Matrix Measure with Application in Quantized Synchronization Analysis of Complex Networks with Delayed Time via the General Intermittent Control ()
1. Introduction
Since its origins in the work of Fujisaka and Yamada [1- 3], Afraimovich, Verichev, and Rabinovich [4], and Pecora and Carrol [5], the study of synchronization of chaotic systems [6-19] is of great practical significance and has received great interest in recent years. In the above literatures, the approach applied to stability analysis is basically the Lyapunov’s method. As we all know, the construction of a proper Lyapunov function usually becomes very skillful, and the Lyapunov’s method does not specifically describe the convergence rate near the equilibrium point of the system. Hence, there is little compatibility among all of the stability criteria obtained so far.
The concept named the matrix measur [20-25] has been applied to the investigation of the existence, uniqueness or stability analysis of the equilibrium. Intermittent control [26-29] has been used for a variety of purposes in engineering fields such as manufacturing, transportation, air-quality control and communication. A wide variety of synchronization or stabilization using the periodically intermittent control method has been studied (see [27- 32]). Compared with continuous control methods [7-14], intermittent control is more efficient when the system output is measured intermittently rather than continuously. All of intermittent control and impulsive control are belong to switch control. But the intermittent control is different from the impulsive control, because impulsive control is activated only at some isolated moments, namely it is of zero duation, while intermittent control has a nonzero control width.
But it should be mentioned that the influence caused by quantization has not been considered in their results. It is well known that in modern networked systems, quantization is an indispensable step that aims at saving limited bandwidth and energy consumption [33]. Quantization cannot be avoided in the digital control setting, and it is indeed a natural way to be inserted into the control design complexity constraints of the controller and communication constraints of the channels which connect the controller and the plant [34]. The important application of quantization in real world can be found in humanmachine interaction, for instance, see [35-37]. Therefore, it is essential and important to investigate the exponential quantized synchronization problem of networks with mixed delays by periodically intermittent control.
Our interest focuses on the class of commonly intermittent controller with time duration, where the control is activated in certain nonzero time intervals, and is off in other time intervals. A special case of such a control law is of the form
where 
denotes the control strength, 
denotes the switching width, and T denotes the control period. The general intermittent controller
where 
is a strictly monotone increasing function on
, has been studied (see [38]).
Moreover, a logarithmic quantizer 
has quantization levels give by
where the quantization densitie is
, and the scaling parameter is
. Then, the quantizer 
is defined as follow
(1)
where
. Based on (1), it is obvious that
and the quantization synchronization error 
(see [39-43]).
In this paper, based on matrix measure and Gronwall inequality, the general intermittent controller
where 
is a strictly monotone increasing function on
,
where 
is a strictly monotone decreasing function on
, is designed. Then the sufficient yet generic criteria for synchronization of complex networks with and without delayed item are obtained.
This paper is organized as follows. In Section 2, some necessary background materials are presented. In Section 3, the state vectors scope estimated via matrix measure are formulated. Section 4 deals with the quantized synchronization. The theoretical results are applied to complex networks, and numerical simulations of delayed neural network systems are shown in this section. Finally, some concluding remarks are given in Section 5.
2. Preliminaries
Let 
be a Banach space endowed with the l2-norm
, i.e.
, where 
is inner product, and 
be a open subset of
. We consider the following system:
(2)
where 
are nonlinear operators defined on
, and 
, and 
is a time-delayed positive constant, and
.
Definition 1 [12,26,28,44] System (2) is called to be exponentially stable on a neighborhood 
of the equilibrium point, if there exist constants
, such that
where 
is any solution of (2) initiated from 
.
Definition 2 [20-25] Suppose that 
is a matrix. Let 
be the matrix measure of 
defined as
where 
is the identity matrix.
Lemma [20-25] The matrix measure 
is well defined for the l2-norm
, the induced matrix measure is given by
where 
denotes all eigenvalues of the matrix
.
3. Estimating the Scope of the State Vectors
We consider the following system:
(3)
(4)
where 
are nonlinear operators defined on
, and 
, and 
is a timedelayed positive constant, and 
.
Theorem 1 For any 
in the system (3), (4), if the operator 
satisfies
, (5)
is bound, where 
is a positive constant. The solutions 
initiated from 
of the system (3) and (4) satisfy
where 
Proof Under the initial conditions 
we have
for any
.
Let 
then
Using Cauchy-Bunyakovsky Inequality and condition (5), we obtain
So
Namely
Using the Gronwall inequality [45,46], we have
that is
4. Synchronization via the General Intermittent Control and Examples
Consider a delayed complex dynamical network consisting of 
linearly coupled nonidentical nodes described by
(6)
where 
is the state vector of the ith node, 
are nonlinear vector functions, 
is the control input of the ith node, and 
is the coupling figuration matrix representing the coupling strength and the topological structure of the complex networks, in which 
if there is connection from node i to node 
, and is zero, otherwise, and the constraint
, is set.
A complex network is said to achieve asymptotical synchronization if
, (7)
where 
is a solution of a real target node, satisfying
.
For our synchronization scheme, let us define error vector and control input 
as follows, respectively:
When 
is a strictly monotone increasing function on n with 
(8)
When 
is a strictly monotone decreasing function on n with 
(9)
In this work, the goal is to design suitable function 
and parameters
, 
and 
satisfying the condition (7). The error system follows from the expression (6), (8) and (9)
(10)
When 
is a strictly monotone increasing function on 
with 
we obtain the following result:
Theorem 2 Suppose that the operator 
in the network (6) satisfies condition (5), and 
is defined as Definition 2,
where the constant
satisfies 
Then the synchronization of network (6) is achieved if the parameters
, 
, 
, 
and 
satisfy
(11)
where 
is the inverse function of the function 
Proof From Theorem 1, the following conclusion is valid:
(12)
for any
;
(13)
for any
.
In the following, we use mathematical induction to prove, for any nonnegative integer
,
(14)
1) For
, from (12) and (13), we can see that
a) For
,
b) For
,
So (14) is true for
.
2) Assume that (14) is true for all
, that is
We will prove (14) is also true when
. From (12) and (13), it is easy to see that
Then, for
, we have
and also, for
, it follows from above results that
From above discussion, we can see that the (14) is always correct for any nonnegative integer
.
When 
is a strictly monotone increasing function on 
and
, it is easy to obtain 
When 
is a strictly monotone increasing function on 
and
, it follows that
then
Therefore
when 
is obtained under the condition (11). So the synchronization of the network (6) is achieved.
When 
is a strictly monotone decreasing function on 
with
, we obtain the following result:
Theorem 3 Suppose that the operator 
in the network (6) satisfies condition (5), and 
is defined as Definition 2, 
where the constant
are the same as Theorem 2. So the synchronization of networks (6) is achieved if the parameters
, and 
satisfy
(15)
where 
is the inverse function of the function 
Proof From Theorem 1, the following conclusion is valid:
(16)
for any
;
(17)
for any
.
From (16) and (17), imitating Theorem 2,we can prove
where
,
,
,
.
when 
is obtained under the condition (15). So the synchronization of network (6) is achieved.
Corollary 1 Supposing that 
, and the rest of restricted conditions are invariable. Then the synchronization of the network (6) is achieved if the parameters
, 
and 
satisfy
Corollary 2 when we add normally distributed white noise randn (size(t)), the result similar to Theorem 2 and Theorem 3 is obtained if the condition (11) or (12) , respectively, is satisfied.
In the simulations of following examples, we always choose 
the matrix
.
Let the initial condition be
Example 1 Consider a delayed system [47]:
(18)
The function
, 
which are the strictly monotone increasing or decreasing function on
, respectively, then they can be clearly seen that the synchronization of network (6), which is composed of system (18), is realized in Figures 1-4 (Excited by parameter white-noise), respectively.
(a)
(b)
Figure 1. Synchronization error when h(n) = 2n + 1n(n + 1). (a) The error xi1 − s1, (i = 1, 2, 3, 4, 5); (b) The error xi2 − s2, (i = 1, 2, 3, 4, 5).
(a)
(b)
Figure 2. Synchronization error when h(n) = 2n + 1n(n + 1), white noise 0.5 (xi − s) randn (size(t)), (i = 1, 2, 3, 4, 5). (a) The error xi1 − s1, (i = 1, 2, 3, 4, 5); (b) The error xi2 − s2, (i = 1, 2, 3, 4, 5).
(a)
(b)
Figure 3. Synchronization error when h(n) = 3/n + (n + 1)/n2. (a) The error xi1 − s1, (i = 1, 2, 3, 4, 5); (b) The error xi2 − s2, (i = 1, 2, 3, 4, 5).
(a)
(b)
Figure 4. Synchronization error when h(n) = 3/n + (n + 1)/n2, white noise 0.5 (xi − s) randn (size(t)), (i = 1, 2, 3, 4, 5). (a) The error xi1 − s1, (i = 1, 2, 3, 4, 5); (b) The error xi2 − s2, (i = 1, 2, 3, 4, 5).
5. Conclusion
Approaches for quantized synchronization of complex networks with delayed time via general intermittent which uses the nonlinear operator named the matrix measure have been presented in this paper. Strong properties of global and exponential synchronization have been achieved in a finite number of steps. Numerical simulations have verified the effectiveness of the method.
NOTES