Adaptive Neural Tracking Control for Discrete-Time Switched Nonlinear Systems with Dead Zone Inputs

Wang, Jidong; Zhu, Lengxue; Si, Xiaoping

doi:https://doi.org/10.1155/2017/5628932

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Abstract Introduction Conclusion Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2017 | Article ID 5628932 | https://doi.org/10.1155/2017/5628932

Adaptive Neural Tracking Control for Discrete-Time Switched Nonlinear Systems with Dead Zone Inputs

Jidong Wang,¹Lengxue Zhu,¹and Xiaoping Si¹

Academic Editor: Sigurdur F. Hafstein

Received13 Dec 2016

Revised26 Feb 2017

Accepted01 Mar 2017

Published23 Mar 2017

Abstract

In this paper, the adaptive neural controllers of subsystems are proposed for a class of discrete-time switched nonlinear systems with dead zone inputs under arbitrary switching signals. Due to the complicated framework of the discrete-time switched nonlinear systems and the existence of the dead zone, it brings about difficulties for controlling such a class of systems. In addition, the radial basis function neural networks are employed to approximate the unknown terms of each subsystem. Switched update laws are designed while the parameter estimation is invariable until its corresponding subsystem is active. Then, the closed-loop system is stable and all the signals are bounded. Finally, to illustrate the effectiveness of the proposed method, an example is employed.

1. Introduction

In the past few decades, intelligent control of uncertain nonlinear systems has attracted much more attention. Based on the universal approximation properties of neural networks (NNs) and fuzzy logic systems (FLSs), they are always used to approximate the unknown system functions or the control inputs. So far, there are a lot of results in intelligent control [1–6]. For example, the NN-based adaptive control methods were proposed for uncertain nonlinear systems in [7, 8] by using backstepping technique. The above results are all about nonlinear systems in continuous-time form. On the contrary, many researchers devoted much effort to study the adaptive control problem of discrete-time systems by using intelligent methods on the basis of these works [9–11].

However, the previous results do not consider the switching phenomenon of the system. Actually, the switching phenomenon often exists in practical systems. We call this class of systems switched systems. Switched systems, which are used to model a wide variety of physical systems, consist of a family of continuous-time or discrete-time subsystems and a switching rule to govern the switching between the subsystems. Generally, the stability and stabilisation problems are the main concerns in the study of switched systems [12–16]. Subsequently, adaptive control and intelligent control of switched systems have been studied more and more [17–22]. But to the best of our knowledge, there are no results on discrete-time switched systems, and let alone results on discrete-time switched nonlinear systems.

In addition, the dead zone input, as one of the most important input nonlinearity, widely exists in lots of practical systems. The existence of the dead zone input may damage the system performance or even destroy the system stability. Thus, the related robust control attracted high attention [8, 23–25]. Since the dead zone parameters are poorly known in most practical systems, adaptive control techniques are naturally used to deal with this problem. An adaptive tracking control strategy was proposed for a nonlinear system with nonsymmetric dead zone input having unknown but bounded parameters in [26].

Motivated by the above discussion, we study the adaptive neural control problem of a class of discrete-time switched nonlinear systems with dead zone inputs under arbitrary switching signals. We use the radial basis function neural networks to approximate the unknown terms of each subsystem. Dead zone inputs of subsystems and switched update laws are designed such that the closed-loop system is stable and all the signals are bounded. Finally, an example is employed to illustrate the effectiveness of the proposed method.

The main contributions of this paper compared with the existing results on switched and nonswitched nonlinear systems are in two aspects:(1)Compared with the existing results [8, 23–25], in which the considered systems are all nonswitched systems or continuous-time switched systems, in this paper, we study a discrete-time switched uncertain nonlinear system.(2)The dead zone input is considered in the discrete-time switched uncertain nonlinear system. As far as we know, there are no results on discrete-time switched systems with dead zone inputs. This is mainly due to the complicated framework of the discrete-time switched nonlinear systems, the existence of the dead zone, and the interaction between the system structure and switching.

2. System Description

In this paper, we consider the following discrete-time switched uncertain nonlinear system:where and are the system state and output, respectively. Moreover, the state of switched system (1) is assumed not to jump at the switching instants, which is a standard assumption in the switched system [27]. is the switching signal taking values in with being the subsystems number of switched system (1). and , are unknown smooth functions. is the dead zone output of the th subsystem, which is described by

In (2), is the dead zone input of the th subsystem. and stand for the right and left slopes of the dead zone characteristic, respectively. and represent the breakpoints of the input nonlinearity.

According to [28], the dead zone can be further expressed aswhere

Assumption 1. The parameters of dead zone are all positive, and they are unknown but bounded, that is, , , and .

From Assumption 1, we can easily know that

Assumption 2 (see [29–31]). , , are strictly positive or negative and the signs of are known. Without loss of generality, we suppose that are positive with and being the lower bound and upper bound, respectively. It implies that .

The control objective of this paper is to design dead zone inputs of subsystems, such that(1)the system state can track a given signal , or equivalently speaking, the output can track a desired trajectory ,(2)all the signals in the closed-loop system are bounded under arbitrary switching signals.

Assumption 3 (see [32]). The desired trajectory of the system state satisfies the condition , . Moreover is a known smooth bounded function.

According to Assumption 2 and system (1), we can obtain that , .

3. Neural Networks

Similar to FLSs, NNs also have the approximation properties. In this paper, we consider the case that there exist unknown functions in the systems. Here, we use RBFNNs to approximate the unknown functions of subsystems.

For any unknown continuous function defined on , there exists NN , such thatwhere is the input variable of the NN, is the weight vector with being the node number of NNs, is the approximation error, and is the basis function. In RBFNNs, , , are chosen as Gaussian functionswhere and , , are the centers and widths of the Gaussian functions, respectively.

According to (7), we have . Thus, we can further obtain that , . This leads to the following inequality:

In this paper, since there are unknown functions in (11), we cannot obtain , , directly. Thus, the neural networks are used to approximate the unknown function , . In fact, the neural networks in this paper are used for identification.

4. Dead Zone Input Design

Define the tracking error at instant asThen, we can get the th state error . Furthermore, we havewhere and are the abbreviations of and , respectively.

Define functions

Because of the existence of the unknown functions in (11), cannot be directly obtained. Here, we use RBFNNs to approximate themwhere are the weight matrices which are assumed to be bounded, that is, , and and are the basis function vector and the approximation errors, respectively. Furthermore, and . is the input vector of RBFNNs.

By adding and subtracting on the right side of equation (10), we can conclude thatwhere and are the shorthand for and , respectively. Denote that and , , are unknown constants. Let be the estimation of and lwt be the estimation error.

According to (13), we design the controllers aswhere are the estimation of . Let the estimation error satisfy .

The switched update laws are designed aswhere and are positive matrices and positive design parameters, respectively.

Based on (14), (13) becomes

5. Stability Analysis

In this section, the stability of switched system (1) and the boundedness of all the signals in the closed-loop system are proved under arbitrary switching signals.

Theorem 4. Consider the discrete-time switched system (1), the dead zone (3), the switched update laws given by (15), and the input of the th dead zone selected as in (14) under Assumptions 1–3. If the design parameters satisfy the conditionsthen the boundedness of all the signals in the closed-loop switched system can be ensured and the state can track the reference signal and the system output can follow the desired trajectory .

Proof. Choose the following common Lyapunov function for system (1) aswhere and .
The first difference of isSubstituting (15) into (19), one haswith being the maximum eigenvalue of .
According to (16), we can conclude thatThen, becomesBased on Assumption 2, it holds that By using Young’s inequality, we have the following inequalities: and we consider the following equations: Then, we havewhere If the design parameters satisfy condition (17), then we can deduce thatIf or or holds, we have . According to the standard Lyapunov extension theorem, the tracking error and the estimation errors and are bounded. Since , , we can easily know that , , are also bounded. Furthermore, because and are bounded, the boundedness of the estimated parameters and is guaranteed by and . Thus, from (14), we know that the dead zone inputs are bounded. Based on the definition of dead zone, one can conclude that the control inputs are also bounded. Therefore, all the signals in the closed-loop system are bounded under arbitrary switching signals.

Remark 5. As we all know that a switched system might be unstable even if all the subsystems are stable, arbitrary switching method can guarantee the stability of the switched system in this case. A sufficient condition for analysing the stability of switched systems under arbitrary switching signals is that there exists a common Lyapunov function. In many cases, it is easy to find a Lyapunov function for each subsystem. However, a common Lyapunov is hard to find for switched systems. In this paper, we have successfully found a common Lyapunov function shown as in (18).

Remark 6. There are many parameters in the design procedure. , , , and , , are parameters of dead zone. They are all positive based on Assumption 1. In addition, , , , and , , are introduced to improve the flexibility of the update laws. In this paper, they are chosen to be positive. But how to choose a set of optimal ones to ensure the satisfactory performance is still an open problem; they need to be flexibly selected. In the following, a constructive suggestion on how to choose these parameters is given. From the above stability analysis, we can see that the bounds of , , and depend on the values of , , , and . Actually, we can select smaller and larger and to obtain smaller bounds of , , and to further get good performance. But, it is important to note that these parameters must satisfy conditions (17). In the simulation, a trial-and-error method is used.

6. Simulation

In this section, we apply the proposed approach to the following discrete-time switched nonlinear system:where , , , , , , , and . The switching signal takes values in .

The dead zone is described as

The parameters of the dead zone inputs are selected as , , , , , , , and .

Define the tracking errors and . The desired signals and are selected as and , respectively. The initial values for the system states are , , , and .

The dead zone input is designed for each subsystem where is the Gaussian function.

The switched update laws are designed for , aswhere , , , , , , , and .

Furthermore, we give the simulation results in Figures 1–7. Figure 1 gives the curves of and . Figure 2 shows the trajectories of and . Figures 3 and 4 display the tracking errors and , respectively. From Figures 1–4, we can observe that a good tracking performance is obtained. Moreover, the dead zone inputs are shown in Figures 5 and 6, which explain the boundedness of the dead zone inputs. Figure 7 is the switching signal.

7. Conclusion

We design a controller of each subsystem for a class of discrete-time switched nonlinear systems with dead zone inputs under arbitrary switching signals in this paper. The unknown functions are approximated by RBFNNs. Switched update laws are designed while the parameter estimation is invariable until its corresponding subsystem is active. Then, the closed-loop system is stable and all the signals are bounded. Due to the complicated framework of the discrete-time switched nonlinear systems and the existence of the dead zone, it brings about great difficulties for controlling such a class of systems.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the Science and Technology Key Projects of Henan Province (162102210080).

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Copyright © 2017 Jidong Wang 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.

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