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Impulsive control of Lurie systems
Computers and Mathematics with Applications 56 (2008) 2806–2813
Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Impulsive control of Lurie systems Yongbin Yu a,∗ , Fengli Zhang a , Qishui Zhong b , Xiaofeng Liao c , Juebang Yu b a
School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, 610054, PR China
b
School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, 610054, PR China
c
College of Computer Science, Chongqing University, Chongqing, 400044, PR China
article
info
Article history: Received 28 February 2007 Received in revised form 3 August 2008 Accepted 10 September 2008 Keywords: Impulsive control Lurie systems Impulsive differential equation Asymptotically stable Globally exponentially stable
a b s t r a c t In this paper the impulsive control scheme of Lurie systems is represented. Using Lyapunov’s direct method and comparison technique, we present novel conditions under which impulsively controlled Lurie systems are asymptotically stable, and give the estimate of the upper bound of impulse interval. In particular, globally exponential stability of impulsive control of Lurie systems is presented. Impulsive controller design with examples is provided to show the feasibility and effectiveness of our method finally. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Impulsive control is a control paradigm based on impulsive differential equations. In practice, there exist many examples of impulsive control systems (see [1–3]). Recently, impulsive systems and impulsive control have been studied by many researchers. Bainov and Simeonov [1], and Lakshmikantham et al. [2] have considered the stability of impulsive systems by using Lyapunov functions which are required to be nonincreasing along the whole sequence of switchings. Yang Tao [4] and Li et al. [5] have presented the concept and principles of impulsive control and obtained some sufficient conditions of a class of nonlinear systems by using the results in [2]. Yang et al. [6] and Sun et al. [7] have presented some interesting applications of impulsive control in Lorenz systems. Yang and Chua [8], Li et al. [9] and Sun et al. [10] have studied impulsive control and synchronization of Chua’s oscillators. However, these proposed impulsive control and impulsive systems do not consider Lurie systems or consider less, which is an important issue of impulsive control. In this paper, we shall consider impulsive control of Lurie systems. Utilizing the methods of Lyapunov functionals and comparison principle, we present novel conditions under which the impulsively controlled Lurie systems are asymptotically stable, and give the estimate of the upper bound of impulse interval. Furthermore, globally exponential stability of impulsive control of Lurie systems is still given. Comparing with [11–18], this paper is focused on impulsive control instead of absolute stability analysis of Lurie systems. The rest of this paper is organized as follows. In the following section, we introduce some preliminary definitions and the theory of impulsive differential equation. In Section 3, the impulsively controlled Lurie systems are studied, and new asymptotical stability theorem and the exponential stability theorem of the impulsively controlled Lurie systems are given. Following the numerical examples to illustrate design of impulsive control Lurie systems in Section 4, finally, some concluding remarks are presented in Section 5.
∗
Corresponding author. E-mail address: [email protected] (Y. Yu).
0898-1221/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2008.09.015
Y. Yu et al. / Computers and Mathematics with Applications 56 (2008) 2806–2813
2807
2. Preliminaries Consider the following Lurie control system
x˙ (t ) = Ax(t ) + bf (σ (t )) σ (t ) = c T x(t )
(1)
where x(t ) ∈ Rn is the state variable, A ∈ Rn×n , b, c ∈ Rn , and c T is the transpose of c. The nonlinearity f (σ ) satisfies f (σ ) ∈ zκ = {f (σ )|f (0) = 0, 0 < σ f (σ ) ≤ κσ 2 , σ 6= 0, f (σ ) ∈ C [R, R]}.
(2)
From (2), we know that
κσ 2 − σ f (σ ) ≥ 0 ⇔ σ (κσ − f (σ )) ≥ 0. Consider a discrete time set {τk } of control instants, where 0 < τ1 < τ2 < · · · < τk < τk+1 < · · · , τk → ∞,
as k → ∞.
Let U (k, x) = ∆x|t =τk , x(τk+ ) − x(τk− ), x(τk+ ) = lim x(t ), + t →τk
x(τk− ) = lim x(t ) − t →τk
be the control law, which is the ‘‘jump’’ in the state variable at the time instant τk , where x(τk− ) = x(τk ) is assumed. Then the impulsive Lurie system is presented by
x˙ (t ) = Ax(t ) + bf (σ (t )), t 6= τk , σ (t ) = c T x(t ), t 6= τk , ∆x+|t =τk = U (k, x), t = τk , x(t0 ) = x0 , t0 ≥ 0
(3)
which is a nonlinear impulsive differential equation according to [2]. Remark: On the one hand, when t 6= τk , the dynamics of the impulsive control system in (3) is governed by the system in (1). We can see that the controlled system is a free system whenever t 6= τk . On the other hand, when t = τk , k = 1, 2, . . ., the state variable is changed from x(τk− ) to x(τk+ ) = x(τk− ) + ∆x|t =τk instantaneously. In order to establish sufficient conditions under which impulsively controlled Lurie system is stable, we need the following basic definitions and theory of impulsive differential equations. Definition 1 ([1]). Let V : R+ × Rn → R+ , then V is said to belong to Class v0 if (1) V is continuous in (τk−1 , τk ] × Rn and for each x ∈ Rn , k = 1, 2, . . . lim
(t ,y)→(τk+ ,x)
V (t , y) = V (τk+ , x)
exists; (2) V is locally Lipschitzian in x. Definition 2. Let V : R+ × Rn → R+ . For (t , x) ∈ (τk−1 , τk ] × Rn , we define the right upper derivative of V (t , x) as D+ V (t , x) = lim sup s→0+
1 s
[V (t + s, x + s(Ax + bf (c T x))) − V (t , x)].
Definition 3. Let x(t , t0 , x0 ) be any solution to (3), and x(t0 ) = x0 . System (3) is called globally exponentially stable if for any scalar δ > 0, there exist scalars ξ > 0 and F (δ) > 0, when kx0 k < δ , such that
kx(t , t0 , x0 )k ≤ F (δ) exp(−ξ (t − t0 )),
∀t ≥ t0 ≥ 0.
We also need to define a comparison system, which plays an important part in impulsive control of Lurie systems.
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Y. Yu et al. / Computers and Mathematics with Applications 56 (2008) 2806–2813
Definition 4 ([2]– Comparison System). Let V ∈ v0 and assume that D+ V (t , x) ≤ g (t , V (t , x)),
t 6= τk
V (t , x + U (k, x)) ≤ ψk (V (t , x)),
(4)
t = τk
where g : R+ × R+ 7→ R is continuous and ψk : R+ 7→ R+ is nondecreasing. Then the following system:
˙ (t ) = g (t , w), w
t 6= τk
w (τk ) = ψk (w (τk )) +
(5)
w (t0 ) = w0 ≥ 0 +
is the comparison system of (3). Let S (ρ) = {x ∈ Rn | kxk < ρ}, where k · k denotes the Euclidean norm on Rn . Then we can obtain the following theorem to give sufficient conditions in a unified way for various stability criteria. Theorem 1 ([2]). Assume that the following three conditions are satisfied. (1) V : R+ × S (ρ) 7→ R+ , ρ > 0, V ∈ v0 , D+ V (t , x) ≤ g (t , V (t , x)), t 6= τk . (2) there exists a ρ0 such that x ∈ S (ρ0 ) implies that x + U (k, x) ∈ S (ρ0 ) for all k and V (t , x + U (k, x)) ≤ ψk (V (t , x)), t = τk , x ∈ S (ρ0 ). (3) there exist α(·) and β(·) ∈ ℵ (class of continuous strictly increasing functions α : R+ → R+ such that α(0) = 0), such that β(kxk) ≤ V (t , x) ≤ α(kxk) on R+ × S (ρ). Then, the stability properties of the trivial solution w = 0 of comparison system (5) imply the corresponding stability properties of the trivial solution x = 0 of the impulsive control system (3).
˙ t )w (t ), λ ∈ C 1 [R+ , R+ ], ψk (w) = dk w (τk ), dk ≥ 0 for all k, then the origin of the impulsive Theorem 2 ([2]). Let g (t , w) = λ( control system (3) is asymptotically stable if conditions λ(τk+1 ) − λ(τk ) ≤ − ln(γ dk ),
for all k, where γ > 1
(6)
and
˙ t) ≥ 0 λ(
(7)
are satisfied. 3. Main results In this section, we study the impulsive control of Lurie system. First, we establish novel sufficient conditions under which impulsively controlled Lurie system is asymptotically stable, and give the estimate of the upper bound of impulse interval. Then, we represent globally exponential stability of the impulsive Lurie system. Theorem 3. Let the n × n matrix P be symmetric and positive definite, and λ1 > 0, λ2 > 0 are respectively, the smallest and the largest eigenvalues of P. Let
Ω=
AT (P + ηκ cc T ) + (P + ηκ cc T )A bT (P + ηκ cc T ) + (ζ − η)c T A
(P + ηκ cc T )b + (ζ − η)AT c 2(ζ − η)c T b
(8)
where ζ > 0, η > 0 are scalars. Then the origin of impulsive Lurie system (3) is asymptotically stable if there exist scalars ek > 0, ε1 > 0 and the following three conditions hold
Ω < 0,
(9)
kx(τk ) + U (k, x)k2 ≤
ek λ 1
λ2 + (ζ + η)κkc k2
ε1 (τk+1 − τk ) ≤ − ln(γ ek ), λ1
kx(τk )k2 ,
where γ ek < 1 and
(10)
γ > 1.
(11)
Proof. Construct the following Lyapunov function V (t , x) = xT Px + 2ζ
σ
Z 0
f (s)ds + 2η
σ
Z
(κ s − f (s))ds = xT Px + 2ζ
0
cTx
Z
f (s)ds + 2η 0
cTx
Z
(κ s − f (s))ds. 0
It yields that
λ1 kxk2 ≤ V (t , x) ≤ [λ2 + (ζ + η)κkc k2 ] kxk2 .
(12)
Y. Yu et al. / Computers and Mathematics with Applications 56 (2008) 2806–2813
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From which one can see that condition 3 of Theorem 1 is also satisfied with β(x) = λ1 x and α(x) = [λ2 + (ζ + η)κkc k2 ]x. When t 6= τk , the right-hand upper derivative of V (t , x) is D+ V (t , x) = x˙ T Px + xT P x˙ + 2ζ f (c T x)c T x˙ + 2η(κ c T x − f (c T x))c T x˙
= [Ax + bf (c T x)]T Px + xT P [Ax + bf (c T x)] + 2ζ f (c T x)c T [Ax + bf (c T x)] + 2η(κ c T x − f (c T x))c T [Ax + bf (c T x)] = xT (AT P + PA)x + (bT Px + xT Pb)f (c T x) + 2ζ f (c T x)c T [Ax + bf (c T x)] + 2η(κ c T x − f (c T x))c T [Ax + bf (c T x)] T A (P + ηκ cc T ) + (P + ηκ cc T )A (P + ηκ cc T )b + (ζ − η)AT c = (xT , f (c T x)) (xT , f (c T x))T bT (P + ηκ cc T ) + (ζ − η)c T A 2(ζ − η)c T b = (xT , f (c T x))Ω (xT , f (c T x))T . For any P in linear matrix inequality (9), we can find ε1 > 0 and ε2 > 0 such that
Ω<
−ε1 I
0
0
−ε2
<
ε1 I
0
0
−ε2
(13)
where I is the identity matrix. According to (12) and (13), we get D+ V (t , x) < (xT , f (c T x))
ε1 I
0
0
−ε2
(xT , f (c T x))T
= ε1 xT x − ε2 f 2 (c T x) ≤ ε1 kxk2 ε1 ≤ V (t , x), t 6= τk . λ1
(14) ε
Hence, condition 1 of Theorem 1 is satisfied with g (t , w) = λ1 w . 1 When t = τk , by condition (10) and (12), we have V (t , x)|t =τ + = V (t , x + U (k, x))|t =τk k
= [x + U (k, x)] P [x + U (k, x)] + 2ζ T
c T (x+U (k,x))
Z 0
f (s)ds + 2η
c T (x+U (k,x))
Z
(κ s − f (s))ds
0
≤ [λ2 + (ζ + η)κkc k2 ]kx + U (k, x)k2 ≤ ek λ1 kxk2 ≤ ek V (t , x).
(15)
Hence, condition 2 of Theorem 1 is satisfied with ψk (w) = ek w . It follows from Theorem 1 that the asymptotic stability of the impulsive Lurie system in (3) is implied by that of the following comparison system:
ε1 w (t ), t 6= τk λ1 w (τk+ ) = ek w (τk )
˙ (t ) = w
(16)
w (t0+ ) = w0 ≥ 0. It follows from Theorem 2 that if
Z
τk+1 τk
ε1 dt ≤ − ln(γ ek ), λ1
γ >1
i.e.,
ε1 (τk+1 − τk ) ≤ − ln(γ ek ), λ1
γ >1
and
ε1 ≥ 0. λ1 are satisfied, then the origin of (3) is asymptotically stable.
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From (11), we know that
τk+1 − τk ≤
−λ1 ln(γ ek ), ε1
γ > 1,
So the above theorem also gives an estimate of the upper bound of impulse interval ∆ , τk+1 − τk , ∆max .
∆max =
−λ1 ln(γ ek ) > 0, ε1
γ −→ 1+ .
From above we can see that ek should satisfy 0 < ek < 1. Theorem 4. System (3) is globally exponentially stable if there exist scalars ι > 0, ek > 0 and the following four conditions hold
Ω < 0,
kx(τk ) + U (k, x)k2 ≤
ek λ 1
λ2 + (ζ + η)κkc k2
θ,
ε1 < ι, λ1
Y
ek ≤ exp(−ι(t − t0 )),
kx(τk )k2 , (17)
and t ≥ t0 ≥ 0.
(18)
τ k
Proof. According to (14) and (15), we shall show that
Y
V (t , x) ≤ V (t0 , x)
ek exp(θ (t − t0 )),
∀t ≥ t0 ≥ 0.
τ k
To do this, we shall consider the case for all t. In detail, the following situations are considered. If t0 < t < τ1 , we have V (t , x) ≤ V (t0+ , x) exp(θ (t − t0+ )) = V (t0 , x) exp(θ (t − t0 )), V (τ1 , x) ≤ V (t0 , x) exp(θ (τ1 − t0 )); when τ1 < t < τ2 , we obtain V (t , x) ≤ V (τ1+ , x) exp(θ (t − τ1+ )) ≤ V (t0 , x)e1 exp(θ (t − t0 )), V (τ2 , x) ≤ V (t0 , x)e1 exp(θ (τ2 − t0 )); when τ2 < t < τ3 , we obtain V (t , x) ≤ V (τ2+ , x) exp(θ (t − τ2+ )) ≤ V (t0 , x)e1 e2 exp(θ (t − t0 )) = V (t0 , x)e1 e2 exp(θ (t − t0 )), V (τ3 , x) ≤ V (t0 , x)e1 e2 exp(θ (τ3 − t0 )); generally, when τk−1 < t < τk , for k = 1, 2, . . ., we have V (t , x) ≤ V (τk+−1 , x) exp(θ (t − τk+−1 )) ≤ V (t0 , x)e1 · · · ek−1 exp(θ (t − t0 )). when τk < t < τk+1 , we have V (t , x) ≤ V (τk+ , x) exp(θ (t − τk+ )) ≤ V (t0 , x)e1 · · · ek exp(θ (t − t0 ))
= V (t0 , x)
Y
ek exp(θ (t − t0 )).
τ k
Therefore, (19) holds.
Applying (18) to (19), and let υ = ι−θ > 0, we can get 2 V (t , x) ≤ V (t0 , x)
Y
ek exp(θ (t − t0 ))
τk
≤ V (t0 , x) exp(−ι(t − t0 )) exp(θ (t − t0 )) = V (t0 , x) exp(−2υ(t − t0 )), t ≥ t0 ≥ 0.
(19)
Y. Yu et al. / Computers and Mathematics with Applications 56 (2008) 2806–2813
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Fig. 1. Asymptotically stable Lurie systems (B1 ).
Arbitrarily choose a scalar δ > 0, and a vector x0 such that kx0 k < δ . According to (12), we can obtain
λ1 kxk2 ≤ V (t , x) ≤ V (t0 , x) exp(−2υ(t − t0 )) ≤ [λ2 + (ζ + η)κkc k2 ]kx0 k2 exp(−2υ(t − t0 )) ≤ [λ2 + (ζ + η)κkc k2 ]δ 2 exp(−2υ(t − t0 )), t ≥ t0 ≥ 0. q λ2 +(ζ +η)κkc k2 Let F (δ) = δ , such that F (δ) > 0. Therefore, we can get λ1 s λ2 + (ζ + η)κkc k2 δ exp (−υ(t − t0 )) = F (δ) exp (−υ(t − t0 )) , kxk ≤ λ1
t ≥ t0 ≥ 0.
According to the Definition 3, system (3) is globally exponentially stable, that is, the Lurie system (1) are impulsively controlled globally exponentially stable. 4. Impulsive controller design with examples In this section, we design impulsive controller of Lurie systems on the basis of the above theoretical results and illustrate the feasibility and effectiveness of our method. We consider the following Lurie control system [19]: x˙ = Ax + bf (c T x)
(20)
where the nonlinearity f (x1 ) =
−α m1 A=
1 0
α −1 −β
(|x1 + 1| − |x1 − 1|) belongs to sector [0, 1], and ! ! 0 −α(m0 − m1 ) 1 1 , 0 b= , c= 0 1 2
!
0
0
(21)
0
Without loss of generality, we assume that the impulsive control law U (k, x) = Bx,
t = τk
where B is the impulsive controller to be determined. Define the parameters α = 9, β = 14, m0 = −71 , m1 = 27 , κ = 1, ζ = 2, η = 3, ε1 = 0.1, γ = 1.3, we can calculate ek = 0.7691 and obtain the following impulsive controller according to the Theorem 3.
−0.2840 B1 =
0 0
0 −0.2840 0
0 0 , −0.2840
−1.7160
!
B2 =
0 0
0 −1.7160 0
0 0 . −1.7160
!
As for the impulsive control matrix B1 , B2 , the simulation results are shown in Figs. 1 and 2 respectively. The following figures illustrate that the impulsively controlled Lurie systems are asymptotically stable.
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Y. Yu et al. / Computers and Mathematics with Applications 56 (2008) 2806–2813
Fig. 2. Asymptotically stable Lurie systems (B2 ).
Fig. 3. Globally exponentially stable Lurie systems (B3 ).
Similar to the above design, we can determine the impulsive controller based on the Theorem 4.
−0.1069 B3 =
0 0
0 −0.1069 0
0 0 , −0.1069
−1.8931
!
B4 =
0 0
0 −1.8931 0
0 0 . −1.8931
!
As for the impulsive control matrix B3 , B4 , the simulation results are shown in Figs. 3 and 4 respectively. The following figures illustrate that the impulsively controlled Lurie systems are globally exponentially stable. 5. Conclusions In this paper, we have presented an impulsive control scheme of Lurie system. First, we use Lyapunov’s direct method and comparison technique to obtain novel conditions under which the Lurie system can be impulsively controlled to the origin. Second, we present the estimate of the upper bound of impulse interval for asymptotically stable control. Then, we give applicable conditions under which impulsively controlled Lurie system is globally exponentially stable. We give impulsive controller design with two examples to illustrate the feasibility and effectiveness of our method finally.
Y. Yu et al. / Computers and Mathematics with Applications 56 (2008) 2806–2813
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Fig. 4. Globally exponentially stable Lurie Systems (B4 ).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
D.D. Bainov, P.S. Simeonov, Systems with Impulse Effect: Stability, Theory and Applications, Halsted, New York, 1989. V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, London, U.K., 1989. T. Yang, Impulsive Control Theory, Springer-Verlag, Berlin, 2001. T. Yang, Impulsive control, IEEE Trans. Automat. Control. 44 (May) (1999) 1081–1083. Z.G. Li, C.Y. Wen, Y.C. Soh, Analysis and design of impulsive control systems, IEEE Trans. Automat. Control. 46 (June) (2001) 894–897. T. Yang, L.B. Yang, C.M. Yang, Impulsive control of Lorenz system, Physica D 110 (1997) 18–24. Jitao Sun, Yinping Zhang, Impulsive control of Lorenz system, in: Proceedings of the 5th World Congress on Intelligent Control and Automation, Hangzhou, PR China, June 15–19, 2004, pp. 71–73. T. Yang, L.O. Chua, Impulsive stabilization for control and synchronization of chaotic systems: Theory and application to secure communication, IEEE Trans. Circuits Syst. 1 44 (1997) 976–988. Z.G. Li, C.Y. Wen, Y.C. Soh, W.X. Xie, The stabilization and synchronization of Chua’s oscillators via impulsive control, IEEE Trans. Circuits Syst. 1 (48) (2001) 1351–1355. J.T. Sun, Y.P. Zhang, Impulsive control and synchronization of Chua’s oscillators, Math. Comput. Simul. 66 (2004) 499–508. A.I. Lur’e, V.N. Postnikov, On the theory of stability of control systems, Prikl. Nat. Meh. 8 (3) (1944). K.S. Narendra, J.H. Taylor, Frequency Domain Criteria for Absolute Stability, Academic Press, New York, 1973. V. Singh, A new approach to the absolute stability of nonlinear feedback systems with slope-restricted nonlinearity, IEEE Trans. Circuits Syst. 32 (1985) 402–403. A. Halanay, V. Rasvan, Absolute stability of feedback systems with several differential nonlinearities, Internat. J. Systems Sci. 22 (1991) 1911–1927. B.J. Xu, X.Z. Liu, X.X. Liao, Absolute stability of Lurie systems with impulsive effects, Comput. Math. Appl. 47 (2004) 419–425. F. Wang, Robust absolute stability of interval Lurie systems with time-delays, in: Proceedings of the Fourth International Conference on Machine Learning and Cybernetics, Guangzhou, PR China, August, 2005, pp. 18–21. J.K. Tian, S.M. Zhong, L.L. Xiong, Delay-dependent absolute stability of Lurie control systems with multiple time-delays, Appl. Math. Comput. (2006). X.X. Liao, P. Yu, Sufficient and necessary conditions for absolute stability of time-delayed Lurie control systems, J. Math. Anal. Appl. 323 (2006) 876–890. J.G. Lu, D.J. Hill, Impulsive synchronization of chaotic Lurie systems by linear static measurement feedback: An LMI approach, IEEE Transactions on Circuits and Systems-II: Express Briefs 54 (8) (2007) 710–714.
Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Impulsive control of Lurie systems Yongbin Yu a,∗ , Fengli Zhang a , Qishui Zhong b , Xiaofeng Liao c , Juebang Yu b a
School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, 610054, PR China
b
School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, 610054, PR China
c
College of Computer Science, Chongqing University, Chongqing, 400044, PR China
article
info
Article history: Received 28 February 2007 Received in revised form 3 August 2008 Accepted 10 September 2008 Keywords: Impulsive control Lurie systems Impulsive differential equation Asymptotically stable Globally exponentially stable
a b s t r a c t In this paper the impulsive control scheme of Lurie systems is represented. Using Lyapunov’s direct method and comparison technique, we present novel conditions under which impulsively controlled Lurie systems are asymptotically stable, and give the estimate of the upper bound of impulse interval. In particular, globally exponential stability of impulsive control of Lurie systems is presented. Impulsive controller design with examples is provided to show the feasibility and effectiveness of our method finally. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Impulsive control is a control paradigm based on impulsive differential equations. In practice, there exist many examples of impulsive control systems (see [1–3]). Recently, impulsive systems and impulsive control have been studied by many researchers. Bainov and Simeonov [1], and Lakshmikantham et al. [2] have considered the stability of impulsive systems by using Lyapunov functions which are required to be nonincreasing along the whole sequence of switchings. Yang Tao [4] and Li et al. [5] have presented the concept and principles of impulsive control and obtained some sufficient conditions of a class of nonlinear systems by using the results in [2]. Yang et al. [6] and Sun et al. [7] have presented some interesting applications of impulsive control in Lorenz systems. Yang and Chua [8], Li et al. [9] and Sun et al. [10] have studied impulsive control and synchronization of Chua’s oscillators. However, these proposed impulsive control and impulsive systems do not consider Lurie systems or consider less, which is an important issue of impulsive control. In this paper, we shall consider impulsive control of Lurie systems. Utilizing the methods of Lyapunov functionals and comparison principle, we present novel conditions under which the impulsively controlled Lurie systems are asymptotically stable, and give the estimate of the upper bound of impulse interval. Furthermore, globally exponential stability of impulsive control of Lurie systems is still given. Comparing with [11–18], this paper is focused on impulsive control instead of absolute stability analysis of Lurie systems. The rest of this paper is organized as follows. In the following section, we introduce some preliminary definitions and the theory of impulsive differential equation. In Section 3, the impulsively controlled Lurie systems are studied, and new asymptotical stability theorem and the exponential stability theorem of the impulsively controlled Lurie systems are given. Following the numerical examples to illustrate design of impulsive control Lurie systems in Section 4, finally, some concluding remarks are presented in Section 5.
∗
Corresponding author. E-mail address: [email protected] (Y. Yu).
0898-1221/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2008.09.015
Y. Yu et al. / Computers and Mathematics with Applications 56 (2008) 2806–2813
2807
2. Preliminaries Consider the following Lurie control system
x˙ (t ) = Ax(t ) + bf (σ (t )) σ (t ) = c T x(t )
(1)
where x(t ) ∈ Rn is the state variable, A ∈ Rn×n , b, c ∈ Rn , and c T is the transpose of c. The nonlinearity f (σ ) satisfies f (σ ) ∈ zκ = {f (σ )|f (0) = 0, 0 < σ f (σ ) ≤ κσ 2 , σ 6= 0, f (σ ) ∈ C [R, R]}.
(2)
From (2), we know that
κσ 2 − σ f (σ ) ≥ 0 ⇔ σ (κσ − f (σ )) ≥ 0. Consider a discrete time set {τk } of control instants, where 0 < τ1 < τ2 < · · · < τk < τk+1 < · · · , τk → ∞,
as k → ∞.
Let U (k, x) = ∆x|t =τk , x(τk+ ) − x(τk− ), x(τk+ ) = lim x(t ), + t →τk
x(τk− ) = lim x(t ) − t →τk
be the control law, which is the ‘‘jump’’ in the state variable at the time instant τk , where x(τk− ) = x(τk ) is assumed. Then the impulsive Lurie system is presented by
x˙ (t ) = Ax(t ) + bf (σ (t )), t 6= τk , σ (t ) = c T x(t ), t 6= τk , ∆x+|t =τk = U (k, x), t = τk , x(t0 ) = x0 , t0 ≥ 0
(3)
which is a nonlinear impulsive differential equation according to [2]. Remark: On the one hand, when t 6= τk , the dynamics of the impulsive control system in (3) is governed by the system in (1). We can see that the controlled system is a free system whenever t 6= τk . On the other hand, when t = τk , k = 1, 2, . . ., the state variable is changed from x(τk− ) to x(τk+ ) = x(τk− ) + ∆x|t =τk instantaneously. In order to establish sufficient conditions under which impulsively controlled Lurie system is stable, we need the following basic definitions and theory of impulsive differential equations. Definition 1 ([1]). Let V : R+ × Rn → R+ , then V is said to belong to Class v0 if (1) V is continuous in (τk−1 , τk ] × Rn and for each x ∈ Rn , k = 1, 2, . . . lim
(t ,y)→(τk+ ,x)
V (t , y) = V (τk+ , x)
exists; (2) V is locally Lipschitzian in x. Definition 2. Let V : R+ × Rn → R+ . For (t , x) ∈ (τk−1 , τk ] × Rn , we define the right upper derivative of V (t , x) as D+ V (t , x) = lim sup s→0+
1 s
[V (t + s, x + s(Ax + bf (c T x))) − V (t , x)].
Definition 3. Let x(t , t0 , x0 ) be any solution to (3), and x(t0 ) = x0 . System (3) is called globally exponentially stable if for any scalar δ > 0, there exist scalars ξ > 0 and F (δ) > 0, when kx0 k < δ , such that
kx(t , t0 , x0 )k ≤ F (δ) exp(−ξ (t − t0 )),
∀t ≥ t0 ≥ 0.
We also need to define a comparison system, which plays an important part in impulsive control of Lurie systems.
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Definition 4 ([2]– Comparison System). Let V ∈ v0 and assume that D+ V (t , x) ≤ g (t , V (t , x)),
t 6= τk
V (t , x + U (k, x)) ≤ ψk (V (t , x)),
(4)
t = τk
where g : R+ × R+ 7→ R is continuous and ψk : R+ 7→ R+ is nondecreasing. Then the following system:
˙ (t ) = g (t , w), w
t 6= τk
w (τk ) = ψk (w (τk )) +
(5)
w (t0 ) = w0 ≥ 0 +
is the comparison system of (3). Let S (ρ) = {x ∈ Rn | kxk < ρ}, where k · k denotes the Euclidean norm on Rn . Then we can obtain the following theorem to give sufficient conditions in a unified way for various stability criteria. Theorem 1 ([2]). Assume that the following three conditions are satisfied. (1) V : R+ × S (ρ) 7→ R+ , ρ > 0, V ∈ v0 , D+ V (t , x) ≤ g (t , V (t , x)), t 6= τk . (2) there exists a ρ0 such that x ∈ S (ρ0 ) implies that x + U (k, x) ∈ S (ρ0 ) for all k and V (t , x + U (k, x)) ≤ ψk (V (t , x)), t = τk , x ∈ S (ρ0 ). (3) there exist α(·) and β(·) ∈ ℵ (class of continuous strictly increasing functions α : R+ → R+ such that α(0) = 0), such that β(kxk) ≤ V (t , x) ≤ α(kxk) on R+ × S (ρ). Then, the stability properties of the trivial solution w = 0 of comparison system (5) imply the corresponding stability properties of the trivial solution x = 0 of the impulsive control system (3).
˙ t )w (t ), λ ∈ C 1 [R+ , R+ ], ψk (w) = dk w (τk ), dk ≥ 0 for all k, then the origin of the impulsive Theorem 2 ([2]). Let g (t , w) = λ( control system (3) is asymptotically stable if conditions λ(τk+1 ) − λ(τk ) ≤ − ln(γ dk ),
for all k, where γ > 1
(6)
and
˙ t) ≥ 0 λ(
(7)
are satisfied. 3. Main results In this section, we study the impulsive control of Lurie system. First, we establish novel sufficient conditions under which impulsively controlled Lurie system is asymptotically stable, and give the estimate of the upper bound of impulse interval. Then, we represent globally exponential stability of the impulsive Lurie system. Theorem 3. Let the n × n matrix P be symmetric and positive definite, and λ1 > 0, λ2 > 0 are respectively, the smallest and the largest eigenvalues of P. Let
Ω=
AT (P + ηκ cc T ) + (P + ηκ cc T )A bT (P + ηκ cc T ) + (ζ − η)c T A
(P + ηκ cc T )b + (ζ − η)AT c 2(ζ − η)c T b
(8)
where ζ > 0, η > 0 are scalars. Then the origin of impulsive Lurie system (3) is asymptotically stable if there exist scalars ek > 0, ε1 > 0 and the following three conditions hold
Ω < 0,
(9)
kx(τk ) + U (k, x)k2 ≤
ek λ 1
λ2 + (ζ + η)κkc k2
ε1 (τk+1 − τk ) ≤ − ln(γ ek ), λ1
kx(τk )k2 ,
where γ ek < 1 and
(10)
γ > 1.
(11)
Proof. Construct the following Lyapunov function V (t , x) = xT Px + 2ζ
σ
Z 0
f (s)ds + 2η
σ
Z
(κ s − f (s))ds = xT Px + 2ζ
0
cTx
Z
f (s)ds + 2η 0
cTx
Z
(κ s − f (s))ds. 0
It yields that
λ1 kxk2 ≤ V (t , x) ≤ [λ2 + (ζ + η)κkc k2 ] kxk2 .
(12)
Y. Yu et al. / Computers and Mathematics with Applications 56 (2008) 2806–2813
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From which one can see that condition 3 of Theorem 1 is also satisfied with β(x) = λ1 x and α(x) = [λ2 + (ζ + η)κkc k2 ]x. When t 6= τk , the right-hand upper derivative of V (t , x) is D+ V (t , x) = x˙ T Px + xT P x˙ + 2ζ f (c T x)c T x˙ + 2η(κ c T x − f (c T x))c T x˙
= [Ax + bf (c T x)]T Px + xT P [Ax + bf (c T x)] + 2ζ f (c T x)c T [Ax + bf (c T x)] + 2η(κ c T x − f (c T x))c T [Ax + bf (c T x)] = xT (AT P + PA)x + (bT Px + xT Pb)f (c T x) + 2ζ f (c T x)c T [Ax + bf (c T x)] + 2η(κ c T x − f (c T x))c T [Ax + bf (c T x)] T A (P + ηκ cc T ) + (P + ηκ cc T )A (P + ηκ cc T )b + (ζ − η)AT c = (xT , f (c T x)) (xT , f (c T x))T bT (P + ηκ cc T ) + (ζ − η)c T A 2(ζ − η)c T b = (xT , f (c T x))Ω (xT , f (c T x))T . For any P in linear matrix inequality (9), we can find ε1 > 0 and ε2 > 0 such that
Ω<
−ε1 I
0
0
−ε2
<
ε1 I
0
0
−ε2
(13)
where I is the identity matrix. According to (12) and (13), we get D+ V (t , x) < (xT , f (c T x))
ε1 I
0
0
−ε2
(xT , f (c T x))T
= ε1 xT x − ε2 f 2 (c T x) ≤ ε1 kxk2 ε1 ≤ V (t , x), t 6= τk . λ1
(14) ε
Hence, condition 1 of Theorem 1 is satisfied with g (t , w) = λ1 w . 1 When t = τk , by condition (10) and (12), we have V (t , x)|t =τ + = V (t , x + U (k, x))|t =τk k
= [x + U (k, x)] P [x + U (k, x)] + 2ζ T
c T (x+U (k,x))
Z 0
f (s)ds + 2η
c T (x+U (k,x))
Z
(κ s − f (s))ds
0
≤ [λ2 + (ζ + η)κkc k2 ]kx + U (k, x)k2 ≤ ek λ1 kxk2 ≤ ek V (t , x).
(15)
Hence, condition 2 of Theorem 1 is satisfied with ψk (w) = ek w . It follows from Theorem 1 that the asymptotic stability of the impulsive Lurie system in (3) is implied by that of the following comparison system:
ε1 w (t ), t 6= τk λ1 w (τk+ ) = ek w (τk )
˙ (t ) = w
(16)
w (t0+ ) = w0 ≥ 0. It follows from Theorem 2 that if
Z
τk+1 τk
ε1 dt ≤ − ln(γ ek ), λ1
γ >1
i.e.,
ε1 (τk+1 − τk ) ≤ − ln(γ ek ), λ1
γ >1
and
ε1 ≥ 0. λ1 are satisfied, then the origin of (3) is asymptotically stable.
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From (11), we know that
τk+1 − τk ≤
−λ1 ln(γ ek ), ε1
γ > 1,
So the above theorem also gives an estimate of the upper bound of impulse interval ∆ , τk+1 − τk , ∆max .
∆max =
−λ1 ln(γ ek ) > 0, ε1
γ −→ 1+ .
From above we can see that ek should satisfy 0 < ek < 1. Theorem 4. System (3) is globally exponentially stable if there exist scalars ι > 0, ek > 0 and the following four conditions hold
Ω < 0,
kx(τk ) + U (k, x)k2 ≤
ek λ 1
λ2 + (ζ + η)κkc k2
θ,
ε1 < ι, λ1
Y
ek ≤ exp(−ι(t − t0 )),
kx(τk )k2 , (17)
and t ≥ t0 ≥ 0.
(18)
τ k
Proof. According to (14) and (15), we shall show that
Y
V (t , x) ≤ V (t0 , x)
ek exp(θ (t − t0 )),
∀t ≥ t0 ≥ 0.
τ k
To do this, we shall consider the case for all t. In detail, the following situations are considered. If t0 < t < τ1 , we have V (t , x) ≤ V (t0+ , x) exp(θ (t − t0+ )) = V (t0 , x) exp(θ (t − t0 )), V (τ1 , x) ≤ V (t0 , x) exp(θ (τ1 − t0 )); when τ1 < t < τ2 , we obtain V (t , x) ≤ V (τ1+ , x) exp(θ (t − τ1+ )) ≤ V (t0 , x)e1 exp(θ (t − t0 )), V (τ2 , x) ≤ V (t0 , x)e1 exp(θ (τ2 − t0 )); when τ2 < t < τ3 , we obtain V (t , x) ≤ V (τ2+ , x) exp(θ (t − τ2+ )) ≤ V (t0 , x)e1 e2 exp(θ (t − t0 )) = V (t0 , x)e1 e2 exp(θ (t − t0 )), V (τ3 , x) ≤ V (t0 , x)e1 e2 exp(θ (τ3 − t0 )); generally, when τk−1 < t < τk , for k = 1, 2, . . ., we have V (t , x) ≤ V (τk+−1 , x) exp(θ (t − τk+−1 )) ≤ V (t0 , x)e1 · · · ek−1 exp(θ (t − t0 )). when τk < t < τk+1 , we have V (t , x) ≤ V (τk+ , x) exp(θ (t − τk+ )) ≤ V (t0 , x)e1 · · · ek exp(θ (t − t0 ))
= V (t0 , x)
Y
ek exp(θ (t − t0 )).
τ k
Therefore, (19) holds.
Applying (18) to (19), and let υ = ι−θ > 0, we can get 2 V (t , x) ≤ V (t0 , x)
Y
ek exp(θ (t − t0 ))
τk
≤ V (t0 , x) exp(−ι(t − t0 )) exp(θ (t − t0 )) = V (t0 , x) exp(−2υ(t − t0 )), t ≥ t0 ≥ 0.
(19)
Y. Yu et al. / Computers and Mathematics with Applications 56 (2008) 2806–2813
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Fig. 1. Asymptotically stable Lurie systems (B1 ).
Arbitrarily choose a scalar δ > 0, and a vector x0 such that kx0 k < δ . According to (12), we can obtain
λ1 kxk2 ≤ V (t , x) ≤ V (t0 , x) exp(−2υ(t − t0 )) ≤ [λ2 + (ζ + η)κkc k2 ]kx0 k2 exp(−2υ(t − t0 )) ≤ [λ2 + (ζ + η)κkc k2 ]δ 2 exp(−2υ(t − t0 )), t ≥ t0 ≥ 0. q λ2 +(ζ +η)κkc k2 Let F (δ) = δ , such that F (δ) > 0. Therefore, we can get λ1 s λ2 + (ζ + η)κkc k2 δ exp (−υ(t − t0 )) = F (δ) exp (−υ(t − t0 )) , kxk ≤ λ1
t ≥ t0 ≥ 0.
According to the Definition 3, system (3) is globally exponentially stable, that is, the Lurie system (1) are impulsively controlled globally exponentially stable. 4. Impulsive controller design with examples In this section, we design impulsive controller of Lurie systems on the basis of the above theoretical results and illustrate the feasibility and effectiveness of our method. We consider the following Lurie control system [19]: x˙ = Ax + bf (c T x)
(20)
where the nonlinearity f (x1 ) =
−α m1 A=
1 0
α −1 −β
(|x1 + 1| − |x1 − 1|) belongs to sector [0, 1], and ! ! 0 −α(m0 − m1 ) 1 1 , 0 b= , c= 0 1 2
!
0
0
(21)
0
Without loss of generality, we assume that the impulsive control law U (k, x) = Bx,
t = τk
where B is the impulsive controller to be determined. Define the parameters α = 9, β = 14, m0 = −71 , m1 = 27 , κ = 1, ζ = 2, η = 3, ε1 = 0.1, γ = 1.3, we can calculate ek = 0.7691 and obtain the following impulsive controller according to the Theorem 3.
−0.2840 B1 =
0 0
0 −0.2840 0
0 0 , −0.2840
−1.7160
!
B2 =
0 0
0 −1.7160 0
0 0 . −1.7160
!
As for the impulsive control matrix B1 , B2 , the simulation results are shown in Figs. 1 and 2 respectively. The following figures illustrate that the impulsively controlled Lurie systems are asymptotically stable.
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Fig. 2. Asymptotically stable Lurie systems (B2 ).
Fig. 3. Globally exponentially stable Lurie systems (B3 ).
Similar to the above design, we can determine the impulsive controller based on the Theorem 4.
−0.1069 B3 =
0 0
0 −0.1069 0
0 0 , −0.1069
−1.8931
!
B4 =
0 0
0 −1.8931 0
0 0 . −1.8931
!
As for the impulsive control matrix B3 , B4 , the simulation results are shown in Figs. 3 and 4 respectively. The following figures illustrate that the impulsively controlled Lurie systems are globally exponentially stable. 5. Conclusions In this paper, we have presented an impulsive control scheme of Lurie system. First, we use Lyapunov’s direct method and comparison technique to obtain novel conditions under which the Lurie system can be impulsively controlled to the origin. Second, we present the estimate of the upper bound of impulse interval for asymptotically stable control. Then, we give applicable conditions under which impulsively controlled Lurie system is globally exponentially stable. We give impulsive controller design with two examples to illustrate the feasibility and effectiveness of our method finally.
Y. Yu et al. / Computers and Mathematics with Applications 56 (2008) 2806–2813
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Fig. 4. Globally exponentially stable Lurie Systems (B4 ).
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