Mathematical Problems in Engineering

Volume 2015, Article ID 436906, 7 pages

http://dx.doi.org/10.1155/2015/436906

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

^{1}Department of Automation, Shanghai Jiao Tong University and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, China^{2}Key Laboratory of Synthetical Automation for Process Industry, Ministry of Education and Northeastern University, Shenyang, Liaoning 110004, China

Received 27 May 2014; Accepted 3 June 2014

Academic Editor: Ligang Wu

Copyright © 2015 Chuanjing Hou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

The hysteresis phenomenon occurs in all the smart material-based actuators and sensors. With such nonlinearity in control systems, it may lead to undesirable inaccuracies or oscillations. On the other hand, actuator failures seem inevitable in practical systems. Thus, failure compensation of hysteretic actuators is an important and challenging problem. Adaptive failure compensation has received great attention in recent years [1–12]. However, available results based on adaptive approaches to address hysteretic actuator failures are very limited [13–16].

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 [13–16] 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.

#### 2. Problem Formulation

Consider a class of nonlinear systems in the following form [15]:where is the relative degree of the system; , are the inputs whose actuators may fail during system operation; , are the state vectors; are unknown constant parameters; are unknown constant parameters with known signs; , , , , and are known smooth functions; denotes bounded disturbance; and is an unknown nonlinear function representing system modeling errors. There exists a known function such that .

The hysteresis nonlinearity can be described by Bouc-Wen model [17, 18]. Considerwhere are weighting parameters, are stiffness coefficients, , are constants; is the input of the th actuator, , describe the shape and amplitude of the th hysteresis, respectively, governs the smoothness of the transition from initial slope to the slope of the asymptote, and , . By Lemma 1 in [17], are bounded.

The actuator failure can be modeled as [8, 9, 13–15]where , , and are all unknown constants and are bounded. For different values of , three types of failures are included:(1), where the actuator works normally; namely, , which is regarded as a failure-free actuator;(2); it implies ; the th actuator is called partial loss of effectiveness (PLOE);(3); it indicates ; the th actuator is called total loss of effectiveness (TLOE).

*Remark 1. *The values of can change only from to some values with . 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 haveTo derive a suitable adaptive control scheme, the following assumptions are made.

*Assumption 2. *When TOLE type of actuator failures up to , 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 is input to state stable with respect to as its input.

Let . Suppose that there are actuators failing at time instants , , and . In other words, all actuators work normally in time interval and no new failure will occur after time . Let the set denote the actuators of total failure in interval and use the set to represent other normal actuators. It can be concluded that .

Our objective is to design a control law for the nonlinear systems with unknown actuator failures, when changes at time instants , , 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.

#### 3. 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 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 i.* Consider ; the th stabilizing function , the th regressor , and the th tuning function are chosen aswhere are positive design parameters and is estimator of the unknown vector .

*Step **.* From (6), the derivative of isIf 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) cannot be bounded by any known function but are known and 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 bywhere is bounded and exists, and is a positive design parameter.

Let

The control law and parameter update laws are obtained as followswhere , , and are the estimates of and , respectively, and are positive definite matrices chosen by users. If is a desired constant vector which can be chosen to satisfythis givesSubstituting (11)–(13) and (8) into (7), we have

#### 4. Stability Analysis

To prepare for the stability analysis, we rewrite the error system aswhere 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 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: .*

*Proof. *For each time interval , , we have a Lyapunov function in the following form:Taking the derivative of (19) yieldsSubstituting (11)–(13) and (8) into (7), we haveFrom , , and bounded , we getwhereFrom (22), we conclude that , , so that , , and are bounded for . It follows from (8) and (10) that and are bounded. Therefore, all closed-loop signals are bounded for . In order to prove the asymptotic tracking, considering the last time interval . From (22), we can obtainBecause of the boundedness of , we have , . By Barbalat’s lemma, can be obtained. This completes the proof.

#### 5. Simulations

We consider a second order nonlinear [15] system with two inputs described aswhere , , , , , , , are unknown constants, , are the outputs of two hysteretic actuators, are the states, and is bounded by . The reference signal is set as . The backlash-like hysteresis is described by (2) with parameters , , , and . The high-gain function is chosen as . For simulation, we consider three actuator failure cases.

*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 is the 3rd order identity matrix.

The simulation results including output , reference output , and tracking error are shown in Figure 1; the actuators outputs 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.