Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
ThB01.5
Synchronization of nonlinear pendulum-like systems Pingli Lu and Ying Yang
Abstract— This paper deals with the chaos synchronization problem for a class of nonlinear pendulum-like systems with multiple equilibria. Based on the method of periodic Lyapunov functions, linear matrix inequality (LMI) formulations are established to guarantee the stable synchronization of both master and slave pendulum-like systems by feedback control technique. Finally, a concrete application to phase-locked loop (PLL) shows the applicability and validity of the proposed approach.
I. I NTRODUCTION Since the synchronization was first introduced by Pecora and Carroll in 1990, chaos synchronization has been a focus topic of intensive research due to its important implications in secure communication, chemical and biological systems. Recently, chaos synchronization of the master-slave Lur’e systems has been investigated by discussing the absolute stability of its error dynamics [1], [2], [3]. However, the results were constructed for investigation of global stability of nonlinear systems with single equilibrium. Meanwhile engineers often have to deal with nonlinear systems with multiple equilibria, for example, phase-locked loops and synchronous machines which are typical nonlinear pendulumlike systems [4], [5], [6], [7], [8], [9]. Interaction among identical and nonidentical pendulum-like systems can lead to synchronized dynamics. The synchronization of two coupled nonlinear pendulum-like systems has not only purely theoretical interest, but also important applied significance. For instance, the synchronization between two coupled phaselocked loops (PLLs) means phase-locking of the constituent PLLs. Therefore, studying the synchronization of nonlinear pendulum-like systems is naturally important and necessary. In this paper, for the first time we employ the feedback control technique to investigate the chaos synchronization behavior of two pendulum-like systems with identical system parameters and different initial conditions. The linear matrix inequality (LMI) conditions, under which the error pendulum-like system is globally asymptotically stabilized at equilibria, are established by the method of periodic Lyapunov functions [4]. A practical application to PLL is given to verify the effectiveness of proposed methods at the end of the paper. The rest of this paper is organized as follows: Section 2 presents some basic results necessary for the successive development, while in Section 3, we give the LMI-based This work is supported by the National Science Foundation 60874011. Pingli Lu is with School of Automation, Beijing Institute of Technology, Beijing, 100081, P.R. China
[email protected] Ying Yang is with State Key Lab for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, Peking University, Beijing, 100871, P.R. China
[email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
criterion which guarantees the globally asymptotically stabilization of solutions for the error pendulum-like system. Section 4 gives a concrete phase-locked loop circuit to demonstrate the applicability and validity of the proposed methods. Section 5 shows our conclusions of this paper. We use the following notations in this paper. Rn×n is the set of n×n real matrices. The superscript T denotes transpose for real matrices. The matrix inequality A < 0(A ≤ 0) means that A = AT is negative (semi-)definite. For matrix A ∈ Rn×n , vectors x ∈ Rn , y ∈ Rn , He A = A + AT , kAk means the induced matrix two-norm, kxk denotes its Euclidean norm, and < x, y > is defined by < x, y >= xT y II. P RELIMINARIES Consider the following pendulum-like system with state delay x(t) ˙ = Ax(t) + Ah x(t − h) + Bϕ (σ (t)) (1) σ˙ (t) = CT x(t) + Dϕ (σ ) where h > 0 is a given constant and A ∈ Rn×n , Ah ∈ Rn×n , B ∈ Rn×m ,C ∈ Rn×m , D ∈ Rm×m respectively. x(t) = λ (t), ∀t ∈ [−h, 0], where λ (t) is the initial condition. A is Hurwitzian, and ϕ : Rm → Rm is a vector-valued function having the components ϕi (σ ) = ϕi (σi ) with σ = [σ1 , σ2 , ..., σm ]T . Each component ϕi : R → R is ∆i -periodic, has a finite number of zeros on [0, ∆i ) and satisfies a local Lipschitz condition. We introduce the following transfer function of the linear part of (1) from the input ϕ to the output σ˙ G(s) = CT (sI − A − Ah e−hs )−1 B + D We also suppose that G(0) 6= 0, (A + Ah )−1 exists, and the equilibrium is (xeq , σeq ) which satisfies Axeq + Ah xeq + Bϕ (σeq ) = 0 and CT xeq + Dϕ (σeq ) = 0 Based on the above assumptions, we have
ϕ (σeq ) = 0 i.e., ϕi (σieq ) = 0. It follows that xeq = 0. Since ϕi is ∆i periodic, the pendulum-like system (1) has infinitely many isolated equilibria. Lemma 2.1 (Gronwall-Bellman Inequality [10]): Suppose that u(t), v(t) are non-negative,continuous functions on [0, +∞], respectively. If there exists a scalar l ≥ 0 such that Z t u(t) ≤ l + u(α )v(α )d α , t ≥0
4222
0
ThB01.5 then the following inequality is satisfied. Rt
u(t) ≤ le 0 v(α )d α Proposition 2.1 ([4]): If the solution x(t) of the pendulum-like system (1) is bounded then the functions ϕi (σi (t))(i = 1, ...m),where σi (t) belongs to a solution of (1), are uniformly continuous on [0, +∞). The validity of this assertion follows from the facts that ϕi (σi ) is locally Lipschitz continuous and σ˙ i (t) is bounded on [0, +∞) which is guaranteed by bounded x(t) and ϕ (σ (t)). Lemma 2.2 ([4]): If α : R+ → R belongs to L2 [0, +∞) and β : R+ → R belongs to L2 [0, +∞) then r(t) =
Z t
synchronization between state x(t) and y(t) is then to obtain the global asymptotic stability of solution (z(t), σ˙ ∆ (t)) for the error pendulum-like system (6). Lemma 3.1: The solution z(t) of the error pendulum-like system (6) is bounded if the system matrix Az is Hurwitzian and kKk is bounded. Proof: From the differential equation (6), the solution z(t) can be expressed as follows z(t) = e
Az t
z(0) −
Z t
eAz (t−τ ) Kz(τ − h)d τ
0
Z t
eAz (t−τ ) (B − JD)η (σ∆ (τ ), σ2 (τ ))d τ (7) The second item of solution z(t) can be simplified in the following +
0
α (t − τ )β (τ )d τ → 0,
t → +∞
0
III. S YNCHRONIZATION OF PENDULUM - LIKE SYSTEMS Consider a general master-slave type of coupled pendulum-like systems: ½ x(t) ˙ = Ax(t) + Bϕ (σ1 (t)) (2) M: σ˙ 1 (t) = CT x(t) + Dϕ (σ1 (t)) ½ y(t) ˙ = Ay(t) + Bϕ (σ2 (t)) + u(t) S: (3) σ˙ 2 (t) = CT y(t) + Dϕ (σ2 (t)) where ϕ (σi (t)), (i = 1, 2) is ∆−periodic nonlinear vector function, the controller u(t) = K[x(t − h) − y(t − h)] + J[σ˙ 1 (t) − σ˙ 2 (t)]
Z t
eAz (t−τ ) Kz(τ − h)d τ = e−Az h
0
Z t
eAz (t−s) Kz(s)ds
(8)
0
where s = τ −h. Since Az is Hurwitzian, there exist q > 0, γ > 0 such that (9) keAz t k ≤ qe−γ t From formulations (8) and (9), the solution z(t) can be written as z(t) ≤ qe−γ t z(0) − qe−Az h
(4)
+q
Z t 0
where h > 0 is a constant time delay, and
Z t
e−γ (t−s) Kz(s)ds
0
e−γ (t−τ ) (B − JD)η (σ∆ (τ ), σ2 (τ ))d τ
Since η (σ∆ (t), σ2 (t)) is a bounded function , there exists a scalar M1 > 0 such that
A, K ∈ Rn×n , B,C, J ∈ Rn×m , D ∈ Rm×m Assume the synchronization error
qe−γ t kz(0)k + q
Z t 0
z(t) = x(t) − y(t)
e−γ (t−τ ) k(B − JD)η kd τ ≤ M1
Accordingly,
from (2)-(4), we have
kz(t)k ≤ M1 + qke−Az h k
T
z˙(t) = (A − JC )z(t) − Kz(t − h) + (B − JD)[ϕ (σ1 (t)) − ϕ (σ2 (t))]
(5)
Z t
e−γ (t−s) kKkkz(s)kds
0
furthermore, utilizing Lemma 2.1, we have
Define
kz(t)k ≤ M1 eqke
σ∆ (t) = σ1 (t) − σ2 (t)
≤ M1 eγ
then
−Az h kkKk R t e−γ (t−s) ds 0
−1 qke−Az h kkKk
Since kKk is bounded and let
ϕ (σ1 (t)) − ϕ (σ2 (t)) = ϕ (σ∆ (t) + σ2 (t)) − ϕ (σ2 (t))
M2 = M1 eγ
and let
−1 qke−Az h kkKk
we obtain
η (σ∆ (t), σ2 (t)) = ϕ (σ∆ (t) + σ2 (t)) − ϕ (σ2 (t)) where η (σ∆ (t), σ2 (t)) is also ∆−periodic nonlinear function about σ∆ due to the periodic property of function ϕ . Further, the equation (5) and the derivative of σ∆ can be written as ½ z˙(t) = Az z(t) − Kz(t − h) + (B − JD)η (σ∆ (t), σ2 (t)) σ˙ ∆ (t) = CT z(t) + Dη (σ∆ (t), σ2 (t)) (6) where Az = A − JCT It is shown that the state equation (6) of the error system has the standard form of pendulum-like system (1). The aim of
kz(t)k ≤ M2 i.e., the solution z(t) of the error pendulum-like system (6) is bounded. This completes the proof. In the following, we will give the LMI characterizations guaranteeing the global asymptotic stability for the error pendulum-like systems (6). Consequently, the synchronization of the master-slave pendulum-like systems (2) and (3) can be realized. ¯ Theorem 3.1: Suppose there exist a scale number h, symmetric matrices P > 0, R > 0, S > 0,Y, M, Z,W,U,V and diagonal matrices κ = diag(κ1 , κ2 , ..., κm ), δ =
4223
ThB01.5 diag(δ1 , δ2 , ..., δm ), ε = diag(ε1 , ε2 , ..., εm ) with δ > 0, ε > 0 such that inequalities (10), (12) hold and matrix A − JCT is Hurwitzian. Ξ11 W T −Y +V K Ξ13 ∗ − HeW − S −Z T ∗ ∗ DT ε D + 21 κ He D + δ ∗ ∗ ∗ ∗ ∗ ∗
where
¯ P + (A − JCT )T U T +V + M T −hY ¯ −K T U T − M T −hW ¯ 0 κ δ
Then, in virtue of (16) and (6), we have the following formulations for any matrices Y,W, Z, M,U and V with appropriate dimensions. Ψ1 =< g(t), Ye [z(t) − z(t − h) −
(12)
νi = R ∆0 i 0
ηi (σ∆ , σ2 )d σ∆
|ηi (σ∆ , σ2 )|d σ∆
, i = 1, 2, ...m
Calculating the derivative of V1 (z(t)) along the solutions of (6) and adding 2Ψ1 from (17) and 2Ψ2 from (18) to it, we have V˙1 (z(t)) =2zT (t)P˙z(t) + 2Ψ1 + 2Ψ2 Z (19) 1 t T g˘ (t)Π1 g(t)d ˘ α = h t−h where
(13)
The controller parameters K, J are given in advance. Then the error pendulum-like system (6) is global asymptotic ¯ i.e., the systems (2)-(4) stable for any h satisfying 0 < h ≤ h, synchronizes. Proof: Introduce the new functions
¸ g(t) g(t) ˘ = z˙(α ) ·
Π1 =
−V (B − JD) + Z T −Z T 0 ∗ ∗ P + ATz U T +V + M T −hY −K T U T − M T −hW (B − JD)T U T −hZ (20) − HeU −hM ∗ 0
He(Y −VAz ) ∗ ∗ ∗ ∗
Fi (σ∆ , σ2 ) = ηi (σ∆ , σ2 ) − νi |ηi (σ∆ , σ2 )| therefore,
Z ∆i 0
Fi (σ∆ , σ2 )d σ∆ = 0
Then V˙2 (z(t)) =
and the function Fi has a mean value zero. We consider the following Lyapunov function
=
V (z(t), σ∆ (t), σ2 (t)) = V1 (z(t)) +V2 (z(t)) +V3 (z(t)) m
+ ∑ κk k=1
Z σk ∆ 0
V3 (z(t)) =
−h t+β
Z t
zT (α )Sz(α )d α
t−h
t
z˙(α )d α = z(t) − z(t − h)
Z t
(21) [˙zT (t)hR˙z(t) − z˙T (α )hR˙z(α )]d α
t−h
Z t
[zT (t)Sz(t) − zT (t − h)Sz(t − h)]d α
(22)
t−h
1 V˙1 (z(t)) + V˙2 (z(t)) + V˙3 (z(t)) = h
(15)
with P > 0, R > 0, S > 0. By the New-Leibniz formula, we have Z t−h
1 h
[˙zT (t)R˙z(t) − z˙T (t + β )R˙z(t + β )]d β
It follows from (19), (21) and (22) that
T
z˙ (α )R˙z(α )d α d β
−h
1 V˙3 (z(t)) = h
(14)
V1 (z(t)) = zT (t)Pz(t) V2 (z(t)) =
Z 0
W T −Y +V K − HeW ∗ ∗ ∗
and
Fk (θ , σ2k )d θ
where Z 0Z t
(17)
g(t) = [zT (t), zT (t − h), η T (σ∆ (t), σ2 (t)), z˙T (t)]T Y · ¸ · ¸ z(t) e W , Ve = V h(t) = ,Y = Z z˙(t) −U M
where ν = diag(ν1 , ν2 , ..., νm ) with R ∆i
z˙(α )d α ] >= 0
t−h
Ψ2 =< h(t),Ve [˙z(t) − Az z(t) + Kz(t − h) − (B − JD)η ] >= 0 (18) where
(10)
(11)
Z t
(16)
where
Z t
t−h
α (23) g˘T (t)Π2 g(t)d ˘
He(Y −VAz ) + S W T −Y +V K ∗ − HeW − S ∗ ∗ Π2 = ∗ ∗ ∗ ∗
4224
−V (B − JD) + Z T −Z T 0 ∗ ∗
ThB01.5 P + ATz U T +V + M T −hY −K T U T − M T −hW (B − JD)T U T −hZ hR − HeU −hM ∗ −hR
Then
(24)
Then there exists a diagonal ρ =diag(ρ1 , ρ2 , ..., ρn ), ρk > 0, k = 1, 2, ...n, 1 h
Z t
n
t−h
α < − ∑ ρk z2k g˘T (t)Πg(t)d ˘ k=1
namely,
m
dV = ϒ1 (t) + ∑ κk Fk (σ∆k (t), σ2k (t))σ˙ ∆k (t) dt k=1 m
[κk ηk (σ∆k (t), σ2k (t))σ˙ ∆k (t) − κk νk |ηk |
=ϒ1 (t) + ∑ k=1
(25)
Hence V (t) −V (0) ≤ −
m
n
k=1
k=1
(
Z t
m
∑
k=1 0
+ ∑ [εk (σ˙ ∆k )2 (t) + δk ηk2 (σ∆k (t), σ2k (t))]
+∑
k=1 0
where (26)
In virtue of condition (12) of the theorem, there exist a δ0k > 0 and ε0k > 0 such that
κk νk |ηk (σ∆k (t), σ2k (t))|σ˙ ∆k (t) + εk (σ˙ ∆k )2 (t) + δk ηk2 ≥ ε0k (σ˙ ∆k )2 (t) + δ0k ηk2
Z +∞ 0
0
ϒ1 (t) + ∑ k=1
0
(28)
Assuming
∑
(29)
(32b) (32c)
Further, we have
σ∆k (t) → σˆ ∆k ,
and substituting the second equation of (6) into (29), we have ϒ1 (t) + ϒ2 (t) =
1 h
Z t
t−h
Ξ13 −Z T Π= DT ε D + 12 κ He D + δ ∗ ∗ P + (A − JCT )T U T +V + M T −hY −K T U T − M T −hW (B − JD)T U T −hZ (30) hR − HeU −hM ∗ −hR −Y +V K − HeW − S ∗ ∗ ∗
¯ the inequality where Ξ11 , Ξ13 are defined in (11). Since h ≤ h, (10) guarantees Π < 0, thus Z t
t−h
α
