Stability Analysis for Digital Redesign of Discrete-Time Switched Systems Using H_∞ Linear Matrix Inequality

Abstract

In this paper, the stability problem for the digital redesign of discrete-time switched systems using H_∞ linear matrix inequality (LMI) is investigated. We propose the switching time approach for digital redesign between controller work and failure, and this switching time will limit the system output within the system capacity. When the controller fails, the overall system will be unstable. Therefore, if the digital redesign controller is not restored in a certain period of time, the system output will exceed the system capacity. To solve this problem, we propose a switching law to determine the switching time between the stable mode (controller work) and the unstable (controller failure) mode; this will limit the overall system states in the unstable mode. In addition, the digital redesign controller has the advantage of faster tracking. After we propose a discrete-time switching system with stable and unstable modes, we use H_∞ linear matrix inequality (LMI) and Lyapunov functions to prove the stability in detail. Finally, the numerical example illustrates the feasibility of the proposed approach.

1. Introduction

Many systems are dynamic systems that exhibit continuous and discrete dynamic behaviours. These systems that include two kinds of dynamic behaviour and interaction are called hybrid systems—for example, embedded systems, communication networks, and aircraft and traffic control. The history of hybrid systems began in the 1950s and attracted attention in the 1990s to implement the development of digital microcontrollers and embedded devices. Switched systems [1,2,3,4,5,6,7,8] are a special class of hybrid systems that might contain continuous-time or discrete-time subsystems and a switching law orchestrating switching between these subsystems and can be viewed as higher-level abstractions of hybrid systems.

In fact, many practical systems have multimodal dynamical subsystems to exhibit diversified behaviour, which might depend on various environments. In recent years, switched system research has seen an increasing trend and has made great efforts; see, e.g., [9,10,11]. The research is always focused on stability analysis and controller design to guarantee switched systems stability and achieve good performance. As we know, there are several methods applied to stability analysis of those systems, namely Lyapunov function [12,13,14,15], linear matrix inequality [16,17,18], and dwell time [19,20,21]. However, the above and recent studies have not discussed the stability analysis of the switching system based on the digital redesign in the stable mode and the unstable mode. This paper is based on the first and third cases and uses the $H_{\infty }$ linear matrix inequality to cooperate with the piecewise Lyapunov method functions and the average dwell time to derive switching laws. In addition, the proposed switching law can be guaranteed to be convergent after the detailed proof in the Appendix A and Appendix B.

In terms of contributions, this paper successfully derives switching laws by $H_{\infty }$ linear matrix inequality for unstable discrete-time switched systems and proves that state error can be bounded. Because state error is bounded, the system performance can also be bounded in a certain range. That tells us whether the controller will work or fail for an unstable discrete-time switched system, and we can know beforehand the approximate bound of system performance, which will be our expectation or not. Finally, the numerical example illustrates the feasibility of the proposed approach, and the robust performance and tracking performance for different Q values are shown in detail.

The stability problem for the digital redesign of discrete-time switched systems using $H_{\infty }$ linear matrix inequality (LMI) is proposed in this paper. The proposed switching law is derived and strictly proves the stability. This paper is organized as follows. The switched system is briefly described along with $H_{\infty }$ linear matrix inequality constraint in Section 2. The prediction-based digital redesign is introduced in Section 3. The main results and proofs are in Section 4, and an illustrative example is presented in Section 5 to demonstrate the effectiveness of the proposed methodology. Finally, the conclusion is addressed in Section 6.

2. Systems Description and Preliminaries

The stability analysis of the switched systems and the controller design have been our focus points in this paper, given the discrete-time switched systems [22] as follows:

$$\{\begin{array}{cc}\hfill x(k+1)& =A_{\sigma (k)}x(k)+B_{\sigma (k)}u(k),x(0)=x_0\hfill \\ \hfill y(k)& =C_{\sigma (k)}x(k),\hfill \end{array}$$

(1)

where $\sigma (k):N^{+}\to I_N=\left\{1,2,\cdots ,N\right\},N>1$, $N^{+}$ denotes the set of all nonnegative integers, and $\sigma (k)$ is called a switching signal. $N$ is the number of subsystems, which must be greater than 1, showing that it is $N$ times that we switch over discrete-time switched systems (1). Every subsystem has its corresponding constant matrix, which gives $A_i$, $B_i$, and $C_i$ ($i\in I_N)$ to express. To simplify, all $k$ represents $kT$, and $k_a-k_b$ represents $(k_a-k_b)T$ ($T$ is the sampling time). Since all $A_i$ or some $A_i$ are stable, discrete-time switched systems, (1) can be exponentially stable under switching laws and satisfy the following inequality:

$$N_{{}_{\sigma }}(0,k)\le N_0+\frac{k}{\tau }, N_0\ge 0, \tau >0,$$

(2)

where $N_{{}_{\sigma }}(0,k)$ denotes the number of switching signals of $\sigma (k)$ on the interval $[0,k)$; $\tau$ is called the average dwell time; and $N_0$ is the chatter bound. For the concept of average dwell time, see [23]. For example, if $N_0=1$, then inequality (2) implies that $\sigma (k)$ cannot switch twice on any interval of length smaller than $\tau$. Switching signals with this property are exactly the switching signals with dwell time $\tau$. Note, also, that $N_0=0$ corresponds to the case of no switching since $\sigma (k)$ cannot switch at all on any interval of length smaller than $\tau$. A detailed description is that consecutive switchings are separated by less than $\tau$, but the average time interval between consecutive switchings is not less than $\tau$. In general, if we discard the first $N_0$ switchings, then the average time interval between consecutive switchings is at least $\tau$. We use the idea of average dwell time to derive a class of switching laws that exponentially stabilize the switched system (1) where both stable and unstable subsystems exist. As we obtain switching laws, we can know the upper and lower working time limits, which relate to controller design in the switched system (1).

For the controller design, we use the method of prediction-based digital redesign, and then our focal point is placed on utilizing $H_{\infty }$ linear matrix inequality to cooperate with the piecewise Lyapunov method functions [24,25] and the preceding concept of average dwell time to derive switching laws. From [26,27,28,29,30], the $H_{\infty }$ linear matrix inequality constraint is as follows:

Lemma 1 ([29]).

Given any transfer function $G_i\equiv D_i+C_i{(zI-A_i)}^{-1}{B}_i$ (not necessarily minimal), we have

$${\Vert G_i\Vert }_{\infty }<r_i,$$

$A_i$ asymptotically stable.

If and only if the following LMI in $P_i$ is feasible:

$$\left[\begin{array}{ccc}{A}_i^T{P}_i{A}_i-P_i& A_i^T{P}_i{B}_i& C_i^T\\ B_i^T{P}_i{A}_i& B_i^T{P}_i{B}_i-r_{i}I& D_i^T\\ C_i& D_i& -r_{i}I\end{array}\right]<0, P_i>0$$

(3)

Lemma 1 stipulates that all $A_i$ must be asymptotically stable. In much research, $A_i$ addresses the situation of asymptotically stable, but when discrete-time switched system (1) is not asymptotically stable, and the controller fails $(u(k)=0)$, how should we deal with this problem? To address this problem, this paper proposes a method to solve and prove that the norm of tracker error $e(k)$ is bounded in a certain range and derives switching laws suitable for unstable discrete-time switched systems (1).

3. The Prediction-Based Digital Redesign

3.1. Optimal Linearization

We apply the optimal linearization methodology [30] for generating optimal local models, which provides a tool for digital redesign for the class of time-varying nonlinear systems. This method is briefly described here and is used in the following.

Consider a class of nonlinear systems as follows:

$$\dot{x}(t)= f(x(t))+ g(x(t)) u(t),$$

(4)

where $f(\cdot ) : {\Re }^n \to {\Re }^n$ and $g(\cdot ) : {\Re }^n \to {\Re }^m$ are nonlinear functions; $x(t) \in {\Re }^n$ is the state vector; and $u(t) \in {\Re }^m$ is the control input. Assume that a local “linear model” $(A_l , B_l)$ at the operation state, $x_l(t) \in {\Re }^n$, is desired in the form

$$\dot{x}(t)= A_l x(t)+ B_l u(t),$$

(5)

Assume that the operating state, $x_l \ne 0$, is not necessarily an equilibrium of the given system (4). One wishes to find two constant matrices $A_l$ and $B_l$ such that

$$f(x)+ g(x) u \approx A_l x+ B_l u \mathrm{for} \mathrm{any} u,$$

(6)

$$f(x_l)+ g(x_l) u_k = A_l x_l+ B_l u \mathrm{for} \mathrm{any} u .$$

(7)

$$g(x_l^{})= B_l.$$

(8)

Therefore, (6) and (7) can be reduced to a brief form as

$$f(x) \approx A_l x,$$

(9)

and

$$f(x_l) = A_l x_l.$$

(10)

To satisfy these, $a_j{}^T$ is denoted as the $j$th row of the matrix $A_l$ so that (9) and (10) can be represented as

$$f_j (x) \approx a_j{}^T x, j= 1 , 2 , \dots , n$$

(11)

and

$$f_j (x_l) = a_j{}^T x_l, j= 1 , 2 , \dots , n,$$

(12)

respectively, where $f_j(\cdot ) : {\Re }^n \to \Re$ is the $j$th component of the vector $f(\cdot )$. Then, expanding the left-hand side of (11) about $x_l$ and neglecting the second- and higher-order terms, we can obtain

$$f_j(x_l)+{[\nabla f_j(x_l)]}^T(x-x_l)\approx a_j{}^{T}x,$$

(13)

where $\nabla f_j(x_l):{\Re }^n\to {\Re }^n$ is the gradient column vector of $f_j$ evaluated at $x_l$. Now, using (12), one can rewrite (13) as

$${[\nabla f_j(x_l)]}^T(x-x_l)\approx a_j{}^T(x-x_l)$$

in which $x$ is arbitrary but should be close to $x_l$ so that the approximation is good. To determine a constant vector, $a_j{}^T$, such that it is as close as possible to ${[\nabla f_j(x_l)]}^T$ and satisfies $a_j{}^T x_l=f_j(x_l)$, our objective can, thus, be formulated as a constrained optimization problem to minimize

$$E \equiv \frac{1}{2} {\Vert \nabla f_j (x_l)- a_j\Vert }_2{}^2 \mathrm{subject} \mathrm{to} a_j{}^T{x}_l = f_j (x_l),$$

(14)

where $a_j{}^T$ is the $j$th row of the matrix $A_l$. Note that this is also a convex-constrained optimization problem. Thus, using the Lagrange multiplier method, the optimal solution is

$$a_j= \nabla f_j (x_l)+ \frac{f_j(x_l)+ x_l{}^T \nabla f_j(x_l)}{{\Vert x_l\Vert }_2{}^2} \cdot x_l \mathrm{for} x_l \ne 0,$$

(15)

where ${\Vert x_l\Vert }_2{}^2=x_l{}^T x_l$ is the square magnitude of the components of $x_l$.

The controllability matrix for the nonlinear system (4) at the operating state $x_l$ is derived from the optimal linear model $A_l, B_l$, resulting in

$$\overline{C}= [{\overline{B}}_l {\overline{A}}_l {\overline{B}}_l {\overline{A}}_l{}^2{\overline{B}}_l \cdots {\overline{A}}_l{}^{n-1}{\overline{B}}_l],$$

(16)

where ${\overline{A}}_l$ and ${\overline{B}}_l$ are constructed via the following rule: the $q$th columns of $A_l$ and $B_l$ are set to zero whenever the $q$th component of $x_l$ is zero.

Remark 1. ([31]).

It is well known that the high-gain property can suppress system uncertainties [31]. For this reason, the high-gain property is adopted in this paper.

3.2. Linear-Quadratic Analogue Tracker Design

Consider a linear-quadratic analogue system described as

$$\begin{array}{l}{\dot{x}}_c (t)= A x_c (t)+ B u_c (t) \\ y_c (t)= C x_c (t)+D u_c (t), x_{ c} (0)= x_0\end{array}$$

(17)

which is assumed to be both controllable and observable, where $x_c(t) \in {\Re }^n$, $u_c(t) \in {\Re }^m$, and $y_c(t) \in {\Re }^p$. The optimal state-feedback control law minimizes the following performance index:

$$J= {\displaystyle {\int }_0^{\infty } \left\{{[C x_c (t)- y_r (t)]}^T Q [C x_c (t)- y_r (t) ]+ u_c^T(t)R u{}_c(t)\right\}dt},$$

(18)

with $Q\ge 0$ and $R>0$ for system (17) with $D=0$. This optimal control [31] is given by

$$u_c(t)= -K_c x_c(t)+ E_c y_r(t).$$

(19)

The resulting closed-loop system becomes

$${\dot{x}}_c(t)= (A- B K_c) x_c(t)+ B E_c y_r(t),$$

(20)

where

$$K_c= R^{-1} B^T P,$$

(21)

$$E_c= -R^{-1} B^T {[{(A- B K_c)}^{ -1}]}^T C^T Q.$$

(22)

Here, $y_r(t)$ is a reference input and $P$ is the solution of the Riccati equation:

$$A^T P+ P A- P B R^{-1}{B}^T P+ C^{T}Q C= 0.$$

(23)

Remark 2. ([31]).

It is noted that the high-gain property can be obtained by choosing a sufficiently high ratio of $Q$ to $R$  in (18) so that the system output can closely track the reference input. However, the high-gain property of the tracker usually yields large control signals, which might cause the system actuator to saturate and give an unsatisfactory system response. To achieve perfect performance, sometimes a search process is needed for the selection of the weighting matrices.

3.3. Observer-Based Linear-Quadratic Analogue Tracker Design

Consider the situation in which the system state of (17) cannot all be measured. Then, the observer can be used to estimate the unmeasured system state. Consider the linear observable continuous-time system in Figure 1, which is described as follows:

$${\dot{\widehat{x}}}_c(t)= A {\widehat{x}}_c(t) + B u_c(t)+ L [y_c(t)- C {\widehat{x}}_c(t)]$$

(24)

where ${\widehat{x}}_c(t)$ is the estimate of $x_c(t)$, and $L\in {\Re }^{n\times p}$ is the observer gain. Let the estimation error be

$${\tilde{x}}_c(t)= x_c(t)- {\widehat{x}}_c(t),$$

(25)

which implies

$${\dot{\tilde{x}}}_c(t)= {\dot{x}}_c(t)- {\dot{\widehat{x}}}_c(t).$$

(26)

Substituting (17) with $D=0$ and (24) into (26) yields

$${\dot{\tilde{x}}}_c(t)= (A- L C) {\tilde{x}}_c(t),$$

(27)

which contrasts (20) and (27) with $y_r(t)= 0$. One can see that

$${(A- L C)}^T= A^T- C^T L^T,$$

(28)

which has the structure as a state-feedback controller. The optimal control gain $K_c$ can be used to select the observer gain $L$ as follows:

$$L= K_c{}^T= P_{ob} C^T R_{ob}{}^{- 1},$$

(29)

where $P_{ob}$ is the solution of the following Riccati equation:

$$A P_{ob}+ P_{ob} A^T- P_{ob} C^T R_{ob}{}^{- 1} C P_{ob}+ Q_{ob}= 0,$$

(30)

in which $Q_{ob}\ge 0$, and $R_{ob}> 0$.

3.4. Digital Redesign of the Linear-Quadratic Analogue Tracker

Let the continuous-time state-feedback controller be

$$u_c(t)= -K_c x_c(t)+ E_c y_r(t),$$

(31)

where $K_c \in {\Re }^{m\times n}$ and $E_c \in {\Re }^{n\times m}$ have been designed to satisfy some specified goals, and $y_r(t)\in {\Re }^m$ is a desired reference input vector. Thus, the analogously controlled system is

$${\dot{x}}_c(t)= A_c x_c(t)+ B E_c y_r(t), x_c(0)=x_{c0}= x_0,$$

(32)

where $A_c= A- B K_c$. Let the state equation of a corresponding discrete-time equivalent model be

$${\dot{x}}_d(t)= A x_d(t)+ B u_d(t), x_d(0)= x_{d0}=x_0,$$

(33)

where $u_d(t)\in {\Re }^m$ is a piecewise-constant input vector, satisfying

$$u_d(t)= u_d(kT), \mathrm{for} kT\le t < (k+ 1) T,$$

and $T > 0$ is the sampling period. Then, the discrete-time state-feedback controller is given by [30]

$$u_d(kT)= -K_d x_d(kT)+ E_d y_r{}^{*}(kT)$$

(34)

where

$$K_d= {(I_m+ K_c H)}^{ - 1} K_c G,$$

(35)

$$E_d= {(I_m+ K_c H)}^{ - 1} E_c,$$

(36)

$$y_r{}^{*}(kT)= y_r(kT+ T),$$

(37)

$$G= e^{A T},$$

(38)

$$H= (G- I_n) A^{- 1} B,$$

(39)

Whenever $A^{- 1}$ does not exist, $H$ can be computed by the following formula [32]:

$$H={\displaystyle \sum _{v=1}^{\infty } \frac{1}{v !} {(AT)}^{ v-1} B T}.$$

The digitally controlled closed-loop system thus becomes

$$\begin{gathered} {\dot{x}}_d(t)= A x_d(t)+ B [-K_d x_d(kT)+ E_d y_r{}^{*}(kT) ] , x_d(0)= x_{d0} , \\ {y}_d(kT)= C x_d(kT), \mathrm{for} kT\le t < (k+ 1) T. \end{gathered}$$

(40)

Remark 3 (sampling period selection).

It is noted that the mapping of a continuous-time system to its corresponding discretized system can be one-to-one if the selected sampling period satisfies the sampling theorem [33].

3.5. Digital Redesign of the Observer-Based Linear-Quadratic Analogue Tracker

Consider the linear observable continuous-time system

$${\dot{\widehat{x}}}_c(t)= A {\widehat{x}}_c(t)+ B u_c(t)+ L [y_c(t)- C {\widehat{x}}_c(t)],$$

(41)

by defining the continuous-time and discrete-time state estimate errors, respectively, as

$$\begin{gathered} {\tilde{x}}_c(t) \equiv x_c(t)- {\widehat{x}}_c(t), \\ {\tilde{x}}_d(kT) \equiv x_d(kT)- {\widehat{x}}_d(kT), \end{gathered}$$

(42)

then

$${\tilde{x}}_d(kT) \approx {\tilde{x}}_c(t)|{}_{ t = k T}.$$

Using the duality once again, the discrete-time state estimation error dynamics can be found as follows:

$${\tilde{x}}_d(kT+ T)= (G- M N) {\tilde{x}}_d(kT),$$

(43)

where

$$G= e^{A T_{ }},$$

(44)

$$M= (G-I_n) A^{- 1}L,$$

(45)

$$N= {(I_{ m}+ C M)}^{ - 1}C G.$$

(46)

Further define

$$L_d= M {(I_m+ C M)}^{ - 1},$$

(47)

then one has

$$M N= (G- I_n) A^{- 1} L {(I_{ m}+ C M)}^{ - 1} C G= L C G.$$

(48)

Since

$$\begin{gathered} x_d(kT+ T)= G x_d(kT)+ H u_d(kT), \\ {y}_d(kT)= C x_d(kT), \end{gathered}$$

(49)

from (49), one has

$$C G x_d(kT)= y_d(kT+ T)- C H u_d(kT).$$

(50)

Substituting (48) into (43) yields

$$\begin{array}{cc}{\tilde{x}}_{ d}(kT+ T)& = x_d(kT+ T)- {\widehat{x}}_{ d}(kT+ T)\\ & = (G- L_{ d} C G)(x_d(kT)- {\widehat{x}}_d(kT))\\ & = (G- L_d C G) x_d(kT)- (G- L_d C G) {\widehat{x}}_d(kT).\end{array}$$

(51)

Substituting (50) into (51), one has

$$\begin{array}{cc}{\tilde{x}}_d(kT+ T)& = x_d(kT+ T)- {\widehat{x}}_d(kT+ T)\\ & = [G x_d(kT)+ H u_d(kT)]- {\widehat{x}}_d(kT+ T)\end{array}$$

(52)

$$\begin{gathered} = G x_d(kT)- L_d [y_d(kT+ T)- C H u_d(kT)]- (G- L_d C G) {\widehat{x}}_d(kT) \\ = G x_d(kT)- L_d y_d(kT+ T)+ L_d C H u_d(kT)- (G- L_d C G) {\widehat{x}}_d(kT). \end{gathered}$$

(53)

From (52) and (53), one has the discrete observer

$$\begin{array}{cc}{\widehat{x}}_d(kT+ T)& =(G-L_d C G) {\widehat{x}}_d(kT)+(I_n-L_d C) H u_d(kT)+L_d y_d(kT+T)\\ & =G_d {\widehat{x}}_d(kT)+H_d u_d(kT)+L_d y_d(kT+T)\end{array}$$

(54)

and

$${\widehat{y}}_d(kT+T)=C {\widehat{x}}_d(kT+T),$$

where

$$L_d=(G-I_n) A^{-1}L {(I_m+C (G-I_n) A^{-1}L)}^{-1},$$

(55)

$$G_d=G-L_d C G,$$

(56)

$$H_d=(I_n-L_d C) H,$$

(57)

$$G=e^{A T_{ }},$$

(58)

$$H=(G-I_n) A^{-1} B.$$

(59)

Considering the practical implementation, the following discrete observer using the current output $y_d(kT)$ and previously estimated state ${\widehat{x}}_d(kT-T)$ to compute the currently estimated state ${\widehat{x}}_d(kT)$ is suggested as an alternative:

$$\begin{array}{c}{\widehat{x}}_d(kT)= G_d {\widehat{x}}_d(kT-T)+H_d u_d(kT-T)+L_d y_d(kT),\\ {\widehat{y}}_d(kT)=C {\widehat{x}}_d(kT).\end{array}$$

(60)

The observer-based digital tracker and observer for the sampled-data optimal linear model are shown in Figure 2.

4. ${\mathit{H}}_{\mathbf{\infty }}$ Linear Matrix Inequality Constraint with An Unstable Discrete-Time System

We consider an unstable discrete-time system

$$\{\begin{array}{cc}\hfill x_d(k+1)& =A_d{x}_d(k)+B_d{u}_d(k),x_d(0)=x_0\hfill \\ \hfill y_d(k)& =C_d{x}_d(k),\hfill \end{array}$$

(61)

where $A_d\in {\Re }^{n\times n}$, $B_d\in {\Re }^{n\times m}$, and $C_d\in {\Re }^{p\times n}$ are system, input, and output matrices, respectively, and $x_d\in {\Re }^n$, $u_d\in {\Re }^m$, and $y_d\in {\Re }^p$ are state, input, and output vectors. In the controller design, we utilize 3.2 small matters (21)~(23) with 3.4 small matters (34)~(39), where

$$G= e^{A T_{ }}=A_d,H= (G- I_n) A^{- 1} B=B_d.$$

(62)

When the controller works, the unstable discrete-time system (61) changes into a stable discrete-time system as follows:

$$\{\begin{array}{cc}\hfill x_d(k+1)& =A_s{x}_d(k)+B_s{y}_r^{*}(k),x_d(0)=x_0\hfill \\ \hfill y_d(k)& =C_s{x}_d(k),\hfill \end{array}$$

(63)

where $A_s=G-H{K}_d$, $B_s=H{E}_d$, and $C_s=C_d$. When the controller fails ($u_d(k)=0$), the unstable discrete-time system (61) becomes

$$\{\begin{array}{cc}\hfill x_d(k+1)& =A_u{x}_d(k),x_d(0)=x_0\hfill \\ \hfill y_d(k)& =C_d{x}_d(k),\hfill \end{array}$$

(64)

where $A_u=A_d$.

If the controller works, $A_s$ is stable, and according to [27,28,34,35] and Lemma 1, the stable discrete-time system (63) will satisfy the following inequalities:

$$A_s^T{P}_s{A}_s-P_s<0,$$

$$\left[\begin{array}{ccc}{A}_s^T{P}_s{A}_s-P_s& A_s^T{P}_s{B}_s& C_s^T\\ B_s^T{P}_s{A}_s& B_s^T{P}_s{B}_s-r_{0}I& D_s^T\\ C_s& D_s& -r_{0}I\end{array}\right]<0,$$

$$P_s>0,$$

(65)

where ${\Vert G_s\Vert }_{\infty }<r_0$, $G_s\equiv D_s+C_s{(zI-A_s)}^{-1}{B}_s$. In inequality (65), the positive scalar ${\lambda }_s$ (${\lambda }_s<1$) can exist and can be rewritten as follows:

$$A_s^T{P}_s{A}_s-{\lambda }_s^2{P}_s<0,$$

$$\left[\begin{array}{ccc}{A}_s^T{P}_s{A}_s-{\lambda }_s^2{P}_s& A_s^T{P}_s{B}_s& C_s^T\\ B_s^T{P}_s{A}_s& B_s^T{P}_s{B}_s-r_{0}I& D_s^T\\ C_s& D_s& -r_{0}I\end{array}\right]<0,$$

$$P_s>0.$$

(66)

Inequalities (66) can be equivalent to

$$A_s^T{\mathsf{\Lambda }}_s{A}_s-{\lambda }_s^2{\mathsf{\Lambda }}_s<0,$$

$$\left[\begin{array}{cc}{A}_s^T{\mathsf{\Lambda }}_s{A}_s-{\lambda }_s^2{\mathsf{\Lambda }}_s+C_s^T{C}_s& A_s^T{\mathsf{\Lambda }}_s{B}_s+C_s^T{D}_s\\ B_s^T{\mathsf{\Lambda }}_s{A}_s+D_s^T{C}_s& B_s^T{\mathsf{\Lambda }}_s{B}_s-r_0^{2}I+D_s^T{D}_s\end{array}\right]<0,$$

$${\mathsf{\Lambda }}_s>0,$$

(67)

where ${\mathsf{\Lambda }}_s=r_0{P}_s$. The proof (66) to (67) is available in Appendix A.

Inequalities (67) can be used in the LMI control toolbox to obtain the solution of ${\mathsf{\Lambda }}_s$. After the solution of ${\mathsf{\Lambda }}_s$, we define the Lyapunov function candidate

$$\begin{gathered} V\left(x_d\left(k\right)\right)={\left[C_s{x}_d(k)-y_r^{*}(k)\right]}^T{P}_x\left[C_s{x}_d(k)-y_r^{*}(k)\right] \\ ={\left[x_d(k)-x_r^{*}(k)\right]}^T{\mathsf{\Lambda }}_s\left[x_d(k)-x_r^{*}(k)\right], \end{gathered}$$

(68)

where $P_x>\alpha I>0 (\alpha >0)$, $y_r^{*}(k)=C_s{x}_r^{*}(k)$, and ${\mathsf{\Lambda }}_s=C_s^T{P}_x{C}_s$. From (68), one obtains

$$\begin{array}{l}V\left(x_d\left(k+1\right)\right)-V\left(x_d\left(k\right)\right)\\ ={\left[x_d(k+1)-x_r^{*}(k)\right]}^{ T}{\mathsf{\Lambda }}_s\left[x_d(k+1)-x_r^{*}(k)\right]-{\left[x_d(k)-x_r^{*}(k)\right]}^{ T}{\mathsf{\Lambda }}_s\left[x_d(k)-x_r^{*}(k)\right]\\ ={\left\{\left[A_s{x}_d(k)+B_s{y}_r^{*}(k)\right]-x_r^{*}(k)\right\}}^T{\mathsf{\Lambda }}_s\left\{\left[A_s{x}_d(k)+B_s{y}_r^{*}(k)\right]-x_r^{*}(k)\right\}-\\ { \left[x_d(k)-x_r^{*}(k)\right]}^{ T}{\mathsf{\Lambda }}_s\left[x_d(k)-x_r^{*}(k)\right]\\ =\left[\begin{array}{cc}{x}_d{}^T(k)& x_r^{*}{}^{ T}(k)\end{array}\right]\left[\begin{array}{cc}{A}_s^T{\mathsf{\Lambda }}_s{A}_s-{\mathsf{\Lambda }}_s& A_s^T{\mathsf{\Lambda }}_s(B_s{C}_s-I)+{\mathsf{\Lambda }}_s\\ {(B_s{C}_s-I)}^T{\mathsf{\Lambda }}_s{A}_s+{\mathsf{\Lambda }}_s& {(B_s{C}_s-I)}^T{\mathsf{\Lambda }}_s(B_s{C}_s-I)-{\mathsf{\Lambda }}_s\end{array}\right]\left[\begin{array}{c}{x}_d(k)\\ x_r^{*}(k)\end{array}\right].\end{array}$$

(69)

By (67) and (69) can be represented as (see Appendix B)

$$V(x_d(k+1))-V(x_d(k))\le ({\lambda }_s^2-1)V(x_d(k))-c{y}_d{}^T(k)y_d(k)+(r_0^2+\omega )y_r^{*}{}^{ T}(k)y_r^{*}(k),$$

(70)

where $c (c>0)$ is a scalar, and $\omega =\alpha (2{\lambda }_s^2+1)$. We can solve the inequality (70) to obtain

$$V(x_d(k))\le {\lambda }_s^{2(k-k_0)}V(x_d(k_0))-{\displaystyle \sum _{j=k_0}^{k-1}{\lambda }_s^{2(k-1-j)}\Gamma (j)},$$

(71)

where $\Gamma (j)=c{y}_d{}^T(j)y_d(j)-(r_0^2+\omega )y_r^{*}{}^{ T}(j)y_r^{*}(j)$, and inequality (71) denotes the Lyapunov function with the controller working over the interval $[k_{2i},k_{2i+1})$.

Assume the controller of the unstable discrete-time system (61) works over the interval $[k_{2i},k_{2i+1})$ and fails over the interval $[k_{2i+1},k_{2i+2})$, where $i=0,1,2,3\dots$, and $k_0=0$. Now, we first consider that working time $k$ is located in the interval $[k_0,k_1)$ and use the following theorem (see [6]):

$${\alpha }_1{\Vert x_d(k)-x_r^{*}(k)\Vert }^2\le V(x_d(k))\le {\alpha }_2{\Vert x_d(k)-x_r^{*}(k)\Vert }^2, \forall x_d,x_r^{*}\in {\Re }^n$$

(72)

where ${\alpha }_1={\lambda }_m({\mathsf{\Lambda }}_s), {\alpha }_2={\lambda }_M({\mathsf{\Lambda }}_s)$, ${\lambda }_m(\cdot ), {\lambda }_M(\cdot )$ denotes the smallest (largest) eigenvalue of a symmetric matrix, and then

$$V(x_d(k))\ge {\alpha }_1{\Vert x_d(k)-x_r^{*}(k)\Vert }^2, V(x_d(k_0))\le {\alpha }_2{\Vert x_d(k_0)-x_r^{*}(k_0)\Vert }^2.$$

(73)

From (71) and (73), we can obtain

$$\begin{array}{ll}{\alpha }_1{\Vert x_d(k)-x_r^{*}(k)\Vert }^2& \le V(x_d(k))\\ & ={\lambda }_s^{2(k-k_0)}V(x_d(k_0))-{\displaystyle \sum _{j=k_0}^{k-1}{\lambda }_s^{2(k-1-j)}\Gamma (j)}\\ & \le {\lambda }_s^{2(k-k_0)}{\alpha }_2{\Vert x_d(k_0)-x_r^{*}(k_0)\Vert }^2-{\displaystyle \sum _{j=k_0}^{k-1}{\lambda }_s^{2(k-1-j)}\Gamma (j)}, \end{array}$$

(74)

and then

$$\Vert x_d(k)-x_r^{*}(k)\Vert \le \sqrt{\mu {\lambda }_s^{2(k-k_0)}{\Vert x_d(k_0)-x_r^{*}(k_0)\Vert }^2-{\displaystyle \sum _{j=k_0}^{k-1}{\lambda }_c{\lambda }_s^{2(k-1-j)}\Gamma (j)}},$$

(75)

where $\mu =\frac{{\alpha }_2}{{\alpha }_1}$ and ${\lambda }_c=\frac{1}{{\alpha }_1}$. If $k\to k_1$, we can obtain

$$\Vert x_d(k_1)-x_r^{*}(k_1)\Vert \le \sqrt{\mu {\lambda }_s^{2(k_1-k_0)}{\Vert x_d(k_0)-x_r^{*}(k_0)\Vert }^2-{\displaystyle \sum _{j=k_0}^{k_1-1}{\lambda }_c{\lambda }_s^{2(k_1-1-j)}\Gamma (j)}}.$$

(76)

When the interval is $[k_0,k_1)$ to $[k_1,k_2)$, the controller works to fail, and a stable discrete-time system (63) will switch to an unstable discrete-time system (64). Thus, we can know

$$x_d(k)=A_u^{k-k_1}{x}_d(k_1).$$

(77)

System (76) is unstable, which causes system performances and states to grow quickly. To solve this problem, we assume there always exists ${\lambda }_u ({\lambda }_u\ge 1)$, such that

$$\Vert A_u^q\Vert \le h_u{\lambda }_u^q,$$

(78)

holds for $\forall q\ge 1$, where $h_u$ is a constant scalar. From (78), we obtain

$$\Vert A_u^{k-k_1}\Vert \le {\lambda }_u^{k-k_1},$$

(79)

here we set $h_u=1$. From (77) and (79), one obtains

$$\Vert x_d(k)\Vert \le {\lambda }_u^{k-k_1}\Vert x_d(k_1)\Vert .$$

(80)

Because

$$\Vert x_d(k_1)\Vert -\Vert x_r^{*}(k_1)\Vert \le \Vert x_d(k_1)-x_r^{*}(k_1)\Vert \le \sqrt{\mu {\lambda }_s^{2(k_1-k_0)}{\Vert x_d(k_0)-x_r^{*}(k_0)\Vert }^2-{\displaystyle \sum _{j=k_0}^{k_1-1}{\lambda }_c{\lambda }_s^{2(k_1-1-j)}\Gamma (j)}},$$

(81)

we obtain

$$\Vert x_d(k_1)\Vert \le \sqrt{\mu {\lambda }_s^{2(k_1-k_0)}{\Vert x_d(k_0)-x_r^{*}(k_0)\Vert }^2-{\displaystyle \sum _{j=k_0}^{k_1-1}{\lambda }_c{\lambda }_s^{2(k_1-1-j)}\Gamma (j)}}+\Vert x_r^{*}(k_1)\Vert .$$

(82)

Combining (80) and (82), we can obtain the following inequality:

$$\begin{array}{l}\Vert x_d(k)\Vert \le {\lambda }_u^{k-k_1}\Vert x_d(k_1)\Vert \\ =\sqrt{\mu {\lambda }_u^{2(k-k_1)}{\lambda }_s^{2(k_1-k_0)}{\Vert x_d(k_0)-x_r^{*}(k_0)\Vert }^2-{\displaystyle \sum _{j=k_0}^{k_1-1}{\lambda }_c{\lambda }_u^{2(k-k_1)}{\lambda }_s^{2(k_1-1-j)}\Gamma (j)}}+{\lambda }_u^{k-k_1}\Vert x_r^{*}(k_1)\Vert \\ \le \sqrt{\mu {\lambda }_u^{2(k-k_1)}{\lambda }_s^{2(k_1-k_0)}{\Vert x_d(k_0)-x_r^{*}(k_0)\Vert }^2-{\displaystyle \sum _{j=k_0}^{k_1-1}{\lambda }_c{\lambda }_u^{2(k-k_1)}{\lambda }_s^{2(k_1-1-j)}\Gamma (j)}}+\Vert x_r^{*}(k)\Vert , \end{array}$$

(83)

and then

$$\Vert x_d(k)-x_r^{*}(k)\Vert \le \sqrt{\mu {\lambda }_u^{2(k-k_1)}{\lambda }_s^{2(k_1-k_0)}{\Vert x_d(k_0)-x_r^{*}(k_0)\Vert }^2-{\displaystyle \sum _{j=k_0}^{k_1-1}{\lambda }_c{\lambda }_u^{2(k-k_1)}{\lambda }_s^{2(k_1-1-j)}\Gamma (j)}}.$$

(84)

If $k\to k_2$, we can obtain

$$\Vert x_d(k_2)-x_r^{*}(k_2)\Vert \le \sqrt{\mu {\lambda }_u^{2(k_2-k_1)}{\lambda }_s^{2(k_1-k_0)}{\Vert x_d(k_0)-x_r^{*}(k_0)\Vert }^2-{\displaystyle \sum _{j=k_0}^{k_1-1}{\lambda }_c{\lambda }_u^{2(k_2-k_1)}{\lambda }_s^{2(k_1-1-j)}\Gamma (j)}}.$$

(85)

Similarly, when the working time $k$ is located in the interval $[k_2,k_3)$, we repeat steps similar to (73)~(75) and utilize (85) to obtain

$$\begin{array}{l}\Vert x_d(k)-x_r^{*}(k)\Vert \le \\ \sqrt{{\mu }^2{\lambda }_u^{2(k_2-k_1)}{\lambda }_s^{2(k-k_2+k_1-k_0)}{\Vert x_d(k_0)-x_r^{*}(k_0)\Vert }^2-{\displaystyle \sum _{j=k_0}^{k_1-1}\mu {\lambda }_c{\lambda }_u^{2(k_2-k_1)}{\lambda }_s^{2(k-k_2+k_1-1-j)}\Gamma (j)-}{\displaystyle \sum _{j=k_2}^{k-1}{\lambda }_c{\lambda }_s^{2(k-1-j)}\Gamma (j)}}.\end{array}$$

(86)

If $k\to k_3$, we can obtain

$$\begin{array}{l}\Vert x_d(k_3)-x_r^{*}(k_3)\Vert \le \\ \sqrt{{\mu }^2{\lambda }_u^{2(k_2-k_1)}{\lambda }_s^{2(k_3-k_2+k_1-k_0)}{\Vert x_d(k_0)-x_r^{*}(k_0)\Vert }^2-{\displaystyle \sum _{j=k_0}^{k_1-1}\mu {\lambda }_c{\lambda }_u^{2(k_2-k_1)}{\lambda }_s^{2(k_3-k_2+k_1-1-j)}\Gamma (j)-}{\displaystyle \sum _{j=k_2}^{k_3-1}{\lambda }_c{\lambda }_s^{2(k_3-1-j)}\Gamma (j)}}.\end{array}$$

(87)

When the working time $k$ is located in the interval $[k_3,k_4)$, we repeat (77)~(84) and utilize (87). We can obtain

$$\begin{array}{l}\Vert x_d(k)-x_r^{*}(k)\Vert \le \\ \sqrt{{\mu }^2{\lambda }_u^{2(k-k_3+k_2-k_1)}{\lambda }_s^{2(k_3-k_2+k_1-k_0)}{\Vert x_d(k_0)-x_r^{*}(k_0)\Vert }^2-{\displaystyle \sum _{j=k_0}^{k_1-1}\mu {\lambda }_c{\lambda }_u^{2(k-k_3+k_2-k_1)}{\lambda }_s^{2(k_3-k_2+k_1-1-j)}\Gamma (j)-{\displaystyle \sum _{j=k_2}^{k_3-1}{\lambda }_c{\lambda }_u^{2(k-k_3)}{\lambda }_s^{2(k_3-1-j)}\Gamma (j)}}}.\end{array}$$

(88)

By repeatedly working, we can find that the error $\Vert x_d(k)-x_r^{*}(k)\Vert$ of the unstable discrete-time switched system (61) is limited whether the controller works or not, and the general form is

$$\begin{array}{l}\Vert x_d(k)-x_r^{*}(k)\Vert \le \sqrt{{\mu }^{I_d(0,k)}{\lambda }_u^{2K_u(0,k)}{\lambda }_s^{2K_s(0,k)}{\Vert x_d(k_0)-x_r^{*}(k_0)\Vert }^2-{\displaystyle \sum _{q=0,2,4\dots }\left({\displaystyle \sum _{j=k_q}^{k-1(k_{q+1}-1)}{\mu }^{I_d(j,k)-1}{\lambda }_c{\lambda }_u^{2K_u(k_q,k)}{\lambda }_s^{2K_s(j,k)}\Gamma (j)}\right)}}, \\ k\ge 0.\end{array}$$

(89)

We set $e(k)=x_d(k)-x_r^{*}(k)$, then (89) can be written as

$$\Vert e(k)\Vert \le \sqrt{{\mu }^{I_d(0,k)}{\lambda }_u^{2K_u(0,k)}{\lambda }_s^{2K_s(0,k)}{\Vert e(k_0)\Vert }^2-{\displaystyle \sum _{q=0,2,4\dots }\left({\displaystyle \sum _{j=k_q}^{k-1(k_{q+1}-1)}{\mu }^{I_d(j,k)-1}{\lambda }_c{\lambda }_u^{2K_u(k_q,k)}{\lambda }_s^{2K_s(j,k)}\Gamma (j)}\right)}}, k\ge 0$$

(90)

where $\Gamma (j)=c{y}_d{}^T(j)y_d(j)-(r_0^2+\omega )y_r^{*}{}^{ T}(j)y_r^{*}(j)$. $K_u(a,b)$ and $K_s(a,b)$ denote all the controller working and failure times, respectively, in the interval $[a,b)$.

We define $N_{\sigma }(a,b)$ as the switched number in the interval $[a,b)$ and $I_d(a,b)$ as the corresponding constant. The relation that $N_{\sigma }(a,b)$ corresponds to $I_d(a,b)$ is as follows:

$$\begin{array}{ccccc}{N}_{\sigma }(a,b):& \boxed{0 1}& \boxed{2 3}& \boxed{4 5}& \cdots \\ & \downarrow & \downarrow & \downarrow & \cdots \\ I_{\sigma }(a,b):& 1& 2& 3& \cdots \end{array}$$

The general solution (90) of all states reflects all conditions where the controller works or fails. In the next study, we prove that all states of the switched system (61) can be bounded by $L_2$ and that the response is limited in a certain range.

Considering (90), set $e(k_0)=0$.

Case 1: When $\mu =1$, because $\Vert e(k)\Vert \ge 0$, from (90), we know that

$${\displaystyle \sum _{q=0,2,4\dots }\left({\displaystyle \sum _{j=k_q}^{k-1(k_{q+1}-1)}{\lambda }_u^{2K_u(k_q,k)}{\lambda }_s^{2K_s(j,k)}\Gamma (j)}\right)}\le 0.$$

(91)

If there exists a positive number ${\lambda }^{*}$, then

$${\lambda }_u^{2K_u(k_q,k)}{\lambda }_s^{2K_s(j,k)}\le {\lambda }^{*2(K_u(k_q,k)+K_s(j,k))}$$

(92)

holds. We set $j=0, k_q=0$ and then

$$\ln \left({\lambda }_u^{2K_u(0,k)}{\lambda }_s^{2K_s(0,k)}\right)\le \ln \left({\lambda }^{*2\left(K_u(0,k)+K_s(0,k)\right)}\right),$$

(93)

because $k=K_s(0,k)+K_u(0,k)$, we obtain the switching law from (93)

$$\frac{K_u(0,k)}{k}\le \frac{\ln {\lambda }^{*}-\ln {\lambda }_s}{\ln {\lambda }_u-\ln {\lambda }_s}.$$

(94)

We observe (91); when $q=0$ (or $q=2,4\cdots$), we sum (91) from $k=1$ to $k=\infty$, then

$${\displaystyle \sum _{k=1}^{\infty }\left({\displaystyle \sum _{j=0}^{k-1}{\lambda }_u^{2K_u(0,k)}{\lambda }_s^{2K_s(j,k)}\Gamma (j)}\right)}={\displaystyle \sum _{j=0}^{\infty }\Gamma (j)\left({\displaystyle \sum _{k=j+1}^{\infty }{\lambda }_u^{2K_u(0,k)}{\lambda }_s^{2K_s(j,k)}}\right)}\le 0.$$

(95)

Because

$${\displaystyle \sum _{k=j+1}^{\infty }{\lambda }_u^{2K_u(0,k)}{\lambda }_s^{2K_s(j,k)}}>0,$$

(96)

then

$${\displaystyle \sum _{j=0}^{\infty }\Gamma (j)}\le 0.$$

(97)

Inequality (97) denotes

$${\displaystyle \sum _{j=0}^{\infty }{y}_d{}^T(k)y_d(k)\le \left(\frac{r_0^2+\omega }{c}\right){\displaystyle \sum _{j=0}^{\infty }{y}_r^{*}{}^{ T}(k)y_r^{*}(k)}}.$$

(98)

This result tells us that an $L_2$ gain $\frac{r_0^2+\omega }{c}$ is achieved for the switching system under the switching law. Thus, the switching system is bounded by $L_2$.

Case 2: When $\mu >1$ and $j=0, k_q=0$, we obtain

$$\begin{array}{ll}{\mu }^{I_d(0,k)-1}{\lambda }_u^{2K_u(0,k)}{\lambda }_s^{2K_s(0,k)}& \le {\mu }^{I_d(0,k)-1}{\lambda }^{*2(K_u(0,k)+K_s(0,k))}\\ & ={\mu }^{I_d(0,k)-1}{\lambda }^{*2k}.\end{array}$$

(99)

If there exists a positive number $\lambda$, then

$${\mu }^{I_d(0,k)-1}{\lambda }^{*2k}\le {\lambda }^{2k}.$$

(100)

We can obtain another switching law as follows:

$$I_d(0,k)\le \frac{2k\left(\ln \lambda -\ln {\lambda }^{*}\right)}{\ln \mu }+1, \tau =\frac{\ln \mu }{2\left(\ln \lambda -\ln {\lambda }^{*}\right)}, N_0=1.$$

(101)

We know that the average time between consecutive switching is larger than or equal to $\tau$ and (90) still holds from (101). Imitating the Case 1 proof, we know that Case 2 is also bounded by $L_2$.

The above discussion is $e(k_0)=0$. If the initial state $e(k_0)\ne 0$, because we care about the same terms (91) and (99), we can obtain the same switching laws (94) and (101). The different part is that the term ${\mu }^{I_d(0,k)}{\lambda }_u^{2K_u(0,k)}{\lambda }_s^{2K_s(0,k)}{\Vert e(k_0)\Vert }^2$ in (90) can be ignored. Combining the above analysis and results, we successfully provide switching laws and know that we can effectively bound the states of the unstable discrete-time system (61) under the switching laws. Inequalities (94) and (101) are our important results, and we combine them with the digital redesign method to simulate an example.

5. An Illustrative Example

In this section, we study an example to apply to this paper’s result. Consider an unstable MIMO continuous-time system as follows:

$$\{\begin{array}{l}{\dot{x}}_c(t)=\left[\begin{array}{ccc}-9.996& -3.41& 2.641\\ 0& -7.985& 5.157\\ 0& 0& 1.043\end{array}\right]x_c(t)+\left[\begin{array}{cc}0.987& 0.016\\ -0.011& 0.99\\ 1.043& 1.043\end{array}\right]u_c(t),x_c\left(0\right)=\left[\begin{array}{ccc}0& 0& 0\end{array}\right], \\ y_c(t)=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]x_c(t).\end{array}$$

(102)

The optimal controller (19) is given by

$$u_c(t)=-K_c{x}_c(t)+E_c{y}_r(t),$$

for $Q={10}^5{I}_2$ and $R=I_2$ where

$$K_c=\left[\begin{array}{ccc}306.266& -4.528& 2.584\\ 1.293& 308.346& 5.105\end{array}\right], \mathrm{and} E_c=\left[\begin{array}{cc}316.069& -1.212\\ 1.215& 316.223\end{array}\right].$$

The corresponding discrete-time model is

$$\{\begin{array}{cc}\hfill x_d(kT+T)& =\left[\begin{array}{ccc}0.368& -0.139& 0.126\\ 0& 0.45& 0.377\\ 0& 0& 1.11\end{array}\right]x_d(kT)+\left[\begin{array}{cc}0.071& 0\\ 0.021& 0.09\\ 0.11& 0.11\end{array}\right]u_d(kT),x_d\left(0\right)=\left[\begin{array}{ccc}0& 0& 0\end{array}\right], \hfill \\ \hfill y_d(kT)& =\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]x_d(kT),\hfill \end{array}$$

(103)

where $T=0.1$ s. The digital redesign controller is

$$u_d(kT)=-K_d{x}_d(kT)+E_d{y}_r^{*}(kT),$$

where

$$K_d=\left[\begin{array}{ccc}4.908& -1.917& 1.753\\ -1.177& 5.193& 3.738\end{array}\right], \mathrm{and} E_d=\left[\begin{array}{cc}13.763& 0.005\\ -3.305& 10.786\end{array}\right].$$

Depending on whether the controller works well or fails, the corresponding switched forms of the system (103) will be similar to (63) or (64), where

$$\begin{array}{l}{A}_s=G-H{K}_d=\left[\begin{array}{ccc}0.0195& -0.002& 0.001\\ 0.002& 0.022& 0.003\\ -0.41& -0.36& 0.505\end{array}\right], B_s=H{E}_d=\left[\begin{array}{cc}0.977& 0.0004\\ -0.008& 0.97\\ 1.15& 1.187\end{array}\right]\\ A_u=A_d=G=\left[\begin{array}{ccc}0.368& -0.139& 0.126\\ 0& 0.45& 0.377\\ 0& 0& 1.11\end{array}\right], C_s=C_d=C=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right].\end{array}$$

To satisfy the $H_{\infty }$ linear matrix inequalities in (66), set $r_0=2$ and choose parameters ${\lambda }_s=0.9$, ${\lambda }_u=3.8$, ${\lambda }^{*}=1.65$, and $\lambda =1.7$. By using the LMI control toolbox, we can obtain $P_s$ and

$${\mathsf{\Lambda }}_s=\left[\begin{array}{ccc}4.2& 0.096& -0.517\\ 0.096& 4.92& -0.632\\ -0.517& -0.632& 0.898\end{array}\right]$$

and then $\mu =\frac{{\lambda }_M({\mathsf{\Lambda }}_s)}{{\lambda }_m({\mathsf{\Lambda }}_s)}\cong 6.9$. According to (94) and (101), one has the switching laws for the controller working and not, respectively, as

Switching law 1:

$$\tau =\frac{\ln \mu }{2\left(\ln \lambda -\ln {\lambda }^{*}\right)}=31.428,\tau T\cong 3.1$$

Switching law 2:

$$\frac{K_u(0,k)}{k}\le \frac{\ln {\lambda }^{*}-\ln {\lambda }_s}{\ln {\lambda }_u-\ln {\lambda }_s}\cong 0.4.$$

Switching law 1 denotes the minimum value of the average time interval, which must be greater than or equal to 3.1, and switching law 2 denotes the unavailability rate of the switched system (103) with controller failure. We use switching laws to simulate the tracking of a switched system (103). Here, we choose 0.4 s for which the switched system (103) controller works and 3.2 s for which the controller fails. Figure 3a,b show the comparisons of the output response and the reference, where the reference $y_r^{*}=\left[\begin{array}{c}\cos (t)\\ 1+\sin (t)\end{array}\right]$, the sampling time $T=0.1$ s, the initial state $x_d(0)=0$, and the simulation time is 30 s.

From Figure 3a,b, we know that the MIMO discrete-time switched system (103) output responses $y_{d1}(k)$ and $y_{d2}(k)$ are bounded in the intervals $[-0.99, 0.99]$ and $[-0.44, 3.25]$. Thus, we can bound the output response in a certain range. The next consideration is the initial state $x_d(0)\ne 0$ in the system (103). Additionally, we use the same conditions ($y_r^{*}$, $T$ and the simulation time) as in Figure 3 to get the simulation results in Figure 4a,b. For system (103), the switching laws are the same whether $x_d(0)=0$ or not, and we set $x_d=\left[\begin{array}{ccc}1& -0.7& 0\end{array}\right]$ to simulate as follows (Figure 4a,b):

From Figure 4a,b, we know that the MIMO discrete-time switched system (103) output responses $y_{d1}(k)$ and $y_{d2}(k)$ are bounded in the intervals $[-0.99, 1]$ and $[-0.7, 3.25]$. Next, for different $Q$ ($R=I_2$) selections, the robust performance and tracking performance are shown in Figure 5, Figure 6, Figure 7 and Figure 8 and

Table 1. The robust performance and tracking performance comparison for different Q values.

Q	Bounded in the Intervals	Tracking Performance
$Q=10\times I_2$	$\begin{array}{l}{y}_{d1}(k): [-0.17, 0.26]\\ y_{d2}(k): [0, 2.38]\end{array}$	poor
$Q={10}^2\times I_2$	$\begin{array}{l}{y}_{d1}(k): [-0.53, 0.53]\\ y_{d2}(k): [-0.03, 2.97]\end{array}$	poor
$Q={10}^3\times I_2$	$\begin{array}{l}{y}_{d1}(k): [-0.9, 0.91]\\ y_{d2}(k): [-0.38, 3.14]\end{array}$	acceptable
$Q={10}^4\times I_2$	$\begin{array}{l}{y}_{d1}(k): [-0.99, 0.99]\\ y_{d2}(k): [-0.45, 3.26]\end{array}$	good
$Q={10}^5\times I_2$	$\begin{array}{l}{y}_{d1}(k): [-0.99, 0.99]\\ y_{d2}(k): [-0.44, 3.25]\end{array}$	good

.

6. Conclusions

This paper successfully derives switching laws by $H_{\infty }$ linear matrix inequality for unstable discrete-time switched systems (61) and proves that state error (90) can be bounded. Because state error (90) is bounded, the system performance can also be bounded in a certain range. That tells us whether the controller will work or fail for an unstable discrete-time switched system (61), and we can know beforehand the approximate bound of system performance, which will be our expectation or not. However, we must note that $P_s$ in the $H_{\infty }$ linear matrix inequality might not have a solution, so it is necessary to choose appropriate parameters. In this paper, when the controller fails, we consider the controller to be zero, which denotes that all control signals are broken, and this is the worst situation. In the future, we will study when the controller fails and only several control factors are lost and use the method in this paper for further discussion.