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

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

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 .

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

Proof.

From system (2.3), we have

Let , , , and note that ; then

Since is generated by (2.3), then

Obviously, as .

From and , we have

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

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

and the corresponding slave system can be constructed as

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