# Global Exponential Stability of Both Continuous-Time and Discrete-Time Switched Positive Time-Varying Delay Systems with Interval Uncertainties and All Unstable Subsystems

Authors: Thanasak Mouktonglang; Suriyon Yimnet
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The global stability problem for a class of linear switched positive time-varying delay systems (LSPTDSs) with interval uncertainties by means of a fast average dwell time (FADT) switching is analyzed in this paper. A distinctive feature of this research is that all subsystems are considered to be unstable. Both the continuous-time and the discrete-time cases of LSPTDSs with interval uncertainties and all unstable subsystems (AUSs) are investigated. By constructing a time-scheduled multiple copositive Lyapunov-Krasovskii functional (MCLKF), novel sufficient conditions are derived within the framework of the FADT switching to guarantee such systems in the case of continuous-time to be globally uniformly exponentially stable. Based on the above approach, the corresponding result is extended to the discrete-time LSPTDSs including both interval uncertainties and AUSs. In addition, new stability criteria in an exponential sense are formulated for the studied systems without interval uncertainties. The efficiency and validity of the theoretical results are shown through simulation examples.


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## Body

## 1. Introduction

One of the crucial topics for studying the behavior of trajectories of dynamical systems under small perturbations of initial conditions is stability analysis [1–18]. The fundamental concept of stability analysis has been widely investigated in various types, such as asymptotic stability, exponential stability, robust stability, practical stability, and instability. Generally, the behavior of the considered dynamical systems depends only on the present state. Nevertheless, many phenomena cannot be explained under the specific constraints arising from the only present state, for instance, fluid and mechanical transmissions, metallurgical processes, and networked communications. Therefore, it is better to consider that the system’s behavior also includes information on the former states. This characteristic is called a time delay [1–3]. On the other hand, the time delay involved in both the continuous-time and discrete-time systems may lead to chaos and instability of the systems. Consequently, the stability problem on the systems with time delay as well as time-varying delay has been intensively analyzed, see [4–11].In real-world systems, there exists a class of dynamical systems that compose of a family of subsystems and a switching rule orchestrating the switching among subsystems. The systems under this mechanism are well known as switched systems [19–21]. The switched systems can be described by the hybrid behavior, which is discovered in the following situations. For instance, a thermostat turning the heat on and off, a server switching between buffers in a queueing network, and the dynamics of a car changing abruptly because of wheels locking and unlocking. Among the many problems of switched systems studied both in theory and practice, stability analysis of switched systems with an appropriate switching law is the primary concern, which has drawn great attention, see [22–29].As a special class of switched systems, switched positive systems (SPSs) can be described by a collection of positive subsystems and a switching signal specifying the switching laws. Because of the existence of positive constraints in the switched systems, numerous results from the study on normal switched systems may not be applicable to SPSs. Besides, several phenomena can be modeled by SPSs, such as compartmental model [30], water-quality model [31], formation flying [32], congestion control [33], wireless power control [34], and network communication using transmission control protocol [35]. Due to the complex dynamics of SPSs and their numerous applications, stability analysis on SPSs has been a significant investigation, and some relevant researches have been reported in [36–44]. In addition, most practical systems often contain the term uncertainties, which refer to the differences or errors between models and reality. However, the existence of even the slight uncertainties in the considered systems can lead to the instability of those systems. Thus, it is essential to study the robust stability and stabilization problems of the systems including uncertainties. A great number of useful results on the uncertainties have been found in many switched systems, such as switched continuous-time systems [45, 46], switched discrete-time systems [47–49], stochastic switched discrete-time systems [50–52], SPSs [35, 53–55], and switched positive delay systems [56, 57].In most of the existing works of literature, the study on the dynamic behavior of each subsystem of the overall switched system can be divided into three cases; (i) all subsystems in the first case are stable, (ii) the considered switched system in the second case composes of stable and unstable subsystems, and (iii) all subsystems interested in the last case are unstable. In general, the stability analysis of switched systems can be accomplished when either all subsystems are stable or there exists at least one stable subsystem for recompensing the state divergence caused by unstable subsystems. Nonetheless, the mentioned idea is inoperative in the case that all subsystems are unstable. Thus, an arisen natural question is how to design a suitable switching law to stabilize the switched systems without a stable subsystem? So far, this issue has been discussed, and it is still a challenging problem. By using suitable switching laws, the (robust) stability of switched systems with AUSs has been investigated and reviewed briefly in the following. In [28], Xiang and Xiao studied the asymptotic stability of switched systems with AUSs by using a discretized Lyapunov function and a dwell time (DT) switching law. Also, the results on exponential stability of switched systems with AUSs were addressed in both the continuous-time case [23] and the discrete-time case [24]. Nevertheless, the problems of time delay, interval uncertainties, and positivity for switched systems with AUSs were not taken into account in mentioned researches. Later, Feng et al. [35] dealt with the problems of stability and robust stability for linear SPSs with AUSs by employing a discretized copositive Lyapunov function and a mode-dependent dwell time (MDDT) switching rule. More recently, Zhang and Sun [32] investigated the practical exponential stability of discrete-time linear SPSs with impulse and AUSs by establishing a switched time-varying vector function and applying a mode-dependent interval dwell time switching law. From the above two results, it should be noted that the existence of the time delay was not considered on the systems. On the other hand, several research articles about switched positive delay systems in the case of all subsystems are unstable were reported in [43, 44, 57]. As mentioned in [43], Liu et al. utilized the multiple discretized copositive Lyapunov-Krasovskii functionals and the DT switching rule to derive the delay-dependent sufficient criteria (DDSC) of the continuous-time and discrete-time LSPTDSs with AUSs. Next, a sufficient criterion guaranteeing the global uniform exponential stability of the only continuous-time LSPTDSs with AUSs based on the time-scheduled MCLKF method combining with the FADT switching law was provided in [44]. However, the interval uncertainties were not regarded in [43, 44]. Furthermore, Rojsiraphisal et al. [57] formulated the robust stability criteria within the framework of the time-scheduled MCLKF tactic and the MDDT switching strategy to ensure the continuous-time LSPTDSs including both interval uncertainties and AUSs to be globally uniformly asymptotically stable.To the best of our knowledge, there is no result on the robust exponential stability of both continuous-time and discrete-time LSPTDSs with interval uncertainties and AUSs in the literature. This observed idea is the motivation of this paper. The main contributions of this study are highlighted as follows:(1)
The global stability problem of continuous-time LSPTDSs including both interval uncertainties and AUSs by adopting the time-scheduled MCLKF technique together with the FADT switching law is studied(2)
Novel DDSC for global uniform exponential stability of the systems is derived(3)
The corresponding results for discrete-time LSPTDSs including both interval uncertainties and AUSs are also provided(4)
Unlike the existing results in [28, 35, 43, 57], the type of stability analysis in this paper is pointed out as the exponential stability analysis instead of the asymptotic stability analysis for the underlying systems. Different from the above results, the FADT switching law, which is less conservative than the DT switching law, is applied to investigate the problem of robust exponential stability for LSPTDSs including both interval uncertainties and AUSs. Moreover, it should be noted that the discrete-time case and interval uncertainties were not studied in [44]The component of this paper is arranged as follows. In the next section, the system descriptions and preliminaries are proposed. In Section3, the main results are presented. In Section 4, the numerical examples are shown to support and validate our theoretical results. Lastly, the conclusions are reported in Section 5.Notations: the following notations are exploited throughout this article. The sets of integers, nonnegative integers, and positive integers are denoted by ℤ, ℕ0, and ℕ, respectively. Set L¯0=0,1,2,⋯,L and N¯=1,2,⋯,N for any L,N∈ℕ. ℝn and ℝ+n refer to the vectors of n-tuples of real and positive real numbers, respectively. The set of all m×n real matrices is represented by ℝm×n. In and AT are the n×n dimensional identity matrix and the transpose of matrix A, respectively. The matrix A is called nonnegative if all entries are nonnegative and defined by A⪰0. For given vector ν∈ℝn, νi1≤i≤n is the ith component of ν. The notation ν⪰0ν≻0 stands for nonnegative (positive) vector, namely, all components of ν are nonnegative (positive) for vector ν∈ℝn. Let ν1=∑i=1nνi and ν2=∑i=1nνi21/2 be the 1-norm and the Euclidean norm of ν∈ℝn, respectively. ω¯ν symbolizes the minimal elements of ν∈ℝn. The floor function x=maxn∈ℤn≤x,x∈ℝ. In addition, t−=limε⟶0+t−ε and t+=limε⟶0+t+ε.


## 2. System Descriptions and Preliminaries

In this section, we consider the linear switched positive time-varying delay system (LSPTDS) with interval uncertainties and AUSs for both the continuous-time and the discrete-time systems.


### 2.1. The Continuous-Time LSPTDS including Both Interval Uncertainties and AUSs

A class of continuous-time linear switched system with time-varying delay can be stated as(1)ẋt=Aσtxt+Dσtxt−αt,xt0+θ=ψθ,θ∈−α^,0,where xt∈ℝn. The switching signal σt is a piecewise constant function of time t, which takes values in the finite set N¯=1,2,⋯,N, N>1 is the number of subsystems or modes of the switched system. Without loss of generality, we presume that σt is continuous from the right everywhere: σt=limχ⟶t+σχ. The switching instants can be defined by a sequence 0≤t0<t1<t2<⋯<tm<tm+1<⋯<+∞, where t0 is the initial time and tm stands for the mth switching instant, m∈ℕ0. We impose that σt=σtm=i,i∈N¯ and the ith subsystem is activated when t∈tm,tm+1. Based on the logical rule of σt at the switching instant tm, system (1) switches from the jth subsystem to the ith subsystem, where σtm−1=j,j∈N¯. αt is the time-varying delay satisfying 0≤αt≤α^ and α̇t≤Δ<1, where α^ and Δ are known constants. As mentioned in [35, 57], the system matrices Ai and Di are supposed to be interval uncertain; namely,
(2)A¯i⪯Ai⪯A¯i,(3)D¯i⪯Di⪯D¯i,where A¯i,D¯i,A¯i,D¯i are the given constant system matrices with appropriate dimensions for all i∈N¯. In addition, ψ·:−α^,0⟶ℝn is a vector-valued initial state with ∥ψ∥α^=sup−α^≤θ≤0∥ψθ∥2.First, we introduce some definitions, lemma, and assumptions helpful in obtaining the main results in the continuous-time system.Definition 1 (see [43]).
System (1) is said to be positive if for any initial condition ψθ⪰0,θ∈−α^,0 and any switching signal σt, the corresponding trajectory xt⪰0 holds for all t≥t0.Definition 2 (see [35]).
A matrixA is said to be a Metzler matrix if all off-diagonal elements are nonnegative.Lemma 3 (see [43]).
System (1) is positive if and only if Ai is Metzler matrices and Di⪰0 for all i∈N¯.Definition 4 (see [44]).
System (1) is said to be globally uniformly exponentially stable (GUES) with switching signal σt if there exist two constants ε>0 and ϖ>0 such that ∥xt∥2≤εe−ϖt−t0∥ψ∥α^ for all t≥t0.In general, actual systems can be modeled by systems in the form of interval uncertainties. Therefore, the assumption of the interval uncertainties for studying the robust exponential stability of system (1) is stated as follows.Assumption 5 (see [35, 57]).
For eachAi and Di in system (1), there are the known Metzler matrices A¯i and the matrices D¯i⪰0 such that Ai∈A¯i,A¯i and Di∈D¯i,D¯i, where A¯i,D¯i,A¯i,D¯i are the given constant system matrices with appropriate dimensions for all i∈N¯.The following assumption is necessary and reasonable to analyze the problem of robust exponential stability for system (1) including interval uncertainties in the case of all subsystems are unstable.Assumption 6 (see [35, 43, 44, 57]).
All subsystems of system (1) are unstable.Besides, it is well-known that the ADT switching law is less conservative and more general than the DT switching law. Especially, the FADT switching law can be utilized to certify the (robust) stability of the switched systems that all subsystems are unstable. Therefore, the definitions of both the DT and FADT switching laws are stated as follows.Definition 7 (see [41, 44]).
For two switching instantstm and tm+1,m∈ℕ0 of system (1), if there exists a constant τm>0 such that τm=tm+1−tm holds for any m∈ℕ0, then, τm is called DT of system (1). Moreover, if there exists a constant τ∗>0 such that τ∗≤infm∈ℕ0τm holds for any m∈ℕ0, then, τ∗ is called minimum DT of system (1).Definition 8 (see [41, 44]).
For anyT>t≥0 and a switching signal σt, let NσT,t be the switching numbers over the interval t,T, if there exist two constants N0≥0,τa>0 such that
(4)NσT,t≥N0+T−tτa,∀T>t≥0,
then, the constantsN0 and τa are called the chattering bound and FADT of system (1), respectively.Remark 9.
Already discussed in [44], the mechanism of both the FADT switching law and the ADT switching law is different. Namely, the FADT switching law imposes that the activation average time of AUSs is neither too large nor too small to stabilize the switched system that all subsystems are unstable, while the ADT switching law can be described that some stable subsystems of the switched system remain sufficiently long ADT to ensure the stability of the considered system. Moreover, from (4), it implies that the FADT switching law satisfies τa≥T−t/NσT,t−N0, which is different from the ADT switching law satisfying τa≤T−t/NσT,t−N0.


### 2.2. The Discrete-Time LSPTDS including Both Interval Uncertainties and AUSs

A class of discrete-time linear switched system with time-varying delay can be described in the form of(5)xk+1=Aσkxk+Dσkxk−dk,xk0+ζ=φζ,ζ=−d^,−d^+1,⋯,−1,0,where xk∈ℝn. σk:ℕ0⟶N¯=1,2,⋯,N is the switching signal, and N>1 is the number of subsystems or modes of the switched system. Given the switching signal σk, we denote the set of switching moments by km:km∈ℕ0 where k0 is the initial time and km<km+1 for m∈ℕ0. For two successive switching moments km and km+1, we impose that σk−1=σkm−1=j and σk=σkm=i, where j,i∈N¯, and the σkmth subsystem is activated when k∈km,km+1. Similarly, with the continuous-time system, the system matrices Ai and Di are presumed to be interval uncertain for all i∈N¯. The time-varying delay dk satisfies d1≤dk≤d2 where d1,d2 are known positive integers, and d^=maxd1,d2. Moreover, φ·:−d^,−d^+1,⋯,−1,0⟶ℝn is a given discrete vector-valued initial state with ∥φ∥d^=maxζ∈−d^,−d^+1,⋯,−1,0∥φζ∥2.The essential definitions, lemma, and assumptions utilized in the discrete-time system are presented in the following.Definition 10 (see[43]).
System (5) is said to be positive if for any initial condition φζ⪰0,ζ=−d^,−d^+1,⋯,−1,0 and any switching signal σk, the corresponding trajectory xk⪰0 holds for all k∈ℕ0.Lemma 11 (see [43]).
System (5) is positive if and only if Ai⪰0 and Di⪰0 hold for all i∈N¯.Definition 12 (see [9]).
System (5) is said to be globally uniformly exponentially stable (GUES) with switching signal σk if there exist two constants ε>0 and 0<ρ<1 such that ∥xk∥2≤ερk−k0∥φ∥d^ for all k≥k0.Assumption 13 (see [35]).
For eachAi and Di in system (5), there are the known constant matrices A¯i⪰0 and D¯i⪰0 such that Ai∈A¯i,A¯i and Di∈D¯i,D¯i, where A¯i,D¯i,A¯i,D¯i are the given constant system matrices with appropriate dimensions for all i∈N¯.Assumption 14 (see [35, 43]).
All subsystems of system (5) are unstable.Definition 15 (see [43, 44]).
For two switching momentskm and km+1,m∈ℕ0 of system (5), if there exists a constant κm>0 such that κm=km+1−km holds for any m∈ℕ0, then, κm is called DT of system (5). Moreover, if there exists a constant κ∗>0 such that κ∗≤minm∈ℕ0κm holds for any m∈ℕ0, then, κ∗ is called minimum DT of system (5).Definition 16 (see [44]).
For anyK>k≥0 and a switching signal σk, let NσK,k be the switching numbers over the interval k,K, if there exist two constants N0≥0,κa>0 satisfying
(6)NσK,k≥N0+K−kκa,∀K>k≥0,
then, the constantsN0 and κa are called the chattering bound and FADT of system (5), respectively.The main purpose of this research is to propose the global stability criteria that ensure the continuous-time system (1) with interval uncertainties and the discrete-time system (5) with interval uncertainties are positive and GUES with respect to the FADT switching law when all modes of the systems are unstable.


## 2.1. The Continuous-Time LSPTDS including Both Interval Uncertainties and AUSs

A class of continuous-time linear switched system with time-varying delay can be stated as(1)ẋt=Aσtxt+Dσtxt−αt,xt0+θ=ψθ,θ∈−α^,0,where xt∈ℝn. The switching signal σt is a piecewise constant function of time t, which takes values in the finite set N¯=1,2,⋯,N, N>1 is the number of subsystems or modes of the switched system. Without loss of generality, we presume that σt is continuous from the right everywhere: σt=limχ⟶t+σχ. The switching instants can be defined by a sequence 0≤t0<t1<t2<⋯<tm<tm+1<⋯<+∞, where t0 is the initial time and tm stands for the mth switching instant, m∈ℕ0. We impose that σt=σtm=i,i∈N¯ and the ith subsystem is activated when t∈tm,tm+1. Based on the logical rule of σt at the switching instant tm, system (1) switches from the jth subsystem to the ith subsystem, where σtm−1=j,j∈N¯. αt is the time-varying delay satisfying 0≤αt≤α^ and α̇t≤Δ<1, where α^ and Δ are known constants. As mentioned in [35, 57], the system matrices Ai and Di are supposed to be interval uncertain; namely,
(2)A¯i⪯Ai⪯A¯i,(3)D¯i⪯Di⪯D¯i,where A¯i,D¯i,A¯i,D¯i are the given constant system matrices with appropriate dimensions for all i∈N¯. In addition, ψ·:−α^,0⟶ℝn is a vector-valued initial state with ∥ψ∥α^=sup−α^≤θ≤0∥ψθ∥2.First, we introduce some definitions, lemma, and assumptions helpful in obtaining the main results in the continuous-time system.Definition 1 (see [43]).
System (1) is said to be positive if for any initial condition ψθ⪰0,θ∈−α^,0 and any switching signal σt, the corresponding trajectory xt⪰0 holds for all t≥t0.Definition 2 (see [35]).
A matrixA is said to be a Metzler matrix if all off-diagonal elements are nonnegative.Lemma 3 (see [43]).
System (1) is positive if and only if Ai is Metzler matrices and Di⪰0 for all i∈N¯.Definition 4 (see [44]).
System (1) is said to be globally uniformly exponentially stable (GUES) with switching signal σt if there exist two constants ε>0 and ϖ>0 such that ∥xt∥2≤εe−ϖt−t0∥ψ∥α^ for all t≥t0.In general, actual systems can be modeled by systems in the form of interval uncertainties. Therefore, the assumption of the interval uncertainties for studying the robust exponential stability of system (1) is stated as follows.Assumption 5 (see [35, 57]).
For eachAi and Di in system (1), there are the known Metzler matrices A¯i and the matrices D¯i⪰0 such that Ai∈A¯i,A¯i and Di∈D¯i,D¯i, where A¯i,D¯i,A¯i,D¯i are the given constant system matrices with appropriate dimensions for all i∈N¯.The following assumption is necessary and reasonable to analyze the problem of robust exponential stability for system (1) including interval uncertainties in the case of all subsystems are unstable.Assumption 6 (see [35, 43, 44, 57]).
All subsystems of system (1) are unstable.Besides, it is well-known that the ADT switching law is less conservative and more general than the DT switching law. Especially, the FADT switching law can be utilized to certify the (robust) stability of the switched systems that all subsystems are unstable. Therefore, the definitions of both the DT and FADT switching laws are stated as follows.Definition 7 (see [41, 44]).
For two switching instantstm and tm+1,m∈ℕ0 of system (1), if there exists a constant τm>0 such that τm=tm+1−tm holds for any m∈ℕ0, then, τm is called DT of system (1). Moreover, if there exists a constant τ∗>0 such that τ∗≤infm∈ℕ0τm holds for any m∈ℕ0, then, τ∗ is called minimum DT of system (1).Definition 8 (see [41, 44]).
For anyT>t≥0 and a switching signal σt, let NσT,t be the switching numbers over the interval t,T, if there exist two constants N0≥0,τa>0 such that
(4)NσT,t≥N0+T−tτa,∀T>t≥0,
then, the constantsN0 and τa are called the chattering bound and FADT of system (1), respectively.Remark 9.
Already discussed in [44], the mechanism of both the FADT switching law and the ADT switching law is different. Namely, the FADT switching law imposes that the activation average time of AUSs is neither too large nor too small to stabilize the switched system that all subsystems are unstable, while the ADT switching law can be described that some stable subsystems of the switched system remain sufficiently long ADT to ensure the stability of the considered system. Moreover, from (4), it implies that the FADT switching law satisfies τa≥T−t/NσT,t−N0, which is different from the ADT switching law satisfying τa≤T−t/NσT,t−N0.


## 2.2. The Discrete-Time LSPTDS including Both Interval Uncertainties and AUSs

A class of discrete-time linear switched system with time-varying delay can be described in the form of(5)xk+1=Aσkxk+Dσkxk−dk,xk0+ζ=φζ,ζ=−d^,−d^+1,⋯,−1,0,where xk∈ℝn. σk:ℕ0⟶N¯=1,2,⋯,N is the switching signal, and N>1 is the number of subsystems or modes of the switched system. Given the switching signal σk, we denote the set of switching moments by km:km∈ℕ0 where k0 is the initial time and km<km+1 for m∈ℕ0. For two successive switching moments km and km+1, we impose that σk−1=σkm−1=j and σk=σkm=i, where j,i∈N¯, and the σkmth subsystem is activated when k∈km,km+1. Similarly, with the continuous-time system, the system matrices Ai and Di are presumed to be interval uncertain for all i∈N¯. The time-varying delay dk satisfies d1≤dk≤d2 where d1,d2 are known positive integers, and d^=maxd1,d2. Moreover, φ·:−d^,−d^+1,⋯,−1,0⟶ℝn is a given discrete vector-valued initial state with ∥φ∥d^=maxζ∈−d^,−d^+1,⋯,−1,0∥φζ∥2.The essential definitions, lemma, and assumptions utilized in the discrete-time system are presented in the following.Definition 10 (see[43]).
System (5) is said to be positive if for any initial condition φζ⪰0,ζ=−d^,−d^+1,⋯,−1,0 and any switching signal σk, the corresponding trajectory xk⪰0 holds for all k∈ℕ0.Lemma 11 (see [43]).
System (5) is positive if and only if Ai⪰0 and Di⪰0 hold for all i∈N¯.Definition 12 (see [9]).
System (5) is said to be globally uniformly exponentially stable (GUES) with switching signal σk if there exist two constants ε>0 and 0<ρ<1 such that ∥xk∥2≤ερk−k0∥φ∥d^ for all k≥k0.Assumption 13 (see [35]).
For eachAi and Di in system (5), there are the known constant matrices A¯i⪰0 and D¯i⪰0 such that Ai∈A¯i,A¯i and Di∈D¯i,D¯i, where A¯i,D¯i,A¯i,D¯i are the given constant system matrices with appropriate dimensions for all i∈N¯.Assumption 14 (see [35, 43]).
All subsystems of system (5) are unstable.Definition 15 (see [43, 44]).
For two switching momentskm and km+1,m∈ℕ0 of system (5), if there exists a constant κm>0 such that κm=km+1−km holds for any m∈ℕ0, then, κm is called DT of system (5). Moreover, if there exists a constant κ∗>0 such that κ∗≤minm∈ℕ0κm holds for any m∈ℕ0, then, κ∗ is called minimum DT of system (5).Definition 16 (see [44]).
For anyK>k≥0 and a switching signal σk, let NσK,k be the switching numbers over the interval k,K, if there exist two constants N0≥0,κa>0 satisfying
(6)NσK,k≥N0+K−kκa,∀K>k≥0,
then, the constantsN0 and κa are called the chattering bound and FADT of system (5), respectively.The main purpose of this research is to propose the global stability criteria that ensure the continuous-time system (1) with interval uncertainties and the discrete-time system (5) with interval uncertainties are positive and GUES with respect to the FADT switching law when all modes of the systems are unstable.


## 3. Main Results

In this section, we apply the time-scheduled MCLKF tactic and the FADT switching strategy to derive novel DDSC guaranteeing the positivity and the robust exponential stability of both the continuous-time system (1) and the discrete-time system (5) with interval uncertainties and AUSs. Besides, we propose the positivity and the exponential stability criteria of both system (1) and system (5) without interval uncertainties.For convenience, we first define important symbols used in our main theorem as follows:(7)D~=d¯kl∈ℝn×n,d¯kl=maxi∈N¯D¯ikl,where D¯ikl refers to the kth row and lth column element of system matrices D¯i,i∈N¯. And
(8)D=dkl∈ℝn×n,dkl=maxi∈N¯Dikl,where Dikl denotes the kth row and lth column element of system matrices Di,i∈N¯.


### 3.1. Global Uniform Exponential Stability of Continuous-Time LSPTDS including Both Interval Uncertainties and AUSs

Theorem 17.
Consider the continuous-time system (1) satisfying Assumption 5 and Assumption 6. Given constants 0<μ<1,λ>0,τ∗>0, and L∈ℕ. If there exist positive vectors νi,q,i∈N¯,q∈L¯0 and constants τa≥τ∗ such that
(9)1−ΔΦi,q+1−ΔA¯iT+1+α^D~T−λ1−ΔInνi,q≺0,(10)1−ΔΦi,q+1−ΔA¯iT+1+α^D~T−λ1−ΔInνi,q+1≺0,(11)D~TΦi,q−νi,q≺0,(12)D~TΦi,q−νi,q+1≺0,(13)1−ΔA¯iT+1+α^D~T−λ1−ΔInνi,L≺0,(14)νi,0−μνj,L⪯0,
hold for anyq=0,1,⋯,L−1, and for any i,j∈N¯,i≠j, then, the continuous-time system (1) is positive and GUES under the switching signal with the FADT satisfying
(15)τ∗≤τa<−lnμλ,where
(16)Φi,q=νi,q+1−νi,qLτ∗,
andD~ is mentioned in (7).Proof.
We divide the proof process into the following two steps.Step 1.
We will prove that system (1) is positive.Using Assumption5, we obtain that Ai is also Metzler matrices and Di⪰0 for all i∈N¯. According to Lemma 3, system (1) is positive.Step 2.
We will prove that system (1) is GUES under the switching signal under the FADT satisfying condition (15).For anyt>0, we suppose that t∈tm,tm+1=tm,tm+τ∗∪tm+τ∗,tm+1,m∈ℕ0. And we divide the interval tm,tm+τ∗ into L segments with equal length h=τ∗/L. In addition, we define Ym,q=tm+qh,tm+q+1h,q=0,1,⋯,L−1, then tm,tm+τ∗=⋃q=0L−1Ym,q.Due to each stable subsystem has been completely replaced by AUSs, some previous researches about the switched systems are no longer applicable. Motivated by the concept utilized in [28, 43, 44, 57], we employ the discretized Lyapunov function and the FADT switching law to stabilize system (1) with interval uncertainties for the case that all subsystems are unstable. First, we establish the following vector function
(17)νit=νitm+qh+ηth=1−ηtνi,q+ηtνi,q+1,t∈Ym,q,q=0,1,⋯,L−1,νi,L,t∈tm+τ∗,tm+1,where i∈N¯,m∈ℕ0,ηt=t−tm−qh/h with 0≤ηt≤1, and νi,q are positive vectors for i∈N¯,q∈L¯0. For t∈Ym,q, we can get
(18)ν̇it=η̇tνi,q+1−η̇tνi,q=νi,q+1−νi,qh,which yields
(19)ν̇it=Φi,q,where Φi,q is defined as in (16). For any i∈N¯, we establish the time-scheduled MCLKF:
(20)Vit,xt=1−ΔxTtνit+∫t−αttxTsD~Tνitds+∫−α^0∫t+wtxTsD~Tνisdsdw.DifferentiatingVit,xt in (20) along the trajectories of system (1), we obtain
(21)V̇it,xt=1−ΔxTtν̇it+xTtAiTνit+xTt−αtDiTνit+∫t−αttxTsD~Tν̇itds+xTtD~Tνit−xTt−αtD~Tνit1−α̇t+α^xTtD~Tνit−∫t−α^txTsD~Tνisds,then
(22)V̇it,xt≤1−ΔxTtν̇it+xTtA¯iTνit+xTt−αtD¯iTνit+∫t−αttxTsD~Tν̇itds+xTtD~Tνit−xTt−αtD~Tνit1−α̇t+α^xTtD~Tνit−∫t−α^txTsD~Tνisds.We observe that(23)V̇it,xt−λVit,xt≤1−ΔxTtν̇it+xTtA¯iTνit+xTt−αtD¯iTνit+∫t−αttxTsD~Tν̇itds+xTtD~Tνit−xTt−αtD~Tνit1−α̇t+α^xTtD~Tνit−∫t−α^txTsD~Tνisds−λ1−ΔxTtνit−λ∫t−αttxTsD~Tνitds−λ∫−α^0∫t+wtxTsD~Tνisdsdw.Along withα̇t≤Δ,0<λ and D¯i⪯D~ for all i∈N¯, one has
(24)V̇it,xt−λVit,xt≤xTt1−Δν̇it+1−ΔA¯iTνit+1+α^D~Tνit−λ1−Δνit+∫t−αttxTsD~Tν̇it−νisds.Whent∈Ym,q⊂tm,tm+τ∗, it leads to
(25)1−Δν̇it+1−ΔA¯iTνit+1+α^D~Tνit−λ1−Δνit=1−ηt1−ΔΦi,q+1−ΔA¯iT+1+α^D~T−λ1−ΔInνi,q+ηt1−ΔΦi,q+1−ΔA¯iT+1+α^D~T−λ1−ΔInνi,q+1,and(26)D~Tν̇it−νis=1−ηsD~TΦi,q−νi,q+ηsD~TΦi,q−νi,q+1.According to conditions (9)–(12), we obtain
(27)V̇it,xt−λVit,xt<0,t∈Ym,q,which implies
(28)V̇it,xt<λVit,xt,t∈⋃q=0L−1Ym,q=tm,tm+τ∗.Whent∈tm+τ∗,tm+1, it yields that
(29)1−Δν̇it+1−ΔA¯iTνit+1+α^D~Tνit−λ1−Δνit=1−ΔA¯iT+1+α^D~T−λ1−ΔInνi,L,(30)D~Tν̇it−νis=−D~Tνi,L≺0.Applying condition (13), we get
(31)V̇it,xt<λVit,xt,t∈tm+τ∗,tm+1.Combining (28) with (31), it is obvious that
(32)V̇it,xt<λVit,xt,t∈tm,tm+1,m∈ℕ0.Integrating both sides of (32) over tm,t for t∈tm,tm+1,m∈ℕ0, it is immediate that
(33)Vit,xt<eλt−tmVitm,xtm.Using condition (14), we have
(34)νitm⪯μνjtm−foralli,j∈N¯,i≠j.From (17) and (20), it can be seen that
(35)Vitm,xtm=1−ΔxTtmνitm+∫tm−αtmtmxTsD~Tνitmds+∫−α^0∫tm+wtmxTsD~Tνisdsdw≤1−ΔxTtm−μνjtm−+∫tm−−αtm−tm−xTsD~Tμνjtm−ds+∫−α^0∫tm−+wtm−xTsD~Tμνjsdsdw,(36)=μVjtm−,xtm−foralli,j∈N¯,i≠j.Based on the relationship between (33) and (36), we can derive
(37)Vσtmt,xt<eλt−tmVσtmtm,xtm≤μeλt−tmVσtm−1tm−,xtm−<μeλt−tmeλtm−tm−1Vσtm−1tm−1,xtm−1⋮<μμ⋯μeλt−tmeλtm−tm−1⋯eλt1−t0Vσt0t0,xt0=μNσt,t0eλt−t0Vσt0t0,xt0.It follows from Definition8 and 0<μ<1 by employing (4), then, we obtain
(38)Vσtmt,xt≤μN0+t−t0/τaeλt−t0Vσt0t0,xt0=eN0lnμeλ+lnμ/τat−t0Vσt0t0,xt0.Without loss of generality, we impose thatVσt0t0,xt0=Vσ00,x0. From (17) and (20), we have
(39)Vσ00,x0≤1−Δ∥xT0∥2∥νσ00∥2+∫−α^0dssup−α^≤s≤0∥xTs∥2∥D~T∥2∥νσ00∥2+∫−α^0∫w0∥νσ0s∥2dsdwsup−α^≤s≤0∥xTs∥2∥D~T∥2≤1−Δ∥xT0∥2∑b∈L¯0∥νσ0,b∥2+α^sup−α^≤s≤0∥xTs∥2∥D~T∥2∑b∈L¯0∥νσ0,b∥2+2∑b∈L¯0∥νσ0,b∥2∫−α^0∫w0dsdwsup−α^≤s≤0∥xTs∥2∥D~T∥2≤1−Δn∥ψ∥α^∑b∈L¯0∥νσ0,b∥2+α^n∥ψ∥α^∥D~T∥2∑b∈L¯0∥νσ0,b∥2+α∧2n∥ψ∥α^∥D~T∥2∑b∈L¯0∥νσ0,b∥2=1−Δ+α^1+α^∥D~T∥2n∑b∈L¯0∥νσ0,b∥2∥ψ∥α^.Moreover, we get(40)1−Δξ∥xt∥2≤Vσtmt,xt,where ξ=mina,b∈N¯×L¯0ω¯νa,b. Substituting (39) and (40) into (38), we obtain
(41)∥xt∥2≤eN0lnμ1−Δξeλ+lnμ/τat−t01−Δ+α^1+α^∥D~T∥2n∑b∈L¯0∥νσ0,b∥2∥ψ∥α^,fort≥t0. If the FADT switching signal satisfies condition (15), then
(42)−λ+lnμτa>0.From (41), for t≥t0, we arrive at
(43)∥xt∥2≤εe−ϖt−t0∥ψ∥α^,where ε=eN0lnμ/1−Δξ1−Δ+α^1+α^∥D~T∥2n∑b∈L¯0∥νσ0,b∥2, and ϖ=−λ+lnμ/τa.By Definition4, we can conclude that system (1) is GUES under the switching signal with the FADT satisfying condition (15).Remark 18.
Inspired by the idea in [35, 43–44, 28, 57], the time-scheduled positive vector function defined in (17) is time-varying rather than a constant, and the time-scheduled MCLKF proposed in (20) is validly established to investigate the problem of robust stability of continuous-time system (1) when each subsystem is unstable. However, our time-scheduled MCLKF shown in Theorem 17 is different from the discretized copositive Lyapunov function, the discretized copositive Lyapunov-Krasovskii functional, the time-scheduled MCLKF, and the discretized quadratic Lyapunov function used in [28, 35, 43, 44], respectively. Although our time-scheduled MCLKF is similar to the time-scheduled MCLKF utilized in [57], our robust stability criteria under the FADT switching technique for the considered system can guarantee for being the global uniform exponential stability rather than for being the global uniform asymptotic stability derived in [57].Remark 19.
According to the result in [44], the FADT switching law is applied to deal with the stabilization problem system (1) with AUSs but without interval uncertainties. This switching law can compensate for the state divergence caused by the unstable subsystem under the appropriate average time of each unstable subsystem; namely, the ADT of AUSs is neither too long nor too short. However, the model in this paper, which includes the interval uncertainties, can also apply this switching law effectively. Therefore, our result obtained from Theorem 17 is more applicable than that of Theorem 1 in [44].Next, another stability result of the continuous-time system (1) without its interval uncertainty will be presented as follows:Corollary 20.
Consider the continuous-time system (1) without its interval uncertainty satisfying Assumption 6. Let Ai be the Metzler matrices and Di⪰0,∀i∈N¯. Given constants 0<μ<1,λ>0,τ∗>0, and L∈ℕ. If there exist positive vectors νi,q,i∈N¯,q∈L¯0, and constants τa≥τ∗ such that
(44)1−ΔΦi,q+1−ΔAiT+1+α^DT−λ1−ΔInνi,q≺0,(45)1−ΔΦi,q+1−ΔAiT+1+α^DT−λ1−ΔInνi,q+1≺0,(46)DTΦi,q−νi,q≺0,(47)DTΦi,q−νi,q+1≺0,(48)1−ΔAiT+1+α^DT−λ1−ΔInνi,L≺0,(49)νi,0−μνj,L⪯0,hold for any q=0,1,⋯,L−1, and for any i,j∈N¯,i≠j, then, the continuous-time system (1) without its interval uncertainty is positive and GUES with the same FADT satisfying (15) where Φi,q and D are defined as in (16) and (8), respectively.Proof.
With the same symbols and vector function (17) in Theorem 17, this corollary can be proved by utilizing the following time-scheduled MCLKF:
(50)Vit,xt=1−ΔxTtνit+∫t−αttxTsDTνitds+∫−α^0∫t+wtxTsDTνisdsdw.
The remainder of the proof is similar to that of Theorem17. Hence, the detail is omitted.Remark 21.
It should be noted that our time-scheduled MCLKF proposed in (50) is similar to the time-scheduled MCLKF used in Corollary 1 of [57]. However, the DDSC obtained from this corollary are derived to ensure the global uniform exponential stability of system (1) without its interval uncertainty under the FADT switching technique rather than the global uniform asymptotic stability of the same system with the MDDT approach derived in [57]. In addition, as mentioned in [44], the ADT switching law includes the DT switching law. Consequently, the result of Theorem 3.1 in [43] is a special case of our theoretical result.In this subsection, we extend the results of the continuous-time LSPTDS including both interval uncertainties and AUSs to the discrete-time case.


### 3.2. Global Uniform Exponential Stability of Discrete-Time LSPTDS including Both Interval Uncertainties and AUSs

Theorem 22.
Consider the discrete-time system (5) satisfying Assumption 13 and Assumption 14. Given constants 0<μ<1,γ>1,κ∗>0, and L∈ℕ. If there exist positive vectors υi,q,i∈N¯,q∈L¯0, and constants κa≥κ∗ such that
(51)A¯iT+d2−d1+1D~TΠi,q+A¯iT+d2−d1+1D~T−γInυi,q≺0,(52)A¯iT+d2−d1+1D~TΠi,q+A¯iT+d2−d1+1D~T−γInυi,q+1≺0,(53)D~TΠi,q+1−γυi,q≺0,(54)D~TΠi,q+1−γυi,q+1≺0,(55)A¯iT+d2−d1+1D~T−γInυi,L≺0,(56)υi,0−μυj,L⪯0,
hold for anyq=0,1,⋯,L−1, and for any i,j∈N¯,i≠j, then, the discrete-time system (5) is positive and GUES under the switching signal with the FADT satisfying
(57)κ∗≤κa<−logγμ,where
(58)Πi,q=υi,q+1−υi,qℏ,
ℏ=κ∗/L and D~ is defined as in (7).Proof.
The proof process of this theorem is similar to the continuous-time system; namely, it can be separated into the following two steps.Step 1.
We will prove that system (5) is positive.Using Assumption13, we obtain that Ai⪰0 and Di⪰0 for all i∈N¯. According to Lemma 11, system (5) is positive.Step 2.
We will prove that system (5) is GUES under the switching signal under the FADT satisfying condition (57).For a givenk∈ℕ, we suppose that k∈km,km+1=km,km+κ^∪km+κ^,km+1,m∈ℕ0,κ^=Lκ∗/L, and the interval km,km+κ^ is split into L segments with equal length ℏ=κ∗/L. We define Ξm,q=km+qℏ,km+q+1ℏs,q=0,1,⋯,L−1, and let km,km+κ^=⋃q=0L−1Ξm,q. Next, we construct the vector function:
(59)υik=υikm+qℏ+r=1−rℏυi,q+rℏυi,q+1,k∈Ξm,q,r∈ℏ¯0q=0,1,⋯,L−1,υi,L,k∈km+κ^,km+1,where i∈N¯,m∈ℕ0, and υi,q are positive vectors for i∈N¯,q∈L¯0. For k∈Ξm,q, we obtain
(60)υik+1−υik=υikm+qℏ+r+1−υikm+qℏ+r=1−r+1ℏυi,q+r+1ℏυi,q+1−1−rℏυi,q+rℏυi,q+1,which yields
(61)υik+1−υik=Πi,q,where Πi,q is defined as in (58). For each subsystem, we establish the time-scheduled MCLKF:
(62)Vik,xk=xTkυik+∑h=k−dkk−1xThsD~Tυik+∑l=−d2+1−d1∑h=k+lk−1xThD~Tυik.ConsideringVik+1,xk+1 in (62) along the trajectories of system (5), we have
(63)Vik+1,xk+1=xTkAiTυik+1+xTk−dkDiTυik+1+∑h=k+1−dk+1kxThD~Tυik+1+∑l=−d2+1−d1∑h=k+1+lkxThD~Tυik+1,then
(64)Vik+1,xk+1≤xTkA¯iTυik+1+xTk−dkD¯iTυik+1+∑h=k+1−dk+1kxThD~Tυik+1+∑l=−d2+1−d1∑h=k+1+lkxThD~Tυik+1.It follows that(65)Vik+1,xk+1−γVik,xk≤xTkA¯iTυik+1+xTk−dkD¯iTυik+1+∑h=k+1−dk+1kxThD~Tυik+1+∑l=−d2+1−d1∑h=k+1+lkxThD~Tυik+1−γxTkυik−γ∑h=k−dkk−1xThD~Tυik−γ∑l=−d2+1−d1∑h=k+lk−1xThD~Tυik.In factD¯i⪯D~ for all i∈N¯. Then
(66)Vik+1,xk+1−γVik,xk≤xTkA¯iTυik+1+xTk−dkD~Tυik+1+∑h=k+1−dk+1kxThD~Tυik+1+∑l=−d2+1−d1∑h=k+1+lkxThD~Tυik+1−γxTkυik−γ∑h=k−dkk−1xThD~Tυik−γ∑l=−d2+1−d1∑h=k+lk−1xThD~Tυik≤xTkA¯iTυik+1+xTk−dkD~Tυik+1+∑h=k−d2+1k−1xThD~Tυik+1+xTkD~Tυik+1−γxTkυik−γ∑h=k−d1+1k−1xThD~Tυik−γxTk−dkD~Tυik+∑l=−d2+1−d1∑h=k+1+lk−1xThD~Tυik+1+∑l=−d2+1−d1xTkD~Tυik+1−γ∑l=−d2+1−d1∑h=k+1+lk−1xThD~Tυik−γ∑l=−d2+1−d1xTk+lD~Tυik.We observe that(67)∑l=−d2+1−d1xTkD~Tυik+1=d2−d1xTkD~Tυik+1,(68)γ∑l=−d2+1−d1xTk+lD~Tυik=γ∑h=k−d2+1k−d1xThD~Tυik.From (66), we arrive at
(69)Vik+1,xk+1−γVik,xk≤xTkA¯iTυik+1+d2−d1+1D~Tυik+1−γυik+xTk−dkD~Tυik+1−γυik+∑h=k−d2+1k−1xThD~Tυik+1−γυik+∑l=−d2+1−d1∑h=k+1+lk−1xThD~Tυik+1−γυik.Whenk∈Ξm,q⊂km,km+κ^, it can be seen that
(70)A¯iTυik+1+d2−d1+1D~Tυik+1−γυik=1−rℏA¯iT+d2−d1+1D~TΠi,q+A¯iT+d2−d1+1D~T−γInυi,q+rℏA¯iT+d2−d1+1D~TΠi,q+A¯iT+d2−d1+1D~T−γInυi,q+1,(71)D~Tυik+1−γυik=1−rℏD~TΠi,q+1−γυi,q+rℏD~TΠi,q+1−γυi,q+1.According to conditions (51)–(54), we obtain
(72)Vik+1,xk+1−γVik,xk<0,k∈Ξm,q,which implies
(73)Vik+1,xk+1<γVik,xk,k∈⋃q=0L−1Ξm,q=km,km+κ^.Fork∈km+κ^,km+1, it yields that
(74)A¯iTυik+1+d2−d1+1D~Tυik+1−γυik=A¯iT+d2−d1+1D~T−γInυi,L,(75)D~Tυik+1−γυik=D~T1−γυi,L≺0.Utilizing condition (55), we get
(76)Vik+1,xk+1<γVik,xk,k∈km+κ^,km+1.Combining (73) with (76), we can easily see that
(77)Vik+1,xk+1<γVik,xk,k∈km,km+1,m∈ℕ0.This implies(78)Vik,xk<γk−kmVikm,xkm,k∈km,km+1,m∈ℕ0.Using (59), (62) and condition (56), we have
(79)Vikm,xkm≤μVjkm−1,xkm−1,km−1∈km−1+κ^,km.Sincekm−1+κ^,km⊂km−1,km, one can claim from (78) that
(80)Vjkm−1,xkm−1<γkm−1−km−1Vjkm−1,xkm−1,km−1∈km−1,km,which implies
(81)Vjkm−1,xkm−1<γkm−km−1Vjkm−1,xkm−1,km−1∈km−1,km,m∈ℕ0.From (78), (79), and (81), we can derive
(82)Vσkmk,xk<γk−kmμγkm−km−1Vjkm−1,xkm−1,then
(83)Vσkmk,xk<μμ⋯μγk−kmγkm−km−1⋯γk1−k0Vσk0k0,xk0=μNσk,k0γk−k0Vσk0k0,xk0.It follows from Definition16 and 0<μ<1 by employing (6), then, we obtain
(84)Vσkmk,xk≤μN0+k−k0/κaγk−k0Vσk0k0,xk0,(85)=γN0logγμγ1+logγμ/κak−k0Vσk0k0,xk0.Without loss of generality, we impose thatVσk0k0,xk0=Vσ00,x0. From (59) and (62), we have
(86)Vσ00,x0≤∥xT0∥2∥υσ00∥2+∑h=−d^−1maxh∈−d^,−d^+1,⋯,−1,0∥xTh∥2∥D~T∥2∥υσ00∥2+∑l=−d2+1−d1∑h=l−1maxh∈−d^,−d^+1,⋯,−1,0∥xTh∥2∥D~T∥2∥υσ00∥2≤∥xT0∥2∑b∈L¯0∥υσ0,b∥2+d^maxh∈−d^,−d^+1,⋯,−1,0∥xTh∥2∥D~T∥2∑b∈L¯0∥υσ0,b∥2+d2−d12d2+d1−1maxh∈−d^,−d^+1,⋯,−1,0∥xTh∥2∥D~T∥2∑b∈L¯0∥υσ0,b∥2≤n∥φ∥d^∑b∈L¯0∥υσ0,b∥2+d^n∥φ∥d^∥D~T∥2∑b∈L¯0∥υσ0,b∥2+d2−d12d2+d1−1n∥φ∥d^∥D~T∥2∑b∈L¯0∥υσ0,b∥2,(87)=1+d^+d2−d12d2+d1−1∥D~T∥2n∑b∈L¯0∥υσ0,b∥2∥φ∥d^.Furthermore, we get(88)ς∥xk∥2≤Vσkmk,xk,where ς=mina,b∈N¯×L¯0ω¯υa,b. Substituting (87) and (88) into (85), we obtain
(89)∥xk∥2≤γN0logγμςγ1+logγμ/κak−k01+d^+d2−d12d2+d1−1∥D~T∥2n∑b∈L¯0∥υσ0,b∥2∥φ∥d^,fork≥k0. According to the FADT (57), it is immediate that
(90)1+logγμκa<0,which yields
(91)0<γ1+logγμ/κa<1.From (89), for k≥k0, we arrive at
(92)∥xk∥2≤ερk−k0∥φ∥d^,where ε=γN0logγμ/ς1+d^+d2−d1/2d2+d1−1∥D~T∥2n∑b∈L¯0∥υσ0,b∥2, and ρ=γ1+logγμ/κa.By Definition12, we can conclude that system (5) is GUES under the switching signal with the FADT satisfying condition (57).Remark 23.
Unlike the discretized copositive Lyapunov function utilized in [35], the time-scheduled MCLKF defined in (62) is constructed specifically for system (3) including both interval uncertainties and AUSs. Furthermore, our time-scheduled MCLKF, which is different from the discretized copositive Lyapunov-Krasovskii functional in [43], and the FADT switching method are applied together to guarantee the global uniform exponential stability of the studied system. Therefore, it is worth noting that here we consider the more general systems, and our theoretical result is more general than those of Theorem 3.4 and Theorem 3.6 in [43].Remark 24.
Apart from the studies in [32] that proposed practical exponential stability criteria for discrete-time linear SPSs with impulse, disturbance, and all modes unstable, our result shown in Theorem 22 focuses on the robust exponential stability of system (5) including both interval uncertainties and AUSs. In addition, it should be noted that the conditions of Theorem 3 and Theorem 4 in [35] ensure the global asymptotic stability of discrete-time linear SPSs with AUSs. Still, the existence of the time-varying delay has not been taken into account yet. Hence, our theoretical result is more general and applicable than those of the results in [32, 35].The last result of the discrete-time system (5) without its interval uncertainty will be shown in the following:Corollary 25.
Consider the discrete-time system (5) without its interval uncertainty satisfying Assumption 14. Let Ai⪰0 and Di⪰0,∀i∈N¯. Given constants 0<μ<1,γ>1,κ∗>0, and L∈ℕ. If there exist positive vectors υi,q,i∈N¯,q∈L¯0, and constants κa≥κ∗ such that
(93)AiT+d2−d1+1DTΠi,q+AiT+d2−d1+1DT−γInυi,q≺0,(94)AiT+d2−d1+1DTΠi,q+AiT+d2−d1+1DT−γInυi,q+1≺0,(95)DTΠi,q+1−γυi,q≺0,(96)DTΠi,q+1−γυi,q+1≺0,(97)AiT+d2−d1+1DT−γInυi,L≺0,(98)υi,0−μυj,L⪯0,
hold for anyq=0,1,⋯,L−1, and for any i,j∈N¯,i≠j, then, the discrete-time system (5) without its interval uncertainty is positive and GUES with the same FADT satisfying (57) where Πi,q and D are defined as in (58) and (8), respectively.Proof.
With the same symbols and vector function (59) in Theorem 22, the proof of this corollary can be achieved by using the time-scheduled MCLKF in the form of
(99)Vik,xk=xTkυik+∑h=k−dkk−1xThDTυik+∑l=−d2+1−d1∑h=k+lk−1xThDTυik.
The proof is very similar to that of Theorem22, and the methodology can easily derive it as above. Therefore, the rest of the proof of this corollary is omitted here.Remark 26.
Compared with the results of Theorem 3.4 and Theorem 3.6 in [43], in this corollary, we use the FADT switching technique to verify the exponential stability criteria for system (3) with AUSs. Thus, our theoretical result is more general than those of the results in [43].


## 3.1. Global Uniform Exponential Stability of Continuous-Time LSPTDS including Both Interval Uncertainties and AUSs

Theorem 17.
Consider the continuous-time system (1) satisfying Assumption 5 and Assumption 6. Given constants 0<μ<1,λ>0,τ∗>0, and L∈ℕ. If there exist positive vectors νi,q,i∈N¯,q∈L¯0 and constants τa≥τ∗ such that
(9)1−ΔΦi,q+1−ΔA¯iT+1+α^D~T−λ1−ΔInνi,q≺0,(10)1−ΔΦi,q+1−ΔA¯iT+1+α^D~T−λ1−ΔInνi,q+1≺0,(11)D~TΦi,q−νi,q≺0,(12)D~TΦi,q−νi,q+1≺0,(13)1−ΔA¯iT+1+α^D~T−λ1−ΔInνi,L≺0,(14)νi,0−μνj,L⪯0,
hold for anyq=0,1,⋯,L−1, and for any i,j∈N¯,i≠j, then, the continuous-time system (1) is positive and GUES under the switching signal with the FADT satisfying
(15)τ∗≤τa<−lnμλ,where
(16)Φi,q=νi,q+1−νi,qLτ∗,
andD~ is mentioned in (7).Proof.
We divide the proof process into the following two steps.Step 1.
We will prove that system (1) is positive.Using Assumption5, we obtain that Ai is also Metzler matrices and Di⪰0 for all i∈N¯. According to Lemma 3, system (1) is positive.Step 2.
We will prove that system (1) is GUES under the switching signal under the FADT satisfying condition (15).For anyt>0, we suppose that t∈tm,tm+1=tm,tm+τ∗∪tm+τ∗,tm+1,m∈ℕ0. And we divide the interval tm,tm+τ∗ into L segments with equal length h=τ∗/L. In addition, we define Ym,q=tm+qh,tm+q+1h,q=0,1,⋯,L−1, then tm,tm+τ∗=⋃q=0L−1Ym,q.Due to each stable subsystem has been completely replaced by AUSs, some previous researches about the switched systems are no longer applicable. Motivated by the concept utilized in [28, 43, 44, 57], we employ the discretized Lyapunov function and the FADT switching law to stabilize system (1) with interval uncertainties for the case that all subsystems are unstable. First, we establish the following vector function
(17)νit=νitm+qh+ηth=1−ηtνi,q+ηtνi,q+1,t∈Ym,q,q=0,1,⋯,L−1,νi,L,t∈tm+τ∗,tm+1,where i∈N¯,m∈ℕ0,ηt=t−tm−qh/h with 0≤ηt≤1, and νi,q are positive vectors for i∈N¯,q∈L¯0. For t∈Ym,q, we can get
(18)ν̇it=η̇tνi,q+1−η̇tνi,q=νi,q+1−νi,qh,which yields
(19)ν̇it=Φi,q,where Φi,q is defined as in (16). For any i∈N¯, we establish the time-scheduled MCLKF:
(20)Vit,xt=1−ΔxTtνit+∫t−αttxTsD~Tνitds+∫−α^0∫t+wtxTsD~Tνisdsdw.DifferentiatingVit,xt in (20) along the trajectories of system (1), we obtain
(21)V̇it,xt=1−ΔxTtν̇it+xTtAiTνit+xTt−αtDiTνit+∫t−αttxTsD~Tν̇itds+xTtD~Tνit−xTt−αtD~Tνit1−α̇t+α^xTtD~Tνit−∫t−α^txTsD~Tνisds,then
(22)V̇it,xt≤1−ΔxTtν̇it+xTtA¯iTνit+xTt−αtD¯iTνit+∫t−αttxTsD~Tν̇itds+xTtD~Tνit−xTt−αtD~Tνit1−α̇t+α^xTtD~Tνit−∫t−α^txTsD~Tνisds.We observe that(23)V̇it,xt−λVit,xt≤1−ΔxTtν̇it+xTtA¯iTνit+xTt−αtD¯iTνit+∫t−αttxTsD~Tν̇itds+xTtD~Tνit−xTt−αtD~Tνit1−α̇t+α^xTtD~Tνit−∫t−α^txTsD~Tνisds−λ1−ΔxTtνit−λ∫t−αttxTsD~Tνitds−λ∫−α^0∫t+wtxTsD~Tνisdsdw.Along withα̇t≤Δ,0<λ and D¯i⪯D~ for all i∈N¯, one has
(24)V̇it,xt−λVit,xt≤xTt1−Δν̇it+1−ΔA¯iTνit+1+α^D~Tνit−λ1−Δνit+∫t−αttxTsD~Tν̇it−νisds.Whent∈Ym,q⊂tm,tm+τ∗, it leads to
(25)1−Δν̇it+1−ΔA¯iTνit+1+α^D~Tνit−λ1−Δνit=1−ηt1−ΔΦi,q+1−ΔA¯iT+1+α^D~T−λ1−ΔInνi,q+ηt1−ΔΦi,q+1−ΔA¯iT+1+α^D~T−λ1−ΔInνi,q+1,and(26)D~Tν̇it−νis=1−ηsD~TΦi,q−νi,q+ηsD~TΦi,q−νi,q+1.According to conditions (9)–(12), we obtain
(27)V̇it,xt−λVit,xt<0,t∈Ym,q,which implies
(28)V̇it,xt<λVit,xt,t∈⋃q=0L−1Ym,q=tm,tm+τ∗.Whent∈tm+τ∗,tm+1, it yields that
(29)1−Δν̇it+1−ΔA¯iTνit+1+α^D~Tνit−λ1−Δνit=1−ΔA¯iT+1+α^D~T−λ1−ΔInνi,L,(30)D~Tν̇it−νis=−D~Tνi,L≺0.Applying condition (13), we get
(31)V̇it,xt<λVit,xt,t∈tm+τ∗,tm+1.Combining (28) with (31), it is obvious that
(32)V̇it,xt<λVit,xt,t∈tm,tm+1,m∈ℕ0.Integrating both sides of (32) over tm,t for t∈tm,tm+1,m∈ℕ0, it is immediate that
(33)Vit,xt<eλt−tmVitm,xtm.Using condition (14), we have
(34)νitm⪯μνjtm−foralli,j∈N¯,i≠j.From (17) and (20), it can be seen that
(35)Vitm,xtm=1−ΔxTtmνitm+∫tm−αtmtmxTsD~Tνitmds+∫−α^0∫tm+wtmxTsD~Tνisdsdw≤1−ΔxTtm−μνjtm−+∫tm−−αtm−tm−xTsD~Tμνjtm−ds+∫−α^0∫tm−+wtm−xTsD~Tμνjsdsdw,(36)=μVjtm−,xtm−foralli,j∈N¯,i≠j.Based on the relationship between (33) and (36), we can derive
(37)Vσtmt,xt<eλt−tmVσtmtm,xtm≤μeλt−tmVσtm−1tm−,xtm−<μeλt−tmeλtm−tm−1Vσtm−1tm−1,xtm−1⋮<μμ⋯μeλt−tmeλtm−tm−1⋯eλt1−t0Vσt0t0,xt0=μNσt,t0eλt−t0Vσt0t0,xt0.It follows from Definition8 and 0<μ<1 by employing (4), then, we obtain
(38)Vσtmt,xt≤μN0+t−t0/τaeλt−t0Vσt0t0,xt0=eN0lnμeλ+lnμ/τat−t0Vσt0t0,xt0.Without loss of generality, we impose thatVσt0t0,xt0=Vσ00,x0. From (17) and (20), we have
(39)Vσ00,x0≤1−Δ∥xT0∥2∥νσ00∥2+∫−α^0dssup−α^≤s≤0∥xTs∥2∥D~T∥2∥νσ00∥2+∫−α^0∫w0∥νσ0s∥2dsdwsup−α^≤s≤0∥xTs∥2∥D~T∥2≤1−Δ∥xT0∥2∑b∈L¯0∥νσ0,b∥2+α^sup−α^≤s≤0∥xTs∥2∥D~T∥2∑b∈L¯0∥νσ0,b∥2+2∑b∈L¯0∥νσ0,b∥2∫−α^0∫w0dsdwsup−α^≤s≤0∥xTs∥2∥D~T∥2≤1−Δn∥ψ∥α^∑b∈L¯0∥νσ0,b∥2+α^n∥ψ∥α^∥D~T∥2∑b∈L¯0∥νσ0,b∥2+α∧2n∥ψ∥α^∥D~T∥2∑b∈L¯0∥νσ0,b∥2=1−Δ+α^1+α^∥D~T∥2n∑b∈L¯0∥νσ0,b∥2∥ψ∥α^.Moreover, we get(40)1−Δξ∥xt∥2≤Vσtmt,xt,where ξ=mina,b∈N¯×L¯0ω¯νa,b. Substituting (39) and (40) into (38), we obtain
(41)∥xt∥2≤eN0lnμ1−Δξeλ+lnμ/τat−t01−Δ+α^1+α^∥D~T∥2n∑b∈L¯0∥νσ0,b∥2∥ψ∥α^,fort≥t0. If the FADT switching signal satisfies condition (15), then
(42)−λ+lnμτa>0.From (41), for t≥t0, we arrive at
(43)∥xt∥2≤εe−ϖt−t0∥ψ∥α^,where ε=eN0lnμ/1−Δξ1−Δ+α^1+α^∥D~T∥2n∑b∈L¯0∥νσ0,b∥2, and ϖ=−λ+lnμ/τa.By Definition4, we can conclude that system (1) is GUES under the switching signal with the FADT satisfying condition (15).Remark 18.
Inspired by the idea in [35, 43–44, 28, 57], the time-scheduled positive vector function defined in (17) is time-varying rather than a constant, and the time-scheduled MCLKF proposed in (20) is validly established to investigate the problem of robust stability of continuous-time system (1) when each subsystem is unstable. However, our time-scheduled MCLKF shown in Theorem 17 is different from the discretized copositive Lyapunov function, the discretized copositive Lyapunov-Krasovskii functional, the time-scheduled MCLKF, and the discretized quadratic Lyapunov function used in [28, 35, 43, 44], respectively. Although our time-scheduled MCLKF is similar to the time-scheduled MCLKF utilized in [57], our robust stability criteria under the FADT switching technique for the considered system can guarantee for being the global uniform exponential stability rather than for being the global uniform asymptotic stability derived in [57].Remark 19.
According to the result in [44], the FADT switching law is applied to deal with the stabilization problem system (1) with AUSs but without interval uncertainties. This switching law can compensate for the state divergence caused by the unstable subsystem under the appropriate average time of each unstable subsystem; namely, the ADT of AUSs is neither too long nor too short. However, the model in this paper, which includes the interval uncertainties, can also apply this switching law effectively. Therefore, our result obtained from Theorem 17 is more applicable than that of Theorem 1 in [44].Next, another stability result of the continuous-time system (1) without its interval uncertainty will be presented as follows:Corollary 20.
Consider the continuous-time system (1) without its interval uncertainty satisfying Assumption 6. Let Ai be the Metzler matrices and Di⪰0,∀i∈N¯. Given constants 0<μ<1,λ>0,τ∗>0, and L∈ℕ. If there exist positive vectors νi,q,i∈N¯,q∈L¯0, and constants τa≥τ∗ such that
(44)1−ΔΦi,q+1−ΔAiT+1+α^DT−λ1−ΔInνi,q≺0,(45)1−ΔΦi,q+1−ΔAiT+1+α^DT−λ1−ΔInνi,q+1≺0,(46)DTΦi,q−νi,q≺0,(47)DTΦi,q−νi,q+1≺0,(48)1−ΔAiT+1+α^DT−λ1−ΔInνi,L≺0,(49)νi,0−μνj,L⪯0,hold for any q=0,1,⋯,L−1, and for any i,j∈N¯,i≠j, then, the continuous-time system (1) without its interval uncertainty is positive and GUES with the same FADT satisfying (15) where Φi,q and D are defined as in (16) and (8), respectively.Proof.
With the same symbols and vector function (17) in Theorem 17, this corollary can be proved by utilizing the following time-scheduled MCLKF:
(50)Vit,xt=1−ΔxTtνit+∫t−αttxTsDTνitds+∫−α^0∫t+wtxTsDTνisdsdw.
The remainder of the proof is similar to that of Theorem17. Hence, the detail is omitted.Remark 21.
It should be noted that our time-scheduled MCLKF proposed in (50) is similar to the time-scheduled MCLKF used in Corollary 1 of [57]. However, the DDSC obtained from this corollary are derived to ensure the global uniform exponential stability of system (1) without its interval uncertainty under the FADT switching technique rather than the global uniform asymptotic stability of the same system with the MDDT approach derived in [57]. In addition, as mentioned in [44], the ADT switching law includes the DT switching law. Consequently, the result of Theorem 3.1 in [43] is a special case of our theoretical result.In this subsection, we extend the results of the continuous-time LSPTDS including both interval uncertainties and AUSs to the discrete-time case.


## 3.2. Global Uniform Exponential Stability of Discrete-Time LSPTDS including Both Interval Uncertainties and AUSs

Theorem 22.
Consider the discrete-time system (5) satisfying Assumption 13 and Assumption 14. Given constants 0<μ<1,γ>1,κ∗>0, and L∈ℕ. If there exist positive vectors υi,q,i∈N¯,q∈L¯0, and constants κa≥κ∗ such that
(51)A¯iT+d2−d1+1D~TΠi,q+A¯iT+d2−d1+1D~T−γInυi,q≺0,(52)A¯iT+d2−d1+1D~TΠi,q+A¯iT+d2−d1+1D~T−γInυi,q+1≺0,(53)D~TΠi,q+1−γυi,q≺0,(54)D~TΠi,q+1−γυi,q+1≺0,(55)A¯iT+d2−d1+1D~T−γInυi,L≺0,(56)υi,0−μυj,L⪯0,
hold for anyq=0,1,⋯,L−1, and for any i,j∈N¯,i≠j, then, the discrete-time system (5) is positive and GUES under the switching signal with the FADT satisfying
(57)κ∗≤κa<−logγμ,where
(58)Πi,q=υi,q+1−υi,qℏ,
ℏ=κ∗/L and D~ is defined as in (7).Proof.
The proof process of this theorem is similar to the continuous-time system; namely, it can be separated into the following two steps.Step 1.
We will prove that system (5) is positive.Using Assumption13, we obtain that Ai⪰0 and Di⪰0 for all i∈N¯. According to Lemma 11, system (5) is positive.Step 2.
We will prove that system (5) is GUES under the switching signal under the FADT satisfying condition (57).For a givenk∈ℕ, we suppose that k∈km,km+1=km,km+κ^∪km+κ^,km+1,m∈ℕ0,κ^=Lκ∗/L, and the interval km,km+κ^ is split into L segments with equal length ℏ=κ∗/L. We define Ξm,q=km+qℏ,km+q+1ℏs,q=0,1,⋯,L−1, and let km,km+κ^=⋃q=0L−1Ξm,q. Next, we construct the vector function:
(59)υik=υikm+qℏ+r=1−rℏυi,q+rℏυi,q+1,k∈Ξm,q,r∈ℏ¯0q=0,1,⋯,L−1,υi,L,k∈km+κ^,km+1,where i∈N¯,m∈ℕ0, and υi,q are positive vectors for i∈N¯,q∈L¯0. For k∈Ξm,q, we obtain
(60)υik+1−υik=υikm+qℏ+r+1−υikm+qℏ+r=1−r+1ℏυi,q+r+1ℏυi,q+1−1−rℏυi,q+rℏυi,q+1,which yields
(61)υik+1−υik=Πi,q,where Πi,q is defined as in (58). For each subsystem, we establish the time-scheduled MCLKF:
(62)Vik,xk=xTkυik+∑h=k−dkk−1xThsD~Tυik+∑l=−d2+1−d1∑h=k+lk−1xThD~Tυik.ConsideringVik+1,xk+1 in (62) along the trajectories of system (5), we have
(63)Vik+1,xk+1=xTkAiTυik+1+xTk−dkDiTυik+1+∑h=k+1−dk+1kxThD~Tυik+1+∑l=−d2+1−d1∑h=k+1+lkxThD~Tυik+1,then
(64)Vik+1,xk+1≤xTkA¯iTυik+1+xTk−dkD¯iTυik+1+∑h=k+1−dk+1kxThD~Tυik+1+∑l=−d2+1−d1∑h=k+1+lkxThD~Tυik+1.It follows that(65)Vik+1,xk+1−γVik,xk≤xTkA¯iTυik+1+xTk−dkD¯iTυik+1+∑h=k+1−dk+1kxThD~Tυik+1+∑l=−d2+1−d1∑h=k+1+lkxThD~Tυik+1−γxTkυik−γ∑h=k−dkk−1xThD~Tυik−γ∑l=−d2+1−d1∑h=k+lk−1xThD~Tυik.In factD¯i⪯D~ for all i∈N¯. Then
(66)Vik+1,xk+1−γVik,xk≤xTkA¯iTυik+1+xTk−dkD~Tυik+1+∑h=k+1−dk+1kxThD~Tυik+1+∑l=−d2+1−d1∑h=k+1+lkxThD~Tυik+1−γxTkυik−γ∑h=k−dkk−1xThD~Tυik−γ∑l=−d2+1−d1∑h=k+lk−1xThD~Tυik≤xTkA¯iTυik+1+xTk−dkD~Tυik+1+∑h=k−d2+1k−1xThD~Tυik+1+xTkD~Tυik+1−γxTkυik−γ∑h=k−d1+1k−1xThD~Tυik−γxTk−dkD~Tυik+∑l=−d2+1−d1∑h=k+1+lk−1xThD~Tυik+1+∑l=−d2+1−d1xTkD~Tυik+1−γ∑l=−d2+1−d1∑h=k+1+lk−1xThD~Tυik−γ∑l=−d2+1−d1xTk+lD~Tυik.We observe that(67)∑l=−d2+1−d1xTkD~Tυik+1=d2−d1xTkD~Tυik+1,(68)γ∑l=−d2+1−d1xTk+lD~Tυik=γ∑h=k−d2+1k−d1xThD~Tυik.From (66), we arrive at
(69)Vik+1,xk+1−γVik,xk≤xTkA¯iTυik+1+d2−d1+1D~Tυik+1−γυik+xTk−dkD~Tυik+1−γυik+∑h=k−d2+1k−1xThD~Tυik+1−γυik+∑l=−d2+1−d1∑h=k+1+lk−1xThD~Tυik+1−γυik.Whenk∈Ξm,q⊂km,km+κ^, it can be seen that
(70)A¯iTυik+1+d2−d1+1D~Tυik+1−γυik=1−rℏA¯iT+d2−d1+1D~TΠi,q+A¯iT+d2−d1+1D~T−γInυi,q+rℏA¯iT+d2−d1+1D~TΠi,q+A¯iT+d2−d1+1D~T−γInυi,q+1,(71)D~Tυik+1−γυik=1−rℏD~TΠi,q+1−γυi,q+rℏD~TΠi,q+1−γυi,q+1.According to conditions (51)–(54), we obtain
(72)Vik+1,xk+1−γVik,xk<0,k∈Ξm,q,which implies
(73)Vik+1,xk+1<γVik,xk,k∈⋃q=0L−1Ξm,q=km,km+κ^.Fork∈km+κ^,km+1, it yields that
(74)A¯iTυik+1+d2−d1+1D~Tυik+1−γυik=A¯iT+d2−d1+1D~T−γInυi,L,(75)D~Tυik+1−γυik=D~T1−γυi,L≺0.Utilizing condition (55), we get
(76)Vik+1,xk+1<γVik,xk,k∈km+κ^,km+1.Combining (73) with (76), we can easily see that
(77)Vik+1,xk+1<γVik,xk,k∈km,km+1,m∈ℕ0.This implies(78)Vik,xk<γk−kmVikm,xkm,k∈km,km+1,m∈ℕ0.Using (59), (62) and condition (56), we have
(79)Vikm,xkm≤μVjkm−1,xkm−1,km−1∈km−1+κ^,km.Sincekm−1+κ^,km⊂km−1,km, one can claim from (78) that
(80)Vjkm−1,xkm−1<γkm−1−km−1Vjkm−1,xkm−1,km−1∈km−1,km,which implies
(81)Vjkm−1,xkm−1<γkm−km−1Vjkm−1,xkm−1,km−1∈km−1,km,m∈ℕ0.From (78), (79), and (81), we can derive
(82)Vσkmk,xk<γk−kmμγkm−km−1Vjkm−1,xkm−1,then
(83)Vσkmk,xk<μμ⋯μγk−kmγkm−km−1⋯γk1−k0Vσk0k0,xk0=μNσk,k0γk−k0Vσk0k0,xk0.It follows from Definition16 and 0<μ<1 by employing (6), then, we obtain
(84)Vσkmk,xk≤μN0+k−k0/κaγk−k0Vσk0k0,xk0,(85)=γN0logγμγ1+logγμ/κak−k0Vσk0k0,xk0.Without loss of generality, we impose thatVσk0k0,xk0=Vσ00,x0. From (59) and (62), we have
(86)Vσ00,x0≤∥xT0∥2∥υσ00∥2+∑h=−d^−1maxh∈−d^,−d^+1,⋯,−1,0∥xTh∥2∥D~T∥2∥υσ00∥2+∑l=−d2+1−d1∑h=l−1maxh∈−d^,−d^+1,⋯,−1,0∥xTh∥2∥D~T∥2∥υσ00∥2≤∥xT0∥2∑b∈L¯0∥υσ0,b∥2+d^maxh∈−d^,−d^+1,⋯,−1,0∥xTh∥2∥D~T∥2∑b∈L¯0∥υσ0,b∥2+d2−d12d2+d1−1maxh∈−d^,−d^+1,⋯,−1,0∥xTh∥2∥D~T∥2∑b∈L¯0∥υσ0,b∥2≤n∥φ∥d^∑b∈L¯0∥υσ0,b∥2+d^n∥φ∥d^∥D~T∥2∑b∈L¯0∥υσ0,b∥2+d2−d12d2+d1−1n∥φ∥d^∥D~T∥2∑b∈L¯0∥υσ0,b∥2,(87)=1+d^+d2−d12d2+d1−1∥D~T∥2n∑b∈L¯0∥υσ0,b∥2∥φ∥d^.Furthermore, we get(88)ς∥xk∥2≤Vσkmk,xk,where ς=mina,b∈N¯×L¯0ω¯υa,b. Substituting (87) and (88) into (85), we obtain
(89)∥xk∥2≤γN0logγμςγ1+logγμ/κak−k01+d^+d2−d12d2+d1−1∥D~T∥2n∑b∈L¯0∥υσ0,b∥2∥φ∥d^,fork≥k0. According to the FADT (57), it is immediate that
(90)1+logγμκa<0,which yields
(91)0<γ1+logγμ/κa<1.From (89), for k≥k0, we arrive at
(92)∥xk∥2≤ερk−k0∥φ∥d^,where ε=γN0logγμ/ς1+d^+d2−d1/2d2+d1−1∥D~T∥2n∑b∈L¯0∥υσ0,b∥2, and ρ=γ1+logγμ/κa.By Definition12, we can conclude that system (5) is GUES under the switching signal with the FADT satisfying condition (57).Remark 23.
Unlike the discretized copositive Lyapunov function utilized in [35], the time-scheduled MCLKF defined in (62) is constructed specifically for system (3) including both interval uncertainties and AUSs. Furthermore, our time-scheduled MCLKF, which is different from the discretized copositive Lyapunov-Krasovskii functional in [43], and the FADT switching method are applied together to guarantee the global uniform exponential stability of the studied system. Therefore, it is worth noting that here we consider the more general systems, and our theoretical result is more general than those of Theorem 3.4 and Theorem 3.6 in [43].Remark 24.
Apart from the studies in [32] that proposed practical exponential stability criteria for discrete-time linear SPSs with impulse, disturbance, and all modes unstable, our result shown in Theorem 22 focuses on the robust exponential stability of system (5) including both interval uncertainties and AUSs. In addition, it should be noted that the conditions of Theorem 3 and Theorem 4 in [35] ensure the global asymptotic stability of discrete-time linear SPSs with AUSs. Still, the existence of the time-varying delay has not been taken into account yet. Hence, our theoretical result is more general and applicable than those of the results in [32, 35].The last result of the discrete-time system (5) without its interval uncertainty will be shown in the following:Corollary 25.
Consider the discrete-time system (5) without its interval uncertainty satisfying Assumption 14. Let Ai⪰0 and Di⪰0,∀i∈N¯. Given constants 0<μ<1,γ>1,κ∗>0, and L∈ℕ. If there exist positive vectors υi,q,i∈N¯,q∈L¯0, and constants κa≥κ∗ such that
(93)AiT+d2−d1+1DTΠi,q+AiT+d2−d1+1DT−γInυi,q≺0,(94)AiT+d2−d1+1DTΠi,q+AiT+d2−d1+1DT−γInυi,q+1≺0,(95)DTΠi,q+1−γυi,q≺0,(96)DTΠi,q+1−γυi,q+1≺0,(97)AiT+d2−d1+1DT−γInυi,L≺0,(98)υi,0−μυj,L⪯0,
hold for anyq=0,1,⋯,L−1, and for any i,j∈N¯,i≠j, then, the discrete-time system (5) without its interval uncertainty is positive and GUES with the same FADT satisfying (57) where Πi,q and D are defined as in (58) and (8), respectively.Proof.
With the same symbols and vector function (59) in Theorem 22, the proof of this corollary can be achieved by using the time-scheduled MCLKF in the form of
(99)Vik,xk=xTkυik+∑h=k−dkk−1xThDTυik+∑l=−d2+1−d1∑h=k+lk−1xThDTυik.
The proof is very similar to that of Theorem22, and the methodology can easily derive it as above. Therefore, the rest of the proof of this corollary is omitted here.Remark 26.
Compared with the results of Theorem 3.4 and Theorem 3.6 in [43], in this corollary, we use the FADT switching technique to verify the exponential stability criteria for system (3) with AUSs. Thus, our theoretical result is more general than those of the results in [43].


## 4. Numerical Simulations

In this section, we intensively provide two numerical examples along with the simulation results to demonstrate the correctness and effectiveness of our theoretical analysis presented in the previous section.Example 1.
The robust stability problem for the continuous-time system (1) comprising of two subsystems is studied in this example. The system data are given as follows:
(100)A¯1=−2.620.660.140.005,A¯1=−2.60.70.150.01,D¯1=0.0040.0020.0010.008,D¯1=0.0050.0030.0020.01,(101)A¯2=0.0350.050.04−1.92,A¯2=0.0360.060.05−1.9,D¯2=0.0010.0010.0010.006,D¯2=0.0030.0020.0020.008,(102)αt=0.15+0.05sint.
Under the given time-varying delay above, we selectα^=0.2 and Δ=0.05. According to Definition 2, one can see that A¯1 and A¯2 are Metzler matrices. Furthermore, it is obvious that D¯1⪰0 and D¯2⪰0. By Assumption 5 and Lemma 3, the studied system is positive. We set the initial state ψθ=510T,θ∈−α^,0, and let the system matrices be
(103)A1=A¯1+A¯12=−2.610.680.1450.0075,D1=D¯1+D¯12=0.00450.00250.00150.009,(104)A2=A¯2+A¯22=0.03550.0550.045−1.91,D2=D¯2+D¯22=0.0020.00150.00150.007.
We first present two figures for the corresponding state responses of two subsystems. From Figures1 and 2, it can be seen that two subsystems are positive and unstable.
As defined in (7), it is obvious that
(105)D~=0.0050.0030.0020.01.
For given sclarsL=1,λ=0.4,μ=0.657, and τ∗=1, we can get a set of feasible solution for Theorem 17:
(106)ν1,0=12.818588.4196,ν1,1=31.7095105.1495,ν2,0=20.607268.8325,ν2,1=20.7160135.1355,
andτa=1.0100. Thus, the continuous-time system (1) is GUES with the FADT switching signal satisfying 1≤τa<1.0502. The corresponding switching signal σt and the state response of the system are illustrated in Figure 3. This shows that our designed switching signal can guarantee the positivity and robust stability of the considered system effectively.Example 2.
To show some advantages of our result, the comparison between the previous result studied in [35] and our result are presented in this example. We consider the two modes of the discrete-time system (5) without time delay and take the same system matrices as given in Example 2 of [35]; namely,
(107)A1=0.6001.1,A2=1.200.050.7.
It is easy to see thatA1⪰0 and A2⪰0. Thus, by Lemma 11, the considered system is positive. For comparison in the view of the numerical simulations, we impose the same initial condition as given in Example 2 of [35]; namely, x0=46T. From the system matrices above, it is obvious that the eigenvalues of A1 are λ1=0.6 and λ2=1.1 and the eigenvalues of A2 are λ1=1.2 and λ2=0.7. Therefore, both modes are positive and unstable which can be seen from Figures 4 and 5.
To stabilize the considered system via the FADT approach, we can chooseL=1,γ=1.3,μ=0.532, and the positive integer κ∗=2. By solving conditions in Corollary 25 with D=0, we get the following feasible solution κa=2.2100,
(108)υ1,0=56.921235.3838,υ1,1=186.233848.0901,υ2,0=98.426825.5626,υ2,1=107.640367.0016.
Consequently, the considered system is GUES with the FADT switching signal satisfying2≤κa<2.4055. It should be pointed out that the proportion between the maximum possible time and the minimum DT of this obtained FADT interval is 1.20275. Meanwhile, the proportion between the upper bound and the lower bound of the obtained mode-dependent dwell time for the first subsystem and the second subsystem in Example 2 of [35] is 1.2 and 1.1, respectively. Moreover, it should be noted that the average of the proportion between the upper bound and the lower bound of the obtained mode-dependent dwell time for both subsystems is 1.15. Hence, we can say that our dwell time interval is wider than that of Example 2 in [35]. This shows that our result does not need to switch too often in order to stabilize the considered system. Finally, the state response of the system and the corresponding switching signal σk are plotted mutually in Figure 6. The considered system can converge to zero under the FADT switching signal.Figure 1
State response of the first subsystem.Figure 2
State response of the second subsystem.Figure 3
State response of the continuous-time system.Figure 4
State response of the first subsystem.Figure 5
State response of the second subsystem.Figure 6
State response of the discrete-time system.


## 5. Conclusions

The global stability problem for both continuous-time and discrete-time LSPTDSs with interval uncertainties in the case of all subsystems are unstable has been intensively studied. By establishing the time-scheduled MCLKFs and applying the FADT switching law, new DDSC under the reasonable assumptions to ensure the global uniform exponential stability of both systems have been derived in the main theorems. Furthermore, novel DDSC of both systems without the interval uncertainties have also been acquired in the corollaries. Finally, two numerical examples have been displayed to validate the effectiveness along with some advantages of obtained theoretical results. In the future, it is interesting to study the global stability of nonlinear switched positive time-varying delay systems with interval uncertainties and all unstable subsystems.



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*Source: 3968850-2022-05-28.xml*