State Tracking of MRAC Systems in the Presence of Controller Temporary Failure Based on a Switching Method

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.


Introduction
There are often parameter, structural, and environment uncertainties in practical systems [1][2][3][4].Model reference adaptive control (MRAC) has been used as an important control approach for such uncertain systems [5,6].Closed-loop signal boundedness and asymptotic tracking can be ensured by a state feedback controller and adaptive laws in MRAC systems [7].
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 [8].The reasons can be roughly classified into two categories: passive and positive ones.A typical passive reason is that the signals are not transmitted perfectly or the controller itself is not available for some reasons.For instance, the packet dropout phenomenon in networked control systems leads to controller failure, which is inevitable because of unreliable transmission paths.In contrast to passive reasons, a typical positive reason is that the controller is purposefully suspended from time to time for an economic or system life consideration [9].Apparently, controller failures may lead to severe performance deterioration of systems.Especially, for adaptive control systems, the controller failure may cause the tracking error divergence due to the uncertainties of systems.Therefore, it is both theoretically and practically important to develop some new techniques to deal with the case of controller temporary failure of adaptive systems.
Recently, there are rapidly growing interests in switched systems and switching control in the control community [10][11][12][13].In the study of stability of switched systems, one of the effective tools is the average dwell time approach [14][15][16].Based on this approach, exponential stability is guaranteed if the unavailability rate of the controller is smaller than a specified constant and the average time interval between controller failures is large enough [9,17,18].In [8], this result was further extended to symmetric linear time-invariant system.The concept of controller failure frequency was first introduced in [9], and the cases of controller temporary failure for a class of time-varying delay systems were analyzed in [19].Interestingly, a nonswitched MRAC system in the presence of controller temporary failure can be viewed as a switched system with a switching signal depending on the time interval between controller failures.Thus, theories and methods of switched systems may be applicable to the study of the state tracking problem for nonswitched MRAC systems with controller temporary failure.However, this issue has been rarely explored so far.

Mathematical Problems in Engineering
In this paper, we study the state tracking problem for MRAC systems in the presence of controller temporary failure.As in [8], the controller temporary failure means that the controller itself is not available or the controller signals are not transmitted perfectly within a certain time interval.Furthermore, we assume that the parameter estimation is "frozen" in the instant of the controller temporary failure until the controller works normally.There are two main issues to be addressed in this paper.One is to describe a tradition MRAC system in the presence of controller temporary failure as an error switched system with two subsystems: the normal error subsystem which stands for the case without controller failure and the unstable error subsystem which describes the case of controller failure.The other issue is the stability analysis for the error switched system.To address the second issue, we analyze the properties of Lyapunov function candidates without switching and introduce the notions of global practical stability, failure frequency, and unavailability rate.
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 2. In Section 3, we present an error switched system.Section 4 gives three lemmas and the main result.An example is given to illustrate the effectiveness of the proposed method in Section 5. Finally, the conclusions are presented in Section 6.
The notation is standard.
The classical state tracking problem is to design a controller such that the state () of the system (1) tracks a given reference state   () generated from the reference model system where   ∈  × is a constant Hurwitz matrix,   ∈  × is a constant input matrix, and () ∈   is a bounded and piecewise continuous reference input.Suppose that there exist matrices , , and  * such that the following matching equations are satisfied: where  * is an unknown matrix due to uncertain constant parameter matrix Θ.
Apply a parameter projection adaptive law where Then, the closed-loop system is where W() =  * − Ŵ().
We now consider the case of controller temporary failure depicted in Figure 1.Controller failures occur when the controller (a) itself is not available or when the signals are not transmitted perfectly on the route (b).Suppose that the time interval of the controller failures is not more than a specified constant   , which means the designed controller can be recovered within a finite time interval [19].Also, the failed controller implies the complete breakdown of the controller (() = 0) in its failure time interval [8].Hence, the system (1) with the controller temporary failure is dominated by the following piecewise differential equations: ẋ () = { Θ () +  () , when the controller works, Θ () , when the controller fails.
We introduce the following definitions which will play key roles in deriving our main results.Definition 1 (see [8]).For any  2 >  1 , denote  + ( 1 ,  2 ) as the total time interval of controller failure during ( 1 ,  2 ), and call the ratio  + ( 1 ,  2 )/( 2 − 1 ) the unavailability rate of the controller in the system.
Definition 2 (see [9]).For any denote the number of control failure in the time interval ) is referred to as the controller failure frequency in the time interval ( 1 ,  2 ).
In this paper, our objective is to develop conditions under which () tracks   () subject to controller temporary failure.

Error Switched System
From the closed-loop system (7), we have a normal error system When controller fails, we obtain an unstable error system In this condition, we choose the adaptive law Remark 3. When controller fails, because the adaptive parameter Ŵ() has no influence on the tracking error (), it is proper that the parameter estimation Ŵ() is "frozen" in the instants of the controller temporary failure until the controller works normally.
When  = 1, the normal error subsystem is active, which corresponds to the case of no controller failure; when  = 2, the unstable error subsystem is active, which denotes that the controller fails.
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 (12) with the switching adaptive law (13).
To analyze the stability of the error switched system (12), we introduce the following definition.

Main Result
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 (12) is globally practically stable.Let  () = [  () , w 1 () , . . .w  () , . . .w where w () ∈   is the th column of W(),  = 1, 2, . . ., ; that is, According to [21], we have Note that the parameter estimates Ŵ are bounded; thus there exists a constant  > 0 defined as Consider the situation of the system (1) without controller failure.A Lyapunov functional candidate of the normal error subsystem of ( 12) is chosen as Differentiating  1 along the trajectory of the normal error subsystem of (12) and the adaptive law (13) gives Mathematical Problems in Engineering The following lemma gives the estimate of the convergence rate of  1 along the trajectory of the normal error subsystem of (12).Lemma 6.Consider the normal error subsystem of (12).For any given  1 > √/ min ( 1 ), denote ( 1 ) = { | ‖‖ <  1 }.If () ∈   /( 1 ), then the inequality holds for any  1 satisfying 0 <  1 ≤ ( 1 /2)(1−(/ 2 1  min ( 1 ))).
Proof.From ( 17), we have It is obvious that From ( 10) and ( 26), we have where Then, it holds that where Therefore, for any  2 > 0, when () ∈   /( 2 ), then the inequality V This completes the proof.
Based on Lemmas 6 and 7, we have the following lemma.

Remark 9.
Because of the uncertainties of the systems, the properties of the Lyapunov function candidates are restricted outside the ball with the radius  0 as described as in Lemmas 6-8.Now, we are in the position to give the main result of this paper.
This completes the proof.
Remark 12.The error switched system ( 12) is globally practically stable if the controller fails only for a short time interval and with a low frequency of occurrence.

Example
In this section, we present an example to demonstrate the effectiveness of the proposed method in this paper.
From Figures 5 and 6, we can conclude that whether ‖( 0 )‖ ≥  0 or not, the states of the system (1) with controller  temporary failure track the reference signal   () well under the switching signal (), and the tracking error ‖()‖ is small in the sense of  * .Simulation illustrates the effectiveness of the proposed method.

Conclusion
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.

Figure 4 :
Figure 4: The norm of the tracking error of (12) without controller failure.