ds hR i +∞ ¯ − Υ] ¯ T R[Θ ¯ − Υ], ¯ − K(s)g 2 (s)ds [Θ
˜= Ω
(11)
¯ is given in (10) and where Ω +∞
K(s)g 2 (s)ds
0
i−1 Z
+∞
K(s)g(s)ω(s)ds.
0
0
Note that the choice of g(s) plays a crucial role in the application of Theorem 1. Given K0 in (10) and
let
R +∞ 0
sK(s)ds,
g(u) = c(K0 u − K1 ), c ∈ R\{0}, u ≥ 0, such that
R +∞ 0
0
K(s)ω(s)ds −
R +∞ 0
sK(s)ω(s)ds.
R +∞ R t
Then, the same arguments in the proof of Lemma 1 and ¯ =Υ ¯ lead to the maximum of the right-hand the choice Θ side of (11) and thus, (9) holds. This concludes the proof.
K1 =
R +∞
Theorem 2 If there exist an n × n matrix R ≻ 0, scalar functions g : [t − θ − h, t] → R, K : [0, +∞) → R+ , a scalar h ≥ 0 and a vector function ω : [t − θ − h, t] → Rn such R +∞that R t the integrations concerned are well defined and 0 t−θ−h K(θ)g(s)dsdθ = 0, where g(s) is not identically zero, then the following inequality holds:
0
hZ
K1 K0
The same methodology to prove Lemma 1 and Theorem 1 can be applied to generalize the inequality (5). We have the following result:
0
¯ = Υ
K(s)ω T (s)Rω(s)ds R +∞ R +∞ ≥ K0−1 0 K(s)ω T (s)dsR 0 K(s)ω(s)ds −1 K2 ˜ T RΩ, ˜ + K2 − K10 Ω
0
T t−θ−h K(θ)ω (s)Rω(s)dsdθ R R +∞ t −1 ≥ K1h K(θ)ω T (s)dsdθR 0 R +∞ R t t−θ−h × 0 t−θ−h K(θ)ω(s)dsdθ hR i−1 +∞ R t + 0 K(θ)g 2 (s)dsdθ ΣT RΣ, t−θ−h
(17)
where
(12)
R +∞ K1h = 0 K(s)(s + h)ds = hK0 + K1 , R +∞ R t Σ= 0 t−θ−h K(θ)g(s)ω(s)dsdθ.
(13)
(18)
Proof: See Appendix A.
K(s)g(s)ds = 0 holds. Then, we find that
(14)
Remark 2 Theorems 1 and 2 refine the inequalities of [22], in which the last terms of the right-hand-side of (9) and (17) are zero. Hence, our new inequalities develop R +∞ more accurate lower bounds of 0 K(s)ω T (s)Rω(s)ds R +∞ R t and 0 K(θ)ω T (s)Rω(s)dsdθ than the ones t−θ−h provided in [22].
(15)
We choose a scalar function
From (13), (14) and Theorem 1, we have the following corollary:
g(u) = −u + t −
R +∞ 0
K(s)g 2 (s)ds = c2
R +∞ 0
K(s)(K0 s − K1 )2 ds
= c2 (K02 K2 − K0 K12 ), R +∞ ¯= Ω K(s)g(s)ω(s)ds i h0 R R +∞ +∞ = c K0 0 sK(s)ω(s)ds − K1 0 K(s)ω(s)ds ,
where
K2 =
R +∞ 0
s2 K(s)ds.
4
hK1h +hK1 +K2 , 2K1h
(19)
R +∞ R t such that 0 K(θ)g(s)dsdθ = 0, where K1 , K2 t−θ−h and K1h are given by (12), (15) and (18), respectively. Hence, we have R +∞ R t 0
=
=
K(θ)g 2 (s)dsdθ t−θ−h K(θ) − s + t −
t−θ−h
R +∞ R t 0
3
h 2
K0 + 2K3 +
hK1h +hK1 +K2 2K1h 3h2 K0 (hK1 +2K2 )−3K22 ∆ ˜ = K1 , 2K1h
2
where x(t) ∈ Rn is the state vector, A, A1 ∈ Rn×n are constant system matrices, and h ≥ 0 represents a fixed time gap. The smooth kernel Γ is given by N −1 − θ
T Γ(θ) = Tθ N (Ne−1)! , where N ≥ 2, N ∈ N, is a shape parameter of the distribution and T > 0 is a scale parameter. The matrices A and A + A1 are not allowed to be Hurwitz. The initial condition is given by φ ∈ C 1 (−∞, 0], where C 1 (−∞, 0] denotes the space of continuously differentiable functions φ : (−∞, 0] → Rn ˙ C < +∞, with the norm kφkC 1 = kφkC + kφk kφkC = sups∈(−∞,0] |φ(s)| < +∞.
dsdθ (20)
and R +∞ R t Σ= 0 K(θ)g(s)ω(s)dsdθ R +∞ R t t−θ−h +hK1 +K2 = 0 K(θ) −s+t− hK1h 2K ω(s)dsdθ t−θ−h 1h R +∞R t Rt = 0 K(θ)ω(s)drdsdθ t−θ−h s R +∞ R t hK1+K2 h − 2 + 2K1h K(θ)ω(s)dsdθ 0 t−θ−h R +∞ R t Rr = 0 K(θ)ω(s)dsdrdθ R t−θ−h t−θ−h Rt ∆ ˜ +∞ 1+K2 − h2 + hK 2K1h 0 t−θ−h K(θ)ω(s)dsdθ = Σ, (21) where R +∞ K3 = 0 s3 K(s)ds.
Following [22] and introducing y(t) =
0
ρ(t) =
R +∞
R0+∞
R0+∞
R0+∞
(24)
−∞ Ψ(t− s)x(s− θ θ N −2 e− T T N (N −2)! .
h)ds =
R +∞ 0
Ψ(θ)x(t −θ− h)dθ,
Ψ(θ)dθ =
1 ∆ T =
Ψ0 , ∆
θΨ(θ)dθ = N − 1 = Ψ1 , ∆
θ2 Ψ(θ)dθ = N (N − 1)T = Ψ2 ,
∆ θ3 Ψ(θ)dθ = (N + 1)N (N − 1)T 2 = Ψ3 , 0 R +∞ ∆ (θ + h)Ψ(θ)dθ = hΨ0 + Ψ1 = N − 1 + Th = 0
(22)
Ψ1h . (25) In the following, we provide two sufficient conditions for the stability of system (24); one is derived by applying (16) and (5), the other is obtained from (16) and (22). 3.1
Stability result I
We consider the following augmented LKF: V (t) = V1 (t) + VG (t) + VH (t), V1 (t) = η T (t)W η(t), R +∞ R t VG (t) = 0 Ψ(θ)xT (s)Gx(s)dsdθ, R +∞ Rt−θ−h R θ+h t VH (t) = 0 ˙ T (s)H x(s)dsdλdθ, ˙ 0 t−λ Ψ(θ)x (26) where W ≻ 0, G ≻ 0, H ≻ 0, η(t) = col{x(t), y(t)}. Since A and A+A1 are not allowed to be Hurwitz, we use
Consider the linear continuous-time systems with gamma-distributed delays: Γ(θ)x(t − θ − h)dθ,
Γ(t− s)x(s− h)ds,
It follows readily that
Stability analysis of continuous-time systems with gamma-distributed delays
0
Rt
Ψ(θ) =
The generalized integral inequality (16) and its double integral extension (22) will be employed for the stability analysis of continuous-time systems with gammadistributed delays in the next section.
R +∞
−∞
where
˜ 1 and Σ ˜ are given by (20) and (21), respectively. where K
x(t) ˙ = Ax(t) + A1
Rt
y(t) ˙ = − T1 y(t) + ρ(t),
1
3
Γ(θ)x(t −θ− h)dθ =
x(t) ˙ = Ax(t) + A1 y(t),
Corollary 2 If there exist an n × n matrix R ≻ 0, a scalar function K : [0, +∞) → R+ , a scalar h ≥ 0 and a vector function ω : [t − θ − h, t] → Rn such that the integrations concerned are well defined, then the following inequality holds: T t−θ−h K(θ)ω (s)Rω(s)dsdθ R R −1 +∞ t ≥ K1h K(θ)ω T (s)dsdθR 0 R +∞ R t t−θ−h × 0 t−θ−h K(θ)ω(s)dsdθ −1 ˜ T ˜ ˜ +K Σ RΣ,
0
the system (23) can be transformed into
From (19)–(21) and Theorem 2, we arrive at the following result:
R +∞ R t
R +∞
(23)
5
with ξ(t) = col{x(t), y(t), ρ(t)}. By applying Corollary 1 we obtain
augmented Lyapunov functionals. The term VH extends the triple integral of [24] for finite delay to infinite delay [22].
R +∞ V˙ G (t) = Ψ0 xT (t)Gx(t)− 0 Ψ(θ)xT (t−θ−h)Gx(t−θ−h)dθ ≤ Ψ0 xT (t)Gx(t) − Ψ−1 ρT (t)Gρ(t) −1 h 0 iT 2 Ψ Ψ1 − Ψ2 − Ψ10 ρ(t) − (N − 1)y(t) G Ψ 0 i h Ψ1 ρ(t) − (N − 1)y(t) × Ψ 0
Remark 3 The recent method of [22] for the stability analysis of system (24) is based on a functional of the form V (t) = V1 (t) + VG (t) + VH (t) + VE (t) + VF (t), R +∞ R t (27) VE (t) = 0 Γ(θ)xT (s)Ex(s)dsdθ, R +∞ Rt−θ−h R θ+h t T VF (t) = 0 Γ(θ)x˙ (s)F x(s)dsdλdθ ˙ 0 t−λ
T T = ξ T (t)[Ψ1h F01 HF01 − Ψ−1 1h F13 HF13 ]ξ(t).
Therefore, (28) guarantees that V˙ (t) ≤ ξ T (t)Ξξ(t) ≤ −β|x(t)|2 for some β > 0, which proves the asymptotic stability.
Proposition 1 If there exist 2n × 2n positive definite matrix W and n × n positive definite matrices G, H such that the following LMI is feasible:
F0 =
A A1 0 0 − T1 I
#
, F1 =
"
I 0 0 0 I 0
#
Stability result II
The stability of system (24) can be alternatively analyzed via a LKF given by ¯ η¯(t), V¯ (t) = V¯1 (t) + VG (t) + VH (t), V¯1 (t) = η¯T (t)W
(28)
¯ ≻ 0, η¯(t) = col{x(t), y(t), ζ(t)}, ζ(t) = where W R +∞ R t Ψ(θ)x(s)dsdθ, VG (t) and VH (t) are given by 0 t−θ−h ˙ = 1 x(t) − ρ(t) and differentiating (26). Noting that ζ(t) T V1 (t) along (24), we have
where Ψ1h is given by (25), Σ = diag{ T1 G, 0, −T G} and "
1 T T T x (t)Gx(t) − T ρ (t)Gρ(t) T GF23 ξ(t). − NT−1 ξ T (t)F23
V˙ H (t) ≤ Ψ1h x˙ T (t)H x(t) ˙ h iT h i −1 −Ψ1h Ψ0 x(t) − ρ(t) H Ψ0 x(t) − ρ(t)
3.2
T T +Ψ1h F01 HF01 − Ψ−1 1h F13 HF13 ≺ 0,
=
− T ρT (t)Gρ(t) h iT h i − NT−1 T ρ(t) − y(t) G T ρ(t) − y(t)
Furthermore, applying (5) we find that
The following proposition is provided for the asymptotic stability of system (24).
N −1 T T F23 GF23
1 T T x (t)Gx(t)
(29)
together with the utilization of the integral inequalities (4) and (5). Compared to (26), the functional (27) has two additional terms VE (t) and VF (t). In the example below, we will show the advantages of our proposed approach (larger stability region in the (T, h) plane and less number of scalar decision variables). The improvement is achieved due to that the application of Corollary 1 leads to one more negative term in the derivative of the LKF.
Ξ = Σ + F1T W F0 + F0T W F1 −
=
,
Ax(t) + A1 y(t)
¯ − 1 y(t) + ρ(t) V¯˙ 1 (t) = 2¯ η T (t)W T 1 T x(t) − ρ(t) T T ¯ ¯ ¯ ¯ ¯ = 2ξ (t)F W F0 ξ(t)
F01 = [A A1 0], F13 = [ T1 I 0 − I], F23 = [0 − I T I],
(30)
1
then system (24) is asymptotically stable.
¯ = col{x(t), y(t), ρ(t), ζ(t)} and with ξ(t)
Proof: Differentiating V1 (t) along (24), we have
I 0 0 0
A
A1
0 0
1 ¯ F¯1 = 0 I 0 0 , F0 = 0 − T I I 0 . 1 0 −I 0 0 0 0 I TI
V˙ 1 (t) = 2η T (t)W η(t) ˙ = 2ξ T (t)F1T W F0 ξ(t)
6
(31)
Table 1 Example 1: maximum allowable value of T for different h
Furthermore, by applying (22) we find that
[max T ]
10−5
0.01
0.15
0.34
0.35
0.36
\ h
V˙ H (t) ≤ Ψ1h x˙ T (t)H x(t) ˙ h iT h i −1 −Ψ1h Ψ0 x(t) − ρ(t) H Ψ0 x(t) − ρ(t)
˜ −1 ϕT (t)Hϕ(t) −Ψ 1 T ¯ T ¯ ˜ −1 ¯ T ¯ ¯ H F¯01−Ψ−1 = ξ¯T (t)[Ψ1h F¯01 1h F13 H F13−Ψ1 F33 H F33 ]ξ(t), (32) where Ψ1h is given in (25) and
Decision variables
[22]
0.274
0.265
0.141
0.005
-
-
22
Prop. 1
0.305
0.296
0.158
0.008
0.002
-
16
Prop. 2
0.322
0.312
0.168
0.014
0.008
0.003
31
[22] and using Propositions 1 and 2, we obtain the maximum allowable values of T that achieve the stability. Fig. 1 presents tradeoff curves between maximal allowable T and h by applying the above three methods. Furthermore, the stability region in the (T, h) plane that preserves the asymptotic stability is depicted in Figs. 2–4 by using the condition in [22], Proposition 1 and Proposition 2, respectively. From Figs. 2–4 we can see that Proposition 1 induces a more dense stability region than [22], but guarantees a little sparser stability region than Proposition 2. Therefore, Figs. 1–4 show that Proposition 1 improves the results in [22] and that the conditions can be further enhanced by Proposition 2.
F¯01 = [A A1 0 0], F¯13 = [ T1 I 0 − I 0], ϕ(t) = ~[Ψ0 x(t) − ρ(t)] − ζ(t) + hρ(t) + (N − 1)y(t) ¯ = F¯33 ξ(t), F¯33 = [ T~ I (N − 1)I (h − ~)I − I], 2 2 ˜ 1 = h3 + 2Ψ3 + 3h Ψ0 (hΨ1 +2Ψ2 )−3Ψ2 , Ψ 2Ψ1h 2T 1 +Ψ2 ~ = h2 + hΨ2Ψ . 1h
Let us now compare the number of scalar decision variables in the LMIs. The LMIs of [22] have {4n2 + 3n}n=2 = 22 variables. Proposition 1 in this paper not only possess a fewer number {3n2 +2n}n=2 = 16 of variables but also lead to less conservative results. In comparison with Proposition 1, Proposition 2 slightly improves the results at the price of {2.5n2 + 2.5n}n=2 = 15 additional decision variables.
(33) Therefore, by combining (29), (30) and (32) we obtain 2 ¯ ≤ −β|x(t)| ¯ ¯ ξ(t) V¯˙ (t) ≤ ξ¯T (t)Ξ for some β¯ > 0, if T ¯ =Σ ¯ + F¯1T W ¯ F¯0 + F¯0T W ¯ F¯1 − N −1 F¯23 Ξ GF¯23 T T ¯T ¯ ˜ −1 ¯ T ¯ H F¯01 − Ψ−1 +Ψ1h F¯01 1h F13 H F13 − Ψ1 F33 H F33 ≺ 0, (34) where ¯ = diag{ 1 G, 0, −T G, 0}, Σ T (35) F¯23 = [0 − I T I 0].
0.4 [22] Proposition 1 Proposition 2
0.35 0.3 0.25 h
We have thus proved the following proposition:
0.2 0.15
Proposition 2 If there exist 3n × 3n positive definite ¯ and n × n positive definite matrices G, H such matrix W that LMI (34) with notations given by (25), (31), (33) and (35) is feasible, then system (24) is asymptotically stable.
0.1 0.05 0
Example 1
4
We illustrate the efficiency of the presented results through an example of two cars on a ring, see [14] and [22] for details. In this example,
A = 0 and A1 =
"
−2 2 2 −2
#
0.05
0.1
0.15
0.2 T
0.25
0.3
0.35
0.4
Fig. 1. Example 1: tradeoff curve between maximal allowable T and h for Propositions 1 and 2 compared with the result of [22]
Next we present an example to illustrate the applicability of the theoretical results. 3.3
0
Extended Jensen summation inequalities with infinite sequences
The objective of this section is to present the discrete counterpart of the results obtained in Section 2 and to provide extended Jensen summation inequality with infinite sequences. We first introduce the following lemma for the discrete counterpart of the integral inequalities (4) and (5):
,
Lemma 2 Assume that there exist an n × n matrix R ≻ 0, a scalar function M (i) ∈ R+ and a vector function
so neither A nor A + A1 is Hurwitz. For the values of h given in Table I and N = 2, by applying the method in
7
hold, where
0.4 condition of [22] 0.35
M1h
h
0.25 0.2 0.15
0.05
0
0.05
0.1
0.15
0.2 T
0.25
0.3
0.35
(38)
The proof of (36) and (37) follows from [22] by using sums instead of integrals and is therefore omitted here. By applying the arguments of Theorem 1 to the discrete case, we obtain the following theorem for the extended Jensen summation inequality with infinite sequences. Note that this result includes (36) as a special case and that the generalization of (37) can be done by the same approach as exploited in Theorem 2.
0.1
0
P+∞
i=0 M (i), P+∞ = i=0 (i + h)M (i).
M0 =
0.3
0.4
Fig. 2. Example 1: stability region by the condition of [22]
0.4 Proposition 1 0.35
Theorem 3 For a given n × n matrix R ≻ 0, scalar functions M (i) : Z+ → R+ , g(i) : Z+ → R and a vector function x(i) : Z+ → Rn , assume that the series conP+∞ cerned are well defined and i=0 M (i)g(i) = 0 with g(i) not identically zero. Then the following inequality holds:
0.3
h
0.25 0.2 0.15 0.1
P+∞
M (i)xT (i)Rx(i) hP iT h P i +∞ +∞ M (i)x(i) ≥ M0−1 R M (i)x(i) i=0 i=0 hP i−1 +∞ 2 T + Π RΠ i=0 M (i)g (i) (39) with M0 given by (38) and
0.05 0
i=0
0
0.05
0.1
0.15
0.2 T
0.25
0.3
0.35
0.4
Fig. 3. Example 1: stability region by Proposition 1 0.4 Proposition 2 0.35
Π=
0.3
h
0.25 0.2
P+∞ i=0
M (i)g(i)x(i).
(40)
Proof: See Appendix B.
0.15
In order to apply Theorem 3 to the stability analysis of time-delay systems, we take
0.1 0.05 0
0
0.05
0.1
0.15
0.2 T
0.25
0.3
0.35
0.4
g(v) = c(M0 v − M1 ), c ∈ R\{0}, v ∈ Z+ ,
Fig. 4. Example 1: stability region by Proposition 2
such that n
x(i) ∈ R such that the series concerned are convergent. Then the inequality M (i)xT (i)Rx(i) hP iT h P i +∞ +∞ ≥ M0−1 R i=0 M (i)x(i) i=0 M (i)x(i) ,
i=0
P+∞ i=0
(36)
M (i)xT (j)Rx(j) iT +∞ Pk−1 −1 ≥ M1h M (i)x(j) R j=k−i−h h P i=0 i +∞ Pk−1 × i=0 j=k−i−h M (i)x(j) , j=k−i−h
hP
M (i)g(i) = 0 is satisfied, where M1 =
P+∞
M (i)g 2 (i) = c2
P+∞
i=0
iM (i).
(42)
2 i=0 M (i)(M0 i − M1 ) c2 (M02 M2 − M0 M12 ),
= P+∞ Π = i=0 M (i)g(i)x(i) i h P+∞ P M (i)x(i) , iM (i)x(i) − M = c M0 +∞ 1 i=0 i=0
and its double summation extension i=0
i=0
Hence, we have
P+∞
P+∞ Pk−1
P+∞
(41)
(43)
(37)
where
M2 =
8
P+∞ i=0
i2 M (i).
(44)
τ
where P(τ ) = e τ !λ . We next derive LMI conditions for the asymptotic stability of (47) via a direct Lyapunov method. −λ
From (41) and (43), Theorem 3 is reduced to the following corollary, which will be employed in the next section for the stability analysis of discrete-time systems with poisson-distributed delays.
Denoting Corollary 3 Given an n × n matrix R ≻ 0, a scalar function M (i) : Z+ → R+ and a vector function x(i) : Z+ → Rn such that the series concerned are well defined, the following inequality holds: P+∞
f (k) =
M (i)x (i)Rx(i) hP iT h P i +∞ +∞ ≥ M0−1 M (i)x(i) R M (i)x(i) i=0 i=0 −1 M12 T ˜ RΠ, ˜ + M2 − Π
(45) where M0 , M1 and M2 are given by (38), (42) and (44), respectively, and
5
P+∞ i=0
M (i)x(i) −
P+∞ i=0
= e−λ Ax(k − h) + e−λ A1 f (k − h) P+∞ + τ =0 Q(τ )x(k − τ − h),
iM (i)x(i).
(48) where Q(τ ) = It is noted that the augmented system (48) has not only distributed but also discrete delays. This is different from augmented system (24) for the case of gamma-distributed delays. Moreover, we find that e−λ λτ +1 (τ +1)! .
Stability analysis of discrete-time systems with poisson-distributed delays
In this section, we will demonstrate the efficiency of the extended Jensen summation inequality (45) through the stability analysis of linear discrete-time systems with poisson-distributed delays. Consider the following system:
P+∞
i=0 (i
P+∞ i=0
P+∞
P+∞
− τ ), k ∈ Z+ , (46) where x(k) ∈ Rn is the state vector, the system matrices A and A1 are constant with appropriate dimensions. We do not allow A and A + A1 to be Schur stable. The initial condition is given as col{x(0), x(−1), x(−2), . . . } = col{φ(0), φ(−1), φ(−2), . . . }. The function p(v), v ∈ Z+ , is a poisson distribution with a fixed time gap h ∈ Z+ : x(k + 1) = Ax(k) + A1
p(v) =
(
τ =0 p(τ )x(k
e−λ λv−h (v−h)!
v ≥ h,
0
v < h,
i=0
P+∞ i=0
P+∞
τ =0 p(τ )x(k
− τ) = =
P+∞
τ =h p(τ )x(k
P+∞
θ=0 p(θ
P+∞
2
2
i Q(i) = λ − λ + 1 − e
(49)
−λ ∆
¯ 2, =Q ∆
¯ 1h . + h)Q(i) = λ+(1 − e−λ )(h−1) = Q
i P2 h ˆx V (k) = xˆT (k)W ˆ(k)+ i=1 VGi (k) + VHi (k) + VSi (k) , P+∞ Pk−1 VG1 (k) = i=0 s=k−i−h P(i)xT (s)G1 x(s), P+∞ Pi+h Pk−1 VH1 (k) = i=0 j=1 s=k−j P(i)η1T (s)H1 η1 (s), P+∞ Pk−1 VG2 (k) = i=0 s=k−i−h Q(i)xT (s)G2 x(s), P+∞ Pi+h Pk−1 VH2 (k) = i=0 j=1 s=k−j Q(i)η1T (s)H2 η1 (s), Pk−1 VS1 (k) = s=k−h xT (s)S1 x(s) P−1 Pk−1 +h j=−h s=k+j η1T (s)R1 η1 (s), Pk−1 VS2 (k) = s=k−h f T (s)S2 f (s) P−1 Pk−1 +h j=−h s=k+j η2T (s)R2 η2 (s),
− τ)
+ h)x(k − θ − h),
τ =0 P(τ )x(k
∆
¯ 1, iQ(i) = λ − 1 + e−λ = Q
Consider system (48) with both distributed and discrete delays. The stability analysis will be based on the following discrete-time LKF:
we arrive at the equivalent system: x(k + 1) = Ax(k)+A1
+ h)P(i) = λ + h,
Q(i) = 1 − e−λ ,
i=0 (i
where λ > 0 is a parameter of the distribution. The mean value of p is h + λ. Due to the fact that P+∞
− τ − h), k ∈ Z+ ,
x(k + 1) = Ax(k) + A1 f (k), P+∞ −λ τ f (k + 1) = τ =0 e τ !λ x(k + 1 − τ − h) P+∞ −λ τ = e−λ x(k + 1 − h) + τ =1 e τ !λ x(k + 1 − τ − h) P+∞ −λ λτ +1 x(k − τ − h) = e−λ x(k + 1 − h) + τ =0 e (τ +1)!
M0
M1 M0
τ =0 P(τ )x(k
the system (47) can be transformed into the following augmented form
T
i=0
˜ = Π
P+∞
−τ − h), k ∈ Z+ , (47)
ˆ ≻ 0, Gi ≻ 0, Hi ≻ 0, Si ≻ 0, Ri ≻ 0, i = 1, 2, where W
9
and
and x ˆ(k) = col{x(k), f (k)},
h i=1 VGi (k + 1) + VHi (k + 1) + VSi (k + 1) i −VGi (k) − VHi (k) − VSi (k) h ˆ + Fˆ T [(λ + h)H1 + Q ¯ 1h H2 + h2 R1 ]Fˆ01 = ξ T (k) Σ 01
P2
η1 (k) = x(k + 1) − x(k), η2 (k) = f (k + 1) − f (k). Here the last two terms VS1 (k) and VS2 (k) are added to compensate the delayed terms x(k − h) and f (k − h) of (48), respectively. Therefore, for system (48) with h = 0, the terms VS1 (k) and VS2 (k) are not necessary. From standard arguments, we arrive at the following result for the asymptotic stability of (48):
T T R2 Fˆ02 H1 Fˆ12 + h2 Fˆ02 −(λ + h)−1 Fˆ12 i −Fˆ T R1 Fˆ13 − Fˆ T R2 Fˆ24 ξ(k) 13
+(1 − e ) q (k)G2 q(k) P+∞ − i=0 Q(i)xT (k − i − h)G2 x(k − i − h) P+∞ Pk−1 − i=0 s=k−i−h Q(i)η1T (s)H2 η1 (s).
Proposition 3 Given a real scalar λ > 0 and an integer h ≥ 0, assume that there exist 2n × 2n positive definite ˆ and n × n positive definite matrices Gi , Hi , matrix W Si , Ri , i = 1, 2, such that the following LMI is satisfied:
(52) Applying the generalized Jensen inequality (45) with infinite sequences, we obtain −
T ˆ = Σ+ ˆ Fˆ0T W ˆ Fˆ0 − Fˆ1T W ˆ Fˆ1 − (λ + h)−1 Fˆ12 Ξ H1 Fˆ12 T ¯ 1h H2 + h2 R1 ]Fˆ01 +Fˆ01 [(λ+h)H1 + Q
Fˆ0 =
A A1
Fˆ1 =
"
I 0 0 00
0 0 e
0
0
−λ
−λ
0 I 0 00
Ae #
0
A1 I
,
e−λ A1 I], Fˆ12 = [I −I 0 0 0], Fˆ15 = [(1 − e−λ )I 0 0 0 −I],
τ =0
P+∞ Pk−1
¯ −1 [(1−e−λ )x(k)−q(k)]T H2 [(1−e−λ )x(k)−q(k)] = −Q 1h T ¯ = −Q−1 ξ T (k)Fˆ15 H2 Fˆ15 ξ(k).
¯1 Q Fˆ25 = [0 − λI 0 0 ( 1−e −λ + 1)I].
1h
(54) Therefore, (51)–(54) yield ∆V (k) = V (k + 1) − V (k) ≤ ˆ ξ T (k)Ξξ(k). Then if (50) holds for given scalars λ > 0 and h ≥ 0, the system (48) is asymptotically stable. Remark 4 The LMI condition in Proposition 3 is derived by employing the generalized Jensen inequality (45). The system (48) can be alternatively analyzed by inequality (36). In this case, (53) is reduced to:
Proof: Define P+∞
(53)
T s=k−i−h Q(i)η1 (s)H2 η1 (s) hi=0 i h iT −1 P+∞ P+∞ Pk−1 ≤− i=0 (i+h)Q(i) i=0 s=k−i−h Q(i)η1 (s) H2 i hP +∞ Pk−1 Q(i)η (s) × 1 s=k−i−h i=0
−
Then the system (48) is asymptotically stable.
q(k) =
Q(i)xT (k − i − h)G2 x(k − i − h)
Furthermore, the application of (37) leads to
, Fˆ01 = [A−I A1 0 0 0],
Fˆ02 = [0 −I e−λ A Fˆ13 = [I 0 −I 0 0], Fˆ24 = [0 I 0 −I 0],
i=0
= −(1 − e−λ )−1 q T (k)G2 q(k) −1 2 T ¯ 2 − Q¯ 1−λ ξ T (k)Fˆ25 G2 Fˆ25 ξ(k). − Q 1−e
(50) ¯ 1, Q ¯ 2 and Q ¯ 1h are given by (49), Σ ˆ = diag{S1 + where Q G1 +(1−e−λ )G2 , −G1 +S2 , −S1 , −S2 , −(1−e−λ )−1 G2 }, #
P+∞
≤ −(1 − e−λ )−1 q T (k)G2 q(k) −1 h ¯ iT 2 Q1 ¯ 2 − Q¯ 1−λ − Q +1 q(k)−λf (k) G2 −λ 1−e i h ¯1−e Q1 × 1−e−λ + 1 q(k) − λf (k)
T T ¯ −1 Fˆ T H2 Fˆ15 − Fˆ13 R1 Fˆ13 +h2 Fˆ02 R2 Fˆ02 − Q 1h 15¯ 2 −1 T T ¯ 2 − Q1−λ R2 Fˆ24 − Q −Fˆ24 Fˆ25 G2 Fˆ25 ≺ 0, 1−e
"
24
−λ −1 T
Q(τ )x(k − τ − h),
ξ(k) = col{x(k), f (k), x(k − h), f (k − h), q(k)}. − By taking difference of V (k) along (48) and applying Jensen inequalities with finite sequences (see e.g., Chapter 6 of [3]), we have ˆ xˆ(k + 1) − x ˆx xˆT (k + 1)W ˆT (k)W ˆ(k) T T ˆ ˆ T ˆ ˆ ˆ ˆ = ξ (k)[F W F0 − F W F1 ]ξ(k) 0
P+∞ i=0
Q(i)xT (k − i − h)G2 x(k − i − h)
≤ −(1 − e−λ )−1 q T (k)G2 q(k). ˆ ˆ It yields Ξ |F25 =0 ≺ 0, which is more conservative than the condition proposed in Proposition 3 since the matrix −1 2 T ¯ 2 − Q¯ 1−λ − Q Fˆ25 G2 Fˆ25 of (50) is negative definite.
(51)
1−e
1
10
ˆ ∈ Rn is a constant vector to be defined and fˆ(r, u) = Θ h i p R R −1 +∞ t K(u) ω(r) − K1h K(θ)ω(s)dsdθ with 0 t−θ−h K1h given by (17). Since R ≻ 0 we have
Remark 5 Both conditions in Proposition 3 and Remark 4 are derived by the use of inequality (37). It is worth noting that the results could be further improved (in the (λ, h) plane preserving the stability) by the discrete counterpart of Theorem 2. 5.1
Example 2
Consider the linear discrete-time system (46) with A=
"
−0.5 0 0
1
#
and A1 =
"
−0.5 0.8 0.5 −0.2
#
R +∞ R t
f T (s, θ)Rf (s, θ)dsdθ R0+∞ Rt−θ−h t ˆT ˆ = 0 t−θ−h f (s, θ)Rf (s, θ)dsdθ h i R +∞ R t −1 ˆ T + 2K1h Θ R 0 t−θ−h K(θ)ω(s)dsdθ R +∞ R t × K(θ)g(s)dsdθ i h R 0 R t−θ−h +∞ t 2 ˆ T RΘ ˆ Θ K(θ)g (s)dsdθ + 0 t−θ−h R R ˆ T R +∞ t −2Θ 0 t−θ−h K(θ)g(s)ω(s)dsdθ,
0 ≤
.
Here neither A nor A + A1 is Schur stable. For h = 0 the values of λ that guarantee the asymptotic stability of the system by Remark 4 and Proposition 3 are shown in Fig. 5, where we can see that the results achieved by Proposition 3 are less conservative than those obtained by Remark 4. It is noted that Proposition 3 and Remark 4 possess the same number {5n2 + 4n}n=2 = 28 of variables.
Rewriting last two terms as sum of squares together R +∞ Rthe t with 0 K(θ)g(s)dsdθ = 0 leads to t−θ−h R +∞ R t 0
Remark 4 Proposition 3
T t−θ−h K(θ)ω (s)Rω(s)dsdθ R R −1 +∞ t T ≥ K1h 0 t−θ−h K(θ)ω (s)dsdθR R +∞ R t × K(θ)ω(s)dsdθ h R 0 R t−θ−h i−1 +∞ t 2 + 0 K(θ)g (s)dsdθ ΣT RΣ t−θ−h hR i R +∞ t 2 ˆ ˆ T ˆ − 0 t−θ−h K(θ)g (s)dsdθ [Θ − Υ] R[Θ
ˆ − Υ], (55)
where Σ is given by (18) and ˆ = Υ 0
0.5
1
1.5 λ
2
2.5
3
ˆ =Υ ˆ implies (17). This Then the inequality (55) with Θ concludes the proof.
Fig. 5. Example 2: stabilizing values of λ
6
i−1 +∞ R t 2 K(θ)g (s)dsdθ 0 t−θ−h R +∞ R t × 0 t−θ−h K(θ)g(s)ω(s)dsdθ.
hR
Conclusions
Appendix B
In this paper, we have provided extended Jensen integral inequalities. For the discrete counterpart we have generalized Jensen summation inequality. Applications to the stability analysis of linear continuous-time systems with gamma-distributed delays and linear discrete-time systems with poisson-distributed delays demonstrated the advantages of these generalized inequalities. In both cases, the considered infinite distributed delays with a gap have stabilizing effects. The future research may include other applications of these developed inequalities.
Proof of Theorem 3p Define a function f (v) for all v ∈ ˜ where Θ ˜ ∈ Rn is a Z+ by f (v) = f˜(v) − M (v)g(v)Θ, h p constant vector to be defined and f˜(v) = M (v) x(v)− i P+∞ M0−1 i=0 M (i)x(i) with M0 given by (38) . Then since R ≻ 0 it follows that P+∞ T f (i)Rf (i) i h i=0 ˜ T R P+∞ M (i)x(i) P+∞ M (i)g(i) = 2M0−1 Θ i=0 i=0 P+∞ ˜T ˜ + i=0 f (i)Rf (i) hP i +∞ 2 ˜ T RΘ ˜ + M (i)g (i) Θ i=0 P +∞ ˜TR −2Θ i=0 M (i)g(i)x(i).
0 ≤
Appendix A Proof of Theorem 2 Following the proof of Theorem 1, we define a function f (r, u) for allpu ∈ [0, +∞), r ∈ ˆ where [t − u − h, t] by f (r, u) = fˆ(r, u) − K(u)g(r)Θ, 11
RepresentingPthe last two terms as sum of squares to+∞ gether with i=0 M (i)g(i) = 0 yields
[13] Y.R. Liu, Z.D. Wang, J.L. Liang, and X.H. Liu. Synchronization and state estimation for discrete-time complex networks with distributed delays. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 38(5):1314–1325, 2008.
P+∞
M (i)xT (i)Rx(i) hP iT h P i +∞ +∞ M (i)x(i) ≥ M0−1 R M (i)x(i) i=0 i=0 hP i−1 +∞ 2 T + M (i)g (i) Π RΠ h Pi=0 i +∞ 2 ˜ − Υ] ˜ T R[Θ ˜ − Υ], ˜ − M (i)g (i) [Θ
i=0
[14] C.I. Morarescu, S.I. Niculescu, and K. Gu. Stability crossing curves of shifted gamma-distributed delay systems. SIAM Journal on Applied Dynamical Systems, 6(2):475–493, 2007. [15] S.I. Niculescu. Delay Effects on Stability: A Robust Control Approach. Springer-Verlag, Berlin, 2001.
i=0
[16] P.G. Park, W.I. Lee, and S.Y. Lee. Auxiliary functionbased integral inequalities for quadratic functions and their applications to time-delay systems. Journal of the Franklin Institute, 352(4):1378–1396, 2015.
(56)
where Π is given by (40) and ˜ = Υ
+∞ hX i=0
+∞ i−1 X M (i)g(i)x(i). M (i)g 2 (i)
[17] J.P. Richard. Time-delay systems: An overview of some recent advances and open problems. Automatica, 39(10):1667–1694, 2003.
i=0
[18] A. Saleh and R. Valenzuela. A statistical model for indoor multipath propagation. IEEE Journal on Selected Areas in Communications, 5(2):128–137, 1987.
˜ = Υ ˜ results in the maximum of the The choice of Θ right-hand side of (56) and thus (39). This concludes the proof.
[19] A. Seuret and F. Gouaisbaut. On the use of the Wirtinger inequalities for time-delay systems. In Proceedings of the 10th IFAC workshop on time-delay systems, Boston, MA, USA, 2012.
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to
Time-Delay
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12
Generalized Jensen Inequalities with Application to Stability Analysis of Systems with Distributed Delays over Infinite Time-Horizons ⋆ Kun Liu a Emilia Fridman b Karl Henrik Johansson c Yuanqing Xia a a
School of Automation, Beijing Institute of Technology, 100081 Beijing, China (e-mails: kunliubit, xia [email protected]). b
c
School of Electrical Engineering, Tel Aviv University, 69978 Tel Aviv, Israel (e-mail: [email protected]).
ACCESS Linnaeus Centre and School of Electrical Engineering, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden (e-mail: [email protected]).
Abstract The Jensen inequality has been recognized as a powerful tool to deal with the stability of time-delay systems. Recently, a new inequality that encompasses the Jensen inequality was proposed for the stability analysis of systems with finite delays. In this paper, we first present a generalized integral inequality and its double integral extension. It is shown how these inequalities can be applied to improve the stability result for linear continuous-time systems with gamma-distributed delays. Then, for the discrete-time counterpart we provide an extended Jensen summation inequality with infinite sequences, which leads to less conservative stability conditions for linear discrete-time systems with poisson-distributed delays. The improvements obtained thanks to the introduced generalized inequalities are demonstrated by examples. Key words: new integral and summation inequalities, gamma-distributed delays, poisson-distributed delays, Lyapunov method.
1
Introduction
Time-delay often appears in many control systems either in the state, the control input, or the measurements. During the last two decades, the stability of time-delay systems has received considerable attention (e.g., [3], [8], [15], [17] and references therein). One of the most popular approaches is the use of Lyapunov-Krasovskii functionals (LKF) to derive stability conditions (e.g., [1], [5], [9], [26]). The choice of the Lyapunov functional and the method of bounding an integral term in the derivative of the LKF are important ways to reduce conservativeness of the stability results. The Jensen inequality [8], has been widely used as an efficient bounding technique, although at a price of an unavoidable conservativeness [7], [12]. The Jensen inequality claims that for any continuous function ω : [a, b] → Rn and n × n positive definite ⋆ This work was partially supported by the National Natural Science Foundation of China (grant no. 61503026, 61440058), the Knut and Alice Wallenberg Foundation, the Swedish Research Council, and the Israel Science Foundation (grant no. 754/10 and 1128/14).
Preprint submitted to *****
matrix R, Rb a
ω T (s)Rω(s)ds ≥
1 b−a
Rb a
ω T (s)dsR
Rb a
ω(s)ds
holds. There is a discrete counterpart, which involves sums instead of integrals [3], [4]. Some recent efforts have been made to overcome the conservativeness induced by the Jensen inequality when applied to the stability analysis of time-delay systems. The bound on the gap of the Jensen inequality was analyzed in [2] by using the Gr¨ uss inequality. Based on the Wirtinger inequality [11], Seuret and Gouaisbaut [19] derived an extended integral inequality, which encompasses Jensen inequality as a particular case. Recently, the inequality they proposed was further refined in [20]. By combining the newly developed integral inequality and an augmented Lyapunov functional, a remarkable result was obtained for systems with constant discrete and distributed delays. Let us recall the inequality provided in [20] (see [21] for the discrete counterpart): for any continuous function ω : [a, b] → Rn and n × n posi-
18 February 2016
Question 1 Is it possible to derive more accurate lower bounds to reduce the conservativeness of integral inequalities (4) and (5)? If so, how much improvements can we obtain by applying the generalized inequalities to the stability analysis of continuous-time systems with gammadistributed delays?
tive definite matrix R, the inequality Rb a
ω T (s)Rω(s)ds ≥
Rb T Rb 1 b−a a ω (s)dsR a 3 + b−a ΩT RΩ
ω(s)ds
(1)
holds, where Ω=
Rb a
ω(s)ds −
2 b−a
RbRs a
a
ω(r)drds.
(2)
We further analyze the discrete-time case. Poissondistribution is widespread in queuing theory [6]. In [18], the experimental data on the arrivals of pulses in indoor environments revealed that each cluster’s timedelay is poisson-distributed (see also [10]). Therefore, we study the stability of linear discrete-time systems with poisson-distributed delays via appropriate Lyapunov functionals. The Lyapunov-based analysis uses the discrete counterpart of integral inequalities (4) and (5), i.e., Jensen inequalities with infinite sequences [13], [25]. The following question corresponds to Question 1 in the discrete case:
To prove (1), a function f (u), u ∈ [a, b], was introduced in [20] as follows: f (u) = z(u) ˙ = ω(u) −
1 b−a
Rb a
ω(s)ds −
a+b−2u (b−a)2 Θ,
(3)
where Θ ∈ R Rn is a constant be defined R b vector to(b−u)(u−a) u and z(u) = a ω(s)ds − u−a ω(s)ds − Θ, b−a a (b−a)2 Rb u ∈ [a, b]. It is noted that a (a + b − 2u)du = 0 plays an important role in the utilization of (3). Since Θ is a constant vector, it is obvious that in (3), a + b − 2u could be replaced by c(a + b − 2u), c ∈ R\{0}, because Rb c(a + b − 2u)du = 0. By using a more general auxa Rb iliary function g(u) with a g(u)du = 0, an extended integral inequality, which included the one proposed in [20] as a particular case, was provided in [16].
Question 2 Is it possible to generalize Jensen summation inequalities with infinite sequences? If so, how much improvements can be achieved by applying the generalized inequalities to the stability analysis of discrete-time systems with poisson-distributed delays? The central aim of the present paper is to answer the above questions. First, we present generalized Jensen integral inequality and its double integral extension, which are over infinite intervals of integration. We show how they can be applied to improve the stability result for linear continuous-time systems with gammadistributed delays. Then, for the discrete counterpart we provide extended Jensen summation inequality with infinite sequences, which leads to less conservative stability conditions for linear discrete-time systems with poisson-distributed delays. In both the continuous-time and discrete-time cases, the considered infinite distributed delays are shown to have stabilizing effects. Following [22], we derive the results via augmented Lyapunov functionals.
Recently, the stability analysis of systems with gammadistributed delays was studied [22]. The Lyapunov-based analysis was based on two kinds of integral inequalities with infinite intervals of integration: given an n × n positive definite matrix R, a scalar h ≥ 0, a vector function ω : [0, +∞) → Rn and a scalar function K : [0, +∞) → R+ such that the integrations concerned are well defined, the following inequalities R +∞
K(s)ω T (s)Rω(s)ds R +∞ R +∞ ≥ K0−1 0 K(s)ω T (s)dsR 0 K(s)ω(s)ds 0
and
R +∞ R t
T t−θ−h K(θ)ω (s)Rω(s)dsdθ R R −1 +∞ t ≥ K1h K(θ)ω T (s)dsdθR R +∞0 R t t−θ−h × 0 t−θ−h K(θ)ω(s)dsdθ, R +∞ R +∞ hold, where K0 = 0 K(s)ds and K1h = 0
(4)
0
(5)
The structure of this paper is as follows. In Section 2 we derive generalized Jensen integral inequalities. Section 3 presents stability results for linear continuous-time systems with gamma-distributed delays to illustrate the efficiency of the proposed inequalities. Sections 4 and 5 discuss the corresponding extended Jensen summation inequality with infinite sequences and its application to the stability analysis of linear discrete-time systems with poisson-distributed delays, respectively. The conclusions and the future work will be stated in Section 6.
K(s)(s+ h)ds. The inequalities (4) and (5) were used in [23] to the stability and passivity analysis for diffusion partial differential equations with infinite distributed delays. To obtain more accurate lower bounds of integral inequalities (4) and (5) over infinite intervals of integration, the method developed in [20] for the integral inequality over finite intervals of integration seems not to be applicable, since the function f of (3) is directly dependent on both the lower limit a and the upper limit b. Therefore, an interesting question arises:
Notations: The notations used throughout the paper are standard. The superscript ‘T ’ stands for matrix transposition, Rn denotes the n dimensional Euclidean space with vector norm | · |, Rn×m is the set of all n × m
2
real matrices, and the notation P ≻ 0, for P ∈ Rn×n means that P is symmetric and positive definite. The symmetric term in a symmetric matrix is denoted by ∗. The symbols R, R+ , Z+ and N denote the set of real numbers, non-negative real numbers, non-negative integers and positive integers, respectively. 2
Rewriting the last two terms as sum of squares yields Rb T Rb 1 ω (s)dsR a ω(s)ds ω T (s)Rω(s)ds ≥ b−a a hR i−1 R Rb b b T + a g 2 (s)ds a g(s)ω (s)dsR a g(s)ω(s)ds hR i b − a g 2 (s)ds [Θ − Υ]T R[Θ − Υ], (8) where Rb a
Extended Jensen integral inequalities
The objective of this section is to provide extended Jensen integral inequalities over infinite intervals. To do so, we first prove the generalized Jensen integral inequality introduced in [16] over finite intervals in a simpler way. Then we extend the method to prove the inequality over infinite intervals. 2.1
Υ=
hZ
b
g 2 (s)ds
a
i−1 Z
b
g(s)ω(s)ds.
a
Since (8) holds independently of the choice of Θ, we may choose Θ = Υ, which leads to the maximum of the righthand side of (8) and thus, (6) holds. This concludes the proof.
Extended Jensen integral inequality over finite intervals
Remark 1 In [16], the proof was more complicated as the corresponding construction of (7) relied on an auxiliary function g¯(u), where g¯(u) satisfies g(u) = g¯(u) − Rb 1 ¯(s)ds. b−a a g
By changing a + b − 2u of (3) to a more general scalar Rb function g(u) with a g(u)du = 0 and g(u) not identically zero, we first present the extended Jensen inequality over finite intervals of integration. Lemma 1 [16] If there exist an n × n matrix R ≻ 0, a scalar function g : [a, b] → R and a vector function ω : [a, b] → Rn such that the integrations concerned are well Rb defined and a g(s)ds = 0, where g(s) is not identically zero, then the following inequality holds:
2.2
Generalized Jensen integral inequalities over infinite intervals
We extend the method used for proving Lemma 1 from finite intervals of integration to infinite ones in the following result.
Rb
Rb T Rb 1 ω T (s)Rω(s)ds ≥ b−a ω (s)dsR a ω(s)ds a hR i−1 R Rb b b T + a g 2 (s)ds a g(s)ω (s)dsR a g(s)ω(s)ds. (6)
a
Theorem 1 For a given n×n matrix R ≻ 0, scalar functions g : [0, +∞) → R, K : [0, +∞) → R+ and a vector function ω : [0, +∞) → Rn , assume that the integrations R +∞ concerned are well defined and 0 K(s)g(s)ds = 0 with g(s) not identically zero. Then the following inequality holds:
Proof: Define a function f (u) for all u ∈ [a, b] by f (u) = ω(u) −
1 b−a
Rb a
ω(s)ds − g(u)Θ,
(7)
R +∞
K(s)ω T (s)Rω(s)ds R +∞ R +∞ ≥ K0−1 0 K(s)ω T (s)dsR 0 K(s)ω(s)ds hR i−1 +∞ ¯ T RΩ, ¯ + 0 K(s)g 2 (s)ds Ω 0
where Θ ∈ Rn is a constant vector to be defined. Then, since R ≻ 0 it follows that 0≤ =
Rb
f T (s)Rf (s)ds a h iT h i Rb Rb Rb 1 1 ω(s)− ω(θ)dθ ω(θ)dθ ds R ω(s)− b−a a b−a a ah iR Rb T b 2 + b−a a ω (s)dsRΘ a g(s)ds i hR Rb b + a g 2 (s)ds ΘT RΘ − 2ΘT R a g(s)ω(s)ds.
Noting that
Rb a
where
R +∞ K0 = 0 K(s)ds, R ¯ = +∞ K(s)g(s)ω(s)ds. Ω
(10)
0
Proof:
g(s)ds = 0, we obtain
Define a function f¯(u) for all u ∈ [0, +∞) by
Rb T Rb 1 ω (s)dsR a ω(s)ds ω T (s)Rω(s)ds ≥ b−a a a h i Rb Rb − a g 2 (s)ds ΘT RΘ + 2ΘT R a g(s)ω(s)ds.
Rb
(9)
f¯(u) =
3
i h p R +∞ ¯ , K(u) ω(u)−K0−1 0 K(s)ω(s)ds− g(u)Θ
¯ ∈ Rn is a constant vector to be defined. Because where Θ R ≻ 0 we have
Corollary 1 For a given n × n matrix R ≻ 0, a scalar function K : [0, +∞) → R+ and a vector function ω : [0, +∞) → Rn , assume that the integrations concerned are well defined. Then the following inequality holds:
R +∞ 0 ≤ 0 f¯T (s)Rf¯(s)ds iT p R +∞hp R +∞ = 0 K(s)ω(s)−K0−1 K(s) 0 K(θ)ω(θ)dθ R i hp p R +∞ × K(s)ω(s)−K0−1 K(s) 0 K(θ)ω(θ)dθ ds h iR R ¯ T R +∞ K(s)ω(s)ds +∞ K(s)g(s)ds + 2K0−1 Θ 0 0 i hR +∞ ¯ T RΘ ¯ + 0 K(s)g 2 (s)ds Θ R ¯ T R +∞ K(s)g(s)ω(s)ds. −2Θ 0
R +∞
(16) where K0 , K1 and K2 are given by (10), (12) and (15), respectively, and
RepresentingR the last two terms as sum of squares to+∞ gether with 0 K(s)g(s)ds = 0 yields R +∞
K(s)ω T (s)Rω(s)ds R +∞ R +∞ ≥ K0−1 0 K(s)ω T (s)dsR 0 K(s)ω(s)ds i−1 hR +∞ ¯ T RΩ ¯ Ω + 0 K(s)g 2 (s)ds hR i +∞ ¯ − Υ] ¯ T R[Θ ¯ − Υ], ¯ − K(s)g 2 (s)ds [Θ
˜= Ω
(11)
¯ is given in (10) and where Ω +∞
K(s)g 2 (s)ds
0
i−1 Z
+∞
K(s)g(s)ω(s)ds.
0
0
Note that the choice of g(s) plays a crucial role in the application of Theorem 1. Given K0 in (10) and
let
R +∞ 0
sK(s)ds,
g(u) = c(K0 u − K1 ), c ∈ R\{0}, u ≥ 0, such that
R +∞ 0
0
K(s)ω(s)ds −
R +∞ 0
sK(s)ω(s)ds.
R +∞ R t
Then, the same arguments in the proof of Lemma 1 and ¯ =Υ ¯ lead to the maximum of the right-hand the choice Θ side of (11) and thus, (9) holds. This concludes the proof.
K1 =
R +∞
Theorem 2 If there exist an n × n matrix R ≻ 0, scalar functions g : [t − θ − h, t] → R, K : [0, +∞) → R+ , a scalar h ≥ 0 and a vector function ω : [t − θ − h, t] → Rn such R +∞that R t the integrations concerned are well defined and 0 t−θ−h K(θ)g(s)dsdθ = 0, where g(s) is not identically zero, then the following inequality holds:
0
hZ
K1 K0
The same methodology to prove Lemma 1 and Theorem 1 can be applied to generalize the inequality (5). We have the following result:
0
¯ = Υ
K(s)ω T (s)Rω(s)ds R +∞ R +∞ ≥ K0−1 0 K(s)ω T (s)dsR 0 K(s)ω(s)ds −1 K2 ˜ T RΩ, ˜ + K2 − K10 Ω
0
T t−θ−h K(θ)ω (s)Rω(s)dsdθ R R +∞ t −1 ≥ K1h K(θ)ω T (s)dsdθR 0 R +∞ R t t−θ−h × 0 t−θ−h K(θ)ω(s)dsdθ hR i−1 +∞ R t + 0 K(θ)g 2 (s)dsdθ ΣT RΣ, t−θ−h
(17)
where
(12)
R +∞ K1h = 0 K(s)(s + h)ds = hK0 + K1 , R +∞ R t Σ= 0 t−θ−h K(θ)g(s)ω(s)dsdθ.
(13)
(18)
Proof: See Appendix A.
K(s)g(s)ds = 0 holds. Then, we find that
(14)
Remark 2 Theorems 1 and 2 refine the inequalities of [22], in which the last terms of the right-hand-side of (9) and (17) are zero. Hence, our new inequalities develop R +∞ more accurate lower bounds of 0 K(s)ω T (s)Rω(s)ds R +∞ R t and 0 K(θ)ω T (s)Rω(s)dsdθ than the ones t−θ−h provided in [22].
(15)
We choose a scalar function
From (13), (14) and Theorem 1, we have the following corollary:
g(u) = −u + t −
R +∞ 0
K(s)g 2 (s)ds = c2
R +∞ 0
K(s)(K0 s − K1 )2 ds
= c2 (K02 K2 − K0 K12 ), R +∞ ¯= Ω K(s)g(s)ω(s)ds i h0 R R +∞ +∞ = c K0 0 sK(s)ω(s)ds − K1 0 K(s)ω(s)ds ,
where
K2 =
R +∞ 0
s2 K(s)ds.
4
hK1h +hK1 +K2 , 2K1h
(19)
R +∞ R t such that 0 K(θ)g(s)dsdθ = 0, where K1 , K2 t−θ−h and K1h are given by (12), (15) and (18), respectively. Hence, we have R +∞ R t 0
=
=
K(θ)g 2 (s)dsdθ t−θ−h K(θ) − s + t −
t−θ−h
R +∞ R t 0
3
h 2
K0 + 2K3 +
hK1h +hK1 +K2 2K1h 3h2 K0 (hK1 +2K2 )−3K22 ∆ ˜ = K1 , 2K1h
2
where x(t) ∈ Rn is the state vector, A, A1 ∈ Rn×n are constant system matrices, and h ≥ 0 represents a fixed time gap. The smooth kernel Γ is given by N −1 − θ
T Γ(θ) = Tθ N (Ne−1)! , where N ≥ 2, N ∈ N, is a shape parameter of the distribution and T > 0 is a scale parameter. The matrices A and A + A1 are not allowed to be Hurwitz. The initial condition is given by φ ∈ C 1 (−∞, 0], where C 1 (−∞, 0] denotes the space of continuously differentiable functions φ : (−∞, 0] → Rn ˙ C < +∞, with the norm kφkC 1 = kφkC + kφk kφkC = sups∈(−∞,0] |φ(s)| < +∞.
dsdθ (20)
and R +∞ R t Σ= 0 K(θ)g(s)ω(s)dsdθ R +∞ R t t−θ−h +hK1 +K2 = 0 K(θ) −s+t− hK1h 2K ω(s)dsdθ t−θ−h 1h R +∞R t Rt = 0 K(θ)ω(s)drdsdθ t−θ−h s R +∞ R t hK1+K2 h − 2 + 2K1h K(θ)ω(s)dsdθ 0 t−θ−h R +∞ R t Rr = 0 K(θ)ω(s)dsdrdθ R t−θ−h t−θ−h Rt ∆ ˜ +∞ 1+K2 − h2 + hK 2K1h 0 t−θ−h K(θ)ω(s)dsdθ = Σ, (21) where R +∞ K3 = 0 s3 K(s)ds.
Following [22] and introducing y(t) =
0
ρ(t) =
R +∞
R0+∞
R0+∞
R0+∞
(24)
−∞ Ψ(t− s)x(s− θ θ N −2 e− T T N (N −2)! .
h)ds =
R +∞ 0
Ψ(θ)x(t −θ− h)dθ,
Ψ(θ)dθ =
1 ∆ T =
Ψ0 , ∆
θΨ(θ)dθ = N − 1 = Ψ1 , ∆
θ2 Ψ(θ)dθ = N (N − 1)T = Ψ2 ,
∆ θ3 Ψ(θ)dθ = (N + 1)N (N − 1)T 2 = Ψ3 , 0 R +∞ ∆ (θ + h)Ψ(θ)dθ = hΨ0 + Ψ1 = N − 1 + Th = 0
(22)
Ψ1h . (25) In the following, we provide two sufficient conditions for the stability of system (24); one is derived by applying (16) and (5), the other is obtained from (16) and (22). 3.1
Stability result I
We consider the following augmented LKF: V (t) = V1 (t) + VG (t) + VH (t), V1 (t) = η T (t)W η(t), R +∞ R t VG (t) = 0 Ψ(θ)xT (s)Gx(s)dsdθ, R +∞ Rt−θ−h R θ+h t VH (t) = 0 ˙ T (s)H x(s)dsdλdθ, ˙ 0 t−λ Ψ(θ)x (26) where W ≻ 0, G ≻ 0, H ≻ 0, η(t) = col{x(t), y(t)}. Since A and A+A1 are not allowed to be Hurwitz, we use
Consider the linear continuous-time systems with gamma-distributed delays: Γ(θ)x(t − θ − h)dθ,
Γ(t− s)x(s− h)ds,
It follows readily that
Stability analysis of continuous-time systems with gamma-distributed delays
0
Rt
Ψ(θ) =
The generalized integral inequality (16) and its double integral extension (22) will be employed for the stability analysis of continuous-time systems with gammadistributed delays in the next section.
R +∞
−∞
where
˜ 1 and Σ ˜ are given by (20) and (21), respectively. where K
x(t) ˙ = Ax(t) + A1
Rt
y(t) ˙ = − T1 y(t) + ρ(t),
1
3
Γ(θ)x(t −θ− h)dθ =
x(t) ˙ = Ax(t) + A1 y(t),
Corollary 2 If there exist an n × n matrix R ≻ 0, a scalar function K : [0, +∞) → R+ , a scalar h ≥ 0 and a vector function ω : [t − θ − h, t] → Rn such that the integrations concerned are well defined, then the following inequality holds: T t−θ−h K(θ)ω (s)Rω(s)dsdθ R R −1 +∞ t ≥ K1h K(θ)ω T (s)dsdθR 0 R +∞ R t t−θ−h × 0 t−θ−h K(θ)ω(s)dsdθ −1 ˜ T ˜ ˜ +K Σ RΣ,
0
the system (23) can be transformed into
From (19)–(21) and Theorem 2, we arrive at the following result:
R +∞ R t
R +∞
(23)
5
with ξ(t) = col{x(t), y(t), ρ(t)}. By applying Corollary 1 we obtain
augmented Lyapunov functionals. The term VH extends the triple integral of [24] for finite delay to infinite delay [22].
R +∞ V˙ G (t) = Ψ0 xT (t)Gx(t)− 0 Ψ(θ)xT (t−θ−h)Gx(t−θ−h)dθ ≤ Ψ0 xT (t)Gx(t) − Ψ−1 ρT (t)Gρ(t) −1 h 0 iT 2 Ψ Ψ1 − Ψ2 − Ψ10 ρ(t) − (N − 1)y(t) G Ψ 0 i h Ψ1 ρ(t) − (N − 1)y(t) × Ψ 0
Remark 3 The recent method of [22] for the stability analysis of system (24) is based on a functional of the form V (t) = V1 (t) + VG (t) + VH (t) + VE (t) + VF (t), R +∞ R t (27) VE (t) = 0 Γ(θ)xT (s)Ex(s)dsdθ, R +∞ Rt−θ−h R θ+h t T VF (t) = 0 Γ(θ)x˙ (s)F x(s)dsdλdθ ˙ 0 t−λ
T T = ξ T (t)[Ψ1h F01 HF01 − Ψ−1 1h F13 HF13 ]ξ(t).
Therefore, (28) guarantees that V˙ (t) ≤ ξ T (t)Ξξ(t) ≤ −β|x(t)|2 for some β > 0, which proves the asymptotic stability.
Proposition 1 If there exist 2n × 2n positive definite matrix W and n × n positive definite matrices G, H such that the following LMI is feasible:
F0 =
A A1 0 0 − T1 I
#
, F1 =
"
I 0 0 0 I 0
#
Stability result II
The stability of system (24) can be alternatively analyzed via a LKF given by ¯ η¯(t), V¯ (t) = V¯1 (t) + VG (t) + VH (t), V¯1 (t) = η¯T (t)W
(28)
¯ ≻ 0, η¯(t) = col{x(t), y(t), ζ(t)}, ζ(t) = where W R +∞ R t Ψ(θ)x(s)dsdθ, VG (t) and VH (t) are given by 0 t−θ−h ˙ = 1 x(t) − ρ(t) and differentiating (26). Noting that ζ(t) T V1 (t) along (24), we have
where Ψ1h is given by (25), Σ = diag{ T1 G, 0, −T G} and "
1 T T T x (t)Gx(t) − T ρ (t)Gρ(t) T GF23 ξ(t). − NT−1 ξ T (t)F23
V˙ H (t) ≤ Ψ1h x˙ T (t)H x(t) ˙ h iT h i −1 −Ψ1h Ψ0 x(t) − ρ(t) H Ψ0 x(t) − ρ(t)
3.2
T T +Ψ1h F01 HF01 − Ψ−1 1h F13 HF13 ≺ 0,
=
− T ρT (t)Gρ(t) h iT h i − NT−1 T ρ(t) − y(t) G T ρ(t) − y(t)
Furthermore, applying (5) we find that
The following proposition is provided for the asymptotic stability of system (24).
N −1 T T F23 GF23
1 T T x (t)Gx(t)
(29)
together with the utilization of the integral inequalities (4) and (5). Compared to (26), the functional (27) has two additional terms VE (t) and VF (t). In the example below, we will show the advantages of our proposed approach (larger stability region in the (T, h) plane and less number of scalar decision variables). The improvement is achieved due to that the application of Corollary 1 leads to one more negative term in the derivative of the LKF.
Ξ = Σ + F1T W F0 + F0T W F1 −
=
,
Ax(t) + A1 y(t)
¯ − 1 y(t) + ρ(t) V¯˙ 1 (t) = 2¯ η T (t)W T 1 T x(t) − ρ(t) T T ¯ ¯ ¯ ¯ ¯ = 2ξ (t)F W F0 ξ(t)
F01 = [A A1 0], F13 = [ T1 I 0 − I], F23 = [0 − I T I],
(30)
1
then system (24) is asymptotically stable.
¯ = col{x(t), y(t), ρ(t), ζ(t)} and with ξ(t)
Proof: Differentiating V1 (t) along (24), we have
I 0 0 0
A
A1
0 0
1 ¯ F¯1 = 0 I 0 0 , F0 = 0 − T I I 0 . 1 0 −I 0 0 0 0 I TI
V˙ 1 (t) = 2η T (t)W η(t) ˙ = 2ξ T (t)F1T W F0 ξ(t)
6
(31)
Table 1 Example 1: maximum allowable value of T for different h
Furthermore, by applying (22) we find that
[max T ]
10−5
0.01
0.15
0.34
0.35
0.36
\ h
V˙ H (t) ≤ Ψ1h x˙ T (t)H x(t) ˙ h iT h i −1 −Ψ1h Ψ0 x(t) − ρ(t) H Ψ0 x(t) − ρ(t)
˜ −1 ϕT (t)Hϕ(t) −Ψ 1 T ¯ T ¯ ˜ −1 ¯ T ¯ ¯ H F¯01−Ψ−1 = ξ¯T (t)[Ψ1h F¯01 1h F13 H F13−Ψ1 F33 H F33 ]ξ(t), (32) where Ψ1h is given in (25) and
Decision variables
[22]
0.274
0.265
0.141
0.005
-
-
22
Prop. 1
0.305
0.296
0.158
0.008
0.002
-
16
Prop. 2
0.322
0.312
0.168
0.014
0.008
0.003
31
[22] and using Propositions 1 and 2, we obtain the maximum allowable values of T that achieve the stability. Fig. 1 presents tradeoff curves between maximal allowable T and h by applying the above three methods. Furthermore, the stability region in the (T, h) plane that preserves the asymptotic stability is depicted in Figs. 2–4 by using the condition in [22], Proposition 1 and Proposition 2, respectively. From Figs. 2–4 we can see that Proposition 1 induces a more dense stability region than [22], but guarantees a little sparser stability region than Proposition 2. Therefore, Figs. 1–4 show that Proposition 1 improves the results in [22] and that the conditions can be further enhanced by Proposition 2.
F¯01 = [A A1 0 0], F¯13 = [ T1 I 0 − I 0], ϕ(t) = ~[Ψ0 x(t) − ρ(t)] − ζ(t) + hρ(t) + (N − 1)y(t) ¯ = F¯33 ξ(t), F¯33 = [ T~ I (N − 1)I (h − ~)I − I], 2 2 ˜ 1 = h3 + 2Ψ3 + 3h Ψ0 (hΨ1 +2Ψ2 )−3Ψ2 , Ψ 2Ψ1h 2T 1 +Ψ2 ~ = h2 + hΨ2Ψ . 1h
Let us now compare the number of scalar decision variables in the LMIs. The LMIs of [22] have {4n2 + 3n}n=2 = 22 variables. Proposition 1 in this paper not only possess a fewer number {3n2 +2n}n=2 = 16 of variables but also lead to less conservative results. In comparison with Proposition 1, Proposition 2 slightly improves the results at the price of {2.5n2 + 2.5n}n=2 = 15 additional decision variables.
(33) Therefore, by combining (29), (30) and (32) we obtain 2 ¯ ≤ −β|x(t)| ¯ ¯ ξ(t) V¯˙ (t) ≤ ξ¯T (t)Ξ for some β¯ > 0, if T ¯ =Σ ¯ + F¯1T W ¯ F¯0 + F¯0T W ¯ F¯1 − N −1 F¯23 Ξ GF¯23 T T ¯T ¯ ˜ −1 ¯ T ¯ H F¯01 − Ψ−1 +Ψ1h F¯01 1h F13 H F13 − Ψ1 F33 H F33 ≺ 0, (34) where ¯ = diag{ 1 G, 0, −T G, 0}, Σ T (35) F¯23 = [0 − I T I 0].
0.4 [22] Proposition 1 Proposition 2
0.35 0.3 0.25 h
We have thus proved the following proposition:
0.2 0.15
Proposition 2 If there exist 3n × 3n positive definite ¯ and n × n positive definite matrices G, H such matrix W that LMI (34) with notations given by (25), (31), (33) and (35) is feasible, then system (24) is asymptotically stable.
0.1 0.05 0
Example 1
4
We illustrate the efficiency of the presented results through an example of two cars on a ring, see [14] and [22] for details. In this example,
A = 0 and A1 =
"
−2 2 2 −2
#
0.05
0.1
0.15
0.2 T
0.25
0.3
0.35
0.4
Fig. 1. Example 1: tradeoff curve between maximal allowable T and h for Propositions 1 and 2 compared with the result of [22]
Next we present an example to illustrate the applicability of the theoretical results. 3.3
0
Extended Jensen summation inequalities with infinite sequences
The objective of this section is to present the discrete counterpart of the results obtained in Section 2 and to provide extended Jensen summation inequality with infinite sequences. We first introduce the following lemma for the discrete counterpart of the integral inequalities (4) and (5):
,
Lemma 2 Assume that there exist an n × n matrix R ≻ 0, a scalar function M (i) ∈ R+ and a vector function
so neither A nor A + A1 is Hurwitz. For the values of h given in Table I and N = 2, by applying the method in
7
hold, where
0.4 condition of [22] 0.35
M1h
h
0.25 0.2 0.15
0.05
0
0.05
0.1
0.15
0.2 T
0.25
0.3
0.35
(38)
The proof of (36) and (37) follows from [22] by using sums instead of integrals and is therefore omitted here. By applying the arguments of Theorem 1 to the discrete case, we obtain the following theorem for the extended Jensen summation inequality with infinite sequences. Note that this result includes (36) as a special case and that the generalization of (37) can be done by the same approach as exploited in Theorem 2.
0.1
0
P+∞
i=0 M (i), P+∞ = i=0 (i + h)M (i).
M0 =
0.3
0.4
Fig. 2. Example 1: stability region by the condition of [22]
0.4 Proposition 1 0.35
Theorem 3 For a given n × n matrix R ≻ 0, scalar functions M (i) : Z+ → R+ , g(i) : Z+ → R and a vector function x(i) : Z+ → Rn , assume that the series conP+∞ cerned are well defined and i=0 M (i)g(i) = 0 with g(i) not identically zero. Then the following inequality holds:
0.3
h
0.25 0.2 0.15 0.1
P+∞
M (i)xT (i)Rx(i) hP iT h P i +∞ +∞ M (i)x(i) ≥ M0−1 R M (i)x(i) i=0 i=0 hP i−1 +∞ 2 T + Π RΠ i=0 M (i)g (i) (39) with M0 given by (38) and
0.05 0
i=0
0
0.05
0.1
0.15
0.2 T
0.25
0.3
0.35
0.4
Fig. 3. Example 1: stability region by Proposition 1 0.4 Proposition 2 0.35
Π=
0.3
h
0.25 0.2
P+∞ i=0
M (i)g(i)x(i).
(40)
Proof: See Appendix B.
0.15
In order to apply Theorem 3 to the stability analysis of time-delay systems, we take
0.1 0.05 0
0
0.05
0.1
0.15
0.2 T
0.25
0.3
0.35
0.4
g(v) = c(M0 v − M1 ), c ∈ R\{0}, v ∈ Z+ ,
Fig. 4. Example 1: stability region by Proposition 2
such that n
x(i) ∈ R such that the series concerned are convergent. Then the inequality M (i)xT (i)Rx(i) hP iT h P i +∞ +∞ ≥ M0−1 R i=0 M (i)x(i) i=0 M (i)x(i) ,
i=0
P+∞ i=0
(36)
M (i)xT (j)Rx(j) iT +∞ Pk−1 −1 ≥ M1h M (i)x(j) R j=k−i−h h P i=0 i +∞ Pk−1 × i=0 j=k−i−h M (i)x(j) , j=k−i−h
hP
M (i)g(i) = 0 is satisfied, where M1 =
P+∞
M (i)g 2 (i) = c2
P+∞
i=0
iM (i).
(42)
2 i=0 M (i)(M0 i − M1 ) c2 (M02 M2 − M0 M12 ),
= P+∞ Π = i=0 M (i)g(i)x(i) i h P+∞ P M (i)x(i) , iM (i)x(i) − M = c M0 +∞ 1 i=0 i=0
and its double summation extension i=0
i=0
Hence, we have
P+∞
P+∞ Pk−1
P+∞
(41)
(43)
(37)
where
M2 =
8
P+∞ i=0
i2 M (i).
(44)
τ
where P(τ ) = e τ !λ . We next derive LMI conditions for the asymptotic stability of (47) via a direct Lyapunov method. −λ
From (41) and (43), Theorem 3 is reduced to the following corollary, which will be employed in the next section for the stability analysis of discrete-time systems with poisson-distributed delays.
Denoting Corollary 3 Given an n × n matrix R ≻ 0, a scalar function M (i) : Z+ → R+ and a vector function x(i) : Z+ → Rn such that the series concerned are well defined, the following inequality holds: P+∞
f (k) =
M (i)x (i)Rx(i) hP iT h P i +∞ +∞ ≥ M0−1 M (i)x(i) R M (i)x(i) i=0 i=0 −1 M12 T ˜ RΠ, ˜ + M2 − Π
(45) where M0 , M1 and M2 are given by (38), (42) and (44), respectively, and
5
P+∞ i=0
M (i)x(i) −
P+∞ i=0
= e−λ Ax(k − h) + e−λ A1 f (k − h) P+∞ + τ =0 Q(τ )x(k − τ − h),
iM (i)x(i).
(48) where Q(τ ) = It is noted that the augmented system (48) has not only distributed but also discrete delays. This is different from augmented system (24) for the case of gamma-distributed delays. Moreover, we find that e−λ λτ +1 (τ +1)! .
Stability analysis of discrete-time systems with poisson-distributed delays
In this section, we will demonstrate the efficiency of the extended Jensen summation inequality (45) through the stability analysis of linear discrete-time systems with poisson-distributed delays. Consider the following system:
P+∞
i=0 (i
P+∞ i=0
P+∞
P+∞
− τ ), k ∈ Z+ , (46) where x(k) ∈ Rn is the state vector, the system matrices A and A1 are constant with appropriate dimensions. We do not allow A and A + A1 to be Schur stable. The initial condition is given as col{x(0), x(−1), x(−2), . . . } = col{φ(0), φ(−1), φ(−2), . . . }. The function p(v), v ∈ Z+ , is a poisson distribution with a fixed time gap h ∈ Z+ : x(k + 1) = Ax(k) + A1
p(v) =
(
τ =0 p(τ )x(k
e−λ λv−h (v−h)!
v ≥ h,
0
v < h,
i=0
P+∞ i=0
P+∞
τ =0 p(τ )x(k
− τ) = =
P+∞
τ =h p(τ )x(k
P+∞
θ=0 p(θ
P+∞
2
2
i Q(i) = λ − λ + 1 − e
(49)
−λ ∆
¯ 2, =Q ∆
¯ 1h . + h)Q(i) = λ+(1 − e−λ )(h−1) = Q
i P2 h ˆx V (k) = xˆT (k)W ˆ(k)+ i=1 VGi (k) + VHi (k) + VSi (k) , P+∞ Pk−1 VG1 (k) = i=0 s=k−i−h P(i)xT (s)G1 x(s), P+∞ Pi+h Pk−1 VH1 (k) = i=0 j=1 s=k−j P(i)η1T (s)H1 η1 (s), P+∞ Pk−1 VG2 (k) = i=0 s=k−i−h Q(i)xT (s)G2 x(s), P+∞ Pi+h Pk−1 VH2 (k) = i=0 j=1 s=k−j Q(i)η1T (s)H2 η1 (s), Pk−1 VS1 (k) = s=k−h xT (s)S1 x(s) P−1 Pk−1 +h j=−h s=k+j η1T (s)R1 η1 (s), Pk−1 VS2 (k) = s=k−h f T (s)S2 f (s) P−1 Pk−1 +h j=−h s=k+j η2T (s)R2 η2 (s),
− τ)
+ h)x(k − θ − h),
τ =0 P(τ )x(k
∆
¯ 1, iQ(i) = λ − 1 + e−λ = Q
Consider system (48) with both distributed and discrete delays. The stability analysis will be based on the following discrete-time LKF:
we arrive at the equivalent system: x(k + 1) = Ax(k)+A1
+ h)P(i) = λ + h,
Q(i) = 1 − e−λ ,
i=0 (i
where λ > 0 is a parameter of the distribution. The mean value of p is h + λ. Due to the fact that P+∞
− τ − h), k ∈ Z+ ,
x(k + 1) = Ax(k) + A1 f (k), P+∞ −λ τ f (k + 1) = τ =0 e τ !λ x(k + 1 − τ − h) P+∞ −λ τ = e−λ x(k + 1 − h) + τ =1 e τ !λ x(k + 1 − τ − h) P+∞ −λ λτ +1 x(k − τ − h) = e−λ x(k + 1 − h) + τ =0 e (τ +1)!
M0
M1 M0
τ =0 P(τ )x(k
the system (47) can be transformed into the following augmented form
T
i=0
˜ = Π
P+∞
−τ − h), k ∈ Z+ , (47)
ˆ ≻ 0, Gi ≻ 0, Hi ≻ 0, Si ≻ 0, Ri ≻ 0, i = 1, 2, where W
9
and
and x ˆ(k) = col{x(k), f (k)},
h i=1 VGi (k + 1) + VHi (k + 1) + VSi (k + 1) i −VGi (k) − VHi (k) − VSi (k) h ˆ + Fˆ T [(λ + h)H1 + Q ¯ 1h H2 + h2 R1 ]Fˆ01 = ξ T (k) Σ 01
P2
η1 (k) = x(k + 1) − x(k), η2 (k) = f (k + 1) − f (k). Here the last two terms VS1 (k) and VS2 (k) are added to compensate the delayed terms x(k − h) and f (k − h) of (48), respectively. Therefore, for system (48) with h = 0, the terms VS1 (k) and VS2 (k) are not necessary. From standard arguments, we arrive at the following result for the asymptotic stability of (48):
T T R2 Fˆ02 H1 Fˆ12 + h2 Fˆ02 −(λ + h)−1 Fˆ12 i −Fˆ T R1 Fˆ13 − Fˆ T R2 Fˆ24 ξ(k) 13
+(1 − e ) q (k)G2 q(k) P+∞ − i=0 Q(i)xT (k − i − h)G2 x(k − i − h) P+∞ Pk−1 − i=0 s=k−i−h Q(i)η1T (s)H2 η1 (s).
Proposition 3 Given a real scalar λ > 0 and an integer h ≥ 0, assume that there exist 2n × 2n positive definite ˆ and n × n positive definite matrices Gi , Hi , matrix W Si , Ri , i = 1, 2, such that the following LMI is satisfied:
(52) Applying the generalized Jensen inequality (45) with infinite sequences, we obtain −
T ˆ = Σ+ ˆ Fˆ0T W ˆ Fˆ0 − Fˆ1T W ˆ Fˆ1 − (λ + h)−1 Fˆ12 Ξ H1 Fˆ12 T ¯ 1h H2 + h2 R1 ]Fˆ01 +Fˆ01 [(λ+h)H1 + Q
Fˆ0 =
A A1
Fˆ1 =
"
I 0 0 00
0 0 e
0
0
−λ
−λ
0 I 0 00
Ae #
0
A1 I
,
e−λ A1 I], Fˆ12 = [I −I 0 0 0], Fˆ15 = [(1 − e−λ )I 0 0 0 −I],
τ =0
P+∞ Pk−1
¯ −1 [(1−e−λ )x(k)−q(k)]T H2 [(1−e−λ )x(k)−q(k)] = −Q 1h T ¯ = −Q−1 ξ T (k)Fˆ15 H2 Fˆ15 ξ(k).
¯1 Q Fˆ25 = [0 − λI 0 0 ( 1−e −λ + 1)I].
1h
(54) Therefore, (51)–(54) yield ∆V (k) = V (k + 1) − V (k) ≤ ˆ ξ T (k)Ξξ(k). Then if (50) holds for given scalars λ > 0 and h ≥ 0, the system (48) is asymptotically stable. Remark 4 The LMI condition in Proposition 3 is derived by employing the generalized Jensen inequality (45). The system (48) can be alternatively analyzed by inequality (36). In this case, (53) is reduced to:
Proof: Define P+∞
(53)
T s=k−i−h Q(i)η1 (s)H2 η1 (s) hi=0 i h iT −1 P+∞ P+∞ Pk−1 ≤− i=0 (i+h)Q(i) i=0 s=k−i−h Q(i)η1 (s) H2 i hP +∞ Pk−1 Q(i)η (s) × 1 s=k−i−h i=0
−
Then the system (48) is asymptotically stable.
q(k) =
Q(i)xT (k − i − h)G2 x(k − i − h)
Furthermore, the application of (37) leads to
, Fˆ01 = [A−I A1 0 0 0],
Fˆ02 = [0 −I e−λ A Fˆ13 = [I 0 −I 0 0], Fˆ24 = [0 I 0 −I 0],
i=0
= −(1 − e−λ )−1 q T (k)G2 q(k) −1 2 T ¯ 2 − Q¯ 1−λ ξ T (k)Fˆ25 G2 Fˆ25 ξ(k). − Q 1−e
(50) ¯ 1, Q ¯ 2 and Q ¯ 1h are given by (49), Σ ˆ = diag{S1 + where Q G1 +(1−e−λ )G2 , −G1 +S2 , −S1 , −S2 , −(1−e−λ )−1 G2 }, #
P+∞
≤ −(1 − e−λ )−1 q T (k)G2 q(k) −1 h ¯ iT 2 Q1 ¯ 2 − Q¯ 1−λ − Q +1 q(k)−λf (k) G2 −λ 1−e i h ¯1−e Q1 × 1−e−λ + 1 q(k) − λf (k)
T T ¯ −1 Fˆ T H2 Fˆ15 − Fˆ13 R1 Fˆ13 +h2 Fˆ02 R2 Fˆ02 − Q 1h 15¯ 2 −1 T T ¯ 2 − Q1−λ R2 Fˆ24 − Q −Fˆ24 Fˆ25 G2 Fˆ25 ≺ 0, 1−e
"
24
−λ −1 T
Q(τ )x(k − τ − h),
ξ(k) = col{x(k), f (k), x(k − h), f (k − h), q(k)}. − By taking difference of V (k) along (48) and applying Jensen inequalities with finite sequences (see e.g., Chapter 6 of [3]), we have ˆ xˆ(k + 1) − x ˆx xˆT (k + 1)W ˆT (k)W ˆ(k) T T ˆ ˆ T ˆ ˆ ˆ ˆ = ξ (k)[F W F0 − F W F1 ]ξ(k) 0
P+∞ i=0
Q(i)xT (k − i − h)G2 x(k − i − h)
≤ −(1 − e−λ )−1 q T (k)G2 q(k). ˆ ˆ It yields Ξ |F25 =0 ≺ 0, which is more conservative than the condition proposed in Proposition 3 since the matrix −1 2 T ¯ 2 − Q¯ 1−λ − Q Fˆ25 G2 Fˆ25 of (50) is negative definite.
(51)
1−e
1
10
ˆ ∈ Rn is a constant vector to be defined and fˆ(r, u) = Θ h i p R R −1 +∞ t K(u) ω(r) − K1h K(θ)ω(s)dsdθ with 0 t−θ−h K1h given by (17). Since R ≻ 0 we have
Remark 5 Both conditions in Proposition 3 and Remark 4 are derived by the use of inequality (37). It is worth noting that the results could be further improved (in the (λ, h) plane preserving the stability) by the discrete counterpart of Theorem 2. 5.1
Example 2
Consider the linear discrete-time system (46) with A=
"
−0.5 0 0
1
#
and A1 =
"
−0.5 0.8 0.5 −0.2
#
R +∞ R t
f T (s, θ)Rf (s, θ)dsdθ R0+∞ Rt−θ−h t ˆT ˆ = 0 t−θ−h f (s, θ)Rf (s, θ)dsdθ h i R +∞ R t −1 ˆ T + 2K1h Θ R 0 t−θ−h K(θ)ω(s)dsdθ R +∞ R t × K(θ)g(s)dsdθ i h R 0 R t−θ−h +∞ t 2 ˆ T RΘ ˆ Θ K(θ)g (s)dsdθ + 0 t−θ−h R R ˆ T R +∞ t −2Θ 0 t−θ−h K(θ)g(s)ω(s)dsdθ,
0 ≤
.
Here neither A nor A + A1 is Schur stable. For h = 0 the values of λ that guarantee the asymptotic stability of the system by Remark 4 and Proposition 3 are shown in Fig. 5, where we can see that the results achieved by Proposition 3 are less conservative than those obtained by Remark 4. It is noted that Proposition 3 and Remark 4 possess the same number {5n2 + 4n}n=2 = 28 of variables.
Rewriting last two terms as sum of squares together R +∞ Rthe t with 0 K(θ)g(s)dsdθ = 0 leads to t−θ−h R +∞ R t 0
Remark 4 Proposition 3
T t−θ−h K(θ)ω (s)Rω(s)dsdθ R R −1 +∞ t T ≥ K1h 0 t−θ−h K(θ)ω (s)dsdθR R +∞ R t × K(θ)ω(s)dsdθ h R 0 R t−θ−h i−1 +∞ t 2 + 0 K(θ)g (s)dsdθ ΣT RΣ t−θ−h hR i R +∞ t 2 ˆ ˆ T ˆ − 0 t−θ−h K(θ)g (s)dsdθ [Θ − Υ] R[Θ
ˆ − Υ], (55)
where Σ is given by (18) and ˆ = Υ 0
0.5
1
1.5 λ
2
2.5
3
ˆ =Υ ˆ implies (17). This Then the inequality (55) with Θ concludes the proof.
Fig. 5. Example 2: stabilizing values of λ
6
i−1 +∞ R t 2 K(θ)g (s)dsdθ 0 t−θ−h R +∞ R t × 0 t−θ−h K(θ)g(s)ω(s)dsdθ.
hR
Conclusions
Appendix B
In this paper, we have provided extended Jensen integral inequalities. For the discrete counterpart we have generalized Jensen summation inequality. Applications to the stability analysis of linear continuous-time systems with gamma-distributed delays and linear discrete-time systems with poisson-distributed delays demonstrated the advantages of these generalized inequalities. In both cases, the considered infinite distributed delays with a gap have stabilizing effects. The future research may include other applications of these developed inequalities.
Proof of Theorem 3p Define a function f (v) for all v ∈ ˜ where Θ ˜ ∈ Rn is a Z+ by f (v) = f˜(v) − M (v)g(v)Θ, h p constant vector to be defined and f˜(v) = M (v) x(v)− i P+∞ M0−1 i=0 M (i)x(i) with M0 given by (38) . Then since R ≻ 0 it follows that P+∞ T f (i)Rf (i) i h i=0 ˜ T R P+∞ M (i)x(i) P+∞ M (i)g(i) = 2M0−1 Θ i=0 i=0 P+∞ ˜T ˜ + i=0 f (i)Rf (i) hP i +∞ 2 ˜ T RΘ ˜ + M (i)g (i) Θ i=0 P +∞ ˜TR −2Θ i=0 M (i)g(i)x(i).
0 ≤
Appendix A Proof of Theorem 2 Following the proof of Theorem 1, we define a function f (r, u) for allpu ∈ [0, +∞), r ∈ ˆ where [t − u − h, t] by f (r, u) = fˆ(r, u) − K(u)g(r)Θ, 11
RepresentingPthe last two terms as sum of squares to+∞ gether with i=0 M (i)g(i) = 0 yields
[13] Y.R. Liu, Z.D. Wang, J.L. Liang, and X.H. Liu. Synchronization and state estimation for discrete-time complex networks with distributed delays. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 38(5):1314–1325, 2008.
P+∞
M (i)xT (i)Rx(i) hP iT h P i +∞ +∞ M (i)x(i) ≥ M0−1 R M (i)x(i) i=0 i=0 hP i−1 +∞ 2 T + M (i)g (i) Π RΠ h Pi=0 i +∞ 2 ˜ − Υ] ˜ T R[Θ ˜ − Υ], ˜ − M (i)g (i) [Θ
i=0
[14] C.I. Morarescu, S.I. Niculescu, and K. Gu. Stability crossing curves of shifted gamma-distributed delay systems. SIAM Journal on Applied Dynamical Systems, 6(2):475–493, 2007. [15] S.I. Niculescu. Delay Effects on Stability: A Robust Control Approach. Springer-Verlag, Berlin, 2001.
i=0
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(56)
where Π is given by (40) and ˜ = Υ
+∞ hX i=0
+∞ i−1 X M (i)g(i)x(i). M (i)g 2 (i)
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i=0
[18] A. Saleh and R. Valenzuela. A statistical model for indoor multipath propagation. IEEE Journal on Selected Areas in Communications, 5(2):128–137, 1987.
˜ = Υ ˜ results in the maximum of the The choice of Θ right-hand side of (56) and thus (39). This concludes the proof.
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