- Email: [email protected]
The sufficient criteria for global synchronization of chaotic power systems under linear state-error feedback control
Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
The sufficient criteria for global synchronization of chaotic power systems under linear state-error feedback control Qian Lin a , Xiaofeng Wu a,b,∗ a
College of Electronic Engineering, Naval University of Engineering, Wuhan 430033, PR China
b
Guangzhou Navy Academy, Guangzhou 510430, PR China
article
info
Article history: Received 3 December 2008 Accepted 20 October 2010 Keywords: Chaos Synchronization Linear state-error feedback control Power system
abstract This paper investigates global complete synchronization of two identical power systems and global robust synchronization of two power systems with parameter mismatch and external disturbance, both under the master–slave linear state-error feedback control. Some criteria for achieving the synchronization via a single-variable linear coupling are derived and formulated in simple algebraic inequalities. These algebraic criteria are further optimized so as to improve their performances. The effectiveness of the new algebraic criteria is illustrated by the numerical examples. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction The existing research has shown that electrical power systems are a class of nonlinear dynamical systems which can demonstrate very complex dynamics as chaos [1,2]. The emergent chaotic behavior of the power systems can produce an undesirable consequence such as a power blackout. In order to prevent the damage generated by the chaotic behavior, the control and synchronization of the chaotic power systems should be studied so as to manipulate and manage the systems [3–6]. The single-machine-infinite-bus (SMIB) system is a kind of non-autonomous electrical power system. Very recently, Shahverdiev et al. [6] investigated the synchronization of two identical chaotic SMIB systems via a single-variable linear coupling, where the coupling strength leading to synchronization was detected in terms of the negativity of the maximal Lyapunov exponent of an error system. However, as indicated in [6,7], the negative maximal Lyapunov exponent is only a necessary but not sufficient condition for the synchronization. How to apply the Lyapunov stability theory to study the sufficient criterion for the synchronization is still an issue to be solved, as said in [6]: ‘‘Unfortunately at present there is no a general recipe to find a proper Lyapunov functional, which makes an analytical investigation of the stability conditions of the synchronization manifolds a formidable task’’. In addition, even though two identical power systems are placed in a synchronization scheme, a parameter mismatch between the systems and external disturbance on the systems often occur because of the inevitable perturbation in the practical synchronization process, which may destroy the synchronization [8,9]. Theoretically, studying the synchronization of two different chaotic systems is more challenging. This paper is thus motivated to systematically investigate global complete synchronization of two identical SMIB power systems and global robust synchronization of two SMIB power systems with parameter mismatch and external disturbance, both under the master–slave linear state-error feedback control. Based on the Lyapunov’s direct method [10], some algebraic criteria for achieving the synchronization via a single-variable linear coupling are derived by choosing the proper Lyapunov
∗ Corresponding address: Guangzhou Navy Academy 075, Shisha Lu, Shijing, Baiyun Qu, Guangzhou 510430, PR China. Tel.: +86 20 86407656; fax: +86 20 61695540. E-mail address: [email protected] (X. Wu). 1468-1218/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.10.009
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
1501
function and using special inequality techniques. These algebraic criteria are then optimized by means of the extremum principle of the continuous function to improve the performances of the criteria. All the obtained algebraic criteria are formulated in simple algebraic inequalities, so they can be conveniently used. Some examples are simulated to illustrate the effectiveness of the new algebraic criteria. The rest of the paper is organized as follows. Section 2 studies the sufficient criteria for global complete synchronization of the identical master–slave power systems. Section 3 studies the sufficient criteria for global robust synchronization of the power systems with parameter mismatch and external disturbance. Some numerical examples are given in Section 4 and the concluding remarks are described in Section 5. Notation. The notation ‖x‖ denotes the Euclidean norm of vector x, while ‖A‖ = A, where ρ(M ) is the spectral radius of matrix M.
ρ(AT A) is the spectral norm of matrix
2. Global complete synchronization of two identical power systems Consider a classical SMIB power system described as follows [6]: M (d2 θ /dt 2 ) + D(dθ /dt ) + Pmax sin θ = Pm , where M is the moment of inertia, D is the damping constant, Pmax is the maximum power of generator, Pm = l sin ωt is the power of the machine. Let x1 = θ , x2 = dθ /dt , c = D/M , β = Pmax /M , h = l/M. Then the power system can be rewritten as: x˙ = Ax + f (x) + m(t ), where x = (x1 , x2 )T , and
[
]
0 A= 0
[
1 , −c
]
0 f ( x) = , −β sin x1
[
]
0 m(t ) = . h sin ωt
(1)
All the parameters c , β, ω, and h are positive. Two identical power systems can constitute a master–slave synchronization scheme via a linear state-error feedback controller u(t ), as follows: Master: x˙ = Ax + f (x) + m(t ), Slave: z˙ = Az + f (z ) + m(t ) + u(t ), Controller: u(t ) = K (x − z ),
(2)
where the state variables x, z ∈ R2 , and K ∈ R2×2 is a constant matrix to be determined, referred to as the coupling matrix. The task here is to design a constant coupling matrix K to achieve global complete synchronization in the sense that for the initial conditions x(0) leading the master system to chaos and any initial conditions z (0) of the slave system, the trajectories x(t ) and z (t ) satisfy lim ‖x(t ) − z (t )‖ = 0.
t →∞
(3)
Defining an error variable e = x − z, one can obtain a dynamical error system: e˙ = (A − K )e + f (x) − f (z ) = (A − K )e + Q (t )e = (A − K + Q (t ))e
(4)
with Q (t ) =
[
0 q(t )
]
0 , 0
q(t ) = −β(sin x1 − sin z1 )/(x1 − z1 ).
(5) (6)
Clearly, scheme (2) achieves global complete synchronization in the sense of (3) provided that the error system (4) is globally uniformly asymptotically stable at e = 0. The following result gives a criterion for the global complete synchronization, which is formulated in a linear matrix inequality (LMI). Theorem 1 ([11]). If there exists a symmetric and positive definite matrix P ∈ R2×2 and a constant coupling matrix K ∈ R2×2 such that for any t ≥ 0,
(A − K + Q (t ))T P + P (A − K + Q (t )) < 0,
(7)
then the master–slave scheme (2) achieves global complete synchronization. Remark 1. Numerical algorithms have been developed to solve the LMI synchronization criterion as (7) [12]. But the algebraic synchronization criteria, as shown in (9) and (15), are specially expected in applications because they can be used conveniently to design the synchronization controller and analyze the synchronizability of the scheme on certain conditions. In what follows, some such algebraic synchronization criteria will be derived.
1502
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
Let us consider a synchronization scheme with the single-variable coupling matrix K = diag{k, 0}, which implies that only a pair of state variables (x1 , z1 ) of the master–slave systems is used in synchronization process, which are linearly fed into the first differential equation of the slave system. Such a coupling thus simplifies the synchronization scheme and reduces the synchronization cost. A known result is that for q(t ) defined in (6), one has [11]
|q(t )| ≤ β,
∀t ≥ 0 .
(8)
Theorem 2. If there exists a single-variable coupling matrix K = diag{k, 0} and a constant µ > 0 such that k > g (µ) = (µβ + 1)2 /4µc ,
(9)
then the master–slave scheme (2) achieves global complete synchronization. Proof. Take a positive definite diagonal matrix P = diag{1, µ} with µ > 0. Then one has
(A − K + Q (t ))T P + P (A − K + Q (t )) =
[
−2k 1 + µq(t )
1 + µq(t ) . −2µc
]
The above symmetric matrix is negative definite if and only if for any t ≥ 0,
− 2k < 0
(10)
4µck − µ2 q2 (t ) − 2µq(t ) − 1 > 0.
(11)
and
Inequality (11) can be rewritten as: k > (µq(t ) + 1)2 /4µc .
(12)
It follows from (8) that inequalities (10) and (12) hold for any t ≥ 0 provided that condition (9) is satisfied.
Remark 2. The synchronization condition (9) depends on the constant µ > 0, which must be selected in applications. Theoretically and also practically, it is always expected that the derived synchronization criterion can provide a precise value of the coupling coefficients k as close as possible to the value determined by the necessary synchronization condition (if any). Such a criterion is regarded as being sharp [13]. Thus, in order to improve the sharpness of criterion (9), one should select a suitable µ such that g (µ) defined by (9) is as small as possible, which can be achieved by the following lemma.
√
√
Lemma 1. Let a and b be positive constants. Then the function δ(u) = au + b/u has minimal value δm = δ( b/a) = 2 ab for variable u > 0. Based on Lemma 1, the function g (µ) defined by (9) can be minimized, which leads to a sharper criterion as follows. Theorem 3. If there exists a single-variable coupling matrix K = diag{k, 0} such that k > β/c ,
(13)
then the master–slave scheme (2) achieves complete global synchronization. In what follows, we consider a dislocated single-variable coupling matrix:
[ K =
0 k
]
0 , 0
(14)
which implies that the state variables (x1 , z1 ) of the master–slave systems are linearly fed into the second equation of the slave system. Theorem 4. If there exists a dislocated single-variable coupling matrix K defined by (14) such that k > β,
(15)
then the master–slave scheme (2) achieves complete global synchronization. Proof. Take a symmetric and positive definite matrix: P =
[ µ1 µ2
µ2 1
]
with µ1 > µ22 and 0 < µ2 ≤ c .
(16)
Then one has
(A − K + Q (t ))T P + P (A − K + Q (t )) =
[
2µ2 (q(t ) − k) µ1 − µ2 c + (q(t ) − k)
] µ1 − µ2 c + (q(t ) − k) . 2µ2 − 2c
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
1503
The above symmetric matrix is negative definite if and only if for any t ≥ 0,
µ2 (q(t ) − k) < 0
(17)
(q(t ) − k)2 + (q(t ) − k) 2µ1 + 2c µ2 − 4µ22 + (µ1 − µ2 c )2 < 0.
(18)
and
For the matrix P defined in (16), inequality (17) has the solution: k − q(t ) > 0,
(19)
and inequality (18) has the solution:
− s1 < k − q(t ) < −s2 , 2 2 2 2 2 2 with s1 = − (µ1 − µ2 ) − (c µ2 − µ2 ) , s2 = − (µ1 − µ2 ) + (c µ2 − µ2 ) , and s2 ≤ s1 ≤ 0.
(20)
Again, if inequality (19) holds for any t ≥ 0, one can always find the proper µ1 and µ2 defined in (16) such that inequality (20) is satisfied for any t ≥ 0. It follows from (8) that inequality (19) holds for any t ≥ 0 if condition (15) is satisfied. 3. Global robust synchronization of two nonidentical power systems Consider a synchronization scheme for the master–slave power systems with parameter mismatches and external disturbances, as follows: Master: x˙ = Ax + f (x) + m(t ), Slave: z˙ = (A + 1A(t ))z + f ′ (z ) + m′ (t ) + d(t ) + u(t ), Controller: u(t ) = K (x − z ),
(21)
where, K ∈ R2×2 is a coupling matrix to be determined, the matrix A and the vector functions f (x) and m(t ) are defined by (1), and
[
0 1A(t ) = 0 m′ (t ) =
[
]
0 , −1c (t )
f (z ) = ′
]
0
(h + 1h(t )) sin ωt
,
[
]
0
−(β + 1β(t )) sin z1 [ ] d1 (t ) d(t ) = ∈ R2 . d2 (t )
,
(22)
Here there exist parameter mismatches 1c (t ), 1β(t ), and 1h(t ), between the master system and the slave one, and an external disturbance d(t ). Assume that all the parameter mismatches and external disturbances are time-varied and bounded, i.e. for any t ≥ 0, ‖1c (t )‖ ≤ δ1 , ‖1β(t )‖ ≤ δ2 , ‖1h(t )‖ ≤ δ3 , and ‖d(t )‖ ≤ δ4 , with the positive constants δi , i = 1, 2, 3, 4. Defining an error variable e = x − z, one can obtain an error-dynamic system: e˙ = (A − K )e − 1A(t )z + f (x) − f ′ (z ) + m(t ) − m′ (t ) − d(t )
= (A + 1A(t ) − K + Q (t ))e + Q1 (t ) − 1A(t )x + M (t ) + D(t ),
(23)
where Q (t ) is defined by (5), and
[
]
0 Q1 (t ) = , 1β sin z1
[
]
0 M (t ) = m(t ) − m (t ) = , −1h sin ωt ′
D(t ) = −d(t ).
(24)
For the synchronization scheme (21), an ideal objective is to achieve global complete synchronization in the sense of (3) by choosing the suitable coupling matrix K . However, such an objective cannot be realized for the non-identical master–slave systems. Therefore, a new concept of synchronization, global robust synchronization with error bounds, should be introduced and described in the following. Definition 1. Scheme (21) achieves global robust synchronization with the bounded error σ if for any finite initial states (x(0), z (0)), there exists a (small) real constant σ > 0 and a finite time T ≥ 0 such that ‖x(t ) − z (t )‖ = ‖e(t )‖ ≤ σ for all t ≥ T. The definition above suggests a synchronization for which the trajectory of error system (23) will globally uniformly converge into a small circle Cσ = {e : ||e|| = σ }. The following assumption is based on the bounded characteristic of chaotic attractors. Assumption 1. The chaotic trajectory of the master power system is bounded, i.e. there exist real constants ρi > 0, (i = 1, 2) such that for any initial condition x0 = x(0), there exists a time T (x0 ) for which
||xi (t , x0 )|| ≤ ρi (i = 1, 2), ∀t ≥ T (x0 ).
(25)
1504
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
Fig. 1. Illustration of the balls B1 , B2 and the ellipsoid E (a1 ) with respect to Theorem 5.
Theorem 5. If there exists a symmetric and positive definite matrix P ∈ R2×2 , a constant coupling matrix K ∈ R2×2 and a positive real number θ > δ1 λmax (P ) such that for any t ≥ 0, P (A − K + Q (t )) + (A − K + Q (t ))T P + 2θ I2 ≤ 0,
(26)
then scheme (21) achieves global robust synchronization with the bounded error σ , where
σ ≤ σ0 (P , θ ) = (ϕλmax (P )/(θ − δ1 λmax (P ))) λmax (P )/λmin (P ),
(27)
ϕ = (δ1 ρ2 + δ2 + δ3 + δ4 ).
(28)
ρ2 is defined by (25), λmax (P ) and λmin (P ) are the maximal and minimal eigenvalue of matrix P, respectively. Proof. Choose a positive definite, decrescent and radially unbounded Lyapunov function as follows: V = eT Pe,
P = P T > 0.
Then by (26), the time derivative of V along the orbits of error system (23) is V˙ = ((A + 1A(t ) − K + Q (t ))e + Q1 (t ) − 1A(t )x + M (t ) + D(t ))T Pe
= ≤ ≤ ≤
+ eT P ((A + 1A(t ) − K + Q (t ))e + Q1 (t ) − 1A(t )x + M (t ) + D(t )) eT ((A − K + Q (t ))T P + P (A − K + Q (t )) + 2θ I2 )e − 2θ eT e + 2eT P 1A(t )e + 2eT P (Q1 (t ) + M (t ) + D(t ) − 1A(t )x) −2θ eT e + 2eT P 1A(t )e + 2eT P (Q1 (t ) + M (t ) + D(t ) − 1A(t )x) −2θ eT e + 2‖e‖2 ‖1A(t )‖ ‖P ‖ + 2‖e‖ ‖P ‖ ‖Q1 (t ) + M (t ) + D(t ) − 1A(t )x‖ −2θ‖e‖2 + 2‖e‖2 ‖1A(t )‖ ‖P ‖ + 2‖e‖ ‖P ‖(‖Q1 (t )‖ + ‖M (t )‖ + ‖D(t )‖ + ‖1A(t )x‖).
For 1A(t ), Q1 (t ), M (t ) and D(t ), defined in (22) and (24), one has
‖1A(t )‖ = ‖1c (t )‖ ≤ δ1 , ‖Q1 (t )‖ ≤ ‖1β(t )‖ ≤ δ2 , ‖M (t )‖ ≤ ‖1h(t )‖ ≤ δ3 , ‖D(t )‖ = ‖−d(t )‖ ≤ δ4 , ‖1A(t )x‖ = ‖1c (t )x2 ‖ = ‖1c (t )‖ ‖x2 ‖ ≤ δ1 ρ2 , where ρ2 is defined by (25). Thus, V˙ ≤ (−2θ + 2δ1 ‖P ‖)‖e‖2 + 2‖e‖ ‖P ‖(δ1 ρ2 + δ2 + δ3 + δ4 ) = 2‖e‖(ϕλmax (P ) − (θ − δ1 λmax (P ))‖e‖) where λmax (P ) and ϕ are defined above. Let r = ϕλmax (P ) and α = θ − δ1 λmax (P ). It is clear that α > 0 for the given condition. Hence V˙ (e) < 0 provided that ‖e‖ > r /α , i.e. V˙ (e) < 0 for any e outside of ball B1 with B1 = e : eT e ≤ r 2 /α 2 ,
as seen in Fig. 1. Thus, if a1 > 0 is selected such that the ellipsoid E (a1 ) = e : eT Pe ≤ a1 ⊃ B1 = e : eT e ≤ r 2 /α 2 ,
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
1505
and is the smallest, then for any e outside of the ellipsoid E (a1 ), it can be confirmed that V˙ (e) < 0 and the trajectory of the error system (23) will uniformly enter the ellipsoid E (a1 ) for any finite initial condition e(0). If a2 > 0 is determined such that the ball B2 (a2 ) = e : eT e ≤ a2 ⊃ E (a1 ),
√
and is the smallest, then the radius a2 of ball B2 (a2 ) is the synchronization error bound which one is able to find out. Since matrix P is positive definite, there exists an orthogonal transformation F ∈ R2×2 such that for a new vector y = (y1 , y2 )T ∈ R2 and e = Fy, e Pe = λ T
2 1 y1
+λ
2 2 y2
=
y21
2 2 2 1 / λ2 , 1/ λ1 + y2
where λi > 0 (i = 1, 2) are the eigenvalues of matrix P. Let λmax (P ) = max{λi , i = 1, 2} and λmin (P ) = min {λi , i = 1, 2}. Then the maximal radius Rmax of ellipsoid E (a1 ) equals Rmax =
√
a1 / λmin (P ) =
√
a2 ,
and the minimal radius Rmin of the ellipsoid E (a1 ) equals Rmin =
√
a1 / λmax (P ) = r /α.
Thus, one has
√
a2 = (r /α) λmax (P )/λmin (P ) = σ0 (P , θ ).
The proof is completed.
Remark 3. It is called σ0 (P , θ ) defined by (27) the estimated synchronization error bound. In applications, ones always expect that the estimated synchronization error bound approaches the real one. Since σ0 (P , θ ) depends upon the choice of matrix P and constant θ , an optimization issue is thus suggested as follows: select matrix P and constant θ such that the synchronization conditions (26) are satisfied and function σ0 (P , θ ) is as small as possible.
p11 p12
p12 p22
> 0, it is easily known that the eigenvalues of the matrix P equal: λ1 (P ) = p11 + p22 + (p11 − p22 )2 + 4p212 2, λ2 (P ) = p11 + p22 − (p11 − p22 )2 + 4p212 2,
For P =
with λ1 > λ2 . Thus, it follows from Eq. (27) that p12 = 0 is necessary to the minimal σ0 (P , θ ). Let µ = p22 /p11 with p22 > 0 and p11 > 0. Then, σ0 (P , θ ) with p12 = 0 can be represented as
µ−1/2 ϕ p11 /(θ − δ1 p11 ) σ0 (P , θ ) = ϕ p11 /(θ − δ1 p11 ) 3/2 µ ϕ p11 /(θ − δ1 µp11 )
0 < µ < 1, µ = 1, µ > 1.
Obviously the above σ0 (P , θ ) is continuous for µ and takes the minimal value at µ = 1. Hence, a necessary condition for the minimal σ0 (P , θ ) is that P = pI2 with p > 0. Based on such a matrix P, an optimized synchronization criterion can be directly obtained from Theorem 5 and described in the following. Theorem 6. If there exists a constant coupling matrix K ∈ R2×2 and a positive real number s(= θ /p) > δ1 such that for any t ≥ 0,
(A − K + Q (t )) + (A − K + Q (t ))T + 2sI2 ≤ 0,
(29)
then the master–slave scheme (21) achieves global robust synchronization with the bounded error σ , where σ ≤ σ0 (s) = ϕ/(s − δ1 ) and ϕ is defined by (28). For the single-variable coupling matrix K = diag{k, 0}, a sharp algebraic criterion for the synchronization can be derived from Theorem 6 and described as follows. Theorem 7. If c > δ1 and the coupling matrix K = diag{k, 0} is selected such that for the positive real number s ∈ (δ1 , c ), k ≥ s + (1 + β)2 /(4(c − s)),
(30)
then scheme (21) achieves global robust synchronization with the bounded error σ , where σ ≤ σ0 (s) = ϕ/(s − δ1 ) and ϕ is defined by (28).
1506
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
Proof. The symmetric matrix defined by (29) can be rewritten as
[ −2k + 2s (A − K + Q (t )) + (A − K + Q (t )) + 2sI2 = 1 + q(t ) T
1 + q(t ) . −2c + 2s
]
It is required that s > δ1 according to Theorem 6. The above matrix is semi-negative definite if and only if for any t ≥ 0 and s > δ1 ,
k > s, c > s, 4(−k + s)(−c + s) − (1 + q(t ))2 ≥ 0,
(31)
or, equivalently, k ≥ s + (1 + q(t ))2 /(4(c − s))
(32)
with δ1 < s < c. It follows from (8) that inequality (32) holds for any t ≥ 0 if condition (30) is satisfied.
Remark 4. In Theorem 7, the estimated synchronization error bound σ0 satisfies that +∞ > σ0 > ϕ/(c − δ1 ). On the other hand, formula (30) can be transformed and represented as k ≥ ϕ/σ0 + δ1 + (1 + β)2 /(4(c − δ1 − ϕ/σ0 )),
(33)
which implies that the magnitude of the gain k resulting in the synchronization is inversely proportional to that of the estimated synchronization error bound σ0 , and the gain k approaches the infinity as σ0 → ϕ/(c − δ1 ). 4. Numerical examples In this section, two examples are given to illustrate the effectiveness of the obtained synchronization criteria. First consider a master–slave synchronization scheme consisting of two identical SMIB power systems via a dislocated single-variable coupling matrix K defined by (14) where the parameters of the power system are c = 0.5, β = 1, ω = 1 and h = 2.45. The master power system is chaotic, as shown in Fig. 2(a). The synchronization condition k > 1 for such a coupling matrix can be obtained by means of criterion (15). Fig. 2(b) shows the evolution of complete synchronization with k = 1.1. The second example deals with a master–slave synchronization scheme where there exist parameter mismatches between two systems and external disturbance on the slave system. The orbits of the master SMIB power system is illustrated in Fig. 2(a), which has the bound ρ2 = 4. The parameter mismatches between the two systems and the external disturbance on the slave system are
1β = 0.06 sin t ,
1h = 0.05 sin 2t ,
d = (0.04 sin 2t , 0)T ,
respectively. When the initial states of the master–slave systems are chosen as x(0) = (−1.5, 1.1) and z (0) = (−7, 4), the trajectories of the master system and the uncontrolled slave one separate randomly and remarkably in the course of time, as shown in Fig. 3. Suppose that the coupling K = diag{k, 0} is applied to the synchronization scheme. Then one can obtain a synchronization condition by criterion (30), as follows: K = diag{k, 0},
k ≥ 1/(0.5 − s) with s ∈ (0, 0.5).
(34)
Taking s = 0.1, one can obtain a synchronization condition k = 2.5 by (34). The simulation verifies that it is possible to synchronize two systems with parameter mismatches and external disturbance via such a coupling up to a small bounded error, which is shown in Fig. 4. In order to illustrate the difference between the estimated synchronization error bound and the real one, take the following coupling matrix: K = diag{k, 0},
k = 1/(0.5 − s) with s ∈ (0, 0.5).
(35)
Let ‖e‖T = maxt ≥T (x1 − z1 ) + (x2 − z2 ) be the real synchronization error bound where T represents a time threshold after which the synchronization error will converge and stabilize in a small area (in the practical simulation, T = 1000 time unit). Choose a s ∈ (0, 0.5). The coupling gain can then be determined by (35). Thus the real synchronization error bound ‖e‖T can be numerically solved based on the master–slave synchronization scheme (21) with such a coupling gain, and the estimated synchronization error bound σ0 can be determined by Theorem 7. According to the method, the curve of the real synchronization error bound and that of the estimated one can be obtained, varied with the parameter s ∈ (0, 0.5), as shown in Fig. 5, which reveals that the difference between the real synchronization error bound and the estimated one is inversely proportional to the value of s ∈ (0, 0.5).
2
2
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
1507
a
b
Fig. 2. Complete synchronization of the master–slave identical power systems. (a) The master system with c = 0.5, β = 1, ω = 1 and h = 2.45; (b) the synchronization evolution for K defined by (14) with k = 1.1, where initial conditions are x(0) = (−1.5, 1.1) and z (0) = (−10, −5).
Fig. 3. The remarkable difference of the trajectories of the isolated master–slave power systems with parameter mismatch and external disturbance, where the initial conditions are x(0) = (−1.5, 1.1) and z (0) = (−7, 4).
5. Conclusion The LMI criterion for global complete synchronization of the master–slave identical power systems under linear stateerror feedback control and that for global robust synchronization of the master–slave power systems with parameter
1508
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
Fig. 4. Synchronization with the bounded error of scheme (21) with K = diag{2.5, 0}, where the initial conditions are x(0) = (−1.5, 1.1) and z (0) = (−7, 4).
Fig. 5. The estimated synchronization error bound and the numerical (real) one for the synchronization condition K = diag{k, 0}, k = 1/(0.5 − s) with s ∈ (0, 0.5). The curves plotted by ‘‘∗’’ and ‘‘.’’ represent the estimated synchronization error bound and numerical one, respectively.
mismatches and external disturbance under the same control have been proposed. Some algebraic criteria for the synchronization via a single-variable linear coupling have been derived. The obtained algebraic criteria are further optimized to improve their performances. The effectiveness of the new criteria is illustrated by numerical examples. Acknowledgement This wok was supported by the National Nature Science Foundation of China under grant No. 60674049. References [1] H.K. Chen, T.N. Lin, J.H. Chen, Dynamic analysis, controlling chaos and chaotification of a SMIB power system, Chaos, Solitons and Fractals 24 (2005) 1307–1315. [2] H. Saadat, Power System Analysis, McGraw Hill, 1999. [3] A. Çela, Y. Hamam, J.R. Nielsen, L.T. Christensen, Nonlinear approach to the control of multimachine power systems, Nonlinear Analysis 30 (1997) 4301–4308. [4] A.M. Harb, M.S. Widyan, Chaos and bifurcation control of SSR in the IEEE second benchmark model, Chaos, Solitons and Fractals 21 (2004) 537–552. [5] A.M. Harb, N. Abeld-Jabbar, Controlling Hopf bifurcation and chaos in a small power system, Chaos, Solitons and Fractals 18 (2003) 1055–1063. [6] E.M. Shahverdiev, L.H. Hashimova, N.T. Hashimova, Chaos synchronization in some power systems, Chaos, Solitons and Fractals 37 (2008) 827–834. [7] S. Boccaletti, J. Kurths, G. Osipov, et al., The synchronization of chaotic systems, Physics Reports 336 (2002) 1–101. [8] X.F. Wu, M.H. Wang, Robust synchronization of chaotic Lur’e systems via replacing variables control, International Journal of Bifurcation and Chaos 16 (2006) 3421–3433. [9] P.F. Curran, L.O. Chua, Absolute stability theory and the synchronization problem, International Journal of Bifurcation and Chaos 7 (6) (1997) 1375–1382.
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
1509
[10] J.E. Slotine, W.P. Li, Applied Nonlinear Control, China Machine Press, Beijing, 2004. [11] X.F. Wu, J.P. Cai, M.H. Wang, Master–slave chaos synchronization criteria for the horizontal platform systems via linear state error feedback control, Journal of Sound and Vibration 295 (2006) 378–387. [12] F.X. Chen, W.D. Zhang, LMI criteria for robust chaos synchronization of a class of chaotic systems, Nonlinear Analysis 67 (2007) 4301–4308. [13] X.F. Wu, G.R. Chen, J.P. Cai, Chaos synchronization of the master–slave generalized Lorenz systems via linear state error feedback control, Physica D 229 (2007) 52–80.
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
The sufficient criteria for global synchronization of chaotic power systems under linear state-error feedback control Qian Lin a , Xiaofeng Wu a,b,∗ a
College of Electronic Engineering, Naval University of Engineering, Wuhan 430033, PR China
b
Guangzhou Navy Academy, Guangzhou 510430, PR China
article
info
Article history: Received 3 December 2008 Accepted 20 October 2010 Keywords: Chaos Synchronization Linear state-error feedback control Power system
abstract This paper investigates global complete synchronization of two identical power systems and global robust synchronization of two power systems with parameter mismatch and external disturbance, both under the master–slave linear state-error feedback control. Some criteria for achieving the synchronization via a single-variable linear coupling are derived and formulated in simple algebraic inequalities. These algebraic criteria are further optimized so as to improve their performances. The effectiveness of the new algebraic criteria is illustrated by the numerical examples. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction The existing research has shown that electrical power systems are a class of nonlinear dynamical systems which can demonstrate very complex dynamics as chaos [1,2]. The emergent chaotic behavior of the power systems can produce an undesirable consequence such as a power blackout. In order to prevent the damage generated by the chaotic behavior, the control and synchronization of the chaotic power systems should be studied so as to manipulate and manage the systems [3–6]. The single-machine-infinite-bus (SMIB) system is a kind of non-autonomous electrical power system. Very recently, Shahverdiev et al. [6] investigated the synchronization of two identical chaotic SMIB systems via a single-variable linear coupling, where the coupling strength leading to synchronization was detected in terms of the negativity of the maximal Lyapunov exponent of an error system. However, as indicated in [6,7], the negative maximal Lyapunov exponent is only a necessary but not sufficient condition for the synchronization. How to apply the Lyapunov stability theory to study the sufficient criterion for the synchronization is still an issue to be solved, as said in [6]: ‘‘Unfortunately at present there is no a general recipe to find a proper Lyapunov functional, which makes an analytical investigation of the stability conditions of the synchronization manifolds a formidable task’’. In addition, even though two identical power systems are placed in a synchronization scheme, a parameter mismatch between the systems and external disturbance on the systems often occur because of the inevitable perturbation in the practical synchronization process, which may destroy the synchronization [8,9]. Theoretically, studying the synchronization of two different chaotic systems is more challenging. This paper is thus motivated to systematically investigate global complete synchronization of two identical SMIB power systems and global robust synchronization of two SMIB power systems with parameter mismatch and external disturbance, both under the master–slave linear state-error feedback control. Based on the Lyapunov’s direct method [10], some algebraic criteria for achieving the synchronization via a single-variable linear coupling are derived by choosing the proper Lyapunov
∗ Corresponding address: Guangzhou Navy Academy 075, Shisha Lu, Shijing, Baiyun Qu, Guangzhou 510430, PR China. Tel.: +86 20 86407656; fax: +86 20 61695540. E-mail address: [email protected] (X. Wu). 1468-1218/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.10.009
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
1501
function and using special inequality techniques. These algebraic criteria are then optimized by means of the extremum principle of the continuous function to improve the performances of the criteria. All the obtained algebraic criteria are formulated in simple algebraic inequalities, so they can be conveniently used. Some examples are simulated to illustrate the effectiveness of the new algebraic criteria. The rest of the paper is organized as follows. Section 2 studies the sufficient criteria for global complete synchronization of the identical master–slave power systems. Section 3 studies the sufficient criteria for global robust synchronization of the power systems with parameter mismatch and external disturbance. Some numerical examples are given in Section 4 and the concluding remarks are described in Section 5. Notation. The notation ‖x‖ denotes the Euclidean norm of vector x, while ‖A‖ = A, where ρ(M ) is the spectral radius of matrix M.
ρ(AT A) is the spectral norm of matrix
2. Global complete synchronization of two identical power systems Consider a classical SMIB power system described as follows [6]: M (d2 θ /dt 2 ) + D(dθ /dt ) + Pmax sin θ = Pm , where M is the moment of inertia, D is the damping constant, Pmax is the maximum power of generator, Pm = l sin ωt is the power of the machine. Let x1 = θ , x2 = dθ /dt , c = D/M , β = Pmax /M , h = l/M. Then the power system can be rewritten as: x˙ = Ax + f (x) + m(t ), where x = (x1 , x2 )T , and
[
]
0 A= 0
[
1 , −c
]
0 f ( x) = , −β sin x1
[
]
0 m(t ) = . h sin ωt
(1)
All the parameters c , β, ω, and h are positive. Two identical power systems can constitute a master–slave synchronization scheme via a linear state-error feedback controller u(t ), as follows: Master: x˙ = Ax + f (x) + m(t ), Slave: z˙ = Az + f (z ) + m(t ) + u(t ), Controller: u(t ) = K (x − z ),
(2)
where the state variables x, z ∈ R2 , and K ∈ R2×2 is a constant matrix to be determined, referred to as the coupling matrix. The task here is to design a constant coupling matrix K to achieve global complete synchronization in the sense that for the initial conditions x(0) leading the master system to chaos and any initial conditions z (0) of the slave system, the trajectories x(t ) and z (t ) satisfy lim ‖x(t ) − z (t )‖ = 0.
t →∞
(3)
Defining an error variable e = x − z, one can obtain a dynamical error system: e˙ = (A − K )e + f (x) − f (z ) = (A − K )e + Q (t )e = (A − K + Q (t ))e
(4)
with Q (t ) =
[
0 q(t )
]
0 , 0
q(t ) = −β(sin x1 − sin z1 )/(x1 − z1 ).
(5) (6)
Clearly, scheme (2) achieves global complete synchronization in the sense of (3) provided that the error system (4) is globally uniformly asymptotically stable at e = 0. The following result gives a criterion for the global complete synchronization, which is formulated in a linear matrix inequality (LMI). Theorem 1 ([11]). If there exists a symmetric and positive definite matrix P ∈ R2×2 and a constant coupling matrix K ∈ R2×2 such that for any t ≥ 0,
(A − K + Q (t ))T P + P (A − K + Q (t )) < 0,
(7)
then the master–slave scheme (2) achieves global complete synchronization. Remark 1. Numerical algorithms have been developed to solve the LMI synchronization criterion as (7) [12]. But the algebraic synchronization criteria, as shown in (9) and (15), are specially expected in applications because they can be used conveniently to design the synchronization controller and analyze the synchronizability of the scheme on certain conditions. In what follows, some such algebraic synchronization criteria will be derived.
1502
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
Let us consider a synchronization scheme with the single-variable coupling matrix K = diag{k, 0}, which implies that only a pair of state variables (x1 , z1 ) of the master–slave systems is used in synchronization process, which are linearly fed into the first differential equation of the slave system. Such a coupling thus simplifies the synchronization scheme and reduces the synchronization cost. A known result is that for q(t ) defined in (6), one has [11]
|q(t )| ≤ β,
∀t ≥ 0 .
(8)
Theorem 2. If there exists a single-variable coupling matrix K = diag{k, 0} and a constant µ > 0 such that k > g (µ) = (µβ + 1)2 /4µc ,
(9)
then the master–slave scheme (2) achieves global complete synchronization. Proof. Take a positive definite diagonal matrix P = diag{1, µ} with µ > 0. Then one has
(A − K + Q (t ))T P + P (A − K + Q (t )) =
[
−2k 1 + µq(t )
1 + µq(t ) . −2µc
]
The above symmetric matrix is negative definite if and only if for any t ≥ 0,
− 2k < 0
(10)
4µck − µ2 q2 (t ) − 2µq(t ) − 1 > 0.
(11)
and
Inequality (11) can be rewritten as: k > (µq(t ) + 1)2 /4µc .
(12)
It follows from (8) that inequalities (10) and (12) hold for any t ≥ 0 provided that condition (9) is satisfied.
Remark 2. The synchronization condition (9) depends on the constant µ > 0, which must be selected in applications. Theoretically and also practically, it is always expected that the derived synchronization criterion can provide a precise value of the coupling coefficients k as close as possible to the value determined by the necessary synchronization condition (if any). Such a criterion is regarded as being sharp [13]. Thus, in order to improve the sharpness of criterion (9), one should select a suitable µ such that g (µ) defined by (9) is as small as possible, which can be achieved by the following lemma.
√
√
Lemma 1. Let a and b be positive constants. Then the function δ(u) = au + b/u has minimal value δm = δ( b/a) = 2 ab for variable u > 0. Based on Lemma 1, the function g (µ) defined by (9) can be minimized, which leads to a sharper criterion as follows. Theorem 3. If there exists a single-variable coupling matrix K = diag{k, 0} such that k > β/c ,
(13)
then the master–slave scheme (2) achieves complete global synchronization. In what follows, we consider a dislocated single-variable coupling matrix:
[ K =
0 k
]
0 , 0
(14)
which implies that the state variables (x1 , z1 ) of the master–slave systems are linearly fed into the second equation of the slave system. Theorem 4. If there exists a dislocated single-variable coupling matrix K defined by (14) such that k > β,
(15)
then the master–slave scheme (2) achieves complete global synchronization. Proof. Take a symmetric and positive definite matrix: P =
[ µ1 µ2
µ2 1
]
with µ1 > µ22 and 0 < µ2 ≤ c .
(16)
Then one has
(A − K + Q (t ))T P + P (A − K + Q (t )) =
[
2µ2 (q(t ) − k) µ1 − µ2 c + (q(t ) − k)
] µ1 − µ2 c + (q(t ) − k) . 2µ2 − 2c
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
1503
The above symmetric matrix is negative definite if and only if for any t ≥ 0,
µ2 (q(t ) − k) < 0
(17)
(q(t ) − k)2 + (q(t ) − k) 2µ1 + 2c µ2 − 4µ22 + (µ1 − µ2 c )2 < 0.
(18)
and
For the matrix P defined in (16), inequality (17) has the solution: k − q(t ) > 0,
(19)
and inequality (18) has the solution:
− s1 < k − q(t ) < −s2 , 2 2 2 2 2 2 with s1 = − (µ1 − µ2 ) − (c µ2 − µ2 ) , s2 = − (µ1 − µ2 ) + (c µ2 − µ2 ) , and s2 ≤ s1 ≤ 0.
(20)
Again, if inequality (19) holds for any t ≥ 0, one can always find the proper µ1 and µ2 defined in (16) such that inequality (20) is satisfied for any t ≥ 0. It follows from (8) that inequality (19) holds for any t ≥ 0 if condition (15) is satisfied. 3. Global robust synchronization of two nonidentical power systems Consider a synchronization scheme for the master–slave power systems with parameter mismatches and external disturbances, as follows: Master: x˙ = Ax + f (x) + m(t ), Slave: z˙ = (A + 1A(t ))z + f ′ (z ) + m′ (t ) + d(t ) + u(t ), Controller: u(t ) = K (x − z ),
(21)
where, K ∈ R2×2 is a coupling matrix to be determined, the matrix A and the vector functions f (x) and m(t ) are defined by (1), and
[
0 1A(t ) = 0 m′ (t ) =
[
]
0 , −1c (t )
f (z ) = ′
]
0
(h + 1h(t )) sin ωt
,
[
]
0
−(β + 1β(t )) sin z1 [ ] d1 (t ) d(t ) = ∈ R2 . d2 (t )
,
(22)
Here there exist parameter mismatches 1c (t ), 1β(t ), and 1h(t ), between the master system and the slave one, and an external disturbance d(t ). Assume that all the parameter mismatches and external disturbances are time-varied and bounded, i.e. for any t ≥ 0, ‖1c (t )‖ ≤ δ1 , ‖1β(t )‖ ≤ δ2 , ‖1h(t )‖ ≤ δ3 , and ‖d(t )‖ ≤ δ4 , with the positive constants δi , i = 1, 2, 3, 4. Defining an error variable e = x − z, one can obtain an error-dynamic system: e˙ = (A − K )e − 1A(t )z + f (x) − f ′ (z ) + m(t ) − m′ (t ) − d(t )
= (A + 1A(t ) − K + Q (t ))e + Q1 (t ) − 1A(t )x + M (t ) + D(t ),
(23)
where Q (t ) is defined by (5), and
[
]
0 Q1 (t ) = , 1β sin z1
[
]
0 M (t ) = m(t ) − m (t ) = , −1h sin ωt ′
D(t ) = −d(t ).
(24)
For the synchronization scheme (21), an ideal objective is to achieve global complete synchronization in the sense of (3) by choosing the suitable coupling matrix K . However, such an objective cannot be realized for the non-identical master–slave systems. Therefore, a new concept of synchronization, global robust synchronization with error bounds, should be introduced and described in the following. Definition 1. Scheme (21) achieves global robust synchronization with the bounded error σ if for any finite initial states (x(0), z (0)), there exists a (small) real constant σ > 0 and a finite time T ≥ 0 such that ‖x(t ) − z (t )‖ = ‖e(t )‖ ≤ σ for all t ≥ T. The definition above suggests a synchronization for which the trajectory of error system (23) will globally uniformly converge into a small circle Cσ = {e : ||e|| = σ }. The following assumption is based on the bounded characteristic of chaotic attractors. Assumption 1. The chaotic trajectory of the master power system is bounded, i.e. there exist real constants ρi > 0, (i = 1, 2) such that for any initial condition x0 = x(0), there exists a time T (x0 ) for which
||xi (t , x0 )|| ≤ ρi (i = 1, 2), ∀t ≥ T (x0 ).
(25)
1504
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
Fig. 1. Illustration of the balls B1 , B2 and the ellipsoid E (a1 ) with respect to Theorem 5.
Theorem 5. If there exists a symmetric and positive definite matrix P ∈ R2×2 , a constant coupling matrix K ∈ R2×2 and a positive real number θ > δ1 λmax (P ) such that for any t ≥ 0, P (A − K + Q (t )) + (A − K + Q (t ))T P + 2θ I2 ≤ 0,
(26)
then scheme (21) achieves global robust synchronization with the bounded error σ , where
σ ≤ σ0 (P , θ ) = (ϕλmax (P )/(θ − δ1 λmax (P ))) λmax (P )/λmin (P ),
(27)
ϕ = (δ1 ρ2 + δ2 + δ3 + δ4 ).
(28)
ρ2 is defined by (25), λmax (P ) and λmin (P ) are the maximal and minimal eigenvalue of matrix P, respectively. Proof. Choose a positive definite, decrescent and radially unbounded Lyapunov function as follows: V = eT Pe,
P = P T > 0.
Then by (26), the time derivative of V along the orbits of error system (23) is V˙ = ((A + 1A(t ) − K + Q (t ))e + Q1 (t ) − 1A(t )x + M (t ) + D(t ))T Pe
= ≤ ≤ ≤
+ eT P ((A + 1A(t ) − K + Q (t ))e + Q1 (t ) − 1A(t )x + M (t ) + D(t )) eT ((A − K + Q (t ))T P + P (A − K + Q (t )) + 2θ I2 )e − 2θ eT e + 2eT P 1A(t )e + 2eT P (Q1 (t ) + M (t ) + D(t ) − 1A(t )x) −2θ eT e + 2eT P 1A(t )e + 2eT P (Q1 (t ) + M (t ) + D(t ) − 1A(t )x) −2θ eT e + 2‖e‖2 ‖1A(t )‖ ‖P ‖ + 2‖e‖ ‖P ‖ ‖Q1 (t ) + M (t ) + D(t ) − 1A(t )x‖ −2θ‖e‖2 + 2‖e‖2 ‖1A(t )‖ ‖P ‖ + 2‖e‖ ‖P ‖(‖Q1 (t )‖ + ‖M (t )‖ + ‖D(t )‖ + ‖1A(t )x‖).
For 1A(t ), Q1 (t ), M (t ) and D(t ), defined in (22) and (24), one has
‖1A(t )‖ = ‖1c (t )‖ ≤ δ1 , ‖Q1 (t )‖ ≤ ‖1β(t )‖ ≤ δ2 , ‖M (t )‖ ≤ ‖1h(t )‖ ≤ δ3 , ‖D(t )‖ = ‖−d(t )‖ ≤ δ4 , ‖1A(t )x‖ = ‖1c (t )x2 ‖ = ‖1c (t )‖ ‖x2 ‖ ≤ δ1 ρ2 , where ρ2 is defined by (25). Thus, V˙ ≤ (−2θ + 2δ1 ‖P ‖)‖e‖2 + 2‖e‖ ‖P ‖(δ1 ρ2 + δ2 + δ3 + δ4 ) = 2‖e‖(ϕλmax (P ) − (θ − δ1 λmax (P ))‖e‖) where λmax (P ) and ϕ are defined above. Let r = ϕλmax (P ) and α = θ − δ1 λmax (P ). It is clear that α > 0 for the given condition. Hence V˙ (e) < 0 provided that ‖e‖ > r /α , i.e. V˙ (e) < 0 for any e outside of ball B1 with B1 = e : eT e ≤ r 2 /α 2 ,
as seen in Fig. 1. Thus, if a1 > 0 is selected such that the ellipsoid E (a1 ) = e : eT Pe ≤ a1 ⊃ B1 = e : eT e ≤ r 2 /α 2 ,
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
1505
and is the smallest, then for any e outside of the ellipsoid E (a1 ), it can be confirmed that V˙ (e) < 0 and the trajectory of the error system (23) will uniformly enter the ellipsoid E (a1 ) for any finite initial condition e(0). If a2 > 0 is determined such that the ball B2 (a2 ) = e : eT e ≤ a2 ⊃ E (a1 ),
√
and is the smallest, then the radius a2 of ball B2 (a2 ) is the synchronization error bound which one is able to find out. Since matrix P is positive definite, there exists an orthogonal transformation F ∈ R2×2 such that for a new vector y = (y1 , y2 )T ∈ R2 and e = Fy, e Pe = λ T
2 1 y1
+λ
2 2 y2
=
y21
2 2 2 1 / λ2 , 1/ λ1 + y2
where λi > 0 (i = 1, 2) are the eigenvalues of matrix P. Let λmax (P ) = max{λi , i = 1, 2} and λmin (P ) = min {λi , i = 1, 2}. Then the maximal radius Rmax of ellipsoid E (a1 ) equals Rmax =
√
a1 / λmin (P ) =
√
a2 ,
and the minimal radius Rmin of the ellipsoid E (a1 ) equals Rmin =
√
a1 / λmax (P ) = r /α.
Thus, one has
√
a2 = (r /α) λmax (P )/λmin (P ) = σ0 (P , θ ).
The proof is completed.
Remark 3. It is called σ0 (P , θ ) defined by (27) the estimated synchronization error bound. In applications, ones always expect that the estimated synchronization error bound approaches the real one. Since σ0 (P , θ ) depends upon the choice of matrix P and constant θ , an optimization issue is thus suggested as follows: select matrix P and constant θ such that the synchronization conditions (26) are satisfied and function σ0 (P , θ ) is as small as possible.
p11 p12
p12 p22
> 0, it is easily known that the eigenvalues of the matrix P equal: λ1 (P ) = p11 + p22 + (p11 − p22 )2 + 4p212 2, λ2 (P ) = p11 + p22 − (p11 − p22 )2 + 4p212 2,
For P =
with λ1 > λ2 . Thus, it follows from Eq. (27) that p12 = 0 is necessary to the minimal σ0 (P , θ ). Let µ = p22 /p11 with p22 > 0 and p11 > 0. Then, σ0 (P , θ ) with p12 = 0 can be represented as
µ−1/2 ϕ p11 /(θ − δ1 p11 ) σ0 (P , θ ) = ϕ p11 /(θ − δ1 p11 ) 3/2 µ ϕ p11 /(θ − δ1 µp11 )
0 < µ < 1, µ = 1, µ > 1.
Obviously the above σ0 (P , θ ) is continuous for µ and takes the minimal value at µ = 1. Hence, a necessary condition for the minimal σ0 (P , θ ) is that P = pI2 with p > 0. Based on such a matrix P, an optimized synchronization criterion can be directly obtained from Theorem 5 and described in the following. Theorem 6. If there exists a constant coupling matrix K ∈ R2×2 and a positive real number s(= θ /p) > δ1 such that for any t ≥ 0,
(A − K + Q (t )) + (A − K + Q (t ))T + 2sI2 ≤ 0,
(29)
then the master–slave scheme (21) achieves global robust synchronization with the bounded error σ , where σ ≤ σ0 (s) = ϕ/(s − δ1 ) and ϕ is defined by (28). For the single-variable coupling matrix K = diag{k, 0}, a sharp algebraic criterion for the synchronization can be derived from Theorem 6 and described as follows. Theorem 7. If c > δ1 and the coupling matrix K = diag{k, 0} is selected such that for the positive real number s ∈ (δ1 , c ), k ≥ s + (1 + β)2 /(4(c − s)),
(30)
then scheme (21) achieves global robust synchronization with the bounded error σ , where σ ≤ σ0 (s) = ϕ/(s − δ1 ) and ϕ is defined by (28).
1506
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
Proof. The symmetric matrix defined by (29) can be rewritten as
[ −2k + 2s (A − K + Q (t )) + (A − K + Q (t )) + 2sI2 = 1 + q(t ) T
1 + q(t ) . −2c + 2s
]
It is required that s > δ1 according to Theorem 6. The above matrix is semi-negative definite if and only if for any t ≥ 0 and s > δ1 ,
k > s, c > s, 4(−k + s)(−c + s) − (1 + q(t ))2 ≥ 0,
(31)
or, equivalently, k ≥ s + (1 + q(t ))2 /(4(c − s))
(32)
with δ1 < s < c. It follows from (8) that inequality (32) holds for any t ≥ 0 if condition (30) is satisfied.
Remark 4. In Theorem 7, the estimated synchronization error bound σ0 satisfies that +∞ > σ0 > ϕ/(c − δ1 ). On the other hand, formula (30) can be transformed and represented as k ≥ ϕ/σ0 + δ1 + (1 + β)2 /(4(c − δ1 − ϕ/σ0 )),
(33)
which implies that the magnitude of the gain k resulting in the synchronization is inversely proportional to that of the estimated synchronization error bound σ0 , and the gain k approaches the infinity as σ0 → ϕ/(c − δ1 ). 4. Numerical examples In this section, two examples are given to illustrate the effectiveness of the obtained synchronization criteria. First consider a master–slave synchronization scheme consisting of two identical SMIB power systems via a dislocated single-variable coupling matrix K defined by (14) where the parameters of the power system are c = 0.5, β = 1, ω = 1 and h = 2.45. The master power system is chaotic, as shown in Fig. 2(a). The synchronization condition k > 1 for such a coupling matrix can be obtained by means of criterion (15). Fig. 2(b) shows the evolution of complete synchronization with k = 1.1. The second example deals with a master–slave synchronization scheme where there exist parameter mismatches between two systems and external disturbance on the slave system. The orbits of the master SMIB power system is illustrated in Fig. 2(a), which has the bound ρ2 = 4. The parameter mismatches between the two systems and the external disturbance on the slave system are
1β = 0.06 sin t ,
1h = 0.05 sin 2t ,
d = (0.04 sin 2t , 0)T ,
respectively. When the initial states of the master–slave systems are chosen as x(0) = (−1.5, 1.1) and z (0) = (−7, 4), the trajectories of the master system and the uncontrolled slave one separate randomly and remarkably in the course of time, as shown in Fig. 3. Suppose that the coupling K = diag{k, 0} is applied to the synchronization scheme. Then one can obtain a synchronization condition by criterion (30), as follows: K = diag{k, 0},
k ≥ 1/(0.5 − s) with s ∈ (0, 0.5).
(34)
Taking s = 0.1, one can obtain a synchronization condition k = 2.5 by (34). The simulation verifies that it is possible to synchronize two systems with parameter mismatches and external disturbance via such a coupling up to a small bounded error, which is shown in Fig. 4. In order to illustrate the difference between the estimated synchronization error bound and the real one, take the following coupling matrix: K = diag{k, 0},
k = 1/(0.5 − s) with s ∈ (0, 0.5).
(35)
Let ‖e‖T = maxt ≥T (x1 − z1 ) + (x2 − z2 ) be the real synchronization error bound where T represents a time threshold after which the synchronization error will converge and stabilize in a small area (in the practical simulation, T = 1000 time unit). Choose a s ∈ (0, 0.5). The coupling gain can then be determined by (35). Thus the real synchronization error bound ‖e‖T can be numerically solved based on the master–slave synchronization scheme (21) with such a coupling gain, and the estimated synchronization error bound σ0 can be determined by Theorem 7. According to the method, the curve of the real synchronization error bound and that of the estimated one can be obtained, varied with the parameter s ∈ (0, 0.5), as shown in Fig. 5, which reveals that the difference between the real synchronization error bound and the estimated one is inversely proportional to the value of s ∈ (0, 0.5).
2
2
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
1507
a
b
Fig. 2. Complete synchronization of the master–slave identical power systems. (a) The master system with c = 0.5, β = 1, ω = 1 and h = 2.45; (b) the synchronization evolution for K defined by (14) with k = 1.1, where initial conditions are x(0) = (−1.5, 1.1) and z (0) = (−10, −5).
Fig. 3. The remarkable difference of the trajectories of the isolated master–slave power systems with parameter mismatch and external disturbance, where the initial conditions are x(0) = (−1.5, 1.1) and z (0) = (−7, 4).
5. Conclusion The LMI criterion for global complete synchronization of the master–slave identical power systems under linear stateerror feedback control and that for global robust synchronization of the master–slave power systems with parameter
1508
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
Fig. 4. Synchronization with the bounded error of scheme (21) with K = diag{2.5, 0}, where the initial conditions are x(0) = (−1.5, 1.1) and z (0) = (−7, 4).
Fig. 5. The estimated synchronization error bound and the numerical (real) one for the synchronization condition K = diag{k, 0}, k = 1/(0.5 − s) with s ∈ (0, 0.5). The curves plotted by ‘‘∗’’ and ‘‘.’’ represent the estimated synchronization error bound and numerical one, respectively.
mismatches and external disturbance under the same control have been proposed. Some algebraic criteria for the synchronization via a single-variable linear coupling have been derived. The obtained algebraic criteria are further optimized to improve their performances. The effectiveness of the new criteria is illustrated by numerical examples. Acknowledgement This wok was supported by the National Nature Science Foundation of China under grant No. 60674049. References [1] H.K. Chen, T.N. Lin, J.H. Chen, Dynamic analysis, controlling chaos and chaotification of a SMIB power system, Chaos, Solitons and Fractals 24 (2005) 1307–1315. [2] H. Saadat, Power System Analysis, McGraw Hill, 1999. [3] A. Çela, Y. Hamam, J.R. Nielsen, L.T. Christensen, Nonlinear approach to the control of multimachine power systems, Nonlinear Analysis 30 (1997) 4301–4308. [4] A.M. Harb, M.S. Widyan, Chaos and bifurcation control of SSR in the IEEE second benchmark model, Chaos, Solitons and Fractals 21 (2004) 537–552. [5] A.M. Harb, N. Abeld-Jabbar, Controlling Hopf bifurcation and chaos in a small power system, Chaos, Solitons and Fractals 18 (2003) 1055–1063. [6] E.M. Shahverdiev, L.H. Hashimova, N.T. Hashimova, Chaos synchronization in some power systems, Chaos, Solitons and Fractals 37 (2008) 827–834. [7] S. Boccaletti, J. Kurths, G. Osipov, et al., The synchronization of chaotic systems, Physics Reports 336 (2002) 1–101. [8] X.F. Wu, M.H. Wang, Robust synchronization of chaotic Lur’e systems via replacing variables control, International Journal of Bifurcation and Chaos 16 (2006) 3421–3433. [9] P.F. Curran, L.O. Chua, Absolute stability theory and the synchronization problem, International Journal of Bifurcation and Chaos 7 (6) (1997) 1375–1382.
Q. Lin, X. Wu / Nonlinear Analysis: Real World Applications 12 (2011) 1500–1509
1509
[10] J.E. Slotine, W.P. Li, Applied Nonlinear Control, China Machine Press, Beijing, 2004. [11] X.F. Wu, J.P. Cai, M.H. Wang, Master–slave chaos synchronization criteria for the horizontal platform systems via linear state error feedback control, Journal of Sound and Vibration 295 (2006) 378–387. [12] F.X. Chen, W.D. Zhang, LMI criteria for robust chaos synchronization of a class of chaotic systems, Nonlinear Analysis 67 (2007) 4301–4308. [13] X.F. Wu, G.R. Chen, J.P. Cai, Chaos synchronization of the master–slave generalized Lorenz systems via linear state error feedback control, Physica D 229 (2007) 52–80.