Finite-Time 𝑙₁-Gain Control for Positive Switched Systems with Time-Varying Delay via Delta Operator Approach

Abstract

This paper is concerned with the problem of finite-time 𝑙₁-gain control for positive switched systems with time-varying delay via delta operator approach. Firstly, sufficient conditions which can guarantee the 𝑙₁-gain finite-time boundedness of the underlying system are given by using the average dwell time approach and constructing an appropriate copositive type Lyapunov-Krasovskii functional in delta domain. Moreover, the obtained conditions can unify some previously suggested relevant results seen in literature of both continuous and discrete systems into the delta operator framework. Then, based on the results obtained, a state feedback controller is designed to ensure that the resulting closed-loop system is finite-time bounded with an 𝑙₁-gain performance. Finally, a numerical example is presented to demonstrate the effectiveness and feasibility of the proposed method.

1. Introduction

Positive systems are dynamical systems where the states and outputs are constrained to be nonnegative whenever the initial conditions and inputs are nonnegative [1, 2]. Such systems are applied in various areas, for instance, biomedicine [3, 4], ecology [5], TCP-like Internet congestion control [6, 7], industrial engineering [8], and so on. Recently, positive switched systems, which consist of a family of positive subsystems and a switching signal coordinating the operation of various subsystems to govern the switching among them, have been highlighted and investigated by many researchers due to the broad applications in communication systems [7, 9], formation flying [10], the viral mutation dynamics under drug treatment [2], and systems theories [11–13]. It has been shown that a linear copositive Lyapunov functional is powerful for the analysis and synthesis of positive systems [14–16].

The delta operator, as a novel method of solving the problem of system instability under fast sampling, has drawn considerable attention in the past three decades. As we know, most discrete-time signals and systems are the results of sampling continuous-time signals and systems. When the sampling period tends to zero, namely, data are taken at high sampling rates, however, all resulting signals and systems tend to be ill-conditioned and thus difficult to deal with using the conventional algorithms. Until Goodwin et al. proposed a delta-operator-based algorithm in [17] to take place of the conventional algorithms, the above problem is avoided. It was shown that delta-operator-based algorithms are numerically better behaved under finite precision implementations for fast sampling [17]. The delta operator model can be regarded as a useful approach to deal with discrete-time systems under high sampling rates through the analysis methods of continuous-time systems [18–20]. Based on significant early investigations such as [21–23], the numerical properties and practical applications of the delta operator model have been extensively investigated [24–26]. The delta operator is defined by (𝑡)=(𝑡)𝛿𝑥𝑑𝑥(𝑥()(𝑡𝑑𝑡,𝑇=0𝑡+𝑇−𝑥))𝑇,𝑇≠0, (1) where 𝑇 is a sampling period. When 𝑇→0, the delta operator model will approach the continuous system before discretization and reflect a quasicontinuous performance.

The stability of positive switched systems has been concerned by many researchers [27–29]. Most of them are based on the concept of Lyapunov stability, which is defined over an infinite-time interval. In practice, however, not only the system stability (generally refers to Lyapunov stability) but also the state trajectory over a fixed short time is needed to be considered. The finite-time stability, which is different from Lyapunov stability, admits that the state does not exceed a certain bound during a fixed finite-time interval. It should be noted that a finite-time stable system may not be stable in the sense of Lyapunov, and a Lyapunov stable system may not be finite-time stable since its transient response may exceed the bound [30]. Some early results on finite-time stability can be found in [31–33]. And recently, many results on finite-time control for switched systems have been presented in the literature [34–36]. The definition of finite-time stability for positive switched systems has been given in [37, 38]. In addition, exogenous disturbances are very common in many kinds of engineering systems, and the outputs will be inevitably affected by the disturbances. It is because of the peculiar nonnegative property of positive systems, the 𝑙₁-gain index [39, 40] can characterize the disturbance rejection property. Then we can limit the effect of disturbance in a prescribed level by means of 𝑙₁-gain. Moreover, it should be noted that delays are universal in practice and their existence may give rise to the deterioration of system performance and instability [41–44]. To the best of our knowledge, few results on the issues of finite-time stability and 𝑙₁-gain finite-time boundedness of positive switched systems with time-varying delay via delta operator approach have been proposed to date, which motivates the present research.

In this paper, we focus our attention on investigating the problem of finite-time 𝑙₁-gain control for a class of positive switched systems with time-varying delay via delta operator approach. The main contributions of this paper are threefold: (1) definitions of finite-time boundedness and 𝑙₁-gain performance are for the first time extended to positive switched delta operator systems with time-varying delay; (2) by applying the average dwell time approach, sufficient conditions for the existence of finite-time boundedness of the underlying system are given. Moreover, the results obtained can degenerate into the corresponding ones both in continuous-time systems and discrete-time systems; (3) a state feedback controller design scheme is proposed to guarantee that the closed-loop system is finite-time bounded with 𝑙₁-gain performance.

The remainder of the paper is organized as follows. The problem formulation and some necessary lemmas are provided in Section 2. In Section 3, 𝑙₁-gain finite-time boundedness analysis and controller design are developed. A numerical example is presented to demonstrate the feasibility of the obtained results in Section 4. Section 5 concludes this paper.

Notations. 𝐴≽0(⪯,≻,≺) means that all entries of matrix 𝐴 are nonnegative (nonpositive, positive, and negative); 𝐴≻𝐵(𝐴≽𝐵) means that 𝐴−𝐵≻0(𝐴−𝐵≽0); 𝐴^𝑇 means the transpose of matrix 𝐴; 𝑅(𝑅₊) is the set of all real (positive real) numbers; 𝑅^𝑛 is the 𝑛-dimensional real vector space; 𝑅^𝑛₊ is the set of all 𝑛-dimensional positive real vectors; 𝑅^𝑚×𝑛 is the set of all 𝑚×𝑛-dimensional real matrices. The vector 1-norm is denoted by ‖𝑥‖=∑^𝑛_𝑘=1|𝑥_𝑘|, where 𝑥_𝑘 is the 𝑘th element of 𝑥∈𝑅^𝑛. Given 𝑣∶𝑅→𝑅^𝑛, the 𝑙₁-norm is defined by ‖𝑣‖_𝐿1=∑^∞_𝑘0‖𝑣(𝑘)‖; 1_𝑛∈𝑅^𝑛 denotes a column vector with 𝑛 rows containing only 1 entry; 𝑙₁[𝑘₀,∞) is the space of absolute integrable vector-valued functions on [𝑘₀,∞); that is, we say 𝑧∶[𝑘₀,∞)→𝑅^𝑘 is in 𝑙₁[𝑘₀,∞) if ∑^∞_𝑘0‖𝑧(𝑘)‖<0.

2. Problem Formulation

Consider the following switched delta operator system with time-varying delay: 𝛿𝑥(𝑘)=𝐴_𝜎(𝑘)𝑥(𝑘)+𝐴_{𝑑𝜎(𝑘)}𝑥𝑘−𝑑_𝑘+𝐷_𝜎(𝑘)𝑤(𝑘),𝑥(𝑘)=𝜑(𝑘),𝑘=−𝑑,−𝑧(𝑘)𝑑+1,…,0,=𝐶_𝜎(𝑘)𝑥(𝑘)+𝐸_𝜎(𝑘)𝑤(𝑘), (2) where 𝑥(𝑘)∈𝑅^𝑛 denotes the state; 𝑧(𝑘)∈𝑅^𝑧 is the controlled output; 𝑘 represents the time 𝑡=𝑘𝑇; 𝑇>0 is the sampling period. 𝑤(𝑘)∈𝑅^𝑤 is the disturbance input satisfying ^𝑇𝑓_𝑘=0‖𝑤(𝑘)‖<𝑑,𝑑≥0, (3) and 𝜎(𝑘)∶[0,∞)→𝑚={1,2,…,𝑚} is the switching signal with 𝑚 representing the number of subsystems. 𝐴_𝑝, 𝐴_𝑑𝑝, 𝐶_𝑝, 𝐷_𝑝, and 𝐸_𝑝 are constant matrices with appropriate dimensions. 𝑑_𝑘 denotes the time-varying discrete delay which satisfies 0≤𝑑≤𝑑_𝑘≤𝑑 for known constants 𝑑 and 𝑑; {𝜑(𝑘),𝑘=−𝑑,−𝑑+1,…,0} is a given discrete vector-valued initial condition. The switch is assumed to only occur at the sampling time in this paper.

Remark 1. Since a delta operator system can be regarded as a quasicontinuous system when 𝑇→0, the term 𝛿𝑥(𝑘) can be applicable in normal continuous-time systems.

Definition 2. System (2) is said to be positive if, for any initial conditions 𝜑(𝑘)≽0, 𝑘=−𝑑,−𝑑+1,…,0, any inputs 𝑤(𝑘)≽0, and any switching signals 𝜎(𝑘), the system satisfies 𝑥(𝑘)≽0 and 𝑧(𝑘)≽0 for all 𝑘≥0.

Lemma 3. System (2) is positive if and only if (𝐼+𝑇𝐴_𝑝)≽0, 𝐴_𝑑𝑝≽0, 𝐷_𝑝≽0, 𝐶_𝑝≽0, and 𝐸_𝑝≽0, for all 𝑝∈𝑚.

Proof. From the definition of delta operator 𝛿, the discrete form of system (2) can be obtained as follows: 𝑥(𝑘+1)=𝐼+𝑇𝐴_𝜎(𝑘)𝑥(𝑘)+𝑇𝐴_{𝑑𝜎(𝑘)}𝑥𝑘−𝑑_𝑘+𝑇𝐷_𝜎(𝑘)𝑤(𝑘),𝑥(𝑘)=𝜑(𝑘),𝑘=−𝑑,−𝑧(𝑘)𝑑+1,…,0,=𝐶_𝜎(𝑘)𝑥(𝑘)+𝐸_𝜎(𝑘)𝑤(𝑘), (4) Combining Lemma 2 in [45] and Lemma 1 in [28], one can obtain the remaining proof easily.

Remark 4. When 𝑇→0, system (2) degenerates to a general continuous-time positive switched system as follows: 𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝑑𝜎(𝑡)}𝑥(𝑡−𝑑(𝑡))+𝐷_𝜎(𝑡)𝑤(𝑡),𝑥(𝑡)=𝜑(𝑡),𝑡∈−𝑧(𝑡)𝑑,0=𝐶_𝜎(𝑡)𝑥(𝑡)+𝐸_𝜎(𝑡)𝑤(𝑡), (5) where 𝑑(𝑡) denotes the time-varying delay which is everywhere time differentiable and satisfies 0≤𝑑≤𝑑(𝑡)≤𝑑, and 𝑑(̇𝑡)≤𝑑_𝑑<1 for known constants 𝑑, 𝑑, and 𝑑_𝑑. Then, according to [39], system (5) is positive if and only if 𝐴_𝑝 are Metzler matrices, and 𝐴_𝑑𝑝≽0, 𝐶_𝑝≽0, 𝐷_𝑝≽0, and 𝐸_𝑝≽0, for all 𝑝∈𝑚.

Definition 5 (see [46]). For any switching signal 𝜎(𝑘) and any 𝑘₂>𝑘₁≥0, let 𝑁_𝜎(𝑘₁,𝑘₂) denote the number of switches of 𝜎(𝑘) over the interval [𝑘₁,𝑘₂). For given 𝜏_𝑎>0 and 𝑁₀≥0, if the inequality 𝑁_𝜎𝑘₁,𝑘₂≤𝑁₀+𝑘₂−𝑘₁𝜏_𝑎 (6) holds, then the positive constant 𝜏_𝑎 is called the average dwell time and 𝑁₀ is called the chattering bound.
Without lose of generality, we choose 𝑁₀=0 in this paper.
In order to obtain the main results, we need to give the definitions of finite-time stability, finite-time boundedness and 𝑙₁-gain finite-time boundedness for positive system (2).

Definition 6 (finite-time stability). For a given time constant 𝑇_𝑓 and two vectors 𝜍≻𝜀≻0, positive switched delta operator system (2) with 𝑤(𝑘)=0 is said to be finite-time stable with respect to (𝜍,𝜀,𝑇_𝑓,𝜎(𝑘)), if sup_{−𝑑≤𝑘≤0}{𝑥^𝑇(𝑘)𝜍}≤1⇒𝑥^𝑇(𝑘)𝜀<1, for all 𝑘∈[0,𝑇_𝑓). If the above condition is satisfied for any switching signals 𝜎(𝑘), positive system (2) is said to be uniformly finite-time stable with respect to (𝜍,𝜀,𝑇_𝑓).

Remark 7. As can be seen from Definition 6, the concept of finite-time stability is different from the one of Lyapunov asymptotic stability. A Lyapunov asymptotically stable switched system may not be finite-time stable because its states may exceed the prescribed bounds during the interval time.

Definition 8 (finite-time boundedness). For a given time constant 𝑇_𝑓 and two vectors 𝜍≻𝜀≻0, positive switched delta operator system (2) is said to be finite-time bounded with respect to (𝜍,𝜀,𝑇_𝑓,𝑑,𝜎(𝑘)), if sup_{−𝑑≤𝑘≤0}{𝑥^𝑇(𝑘)𝜍}≤1⇒𝑥^𝑇(𝑘)𝜀<1, for all 𝑘∈[0,𝑇_𝑓).

Definition 9 (𝑙₁-gain finite-time boundedness). For a given time constant 𝑇_𝑓 and two vectors 𝜍≻𝜀≻0, positive switched delta operator system (2) is said to be 𝑙₁-gain finite-time bounded with respect to (𝜍,𝜀,𝑇_𝑓,𝑑,𝜎(𝑘)), if the following conditions are satisfied.(1)Positive switched delta operator system (2) is finite-time bounded with respect to (𝜍,𝜀,𝑇_𝑓,𝑑,𝜎(𝑘)).(2)Under zero-initial condition, that is, 𝜑(𝑘)=0, 𝑘=−𝑑,−𝑑+1,…,0, the output 𝑧(𝑘) satisfies ^𝑇𝑓_𝑘=01+𝑇𝜌^−𝑘‖𝑧(𝑘)‖≤𝛾^𝑇𝑓_𝑘=0‖𝑤(𝑘)‖, (7) where 𝜌>0 and 𝛾>0 are constants.

Remark 10. In Definition 9, as proposed in [39], 𝑙₁-gain performance characterizes system’s suppression to exogenous disturbance.

The aim of this paper is to find a class of switching signals 𝜎(𝑘) and determine a state feedback controller 𝑢(𝑘)=𝐾_𝜎(𝑘)𝑥(𝑘) for the following positive switched delta operator system with time-varying delay: 𝛿𝑥(𝑘)=𝐴_𝜎(𝑘)𝑥(𝑘)+𝐴_{𝑑𝜎(𝑘)}𝑥𝑘−𝑑_𝑘+𝐵_𝜎(𝑘)𝑢(𝑘)+𝐷_𝜎(𝑘)𝑤(𝑘),𝑥(𝑘)=𝜑(𝑘),𝑘=−𝑑,−𝑧(𝑘)𝑑+1,…,0,=𝐶_𝜎(𝑘)𝑥(𝑘)+𝐸_𝜎(𝑘)𝑤(𝑘) (8) such that the resulting closed-loop system is positive and finite-time bounded with 𝑙₁-gain performance.

3. Main Results

3.1. Finite-Time Stability and Boundedness Analysis

This section will focus on the problem of finite-time boundedness for positive switched delta operator system (2).

Theorem 11. Consider positive system (2); for a given positive constant 𝛼>0, a sampling period 𝑇, a time constant 𝑇_𝑓, and two vectors 𝜍≻𝜀≻0, if there exist positive vectors 𝜈_𝑝 and 𝜐_𝑝, for all 𝑝∈𝑚, and positive constants 𝜉₁, 𝜉₂, and 𝜉₃, such that the following inequalities hold: 𝐴^𝑇_𝑝𝜈_𝑝−𝛼𝜈_𝑝+(1+𝑇𝛼)𝑑−𝑑+1𝜐_𝑝⪯0,𝐴^𝑇_𝑑𝑝𝜈_𝑝−(1+𝑇𝛼)^𝑑+1𝜐_𝑝⪯0,𝜉₁𝜀≺𝜈_𝑝≺𝜉₂𝜍,𝜐_𝑝≺𝜉₃𝜍,(1+𝑇𝛼)^𝑇𝑓𝜉₂+𝜉₃𝑇𝑑(1+𝑇𝛼)^𝑑𝑑−𝑑+1+𝑇𝑑𝜛)<𝜉₁, (9)(10)(11)(12) where 𝜈_𝑝=[𝜈_𝑝1,𝜈_𝑝2,…,𝜈_𝑝𝑛]^𝑇 and 𝜐_𝑝=[𝜐_𝑝1,𝜐_𝑝2,…,𝜐_𝑝𝑛]^𝑇, then the system is finite-time bounded with respect to (𝜍,𝜀,𝑇_𝑓,𝑑,𝜎(𝑘)) under the following average dwell time scheme:𝜏_𝑎>𝜏^∗_𝑎=𝑇_𝑓ln𝜇ln𝜉₁−ln(1+𝑇𝛼)^𝑇𝑓𝜉₂+𝜉₃𝑇𝑑(1+𝑇𝛼)^𝑑𝑑−𝑑,+1+𝑇𝑑𝜛 (13)where 𝜛=max_{(𝑟,𝑝)∈𝑤×𝑚}(𝜛_𝑝𝑟), 𝑤={1,2,…,𝑤}, 𝜛_𝑝𝑟 is the 𝑟th element of the vector 𝜛_𝑝=𝜉₂𝐷^𝑇_𝑝𝜍, and 𝜇≥1 satisfies 𝜈_𝑝⪯𝜇𝜈_𝑞,𝜐_𝑝⪯𝜇𝜐_𝑞,∀𝑝,𝑞∈𝑚. (14)

Proof. Choose the following piecewise copositive type Lyapunov functional for positive system (2): 𝑉(𝑘,𝑥(𝑘))=𝑉_𝜎(𝑘)(𝑘,𝑥(𝑘)). (15) The form of each 𝑉_𝑝(𝑘,𝑥(𝑘))(∀𝑝∈𝑚) is given by 𝑉_𝑝(𝑘,𝑥(𝑘))=𝑉_𝑝1(𝑘,𝑥(𝑘))+𝑉_𝑝2(𝑘,𝑥(𝑘))+𝑉_𝑝3(𝑘,𝑥(𝑘)), (16) where 𝑉_𝑝1(𝑘,𝑥(𝑘))=𝑥^𝑇(𝑘)𝜈_𝑝,𝑉_𝑝2(𝑘,𝑥(𝑘))=𝑇^𝑘−1_{𝑠=𝑘−𝑑𝑘}(1+𝑇𝛼)^𝑘−𝑠𝑥^𝑇(𝑠)𝜐_𝑝,𝑉_𝑝3(𝑘,𝑥(𝑘))=𝑇^−𝑑_{𝑙=−𝑑+1}^𝑘−1_{𝑠=𝑘+𝑙}(1+𝑇𝛼)^𝑘−𝑠𝑥^𝑇(𝑠)𝜐_𝑝,∀𝑝∈𝑚. (17)
For simplicity, 𝑉_𝑝(𝑘,𝑥(𝑘)) is written as 𝑉_𝑝(𝑘) (correspondingly, 𝑉(𝑘,𝑥(𝑘)) is written as 𝑉(𝑘)) in the later section of the paper.
Along the state decay of positive system (2), the Lyapunov function in delta domain has the following form: 𝛿𝑉_𝑝1((𝑘𝑥𝑘,𝑥))=𝛿^𝑇(𝑘)𝜈_𝑝=𝛿𝑥^𝑇(𝑘)𝜈_𝑝=𝑥^𝑇(𝑘)𝐴^𝑇_𝑝𝜈_𝑝+𝑥^𝑇𝑘−𝑑_𝑘𝐴^𝑇_𝑑𝑝𝜈_𝑝+𝑤^𝑇(𝑘)𝐷^𝑇_𝑝𝜈_𝑝𝛿𝑉_𝑝2((𝑘=1𝑘,𝑥))𝑇𝑉_𝑝2(𝑘+1)−𝑉_𝑝2(𝑘=1)𝑇𝑇^(𝑘+1)−1_{𝑠=𝑘+1−𝑑𝑘+1}(1+𝑇𝛼)^{𝑘+1−𝑠}𝑥^𝑇(𝑠)𝜐_𝑝−𝑇^𝑘−1_{𝑠=𝑘−𝑑𝑘}(1+𝑇𝛼)^𝑘−𝑠𝑥^𝑇(𝑠)𝜐_𝑝≤𝑇𝛼^𝑘−1_{𝑠=𝑘−𝑑𝑘}(1+𝑇𝛼)^𝑘−𝑠𝑥^𝑇(𝑠)𝜐_𝑝+(1+𝑇𝛼)𝑥^𝑇(𝑘)𝜐_𝑝−(1+𝑇𝛼)^𝑑+1𝑥^𝑇𝑘−𝑑_𝑘𝜐_𝑝+^𝑘−𝑑_{𝑠=𝑘+1−𝑑}(1+𝑇𝛼)^{𝑘+1−𝑠}𝑥^𝑇(𝑠)𝜐_𝑝𝛿𝑉_𝑝3((𝑘=1𝑘,𝑥))𝑇𝑉_𝑝3(𝑘+1)−𝑉_𝑝3(𝑘=1)𝑇𝑇^−𝑑_{𝑙=−𝑑+1}^(𝑘+1)−1_{𝑠=𝑘+1+𝑙}(1+𝑇𝛼)^{𝑘+1−𝑠}𝑥^𝑇(𝑠)𝜐_𝑝−𝑇^−𝑑_{𝑙=−𝑑+1}^𝑘−1_{𝑠=𝑘+𝑙}(1+𝑇𝛼)^𝑘−𝑠𝑥^𝑇(𝑠)𝜐_𝑝=𝑇𝛼^−𝑑_{𝑙=−𝑑+1}^𝑘−1_{𝑠=𝑘+𝑙}(1+𝑇𝛼)^𝑘−𝑠𝑥^𝑇(𝑠)𝜐_𝑝+(1+𝑇𝛼)𝑑−𝑑𝑥^𝑇(𝑘)𝜐_𝑝−^𝑘−𝑑_{𝑠=𝑘+1−𝑑}(1+𝑇𝛼)^{𝑘+1−𝑠}𝑥^𝑇(𝑠)𝜐_𝑝. (18) According to (18), we have 𝛿𝑉_𝑝(𝑘,𝑥(𝑘))−𝛼𝑉_𝑝(𝑘,𝑥(𝑘))≤𝑥^𝑇(𝑘)𝐴^𝑇_𝑝𝜈_𝑝+𝑥^𝑇𝑘−𝑑_𝑘𝐴^𝑇_𝑑𝑝𝜈_𝑝+𝑤^𝑇(𝑘)𝐷^𝑇_𝑝𝜈_𝑝−𝛼𝑥^𝑇(𝑘)𝜈_𝑝+(1+𝑇𝛼)𝑥^𝑇(𝑘)𝜐_𝑝−(1+𝑇𝛼)^𝑑+1𝑥^𝑇𝑘−𝑑_𝑘𝜐_𝑝+^𝑘−𝑑_{𝑠=𝑘+1−𝑑}(1+𝑇𝛼)^{𝑘+1−𝑠}𝑥^𝑇(𝑠)𝜐_𝑝+(1+𝑇𝛼)𝑑−𝑑𝑥^𝑇(𝑘)𝜐_𝑝−^𝑘−𝑑_{𝑠=𝑘+1−𝑑}(1+𝑇𝛼)^{𝑘+1−𝑠}𝑥^𝑇(𝑠)𝜐_𝑝≤𝑥^𝑇(𝑘)𝐴^𝑇_𝑝𝜈_𝑝−𝛼𝜈_𝑝+(1+𝑇𝛼)𝑑−𝑑+1𝜐_𝑝+𝑥^𝑇𝑘−𝑑_𝑘𝐴^𝑇_𝑑𝑝𝜈_𝑝−(1+𝑇𝛼)^𝑑+1𝜐_𝑝+𝑤^𝑇(𝑘)𝐷^𝑇_𝑝𝜈_𝑝. (19) According to (9)–(11), we can easily obtain 𝛿𝑉_𝑝(𝑘)−𝛼𝑉_𝑝(𝑘)≤𝑤^𝑇(𝑘)𝐷^𝑇_𝑝𝜈_𝑝≤𝑤^𝑇(𝑘)𝐷^𝑇_𝑝𝜍𝜉₂. (20) Denoting 𝜛_𝑝=𝐷^𝑇_𝑝𝜍𝜉₂, it follows from (20) that 𝛿𝑉_𝑝(𝑘)−𝛼𝑉_𝑝(𝑘)≤𝑤^𝑇(𝑘)𝜛_𝑝⇒𝛿𝑉_𝑝(𝑘)=𝑉_𝑝(𝑘+1)−𝑉_𝑝(𝑘)𝑇≤𝛼𝑉_𝑝(𝑘)+𝑤^𝑇(𝑘)𝜛_𝑝⇒𝑉_𝑝(𝑘+1)≤𝑉_𝑝(𝑘)+𝑇𝛼𝑉_𝑝(𝑘)+𝑇𝑤^𝑇(𝑘)𝜛_𝑝⇒𝑉_𝑝(𝑘+1)≤(1+𝑇𝛼)𝑉_𝑝(𝑘)+𝑇𝑤^𝑇(𝑘)𝜛_𝑝. (21) Then, along the trajectory of positive system (2), for 𝑘∈[𝑘_𝑖,𝑘_𝑖+1), 𝑉_𝜎(𝑘)(𝑘)≤(1+𝑇𝛼)^{𝑘−𝑘𝑖}𝑉_{𝜎(𝑘𝑖)}𝑘_𝑖+^𝑘−1_{𝑠=𝑘𝑖}(1+𝑇𝛼)^{𝑘−𝑠−1}𝑇𝑤^𝑇(𝑠)𝜛_𝑝. (22) From (14) and (16), we can obtain 𝑉_{𝜎(𝑘𝑖)}𝑘_𝑖≤𝜇𝑉_𝜎(𝑘⁻_𝑖)𝑘⁻_𝑖. (23)
Let 𝑀 be the switching number of 𝜎(𝑘) over [0,𝑇_𝑓), and denote 𝑘₁,𝑘₂,𝑘₃,…,𝑘_𝑀 as the switching instants over the interval [0,𝑇_𝑓). Then, for 𝑘∈[0,𝑇_𝑓), we obtain from (22), and (23) and the relation 𝑁_𝜎(𝑘)(0,𝑘)≤𝑘/𝜏_𝑎 that 𝑉(𝑘)≤(1+𝑇𝛼)^{(𝑘−𝑘𝑀)}𝑉_{𝜎(𝑘𝑀)}𝑘_𝑀+^𝑘−1_{𝑠=𝑘𝑀}(1+𝑇𝛼)^{(𝑘−𝑠−1)}𝑇𝑤^𝑇(𝑠)𝜛_{𝜎(𝑘𝑀)}≤𝜇(1+𝑇𝛼)^{(𝑘−𝑘𝑀)}𝑉_𝜎(𝑘⁻_𝑀)𝑘⁻_𝑀+^𝑘−1_{𝑠=𝑘𝑀}(1+𝑇𝛼)^{(𝑘−𝑠−1)}𝑇𝑤^𝑇(𝑠)𝜛_{𝜎(𝑘𝑀)}≤𝜇(1+𝑇𝛼)^{(𝑘−𝑘𝑀−1)}𝑉_{𝜎(𝑘𝑀−1)}𝑘_𝑀−1+𝜇^𝑘𝑀−1_{𝑠=𝑘𝑀−1}(1+𝑇𝛼)^{(𝑘−𝑠−1)}𝑇𝑤^𝑇(𝑠)𝜛_{𝜎(𝑘𝑀−1)}+^𝑘−1_{𝑠=𝑘𝑀}(1+𝑇𝛼)^{(𝑘−𝑠−1)}𝑇𝑤^𝑇(𝑠)𝜛_{𝜎(𝑘𝑀)}≤⋯≤𝜇^𝑀(1+𝑇𝛼)^𝑘𝑉_𝜎(0)(0)+𝜇^{𝑀𝑘1−1}_𝑠=0(1+𝑇𝛼)^{(𝑘−𝑠−1)}𝑇𝑤^𝑇(𝑠)𝜛_𝜎(0)+𝜇^{𝑘𝑀−12−1}_𝑠=𝑘1(1+𝑇𝛼)^{(𝑘−𝑠−1)}𝑇𝑤^𝑇(𝑠)𝜛_𝜎(𝑘1)+⋯+^𝑘−1_{𝑠=𝑘𝑀}(1+𝑇𝛼)^{(𝑘−𝑠−1)}𝑇𝑤^𝑇(𝑠)𝜛_{𝜎(𝑘𝑀)}=𝜇^𝑀(1+𝑇𝛼)^𝑇𝑓𝑉_𝜎(0)(0)+^𝑘−1_𝑠=0𝜇^{𝑁𝜎(𝑘)(𝑠,𝑘)}(1+𝑇𝛼)^{(𝑘−𝑠−1)}𝑇𝑤^𝑇(𝑠)𝜛_𝜎(𝑠)≤𝜇^𝑀(1+𝑇𝛼)^𝑇𝑓𝑉_𝜎(0)(0)+𝜇^𝑀(1+𝑇𝛼)^{𝑇𝑓𝑘−1}_𝑠=0𝑇𝑤^𝑇(𝑠)𝜛_𝜎(𝑠)≤𝜇^𝑀(1+𝑇𝛼)^𝑇𝑓𝑉_𝜎(0)(0)+𝑇𝑑𝜛. (24) Considering the definition of 𝑉_𝜎(𝑘)(𝑘), it yields that 𝑉_𝜎(𝑘)(𝑘)≥𝜉₁𝑥^𝑇(𝑘)𝑉𝜀,_𝜎(0)(0)≤𝜉₂𝑥^𝑇(0)𝜍+𝜉₃𝑇𝑑(1+𝑇𝛼)^𝑑×sup_{−𝑑≤𝜃≤−1}𝑥^𝑇(𝜃)𝜍+𝑑−𝑑sup_{−𝑑+1≤𝜃≤−1}𝑥^𝑇(𝜃)𝜍≤𝜉₂+𝜉₃𝑇𝑑(1+𝑇𝛼)^𝑑𝑑−𝑑+1sup_{−𝑑≤𝜃≤0}𝑥^𝑇(𝜃)𝜍≤𝜉₂+𝜉₃𝑇𝑑(1+𝑇𝛼)^𝑑𝑑−𝑑+1. (25) Combining (24)-(25), we obtain 𝑥^𝑇(𝑘)1𝜀≤𝜉₁𝜇^1/𝜏𝑎(1+𝑇𝛼)^𝑇𝑓×𝜉₂+𝜉₃𝑇𝑑(1+𝑇𝛼)^𝑑𝑑−𝑑+1+𝑇𝑑𝜛. (26) Substituting (13) into (26) leads to 𝑥^𝑇(𝑘)𝜀<1. (27)
According to Definition 8, we can conclude that the positive switched delta operator system (2) is finite-time bounded with respect to (𝜍,𝜀,𝑇_𝑓,𝑑,𝜎(𝑘)).
The proof is completed.

Remark 12. It should be noted that there is no requirement of negative definitiveness on 𝛿𝑉(𝑘), which is different from the case of classical Lyapunov stability.

Remark 13. When 𝜇=1 in (14), which leads to 𝜈_𝑝=𝜈_𝑞, 𝜐_𝑝=𝜐_𝑞, for all 𝑝,𝑞∈𝑚, and 𝜏^∗_𝑎=0 by (13), system (2) possesses a common co-positive type Lyapunov-Krasovskii functional, and the switching signal can be arbitrary.
When 𝑤(𝑘)=0, the result on finite-time stability can be obtained as follows.

Corollary 14. Consider positive system (2) with 𝑤(𝑘)=0; for a given positive constant 𝛼>0, a sampling period 𝑇, a time constant 𝑇_𝑓, and two vectors 𝜍≻𝜀≻0, if there exist positive vectors 𝜈_𝑝 and 𝜐_𝑝, for all 𝑝∈𝑚, and positive constants 𝜉₁, 𝜉₂, and 𝜉₃, such that (9)–(11) and the following inequality hold: (1+𝑇𝛼)^𝑇𝑓𝜉₂+𝜉₃𝑇𝑑(1+𝑇𝛼)^𝑑𝑑−𝑑+1<𝜉₁, (28) then, the system is finite-time stable with respect to (𝜍,𝜀,𝑇_𝑓,𝜎(𝑘)) under the following average dwell time scheme𝜏_𝑎>𝜏^∗_𝑎=𝑇_𝑓ln𝜇ln𝜉₁−ln(1+𝑇𝛼)^𝑇𝑓𝜉₂+𝜉₃𝑇𝑑(1+𝑇𝛼)^𝑑𝑑−𝑑,+1 (29)where 𝜇≥1 satisfies (14).

Remark 15. As stated in [38], the general idea of finite-time stability concerns the boundedness of the state of a system over a finite interval for given initial conditions. On the other hand, the idea of finite-time boundedness concerns the behaviors of the state in the presence of both given initial conditions and external disturbances. It is easy to see from Definitions 6 and 8 that finite-time stability can be regarded as a special case by setting 𝑑=0.

Remark 16. When the sampling period 𝑇→0, positive system (2) becomes a continuous-time positive system as (5). In this case, Theorem 11 will generate to Theorem 1 proposed in [38].

Let 𝐴_𝜎(𝑘)=𝐴_𝜎(𝑘)+𝐼. When the sampling period 𝑇=1, positive system (2) becomes a discrete-time positive system as follows: 𝑥(𝑘+1)=𝐴_𝜎(𝑘)𝑥(𝑘)+𝐴_{𝑑𝜎(𝑘)}𝑥𝑘−𝑑_𝑘+𝐷_𝜎(𝑘)𝑤(𝑘),𝑥(𝑘)=𝜑(𝑘),𝑘=−𝑑,−𝑧(𝑘)𝑑+1,…,0,=𝐶_𝜎(𝑘)𝑥(𝑘)+𝐸_𝜎(𝑘)𝑤(𝑘), (30) where 𝐴_𝑝≽0, 𝐴_𝑑𝑝≽0, 𝐶_𝑝≽0, and 𝐸_𝑝≽0, for all 𝑝∈𝑚. Then we have the following result.

Corollary 17. Consider positive system (30); for a given positive constant 𝛼>0, a time constant 𝑇_𝑓, and two vectors 𝜍≻𝜀≻0, if there exist positive vectors 𝜈_𝑝 and 𝜐_𝑝, for all 𝑝∈𝑚, and positive constants 𝜉₁, 𝜉₂, and 𝜉₃, such that (11) and the following inequalities hold: 𝐴^𝑇_𝑝𝜈_𝑝−𝛼𝜈_𝑝+(1+𝛼)𝑑−𝑑+1𝜐_𝑝𝐴⪯0,_𝑇𝑑𝑝𝜈_𝑝−(1+𝛼)^𝑑+1𝜐_𝑝()⪯0,1+𝛼^𝑇𝑓𝜉₂+𝜉₃𝑑(1+𝛼)^𝑑𝑑−𝑑+1+𝑑𝜛<𝜉₁, (31) then the system is finite-time bounded with respect to (𝜍,𝜀,𝑇_𝑓,𝑑,𝜎(𝑘)) under the following average dwell time scheme:𝜏_𝑎>𝜏^∗_𝑎=𝑇_𝑓ln𝜇ln𝜉₁−ln(1+𝛼)^𝑇𝑓𝜉₂+𝜉₃𝑑(1+𝛼)^𝑑𝑑−𝑑,+1+𝑑𝜛 (32)where 𝜛 has been defined in Theorem 11 and 𝜇≥1 satisfies (14).

3.2. 𝑙₁-Gain Finite-Time Boundedness Analysis

In this section, we will consider the problem of 𝑙₁-gain finite-time boundedness for positive system (2).

Theorem 18. Consider positive system (2); for a given positive constant 𝛼>0, a sampling period 𝑇, a given time constant 𝑇_𝑓 and two vectors 𝜍≻𝜀≻0, if there exist positive vectors 𝜈_𝑝 and 𝜐_𝑝, for all 𝑝∈𝑚, and positive constants 𝛾, 𝜉₁, 𝜉₂ and 𝜉₃, such that (10), (11), and the following inequalities hold: 𝐴^𝑇_𝑝𝜈_𝑝−𝛼𝜈_𝑝+(1+𝑇𝛼)𝑑−𝑑+1𝜐_𝑝+𝑐_𝑝̃⪯0,𝐷^𝑇_𝑝𝜈_𝑝+𝑒_𝑝̃−𝛾1_𝑤⪯0,𝜛<𝛾,(1+𝑇𝛼)^𝑇𝑓𝜉₂+𝜉₃𝑇𝑑(1+𝑇𝛼)^𝑑𝑑−𝑑+1+𝑇𝑑𝛾<𝜉₁, (33)(34)(35) where 𝜛 has been defined in Theorem 11, 𝜈_𝑝=[𝜈_𝑝1,𝜈_𝑝2,…,𝜈_𝑝𝑛]^𝑇, 𝜐_𝑝=[𝜐_𝑝1,𝜐_𝑝2,…,𝜐_𝑝𝑛]^𝑇, 1_𝑤=[1,1,…,1_𝑤]^𝑇, 𝑐_𝑝̃=[‖𝑐_𝑝1‖,‖𝑐_𝑝2‖,…,‖𝑐_𝑝𝑛‖]^𝑇, 𝑐_𝑝𝑗 represents the 𝑗th column of matrix 𝐶_𝑝, 𝑗∈𝑛={1,2,…,𝑛}. 𝑒_𝑝̃=[‖𝑒_𝑝1‖,‖𝑒_𝑝2‖,…,‖𝑒_𝑝𝑤‖]^𝑇, 𝑒_𝑝𝑗 represents the 𝑗th column of matrix 𝐸_𝑝, and  𝑗∈𝑤={1,2,…,𝑤}, then the system is 𝑙₁-gain finite-time bounded with respect to (𝜍,𝜀,𝑇_𝑓,𝑑,𝜎(𝑘)) under the following average dwell time scheme: 𝜏_𝑎>𝜏^∗_𝑎=max𝑇_𝑓×ln𝜇ln𝜉₁−ln1+𝑇𝛼^𝑇𝑓×𝜉₂+𝜉₃𝑇𝑑(1+𝑇𝛼)^𝑑𝑑−𝑑+1+𝑇𝑑𝛾⁻¹,ln𝜇(),ln1+𝑇𝛼 (36) where 𝜇≥1 satisfies (14).

Proof. Equation (9) can be directly derived from (33). By Theorem 11, we can easily obtain from (10)-(11) and (33)–(35) that system (2) is finite-time bounded with respect to (𝜍,𝜀,𝑇_𝑓,𝑑,𝜎(𝑘)). To show the weighted 𝑙₁-gain performance, we choose the piecewise co-positive type Lyapunov-Krasovskii functional (15). Following the proof line of Theorem 11, we can get from (10), (33), and (34) that 𝛿𝑉_𝑝(𝑘)−𝛼𝑉_𝑝(𝑘)+‖𝑧(𝑘)‖−𝛾‖𝑤(𝑘)‖<0. (37)
Let Λ(𝑠)=𝑇‖𝑧(𝑠)‖−𝑇𝛾‖𝑤(𝑠)‖; then, for 𝑘∈[𝑘_𝑖,𝑘_𝑖+1), (37) gives rise to 𝑉(𝑘)≤(1+𝑇𝛼)^{(𝑘−𝑘𝑖)}𝑉_{𝜎(𝑘𝑖)}𝑘_𝑖−^𝑘−1_{𝑠=𝑘𝑖}(1+𝑇𝛼)^{(𝑘−𝑠−1)}Λ(𝑠). (38) Following the proof line of (24), for any 𝑘∈[0,𝑇_𝑓), we can obtain 𝑉_𝜎(𝑘)(𝑘)≤𝜇^{𝑁𝜎(𝑘)(0,𝑘)}(1+𝑇𝛼)^𝑘𝑉_𝜎(0)(0)−^𝑘−1_𝑠=0𝜇^{𝑁𝜎(𝑘)(𝑠,𝑘)}(1+𝑇𝛼)^{(𝑘−𝑠−1)}Λ(𝑠). (39) Under the zero initial condition, we have from (39) that 0≤−^𝑘−1_𝑠=0𝜇^{𝑁𝜎(𝑘)(𝑠,𝑘)}(1+𝑇𝛼)^{(𝑘−𝑠−1)}Λ(𝑠). (40) Namely, ^𝑘−1_𝑠=0𝜇^{𝑁𝜎(𝑘)(𝑠,𝑘)}(1+𝑇𝛼)^{(𝑘−𝑠−1)}‖𝑧(𝑠)‖≤𝛾^𝑘−1_𝑠=0𝜇^{𝑁𝜎(𝑘)(𝑠,𝑘)}(1+𝑇𝛼)^{(𝑘−𝑠−1)}‖𝑤(𝑠)‖. (41) Multiplying both sides of (41) by 𝜇^{−𝑁𝜎(𝑘)(0,𝑘)} yields ^𝑘−1_𝑠=0𝜇^{−𝑁𝜎(𝑘)(0,𝑠)}(1+𝑇𝛼)^{(𝑘−𝑠−1)}‖𝑧(𝑠)‖≤𝛾^𝑘−1_𝑠=0𝜇^{−𝑁𝜎(𝑘)(0,𝑠)}(1+𝑇𝛼)^{(𝑘−𝑠−1)}‖𝑤(𝑠)‖. (42)
Noticing that 𝑁_𝜎(𝑘)(0,𝑠)≤𝑠/𝜏_𝑎 and 𝜏_𝑎>ln𝜇/ln(1+𝑇𝛼), we have 𝜇^{−𝑁𝜎(𝑘)(0,𝑠)}≥(1+𝑇𝛼)^−𝑠. (43) Combining (42) and (43) leads to ^𝑘−1_𝑠=0(1+𝑇𝛼)^−𝑠(1+𝑇𝛼)^{(𝑘−𝑠−1)}‖𝑧(𝑠)‖≤𝛾^𝑘−1_𝑠=0(1+𝑇𝛼)^{(𝑘−𝑠−1)}‖𝑤(𝑠)‖. (44) Let 𝑘−1=𝑇_𝑓; then multiplying both sides of (44) by (1+𝑇𝛼)^−𝑇𝑓 leads to ^𝑇𝑓_𝑠=0(1+𝑇𝛼)^−2𝑠‖𝑧(𝑠)‖≤𝛾^𝑇𝑓_𝑠=0‖𝑤(𝑠)‖. (45)
Setting (1+𝑇𝛼)²=1+𝑇𝜌, we can obtain that positive system (2) is 𝑙₁-gain finite-time bounded with respect to (𝜍,𝜀,𝑇_𝑓,𝑑,𝜎(𝑘)).
The proof is completed.

3.3. 𝑙₁-Gain Controller Design

In this section, we are interested in designing a state feedback controller 𝑢(𝑘)=𝐾_𝜎(𝑘)𝑥(𝑘) for positive switched delta operator system (8) such that the corresponding closed-loop system 𝛿𝑥(𝑘)=𝐴_𝜎(𝑘)+𝐵_𝜎(𝑘)𝐾_𝜎(𝑘)𝑥(𝑘)+𝐴_{𝑑𝜎(𝑘)}𝑥𝑘−𝑑_𝑘+𝐷_𝜎(𝑘)𝑤(𝑘),𝑥(𝑘)=𝜑(𝑘),𝑘=−𝑑,−𝑧(𝑘)𝑑+1,…,0,=𝐶_𝜎(𝑘)𝑥(𝑘)+𝐸_𝜎(𝑘)𝑤(𝑘), (46) is 𝑙₁-gain finite-time bounded with respect to (𝜍,𝜀,𝑇_𝑓,𝑑,𝜎(𝑘)). By Lemma 3, in order to guarantee the positivity of system (46), 𝐼+𝑇(𝐴_𝑝+𝐵_𝑝𝐾_𝑝)≽0, for all 𝑝∈𝑚, should be satisfied.

Theorem 19. Consider positive system (8); for a given positive constant 𝛼>0, a sampling period 𝑇, a given time constant 𝑇_𝑓, and two vectors 𝜍≻𝜀≻0, if there exist positive vectors 𝜈_𝑝 and 𝜐_𝑝, for all 𝑝∈𝑚, and positive constants 𝛾, 𝜉₁, 𝜉₂, and 𝜉₃, such that (10)-(11), (34)-(35), and the following conditions are satisfied: 𝐴^𝑇_𝑝𝜈_𝑝−𝛼𝜈_𝑝+𝑔_𝑝+(1+𝑇𝛼)𝑑−𝑑+1𝜐_𝑝+𝑐_𝑝̃⪯0,𝐼+𝑇𝐴_𝑝+𝐵_𝑝𝐾_𝑝≽0, (47)(48) where 𝑔_𝑝=𝐾^𝑇_𝑝𝐵^𝑇_𝑝𝜈_𝑝, then the resulting closed-loop system (46) is 𝑙₁-gain finite-time bounded with respect to (𝜍,𝜀,𝑇_𝑓,𝑑,𝜎(𝑘)) under the average dwell time scheme (36), where 𝜇≥1 satisfies (14).

Proof. Denote 𝑔_𝑝=𝐾^𝑇_𝑝𝐵^𝑇_𝑝𝜈_𝑝. Following the proof line of Theorem 18, one can exactly obtain Theorem 19. It is omitted here.
The proof is completed.

Based on Theorem 19, we are now in a position to present an effective algorithm for constructing the desired controller.

Algorithm 20.

Step 1. Input the matrices 𝐴_𝑝, 𝐴_𝑑𝑝, 𝐵_𝑝, 𝐶_𝑝, 𝐷_𝑝, and 𝐸_𝑝.

Step  2. Choose the parameters 𝛼>0 and 𝛾>0. By solving (10)-(11), (34)-(35), and (47), one can obtain the solutions of 𝜈_𝑝, 𝜐_𝑝, 𝜗_𝑝, and 𝑔_𝑝.

Step  3. By the equation 𝑔_𝑝=𝐾^𝑇_𝑝𝐹^𝑇_𝑝𝜈_𝑝 with the obtained 𝑔_𝑝 and 𝜈_𝑝, one can get the gain matrices 𝐾_𝑝.

Step  4. Check condition (48) in Theorem 19. If it holds, go to Step 5; otherwise, adjust the parameter 𝛼 and return to Step 2.

Step  5. Construct the feedback controller 𝑢(𝑘)=𝐾_𝑝𝑥(𝑘), where 𝐾_𝑝 are the gain matrices.

4. Numerical Example

Consider positive switched delta operator system (2) consisting of two subsystems described by

subsystem 1: 𝐴₁=−412−3.5,𝐴_𝑑1=0.20.10.10.2,𝐵₁=,𝐶0.20.1₁=0.10.2,𝐷₁=0.10.2,𝐸₁=[0.1], (49)

subsystem 2: 𝐴₂=−4.522−3,𝐴_𝑑2=0.30.20.10.2,𝐵₂=,𝐶0.10.2₂=0.10.1,𝐷₂=0.20.1,𝐸₂=[0.2]. (50)

Choosing 𝑇=0.2, 𝑇_𝑓=15, 𝛼=0.4, 𝛾=10, 𝑑=2.0, 𝑑=0, 𝑑=0.01, 𝜍=36^𝑇, and 𝜀=0.010.01^𝑇, then, by solving (10)-(11), (34)-(35), and (47) in Theorem 19, we can obtain the following solutions: 𝜈₁=17.962020.8488,𝜐₁=7.365611.2371,𝑔₁=,𝜈6.685413.3700₂=17.829424.6985,𝜐₂=8.31059.9876,𝑔₂=,5.47067.9281 (51) and the state feedback gain matrices can be given by 𝑔_𝑝=𝐾^𝑇_𝑝𝐵^𝑇_𝑝𝜈_𝑝 as follows: 𝐾₁=1.17762.3550,𝐾₂=0.81381.1793. (52)

Obviously, condition (48) is satisfied. From (14) and (36), we have 𝜇=1.1961 and 𝜏^∗_𝑎=2.3272. Choosing 𝜏_𝑎=3, the simulation results are shown in Figures 1–4, where the initial conditions are 𝑥(0)=0.10.12^𝑇 and  𝑥(𝜃)=00^𝑇, 𝜃=−2,−1,0, and the exogenous disturbance input is 𝑤(𝑘)=0.001𝑒^−0.5𝑘 which satisfies (3). The switching signal with average dwell time 𝜏_𝑎=3 is shown in Figure 1 and the state response of the closed-loop system is given in Figure 2. The corresponding control law is given in Figures 3 and 4 plots the evolution of 𝑥^𝑇(𝑘)𝜀.

Figure 1 

Switching signal.

Figure 2 

State response of the closed-loop system.

Figure 3 

State feedback control law .

Figure 4 

The evolution of .

It is easy to see from Figures 1–4 that 𝑥^𝑇(𝑘)𝜀<1, for all 𝑘∈[0,𝑇_𝑓), is satisfied for the given time constant 𝑇_𝑓=15. Thus, according to Definition 9, the closed-loop system is 𝑙₁-gain finite-time bounded with respect to (𝜍,𝜀,𝑇_𝑓,𝑑,𝜎(𝑘)).

5. Conclusions

In this paper, finite-time boundedness and 𝑙₁-gain finite-time boundedness for a class of positive switched systems with time-varying delays via delta operator approach have been investigated. By constructing a co-positive type Lyapunov-Krasovskii functional and using the average dwell time approach, we have proposed sufficient conditions to ensure the finite-time stability and 𝑙₁-gain finite-time boundedness of the considered system. Based on the results obtained, a state feedback controller and a class of switching signals with the average dwell time are designed to guarantee that the corresponding closed-loop system is finite-time stable with an 𝑙₁-gain performance. Finally, a numerical example is presented to demonstrate the feasibility of the obtained results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant no. 61273120.