Parameter Identification and Synchronization of Dynamical System by Introducing an Auxiliary Subsystem

Abstract

We propose a novel approach of parameter identification using the adaptive synchronized observer by introducing an auxiliary subsystem, and some sufficient conditions are given to guarantee the convergence of synchronization and parameter identification. We also demonstrate the mean convergence of synchronization and parameters identification under the influence of noise. Furthermore, in order to suppress the influence of noise, we complement a filter in the output. Numerical simulations on Lorenz and Chen systems are presented to demonstrate the effectiveness of the proposed approach.

1. Introduction

Since the pioneering work of Pecora and Carroll [1], chaos synchronization has become an active research subject due to its potential applications in physics, chemical reactions, biological networks, secure communication, control theory, and so forth [2–12]. An important application of synchronization is in adaptive parameter estimation methods where parameters in a model are adjusted dynamically in order to minimize the synchronization error [13–15]. To achieve system synchronization and parameter convergence, there are two general approaches based on the typical Lyapunov's direct method [2–9] or LaSalle's principle [10]. When adaptive synchronization methods are applied to identify the uncertain parameters, some restricted conditions on dynamical systems, such as persistent excitation (PE) condition [11, 15] or linear independence (LI) conditions [10], should be matched to guarantee that the estimated parameters converge to the true values [12].

In the following, we explore a novel method for parameter estimation by introducing an auxiliary subsystem in adaptive synchronized observer instead of Lyapunov's direct method and LaSalle's principle. It will be shown that through harnessing the auxiliary subsystem, parameters can be well estimated from a time series of dynamical systems based on adaptive synchronized observer. Moreover, noise plays an important role in parameter identification. However, little attention has been given to this point. Here we demonstrate the mean convergence of synchronization and parameters identification under the influence of noise. Furthermore, we implement a filter to recover the performance of parameter identification suppressing the influence of the noise.

2. Parameter Identification Method

In the master-slave framework, consider the following master system:

(2.1)

where is the state vector, is the unique unknown parameter to be identified, and are the nonlinear functions of the state vector in the th equation.

In order to obtain our main results, the auxiliary subsystem is needed

(2.2)

where is a positive constant.

Lemma 2.1.

If is bounded and does not converge to zero as , then the state of system (2.2) is bounded and does not converge to zero, when .

Proof.

If is bounded, we can easily know that is bounded [16]. We suppose that the state of system (2.2) converges to zero, when . According to LaSalle principle, we have the invariant set , then ; therefore, from system (2.2), we get as . This contradicts the condition that does not converge to zero as . Therefore, the state does not converge to zero, when .

Based on observer theory, the following response system is designed to synchronize the state vector and identify the unknown parameters.

Theorem 2.2.

If Lemma 2.1 holds, then the following response system (2.3) is an adaptive synchronized observer for system (2.1), in the sense that for any set of initial conditions, and as .

(2.3)

where are the observed state and estimated parameter of and , respectively, and and are positive constants.

Proof.

From system (2.3), we have

(2.4)

Let , , , and note that ; then

(2.5)

Since is generated by (2.3), then

(2.6)

Obviously, as .

From and , we have

(2.7)

Let us focus on the homogeneous part of system (2.7), which is

(2.8)

The solution of system (2.8) is . From the lemma, we know that does not converge to zero. According to Barbalat theorem, we have as ; correspondingly, as , that is, the system is asymptotically stable.

Now from the exponential convergence of in system (2.6) and asymptotical convergence of in system (2.8), we obtain that in system (2.7) are asymptotical convergent to zero.

Finally, from , , and being bounded, we conclude that are global asymptotical convergence.

The proof of Theorem 2.2 is completed.

Note.

When and is the offset, in this condition no matter is in stable, periodic, or chaotic state, we could use system (2.3) to estimate and synchronize the system (2.1).

Note.

When the system is in stable state, parameter estimation methods based on adaptive synchronization cannot work well [10]. For this paper, when the system is in stable state, such that as , which leads to the lemma not being hold, so system (2.3) cannot be directly applied to identify the parameters. Here, we supplement auxiliary signal in drive system (2.1), such that does not converge to zero as . Then the master system becomes

(2.9)

and the corresponding slave system can be constructed as

(2.10)

In doing so, synchronization of the system and parameters estimation can be achieved.

3. Application of the Above-Mentioned Scheme

To demonstrate and verify the performance of the proposed method, numerical simulations are presented here. We take Lorenz system as the master system [17], which is described by

(3.1)

where the parameters , , and are unknown, and all the states are measurable. When , , , Lorenz system is chaotic.

We construct the slave systems as follows:

(3.2)

When the Lorenz system is in chaotic state, all states of are not convergent to zero as (see Figure 1(a)). Then according to Theorem 2.2, we realize that not only the synchronization can be achieved but also the unknown parameters , , and can be estimated at the same time.

Figure 1

(a) The curves of ; (b) Identified results of , , versus time.

Figure 1(a) shows the curves of . All parameters , , and are estimated accurately and depicted in Figure 1(b). Figures 2(a)–2(c) display the results of synchronization for systems (3.1) and (3.2), where the initial conditions of simulation are , and

Figure 2

(a) The curve of ; (b) The curve of ; (c) The curve of .

When , , and , the states of Lorenz system are not chaotic but convergent to a fixed point. Figure 3(a) shows the curves of . In this case, as displayed in Figure 3(a), convergence to zero as . Figure 3(b) depicts the estimated results of parameters , , and . From Figure 3(b), we can see that parameters , and have been estimated accurately. However, the parameter cannot be estimated well. According to the analysis of Note 2, we add an auxiliary signal in the first subsystem of master system (3.1) and we obtain , such that all states of do not converge to zero as . The curves of are shown in Figure 4(a). Correspondingly, we add signal in the first subsystem of slave system (3.2) and we have ; then all parameters , , and are estimated accurately and depicted in Figure 4(b).

Figure 3

(a) The curves of ; (b) Identified results of , , versus time.

Figure 4

(a) The curves of ; (b) Identified results of , , versus time.

In recent years, more novel chaotic systems are found such as Chen system [18], Lü system [19], and Liu system [20]. Let us consider the identification problem for Chen system. We take Chen system as the master system, which is described by

(3.3)

where the parameters , , and are unknown, and all the states are measurable. When , , and , Chen system is chaotic.

We construct the slave systems as follows:

(3.4)

Figures 5 and 6 show the synchronization error and identification results, respectively, and where , and .

Figure 5

The curves of ,  , and .

Figure 6

Identified results of , , versus time.

From the simulation results of Lorenz and Chen system above, we can see that the unknown parameters could be identified. It indicates that the proposed parameter identifier in this paper could be used as an effective parameter estimator.

4. Parameter Identification in the Presence of Noise

Noise plays an important role in synchronization and parameters identification of dynamical systems. Noise usually deteriorates the performance of parameter identification and results in the drift of parameter identification around their true values. Here we consider the influence of noise. Suppose that there are addition noise in drive system (2.1).

(4.1)

where is the zero mean, bounded noise.

Theorem 4.1.

If the above lemma is hold and is independent to , and , using the synchronized observer (2.3), then for any set of initial conditions, and converge to zero asymptotically as , where and are mean values of and , respectively.

Proof.

Similarly with the proof of Theorem 2.2, let ; then

(4.2)

We have ; then

(4.3)

is independent to , and , and note that ; then

(4.4)

So similarly we have , , and therefore, as .

From Theorem 4.1, we know that that as , which means that the estimated values for unknown parameters will fluctuate around their true values. As an illustrating example, we revisit the Lorenz system (3.1) and its slave systems (3.2), and we assume all the subsystems (3.1) are disturbed by uniformly distributed random noise with amplitude ranging from to . Figure 7(a) shows that the estimated parameters , , and fluctuate around their true values.

Figure 7

(a) Identified results of , , in presence of noises; (b) Identified results of , , in presence of noises and with filters.

To suppress the estimation fluctuation caused by the noise, it is suitable to use mean filters. Here we introduce the following filter:

(4.5)

It is clear to see from Figure 7(b) that unknown parameters , , and can be identified with high accuracy even in the presence of large random noise.

5. Conclusions

In this paper, we propose a novel approach of identifying parameters by the adaptive synchronized observer, and a filter in the output is introduced to suppress the influence of noise. In our method, Lyapunov's direct method and LaSalle's principle are not needed. Considerable simulations on Lorenz and Chen systems are employed to verify the effectiveness and feasibility of our approach.