Exponential Stability and ℒ₁-Gain Performance for Positive Sampled-Data Control Systems

Abstract

This paper aims to explore the application of sampling control technology in positive Markov jump systems (PMJSs), focusing on the exponential stability in mean and ${\mathcal{L}}_1$-gain performance of the system. We first establish a PMJS model based on sampling control and conduct a detailed analysis of its exponential stability in mean. Subsequently, combined with ${\mathcal{L}}_1$-gain theory, we propose a sufficient condition for PMJSs with a prescribed ${\mathcal{L}}_1$-gain performance level, which ensures the robustness and satisfaction of performance indicators of the system in a randomly switching environment. From the sufficient conditions we have given, we can indeed see the effect of the sampling period on the system performance. Based on this sufficient condition, we further design the state feedback controller, and give the feedback gain solving algorithm based on linear programming method. A simple simulation example verifies the correctness and effectiveness of the results. The main contribution of this paper is to introduce sampling control technology into the research of PMJSs and propose a complete theoretical framework and analysis method, providing new theoretical support and practical application value for the sampling control of PMJSs.

1. Introduction

In recent decades, Markov jump systems (MJSs) ([1,2,3,4]) have garnered widespread attention due to their effectiveness in modeling stochastic switching phenomena. MJSs have demonstrated robust applicability across various fields, such as biological system modeling, financial engineering, networked control, and communication systems. However, traditional studies on MJSs often assume that the system states are continuous and the control inputs are continuously available. With the advancement of digital technology, sampled-data control systems ([5,6,7,8]) have emerged as a common implementation approach, especially considering energy efficiency and hardware costs. Therefore, introducing sampling control techniques into Markov jump systems, particularly positive Markov jump systems (PMJSs) ([9,10]), has become an important area of research.

PMJSs possess a special structure characterized by non-negative state variables, which imposes non-negativity requirements on both system outputs and state variables. This feature provides a unique advantage for PMJSs in capturing certain dynamic behaviors in real-world systems, such as in telecommunication networks [11], pest structured population dynamic systems [12], SIS Epidemiological Models [13], and so on. However, the positivity constraints of PMJSs introduce additional challenges for control and stability analysis. In [14], a stability analysis and synthesis of PMJSs in the p-th moment sense, including stability, strong stability, observability, and detectability, were conducted. In [15], the authors studied the problems of exponential stability and ${\mathcal{L}}_1$-gain analysis for positive time-delay MJSs with switching transition rates. In [16], ${\mathcal{L}}_1$-stochastic stability and ${\mathcal{L}}_1$-gain performance of positive Markov jump linear systems (PMJLSs) with time-delays were studied. Some necessary and sufficient conditions were provided. In [17,18], sufficient and necessary conditions of stochastic stability were obtained for a PMJLS, and a state feedback controller was designed to stabilize the PMJLS by solving linear programming problems. In [19,20], sampled-data-based event-triggered control was studied for PMJSs. Event-driven control requires the monitoring and handling of various possible events, which makes the design and implementation of the system more complex. In contrast, sampling control can utilize mature control theories and methods for analysis and design.

For the study of sampling control systems, Lamnabhi-Lagarrigue et al. comprehensively outlined three prevalent methodologies for the investigation of sampled-data control systems in [21,22]. The initial approach, termed digital redesign, entails the sequential steps of conceiving a continuous-time controller followed by its discrete-time implementation. The second approach, known as discrete design, hinges on the utilization of the discrete-time model of the controlled object to formulate a discrete-time controller. The third approach, designated as direct design, is predicated on the precise sampled-data model of the controlled object, with the objective of fulfilling the system’s stability requisites and attaining the desired performance benchmarks. In contrast to the other two methodologies, notwithstanding the fact that the direct design approach mandates the most advanced and intricate design techniques and poses a relatively greater degree of implementation complexity, it obviates the need for approximation procedures and is thus more proficient in safeguarding the stability and performance characteristics of the system. Because of the importance and practicability of the sampling control strategy, the research on this kind of system has been quite extensive and in-depth. In [23], Hu, Shi, and Sun investigated the stability and ${\mathcal{L}}_1$-gain characterization of positive sampled-data systems, in the deterministic sense. There was no random phenomenon in the system studied.

In this paper, we will apply sampling control to positive Markov jump systems, study their exponential stability, and analyze their ${\mathcal{L}}_1$-gain performance in a random sense, based on an accurate sampling model (linear system with time-varying delay). Instead of the usual requirement [24] that the derivative of time delay has to belong to (0,1) in the study of the stability of time-varying delay systems, the derivative of time delay in the system we study here is equal to 1.

The main contribution of this paper lies in the application of sampling control techniques to the study of PMJSs, thereby exploring and providing exponential stability analysis and ${\mathcal{L}}_1$-gain performance analysis for the system. Specifically, we first establish a sampled-data model of a PMJS and analyze the exponential stability of the system based on this model. Secondly, we present sufficient conditions of exponential stability with a weighted ${\mathcal{L}}_1$-gain property for the positive system, ensuring the robustness and performance requirements of the system in a randomly switching environment. Then, a ${\mathcal{L}}_1$ sampled-data control law is designed to maintain the systems’ positivity and attain the exponential stability with ${\mathcal{L}}_1$-gain performance. In the process of the controller design, we present an algorithm based on the linear programming method to calculate the feedback gain. A simulation example is provided to verify our results.

2. Preliminaries

Throughout this paper, ${\mathbb{R}}_{+}$, ${\mathbb{R}}^n$, and ${\mathbb{R}}^{n\times m}$ denote the set of all non-negative real numbers, n-dimensional real space, and $n\times m$ dimensional real matrix space, respectively. For vector $x\in {\mathbb{R}}^n$, $\left|x\right|$ denotes the Euclidean norm $\left|x\right|={\left({\sum }_{i=1}^n{x}_i^2\right)}^{1/2}$. ${\parallel ·\parallel }_p$ denotes the p-norm on ${\mathbb{R}}^n$. The transpose of vectors and matrices are denoted by the superscript T. Unless otherwise specified, all vectors are column vectors. For vectors $x,y\in {\mathbb{R}}^n$, we write $x⪰y(x⪯y)$ if $x_i\ge y_i(x_i\le y_i)$ for $i\in {\mathcal{I}}_n=:\{1,2,\cdots ,n\}$; $x>y(x<y)$ if $x⪰y(x⪯y)$ and $x\ne y$; $x\succ y(x\prec y)$ if $x_i>y_i(x_i<y_i)$ for $i\in {\mathcal{I}}_n$. ${\mathcal{C}}^i$ denotes all the ith continuous differential functions; ${\mathcal{C}}^{i,k}$ denotes all the functions with ith continuously differentiable first component and kth continuously differentiable second component. $E\left\{x\right\}$ denotes the expectation of stochastic variable x. A function $\phi \left(u\right)$ is said to belong to the class $\mathcal{K}$ if $\phi \in \mathcal{C}({\mathbb{R}}_{+},{\mathbb{R}}_{+})$, $\phi \left(0\right)=0$ and $\phi \left(u\right)$ is strictly increasing in u. ${\mathcal{K}}_{\infty }$ is the subset of $\mathcal{K}$ functions that are unbounded. A function $\beta :{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is of class $\mathcal{KL}$, if $\beta (·,t)$ is of class $\mathcal{K}$ in the first argument for each fixed $t\ge 0$, and $\beta (s,t)$ decreases to 0 as $t\to +\infty$ for each fixed $s\ge 0$. Finally, we denote the composition of two functions $\phi :A\to B$ and $\psi :B\to C$ by $\psi \circ \phi :A\to C$.

Consider the system

$$\begin{array}{ccc}\hfill \dot{x}\left(t\right)& =& A\left(\sigma \left(t\right)\right)x\left(t\right)+B_1\left(\sigma \left(t\right)\right)w\left(t\right)+B_2\left(\sigma \left(t\right)\right)u\left(t\right),\hfill \\ \hfill z\left(t\right)& =& C\left(\sigma \right(t\left)\right)x\left(t\right)+D\left(\sigma \right(t\left)\right)w\left(t\right),\hfill \end{array}$$

(1)

where $x\left(t\right)\in {\mathbb{R}}^n$ and $z\left(t\right)\in {\mathbb{R}}^{n_z}$ are the state and output, respectively; $w\left(t\right)\in {\mathbb{R}}^{n_w}$ is the disturbance input, which belongs to $L_1[0,\infty )$; and $u\left(t\right)\in {\mathbb{R}}^{n_u}$ is the control input. The Markov process $\sigma \left(t\right):[0,\infty )\to \mathcal{P}=\{1,2,\cdots ,N\}$ involves the following probability transitions:

$$\begin{array}{ccc}\hfill Pr\left(\sigma \right(t+\Delta )=q|\sigma \left(t\right)=p)& =& \left\{\begin{array}{cc}{\pi }_{pq}\Delta +o(\Delta ),\hfill & p\ne q;\hfill \\ 1+{\pi }_{pp}\Delta +o(\Delta ),\hfill & p=q,\hfill \end{array}\right.\hfill \end{array}$$

where ${\pi }_{pq}\ge 0$ is the transition rate from p to $q(p\ne q)$, ${\pi }_{pp}=-{\sum }_{q=1,q\ne p}^N{\pi }_{pq}$, $\Delta >0$, and $o(\Delta )$ is an infinitesimal of higher order than $\Delta$. The transition rate matrix is denoted by $\Pi ={\left({\pi }_{pq}\right)}_{n\times n}$. $t_0=0$ is the initial time. For $\sigma \left(t\right)=p\in \mathcal{P}$, $A\left(\sigma \left(t\right)\right),B_1\left(\sigma \left(t\right)\right),B_2\left(\sigma \left(t\right)\right),C\left(\sigma \left(t\right)\right)$ and $D\left(\sigma \right(t\left)\right)$ are denoted by $A_p,B_{1p},B_{2p},C_p$ and $D_p$, which are constant matrices of appropriate dimensions. $x(t,x_0,{\sigma }_0)$ represents the state at time t with initial time $t_0=0$, initial state $x\left(0\right)={(x_{01},\cdots ,x_{0n})}^T$, and initial mode $\sigma \left(0\right)={\sigma }_0$. When the context makes it clear, we use $x\left(t\right)$ instead of $x(t,x_0,{\sigma }_0)$. State $x\left(t\right)$ is assumed to not jump at switching instants.

Definition 1. 

System (1) is said to be positive if, for any initial state $x_0⪰0$, $w\left(t\right)⪰0$ and $u\left(t\right)⪰0$, the corresponding trajectory $x\left(t\right)⪰0$, and $z\left(t\right)⪰0$ hold for all $t\ge 0$.

A matrix $A\in {\mathbb{R}}^{n\times n}$ is called the Metzler matrix if all its off-diagonal entries are non-negative.

It is well known ([3,25]) that system (1) is positive if and only if for any $p\in \mathcal{P}$, $A_p$ is Metzler and $B_{1p}\in {\mathbb{R}}_{+}^{n\times n_w},B_{2p}\in {\mathbb{R}}_{+}^{n\times n_u},C_p\in {\mathbb{R}}_{+}^{n_z\times n}$ and $D_p\in {\mathbb{R}}_{+}^{n_z\times n_w}$.

Definition 2 

([15]). System (1) with $w\left(t\right)=0$ and $u\left(t\right)=0$ is said to be exponentially stable in mean if the solution $x\left(t\right)$ satisfies

$$\begin{array}{c}\hfill E\{\parallel x\left(t\right)\parallel \}\le \alpha \parallel x_0\parallel e^{-\beta (t-t_0)},\forall t\ge t_0,\end{array}$$

for constants $\alpha >0$ and $\beta >0$.

Definition 3 

([26]). For $\lambda >0$ and $\gamma >0$, system (1) with $u\left(t\right)=0$ is said to have a prescribed ${\mathcal{L}}_1$-gain performance level $\gamma >0$ if the following conditions are satisfied:

(a) 

System (1) is exponentially stable in mean when $w\left(t\right)=0$.

(b) 

Under zero initial condition, i.e., $x_0=0$, system (1) satisfies

$$\begin{array}{c}\hfill E\{{\int }_{t_0}^{\infty }{e}^{-\lambda (t-t_0)}{\parallel z\left(t\right)\parallel }_{1}dt\}\le \gamma E\{{\int }_{t_0}^{\infty }{\parallel w\left(t\right)\parallel }_{1}dt\},w\left(t\right)\ne 0.\end{array}$$

Remark 1. 

Our Definition 3 generalizes the corresponding definition in [26] to the stochastic case. The ${\mathcal{L}}_1$-gain performance characterizes the system’s suppression to exogenous disturbance. The smaller the value of γ, the better the performance of the system, i.e., the lesser the effect of the disturbance input to the control output.

For the control signal $u\left(t\right)$ in system (1), it is assumed to be generated by a zero-order hold function with a sequence of hold times $0=t_0<t_1<\cdots <t_k<\cdots$

$$u\left(t\right)=K_{\sigma \left(t\right)}x\left(t_k\right),t_k\le t<t_{k+1},$$

where ${lim}_{k\to \infty }{t}_k=\infty$ and $t_{k+1}-t_k\le \tau (\mathrm{a}\mathrm{positive}\mathrm{constant}),\forall k\ge 0$.

The aim of this paper is to design a sate feedback controller of the above form such that the resulting closed-loop system realizes positivity and exponential stability in mean with ${\mathcal{L}}_1$-gain performance.

Similarly to [27], we first represent $t_k=t-\tau \left(t\right)$, where $\tau \left(t\right)=t-t_k$. Then, the digital control law can be seen as a continuous controller with a time-varying piecewise-continuous delay. And, the system (1) can be reformed as the following time delay PMJLS:

$$\begin{array}{ccc}\hfill \dot{x}\left(t\right)& =& A\left(\sigma \left(t\right)\right)x\left(t\right)+B_1\left(\sigma \left(t\right)\right)w\left(t\right)+B_2\left(\sigma \left(t\right)\right)K_{\sigma \left(t\right)}x(t-\tau \left(t\right)),\hfill \\ \hfill z\left(t\right)& =& C\left(\sigma \right(t\left)\right)x\left(t\right)+D\left(\sigma \right(t\left)\right)w\left(t\right),\hfill \end{array}$$

(2)

where $\tau \left(t\right)\in [0,\tau ]$ and $\dot{\tau }\left(t\right)=1$ for $t\ne t_k$.

From Lemma 1 in [28], the following lemma can be obtained.

Lemma 1. 

System (2) is positive if and only if for any $p\in \mathcal{P}$, $A_p$ is Metzler and $B_{1p}\in {\mathbb{R}}_{+}^{n\times n_w},B_{2p}{K}_p\in {\mathbb{R}}_{+}^{n\times n_u},C_p\in {\mathbb{R}}_{+}^{n_z\times n}$ and $D_p\in {\mathbb{R}}_{+}^{n_z\times n_w}$.

Remark 2. 

In [24], for a class of switched positive T-S fuzzy systems with time-varying delays, in the deterministic sense, the stability analysis and ${\mathcal{L}}_1$-gain controller synthesis problem were also considered. The system they studied is similar to our system (2). Similar time-delay systems are also often found in other studies. However, since the time delay in system (2) is caused by sampling, its derivative is 1, which does not conform to the usual assumption about the time delay in the system.

3. Main Results

In this section, the exponential stability analysis, ${\mathcal{L}}_1$-gain performance analysis, and control synthesis will be performed for the time delay PMJLS (2).

3.1. Stability Analysis

By applying the co-positive-type Lyapunov–Krasovskii functional method, sufficient conditions for exponential stability in the mean of the resulting closed-loop system (2) with $w\left(t\right)=0$ are provided in the following theorem.

Theorem 1. 

For a given scalar $\lambda >0$, the positive system (2) with $w\left(t\right)=0$ is exponentially stable in mean if there exists a set of vectors $v_i,{\rho }_{1,i},{\rho }_{2,i},{\eta }_i,{\rho }_1,{\rho }_2\in {\mathbb{R}}_{+}^n$$(i\in \mathcal{S})$, such that

$$\begin{array}{c}\hfill A_i^T{v}_i+\lambda v_i+{\rho }_{1,i}+{\rho }_{2,i}+\tau {\rho }_1+\tau {\rho }_2+{\eta }_i+\sum _{j=1}^N{\pi }_{ij}{v}_j\prec 0,\end{array}$$

(3)

$$\begin{array}{c}\hfill K_i^T{B}_{2i}^T{v}_i-{\eta }_i\prec 0,\end{array}$$

(4)

where

$$\begin{array}{c}\hfill \sum _{j=1}^N{\pi }_{ij}{\rho }_{1,j}⪯{\rho }_1,\sum _{j=1}^N{\pi }_{ij}{\rho }_{2,j}⪯{\rho }_2.\end{array}$$

(5)

Proof. 

First, for system (2) with $w\left(t\right)=0$, we choose the co-positive-type Lyapunov–Krasovskii functional candidate as follows:

$$\begin{array}{c}\hfill V(t,x\left(t\right),i)=V_1(t,x\left(t\right),i)+V_2(t,x\left(t\right),i)+V_3(t,x\left(t\right),i)+V_4(t,x\left(t\right),i),\end{array}$$

(6)

where i means that at time t, the Markovian mode $\sigma \left(t\right)=i\in \mathcal{P}$,

$$\begin{array}{ccc}\hfill V_1(t,x\left(t\right),i)& =& x^T\left(t\right)v_i,\hfill \\ \hfill V_2(t,x\left(t\right),i)& =& {\int }_{t-\tau \left(t\right)}^t{e}^{\lambda (s-t)}{x}^T\left(s\right){\rho }_{1,i}ds,\hfill \\ \hfill V_3(t,x\left(t\right),i)& =& {\int }_{t-\tau }^t{e}^{\lambda (s-t)}{x}^T\left(s\right){\rho }_{2,i}ds,\hfill \\ \hfill V_4(t,x\left(t\right),i)& =& {\int }_{-\tau }^0{\int }_{t+\theta }^t{e}^{\lambda (s-t)}{x}^T\left(s\right)({\rho }_1+{\rho }_2)dsd\theta ,\hfill \end{array}$$

$v_i,{\rho }_{1,i},{\rho }_{2,i},{\rho }_1,{\rho }_2\in {\mathbb{R}}_{+}^n$, and

$$\sum _{j=1}^N{\pi }_{ij}{\rho }_{1,j}\le {\rho }_1,\sum _{j=1}^N{\pi }_{ij}{\rho }_{2,j}\le {\rho }_2.$$

Applying the infinitesimal operator $\mathcal{L}$ ([29]) to (6) along the solution of the system (2) with $w\left(t\right)=0$, we have

$$\begin{array}{ccc}& & \mathcal{L}V(t,x(t),i)\hfill \\ & =& x^T\left(t\right)[A_i^T{v}_i+\lambda v_i+{\rho }_{1,i}+{\rho }_{2,i}+\tau {\rho }_1+\tau {\rho }_2+\sum _{j=1}^N{\pi }_{ij}{v}_j]\hfill \\ & & +x^T(t-\tau \left(t\right))K_i^T{B}_{2i}^T{v}_i-(1-\dot{\tau }\left(t\right))e^{-\lambda \tau \left(t\right)}{x}^T(t-\tau \left(t\right)){\rho }_{1,i}-e^{-\lambda \tau }{x}^T(t-\tau ){\rho }_{2,i}\hfill \\ & & +\sum _{j=1}^N{\pi }_{ij}{\int }_{t-\tau \left(t\right)}^t{e}^{\lambda (s-t)}{x}^T\left(s\right){\rho }_{1,j}ds+\sum _{j=1}^N{\pi }_{ij}{\int }_{t-\tau }^t{e}^{\lambda (s-t)}{x}^T\left(s\right){\rho }_{2,j}ds\hfill \\ & & -{\int }_{t-\tau }^t{e}^{\lambda (s-t)}{x}^T\left(s\right)({\rho }_1+{\rho }_2)ds-\lambda V(t,x\left(t\right),i).\hfill \end{array}$$

Using Leibniz–Newton’s formula, we have

$$\begin{array}{c}\hfill {\int }_{t-\tau \left(t\right)}^t\dot{x}\left(s\right)ds=x\left(t\right)-x(t-\tau \left(t\right))\end{array}$$

and

$$\begin{array}{c}\hfill {\int }_{t-\tau \left(t\right)}^t\dot{x}\left(s\right)ds={\int }_{t-\tau \left(t\right)}^t(A_{i}x\left(s\right)+B_{2i}{K}_{i}x(s-\tau \left(s\right)))ds.\end{array}$$

Then, for any vector ${\eta }_i\in {\mathbb{R}}_{+}^n$, according to the above two equalities, we obtain

$$\begin{array}{c}\hfill {[x\left(t\right)-x(t-\tau \left(t\right))-{\int }_{t-\tau \left(t\right)}^t(A_{i}x\left(s\right)+B_{2i}{K}_{i}x(s-\tau \left(s\right)))ds]}^T{\eta }_i=0.\end{array}$$

According to $-{\int }_{t-\tau }^t{e}^{\lambda (s-t)}{x}^T\left(s\right){\rho }_{1}ds\le -{\int }_{t-\tau \left(t\right)}^t{e}^{\lambda (s-t)}{x}^T\left(s\right){\rho }_{1}ds$, $\dot{\tau }\left(t\right)=1$ and inequality (5), we have

$$\begin{array}{ccc}& & \mathcal{L}V(t,x(t),i)\hfill \\ & \le & x^T\left(t\right)[A_i^T{v}_i+\lambda v_i+{\rho }_{1,i}+{\rho }_{2,i}+\tau {\rho }_1+\tau {\rho }_2+{\eta }_i+\sum _{j=1}^N{\pi }_{ij}{v}_j]\hfill \\ & & +x^T(t-\tau \left(t\right))[K_i^T{B}_{2i}^T{v}_i-{\eta }_i]-e^{-\lambda \tau }{x}^T(t-\tau ){\rho }_{2,i}\hfill \\ & & -{[{\int }_{t-\tau \left(t\right)}^t{A}_{i}x\left(s\right)+B_{2i}{K}_{i}x(s-\tau \left(s\right))ds]}^T{\eta }_i-\lambda V(t,x\left(t\right),i)\hfill \\ & \le & x^T\left(t\right)[A_i^T{v}_i+\lambda v_i+{\rho }_{1,i}+{\rho }_{2,i}+\tau {\rho }_1+\tau {\rho }_2+{\eta }_i+\sum _{j=1}^N{\pi }_{ij}{v}_j]\hfill \\ & & +x^T(t-\tau \left(t\right))[K_i^T{B}_{2i}^T{v}_i-{\eta }_i]-\lambda V(t,x\left(t\right),i).\hfill \end{array}$$

From inequalities (3)–(4), we have

$$\begin{array}{c}\hfill \mathcal{L}V(t,x(t),i)\le -\lambda V(t,x(t),i).\end{array}$$

(7)

Then, by Dynkin’s formula and the Gronwall–Bellman lemma, similarly as in the proof of Theorem 1 in [2], it follows that for $t\ge t_0$, we have

$$\begin{array}{c}\hfill E\left\{V(t,x\left(t\right),\sigma \left(t\right))\right|x_0,r_0\}\le e^{-\lambda (t-t_0)}V(t_0,x\left(t_0\right),r_0).\end{array}$$

(8)

Considering the definition of $V(t,x(t),\sigma (t\left)\right)$ and denoting $p_1={min}_{(i,j)\in \mathcal{P}\times {\mathcal{I}}_n}\left\{v_{ij}\right\}$, $p_2={max}_{(i,j)\in \mathcal{P}\times {\mathcal{I}}_n}\left\{v_{ij}\right\}$, $p_3={max}_{(i,j)\in \mathcal{P}\times {\mathcal{I}}_n}\left\{{\rho }_{1,ij}\right\}$, $p_4={max}_{(i,j)\in \mathcal{P}\times {\mathcal{I}}_n}\left\{{\rho }_{2,ij}\right\}$, $p_5={max}_{(i,j)\in \mathcal{P}\times {\mathcal{I}}_n}\left\{{\rho }_{1,j}\right\}$, and $p_6=$${max}_{(i,j)\in \mathcal{P}\times {\mathcal{I}}_n}\left\{{\rho }_{2,j}\right\}$, the following inequalities hold:

$$\begin{array}{c}\hfill V(t,x\left(t\right),i)\ge p_1\parallel x\left(t\right)\parallel ,\end{array}$$

(9)

$$\begin{array}{ccc}\hfill & & V(t_0,x_0,{\sigma }_0)\hfill \\ \hfill & \le & p_2\parallel x_0\parallel +p_3{\int }_{t_0-\tau \left(t_0\right)}^{t_0}{e}^{\lambda (s-t_0)}\parallel x_0\parallel ds+p_4{\int }_{t_0-\tau }^{t_0}{e}^{\lambda (s-t_0)}\parallel x_0\parallel ds\hfill \\ \hfill & & +p_5{\int }_{-\tau }^0{\int }_{t_0+\theta }^{t_0}{e}^{\lambda (s-t_0)}\parallel x_0\parallel dsd\theta \hfill \\ & \le & [p_2+(p_3+p_4){\displaystyle \frac{\tau }{\lambda }}(1-e^{-\lambda \tau })+p_5{\displaystyle \frac{1}{\lambda }}(\tau -{\displaystyle \frac{1}{\lambda }}+{\displaystyle \frac{e^{-\lambda \tau }}{\lambda }})]\parallel x_0\parallel .\hfill \end{array}$$

(10)

Combining (9), (10), and (8) gives

$$\begin{array}{c}\hfill E[\parallel x\left(t\right)\parallel ]\le \tilde{\alpha }{e}^{-\lambda (t-t_0)}\parallel x_0\parallel ,\end{array}$$

where $\tilde{\alpha }={\displaystyle \frac{p_2+(p_3+p_4)\frac{\tau }{\lambda }(1-e^{-\lambda \tau })+p_5\frac{1}{\lambda }(\tau -\frac{1}{\lambda }+\frac{e^{-\lambda \tau }}{\lambda })}{p_1}}$. Therefore, if inequalities (3)–(5) hold, the positive system (2) with $w\left(t\right)=0$ is exponentially stable in mean. □

Remark 3. 

From our Theorem 1, we can find from Formula (3) that the maximum value τ of the sampling period will affect the existence and value of parameters $v_i$ and so on, thus affecting the stability of the system.

3.2. ${\mathcal{L}}_1$-Gain Performance Analysis

The following theorem establishes the sufficient conditions of exponential stability in mean with weighted ${\mathcal{L}}_1$-gain property for positive system (2).

Theorem 2. 

For a given scalar $\lambda >0$, if there exists a set of vectors $v_i,{\rho }_{1,i},{\rho }_{2,i},{\eta }_i,{\rho }_1,{\rho }_2\in {\mathbb{R}}_{+}^n$$(i\in \mathcal{S})$, such that

$$\begin{array}{c}\hfill A_i^T{v}_i+\lambda v_i+{\rho }_{1,i}+{\rho }_{2,i}+\tau {\rho }_1+\tau {\rho }_2+{\eta }_i+\sum _{j=1}^N{\pi }_{ij}{v}_j+C_i^T\mathbf{1}\prec 0,\end{array}$$

(11)

$$\begin{array}{c}\hfill K_i^T{B}_{2i}^T{v}_i-{\eta }_i\prec 0,\end{array}$$

(12)

$$\begin{array}{c}\hfill B_{1i}^T{v}_i+D_i^T\mathbf{1}-\gamma \mathbf{1}\prec 0,\end{array}$$

(13)

where

$$\begin{array}{c}\hfill \sum _{j=1}^N{\pi }_{ij}{\rho }_{1,j}⪯{\rho }_1,\sum _{j=1}^N{\pi }_{ij}{\rho }_{2,j}⪯{\rho }_2,\end{array}$$

(14)

then the positive system (2) has a prescribed ${\mathcal{L}}_1$-gain performance level $\gamma >0$.

Proof. 

By Theorem 1, the exponential stability in mean of the positive system (2) with $w\left(t\right)=0$ is ensured if (3)–(5) hold. To prove the ${\mathcal{L}}_1$-gain performance of positive system (2), we also choose the co-positive-type Lyapunov–Krasovskii functional candidate (6):

$$\begin{array}{ccc}& & \mathcal{L}V(t,x\left(t\right),i)+{\parallel z\left(t\right)\parallel }_1-\gamma {\parallel w\left(t\right)\parallel }_1\hfill \\ & =& x^T\left(t\right)[A_i^T{v}_i+\lambda v_i+{\rho }_{1,i}+{\rho }_{2,i}+\tau {\rho }_1+\tau {\rho }_2+\sum _{j=1}^N{\pi }_{ij}{v}_j+C_i^T\mathbf{1}]\hfill \\ & & +x^T(t-\tau \left(t\right))[K_i^T{B}_{2i}^T{v}_i-{\eta }_i]-e^{-\lambda \tau }{x}^T(t-\tau ){\rho }_{2,i}\hfill \\ & & -{[{\int }_{t-\tau \left(t\right)}^t(A_{i}x\left(s\right)+B_{2i}{K}_{i}x(s-\tau \left(s\right))+B_{1i}w\left(s\right))ds]}^T{\eta }_i\hfill \\ & & +w^T\left(t\right)(B_{1i}^T{v}_i+D_i^T\mathbf{1}-\gamma \mathbf{1})-\lambda V(t,x\left(t\right),i).\hfill \end{array}$$

For zero initial condition, inequalities (11)–(14) imply that

$$\begin{array}{c}\hfill \mathcal{L}V(t,x\left(t\right),i)+{\parallel z\left(t\right)\parallel }_1-\gamma {\parallel w\left(t\right)\parallel }_1<0.\end{array}$$

Under zero initial condition, from Dynkin’s formula,

$$\begin{array}{c}\hfill E\left\{{\int }_{t_0}^t\mathcal{L}V(t,x\left(t\right),i)\right\}=E\left\{V(t,x\left(t\right),i)\right\}-E\left\{V(t_0,x_0,{\sigma }_0)\right\}=E\left\{V(t,x\left(t\right),i)\right\}.\end{array}$$

Therefore,

$$\begin{array}{ccc}& & E\{{\int }_{t_0}^t\parallel z\left(t\right){\parallel }_1-\gamma \parallel w\left(t\right){\parallel }_{1}dt\}\hfill \\ & =& E\{{\int }_{t_0}^t{\parallel z\left(t\right)\parallel }_1-{\gamma \parallel w\left(t\right)\parallel }_{1}dt+\mathcal{L}V(t,x\left(t\right),i)\}-E\left\{{\int }_{t_0}^t\mathcal{L}V(t,x\left(t\right),i)\right\}\hfill \\ & =& E\{{\int }_{t_0}^t\parallel z\left(t\right){\parallel }_1-\gamma \parallel w\left(t\right){\parallel }_{1}dt+\mathcal{L}V(t,x\left(t\right),i)\}-E\left\{V(t,x\left(t\right),i)\right\}\hfill \\ & \le & E\{{\int }_{t_0}^t\parallel z\left(t\right){\parallel }_1-\gamma \parallel w\left(t\right){\parallel }_{1}dt+\mathcal{L}V(t,x\left(t\right),i)\}<0.\hfill \end{array}$$

Furthermore, it means that

$$\begin{array}{c}\hfill E\{{\int }_{t_0}^{\infty }{\parallel z\left(t\right)\parallel }_{1}dt\}=\underset{t\to \infty }{lim}E\{{\int }_{t_0}^t{\parallel z\left(t\right)\parallel }_{1}dt|(x_0,{\sigma }_0)\}\le \gamma E\{{\int }_{t_0}^{\infty }\parallel w\left(t\right){\parallel }_{1}dt\},\end{array}$$

which means that the ${\mathcal{L}}_1$-gain performance is achieved. Therefore, the positive system (2) has a prescribed ${\mathcal{L}}_1$-gain performance level $\gamma >0$. □

3.3. Controller Design

In this subsection, based on Lemma 1 and the ${\mathcal{L}}_1$-gain analysis of Theorem 2, an ${\mathcal{L}}_1$ sampled-data control law is designed to maintain the systems’ positivity and attain the exponential stability with ${\mathcal{L}}_1$-gain performance in mean. By Lemma 1, the designed feedback gains should satisfy

$$\begin{array}{c}\hfill B_{2p}{K}_p\succ 0\end{array}$$

(15)

to maintain the positivity of system (2). Combining this with Theorem 2, based on the linear programming method, an effective algorithm for solving the feedback gain is presented here to make the positive system (2) have a prescribed ${\mathcal{L}}_1$-gain performance level $\gamma >0$.

Algorithm. Step (1): Input the matrices $A_p$, $B_{1p}$, $B_{2p}$, $C_p$ and $D_p$, system parameters ${\pi }_{ij}$ and $\tau$, set parameters $\lambda$ and $\gamma$.

Step (2): Solve the linear matrix inequalities (11), (13) and (14) to obtain $v_p$, ${\rho }_{1,p}$, ${\rho }_{2,p}$, ${\eta }_p$, ${\rho }_1$, ${\rho }_2$ (Using linear programming method).

Step (3): Compute the gain matrices $K_p$ by (12) and (15), by adjusting the parameter $\lambda$ to make (12) and (15) hold.

Step (4): Construct the ${\mathcal{L}}_1$ sampled-data control law $u\left(t\right)=K_{p}x\left(t_k\right)$, where $K_p,p\in \mathcal{P}$ are the gain matrices, $t_k$ is the sampling time.

4. Numerical Example

In this section, we will give a simple numerical simulation example to verify the effectiveness of the control feedback gain algorithm in Section 3.3. Since this algorithm is based on Theorems 1 and 2, this example also verifies the validity of our system stability and ${\mathcal{L}}_1$-gain performance judging theorems.

Example 1. 

Consider system (2) with two subsystems and and the following system matrices:

$$\begin{array}{ccc}& A_1=\left(\begin{array}{cc}-3& 8\\ 1& -3\end{array}\right),B_{11}=\left(\begin{array}{c}0.1\\ 0.1\end{array}\right),B_{21}=\left(\begin{array}{cc}0.1& 0.1\\ 0.0& 0.0\end{array}\right),& \\ & C_1=\left(\begin{array}{cc}0.2& 0.3\end{array}\right),D_1=\left(\begin{array}{c}0.4\end{array}\right),& \\ & A_2=\left(\begin{array}{cc}-3& 1\\ 1& -4\end{array}\right),B_{12}=\left(\begin{array}{c}0.2\\ 0.1\end{array}\right),B_{22}=\left(\begin{array}{cc}0.1& 0.1\\ 0.0& 0.0\end{array}\right),& \\ & C_2=\left(\begin{array}{cc}0.1& 0.2\end{array}\right),D_2=\left(\begin{array}{c}0.2\end{array}\right).\end{array}$$

The transition rate matrix of the Markov process $\sigma \left(t\right)$ is taken as $\Pi =\left(\begin{array}{cc}-1.2& 1.2\\ 2& -2\end{array}\right)$. Let the maximum sampling period be 2, and take parameters $\lambda =0.1$ and $\gamma =1$. Disturbance input $w\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{100}{{(t+1)}^{\frac{4}{3}}}},\hfill & \mathrm{when}\sigma \left(t\right)=1;\hfill \\ 100e^{-0.02t},\hfill & \mathrm{when}\sigma \left(t\right)=2.\hfill \end{array}\right.$ By the above algorithm (Step (1) and Step (2)), using the linear programming method, we can obtain that $v_1=\left(\begin{array}{c}0.3757\\ 0.9438\end{array}\right),v_2=\left(\begin{array}{c}0.2721\\ 0.4118\end{array}\right),{\rho }_{1,1}={\rho }_{1,2}={\rho }_{2,1}={\rho }_{2,2}={\eta }_1={\eta }_2={\rho }_1={\rho }_2=\left(\begin{array}{c}0.01\\ 0.01\end{array}\right).$ Then, by Step (3) and Step (4), solving (12) and (15), we can take $K_1={\displaystyle \frac{1}{4}}$, $K_2={\displaystyle \frac{1}{3}}$. If we take 1 as the sample period, and under the Markov jump process shown in Figure 1, the system state will evolve according to Figure 2. In Figure 2, the system state is indeed bounded.

5. Conclusions

This paper has presented a comprehensive investigation into the exponential stability and ${\mathcal{L}}_1$-gain performance for positive Markov jump sampled-data control systems. The introduction of sampling control technology in positive Markov jump systems (PMJSs) has led to significant insights and contributions. After establishing a PMJS model based on sampling control, we conducted a meticulous analysis of its exponential stability. This analysis laid the foundation for understanding the behavior of the system. Subsequently, for the ${\mathcal{L}}_1$-gain theory, we proposed a sufficient condition which ensures the robustness and satisfaction of the prescribed performance of the system in a randomly switching environment. The main contribution of this work lies in introducing sampling control technology into the research of PMJSs and providing a complete theoretical framework and analysis method. This not only enriches the theoretical knowledge in the field of control systems but also offers new practical application value for the sampling control of PMJSs. Future research could explore extensions of this work to more complex systems and investigate additional performance criteria to further enhance the effectiveness and applicability of sampling control in PMJSs.