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.