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The state tracking problem for a class of model reference adaptive control (MRAC) systems in the presence of controller temporary failures is studied. Due to the controller temporary failure, the considered system is viewed as an error switched system. The properties of Lyapunov function candidates without switching are described. Then the notion of global practical stability of switched systems is presented, and sufficient conditions for global practical stability of the error system under the restrictions of controller failure frequency and unavailability rate are provided. An example is presented to demonstrate the feasibility and effectiveness of the proposed method.

There are often parameter, structural, and environment uncertainties in practical systems [

On the other hand, controller temporary failures are often encountered in real control systems due to various environment factors during operation. Some motivations of studying controller failures are summarized in [

Recently, there are rapidly growing interests in switched systems and switching control in the control community [

In this paper, we study the state tracking problem for MRAC systems in the presence of controller temporary failure. As in [

The results in this paper have three features. First of all, MRAC systems in the presence of controller temporary failure are first considered. Secondly, the state tracking problem is studied from a switched system point of view. Finally, the global practical stability criterion is given for the considered system under the condition of controller failure frequency and unavailability rate.

The organization of the paper is as follows. The state tracking problem in the presence of controller failure is formulated in Section

The notation is standard. Consider the following:

: the largest (smallest) eigenvalue of matrix

Consider a system

The classical state tracking problem is to design a controller such that the state

Suppose that there exist matrices

Define the tracking error

Apply a parameter projection adaptive law

From [

We now consider the case of controller temporary failure depicted in Figure

Controller failures occur in MRAC systems.

We introduce the following definitions which will play key roles in deriving our main results.

For any

For any

In this paper, our objective is to develop conditions under which

From the closed-loop system (

When controller fails, we obtain an unstable error system

In this condition, we choose the adaptive law

When controller fails, because the adaptive parameter

Based on (

Meanwhile, we have a switching adaptive law of the following form:

When

Therefore, the problem of state tracking in the presence of controller temporary failure can be handled by means of analyzing the stability of the error switched system (

To analyze the stability of the error switched system (

Consider system (

Unlike the

In this section, firstly, we give three lemmas to analyze the properties of Lyapunov function candidates without switching. Secondly, we present a theorem to give some conditions under which the error switched system (

According to [

Note that the parameter estimates

Consider the situation of the system (

Differentiating

The following lemma gives the estimate of the convergence rate of

Consider the normal error subsystem of (

From (

It is obvious that

From (

Since

With the help of (

This completes the proof.

When the controller fails, for the unstable error subsystem of (

Differentiating

Then, in the following lemma, we estimate the divergence rate of

Consider the unstable error subsystem of (

From (

It is obvious that

From (

Since

Then, it holds that

Therefore, for any

This completes the proof.

Based on Lemmas

Form (

Denote that

For any

This completes the proof.

Furthermore, according to Lemmas

For (

Without loss of generality, for

When

Because of the uncertainties of the systems, the properties of the Lyapunov function candidates are restricted outside the ball with the radius

Now, we are in the position to give the main result of this paper.

Consider the error switched system (

For any given

For

For

When the initial error

We first prove (a). Consider

With the help of

Note that

Because of

Then, we prove (b). For

Thus, from Lemmas

If

If

By Definition

Therefore, when

Finally, we prove (c). If

This completes the proof.

When the initial error

The error switched system (

In this section, we present an example to demonstrate the effectiveness of the proposed method in this paper.

Consider the system (

The reference state

Choose

Switching signal.

When

The norm of the tracking error of (

The norm of the tracking error of (

Simulations are carried out for

The norm of the tracking error of (

The norm of the tracking error of (

From Figures

This paper has considered the state tracking problem for a class of MRAC systems in the presence of controller temporary failure. A key point is to describe such a system as an error switched system. The properties of Lyapunov function candidates without switching have been given. Then, the global practical stability of the error switched system can be ensured by the proposed scheme, providing that the controller suffers from failures only for a relatively short time interval and with a low frequency of occurrence. It is an interesting topic to extend the results for the output tracking problem of adaptive systems.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Chinese National Fundamental Research Program under Grant 2009CB320601 and National Natural Science Foundation of China under Grants 61233002 and 61174073.