Asymptotic Tracking Control for a Class of Nonlinear Systems with Unknown Failures of Hysteretic Actuators

An adaptive failure compensation controller for a class of nonlinear systems preceded by hysteretic actuators is proposed in this paper. Three types of high-gain functions are constructed to counteract the effects of the hysteresis, bounded modeling errors, and bounded disturbances. It is shown that the proposed controller not only ensures bounded signals and asymptotic tracking but also avoids possible chattering, despite the presence of unknown hysteretic actuator failures. Simulation results verify the desired failure compensation performance.

To address such a challenging problem, it is important to find an appropriate model for the hysteresis.As mentioned in [17], several models were proposed such as Duhen model, Preisach model, Prandtl-Ishlinskii hysteresis operator, Bouc-Wen differential model [17,18].The Bouc-Wen differential model is one of the most widely accepted models of the hysteresis.Actually, it can be shown that the hysteresis model presented in [13][14][15][16] is a special case of the Bouc-Wen hysteresis model.
In [13], an adaptive failure compensation scheme for a class of nonlinear systems was studied, where control gains are constants.To avoid possible chattering, the sign (⋅) functions were involved in the backstepping controller.When the control gains are nonlinear functions of system states, the effects of the actuator hysteresis can no longer be assumed bounded as in [13].How to handle such effects is a challenging issue especially when there are possible actuator failures.In [14,15], the idea is to separate such effects into two parts by applying Young's inequality.It is noted that the control methods in [14,15] are very complicated and the tracking error cannot asymptotically converge to zero but to the so-called predefined bound.Furthermore, the existence of the estimators increases the order of the closed-loop system.
In this paper, we develop a backstepping [19,20] adaptive compensation controller for a class of nonlinear systems preceded by hysteretic actuators described by Bouc-Wen model.Three types of high-gain functions are incorporated into the controller to counteract the effects of the hysteresis, bounded modeling errors, and bounded disturbances, respectively.In our design, the sign (⋅) function and a priori knowledge on the bounds of control gains are not required.Besides showing the stability of the closed-loop system, the tracking error is also ensured to achieve zero asymptotically.
The rest of the paper is outlined as follows.In Section 2, the control problem is formulated.In Section 3, a robust adaptive compensation scheme with high-gain functions is proposed.In Section 4, the stability analysis is presented.Simulation results are presented to show the proposed scheme is effective in Section 5. Finally, this paper is concluded in Section 6.

Problem Formulation
Consider a class of nonlinear systems in the following form [15]: . . .
Remark 1.The values of   can change only from   = 1 to some values with 0 ≤   < 1.This means that possible changes from normal to any one of the failure cases are unidirectional.
The uniqueness of   indicates that a failure occurs only once on the th actuator.Substituting ( 2), ( 3) into (1), we have To derive a suitable adaptive control scheme, the following assumptions are made.
Assumption 2. When TOLE type of actuator failures up to − 1, the remaining actuators can still achieve a desired control objective.
Remark 3. Note that all actuators are allowed to have partial loss of effectiveness simultaneously.Assumption 4. The zero dynamics of ξ = (, ) +   (, ) is input to state stable with respect to  as its input.
Let  0 = 0. Suppose that there are   (0 ≤   ≤ ) actuators failing at time instants   ,  = 1, 2, . . ., , and In other words, all actuators work normally in time interval [ 0 ,  1 ) and no new failure will occur after time   .Let the set   denote the actuators of total failure in interval [  ,  +1 ) and use the set   to represent other normal actuators.It can be concluded that   ∪   = {1, 2, . . ., }.
Our objective is to design a control law   () for the nonlinear systems with  unknown actuator failures, when  changes at time instants   ,  = 1, 2, . . ., , such that the output () asymptotically tracks a given reference signal   () with up to th order derivatives bounded and that all closed-loop signals are bounded.

Adaptive Compensation Control Schemes
The backstepping technique [19,20] is applied to derive an adaptive actuator failure compensation controller.The following change of coordinates is required: where  1 is the tracking error and   is the th stabilizing function.To illustrate the backstepping procedures, only the last step of the design is elaborated in details.
Step .From (6), the derivative of   is If the effects of hysteresis are treated as disturbances, it should be noted that the disturbances in ( 7) can be classified into three types: (1) () is bounded by an unknown constant ; (2) (, ) is bounded by a known function; (3) ∑  =1     ()  cannot be bounded by any known function but   () ( = 1, . . ., ) are known and     ( = 1, 2 . . ., ) are bounded.According to their different characteristics, three high-gain functions will be proposed to counteract the effects of the disturbances.At the final step, the stabilizing function   is given by where () is bounded and lim  → ∞ ∫  0  −()  exists, and   is a positive design parameter. Let The control law and parameter update laws are obtained as follows where this gives Substituting ( 11)-( 13) and ( 8) into (7), we have

Stability Analysis
To prepare for the stability analysis, we rewrite the error system as ż =   (, )  +   (, )  ã +  ()   , where the system matrices   (, ),   (, ), (), and   are given by The closed-loop system has the following desired properties.

Theorem 5. With the ISS of the zero dynamics of system
(1) and m hysteretic actuators modeled in (2) with possible unknown failures by (3), the controller (11) with the adaptive laws (12) and (13) ensures the boundedness of the closed-loop signals and the asymptotic output tracking: lim  → ∞ (−  ) → 0.
Case 1.There are no actuator failures.
By Theorem 5, we can obtain the actual control law and the update laws.The initial conditions are set as follows: where  3 is the 3rd order identity matrix.
The simulation results including output (), reference output   (), and tracking error () are shown in Figure 1; the actuators outputs  1 ,  2 are shown in Figure 2. The system responses are as expected.At the beginning, there is a transient response in tracking errors.But, as time goes on, the tracking errors become smaller and ultimately vanish.The proposed controller guarantees that asymptotic tracking is achieved.
Case 2. Actuator  1 is stuck at  1 = 45 from  = 3 s, thus undergoing a TLOE type of failure.By Theorem 5, we can obtain the actual control law and the update laws.The initial conditions are set as follows.
The other parameters are the same as those in Case 1.
The simulation results including output (), reference output   (), and tracking error () are shown in Figure 3; the actuators outputs  1 ,  2 are shown in Figure 4.The system responses are as expected.When one of the actuators fails, there is a transient response in tracking errors.But, as time goes on, the tracking errors become smaller and ultimately vanish.The proposed controller guarantees that asymptotic tracking is achieved.Case 3. Actuator  2 is stuck at  2 = 40 from  = 3 s and actuator  1 loses 60% from  = 6 s.Thus,  1 undergoes a PLOE type of failure while  2 is a TLOE type of failure.
The other parameters are the same as those in Case 1.The simulation results including output (), reference output   (), and tracking error () are shown in Figure 5; the actuators outputs  1 ,  2 are shown in Figure 6.The system responses are as expected.When the actuators fail, there is a transient response in tracking errors.But, as time goes on, the tracking errors become smaller and ultimately vanish.The proposed controller guarantees that asymptotic tracking is achieved.

Conclusions
This paper presents an adaptive failure compensation controller for a class of uncertain nonlinear systems dominated by the hysteresis actuator nonlinearity.We propose three types of high-gain functions to deal with the unknown bounded disturbances, unknown modeling errors, and unknown actuator failures.It has been shown that the tracking errors can converge to zero asymptotically while all the closed-loop signals remain bounded.Furthermore, the proposed scheme can avoid possible chattering.Simulation results illustrate the effectiveness of our proposed scheme.