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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.

With the development of smart materials, some smart materials-based actuators, such as piezoceramic actuators [

For the modeling method of the hysteresis, it can be roughly classified as differential equation-based hysteresis models, such as Backlash-like model [

So far, the control design work for the systems in presence of hysteresis nonlinearities has also been paid more attention [

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 [

The rest of this paper is organized as follows. In Section

Consider the following class of uncertain nonlinear systems in strict-feedback form with unknown hysteresis input:

The control objective is to design a control law

The following assumptions of the system (

The desired trajectory

The signs of

The disturbances

It should be mentioned that the knowledge of

In this paper, the Duhem model is used to describe the hysteresis nonlinearity, which is defined by [

In order to get the analytic expression of the hysteresis output

By selecting suitable shape functions, Duhem model can describe the different characteristics of the hysteresis nonlinearities. For example, choose

Hysteresis curves described by

Hysteresis curves described by

Under the previous three conditions, the Duhem model (

For

When

When

According to the analysis in [

According to the previous proof,

Hysteresis curves described by Backlash-like model.

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 (

The function

According to Condition

Let

Following the DSC procedure, the coordinate transformation is made as follows:

The first-order low pass filters and the boundary filter errors

Considering the first equation in (

Define the Lyapunov function candidate

Note that the following inequalities hold [

Based on (

The virtual control law

Substituting (

By using the following inequalities

Considering (

Define the Lyapunov function candidate

Based on (

The virtual control law

where

Considering the following inequalities

The actual control law

Define the Lyapunov function candidate

Based on (

Similar to (

Similarly, the following inequalities will be utilized:

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

From (

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.

Consider the closed-loop system consisting of the plant (

Define the set

From Condition

To demonstrate the effectiveness of the proposed control algorithm, in this section, one second-order nonlinear system with unknown Duhem hysteresis is considered:

In this simulation, the initial values of adaptive laws are selected as

The simulation results are shown in Figures

Tracking performance of the closed-loop system.

Control input

Variable

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.

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

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).