Adaptive fuzzy logic system–based fault tolerant control for a class of discrete-time switched nonlinear systems

Abstract

An adaptive fuzzy-based fault tolerant control method is first proposed for the discrete-time switched nonlinear systems. In this article, the actuator fault is considered, which contains the loss of effectiveness fault and bias fault. Both the effectiveness factor and the bias signal are unknown but bounded. Moreover, there are unknown internal dynamics in the considered system. In order to solve this problem, the fuzzy logic systems are exploited to provide an approximate construction for the unknown dynamics. Then, under arbitrary switching signals, the designed controller and adaptive laws ensure the boundedness of signals appearing in the considered system, and the system states can track the reference signals. Finally, an example shows the efficiency of the developed approach.

Introduction

Faults, which can bring large damages, and may lead to personnel casualties, often exist in practical engineering. Thus, fault tolerant control (FTC) is a significant topic. Actuator faults as a class of common faults have been paid wide attention and achieved a lot of results.^1–7 Two classes of actuator faults are always considered: the actuator loss of effectiveness (LOE)^8,9 and the actuator bias fault. ¹⁰ When the parameters of the actuator fault are known, it is much easy to design the fault tolerant controller, and if the parameters of the actuator fault are unknown, it will bring great obstacle for the fault tolerant controller design. In Benosman and Lum, ¹¹ the actuator LOE fault was considered, and a passive fault tolerant controller was proposed. Literature ¹² concentrated on the actuator fault tolerant problem for the discrete-time networked control systems.

In addition, a lot of practice plants cannot be simply considered as single-model systems. In other words, multiple-model systems (like switched systems) are usually used to describe the characters of real systems, such as gene regulatory networks and switching power converters. During the past decades, the study of switched systems has been attracted the attention of many researchers.^13–18 Switched systems can be regarded as a class of special hybrid systems. In general, it possesses several subsystems and a designed switching rule. In recent years, the FTC approach for switched systems has achieved a few results.^19–23 However, these results are all for continuous-time switched systems. Besides, most of them are obtained for switched linear systems. As far as we know, there do not exist the results of FTC for discrete-time switched systems, let alone for the discrete-time switched nonlinear systems. This inspires us to study this article.

In continuous-time switched systems, the switching may take place at any time instants. However, in discrete-time switched systems, there is a restriction on the switching instants. That is, the switching must occur at a certain sampling point. Moreover, less analysis schemes can be employed to solve the FTC problem for discrete-time switched nonlinear systems. It will bring a large amount of obstacle for designing controllers.

Moreover, there often exist uncertainties in the control systems.^24–38 The existence of unknown functions is the most common. Usually, there are three methods to handle the unknown functions: fuzzy logic systems (FLSs),^26,27 generalized fuzzy hyperbolic models (GFHMs), and neural networks (NNs). That is to say, many great results of fuzzy-based (or NN based) control^28–33 have been proposed. In He et al., ³⁵ an adaptive NN controller was set up for a class of robotic manipulator systems with unknown dead-zone input and predefined output constraint. In Cheng et al., ³⁸ a model predictive control method was presented for the tracking problem of piezoelectric actuators, and a nonlinear autoregressive-moving-average with exogenous inputs model of piezoelectric actuators was implemented by NNs.

Motivated by the above analysis, this work studies the adaptive FTC design problem for a class of discrete-time nonlinear switched systems suffered from the actuator fault, in which both the LOE fault and bias fault are considered. The parameters of the fault are all unknown but bounded, and then, the adaptive scheme is used to deal with the unknown parameter problem. Furthermore, the unknown internal dynamics are taken into consideration. Here, the FLSs are utilized to approximate the unknown functions. The designed controller and adaptive laws guarantee that the states of the system can track the given signals as much as possible and all the signals appearing in the design are bounded under arbitrary switching signals. To explain the effectiveness of the developed method in this work, an example is used.

The main contributions of this article compared with the existing results on switched and nonswitched nonlinear systems are in two aspects: (1) compared with the results in previous works,^26,36,37 in which the continuous-time switched systems were considered, in this article, the discrete-time system is studied. (2) In addition, the FTC problem of discrete-time switched nonlinear system is examined. To the best of our knowledge, there was hardly any result of FTC for discrete-time switched systems, let alone for the discrete-time switched nonlinear systems.

Problem formulations and preliminaries

We consider the following discrete-time nonlinear switched system in the form of

$$\{\begin{array}{c}{x}_1(k+1)=x_2(k)\hfill \\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{x_{n-1}(k+1)=x_n(k)}\\ \multicolumn{1}{c}{x_n(k+1)=f_{\sigma }(x(k))+g_{\sigma }(x(k))u_{\sigma }(k)}\\ \multicolumn{1}{c}{y(k)=x_1(k)}\end{array}$$

(1)

where $x(k)=[x_1(k),\dots ,x_n(k){]}^T$ stands for the state vector, $\sigma :R_{+}\to I_M=\{1,2,\dots ,M\}$ represents the switching signal, and M is the number of subsystems, $u_i(k)(i\in I_M)$ is the input of the ith subsystem, $f_i(x(k))$ and $g_i(x(k))$ are unknown but smooth functions, and y is the output of system (1). Furthermore, we assume that the solution of equation (1) is continuous everywhere. This is a commonly used assumption for switched systems.^18–20

The considered actuator fault is expressed as ³⁹

$$u_i(k)={\alpha }_i{u}_i^0(k)+{\beta }_i(k)$$

(2)

where $u_i^0(k)$ is the actual input and ${\alpha }_i\in [{\underset{¯}{\alpha }}_i,1]$ and ${\beta }_i(k)$ represent the unknown effectiveness factor and the unknown bias signal, respectively. ${\underset{¯}{\alpha }}_i\in (0,1]$ is the lower bounded of ${\alpha }_i$ and ${\underset{¯}{\alpha }}_i=1$ means that the corresponding actuator is in fault-free case.

• Remark 1. In this work, the actuator fault is considered. The actuator LOE fault and the actuator bias fault are the two kinds of faults which are studied most. The actuator LOE fault is of the form $u_i(k)={\alpha }_i{u}_i^0(k)$ , where ${\alpha }_i$ is the effectiveness factor satisfying $0<{\alpha }_i⩽1$ . $u_i(k)=u_i^0(k)+{\beta }_i(k)$ stands for the actuator bias fault, where ${\beta }_i(k)$ represents the bias signal. In this article, the considered actuator fault shown in equation (2) contains both the LOE and bias fault.

Thus, the switched system (1) with actuator fault is further described as

$$\{\begin{array}{c}{x}_1(k+1)=x_2(k)\hfill \\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{x_{n-1}(k+1)=x_n(k)}\\ \multicolumn{1}{c}{x_n(k+1)=f_{\sigma }(x(k))+g_{\sigma }(x(k))({\alpha }_{\sigma }{u}_{\sigma }^0(k)+{\beta }_{\sigma }(k))}\\ \multicolumn{1}{c}{y(k)=x_1(k)}\end{array}$$

(3)

The control goal is to structure subsystem controllers so that under arbitrary switching signals, the system state $x(k)$ tracks a given signal $x_r(k)=[x_{r1}(k),\dots ,x_{\mathrm{rn}}(k){]}^T$ , and all the variables in the resulting system are bounded.

Preparing for the stability analysis, the following assumptions are required.

• Assumption 1. In this article, the bias signal ${\beta }_i(k)$ is unknown but bounded, that is, $|{\beta }_i(k)|⩽{\overline{\beta }}_i$ with ${\overline{\beta }}_i$ being a known constant.

• Assumption 2. The desired signals $x_{\mathrm{rm}}(k),m=1,\dots ,n$ satisfy the condition $x_{\mathrm{rm}}(k)=x_{r,m-1}(k+1)$ .

• Assumption 3.The gain function $g_i(x(k))$ is strictly negative or positive. Here, $g_i(x(k))$ is assumed to be positive and satisfy $0<{\underset{¯}{g}}_i<g_i(x(k))<{\overline{g}}_i$ , where ${\underset{¯}{g}}_i$ and ${\overline{g}}_i$ are the lower bound and the upper bound of $g_i(x(k))$ , respectively.

Description of FLSs

Since there exist unknown functions in the considered system (1), the FLSs are exploited to set up an approximate construction for the unknown functions. We all know that the knowledge base in an FLS contains a batch of IF-THEN rules. In fact, they are described as follows

$R_i$ : if $x_1$ is $F_1^i$ and $x_2$ is $F_2^i$ and … and $x_n$ is $F_n^i$ , then y is $G^i,i=1,2,\dots ,l$ , where $x=[x_1,x_2,\dots ,x_n{]}^T$ and y are the input and output of the FLS, respectively. $F_j^i$ and $G^i$ are fuzzy sets. l denotes the number of the fuzzy rules. ²⁷

Through singleton function, center average defuzzification, and product inference, a FLS can be described as ³²

$$y(x)=\frac{\sum _{i=1}^l{\overline{y}}_i\underset{j=1}{\overset{n}{\Pi }}{\mu }_{F_j^i}(x_j)}{\sum _{i=1}^l[\underset{j=1}{\overset{n}{\Pi }}{\mu }_{F_j^i}(x_j)]}$$

(4)

where ${\mu }_{F_j^i}(x_j)$ represents the membership functions of fuzzy sets $F_j^i$ . ${\overline{y}}_i=\underset{y\in R}{max}{\mu }_{G^i(y)}$ with ${\mu }_{G^i(y)}$ being the membership functions of fuzzy sets $G^i$ . ²⁶

Choose the following fuzzy basis functions

$${\phi }_i(x)=\frac{\underset{j=1}{\overset{n}{\Pi }}{\mu }_{F_j^i}(x_j)}{\sum _{i=1}^l[\underset{j=1}{\overset{n}{\Pi }}{\mu }_{F_j^i}(x_j)]}$$

(5)

Let $\omega =[{\omega }_1,\dots ,{\omega }_l{]}^T=[{\overline{y}}_1,\dots ,{\overline{y}}_l{]}^T$ and $\phi (x)=[{\phi }_1(x),\dots ,{\phi }_{\ell }(x){]}^T$ , the FLS (4) can be expressed as

$$y(x)={\omega }^T\phi (x)$$

Thus, an unknown continuous function $\mathrm{\Gamma }(x)$ , which is defined on $\mathrm{\Omega }$ , ²⁶ can be approximated by FLSs as

$$\mathrm{\Gamma }(x)={\omega }^{*T}\phi (x)+\epsilon (x)$$

(6)

where

$$\begin{array}{c}{\omega }^{*}=\underset{\omega \in {\mathrm{\Omega }}_{\omega }}{\arg \min }[\underset{x\in \mathrm{\Omega }}{\sup }|\hat{\mathrm{\Gamma }}(x|\omega )-\mathrm{\Gamma }\left(x\right)|]\hfill \\ \epsilon =\mathrm{\Gamma }\left(x\right)-\hat{\mathrm{\Gamma }}(x|{\omega }^{*})\hfill \end{array}$$

with ${\mathrm{\Omega }}_{\omega }$ being the compact region of $\omega$ . Besides, $\epsilon$ is the approximation error.

Controller design

Define $z_m(k)=x_m(k)-x_{\mathrm{rm}}(k)$ as the state tracking error. From Assumption 2, it holds that $x_{\mathrm{rm}}(k)=x_{r,m-1}(k+1)=x_{r,m-2}(k+2)=\cdots =x_{r,1}(k+m-1)$ . Then, one has $z_n(k)=x_n(k)-x_{r,1}(k+n-1)$ , which implies that

$$\begin{array}{cc}\multicolumn{1}{c}{z_n(k+1)}& =x_n(k+1)-x_{r,1}(k+n)\hfill \\ \multicolumn{1}{c}{}& =f_{\sigma }(x(k))+g_{\sigma }(x(k))({\alpha }_{\sigma }{u}_{\sigma }^0(k)+{\beta }_{\sigma }(k))\hfill \\ \multicolumn{1}{c}{}& -x_{r,1}(k+n)\hfill \end{array}$$

(7)

• Remark 2. It is worth noting that the system state tracking problem can be transformed to the output tracking problem in this article. According to Assumption 2, we know that $x_{\mathrm{rm}}(k)=x_{r,m-1}(k+1),m=1,\dots ,n$ . Then, the state-tracking errors satisfy that $z_m(k)=x_m(k)-x_{\mathrm{rm}}(k)=x_m(k)-x_{r,m-1}(k+1)=\cdots =x_m(k)-x_{r,1}(k+m-1)$ .

Choose the following desired controllers

$$u_i^0(k)=-\frac{1}{g_i(x(k)){\alpha }_i(k)}(f_i(x(k))-x_{r,1}(k+n)),i\in I_M$$

(8)

Since there exist unknown functions $f_i(x(k))$ and $g_i(x(k))$ in equation (8), $u_i^0(k)$ cannot be used in applications. In order to solve this problem, $u_i^0(k)$ is approximated by FLSs as

$$u_i^0(k)={\omega }_i^{*T}\phi (Z(k))+{\epsilon }_i(Z(k))$$

(9)

• Assumption 4. The approximation error ${\epsilon }_i(Z(k))$ is bounded, that is, $|{\epsilon }_i(Z(k))|⩽{\overline{\epsilon }}_i$ , where ${\overline{\epsilon }}_i$ is a constant.

Then, subtracting and adding the term $g_i(x(k)){\alpha }_i(k)u_i^0(k)$ on the right side of equation (7), and based on equations (8) and (9), the error system (7) can be rewritten as

$$\begin{array}{c}\multicolumn{1}{c}{z_n(k+1)=f_{\sigma }(x(k))+g_{\sigma }(x(k))({\alpha }_{\sigma }{u}_{\sigma }^0(k)+{\beta }_{\sigma }(k))}\\ \multicolumn{1}{c}{-x_{r,1}(k+n)+g_{\sigma }(x(k)){\alpha }_{\sigma }(k)u_{\sigma }^0(k)}\\ \multicolumn{1}{c}{-g_{\sigma }(x(k)){\alpha }_{\sigma }(k)u_{\sigma }^0(k)}\\ \multicolumn{1}{c}{=g_{\sigma }(x(k)){\alpha }_{\sigma }(k)[u_{\sigma }^0(k)-{\omega }_i^{*T}\phi (Z(k))}\\ -{\epsilon }_{\sigma }(Z(k))+{\beta }_{\sigma }(k)/{\alpha }_{\sigma }(k)]\hfill \end{array}$$

(10)

Let ${\beta }_i(k)/{\alpha }_i(k)⩽{\overline{\beta }}_i/{\underset{¯}{\alpha }}_i\stackrel{\Delta }{=}{\delta }_i^{*}$ , where ${\delta }_i^{*}$ is an unknown constant.

Design the following subsystem controller and adaptive laws

$$u_i^0(k)={\hat{\omega }}_i^T(k)\phi (Z(k))+{\hat{\delta }}_i(k)$$

(11)

$${\hat{\omega }}_i(k+1)=\{\begin{array}{c}{\hat{\omega }}_i(k)-{\tau }_{\omega i}[{\gamma }_{\omega i}{\hat{\omega }}_i(k)\hfill \\ \multicolumn{1}{c}{+\phi (Z(k))z_n(k+1)],\sigma =i}\\ \multicolumn{1}{c}{{\hat{\omega }}_i(k),\sigma \ne i}\end{array}$$

(12)

$${\hat{\delta }}_i(k+1)=\{\begin{array}{c}{\hat{\delta }}_i(k)-{\tau }_{\delta i}[{\gamma }_{\delta i}{\hat{\delta }}_i(k)+z_n(k+1)],\sigma =i\hfill \\ \multicolumn{1}{c}{{\hat{\delta }}_i(k),\sigma \ne i}\end{array}$$

(13)

where ${\hat{\omega }}_i(k)$ and ${\hat{\delta }}_i(k)$ are the estimations of ${\omega }_i^{*}(k)$ and ${\delta }_i^{*}$ , respectively. ${\gamma }_{\omega i}$ and ${\gamma }_{\delta i}$ are the design parameters.

Substituting equation (11) into equation (10), we have

$$\begin{array}{cc}\multicolumn{1}{c}{z_n(k+1)=}& g_{\sigma }(x(k)){\alpha }_{\sigma }(k)[{\tilde{\omega }}_{\sigma }^T(k)\phi (Z(k))+{\tilde{\delta }}_{\sigma }(k)\hfill \\ \multicolumn{1}{c}{}& -{\epsilon }_{\sigma }(Z(k))+{\delta }_{\sigma }^{*}+{\beta }_{\sigma }(k)/{\alpha }_{\sigma }(k)]\hfill \end{array}$$

(14)

where ${\tilde{\omega }}_i(k)$ and ${\tilde{\delta }}_i(k)$ are estimation errors with ${\tilde{\omega }}_i(k)={\hat{\omega }}_i(k)-{\omega }_i^{*}$ and ${\tilde{\delta }}_i(k)={\hat{\delta }}_i(k)-{\delta }_i^{*}$ , respectively.

Main Result

The main result will be shown in this section. First, we give some definitions of notations, which will be used in the theorem proving

$$B_1=1-3{\tau }_{\omega i}-l{\tau }_{\omega i}-\overline{g}\overline{\alpha }l{\tau }_{\omega i}-{\tau }_{\delta i}{\tau }_{\omega i}-\overline{g}\overline{\alpha }{\tau }_{\delta i}$$

(15)

$$B_2=1-{\gamma }_{\omega i}{\tau }_{\omega i}-\overline{g}\overline{\alpha }{\gamma }_{\omega i}{\tau }_{\omega i}$$

(16)

$$B_3=1-{\gamma }_{\delta i}{\tau }_{\delta i}-\frac{\overline{g}\overline{\alpha }{\gamma }_{\delta i}{\tau }_{\delta i}}{{\tau }_{\omega i}}$$

(17)

$$A=\underset{i\in I_M}{max}\{{\gamma }_{\omega i}\parallel {\omega }_i^{*}{\parallel }^2+{\gamma }_{\delta i}{\delta }_i^{*2}+\frac{\overline{g}\overline{\alpha }}{{\tau }_{\omega i}}{\overline{\epsilon }}_i^2+2\frac{\overline{g}\overline{\alpha }}{{\tau }_{\omega i}}{\delta }_i^{*2}\}$$

(18)

Moreover, the design parameters ${\tau }_{\omega i}$ , ${\gamma }_{\omega i}$ , and ${\tau }_{\delta i}$ are required to satisfy the following conditions

$${\tau }_{\omega i}<\frac{1-\overline{g}\overline{\alpha }{\tau }_{\delta i}}{3+l+l\overline{g}\overline{\alpha }+{\tau }_{\delta i}}$$

(19)

$${\gamma }_{\omega i}<\frac{1}{{\tau }_{\omega i}(1+\overline{g}\overline{\alpha })}$$

(20)

$${\tau }_{\delta i}<\frac{{\tau }_{\omega i}}{{\gamma }_{\delta i}({\tau }_{\omega i}+\overline{g}\overline{\alpha })}$$

(21)

Theorem 1. Consider the discrete-time switched nonlinear system (1) with actuator faults (2) satisfying Assumptions 1–4. Design the subsystem controller and adaptive laws as in equations (11), (12), and (13). If the conditions (19)–(21) are satisfied, all the variables in the resulting system are bounded, and the state $x(k)$ can track the desired reference signal $x_r(k)$ .

Proof: Choose the common Lyapunov function

$$L(k)=L_1(k)+L_2(k)+L_3(k)$$

(22)

where $L_1(k)=(1/\overline{g}\overline{\alpha })z_n^2(k)$ , $L_2(k)=\sum _{i=1}^M(1/{\tau }_{\omega i}){\tilde{\omega }}_i^T(k){\tilde{\omega }}_i(k)$ , and $L_3(k)=\sum _{i=1}^M(1/{\tau }_{\delta i}){\tilde{\delta }}_i^2(k)$ with $\overline{g}=\underset{i\in I_M}{max}{\overline{g}}_i$ and $\overline{\alpha }=\underset{i\in I_M}{max}{\overline{\alpha }}_i$ .

It is easy to obtain the first difference of $L_1(k)$

$$\mathrm{\Delta }{L}_1(k)=\frac{1}{\overline{g}\overline{\alpha }}{z}_n^2(k+1)-\frac{1}{\overline{g}\overline{\alpha }}{z}_n^2(k)$$

(23)

Subtracting ${\omega }_i^{*}$ on both sides of equation (12), we can deduce that

$${\tilde{\omega }}_i(k+1)=\{\begin{array}{c}{\tilde{\omega }}_i(k)-{\tau }_{\omega i}[{\gamma }_{\omega i}{\hat{\omega }}_i(k)\hfill \\ \multicolumn{1}{c}{+\phi (Z(k))z_n(k+1)],\sigma =i}\\ \multicolumn{1}{c}{{\tilde{\omega }}_i(k),\sigma \ne i}\end{array}$$

Then, we have

$$\begin{array}{cc}\multicolumn{1}{c}{\Delta L_2(k)=}& \sum _{i=1}^M\frac{1}{{\tau }_{\omega i}}{\tilde{\omega }}_i^T(k+1){\tilde{\omega }}_i(k+1)-\sum _{i=1}^M\frac{1}{{\tau }_{\omega i}}{\tilde{\omega }}_i^T(k){\tilde{\omega }}_i(k)\hfill \\ \multicolumn{1}{c}{=}& -2{\tilde{\omega }}_{\sigma }^T(k)[{\gamma }_{\omega \sigma }{\hat{\omega }}_{\sigma }(k)+\phi (Z(k))z_n(k+1)]\hfill \\ \multicolumn{1}{c}{}& +{\tau }_{\omega \sigma }{[{\gamma }_{\omega \sigma }{\hat{\omega }}_{\sigma }(k)+\phi (Z(k))z_n(k+1)]}^T\hfill \\ \multicolumn{1}{c}{}& \times [{\gamma }_{\omega \sigma }{\hat{\omega }}_{\sigma }(k)+\phi (Z(k))z_n(k+1)]\hfill \\ \multicolumn{1}{c}{=}& -2{\gamma }_{\omega \sigma }{\tilde{\omega }}_{\sigma }^T(k){\hat{\omega }}_{\sigma }(k)-2{\tilde{\omega }}_{\sigma }^T(k)\phi (Z(k))\hfill \\ \multicolumn{1}{c}{}& \times z_n(k+1)+{\gamma }_{\omega \sigma }^2{\tau }_{\omega \sigma }{\hat{\omega }}_{\sigma }^T(k){\hat{\omega }}_{\sigma }(k)\hfill \\ \multicolumn{1}{c}{}& +2{\gamma }_{\omega \sigma }{\tau }_{\omega \sigma }{\hat{\omega }}_{\sigma }^T(k)\phi (Z(k))z_n(k+1)\hfill \\ \multicolumn{1}{c}{}& +{\tau }_{\omega \sigma }{z}_n(k+1){\phi }^T(Z(k))\phi (Z(k))z_n(k+1)s\hfill \end{array}$$

From equation (14), we have

$$\begin{array}{cc}\multicolumn{1}{c}{{\tilde{\omega }}_{\sigma }^T(k)\phi (Z(k))}& =\frac{z_n(k+1)}{g_{\sigma }(x(k)){\alpha }_{\sigma }(k)}-{\tilde{\delta }}_{\sigma }(k)\hfill \\ \multicolumn{1}{c}{}& +{\epsilon }_{\sigma }(Z(k))-{\delta }_{\sigma }^{*}-{\beta }_{\sigma }(k)/{\alpha }_{\sigma }(k)\hfill \end{array}$$

Then, $\mathrm{\Delta }{L}_2(k)$ becomes

$$\begin{array}{cc}\multicolumn{1}{c}{\mathrm{\Delta }{L}_2(k)=}& -2{\gamma }_{\omega \sigma }{\tilde{\omega }}_{\sigma }^T(k){\hat{\omega }}_{\sigma }(k)-\frac{2z_n^2(k+1)}{g_{\sigma }(x(k)){\alpha }_{\sigma }(k)}\hfill \\ \multicolumn{1}{c}{}& +2{\tilde{\delta }}_{\sigma }(k)z_n(k+1)-2{\epsilon }_{\sigma }(Z(k))z_n(k+1)\hfill \\ \multicolumn{1}{c}{}& +2{\delta }_{\sigma }^{*}{z}_n(k+1)+2{\beta }_{\sigma }(k)/{\alpha }_{\sigma }(k)z_n(k+1)\hfill \\ \multicolumn{1}{c}{}& +{\gamma }_{\omega \sigma }^2{\tau }_{\omega \sigma }{\hat{\omega }}_{\sigma }^T(k){\hat{\omega }}_{\sigma }(k)+2{\gamma }_{\omega \sigma }{\tau }_{\omega \sigma }{\hat{\omega }}_{\sigma }^T(k)\hfill \\ \multicolumn{1}{c}{}& \times \phi (Z(k))z_n(k+1)+{\tau }_{\omega \sigma }{z}_n(k+1)\hfill \\ \multicolumn{1}{c}{}& \times {\phi }^T(Z(k))\phi (Z(k))z_n(k+1)\hfill \end{array}$$

Consider the following facts

$$\begin{array}{c}{\tilde{\omega }}_i^T(k){\hat{\omega }}_i(k)={\tilde{\omega }}_i^T(k){\tilde{\omega }}_i(k)+{\hat{\omega }}_i^T(k){\hat{\omega }}_i(k)-{\omega }_i^{*T}{\omega }_i^{*}\hfill \\ -\frac{2}{g_i(x(k)){\alpha }_i(k)}{z}_n^2(k+1)⩽-\frac{2}{\overline{g}\overline{\alpha }}{z}_n^2(k+1)\hfill \\ -2{\epsilon }_i(Z(k))z_n(k+1)⩽\frac{\overline{g}\overline{\alpha }}{{\tau }_{\omega i}}{\overline{\epsilon }}_i^2+\frac{{\tau }_{\omega i}}{\overline{g}\overline{\alpha }}{z}_n^2(k+1)\hfill \\ 2{\delta }_i^{*}{z}_n(k+1)⩽\frac{\overline{g}\overline{\alpha }}{{\tau }_{\omega i}}{\delta }_i^{*2}+\frac{{\tau }_{\omega i}}{\overline{g}\overline{\alpha }}{z}_n^2(k+1)\hfill \\ 2\frac{{\beta }_i(k)}{{\alpha }_i(k)}{z}_n(k+1)⩽\frac{\overline{g}\overline{\alpha }}{{\tau }_{\omega i}}{\delta }_i^{*2}+\frac{{\tau }_{\omega i}}{\overline{g}\overline{\alpha }}{z}_n^2(k+1)\hfill \\ 2{\gamma }_{\omega i}{\tau }_{\omega i}{\hat{\omega }}_i^T(k)\phi (Z(k))z_n(k+1)\hfill \\ ⩽\overline{g}\overline{\alpha }{\tau }_{\omega i}{\gamma }_{\omega i}^2\parallel {\hat{\omega }}_i(k){\parallel }^2+\frac{l{\tau }_{\omega i}}{\overline{g}\overline{\alpha }}{z}_n^2(k+1)\hfill \\ {\tau }_{\omega i}{z}_n(k+1){\phi }^T(Z(k))\phi (Z(k))z_n(k+1)⩽l{\tau }_{\omega i}{z}_n^2(k+1)\hfill \end{array}$$

then, $\mathrm{\Delta }{L}_2(k)$ can be rewritten as

$$\begin{array}{c}\mathrm{\Delta }{L}_2(k)⩽-{\gamma }_{\omega \sigma }{\tilde{\omega }}_{\sigma }^T(k){\tilde{\omega }}_{\sigma }(k)-{\gamma }_{\omega \sigma }{\hat{\omega }}_{\sigma }^T(k){\hat{\omega }}_{\sigma }(k)+{\gamma }_{\omega \sigma }{\omega }_{\sigma }^{*T}{\omega }_{\sigma }^{*}\hfill \\ -\frac{2}{\overline{g}\overline{\alpha }}{z}_n^2(k+1)+2{\tilde{\delta }}_{\sigma }(k)z_n(k+1)+\frac{\overline{g}\overline{\alpha }}{{\tau }_{\omega \sigma }}{\overline{\epsilon }}_{\sigma }^2\hfill \\ +\frac{{\tau }_{\omega \sigma }}{\overline{g}\overline{\alpha }}{z}_n^2(k+1)+2\frac{\overline{g}\overline{\alpha }}{{\tau }_{\omega \sigma }}{\delta }_{\sigma }^{*2}+2\frac{{\tau }_{\omega \sigma }}{\overline{g}\overline{\alpha }}{z}_n^2(k+1)\hfill \\ +{\gamma }_{\omega \sigma }^2{\tau }_{\omega \sigma }{\hat{\omega }}_{\sigma }^T(k){\hat{\omega }}_{\sigma }(k)+\overline{g}\overline{\alpha }{\tau }_{\omega \sigma }{\gamma }_{\omega \sigma }^2\parallel {\hat{\omega }}_{\sigma }(k){\parallel }^2\hfill \\ +\frac{l{\tau }_{\omega \sigma }}{\overline{g}\overline{\alpha }}{z}_n^2(k+1)+l{\tau }_{\omega \sigma }{z}_n^2(k+1)\hfill \\ ⩽(-\frac{2}{\overline{g}\overline{\alpha }}+\frac{3{\tau }_{\omega \sigma }}{\overline{g}\overline{\alpha }}+\frac{l{\tau }_{\omega \sigma }}{\overline{g}\overline{\alpha }}+l{\tau }_{\omega \sigma })z_n^2(k+1)\hfill \\ +(-{\gamma }_{\omega \sigma }+{\gamma }_{\omega \sigma }^2{\tau }_{\omega \sigma }+\overline{g}\overline{\alpha }{\tau }_{\omega \sigma }{\gamma }_{\omega \sigma }^2)\parallel {\hat{\omega }}_{\sigma }(k){\parallel }^2\hfill \\ -{\gamma }_{\omega \sigma }\parallel {\tilde{\omega }}_{\sigma }(k){\parallel }^2+2{\tilde{\delta }}_{\sigma }(k)z_n(k+1)\hfill \\ +{\gamma }_{\omega \sigma }\parallel {\omega }_{\sigma }^{*}{\parallel }^2+\frac{\overline{g}\overline{\alpha }}{{\tau }_{\omega \sigma }}{\overline{\epsilon }}_{\sigma }^2+2\frac{\overline{g}\overline{\alpha }}{{\tau }_{\omega \sigma }}{\delta }_{\sigma }^{*2}\hfill \end{array}$$

(24)

Similar to $\mathrm{\Delta }{L}_2(k)$ , $\mathrm{\Delta }{L}_3(k)$ can be obtained as

$$\begin{array}{cc}\multicolumn{1}{c}{\mathrm{\Delta }{L}_3(k)}& =\sum _{i=1}^M\frac{1}{{\tau }_{\delta i}}{\tilde{\delta }}_i^2(k+1)-\sum _{i=1}^M\frac{1}{{\tau }_{\delta i}}{\tilde{\delta }}_i^2(k)\hfill \\ \multicolumn{1}{c}{}& \begin{array}{c}⩽(\frac{{\tau }_{\delta \sigma }{\tau }_{\omega \sigma }}{\overline{g}\overline{\alpha }}+{\tau }_{\delta \sigma })z_n^2(k+1)\hfill \\ \multicolumn{1}{c}{+({\gamma }_{\delta \sigma }^2{\tau }_{\delta \sigma }+\frac{\overline{g}\overline{\alpha }{\tau }_{\delta \sigma }{\gamma }_{\delta \sigma }^2}{{\tau }_{\omega \sigma }}-{\gamma }_{\delta \sigma })}\end{array}\hfill \\ \multicolumn{1}{c}{}& {\hat{\delta }}_{\sigma }^2(k)+{\gamma }_{\delta \sigma }{\delta }_{\sigma }^{*2}\hfill \\ \multicolumn{1}{c}{}& -2{\tilde{\delta }}_{\sigma }(k)z_n(k+1)-{\gamma }_{\delta \sigma }{\tilde{\delta }}_{\sigma }^2(k)\hfill \end{array}$$

(25)

Combining equations (23), (24), and (25), we can conclude that

$$\begin{array}{c}\mathrm{\Delta }L(k)⩽-\frac{B_1}{\overline{g}\overline{\alpha }}{z}_n^2(k+1)-{\gamma }_{\omega \sigma }{B}_2\parallel {\hat{\omega }}_{\sigma }(k){\parallel }^2-{\gamma }_{\delta \sigma }{B}_3{\hat{\delta }}_{\sigma }^2(k)\hfill \\ -\frac{1}{\overline{g}\overline{\alpha }}{z}_n^2(k)-{\gamma }_{\omega \sigma }\parallel {\tilde{\omega }}_{\sigma }(k){\parallel }^2-{\gamma }_{\delta \sigma }{\tilde{\delta }}_{\sigma }^2(k)+A\hfill \end{array}$$

where the definitions of $B_1$ , $B_2$ , $B_3$ , and A have been given in equations (15)–(18).

If conditions (19), (20), and (21) are satisfied, then $\Delta L(k)$ can be further expressed as

$$\begin{array}{cc}\multicolumn{1}{c}{\mathrm{\Delta }L(k)}& ⩽-\frac{1}{\overline{g}\overline{\alpha }}{z}_n^2(k)-{\gamma }_{\omega \sigma }\parallel {\tilde{\omega }}_{\sigma }(k){\parallel }^2\hfill \\ \multicolumn{1}{c}{}& -{\gamma }_{\delta \sigma }{\tilde{\delta }}_{\sigma }^2(k)+A\hfill \end{array}$$

(26)

On the light of equation (26), we can know that if $|z_n(k)|>\sqrt{\overline{g}\overline{\alpha }A}$ or $\parallel {\tilde{\omega }}_i(k)\parallel >\sqrt{A/{\gamma }_{\omega i}}$ or $|{\tilde{\delta }}_i(k)|>\sqrt{A/{\gamma }_{\delta i}}$ is satisfied, then $\mathrm{\Delta }L(k)<0$ . Thus, $z_n(k)$ , ${\tilde{\omega }}_i(k)$ , and ${\tilde{\delta }}_i(k)$ are bounded based on the Lyapunov extension theorem. Because $z_m(k)=z_n(k-n+m),m=1,\dots ,n$ , $z_m(k)$ is also bounded. According to ${\tilde{\omega }}_i(k)={\hat{\omega }}_i(k)-{\omega }_i^{*}$ , ${\tilde{\delta }}_i(k)={\hat{\delta }}_i(k)-{\delta }_i^{*}$ , and the boundedness of ${\omega }_i^{*}$ and ${\delta }_i^{*}$ , ${\hat{\omega }}_i(k)$ and ${\hat{\delta }}_i(k)$ are also bounded. Therefore, all the signals are bounded under arbitrary switching signals.

The proof of Theorem 1 is completed.

• Remark 3. In this article, a class of switched adaptive laws are designed. That is to say, different update laws are applied for active and inactive periods of each subsystem, which implies that the estimation value of one subsystem is updated in a way when it is active, and in another way, when it is inactive. It is not difficult to know that this kind of switched adaptive laws is more suitable for switched systems than the nonswitched one.

• Remark 4. The parameters of the actuator fault in this paper are all unknown but bounded. In order to solve this problem, the adaptive method is used. Let ${\beta }_i(k)/{\alpha }_i(k)⩽{\overline{\beta }}_i/{\underset{¯}{\alpha }}_i\stackrel{\mathrm{\Delta }}{=}{\delta }_i^{*}$ . It is obvious that ${\delta }_i^{*}$ is an unknown constant. Then, using the adaptive method, define ${\hat{\delta }}_i(k)$ as the estimation of ${\delta }_i^{*}$ and design adaptive law (13) to regulate ${\hat{\delta }}_i(k)$ . Thus, this problem is solved.

• Remark 5. Compared with the existing results, the following aspects should be emphasized:

• Unlike the methods in previous works^4,7 and Kang et al., ¹⁰ in which the adaptive FTC problem for nonswitched systems were considered, this article concentrates on the case for switched system. Due to the complicity of the switched systems, it brings a great obstacle in the design of controller.

• Although the FTC issue for switched systems were taken into account in previous works,^19–23 they all focus on the continuous-time switched systems (especially for the continuous-time switched linear systems). The FTC problem for discrete-time switched nonlinear systems is studied in this article. For continuous switched systems, the switching may take place at any time instants. However, in discrete-time switched systems, the switching must occur at a certain sampling point. Furthermore, less analysis schemes can be employed to solve the FTC problem for discrete-time switched nonlinear systems. This makes it more difficult for designing controllers.

Simulation

In order to explain the effectiveness of the proposed method, we apply it to the following system

$$\{\begin{array}{c}{x}_1(k+1)=x_2(k)\hfill \\ \multicolumn{1}{c}{x_2(k+1)=f_{\sigma }(x(k))+g_{\sigma }(x(k))u_{\sigma }(k)}\\ \multicolumn{1}{c}{y(k)=x_1(k)}\end{array}$$

where

$$\begin{array}{c}{f}_1(x(k))=\frac{x_1(k)}{x_1(k)+x_2^2(k)}\hfill \\ f_2(x(k))=\frac{x_1(k)x_2(k)}{x_1(k)+x_2^2(k)}\hfill \\ g_1(x(k))=x_1(k)x_2(k)\hfill \\ g_2(x(k))=x_1(k)+x_2(k)\hfill \end{array}$$

The switching signal is $\sigma =\{1,2\}$ .

Consider the following actuator fault

$$u_1(k)=\{\begin{array}{c}{u}_1^0(k),\mathrm{if}0⩽k<80\hfill \\ \multicolumn{1}{c}{0.7u_1^0(k)+\sin (k),\mathrm{if}k\ge 80}\end{array}$$

$$u_2(k)=\{\begin{array}{c}{u}_2^0(k),\mathrm{if}0⩽k<80\hfill \\ \multicolumn{1}{c}{0.2u_2^0(k)+\cos (k),\mathrm{if}k\ge 80}\end{array}$$

Select the given signals as

$$x_{r1}(k)=0.32\sin (0.2k)+0.32\cos (0.2k)$$

and

$$x_{r2}(k)=0.22\sin (0.2k)+0.22\cos (0.2k).$$

The FLSs are employed to approximate $u_i^0(k)$ , in which the fuzzy membership functions are chosen as

$${\mu }_{F_m^l}(x_m)=\exp \left(\frac{{(x_m-3+2l)}^2}{2}\right),m=1,2,l=1,2,3$$

The initial values of the system states and the adaptive laws are $x_1(0)=x_2(0)=0$ , ${\omega }_1(0)=0.01$ , ${\omega }_2(0)=0.05$ , and ${\delta }_1(0)={\delta }_2(0)=0.05$ , and the related parameters in this simulation are selected as ${\tau }_{\omega 1}=0.65$ , ${\tau }_{\omega 2}=0.35$ , ${\gamma }_{\omega 1}=0.2$ , ${\gamma }_{\omega 2}=0.4$ , ${\tau }_{\delta 1}=0.7$ , ${\tau }_{\delta 2}=0.6$ , ${\gamma }_{\delta 1}=0.5$ , and ${\gamma }_{\delta 2}=0.6$ .

Figures 1–4 show the simulation results. Figure 1 is the switching signal. Figure 2 displays the trajectories of $x_1$ and $x_{r1}$ . Figure 2 presents the curves of $x_2$ and $x_{r2}$ . Observing Figures 2 and 3, we can easily know that a good tracking performance is obtained. That is to say the system states can well track the reference signals. The control input is shown in Figure 4. Before the 80th second, there is no fault, and the curve of the control signal is bounded. However, after the actuator fault occurs at the 80th second, the trajectory of the control signal changes, but still bounded, obviously. Thus, Theorem 1 is well illustrated.

Figure 1.

Switching signal $\sigma$ .

Figure 2.

The trajectories of $x_1$ and $x_{r1}$ .

Figure 3.

The trajectories of $x_2$ and $x_{r2}$ .

Figure 4.

Control input u.

Conclusion

The actuator FTC problem for a class of discrete-time switched nonlinear systems with unknown functions is coped with in this article. The FLSs are utilized to approximate the unknown functions. Both the actuator LOE and bias faults are taken into account, and all the parameters of fault are unknown but bounded, which is solved by adaptive technique. Design the subsystem controllers and adaptive laws in order to ensure that all the signals are bounded under arbitrary switching signals, and the system states can track the reference signals. Finally, the simulation example well shows the effectiveness of the developed method. In the future, we will try our best to discuss the continuity of solution rather than assume that in a switched system.