Abstract and Applied Analysis

Volume 2014 (2014), Article ID 640249, 10 pages

http://dx.doi.org/10.1155/2014/640249

## Robust Adaptive Dynamic Surface Control for a Class of Nonlinear Dynamical Systems with Unknown Hysteresis

^{1}The Key Laboratory of Autonomous System and Network Control, Ministry of Education, China^{2}The College of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China^{3}The Department of Electronic and Information Systems, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama, Saitama 337-8570, Japan

Received 17 June 2013; Accepted 19 November 2013; Published 2 January 2014

Academic Editor: Massimo Furi

Copyright © 2014 Yong-Hua Liu 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

The output tracking problem for a class of uncertain strict-feedback nonlinear systems with unknown Duhem hysteresis input is investigated. In order to handle the undesirable effects caused by unknown hysteresis, the properties in respect to Duhem model are used to decompose it as a nonlinear smooth term and a nonlinear bounded “disturbance-like” term, which makes it possible to deal with the unknown hysteresis without constructing inverse in the controller design. By combining robust control and dynamic surface control technique, an adaptive controller is proposed in this paper to avoid “the explosion complexity” in the standard backstepping design procedure. The negative effects caused by the unknown hysteresis can be mitigated effectively, and the semiglobal uniform ultimate boundedness of all the signals in the closed-loop system is obtained. The effectiveness of the proposed scheme is validated through a simulation example.

#### 1. Introduction

With the development of smart materials, some smart materials-based actuators, such as piezoceramic actuators [1], magnetostrictive actuators, and shape memory alloys, are becoming increasingly important in the application areas of aerospace, manufacturing, defense, and civil infrastructure systems [2–5], because of their excellent performance, for example, high precision, fast response, and flexible actuating ability [6–8]. However, a class of nonsmooth nonlinearities, hystereses, with multibranching and nondifferential properties, widely occur in these smart materials-based actuators. When the system is preceded by these actuators, the existence of the hysteresis behaviour in these actuators will degrade the system performance, causing undesirable inaccuracy. The hysteresis nonlinearities are the nature properties of these smart materials, which cannot be cancelled by the improvement of the smart materials. Therefore, how to mitigate the negative effects caused by the hysteresis nonlinearities from control view becomes one important research topic in this area. Due to the nonsmooth nature of hysteresis, most common control approaches developed for nonlinear systems may not be applicable to hysteretic systems directly, which attracted significant attention in the modeling of hysteresis nonlinearities and the hysteretic systems controller design.

For the modeling method of the hysteresis, it can be roughly classified as differential equation-based hysteresis models, such as Backlash-like model [9], Bouc-Wen model [10, 11], and Duhem model [10, 12], and operator-based hysteresis models, such as Preisach model, Krasnosel’skii-Pokrovskii model, and Prandtl-Ishlinskii model [13–15]. As a differential-equation based hysteresis model, Duhem model can represent numerous hysteresis shapes including saturation and asymmetric properties by choosing different shape functions. However, the output analytical expression of Duhem model is difficult to obtain directly since the output depends on the solution of the differential equation, which may cause a new difficulty for the controller design.

So far, the control design work for the systems in presence of hysteresis nonlinearities has also been paid more attention [16–19]. Generally, two control approaches are used to mitigate the negative effects of hysteresis in the literature. The common one is to construct a hysteresis inverse model to cancel the adverse effects of hysteresis completely or approximately, such as [20, 21]. The main advantage of this inverse control approach is to compensate the effects of hysteresis nonlinearities directly. However, the construction of the inverse hysteresis will increase the complexity of the control systems and may limit the application in the industrial systems. Also, the compensation error depends on the hysteresis modeling parameters; therefore, it is difficult to get the analytical expression of the compensation error. Alternatively, another method is to fuse the hysteresis models with control methods without constructing the hysteresis inverse [9, 22–24], which can be applied in the real-time systems conveniently. For this control structure without inverse, the key point is to explore the characteristics of the hysteresis model and then investigate the suitable control methods to mitigate the effects caused by hysteresis.

Synthesizing the hysteresis modeling methods and control approaches, the output tracking problem for a class of uncertain nonlinear systems in strict-feedback form with unknown Duhem hysteresis is discussed. For the Duhem model, one adaptive robust controller for a class of nonlinear systems was discussed in [25]. Still following the line, the robust adaptive control method for a class of uncertain nonlinear systems in strict-feedback form is investigated in this paper. In order to mitigate the design difficulty caused by the smooth function term in the uncertain nonlinear systems, the mean value theorem and a Nussbaum function lemma are used. The proposed dynamic surface control (DSC) approach [26] without hysteresis inverse avoids “the explosion complexity” in the standard backstepping design, mitigates the negative effects arising from the unknown hysteresis, and ensures the semiglobal uniform ultimate boundedness of all the signals in the closed-loop system.

The rest of this paper is organized as follows. In Section 2, the control problem is formulated. Duhem hysteresis model is introduced in Section 3. In Section 4, an adaptive dynamic surface controller is developed for a class of nonlinear systems in strict-feedback form with unknown Duhem hysteresis, and the stability analysis is given as well. Computer simulations are shown to verify the effectiveness of the proposed scheme in Section 5. Section 6 concludes the paper.

#### 2. Problem Statement

Consider the following class of uncertain nonlinear systems in strict-feedback form with unknown hysteresis input: where , ; and are the system states and output; , are unknown constant parameters, denote the unknown disturbances, are known smooth functions, ; and is the output of the hysteresis nonlinearity with the actual input .

The control objective is to design a control law in (1), forcing the output to track a given desired trajectory , while all the signals of closed-loop system are uniformly bounded.

The following assumptions of the system (1) are made.

*Assumption 1. *The desired trajectory is continuous and its first-order derivative and second-order derivative are bounded and available; that is, there exists a positive constant , such that .

*Assumption 2. *The signs of are known, and there exist unknown positive constants and such that . Without loss of generality, it can be assumed that , .

*Assumption 3. *The disturbances , , satisfy
where are known nonnegative smooth functions and are unknown nonnegative constants.

*Remark 4. *It should be mentioned that the knowledge of and is not required to be known, which is only used in the analysis of the latter stability proof.

#### 3. Hysteresis Model

In this paper, the Duhem model is used to describe the hysteresis nonlinearity, which is defined by [14] where and are the hysteresis input and output, respectively; is a constant; and and are shape functions of .

In order to get the analytic expression of the hysteresis output , the following three conditions [10, 27, 28] are used for Duhem model.

*Condition 1. * is a piecewise smooth, monotone increasing, odd function of , with a derivative , that obtains a finite limit .

*Condition 2. * is a piecewise continuous, even function of , with a finite limit satisfying

*Condition 3. * for all finite .

*Remark 5. *By selecting suitable shape functions, Duhem model can describe the different characteristics of the hysteresis nonlinearities. For example, choose and with different shape function satisfying three properties; the described hysteresis curves are shown in Figures 1 and 2.

Under the previous three conditions, the Duhem model (3) can be solved explicitly for piecewise monotone as [14] where

For , if is the solution of (5) with initial values , one has then it can be deduced that is bounded [14] easily. For simplicity, let denote the bound of , where is a positive constant.

*Remark 6. *When and is a constant , the Duhem model can be expressed as

When , the Duhem model becomes the Backlash-like model defined in [9]. According to the above analysis, it is obvious that the Backlash-like model is a special case of the Duhem model. However, it should be noted that when and , Conditions 1 and 2 are not satisfied necessarily for the Duhem model. Similarly, (8) can be solved explicitly for the Backlash-like model: with

According to the analysis in [9], it has so the disturbance term is still bounded.

According to the previous proof, still can be used to denote the bound of defined for Backlash-like model. As a comparison, when , , and the input , the Duhem model can be reexpressed as the Backlash-like model; then the curve of the Backlash-like model is shown in Figure 3.

#### 4. Adaptive DSC Design and Stability Analysis

In this section, the procedure for the design of adaptive dynamic surface controller and system stability will be given. Considering the characteristics of the hysteresis nonlinearities existing in the actual controlled plant, the following assumption is made for the hysteresis model (3).

*Assumption 7. *The function of Duhem hysteresis (3) is a smooth and strictly increasing function.

According to Condition 1 of Duhem model, . Combining the derivative form of mean value theorem and Assumption 7, there exists such that Then is rewritten as then the system (1) is expressed as Since the sign of the control gain is unknown, one useful lemma is given as follows.

Lemma 8 (see [29]). *Let , be the smooth functions defined on with , for all , and let be an ever smooth Nussbaum-type function. If the following inequalities hold
**
where represents some suitable constant, is a positive constant, and is a time-varying parameter which takes values in the unknown closed intervals , with , then , , and must be bounded on .*

##### 4.1. Adaptive DSC Design

Following the DSC procedure, the coordinate transformation is made as follows: where are output of the filter (17).

The first-order low pass filters and the boundary filter errors are defined as where are the filter time constant and are the filter input, which are also the virtual control law for the th subsystem specified hereinafter, .

*Step 1. *Considering the first equation in (14) and invoking (16)–(18), the time derivative of is given by

Define the Lyapunov function candidate
where , , and with , , and as the estimates of , , and , respectively. , , and are positive design parameters.

Note that the following inequalities hold [30]: where , .

Based on (19) and (21), it has

The virtual control law and the adaptive laws , , and are designed as where , , , and are positive design parameters.

Substituting (23) into (22), we obtain

By using the following inequalities we have

*Step i (). *Considering (17) and (18), and , it has

Define the Lyapunov function candidate where , , and with , , and as the estimates of , , and , respectively. , , and are positive design parameters.

Based on (21) and (28), the time derivative of is given by

The virtual control law and the adaptive update laws , , and are designed as

where , , , and are positive design parameters.

Considering the following inequalities we have

*Step n. *The actual control law will be designed in this step. Considering and , the time derivative of is given by

Define the Lyapunov function candidate where , , and with , , and as the estimates of , , and , respectively. , , and are positive design parameters.

Based on (37), we have

Similar to (21), the following inequalities are used: we have The actual control law and the adaptive laws , , , and are designed as where , , , and are positive design parameters.

Similarly, the following inequalities will be utilized: then, we obtain

##### 4.2. Stability Analysis

In this subsection, the uniform ultimate boundedness of all signals in the closed-loop system will be proven.

From (27) and (31), we have where are the continuous functions, .

To establish the boundedness of the closed-loop system, the following Lyapunov function candidate is defined as

The main results can be summarized as follows.

Theorem 9. *Consider the closed-loop system consisting of the plant (1), the controller (42), and adaptation laws (43) under Assumptions 1–7. If for any , there exist the appropriate design parameters , , , , , , , , and , such that all signals in the closed-loop system are semiglobally uniformly ultimately bounded.*

*Proof. *Define the set . From Assumption 1 and for any , the set and are compact in and . Thus, have a maximum value , , on . Equation (46) can be further derived as
The derivative of with respect to follows from (45), (46), and (48) that
where
Multiplying both sides of (49) by yields
Integrating (51) over , it is deduced that
where .

From Condition 1 of Duhem model and Assumption 7, it is easily concluded that , and is a bounded smooth even function of , and hence is a nonzero bounded time-varying function. By using Lemma 8, it implies that , , are all bounded on . From proposition 2 in [31], , it can be concluded that all error signals , , , , in the closed-loop system are semiglobally uniformly ultimately bounded.

#### 5. Numerical Example

To demonstrate the effectiveness of the proposed control algorithm, in this section, one second-order nonlinear system with unknown Duhem hysteresis is considered: where , , , and are unknown parameters, and are unknown disturbances, and is the output of the unknown hysteresis described by the Duhem model as (3). In the simulation, , , , , , , , and . Correspondingly, , , , and . For the Duhem model, , , and . The objective is to make the output of system (53) to track the desired trajectory .

In this simulation, the initial values of adaptive laws are selected as , , , , , and . In addition, the design parameters are chosen as , , , , , , , and . The Nussbaum function is chosen as with . The initial states of (53) are chosen as , .

The simulation results are shown in Figures 4, 5, 6, 7, 8, and 9. From Figure 4, it is observed that the good tracking performance is achieved under the proposed approach. Figure 5 shows the control input and the hysteresis output . Figures 6, 7, 8, and 9 show the response curves of adaptive parameters , , , , , , and . From these results, the proposed scheme can mitigate the detrimental effects of the unknown hysteresis and guarantee the boundedness of the closed-loop system.

#### 6. Conclusion

In this paper, the adaptive DSC approach for a class of uncertain nonlinear systems in strict-feedback form with unknown Duhem hysteresis is discussed. How to utilize the properties of the hysteresis model and design the related control approach is the main task for this topic. To overcome the design difficulties of Duhem model, three conditions are used to get the analytical output expression of Duhem model. By using DSC technique, the “explosion complexity” in the standard backstepping design procedure is improved. For the last recursive step arising from the unknown hysteresis, the nonlinear smooth term of Duhem model is considered in the robust controller design by using mean value theorem and Nussbaum function lemma. Under the proposed approach, all the signals in the closed-loop system are uniformly semiglobally bounded, and a numerical example is shown to verify the effectiveness.

#### Conflict of Interests

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

#### Acknowledgments

The work was partially supported by the Funds for Natural Science Foundation of China under Grants 61074097, 61105081, 61228301, and U1201244, the Program of Pearl River Young Talents of Science and Technology in Guangzhou (2013J2200100), the Integration of Industry, Education, and Research of Guangdong Province (2012B091100039).

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