Observer-based secure control for discrete-time singular systems subject to DoS attacks

Abstract

This work proposes the problem of observer-based secure control for discrete-time singular systems subject to denial-of-service (DoS) attacks. The non-periodic DoS attacks that destroy the system’s performance are considered. To estimate the states and deal with the effects of DoS attacks, a new switched observer-based control method is established, which models the closed-loop system as a switched system. Firstly, some sufficient criteria for guaranteeing the exponential admissibility with $H_{\infty }$ performance of the corresponding switched systems are derived. Secondly, compared with the existing results, a novel method is given to design the controller gain, where the condition that the measured output is full rank is remove. Finally, two examples are provided to verify the correctness of the proposed approach.

Introduction

In practice, many systems can be modeled by differential or algebraic equations. However, some systems, such as economic systems, biological systems and robotics systems, ¹ need to be modeled by the above two equations. To study those systems, singular systems have been proposed by scholars. For those systems, the study of singular systems is more difficult because the stability problem is considered, while the regularity and the non-impulsiveness (for continuous systems) or the causality (for discrete systems) are discussed. Over the past few decades, singular systems have received attention due to their extensive applications and many results about singular systems have been reported, such as saturation control,^2,3 stability analysis,^4,5 security control ⁶ and sliding mode control (SMC).^7–9 The analysis and design issue ² was studied for singular systems under saturated feedback, where the largest invariant ellipsoid was given by solving optimization problem. The robust $H_{\infty }$ control issue ⁵ was discussed for discrete singular systems. Lin et al. ⁶ focused on the asymptotic stabilization problem of singular systems with quantization and DoS attacks.

A large amount of existing literature is based on the assumption that the states of the system are fully available. However, this assumption is very difficult to satisfy in practice because of limited control techniques. To conquer this issue, the methods to achieve the stability of the system through the output information of the system are proposed. The methods contain dynamic output feedback (DOF) control,^10,11 observer-based output feedback (OBOF) control^12–14 and static output feedback (SOF) control.^15,16 The DOF control design issue ¹⁰ was discussed for singular hybrid systems under input saturation, where the transition rates are partly unknown. Nagamani et al. ¹⁴ studied the issue of observer-based SMC for nonlinear systems, where an event-triggered method was adopted to save the network resources. The SOF control problem ¹⁵ was proposed for semi-MJSs, where a novel multivariate Lyapunov function was established to discuss the stability of the system. However, the above papers required that the output matrix $\mathcal{C}$ is column full rank and this condition limits the scope of application.

Due to the shared nature of communication networks, the transmitted data is vulnerable to hacker attacks. At present, the deception attacks and the DoS attacks are two common categories of network attacks. For the former, it mainly changes the authenticity of the data by injecting false data into it, thereby affecting the judgment of the control center. Moreover, it is difficult to detect because the pattern and time of the attacks are designed by the attackers. For the latter, it mainly cuts off the data transmission, thereby causing system instability. Compared with the deception attacks, the DoS attacks can destroy the system performance without the transmitted information, which is the reason that the DoS attacks receive special attention. To alleviate the impact of the cyber attacks on control systems, many security control methods have been proposed.^17–25 The consensus tracking control issue ²⁶ was presented for multiagent systems with deception attacks, where a hierarchical algorithm was established to decompose the multi-agent consensus tracking problem into a single-agent tracking problem. Wang et al. ²⁰ concentrated on the issue of filter design for SMJSs, where a stochastic variable was adopted to describe whether DoS attacks occur. The asynchronous security tolerant control problem ²² was considered for MJSS, where deception attack and DoS attack were cogitated simultaneously. In Zhu et al., ²³ Zhang et al., ²⁴ and Zeng et al., ²⁵ the secure control problem was discussed for singular systems subject to DoS attacks. From the existing works, most papers considered the secure control problem for continuous singular systems, few papers discussed the secure control problem for discrete-time singular systems subject to DoS attacks, so the observer-based secure control problem remains a challenge for discrete-time singular systems with nonperiodic DoS attacks.

Summarizing the above discussions, the secure control problem is discussed for discrete-time singular systems by observer control method in this work. The main contributions of this work can be summarized as follows:

• Unlike the existing result in Wang et al., ²⁰ modeling DoS attacks with a random variable, the non-periodic DoS attacks that make the communication network unreliable are considered in this paper.

• Some conditions for the exponential admissibility with $H_{\infty }$ performance of the corresponding switched systems are established. In contrast to Park et al. ²⁷ and Chen et al. ²⁸ the proof process is simplified by adopting the relationship between the exponential stability and $\mathcal{E}$ -exponential stability. A novel method is given to solve the controller. Moreover, in contrast to Regaieg et al., ¹³ which requires the measured output to be of row full rank, a novel method is established to obtain the controller gains, which can remove this restrictive condition.

This work is arranged as follows. In Section II, the preliminaries are introduced. In Section III, the main results are presented. Section IV demonstrates the methodology through two examples. Conclusions are given in Section V.

Notations	Explanations
$A>0$	$A$ is a positive definite
$A^T$	the transpose of $A$
$Z^{+}$	positive integer
${\mathcal{L}}_2[0,\infty )$	square-integrable space
${\mathbb{R}}^{m\times n}$	$m\times n$ -dimensional Euclidean space
${\mathbb{R}}^n$	$n$ -dimensional Euclidean space

Problem formulation

Consider the following discrete-time singular systems

$$\{\begin{array}{c}\mathcal{E}x(k+1)=\mathcal{A}x\left(k\right)+\mathcal{B}u\left(k\right)+\mathcal{D}\omega \left(k\right)\hfill \\ \multicolumn{1}{c}{y\left(k\right)=\mathcal{C}x\left(k\right)}\end{array}$$

(1)

where $x\left(k\right)\in {\mathbb{R}}^n$ stands for the state, $u\left(k\right)\in {\mathbb{R}}^u$ is the input, $y\left(k\right)\in {\mathbb{R}}^m$ is the measured output, $\omega \left(k\right)$ is the external disturbance with $\omega \left(k\right)\in {\mathcal{L}}_2[0,\infty )$ . Matrix $\mathcal{E}$ satisfies rank $\left(\mathcal{E}\right)=t<n$ . The constant matrices $\mathcal{A},\mathcal{B},\mathcal{D}$ and $\mathcal{C}$ have appropriate dimensions.

In practice, the transmitted signals are often inevitably attacked because the network transmission channels are open to the public. In this paper, the aperiodic DoS attacks are considered and occur on the sensor-to-observer channel. Assume that $[O_s^{on},O_s^{off}),\left(s\in Z^{+}\right)$ shows the active time of DoS attacks. $o(k_p,k_q)$ represents the number of DoS attacks that occur within the time interval $[k_p,k_q)$ . ${\Gamma }_1(k_p,k_q)$ is the total duration of DoS attack on the time interval $[k_p,k_q)$ . Inspired by Zhang et al. ¹⁸ and Zhao et al. ¹⁹ the following related assumptions are given.

• Assumption 1. ¹⁹ (DoS frequency) There exist constants $a_1\ge 0$ and $a_2\ge 0$ such that

$$o(k_p,k_q)\le a_1+(k_q-k_p)/a_2.$$

(2)

• Assumption 2. ¹⁹ (DoS duration) There exist constants $b_1\ge 0$ and $b_2\ge 0$ such that

$${\Gamma }_1(k_p,k_q)\le b_1+(k_q-k_p)/b_2.$$

(3)

• Remark 1. In practice, although the attackers can usually manipulate the transmitted data, the energy of the injected attack information is limited due to physical constraints. Therefore, the above assumptions are reasonable.

In this paper, in order to estimate the system states, the following observer is established:

for $k\notin [O_s^{\mathrm{on}},O_s^{\mathrm{off}})$ , that is, the DoS attacks are not activated,

$$\{\begin{array}{c}\mathcal{E}\tilde{x}(k+1)=\mathcal{A}\tilde{x}\left(k\right)+\mathcal{B}u\left(k\right)+{\mathcal{F}}_1[y\left(k\right)-\tilde{y}\left(k\right)]\hfill \\ u\left(k\right)={\mathcal{K}}_1\tilde{x}\left(k\right)\hfill \\ \tilde{y}\left(k\right)=\mathcal{C}\tilde{x}\left(k\right)\hfill \end{array}$$

(4)

for $k\in [O_s^{\mathrm{on}},O_s^{\mathrm{off}})$ , that is, the DoS attacks are activated,

$$\{\begin{array}{c}\mathcal{E}\tilde{x}(k+1)=\mathcal{A}\tilde{x}\left(k\right)+\mathcal{B}u\left(k\right)-{\mathcal{F}}_2\tilde{y}\left(k\right)\hfill \\ u\left(k\right)={\mathcal{K}}_2\tilde{x}\left(k\right)\hfill \\ \tilde{y}\left(k\right)=\mathcal{C}\tilde{x}\left(k\right)\hfill \end{array}$$

(5)

where ${\mathcal{K}}_1,{\mathcal{K}}_2,{\mathcal{F}}_1$ and ${\mathcal{F}}_2$ are the observer gains to be determined.

Define $\xi \left(k\right)=x\left(k\right)-\tilde{x}\left(k\right)$ and combining (1) and (5), the following closed-loop systems can be obtain for $k\notin [O_s^{\mathrm{on}},O_s^{\mathrm{off}})$ , that is, the DoS attacks are not activated,

$$\{\begin{array}{c}\overline{\mathcal{E}}\xi (k+1)={\overline{\mathcal{A}}}_1\xi \left(k\right)+\overline{\mathcal{D}}\omega \left(k\right)\hfill \\ \tilde{y}\left(k\right)=\overline{\mathcal{C}}\xi \left(k\right)\hfill \end{array}$$

(6)

where $\xi \left(k\right)=[{\tilde{x}}^T\left(k\right)e^T\left(k\right){]}^T$ ,

$$\begin{array}{c}{\overline{\mathcal{A}}}_1=\left[\begin{array}{cc}\mathcal{A}+\mathcal{B}{\mathcal{K}}_1\hfill & {\mathcal{F}}_1\mathcal{C}\hfill \\ \multicolumn{1}{c}{0}& \mathcal{A}-{\mathcal{F}}_1\mathcal{C}\hfill \end{array}\right],\hfill \\ \overline{\mathcal{D}}=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{\mathcal{D}}\end{array}\right],\overline{\mathcal{C}}=\left[\begin{array}{cc}\mathcal{C}\hfill & \mathcal{C}\hfill \end{array}\right].\hfill \end{array}$$

for $k\in [O_s^{\mathrm{on}},O_s^{\mathrm{off}})$ , that is, the DoS attacks are activated,

$$\{\begin{array}{c}\overline{\mathcal{E}}\xi (k+1)={\overline{\mathcal{A}}}_2\xi \left(k\right)+\overline{\mathcal{D}}\omega \left(k\right)\hfill \\ \tilde{y}\left(k\right)=\overline{\mathcal{C}}\xi \left(k\right)\hfill \end{array}$$

(7)

where $\xi \left(k\right)=[{\tilde{x}}^T\left(k\right),e^T\left(k\right){]}^T$ ,

$$\begin{array}{c}{\overline{\mathcal{A}}}_2=\left[\begin{array}{cc}\mathcal{A}+\mathcal{B}{\mathcal{K}}_2-{\mathcal{F}}_2\mathcal{C}\hfill & 0\hfill \\ \multicolumn{1}{c}{{\mathcal{F}}_2\mathcal{C}}& \mathcal{A}\hfill \end{array}\right],\hfill \\ \overline{\mathcal{D}}=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{\mathcal{D}}\end{array}\right],\overline{\mathcal{C}}=\left[\begin{array}{cc}\mathcal{C}\hfill & \mathcal{C}\hfill \end{array}\right].\hfill \end{array}$$

• Lemma 1. ² If there exist a constant $\alpha$ , matrices $T<0$ , $X$ , $Y$ and $Z$ satisfying

$$\left[\begin{array}{cc}T\hfill & *\hfill \\ \multicolumn{1}{c}{X^T+\alpha Y}& -\alpha Z-\alpha Z^T\hfill \end{array}\right]<0$$

then

$$T+X{Z}^{-1}Y+Y^T{Z}^{-T}{X}^T<0.$$

• Definition 1. ²⁹ The switched singular systems $\mathcal{E}x(k+1)={\mathcal{A}}_{i}x\left(k\right)$ are said to be:

• $\left(i\right)$ regular if det $(s\mathcal{E}-{\mathcal{A}}_i)\ne 0$ ;

• $\left(\mathrm{ii}\right)$ causal if deg $(\det (s\mathcal{E}-{\mathcal{A}}_i))=$ rank $\left(\mathcal{E}\right)$ ;

• $\left(\mathrm{iii}\right)$ exponentially stable (ES) if there exist ${\alpha }_1>0$ and ${\alpha }_2>0$ such that for $k>0$

$$\Vert x\left(k\right){\Vert }_2\le {\alpha }_1{e}^{-{\alpha }_{2}k}\Vert x\left(0\right){\Vert }_2;$$

(8)

• $\left(\mathrm{iv}\right)\mathcal{E}$ -exponentially stable ( $\mathcal{E}$ -ES) if there exist ${\hat{\alpha }}_1>0$    and ${\hat{\alpha }}_2>0$ such that for $k>0$

$$\Vert \mathcal{E}x\left(k\right){\Vert }_2\le {\hat{\alpha }}_1{e}^{-{\hat{\alpha }}_{2}k}\Vert \mathcal{E}x\left(0\right){\Vert }_2;$$

(9)

• $\left(v\right)$ exponentially admissible (EA) if (1)-(3) are met;

• $\left(\mathrm{vi}\right)\mathcal{E}$ -exponentially admissible ( $\mathcal{E}$ -EA) if (i), (ii) and (iv) are met;

• $\left(\mathrm{vii}\right)$ condition (v) and condition (vi) are equivalent.

• Remark 2. In, ²⁹ the relationship between the exponential stability and the $\mathcal{E}$ -exponential stability firstly was discussed by Guan et al. The conclusion brings great convenience to the proof of our paper.

The main objectives of our work are as follows:

• (1) the systems (6) and (7) are EA;

• (2) under zero initial condition and $\omega \left(k\right)\ne 0$ , $H_{\infty }$ performance can be obtain:

$$\sum _{k=0}^{\infty }[y^T\left(k\right)y\left(k\right)-{\hat{\gamma }}^2{\omega }^T\left(k\right)\omega \left(k\right)]<0$$

(10)

where $\hat{\gamma }>0$ .

Main results

In this section, some sufficient conditions are derived to make sure that the systems (6) and (7) are EA with ${\mathcal{H}}_{\infty }$ performance.

• Theorem 1. For the given scalars $a_1>0,a_2>0,b_1>0,b_2>1,\gamma >0$ , the systems (6) and (7) are EA with $H_{\infty }$ performance ${\hat{\gamma }}^2={\gamma }^2\psi$ , if there exist scalars $c>1,d_1\in (0,1),d_2>1,{\overline{d}}_3$ and matrices ${\mathcal{P}}_1>0,{\mathcal{P}}_2>0,{\mathcal{M}}_1$ and ${\mathcal{M}}_2$ such that

$${\mathcal{P}}_1<c{\mathcal{P}}_2,{\mathcal{P}}_2<c{\mathcal{P}}_1$$

(11)

$$0<\frac{\ln {\overline{d}}_2-\ln {\overline{d}}_3}{\ln {\overline{d}}_3-\ln {\overline{d}}_1}\le b_2-1$$

(12)

$$a_2\ln {\overline{d}}_3+\ln c<0$$

(13)

$$\left[\begin{array}{cccc}{\mathrm{\Xi }}_1\hfill & {\mathcal{M}}_1^T{\mathcal{W}}^T\overline{\mathcal{D}}\hfill & {\overline{\mathcal{A}}}_1^T{P}_1\hfill & {\overline{\mathcal{C}}}^T\hfill \\ \multicolumn{1}{c}{*}& -{\gamma }^{2}I\hfill & {\overline{\mathcal{D}}}^T{P}_1\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -P_1\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -I\hfill \end{array}\right]<0$$

(14)

$$\left[\begin{array}{cccc}{\mathrm{\Xi }}_2\hfill & {\mathcal{M}}_2^T{\mathcal{W}}^T\overline{\mathcal{D}}\hfill & {\overline{\mathcal{A}}}_2^T{P}_2\hfill & {\overline{\mathcal{C}}}^T\hfill \\ \multicolumn{1}{c}{*}& -{\gamma }^{2}I\hfill & {\overline{\mathcal{D}}}^T{P}_2\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -P_2\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -I\hfill \end{array}\right]<0$$

(15)

where

$$\begin{array}{c}{\mathrm{\Xi }}_1=-{\overline{d}}_1{\overline{\mathcal{E}}}^T{\mathcal{P}}_1\overline{\mathcal{E}}+{\mathcal{M}}_1^T{\mathcal{W}}^T{\overline{\mathcal{A}}}_1+{\overline{\mathcal{A}}}_1^T\mathcal{W}{\mathcal{M}}_1,\hfill \\ {\mathrm{\Xi }}_2=-{\overline{d}}_2{\overline{\mathcal{E}}}^T{\mathcal{P}}_2\overline{\mathcal{E}}+{\mathcal{M}}_2^T{\mathcal{W}}^T{\overline{\mathcal{A}}}_2+{\overline{\mathcal{A}}}_2^T\mathcal{W}{\mathcal{M}}_2,\hfill \\ {\overline{d}}_1=1-d_1,{\overline{d}}_2=1+d_2,\psi =\frac{{\overline{d}}_2^{b_1}{c}^{\frac{a_1}{a_2}}}{{\overline{d}}_1^{b_1}}\frac{1-{\overline{d}}_1}{1-{\overline{d}}_3{c}^{a_2}}\hfill \end{array}$$

and $\mathcal{W}$ is a constant matrix satisfying ${\overline{\mathcal{E}}}^T\mathcal{W}=0$ .

Proof. First of all, the regular and causal of the systems (6) and (7) will be proved.

Because of rank $\left(\mathcal{E}\right)=t<n$ , the invertible matrices $Q_{1i}$ and $Q_{2i}$ satisfy

$$\begin{array}{c}{Q}_{1i}\overline{\mathcal{E}}{Q}_{2i}=\left[\begin{array}{cc}I\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right],Q_{1i}{\overline{\mathcal{A}}}_i{Q}_{2i}=\left[\begin{array}{cc}{\overline{\mathcal{A}}}_{11i}\hfill & {\overline{\mathcal{A}}}_{12i}\hfill \\ \multicolumn{1}{c}{{\overline{\mathcal{A}}}_{21i}}& {\overline{\mathcal{A}}}_{22i}\hfill \end{array}\right],\hfill \\ {\mathcal{W}}^T{Q}_{1i}^{-1}=\left[\begin{array}{cc}{\mathcal{W}}_{1i}\hfill & {\mathcal{W}}_{2i}\hfill \end{array}\right],{\mathcal{M}}_i{Q}_{2i}=\left[\begin{array}{cc}{\mathcal{M}}_{1i}\hfill & {\mathcal{M}}_{2i}\hfill \end{array}\right].\hfill \end{array}$$

It can be seen from ${\mathcal{E}}^T\mathcal{W}=0$ that ${\mathcal{W}}_{1i}=0$ .

For $k\notin [O_s^{\mathrm{on}},O_s^{\mathrm{off}})$ , that is, the DoS attacks are not activated.

From (14), one has

$$-{\overline{d}}_1{\overline{\mathcal{E}}}^T{\mathcal{P}}_1\overline{\mathcal{E}}+{\mathcal{M}}_1^T{\mathcal{W}}^T{\overline{\mathcal{A}}}_1+{\overline{\mathcal{A}}}_1^T\mathcal{W}{\mathcal{M}}_1<0.$$

(16)

Then, (16) pre- and post-multiplying $Q_{21}^T$ and $Q_{21}$ , respectively, one can obtain

$$\left[\begin{array}{cc}★\hfill & ★\hfill \\ \multicolumn{1}{c}{★}& {\overline{\mathcal{A}}}_{221}^T{\mathcal{W}}_{21}^T{\mathcal{M}}_{11}+{\mathcal{M}}_{11}^T{\mathcal{W}}_{12}{\overline{\mathcal{A}}}_{221}\hfill \end{array}\right]<0$$

(17)

which implies that ${\overline{\mathcal{A}}}_{221}$ is nonsingular.

For $k\in [O_s^{\mathrm{on}},O_s^{\mathrm{off}})$ , one can get that ${\overline{\mathcal{A}}}_{222}$ is nonsingular. Thus, the systems (6) and (7) are regular and causal.

For $k\notin [O_s^{\mathrm{on}},O_s^{\mathrm{off}})$ , that is, the DoS attacks are not activated, the following Lyapunov functional is chosen as:

$${\mathcal{V}}_1\left(k\right)={\xi }^T\left(k\right){\overline{\mathcal{E}}}^T{\mathcal{P}}_1\overline{\mathcal{E}}\xi \left(k\right).$$

(18)

Define $\mathrm{\Delta }\mathcal{V}\left(k\right)=\mathcal{V}(k+1)-\mathcal{V}\left(k\right)$ , one can obtain

$$\begin{array}{c}\mathrm{\Delta }{\mathcal{V}}_1\left(k\right)={\xi }^T(k+1){\overline{\mathcal{E}}}^T{\mathcal{P}}_1\overline{\mathcal{E}}\xi (k+1)\hfill \\ -{\xi }^T\left(k\right){\overline{\mathcal{E}}}^T{\mathcal{P}}_1\overline{\mathcal{E}}\xi \left(k\right).\hfill \end{array}$$

(19)

Since ${\overline{\mathcal{E}}}^T\mathcal{W}=0$ , one has

$$2{\xi }^T(k+1){\overline{\mathcal{E}}}^T\mathcal{W}{\mathcal{M}}_1\xi \left(k\right)=0.$$

(20)

Considering $H_{\infty }$ performance and combining (19) and (20), one can get

$$\begin{array}{c}\mathrm{\Delta }{\mathcal{V}}_1\left(k\right)+d_1{\mathcal{V}}_1\left(k\right)+z^T\left(k\right)z\left(k\right)-{\gamma }^2{\omega }^T\left(k\right)\omega \left(k\right)\hfill \\ \le {\psi }^T\left(k\right)\Pi \psi \left(k\right)\hfill \end{array}$$

(21)

where

$$\begin{array}{c}\psi \left(k\right)={[{\xi }^T\left(k\right){\omega }^T\left(k\right)]}^T,\Pi ={\Pi }_1+{\Pi }_2^T{\mathcal{P}}_1{\Pi }_2+{\Pi }_3^T{\Pi }_3,\hfill \\ {\Pi }_1=\left[\begin{array}{cc}-{\overline{d}}_1{\overline{\mathcal{E}}}^T{\mathcal{P}}_1\overline{\mathcal{E}}+{\mathcal{M}}_1^T{\mathcal{W}}^T{\overline{\mathcal{A}}}_1+{\overline{\mathcal{A}}}_1^T\mathcal{W}{\mathcal{M}}_1\hfill & *\hfill \\ \multicolumn{1}{c}{{\left({\mathcal{M}}_1^T{\mathcal{W}}^T\overline{\mathcal{D}}\right)}^T}& -{\gamma }^{2}I\hfill \end{array}\right],\hfill \\ {\Pi }_2=\left[\begin{array}{cc}{\overline{\mathcal{A}}}_1\hfill & \overline{\mathcal{D}}\hfill \end{array}\right],{\Pi }_3=\left[\begin{array}{cc}\overline{\mathcal{C}}\hfill & 0\hfill \end{array}\right].\hfill \end{array}$$

By Schur complement and (14), one has

$$\begin{array}{c}\mathrm{\Delta }{\mathcal{V}}_1\left(k\right)\le -d_1{\mathcal{V}}_1\left(k\right)-z^T\left(k\right)z\left(k\right)+{\gamma }^2{\omega }^T\left(k\right)\omega \left(k\right).\hfill \end{array}$$

(22)

For $k\in [O_s^{\mathrm{on}},O_s^{\mathrm{off}})$ , that is, the DoS attacks are activated, the following Lyapunov functional is chosen as:

$${\mathcal{V}}_2\left(k\right)={\xi }^T\left(k\right){\overline{\mathcal{E}}}^T{\mathcal{P}}_2\overline{\mathcal{E}}\xi \left(k\right).$$

(23)

Similar to (18), one can obtain

$$\begin{array}{c}\mathrm{\Delta }{\mathcal{V}}_2\left(k\right)={\xi }^T(k+1){\overline{\mathcal{E}}}^T{\mathcal{P}}_2\overline{\mathcal{E}}\xi (k+1)\hfill \\ \multicolumn{1}{c}{-{\xi }^T\left(k\right){\overline{\mathcal{E}}}^T{\mathcal{P}}_2\overline{\mathcal{E}}\xi \left(k\right).}\end{array}$$

(24)

Since ${\overline{\mathcal{E}}}^T\mathcal{W}=0$ , one has

$$2{\xi }^T(k+1){\overline{\mathcal{E}}}^T\mathcal{W}{\mathcal{M}}_2\xi \left(k\right)=0.$$

(25)

Considering $H_{\infty }$ performance and combining (24) and (25), one can get

$$\begin{array}{c}\mathrm{\Delta }{\mathcal{V}}_2\left(k\right)-d_2{\mathcal{V}}_1\left(k\right)+z^T\left(k\right)z\left(k\right)-{\gamma }^2{\omega }^T\left(k\right)\omega \left(k\right)\hfill \\ \multicolumn{1}{c}{\le {\psi }^T\left(k\right)\overline{\Pi }\psi \left(k\right)}\end{array}$$

(26)

where

$$\begin{array}{c}\psi \left(k\right)={[{\xi }^T\left(k\right){\omega }^T\left(k\right)]}^T,\overline{\Pi }={\overline{\Pi }}_1+{\overline{\Pi }}_2^T{\mathcal{P}}_2{\overline{\Pi }}_2+{\overline{\Pi }}_3^T{\overline{\Pi }}_3,\hfill \\ {\Pi }_1=\left[\begin{array}{cc}-{\overline{d}}_2{\overline{\mathcal{E}}}^T{\mathcal{P}}_2\overline{\mathcal{E}}+{\mathcal{M}}_2^T{\mathcal{W}}^T{\overline{\mathcal{A}}}_2+{\overline{\mathcal{A}}}_2^T\mathcal{W}{\mathcal{M}}_2\hfill & *\hfill \\ \multicolumn{1}{c}{{\left({\mathcal{M}}_2^T{\mathcal{W}}^T\overline{\mathcal{D}}\right)}^T}& -{\gamma }^{2}I\hfill \end{array}\right],\hfill \\ {\Pi }_2=\left[\begin{array}{cc}{\overline{\mathcal{A}}}_1\hfill & \overline{\mathcal{D}}\hfill \end{array}\right],{\Pi }_3=\left[\begin{array}{cc}\overline{\mathcal{C}}\hfill & 0\hfill \end{array}\right].\hfill \end{array}$$

By Schur complement and (15), one has

$$\begin{array}{c}\mathrm{\Delta }{\mathcal{V}}_2\left(k\right)\le d_2{\mathcal{V}}_1\left(k\right)-z^T\left(k\right)z\left(k\right)+{\gamma }^2{\omega }^T\left(k\right)\omega \left(k\right).\hfill \end{array}$$

(27)

From (22) and (27), one can derive that

$${\mathcal{V}}_1(k+1)\le {\overline{d}}_1{\mathcal{V}}_1\left(k\right)-z^T\left(k\right)z\left(k\right)+{\gamma }^2{\omega }^T\left(k\right)\omega \left(k\right)$$

(28)

$${\mathcal{V}}_2(k+1)\le {\overline{d}}_2{\mathcal{V}}_2\left(k\right)-z^T\left(k\right)z\left(k\right)+{\gamma }^2{\omega }^T\left(k\right)\omega \left(k\right)$$

(29)

where ${\overline{d}}_1=1-d_1,{\overline{d}}_2=1+d_2$ .

From the definition of Lyapunov functional and (13), we have

$${\mathcal{V}}_1\left(k\right)\le c{\mathcal{V}}_2\left(k\right),{\mathcal{V}}_2\left(k\right)\le c{\mathcal{V}}_1\left(k\right).$$

(30)

Next, there are two situations to consider, namely, $k\notin [O_s^{\mathrm{on}},O_s^{\mathrm{off}})$ and $k\in [O_s^{\mathrm{on}},O_s^{\mathrm{off}})$ . Suppose $k\notin [O_s^{\mathrm{on}},O_s^{\mathrm{off}})$ , with the help of (28)–(30), one can obtain

$$\begin{array}{c}\multicolumn{1}{c}{\mathcal{V}\left(k\right)\le {\overline{d}}_1{\mathcal{V}}_1(k-1)+{\gamma }^2{\omega }^T(k-1)\omega (k-1)}\\ -z^T(k-1)z(k-1)\hfill \\ \le {\overline{d}}_1^2{\mathcal{V}}_1(k-2)+{\overline{d}}_1{\gamma }^2{\omega }^T(k-2)\omega (k-2)\hfill \\ -{\overline{d}}_1{z}^T(k-2)z(k-2)\hfill \\ +{\gamma }^2{\omega }^T(k-1)\omega (k-1)-z^T(k-1)z(k-1)\hfill \\ ⋮\hfill \\ \le {\overline{d}}_1^{k-O_s^{\mathrm{on}}}{\mathcal{V}}_1\left(O_s^{\mathrm{on}}\right)+{\overline{d}}_1^{k-O_s^{\mathrm{on}}-1}[{\gamma }^2{\omega }^T(k-O_s^{\mathrm{on}})\hfill \\ \times \omega (k-O_s^{\mathrm{on}})-z^T(k-O_s^{\mathrm{on}})z(k-O_s^{\mathrm{on}})]\hfill \\ +{\overline{d}}_1^{k-O_s^{\mathrm{on}}-2}[{\gamma }^2{\omega }^T(k-O_s^{\mathrm{on}}+1)\omega (k-O_s^{\mathrm{on}}+1)\hfill \\ -z^T(k-O_s^{\mathrm{on}}+1)z(k-O_s^{\mathrm{on}}+1)]+\cdots \hfill \\ +{\gamma }^2{\omega }^T(k-1)\omega (k-1)-z^T(k-1)z(k-1)\hfill \\ \le {\overline{d}}_1^{k-O_s^{\mathrm{on}}}c{\mathcal{V}}_2\left(O_s^{\mathrm{on}}\right)+{\overline{d}}_1^{k-O_s^{\mathrm{on}}-1}[{\gamma }^2{\omega }^T(k-O_s^{\mathrm{on}})\hfill \\ \times \omega (k-O_s^{\mathrm{on}})-z^T(k-O_s^{\mathrm{on}})z(k-O_s^{\mathrm{on}})\hfill \\ +{\overline{d}}_1^{k-O_s^{\mathrm{on}}-2}[{\gamma }^2{\omega }^T(k-O_s^{\mathrm{on}}+1)\omega (k-O_s^{\mathrm{on}}+1)\hfill \\ -z^T(k-O_s^{\mathrm{on}}+1)z(k-O_s^{\mathrm{on}}+1)]+\cdots \hfill \\ +{\gamma }^2{\omega }^T(k-1)\omega (k-1)-z^T(k-1)z(k-1)\hfill \end{array}$$

$$\begin{array}{c}\le {\overline{d}}_1^{k-O_s^{\mathrm{on}}}{\overline{d}}_{2}c{\mathcal{V}}_2(O_s^{\mathrm{on}}-1)\hfill \\ +{\overline{d}}_1^{k-O_s^{\mathrm{on}}-1}c[{\gamma }^2{\omega }^T(k-O_s^{\mathrm{on}}-1)\omega (k-O_s^{\mathrm{on}}-1)\hfill \\ -z^T(k-O_s^{\mathrm{on}}-1)z(k-O_s^{\mathrm{on}}-1)]\hfill \\ +{\overline{d}}_1^{k-O_s^{\mathrm{on}}-1}[{\gamma }^2{\omega }^T(k-O_s^{\mathrm{on}})\omega (k-O_s^{\mathrm{on}})\hfill \\ -z^T(k-O_s^{\mathrm{on}})z(k-O_s^{\mathrm{on}})]\hfill \\ +{\overline{d}}_1^{k-O_s^{\mathrm{on}}-2}[{\gamma }^2{\omega }^T(k-O_s^{\mathrm{on}}+1)\omega (k-O_s^{\mathrm{on}}+1)\hfill \\ -z^T(k-O_s^{\mathrm{on}}+1)z(k-O_s^{\mathrm{on}}+1)]+\cdots \hfill \\ +{\gamma }^2{\omega }^T(k-1)\omega (k-1)-z^T(k-1)z(k-1)\hfill \\ ⋮\hfill \\ \le {\overline{d}}_1^{k-{\Gamma }_1(0,k)}{\overline{d}}_2^{{\Gamma }_1(0,k)}{c}^{o(0,k)}\mathcal{V}\left(0\right)\hfill \\ +\sum _{q=0}^{k-1}{\overline{d}}_1^{k-p-1}{\overline{d}}_1^{-{\Gamma }_1(q,k)}{\overline{d}}_2^{{\Gamma }_1(q,k)}{c}^{o(q,k)}\hfill \\ \times [{\gamma }^2{\omega }^T\left(k\right)\omega \left(k\right)-z^T\left(k\right)z\left(k\right)].\hfill \end{array}$$

(31)

As to $k\in [O_s^{\mathrm{on}},O_s^{\mathrm{off}})$ , the same result as (31) can be get, the proof process is omitted.

Next, the exponential stability of the systems (6) and (7) are considered.

Let $\omega \left(k\right)=0$ and (31) with Assumption 1 gets

$$\begin{array}{c}\mathcal{V}\left(k\right)\le {\overline{d}}_1^{k-{\Gamma }_1(0,k)}{\overline{d}}_2^{{\Gamma }_1(0,k)}{c}^{o(0,k)}\mathcal{V}\left(0\right)\hfill \\ \multicolumn{1}{c}{=e^{(k-{\Gamma }_1(0,k))\ln {\overline{d}}_1+{\Gamma }_1(0,k)\ln {\overline{d}}_2+o(0,k)\ln c}\mathcal{V}\left(0\right)}\\ \multicolumn{1}{c}{\le e^{(k-{\Gamma }_1(0,k))\ln {\overline{d}}_1+{\Gamma }_1(0,k)\ln {\overline{d}}_2+(a_1+\frac{k}{a_2})\ln c}\mathcal{V}\left(0\right).}\end{array}$$

(32)

By using Assumption 2, we have

$$\begin{array}{c}[k-{\Gamma }_1(0,k)]\ln {\overline{d}}_1+{\Gamma }_1(0,k)\ln {\overline{d}}_2\hfill \\ \multicolumn{1}{c}{\le (k-b_1-\frac{k}{b_2})\ln {\overline{d}}_1+(k+\frac{k}{b_2})\ln {\overline{d}}_2}\\ \multicolumn{1}{c}{=b_1(\ln {\overline{d}}_2-\ln {\overline{d}}_1)+k[(1-\frac{1}{b_2})\ln {\overline{d}}_1+\frac{1}{b_2}\ln {\overline{d}}_2].}\end{array}$$

(33)

In addition, (14) indicates that

$$(1-\frac{1}{b_2})\ln {\overline{d}}_1+\frac{1}{b_2}\ln {\overline{d}}_2\le \ln {\overline{d}}_3.$$

(34)

Combining (33) and (34), we get

$$\begin{array}{c}[k-{\Gamma }_1(0,k)]\ln {\overline{d}}_1+{\Gamma }_1(0,k)\ln {\overline{d}}_2\hfill \\ \multicolumn{1}{c}{\le b_1(\ln {\overline{d}}_2-\ln {\overline{d}}_1)+\ln {\overline{d}}_3.}\end{array}$$

(35)

Substituting (35) into (32), one has

$$\begin{array}{c}\mathcal{V}\left(k\right)\le e^{[b_1(\ln {\overline{d}}_2-\ln {\overline{d}}_1)+(a_1+\frac{k}{a_2})]}{e}^{(\ln {\overline{d}}_3+\frac{\ln c}{a_2})k}\mathcal{V}\left(0\right)\hfill \\ \multicolumn{1}{c}{\le \frac{{\overline{d}}_2^{b_1}{c}^{\frac{a_1}{a_2}}}{{\overline{d}}_1^{b_1}}\left({\overline{d}}_3{c}^{\frac{1}{a_2}}\right)\mathcal{V}\left(0\right).}\end{array}$$

(36)

Define

$$\begin{array}{c}{f}_1=\min \{{\lambda }_{\min }\{{\mathcal{P}}_1\},{\lambda }_{\min }\{{\mathcal{P}}_2\}\},\hfill \\ \multicolumn{1}{c}{f_2=\max \{{\lambda }_{\max }\{{\mathcal{P}}_1\},{\lambda }_{\max }\{{\mathcal{P}}_2\}\}.}\end{array}$$

(37)

From (36), one can obtain

$$f_1\Vert \mathrm{Ex}\left(k\right){\Vert }^2\le \mathcal{V}\left(k\right),\mathcal{V}\left(0\right)\le f_2\Vert \mathrm{Ex}\left(0\right){\Vert }^2.$$

(38)

Combining (36) and (38) yields

$$\begin{array}{c}{\Vert \overline{\mathcal{E}}x\left(k\right)\Vert }^2\le \frac{f_2}{f_1}\frac{{\overline{d}}_2^{b_1}{c}^{\frac{a_1}{a_2}}}{{\overline{d}}_1^{b_1}}\left({\overline{d}}_3{c}^{\frac{1}{a_2}}\right){\Vert \overline{\mathcal{E}}x\left(0\right)\Vert }^2.\hfill \end{array}$$

(39)

From (15), one can get ${\overline{d}}_3{c}^{\frac{1}{a_2}}<1$ . According to Define 1, the exponential stability of the systems (6) and (7) are proved.

Finally, $H_{\infty }$ performance is proved. Under zero initial condition and $\omega \left(k\right)\ne 0$ , it can be obtained from (31)

$$\begin{array}{c}\mathcal{V}\left(k\right)\le \sum _{q=0}^{k-1}{\overline{d}}_1^{k-p-1-{\Gamma }_1(q,k)}{\overline{d}}_2^{{\Gamma }_1(q,k)}{c}^{o(q,k)}\hfill \\ \multicolumn{1}{c}{\times [{\gamma }^2{\omega }^T\left(q\right)\omega \left(q\right)-z^T\left(q\right)z\left(q\right)].}\end{array}$$

(40)

Since $\mathcal{V}\left(k\right)>0$ , (40) provides

$$\begin{array}{c}\sum _{q=0}^{\hat{k}}{\overline{d}}_1^{\hat{k}-p-{\Gamma }_1(q,\hat{k})}{\overline{d}}_2^{{\Gamma }_1(q,\hat{k})}{c}^{o(q,\hat{k})}{z}^T\left(q\right)z\left(q\right)\hfill \\ \multicolumn{1}{c}{=\sum _{q=0}^{\hat{k}}{e}^{(\hat{k}-q-{\Gamma }_1(q,\hat{k}))\ln {\overline{d}}_1+{\Gamma }_1(q,\hat{k})\ln {\overline{d}}_2+o(q,\hat{k})\ln c}{z}^T\left(q\right)z\left(q\right)}\\ \multicolumn{1}{c}{\le {\gamma }^2\sum _{q=0}^{\hat{k}}{\overline{d}}_1^{\hat{k}-p-{\Gamma }_1(q,\hat{k})}{\overline{d}}_2^{{\Gamma }_1(q,\hat{k})}{c}^{o(q,\hat{k})}{\omega }^T\left(q\right)\omega \left(q\right)}\\ \multicolumn{1}{c}{={\gamma }^2\sum _{q=0}^{\hat{k}}{e}^{(\hat{k}-q-{\Gamma }_1(q,\hat{k}))\ln {\overline{d}}_1+{\Gamma }_1(q,\hat{k})\ln {\overline{d}}_2+o(q,\hat{k})\ln c}}\\ \multicolumn{1}{c}{\times {\omega }^T\left(q\right)\omega \left(q\right)}\end{array}$$

(41)

where $\hat{k}=k-1$ .

Notice that $d_1<1,d_2>1,c>1$ , the left side of (41) gives

$$\begin{array}{c}\sum _{q=0}^{\hat{k}}{e}^{(\hat{k}-q-{\Gamma }_1(q,\hat{k}))\ln {\overline{d}}_1+{\Gamma }_1(q,\hat{k})\ln {\overline{d}}_2+o(q,\hat{k})\ln c}{z}^T\left(q\right)z\left(q\right)\hfill \\ \multicolumn{1}{c}{\ge \sum _{q=0}^{\hat{k}}{e}^{(\hat{k}-q)\ln {\overline{d}}_1}{z}^T\left(q\right)z\left(q\right)}\\ \multicolumn{1}{c}{\ge \sum _{q=0}^{\hat{k}}{\overline{d}}_1^{\hat{k}-q}{z}^T\left(q\right)z\left(q\right).}\end{array}$$

(42)

Considering (40), (42) and Assumption 1, one deduces

$$\begin{array}{c}{\gamma }^2\sum _{q=0}^{\hat{k}}{e}^{(\hat{k}-q-{\Gamma }_1(q,\hat{k}))\ln {\overline{d}}_1+{\Gamma }_1(q,\hat{k})\ln {\overline{d}}_2+o(q,\hat{k})\ln c}{\omega }^T\left(q\right)\omega \left(q\right)\hfill \\ \le {\gamma }^2\sum _{q=0}^{\hat{k}}{e}^{[a_1(\ln {\overline{d}}_2-\ln {\overline{d}}_1)+a_1\frac{\ln c}{a_2})]}{e}^{(\ln {\overline{d}}_3+\frac{\ln c}{a_2})(\hat{k}-p)}\hfill \\ \times {\omega }^T\left(q\right)\omega \left(q\right)\hfill \\ ={\gamma }^2\frac{{\overline{d}}_2^{b_1}{c}^{\frac{a_1}{a_2}}}{{\overline{d}}_1^{b_1}}\sum _{q=0}^{\hat{k}}{\left({\overline{d}}_3{c}^{\frac{1}{a_2}}\right)}^{\hat{k}-p}{\omega }^T\left(q\right)\omega \left(q\right).\hfill \end{array}$$

(43)

From (41) and (43), it can be derived that

$$\begin{array}{c}\sum _{q=0}^{\hat{k}}{\overline{d}}_1^{\hat{k}-q}{z}^T\left(q\right)z\left(q\right)\hfill \\ \multicolumn{1}{c}{\le {\gamma }^2\frac{{\overline{d}}_2^{b_1}{c}^{\frac{a_1}{a_2}}}{{\overline{d}}_1^{b_1}}\sum _{q=0}^{\hat{k}}{\left({\overline{d}}_3{c}^{\frac{1}{a_2}}\right)}^{\hat{k}-p}{\omega }^T\left(q\right)\omega \left(q\right).}\end{array}$$

(44)

Adding $\hat{k}$ from 0 to $\infty$ on both sides of (44), we have

$$\begin{array}{c}\sum _{\hat{k}=0}^{\infty }\sum _{q=0}^{\hat{k}}{\overline{d}}_1^{\hat{k}-q}{z}^T\left(q\right)z\left(q\right)\hfill \\ \multicolumn{1}{c}{\le {\gamma }^2\frac{{\overline{d}}_2^{b_1}{c}^{\frac{a_1}{a_2}}}{{\overline{d}}_1^{b_1}}\sum _{\hat{k}=0}^{\infty }\sum _{q=0}^{\hat{k}}{\left({\overline{d}}_3{c}^{\frac{1}{a_2}}\right)}^{\hat{k}-p}{\omega }^T\left(q\right)\omega \left(q\right).}\end{array}$$

(45)

From (45), one has

$$\begin{array}{c}\sum _{\hat{p}=0}^{\infty }\sum _{\hat{k}=p}^{\infty }{\overline{d}}_1^{\hat{k}-q}{z}^T\left(q\right)z\left(q\right)\hfill \\ \multicolumn{1}{c}{\le {\gamma }^2\frac{{\overline{d}}_2^{b_1}{c}^{\frac{a_1}{a_2}}}{{\overline{d}}_1^{b_1}}\sum _{\hat{p}=0}^{\infty }\sum _{\hat{k}=p}^{\infty }{\left({\overline{d}}_3{c}^{\frac{1}{a_2}}\right)}^{\hat{k}-p}{\omega }^T\left(q\right)\omega \left(q\right).}\end{array}$$

(46)

Due to ${\overline{d}}_1<1$ and ${\overline{d}}_3{c}^{\frac{1}{a_2}}<1$ , one gets

$$\begin{array}{c}\sum _{p=0}^{\infty }{z}^T\left(q\right)z\left(q\right)\le {\hat{\gamma }}^2\sum _{p=0}^{\infty }{z}^T\left(q\right)z\left(q\right){\omega }^T\left(q\right)\omega \left(q\right)\hfill \end{array}$$

(47)

where ${\hat{\gamma }}^2={\gamma }^2\frac{{\overline{d}}_2^{b_1}{c}^{\frac{a_1}{a_2}}}{{\overline{d}}_1^{b_1}}\frac{1-{\overline{d}}_1}{1-{\overline{d}}_3{c}^{a_2}}$ . This completes the proof.

• Remark 3. In Park et al. ²⁷ and Chen et al. ²⁸ the authors discussed the exponential admissibility of singular systems. One can find that the proof process is very complex and the calculation is very huge. In this paper, the proof process is simplified by adopting the method described in Guan et al. ²⁹

Next, a novel method is developed to obtain the observer gains.

• Theorem 2. For the given scalars $a_1>0,a_2>0,b_1>0,b_2>1,\gamma >0$ , the systems (6) and (7) are EA with $H_{\infty }$ performance ${\hat{\gamma }}^2={\gamma }^2\psi$ if there existscalars $c>1,d_1\in (0,1),d_2>1,{\overline{d}}_3$ and matrices ${\mathcal{P}}_1=\mathrm{diag}\{{\mathcal{P}}_{11},{\mathcal{P}}_{12}\}>0,{\mathcal{P}}_2=\mathrm{diag}\{{\mathcal{P}}_{21},{\mathcal{P}}_{22}\}>0,{\mathcal{M}}_1=\mathrm{diag}\{{\mathcal{M}}_{11},{\mathcal{M}}_{12}\}$ and ${\mathcal{M}}_2=\mathrm{diag}\{{\mathcal{M}}_{21},{\mathcal{M}}_{22}\}$ such that

$$\begin{array}{cc}{\mathcal{P}}_1<c{\mathcal{P}}_2,\hfill & {\mathcal{P}}_2<c{\mathcal{P}}_1\hfill \end{array}$$

(48)

$$0<\frac{\ln {\overline{d}}_2-\ln {\overline{d}}_3}{\ln {\overline{d}}_3-\ln {\overline{d}}_1}\le b_2-1$$

(49)

$$a_2\ln {\overline{d}}_3+\ln c<0$$

(50)

$$\left[\begin{array}{ccc}{\mathrm{\Xi }}_{11}\hfill & {\mathrm{\Xi }}_{12}\hfill & {\mathrm{\Xi }}_{13}\hfill \\ \multicolumn{1}{c}{*}& {\mathrm{\Xi }}_{22}\hfill & {\mathrm{\Xi }}_{23}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\mathrm{\Xi }}_{33}\hfill \end{array}\right]<0$$

(51)

$$\left[\begin{array}{ccc}{\hat{\mathrm{\Xi }}}_{11}\hfill & {\hat{\mathrm{\Xi }}}_{12}\hfill & {\hat{\mathrm{\Xi }}}_{13}\hfill \\ \multicolumn{1}{c}{*}& {\hat{\mathrm{\Xi }}}_{22}\hfill & {\hat{\mathrm{\Xi }}}_{23}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\hat{\mathrm{\Xi }}}_{33}\hfill \end{array}\right]<0$$

(52)

where

$$\begin{array}{c}{\mathrm{\Xi }}_{11}=\left[\begin{array}{ccc}{\theta }_{11}\hfill & W_1^T{S}_1\mathcal{C}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& {\theta }_{12}\hfill & M_{12}^2{W}_2^T\mathcal{D}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -{\gamma }^{2}I\hfill \end{array}\right],\hfill \\ {\mathrm{\Xi }}_{12}=\left[\begin{array}{ccc}{\mathcal{A}}^T{\mathcal{P}}_{11}+Y_1^T{\mathcal{B}}^T\hfill & 0\hfill & {\mathcal{C}}^T\hfill \\ \multicolumn{1}{c}{{\mathcal{C}}^T{S}_1^T}& {\mathcal{A}}^T{\mathcal{P}}_{12}-{\mathcal{C}}^T{S}_1^T\hfill & {\mathcal{C}}^T\hfill \\ \multicolumn{1}{c}{0}& {\mathcal{D}}^T{\mathcal{P}}_{12}\hfill & 0\hfill \end{array}\right],\hfill \\ {\mathrm{\Xi }}_{22}=\left[\begin{array}{ccc}-{\mathcal{P}}_{11}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -{\mathcal{P}}_{12}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -I\hfill \end{array}\right],\hfill \\ {\mathrm{\Xi }}_{13}=\left[\begin{array}{cc}{\theta }_{13}+\epsilon Y_1^T\hfill & {\theta }_{14}\hfill \\ \multicolumn{1}{c}{0}& {\theta }_{15}+\epsilon {\mathcal{C}}^T{S}_1^T\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right],\hfill \\ {\mathrm{\Xi }}_{23}=\left[\begin{array}{cc}{\theta }_{16}\hfill & {\theta }_{17}\hfill \\ \multicolumn{1}{c}{0}& {\theta }_{18}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right],\hfill \\ {\mathrm{\Xi }}_{33}=\left[\begin{array}{cc}-\epsilon X_1-X_1^T\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -\epsilon Q_1-Q_1^T\hfill \end{array}\right],\hfill \\ {\hat{\mathrm{\Xi }}}_{11}=\left[\begin{array}{ccc}{\hat{\theta }}_{11}\hfill & {\mathcal{C}}^T{S}_2^T{W}_2\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& {\hat{\theta }}_{12}\hfill & M_{22}^2{W}_2^T\mathcal{D}\hfill \\ *\hfill & *\hfill & -{\gamma }^{2}I\hfill \end{array}\right],\hfill \\ {\hat{\mathrm{\Xi }}}_{12}=\left[\begin{array}{ccc}{\mathcal{A}}^T{\mathcal{P}}_{21}+Y_2^T{\mathcal{B}}^T-{\mathcal{C}}^T{S}_2^T\hfill & {\mathcal{C}}^T{S}_2^T\hfill & {\mathcal{C}}^T\hfill \\ \multicolumn{1}{c}{0}& {\mathcal{A}}^T{\mathcal{P}}_{22}\hfill & {\mathcal{C}}^T\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\right],\hfill \\ {\hat{\mathrm{\Xi }}}_{22}=\left[\begin{array}{ccc}-{\mathcal{P}}_{21}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -{\mathcal{P}}_{22}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -I\hfill \end{array}\right],\hfill \end{array}$$

$$\begin{array}{c}{\hat{\mathrm{\Xi }}}_{13}=\left[\begin{array}{cc}{\hat{\theta }}_{13}+\epsilon Y_2^T\hfill & {\hat{\theta }}_{14}+\epsilon {\mathcal{C}}^T{S}_2^T\hfill \\ \multicolumn{1}{c}{0}& {\hat{\theta }}_{15}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right],\hfill \\ {\hat{\mathrm{\Xi }}}_{23}=\left[\begin{array}{cc}{\hat{\theta }}_{16}\hfill & {\hat{\theta }}_{17}\hfill \\ \multicolumn{1}{c}{0}& {\hat{\theta }}_{18}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right],\hfill \\ {\hat{\mathrm{\Xi }}}_{33}=\left[\begin{array}{cc}-\epsilon X_2-X_2^T\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -\epsilon Q_2-Q_2^T\hfill \end{array}\right],\hfill \\ {\theta }_{11}=-{\overline{d}}_1{\mathcal{E}}^T{\mathcal{P}}_{11}\mathcal{E}+M_{11}^T{W}_1^T\mathcal{A}+{\mathcal{A}}^T{W}_1{M}_{11}\hfill \\ +W_1^T\mathcal{B}{Y}_1+Y_1^T{\mathcal{B}}^T{W}_1,\hfill \\ {\theta }_{12}=-{\overline{d}}_1{\mathcal{E}}^T{\mathcal{P}}_{12}\mathcal{E}+M_{12}^T{W}_2^T\mathcal{A}+{\mathcal{A}}^T{W}_2{M}_{12}\hfill \\ -W_2^T{S}_1\mathcal{C}-{\mathcal{C}}^T{S}_1^T{W}_2,\hfill \\ {\theta }_{13}=M_{11}^T{W}_1^T\mathcal{B}-W_1^T\mathcal{B}{X}_1,{\theta }_{14}=M_{11}^T{W}_1^T-W_1^T{Q}_1\hfill \\ {\theta }_{15}=-M_{12}^T{W}_2^T+W_2^T{Q}_1,{\theta }_{16}={\mathcal{P}}_{11}\mathcal{B}-\mathcal{B}{X}_1,\hfill \\ {\theta }_{17}={\mathcal{P}}_{11}-Q_1,{\theta }_{18}=-{\mathcal{P}}_{12}+Q_1,\hfill \\ {\hat{\theta }}_{11}=-{\overline{d}}_2{\mathcal{E}}^T{\mathcal{P}}_{21}\mathcal{E}+M_{21}^T{W}_1^T\mathcal{A}+{\mathcal{A}}^T{W}_1{M}_{21}\hfill \\ +W_1^T\mathcal{B}{Y}_2+Y_2^T{\mathcal{B}}^T{W}_1-W_1^T{S}_2\mathcal{C}-{\mathcal{C}}^T{S}_2^T{W}_1,\hfill \\ {\hat{\theta }}_{12}=-{\overline{d}}_2{\mathcal{E}}^T{\mathcal{P}}_{22}\mathcal{E}+M_{22}^T{W}_2^T\mathcal{A}+{\mathcal{A}}^T{W}_2{M}_{22},\hfill \\ {\hat{\theta }}_{13}=M_{21}^T{W}_1^T\mathcal{B}-W_1^T\mathcal{B}{X}_2,\hfill \\ {\hat{\theta }}_{14}=-M_{21}^T{W}_1^T+W_1^T{Q}_2,\hfill \\ {\hat{\theta }}_{15}=M_{22}^T{W}_2^T-W_2^T{Q}_2,{\hat{\theta }}_{16}={\mathcal{P}}_{21}\mathcal{B}-\mathcal{B}{X}_2,\hfill \\ {\hat{\theta }}_{17}=-{\mathcal{P}}_{21}+Q_2,{\hat{\theta }}_{18}={\mathcal{P}}_{22}-Q_2,\hfill \\ {\overline{d}}_1=1-d_1,{\overline{d}}_2=1+d_2,\psi =\frac{{\overline{d}}_2^{b_1}{c}^{\frac{a_1}{a_2}}}{{\overline{d}}_1^{b_1}}\frac{1-{\overline{d}}_1}{1-{\overline{d}}_3{c}^{a_2}}\hfill \end{array}$$

and $\mathcal{W}=\mathrm{diag}\{{\mathcal{W}}_1,{\mathcal{W}}_2\}$ is a constant matrix with satisfying ${\mathcal{E}}^T{\mathcal{W}}_1={\mathcal{E}}^T{\mathcal{W}}_2=0$ .

Moreover, the observer gains are given as

$${\mathcal{K}}_1=X_1^{-1}{Y}_1,{\mathcal{K}}_2=X_2^{-1}{Y}_2,$$

(53)

$${\mathcal{F}}_1=Q_1^{-1}{S}_1,{\mathcal{F}}_2=Q_2^{-1}{S}_2.$$

(54)

Proof. Define

$${\mathcal{K}}_1=X_1^{-1}{Y}_1,{\mathcal{K}}_2=X_2^{-1}{Y}_2,$$

(55)

$${\mathcal{F}}_1=Q_1^{-1}{S}_1,{\mathcal{F}}_2=Q_2^{-1}{S}_2.$$

(56)

$${\mathcal{P}}_1=\left[\begin{array}{cc}{\mathcal{P}}_{11}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\mathcal{P}}_{12}\hfill \end{array}\right]>0,$$

(57)

$${\mathcal{P}}_2=\left[\begin{array}{cc}{\mathcal{P}}_{21}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\mathcal{P}}_{22}\hfill \end{array}\right]>0,$$

(58)

$${\mathcal{M}}_1=\left[\begin{array}{cc}{\mathcal{M}}_{11}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\mathcal{M}}_{12}\hfill \end{array}\right],$$

(59)

$${\mathcal{M}}_2=\left[\begin{array}{cc}{\mathcal{M}}_{21}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\mathcal{M}}_{22}\hfill \end{array}\right],$$

(60)

$$\mathcal{W}=\left[\begin{array}{cc}{\mathcal{W}}_1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\mathcal{W}}_2\hfill \end{array}\right].$$

(61)

Substituting (55)–(61) into (51) and (52), and applying Lemma 1, we can get (14) and (15). The proof is completed.

• Remark 4. In Wang et al., ²⁰ in order to deal with the coupling term $P\mathcal{F}\mathcal{C}$ , the authors required that $\mathcal{C}$ is a column full rank matrix. This condition limits our application. In this paper, Lemma 1 is adopted to deal with the coupling term $P\mathcal{F}\mathcal{C}$ and the method can remove the above restriction.

Simulation

Two examples are given to illustrate the correctness of the proposed approach.

• Example 1. Consider the following systems with the parameters:

$$\begin{array}{c}\begin{array}{c}\mathcal{E}=\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\right],\mathcal{A}=\left[\begin{array}{ccc}0.4\hfill & 0.5\hfill & -0.3\hfill \\ \multicolumn{1}{c}{-0.5}& 0.2\hfill & 0.65\hfill \\ \multicolumn{1}{c}{0.5}& 0.6\hfill & -0.3\hfill \end{array}\right],\hfill \\ \mathcal{B}=\left[\begin{array}{c}0.3\hfill \\ \multicolumn{1}{c}{0.5}\\ \multicolumn{1}{c}{0.6}\end{array}\right],\mathcal{C}=\left[\begin{array}{ccc}-1\hfill & 0.3\hfill & 0.3\hfill \\ \multicolumn{1}{c}{1}& -1\hfill & 1.2\hfill \\ \multicolumn{1}{c}{0.5}& 0\hfill & 0.17\hfill \end{array}\right],\hfill \\ \mathcal{W}=\left[\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \\ \multicolumn{1}{c}{1}& 2\hfill & 2\hfill \end{array}\right],\mathcal{D}=\left[\begin{array}{c}0.1\hfill \\ \multicolumn{1}{c}{0.15}\\ \multicolumn{1}{c}{0.2}\end{array}\right].\hfill \end{array}\hfill \end{array}$$

Assume that $e=1,a_1=5,a_2=8,b_1=5,b_2=5,c=4,d_1=0.2,d_2=1.5,d_3=0.82,\gamma =10$ , and the external disturbance $\nu \left(k\right)={\mathrm{e}}^{-0.1k}$ . By Theorem 2, the controller gains are given

$$\begin{array}{c}\multicolumn{1}{c}{\begin{array}{c}{\mathcal{K}}_1=\left[\begin{array}{ccc}-1.8601\hfill & -1.5834\hfill & -1.4152\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{{\mathcal{F}}_1=\left[\begin{array}{ccc}0.0791\hfill & -0.1656\hfill & 0.7391\hfill \\ \multicolumn{1}{c}{0.1945}& -0.1393\hfill & 0.8155\hfill \\ \multicolumn{1}{c}{0.3776}& -0.4081\hfill & 2.1857\hfill \end{array}\right],}\\ \multicolumn{1}{c}{{\mathcal{K}}_2=\left[\begin{array}{ccc}-0.4151\hfill & -0.9490\hfill & -0.8836\hfill \end{array}\right],}\\ \multicolumn{1}{c}{{\mathcal{F}}_1=\left[\begin{array}{ccc}-0.4975\hfill & -0.5233\hfill & 1.1548\hfill \\ \multicolumn{1}{c}{-0.1836}& 0.3746\hfill & 0.0478\hfill \\ \multicolumn{1}{c}{-0.0354}& 0.0521\hfill & -0.3084\hfill \end{array}\right].}\end{array}}\end{array}$$

Supposes ${\xi }_0=[-1.21.10.550.10.10.1{]}^T$ . Figure 1 records the states of the open-loop system and the open-loop system is unstable. Figures 2 –4 show the states of the system and its estimation. Figure 5 depicts the evaluation of DoS attacks. The authors ³⁰ required that $\mathcal{C}$ is a column full rank matrix and the singular value decomposition was adopted to deal with the coupling term $P\mathcal{F}\mathcal{C}$ . These requirements reduce practical applicability and increase computational complexity. In our work, these requirements are remove by Lemma 1.

• Example 2. The discretized water pollution model is considered, which can modeled as a discrete-time singular system. ³¹ The model parameters are as follows

$$\begin{array}{c}\mathcal{E}=\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\right],\mathcal{A}=\left[\begin{array}{ccc}-0.15\hfill & 0\hfill & 1.21\hfill \\ \multicolumn{1}{c}{0}& 0.23\hfill & 0.14\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 3.00\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{\mathcal{B}=\left[\begin{array}{c}-0.52\hfill \\ \multicolumn{1}{c}{1.5}\\ \multicolumn{1}{c}{0}\end{array}\right],\mathcal{C}=\left[\begin{array}{ccc}1.21\hfill & 0.98\hfill & -0.23\hfill \\ \multicolumn{1}{c}{0.88}& 0.88\hfill & 0.36\hfill \\ \multicolumn{1}{c}{1.11}& 0.74\hfill & -0.19\hfill \end{array}\right],}\\ \mathcal{W}=\left[\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \\ \multicolumn{1}{c}{2}& 1\hfill & 1\hfill \end{array}\right],\mathcal{D}=\left[\begin{array}{c}0.15\hfill \\ \multicolumn{1}{c}{0.15}\\ \multicolumn{1}{c}{0.2}\end{array}\right].\hfill \end{array}$$

Figure 1.

The states of the open-loop system of Example 1.

Figure 2.

The states of $x_1\left(k\right)$ and its estimation of Example 1.

Figure 3.

The states of $x_2\left(k\right)$ and its estimation of Example 1.

Figure 4.

The states of $x_3\left(k\right)$ and its estimation of Example 1.

Figure 5.

Evaluation of DoS attacks of Example 1.

Assume $e=1,a_1=5,a_2=7.8,b_1=5.1,b_2=5,c=3.8,d_1=0.2,d_2=2.4,d_3=0.8,\gamma =10$ , and the disturbance $\omega \left(t\right)=e^{-0.1k}$ . By Theorem 2, the controller gains are given

$$\begin{array}{c}\multicolumn{1}{c}{\begin{array}{c}{\mathcal{K}}_1=\left[\begin{array}{ccc}-0.0230\hfill & -0.1232\hfill & 0.0928\hfill \end{array}\right],\hfill \\ {\mathcal{F}}_1=\left[\begin{array}{ccc}-1.5077\hfill & 1.0850\hfill & 0.6912\hfill \\ \multicolumn{1}{c}{0.3481}& 0.1666\hfill & -0.5214\hfill \\ \multicolumn{1}{c}{-7.4691}& 4.1116\hfill & 5.3514\hfill \end{array}\right],\hfill \\ {\mathcal{K}}_2=\left[\begin{array}{ccc}-0.0330\hfill & -0.0564\hfill & 0.0558\hfill \end{array}\right],\hfill \\ {\mathcal{F}}_2=\left[\begin{array}{ccc}-3.8331\hfill & 2.1501\hfill & 2.6022\hfill \\ \multicolumn{1}{c}{-0.9682}& 0.5419\hfill & 0.6827\hfill \\ \multicolumn{1}{c}{-8.3474}& 4.3612\hfill & 6.1135\hfill \end{array}\right].\hfill \end{array}}\end{array}$$

Supposes ${\xi }_0=[-1.21.150.580.10.050.05{]}^T$ . Figures 6 –8 show the states of the system and its estimation. Figure 9 depicts the evaluation of DoS attacks. Above all, these results illustrate that the proposed approach is effective.

Figure 6.

The states of $x_1\left(k\right)$ and its estimation of Example 2.

Figure 7.

The states of $x_2\left(k\right)$ and its estimation of Example 2.

Figure 8.

The states of $x_2\left(k\right)$ and its estimation of Example 2.

Figure 9.

Evaluation of DoS attacks of Example 2.

Conclusions

The problem of observer-based secure controller design has been studied for discrete-time singular systems with DoS attacks in this paper. The non-periodic DoS attacks which make the system unstable are considered. A switched observer-based control method is established to estimate the states. Firstly, some sufficient conditions are given for the corresponding switched systems to be exponentially admissible with $H_{\infty }$ performance. Secondly, a novel method is given to design the controller gain, which can remove the condition that the measured output matrix is column full rank. At last, two examples are given to illustrate the effectiveness of the design method. Although the observer-based secure control issue was discussed for linear singular systems, many systems are modeled as nonlinear systems in engineering, such as SMJSs and singular T-S fuzzy systems. The observer-based secure SMC issue will be considered for SMJSs and singular T-S fuzzy systems in the future.