Leader-Following Protocol Design for Switched Multiagent Systems with Randomly Occurring Self-Delay

Abstract

This paper designs a delay-dependent leader-following protocol for multiagent systems with both communication delay and randomly occurring self-delay. Based on the randomly occurring mode of self-delay, a new model of switched multi-agent systems which have not been introduced yet is constructed. By constructing a newly piecewise Lyapunov-Krasovskii functional, a leader-following stability criterion of the switched multi-agent systems is derived by the framework of linear matrix inequalities (LMIs) and the average dwell time with the randomly occurring mode. Based on the result of the derived stability criterion, a designing leader-following protocol for the system will be proposed. One numerical example is included to show the effectiveness of the proposed method.

1. Introduction

During the last few years, Markovian jumping mode [1, 2], the randomly occurring mode with probabilistic property [3–5], the average dwell time approach [6–9], the indicator function definition with arbitrary mode [10], and other changing or switching signal modes are being put to treat in the various systems with changing or switching parameters. Among them, one pays close attention to the average dwell time approach with randomly occurring mode for the following two reasons.(i)It is a well-known fact that when all linear systems in the family 𝑥̇(𝑡)=𝐴_𝑝𝑥(𝑡), 𝑝∈𝒫={1,2,…,𝑚} are stable, the switched linear system 𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡), where 𝜎(𝑡) is switching signal, is stable if the dwell time is sufficiently large [6]. Thus, a more general result of dwell time is the average dwell time.(ii)Switching signals are generated based on the randomly occurring mode with probabilistic property. Use of probabilistic property vouches for the simulation condition in corresponding to ever-changing practice situation.

On the other hand, multi-agent systems (MASs) [11] have received considerable attention due to their extensive applications in many fields such as distributed sensor networks [12], vehicle systems [13, 14], groups of mobile autonomous agent [15], multi-agent robotic system [16], and other applications [17–19]. Before handling this system, since modern systems use information between each agent in networks, these days, we need to pay keen attention to the three following considerations.(i)A prime concern in this system is the agreement of a group of agent on their states of leader by interaction. Namely, this problem is a consensus problem with the leader.(ii)During the information exchange between each agent in networks, there exists communication delay. Also, in implementation of many practical systems such as aircraft and electric circuits, there exist occasionally stochastic perturbations. The perturbations have influence on the random occurrence of self-delay.(iii)It is well known that the time-delay often causes undesirable dynamic behaviors such as performance degradation and instability of various systems. Therefore, the study on stability analysis for systems with time-delay has been widely investigated [20–24]. Moreover, the analysis for systems with time-delay can be classified into two categories, namely, delay-dependent analysis and delay-independent analysis. Here, delay-dependent analysis has been paid more attention than delay-independent one because the sufficient conditions for delay-dependent analysis make use of the information on the size of time delay [20]. That is, the former is less conservative than the latter particularly when the size of time-delay is small.

In this regard, the MASs are being put to use in the consensus problem for time-delay which occurs due to the finite speed of information processing and/or amplifier switching in the implementation of this system. Thus, it is necessary to study the problems for MASs with time-delay [25–28]. However, the above-mentioned literature has mainly addressed the consensus conditions of the MASs with only communication delay. In the point of practical systems, the randomly occurring self-delay is no less important than the communication delay as a considerable factor affecting dynamics in the fields of network science and communication systems applications. Unfortunately, to the best of authors’ knowledge, the consensus problem for MASs with randomly occurring self-delay has not been tackled in any other literature, despite its potential and practical importance.

With this motivation mentioned above, in Section 2, based on the randomly occurring mode of self-delay, a new model of switched MASs is constructed for the first time. In Section 3, the problem to get a delay-dependent leader-following protocol criterion for the switched MASs with both communication delay and randomly occurring self-delay, which are interval time-varying delays, is considered. Here, delay-dependent stability or stabilization of system with interval time-varying delays has been a focused topic of theoretical and practical importance [29] in very recent years. The interval time-varying delays mean that the lower bounds of time-delay which guarantees the stability of system is not restricted to be zero. A networked control system is a typical example of dynamic systems with interval time-varying delays. So, the gist of the main points can be summarized as follows.(i)It is assumed that the self-delay is of probabilistic property with Bernoulli sequence, which is more realistic in practical systems. For the details, see Assumption 2.(ii)Switching signals are generated with not only the average dwell time, 𝜏_𝑎, but also the occurrence probabilistic of self-delay, Pr{𝜌(𝑡)=1}. Thus, in this paper, the new model of switched MASs is simulated with the probabilistic property with Bernoulli sequence. For the details, see the new model of switched MASs (11) in next section.

By construction of a newly piecewise Lyapunov-Krasovskii functional and utilization of some approaches, which are the average dwell time approach [6] and reciprocally convex approach [24] with some added decision variables, a new leader-following protocol criterion is derived in Theorem 12 with the LMI framework. The LMIs can be formulated as convex optimization algorithms which are amenable to computer solution [30]. Next, based on the results of Theorem 12, a leader-following protocol for the MASs with both communication delay and randomly occurring self-delay will be designed in Theorem 15. Finally, one numerical example is included to show the effectiveness of the proposed methods.

Notations. The notations used throughout this paper are fairly standard. ℝ^𝑛 is the 𝑛-dimensional Euclidean space, and ℝ^𝑚×𝑛 denotes the set of all 𝑚×𝑛 real matrices. For real symmetric matrices 𝑋 and 𝑌, 𝑋>𝑌 (resp., 𝑋≥𝑌) means that the matrix 𝑋−𝑌 is positive (resp., nonnegative) definite. 𝑋^⟂ denotes a basis for the null-space of 𝑋. 𝐼_𝑛, 0_𝑛, and 0_𝑚⋅𝑛 denote 𝑛×𝑛 identity matrix and 𝑛×𝑛 and 𝑚×𝑛 zero matrices, respectively. ‖⋅‖ refers to the Euclidean vector norm or the induced matrix norm. 𝜆_max{⋅} and 𝜆_min{⋅}, respectively, denote the maximal and minimal eigenvalues of a given square matrix. diag{⋯} denotes the block diagonal matrix. Pr{𝐴} means the occurrence probability of the event 𝐴. ⊗ denotes the notation of Kronecker product.

2. Problem Statements

The interaction topology of a network of agents is represented using a directed graph (digraph) 𝒢=(𝒱,ℰ,𝒜) with the set of nodes 𝒱={1,2,…,𝑁} and edges ℰ={(𝑖,𝑗)∶𝑖,𝑗∈𝒱}⊂𝒱×𝒱. An adjacency matrix 𝒜=[𝑎_𝑖𝑗]_𝑁×𝑁 of the digraph 𝒢 is the matrix with nonnegative elements satisfying 𝑎_𝑖𝑖=0 and 𝑎_𝑖𝑗≥0. If there is an edge between 𝑖 and 𝑗, then the elements of matrix 𝐴 are described as 𝑎_𝑖𝑗>0⇔(𝑖,𝑗)∈ℰ. The digraph 𝒢 is said to be undirected if (𝑖,𝑗)∈ℰ⇔(𝑗,𝑖)∈ℰ. A set of neighbors of agent 𝑖 is denoted by 𝒩_𝑖={𝑗∈𝒱∶(𝑖,𝑗)∈ℰ}. A degree of node 𝑖 is denoted by deg(𝑖)=∑_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗. A degree matrix of digraph 𝒢 is diagonal and defined as 𝒟=diag{deg(1),…,deg(𝑁)}. The Laplacian matrix ℒ of graph 𝒢 is defined as ℒ=𝒟−𝒜. More details can be seen in [31].

Consider the following MASs with the dynamics of agent 𝑖:𝑝̇_𝑖(𝑡)=𝑢_𝑖(𝑡),∀𝑖∈𝒱, (1) where 𝑁 is the number of agents, 𝒱={1,2,…,𝑁} is the index set, 𝑛 is the number of states of agent 𝑖, and 𝑝_𝑖(𝑡)∈ℝ^𝑛 and 𝑢_𝑖(𝑡)∈ℝ^𝑛 are the position vector and the leader-following protocol vector of agent 𝑖, respectively.

According to the work [32], a leader-following algorithm in agent can be described as 𝑢_𝑖(𝑡)=−𝐾_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑝_𝑖(𝑡)−𝑝_𝑗(𝑡)−𝐾𝑏_𝑖𝑝_𝑖(𝑡)−𝑝₀, (2) where 𝐾∈ℝ^𝑛×𝑛 is a protocol gain matrix, 𝑝₀∈ℝ^𝑛 is the position vector of leader, 𝑎_𝑖𝑗 and 𝑏_𝑖 are the interconnection weights defining 𝑎_𝑖𝑗>0 if agent 𝑖 is connected to agent 𝑗, 𝑎_𝑖𝑗=0, otherwise, and 𝑏_𝑖>0 if agent 𝑖 is connected to the leader, 𝑏_𝑖=0, otherwise.

Remark 1. The agents of the MASs (1) with the leader-following algorithm (2) can be known in the sense that each agent needs information from its local neighborhood. In practice, 𝑁-unmanned vehicles or 𝑁-soccer robots move in a 𝑛-dimensional plan with identical first-order dynamics. For example, the latitude and longitude coordinates of four vehicles shown in Figure 1, mean that 𝑁=4 and 𝑛=2.

Figure 1 

4 vehicles in 2-dimensional plan.

Hence, the protocol gain 𝐾 is of the form of matrix, which is general or flexible, rather than the scalar.

Here, it is assumed that communication delay and self-delay are as described in Figure 2.

Figure 2 

Concept for communication delay and self-delay.

Assumption 2. It is assumed that the self-delay is randomly occurring. This means that 𝜌(𝑡) is a probabilistic process representing the self-delay occurring process; that is, let 𝜌(𝑡) be a Bernoulli distributed sequence defined by 𝜌(𝑡)=1,iftheself-delayoccurs,0,iftheself-delaynotoccurs, (3) where 𝜌(𝑡) satisfies Pr{𝜌(𝑡)=1}=𝔼{𝜌(𝑡)}=𝜌₀, Pr{𝜌(𝑡)=0}=1−𝜌₀.

Remark 3. Commonly, in [33–35], the consensus problems with self-delay and without self-delay are addressed separately. However, we cannot regard that problem as something separate. Therefore, in this paper, the stochastic occurrence rate of self-delay is introduced in the consensus problem.

From algorithm (2) with Assumption 2, a leader-following algorithm with probabilistic self-delay can be 𝑢_𝑖(𝑡)=−𝐾_{𝜌(𝑡)𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑝_𝑖𝑡−𝜌(𝑡_Term1)ℎ(𝑡)−𝑝_𝑗(𝑡−ℎ(𝑡))_Term2−𝐾_𝜌(𝑡)𝑏_𝑖𝑝_𝑖(𝑡)−𝑝₀, (4) where 𝐾_𝜌(𝑡) is a protocol gain matrix at 𝜌(𝑡), the terms 1 and 2 are, respectively, the self-delay and the communication delay, and the time-delay satisfies 0<ℎ_𝑚≤ℎ(𝑡)≤ℎ_𝑀 and ℎ̇(𝑡)≤ℎ_𝐷, where ℎ_𝑚, ℎ_𝑀, and ℎ_𝐷 are known constants.

Remark 4. It is common for the Bernoulli process to be changed to just random variable which belongs to the index set of two elements {0,1}, with Pr{𝜌(𝑡)=1}=𝜌₀. Hence, the Bernoulli process seems to be like an on-off switch; that is, it has two possible configurations. On the other hand, the self-delay occurs by the calculational time delay or the processing limit time for the information in each agent. It depends on the real situation, and since the situation is random, the self-delay is explained with the Bernoulli process. Also, the proposed method in this work can also be extended to the MASs with randomly occurring communication delay given by 𝑢_𝑖(𝑡)=−𝐾_{𝜌(𝑡)𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑝_𝑖(𝑡)−𝑝_𝑗𝑡−𝜌(𝑡)ℎ(𝑡)−𝐾_𝜌(𝑡)𝑏_𝑖𝑝_𝑖(𝑡)−𝑝₀ (5) or 𝑢_𝑖(𝑡)=−𝐾_{𝜌(𝑡)𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑝_𝑖(𝑡−ℎ(𝑡))−𝑝_𝑗𝑡−𝜌(𝑡)ℎ(𝑡)−𝐾_𝜌(𝑡)𝑏_𝑖𝑝_𝑖(𝑡)−𝑝₀. (6) This work will be carried out in our future research.

Remark 5. In practice, the self- and communication delay can have different or the same values. If we consider the systems with these delays of different value, then the analysis for the systems would be very complex. For the details, let us define each range of these delays as 𝑑¹_𝐿≤𝑑₁(𝑡)≤𝑑¹_𝑈𝑑,forselfdelay,_2𝐿≤𝑑₂(𝑡)≤𝑑²_𝑈,forcommunicationdelay. (7) Here, to allow for simple analysis, the time-delay ℎ(𝑡) is set in the following range: min_𝑖=1,2𝑑^𝑖_𝐿≤ℎ(𝑡)≤max_𝑖𝑖=1,2𝑑^𝑖_𝑈. (8) Therefore, each interval of 𝑑_𝑖(𝑡) belongs to the interval of ℎ(𝑡) even if the intervals of 𝑑_𝑖(𝑡) are different. Thus, in this paper, the analysis results for the concerned system with same time-delay ℎ(𝑡) in self- and communication delay are guaranteed to the ones for the system with both these delays 𝑑_𝑖(𝑡), like the analysis for convex-hull. Moreover, in this paper, ℎ_𝑚=min_𝑖=1,2{𝑑^𝑖_𝐿} and ℎ_𝑀=max_𝑖=1,2{𝑑^𝑖_𝑈}.

Remark 6. Such time-delays can coexist in the different terms of leader. It should be pointed out that the proposed method in this paper can also be extended to the network with time-delay between agents and leader. This work will be done for our future research topic.
Let us define 𝑥_𝑖(𝑡)=𝑝_𝑖(𝑡)−𝑝₀. From (1) and (4), the MASs with the error dynamics for agent 𝑖 can be rewritten as the matrix form by 𝑥̇(𝑡)=−ℬ⊗𝐾_𝜌(𝑡)𝑥(𝑡)+𝒜⊗𝐾_𝜌(𝑡)𝑥((𝑡−𝑡−ℎ))𝒟⊗𝐾_𝜌(𝑡)𝑥𝑡−𝜌(𝑡)ℎ(𝑡), (9) where 𝒜=𝑎_{𝑖𝑗𝑁×𝑁},ℬ=diag𝑏₁,𝑏₂,…,𝑏_𝑁,𝒟=diag_𝑗∈𝒩1𝑎_1𝑗,_𝑗∈𝒩2𝑎_2𝑗,…,_{𝑗∈𝒩𝑁}𝑎_𝑁𝑗. (10)

Remark 7. The notation 𝜌(𝑡) in system (9) is used to express the randomly occurring self-delay with Bernoulli distributed sequence. Instead of 𝜌(𝑡), a notation 𝜎(𝑡) will be utilized as the switching signal when the system (9) is transformed into a new model of switched multi-agent systems. Moreover, compared with the existing works for the switched system, in this work, the switching rule between each mode is based on the probability property with Bernoulli distributed sequence.
For each 𝜌(𝑡), the system (9) can be rewritten as the following switched system: 𝑥̇(𝑡)=𝐹_𝜎(𝑡)⊗𝐾_𝜎(𝑡)𝑥(𝑡)+𝐺_𝜎(𝑡)⊗𝐾_𝜎(𝑡)𝑥(𝑡−ℎ(𝑡)), (11) where ℐ={1,2,…,𝑁} is the index set and 𝜎(𝑡)∶ℝ→ℐ is the switching signal which is deterministic, piecewise constant, and right continuous. Correspondingly, the switching sequence {(𝑖₀,𝑡₀),(𝑖₁,𝑡₁),…,(𝑖_𝑙,𝑡_𝑙),…,∣𝑖_𝑙∈ℐ,𝑙=0,1,…} with 𝑡₀=0 means that the 𝑖_𝑙th subsystem is activated when 𝑡∈[𝑡_𝑙,𝑡_𝑙+1), and, by understanding the considerations mentioned above, 𝐹₁=−ℬ,𝐺₁(𝑡)𝜌(𝑡),𝐹=𝒜−𝒟,for𝜎=1=1₂=−(ℬ+𝒟),𝐺₂=𝒜,for𝜎(𝑡)=2𝜌(𝑡)=0. (12) Also, the index set is defined as ℐ={1,2}, and, for each switching signal 𝜎(𝑡), we denote the matrices associated with mode 𝑙∈ℐ by 𝐹_𝜎(𝑡)=𝐹_𝑙, 𝐺_𝜎(𝑡)=𝐺_𝑙, and 𝐾_𝜎(𝑡)=𝐾_𝑙.

Remark 8. By representing the system (9) with probabilistic self-delay as the switching system (11), a deriving process of the criterion can be simple in analyzing the system (9) by utilizing the average dwell time approach [6]. Moreover, since the mathematical expectation operator 𝔼{⋅} used in [3–5] is not needed, the deriving work for the leader-following criterion for MASs (1) with the leader-following algorithm (4) will be simple and easy. It should be noted that this approach has not been proposed yet.
The aim of this paper is to design the gains 𝐾_𝑙, ∀𝑙∈ℐ, which guarantees the leader-following protocol for the MASs with both communication delay and randomly occurring self-delay, in other words, the design of exponential stabilization controller gain 𝐾_𝑙 for the switched multi-agent system (11). This means that the protocol 𝑢_𝑖(𝑡) solves this leader-following problem, if and only if the states of agents satisfy lim_𝑡→∞‖𝑥_𝑖(𝑡)‖=0, ∀𝑖∈𝒱. Therefore, the leader-following protocol gains 𝐾_𝑙 will be obtained through a designing of the exponential stabilization controller gains 𝐾_𝑙 for the switched multi-agent system (11). In order to do this, the following definitions and lemmas will be utilized.

Definition 9 (see [7–9]). System (11) is said to be exponentially stable under switching signal 𝜎(𝑡) if the solution 𝑥(𝑡) of system (11) satisfies ‖𝑥(𝑡)‖≤𝛽𝑒^{−𝛾(𝑡−𝑡0)}𝑥𝑡_0𝑐1,∀𝑡≥𝑡₀, (13) for constants 𝛽≥1 and 𝛾>0, where ‖𝑥(𝑡)‖_𝑐1=sup_{−ℎ𝑀≤𝑠≤0}{‖𝑥(𝑡+𝑠)‖,‖𝑥̇(𝑡+𝑠)‖}.

Definition 10 (see [6]). For any 𝑡_𝑓>𝑡_𝑖≥0, let 𝑁_𝜎(𝑡,𝑇) denote the switching number of 𝜎 on an interval (𝑡_𝑖,𝑡_𝑓), if 𝑁_𝜎(𝑡,𝑇)≤𝑁₀+(𝑇−𝑡)/𝜏_𝑎 holds for given 𝑁₀≥0 and 𝜏_𝑎>0. Then the constant 𝜏_𝑎 is called the average dwell time and 𝑁₀ is the chatter bound. Without loss of generality, we choose 𝑁₀=0 in this paper.

Lemma 11 (Finsler’s lemma; see [36]). Let 𝜁∈ℝ^𝑛, Φ=Φ^𝑇∈ℝ^𝑛×𝑛, and 𝐵∈ℝ^𝑚×𝑛 such that rank(𝐵)<𝑛. The following statements are equivalent:(i)𝜁^𝑇Φ𝜁<0, ∀𝐵𝜁=0, 𝜁≠0, (ii)𝐵^⟂𝑇Φ𝐵^⟂<0.

3. Main Results

In this section, two theorems will be introduced. In Theorem 12, a leader-following stability criterion for system (11) will be introduced. Then, based on the result of Theorem 12, a leader-following protocol design for system (11) will be introduced in Theorem 15. Before introducing Theorem 12, some notations will be defined for simplicity of matrix representation. 𝜗_𝑖∈ℝ^{5𝑁𝑛×𝑁𝑛}(𝑖=1,2,…,5) are defined as block entry matrices; for example, 𝜗₃=[0_{𝑁𝑛⋅2𝑁𝑛},𝐼_𝑁𝑛,0_{𝑁𝑛⋅2𝑁𝑛}]^𝑇. The notations of several matrices are defined as 𝜁^𝑇(𝑡)=𝑥^𝑇(𝑡),𝑥^𝑇𝑡−ℎ_𝑚,𝑥^𝑇(𝑡−ℎ(𝑡)),𝑥^𝑇𝑡−ℎ_𝑀,𝑥̇^𝑇(𝑠,Υ_𝑙)=𝐹_𝑙⊗𝐾_𝑙𝜗^𝑇₁+𝐺_𝑙⊗𝐾_𝑙𝜗^𝑇₃−𝜗^𝑇₅,Ψ_1𝑙=𝐼_𝑁⊗𝑅_1𝑙−𝐼_𝑁⊗𝑅_1𝑙−𝐼_𝑁⊗𝑅^𝑇_1𝑙𝐼_𝑁⊗𝑅_1𝑙,Ψ_2𝑙=𝐼_𝑁⊗𝑅_2𝑙𝐼_𝑁⊗𝑊_𝑙𝐼_𝑁⊗𝑊^𝑇_𝑙𝐼_𝑁⊗𝑅_2𝑙,𝒮_𝑖𝑙=𝐼_𝑁⊗𝑆_𝑖1𝑙𝐼_𝑁⊗𝑆_𝑖2𝑙𝐼_𝑁⊗𝑆^𝑇_𝑖2𝑙𝐼_𝑁⊗𝑆_𝑖3𝑙(),𝒵𝑖=1,2_𝑖𝑙=0_𝑁𝑛𝐼_𝑁⊗𝑍_𝑖𝑙𝐼_𝑁⊗𝑍^𝑇_𝑖𝑙0_𝑁𝑛(),Ξ𝑖=1,2,3_1𝑙=𝜗₁𝐼_𝑁⊗𝑃_𝑙𝜗^𝑇₅+𝜗₅𝐼_𝑁⊗𝑃_𝑙𝜗^𝑇₁+𝜗₁𝐼_𝑁⊗𝛼𝑃_𝑙𝜗^𝑇₁,Ξ_2𝑙=𝜗₁𝐼_𝑁⊗𝑄_1𝑙+𝑄_2𝑙𝜗^𝑇₁−𝜗₂𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑚}𝑄_1𝑙−𝑄_3𝑙𝜗^𝑇₂−𝜗₃𝐼_𝑁⊗1−ℎ_𝐷𝑒^{−𝛼ℎ𝑀}𝑄_2𝑙𝜗^𝑇₃−𝜗₄𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑄_3𝑙𝜗^𝑇₄,Ξ_3𝑙=𝜗₅𝐼_𝑁⊗ℎ²_𝑚𝑅_1𝑙+ℎ_𝑀−ℎ_𝑚²𝑅_2𝑙𝜗^𝑇₅−𝜗₁,𝜗₂𝑒^{−𝛼ℎ𝑚}Ψ_1𝑙𝜗₁,𝜗₂^𝑇−𝜗₂−𝜗₃,𝜗₃−𝜗₄𝑒^{−𝛼ℎ𝑀}Ψ_2𝑙×𝜗₂−𝜗₃,𝜗₃−𝜗₄^𝑇,Ξ_4𝑙=𝜗₁,𝜗₅ℎ_𝑚𝒮_1𝑙+ℎ_𝑀−ℎ_𝑚𝒮_2𝑙𝜗₁,𝜗₅^𝑇,Θ_𝑙=𝜗₁𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑚}𝑍_1𝑙𝜗^𝑇₁−𝜗₂𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑚}𝑍_1𝑙−𝑒^{−𝛼ℎ𝑀}𝑍_2𝑙𝜗^𝑇₂−𝜗₃𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑍_2𝑙−𝑍_3𝑙𝜗^𝑇₃−𝜗₄𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑍_3𝑙𝜗^𝑇₄,Ξ_𝑙=⁴_𝑖=1Ξ_𝑖𝑙+Θ_𝑙, (14) where the subscript 𝑙 means the switching signal 𝜎(𝑡) at 𝑙∈ℐ={1,2}; for example, Ξ₁=Ξ_𝜎(𝑡)=1.

Now, the first result is given by Theorem 12.

Theorem 12. For given scalars 0<ℎ_𝑚≤ℎ_𝑀, ℎ_𝐷<1, 𝛼>0 and the gains 𝐾₁, 𝐾₂, the agents in the system (11) converge to the position of leader exponentially, if there exist positive definite matrices 𝑃_𝑙∈ℝ^𝑛×𝑛, 𝑄_𝑖𝑙∈ℝ^𝑛×𝑛(𝑖=1,2,3), 𝑅_𝑖𝑙∈ℝ^𝑛×𝑛(𝑖=1,2), and 𝑆_𝑖𝑗𝑙∈ℝ^𝑛×𝑛(𝑖=1,2,𝑗=1,3), any matrices 𝑆_𝑖2𝑙∈ℝ^𝑛×𝑛(𝑖=1,2), 𝑊_𝑙∈ℝ^𝑛×𝑛, and any symmetric matrices 𝑍_𝑖𝑙∈ℝ^𝑛×𝑛(𝑖=1,2,3) satisfying the following LMIs for 𝑙=1,2: Υ^⟂_𝑙^𝑇Ξ_𝑙Υ^⟂_𝑙Ψ<0,_2𝑙𝒮≥0,_1𝑙+𝒵_1𝑙𝒮≥0,_2𝑙+𝒵_2𝑙𝒮≥0,_2𝑙+𝒵_3𝑙≥0. (15)(16)(17)(18)(19)

Then, system (11) is exponentially stable for any switching signal with the average dwell time satisfying 𝜏_𝑎>ln𝜇𝛼, (20) where 𝜇≥1 and satisfies the following for all 𝑖,𝑗∈ℐ: 𝑃_𝑖≤𝜇𝑃_𝑗,𝑄_𝑝𝑖≤𝜇𝑄_𝑝𝑗,𝑅_𝑞𝑖≤𝜇𝑅_𝑞𝑗,𝑆_𝑞1𝑖𝑆_𝑞2𝑖𝑆^𝑇_𝑞2𝑖𝑆_𝑞3𝑖𝑆≤𝜇^𝑞1𝑗𝑆_𝑞2𝑗𝑆^𝑇_𝑞2𝑗𝑆_𝑞3𝑗𝑝=1,2,3,𝑞=1,2. (21) Moreover, an estimate of state decay is given by ‖𝑥(𝑡)‖≤𝑏𝑎𝑒^{−((𝛼−ln𝜇/𝜏𝑎)/2)𝑡}‖𝑥(0)‖_𝑐1, (22) where 𝑎=min_𝑙=1,2𝜆_min𝑃_𝑙,𝑏=max_𝑙=1,2𝜆_max𝑃_𝑙+max_𝑙=1,2ℎ_𝑚𝜆_max𝑄_1𝑙+ℎ_𝑀max_𝑙=1,2𝜆_max𝑄_2𝑙+ℎ_𝑀−ℎ_𝑚×max_𝑙=1,2𝜆_max𝑄_3𝑙+ℎ³_𝑚2max_𝑙=1,2𝜆_max𝑅_1𝑙+ℎ_𝑀−ℎ_𝑚³2max_𝑙=1,2𝜆_max𝑅_2𝑙+ℎ²_𝑚2max_𝑙=1,2𝜆_max𝑆_11𝑙𝑆_12𝑙𝑆^𝑇_12𝑙𝑆_13𝑙+ℎ_𝑀−ℎ_𝑚²2max_𝑙=1,2𝜆_max𝑆_21𝑙𝑆_22𝑙𝑆^𝑇_22𝑙𝑆_23𝑙. (23)

Proof. Let us consider the following Lyapunov-Krasovskii functional candidate as 𝑉_𝜎(𝑡)(𝑥(𝑡))=𝑉_1𝜎(𝑡)+𝑉_2𝜎(𝑡)+𝑉_3𝜎(𝑡)+𝑉_4𝜎(𝑡), (24) where 𝑉_1𝜎(𝑡)=𝑥(𝑡)𝐼_𝑁⊗𝑃_𝜎(𝑡)𝑥(𝑡),𝑉_2𝜎(𝑡)=^𝑡_{𝑡−ℎ𝑚}𝑥^𝑇(𝑠)𝐼_𝑁⊗𝑒^{𝛼(𝑠−𝑡)}𝑄_1𝜎(𝑡)𝑥(𝑠)+𝑑𝑠_{𝑡(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)𝐼_𝑁⊗𝑒^{𝛼(𝑠−𝑡)}𝑄_2𝜎(𝑡)𝑥(𝑠)+𝑑𝑠_{𝑡−ℎ𝑚𝑡−ℎ𝑀}𝑥^𝑇(𝑠)𝐼_𝑁⊗𝑒^{𝛼(𝑠−𝑡)}𝑄_3𝜎(𝑡)𝑥(𝑠)𝑉𝑑𝑠,_3𝜎(𝑡)=ℎ_𝑚^𝑡_{𝑡−ℎ𝑚}^𝑡_𝑠𝑥̇^𝑇(𝑢)𝐼_𝑁⊗𝑒^{𝛼(𝑢−𝑡)}𝑅_1𝜎(𝑡)𝑥̇(𝑢)+ℎ𝑑𝑢𝑑𝑠_𝑀−ℎ_𝑚×^{𝑡−ℎ𝑚}_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥̇^𝑇(𝑢)𝐼_𝑁⊗𝑒^{𝛼(𝑢−𝑡)}𝑅_2𝜎(𝑡)𝑥̇(𝑢)𝑉𝑑𝑢𝑑𝑠,_4𝜎(𝑡)=^𝑡_{𝑡−ℎ𝑚}^𝑡_𝑠𝜉^𝑇(𝑢)𝐼_𝑁⊗𝑒^{𝛼(𝑢−𝑡)}𝒮_1𝜎(𝑡)𝜉(𝑢)+𝑑𝑢𝑑𝑠_{𝑡−ℎ𝑚𝑡−ℎ𝑀𝑡𝑠}𝜉^𝑇(𝑢)𝐼_𝑁⊗𝑒^{𝛼(𝑢−𝑡)}𝒮_2𝜎(𝑡)𝜉(𝑢)𝑑𝑢𝑑𝑠, (25) and 𝜉^𝑇(𝑡)=[𝑥^𝑇(𝑡),𝑥̇^𝑇(𝑡)].
The time-derivative of 𝑉_1𝜎(𝑡) and 𝑉_2𝜎(𝑡) can be calculated as 𝑉̇_1𝜎(𝑡)+𝛼𝑉_1𝜎(𝑡)=2𝑥^𝑇(𝑡)𝐼_𝑁⊗𝑃_𝜎(𝑡)𝑥̇(𝑡)+𝛼𝑉_1𝜎(𝑡)=𝜁^𝑇(𝑡)Ξ_1𝜎(𝑡)𝜁(𝑡),𝑉̇_2𝜎(𝑡)+𝛼𝑉_2𝜎(𝑡)≤𝑥^𝑇(𝑡)𝐼_𝑁⊗𝑄_1𝜎(𝑡)+𝑄_2𝜎(𝑡)𝑥(𝑡)−𝑥^𝑇𝑡−ℎ_𝑚𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑚}𝑄_1𝜎(𝑡)−𝑄_3𝜎(𝑡)×𝑥𝑡−ℎ_𝑚−𝑥^𝑇((𝑡×𝐼𝑡−ℎ))_𝑁⊗1−ℎ_𝐷𝑒^{−𝛼ℎ𝑀}𝑄_2𝜎(𝑡)𝑥(𝑡−ℎ(𝑡))−𝑥^𝑇𝑡−ℎ_𝑀𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑄_3𝜎(𝑡)×𝑥𝑡−ℎ_𝑀=𝜁^𝑇(𝑡)Ξ_2𝜎(𝑡)𝜁(𝑡). (26)(27) By use of Jensen’s inequality [37], an upper bound of 𝑉̇_3𝜎(𝑡) is obtained as 𝑉̇_3𝜎(𝑡)+𝛼𝑉_3𝜎(𝑡)≤𝑥̇^𝑇(𝑡)𝐼_𝑁⊗ℎ²_𝑚𝑅_1𝜎(𝑡)+ℎ_𝑀−ℎ_𝑚²𝑅_2𝜎(𝑡)𝑥̇(𝑡)−^𝑡_{𝑡−ℎ𝑚}𝑥̇^𝑇(𝑠)𝑑𝑠𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑚}𝑅_1𝜎(𝑡)×^𝑡_{𝑡−ℎ𝑚}𝑥̇(𝑠)𝑑𝑠−Φ(𝑡), (28) where 𝜛(𝑡)=(ℎ_𝑀−ℎ(𝑡))/(ℎ_𝑀−ℎ_𝑚) and Φ(𝑡)=11−𝜛(𝑡)^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑅_2𝜎(𝑡)×^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)1𝑑𝑠+𝜛(𝑡)^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥̇^𝑇(𝑠)×𝐼𝑑𝑠_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑅_2𝜎(𝑡)^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥̇(𝑠)𝑑𝑠. (29) Here, when ℎ_𝑚<ℎ(𝑡)<ℎ_𝑀, since 𝜛(𝑡) satisfies 0<𝜛(𝑡)<1, by reciprocally convex approach [24], the following inequality holds for any matrices 𝑊_𝜎(𝑡): 𝜛(𝑡)0<1−𝜛(𝑡)^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑅_2𝜎(𝑡)×^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)𝑑𝑠−^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)×𝐼𝑑𝑠_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑊_𝜎(𝑡)^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥̇(𝑠)−𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}𝑥̇^𝑇(𝑠)𝑑𝑠𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑊^𝑇_𝜎(𝑡)×^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)(𝑡)𝑑𝑠1−𝜛𝜛(𝑡)×^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥̇^𝑇(𝑠)𝑑𝑠𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑅_2𝜎(𝑡)×^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥̇(𝑠)𝑑𝑠, (30) which implies that −Φ(𝑡)<−^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑅_2𝜎(𝑡)×^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)−𝑑𝑠_{𝑡−ℎ𝑚(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑊_𝜎(𝑡)×^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥̇(𝑠)𝑑𝑠−^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥̇^𝑇(𝑠)×𝐼𝑑𝑠_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑊^𝑇_𝜎(𝑡)×^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)𝑑𝑠−^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥̇^𝑇(𝑠)×𝐼𝑑𝑠_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑅_2𝜎(𝑡)×^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥̇(𝑠)𝑑𝑠. (31) It should be noted that when ℎ(𝑡)=ℎ_𝑚 or ℎ(𝑡)=ℎ_𝑀, we get ∫^{𝑡−ℎ𝑚}_{𝑡−ℎ(𝑡)}𝑥̇(𝑠)𝑑𝑠=∫^{𝑡−ℎ𝑚}_{𝑡−ℎ𝑚}𝑥̇(𝑠)𝑑𝑠=0_𝑁𝑛⋅1 or ∫^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥̇(𝑠)𝑑𝑠=∫^{𝑡−ℎ𝑀}_{𝑡−ℎ𝑀}𝑥̇(𝑠)𝑑𝑠=0_𝑁𝑛⋅1, respectively. Thus, if inequality (16) holds, then, an upper bound of the 𝑉̇_3𝜎(𝑡) can be rebounded as 𝑉̇_3𝜎(𝑡)+𝛼𝑉_3𝜎(𝑡)≤𝜁^𝑇(𝑡)Ξ_3𝜎(𝑡)𝜁(𝑡). (32) Next, 𝑉̇_4𝜎(𝑡) is obtained as 𝑉̇_4𝜎(𝑡)+𝛼𝑉_4𝜎(𝑡)=𝜉^𝑇(𝑡)ℎ_𝑚𝒮_1𝜎(𝑡)+ℎ_𝑀−ℎ_𝑚𝒮_2𝜎(𝑡)𝜉(𝑡)−^𝑡_{𝑡−ℎ𝑚}𝜉^𝑇(𝑠)𝑒^{−𝛼ℎ𝑚}𝒮_1𝜎(𝑡)𝜉(𝑠)−𝑑𝑠_{𝑡−ℎ𝑚(𝑡)𝑡−ℎ}𝜉^𝑇(𝑠)𝑒^{−𝛼ℎ𝑀}𝒮_2𝜎(𝑡)𝜉(𝑠)−𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}𝜉^𝑇(𝑠)𝑒^{−𝛼ℎ𝑀}𝒮_2𝜎(𝑡)𝜉(𝑠)𝑑𝑠. (33) Before estimating 𝑉̇_4𝜎(𝑡), inspired by the work of [38], the following zero equality with any symmetric matrices 𝑍_1𝜎(𝑡), 𝑍_2𝜎(𝑡), and 𝑍_3𝜎(𝑡) holds as a less conservative tool 0=𝑥^𝑇(𝑡)𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑚}𝑍_1𝜎(𝑡)𝑥(𝑡)−𝑥^𝑇𝑡−ℎ_𝑚𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑚}𝑍_1𝜎(𝑡)𝑥𝑡−ℎ_𝑚−2^𝑡_{𝑡−ℎ𝑚}𝑥^𝑇(𝑠)𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑚}𝑍_1𝜎(𝑡)𝑥̇(𝑠)𝑑𝑠+𝑥^𝑇𝑡−ℎ_𝑚𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑍_2𝜎(𝑡)𝑥𝑡−ℎ_𝑚−𝑥^𝑇(𝑡−ℎ(𝑡))𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑍_2𝜎(𝑡)𝑥(𝑡−ℎ(𝑡))−2^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑍_2𝜎(𝑡)𝑥̇(𝑠)𝑑𝑠+𝑥^𝑇(𝑡−ℎ(𝑡))𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑍_3𝜎(𝑡)𝑥(𝑡−ℎ(𝑡))−𝑥^𝑇𝑡−ℎ_𝑀𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑍_3𝜎(𝑡)𝑥𝑡−ℎ_𝑀−2^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥^𝑇(𝑠)𝐼_𝑁⊗𝑒^{−𝛼ℎ𝑀}𝑍_3𝜎(𝑡)𝑥̇(𝑠)𝑑𝑠=𝜁^𝑇(𝑡)Θ_𝜎(𝑡)𝜁(𝑡)−^𝑡_{𝑡−ℎ𝑚}𝜉^𝑇(𝑠)𝑒^{−𝛼ℎ𝑚}𝒵_1𝜎(𝑡)𝜉(𝑠)−𝑑𝑠_{𝑡−ℎ𝑚(𝑡)𝑡−ℎ}𝜉^𝑇(𝑠)𝑒^{−𝛼ℎ𝑀}𝒵_2𝜎(𝑡)𝜉(𝑠)−𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}𝜉^𝑇(𝑠)𝑒^{−𝛼ℎ𝑀}𝒵_3𝜎(𝑡)𝜉(𝑠)𝑑𝑠. (34) By adding (34) to (33), 𝑉̇_4𝜎(𝑡) can be estimated as 𝑉̇_4𝜎(𝑡)+𝛼𝑉_4𝜎(𝑡)=𝜁^𝑇(𝑡)Ξ_4𝜎(𝑡)+Θ_𝜎(𝑡)𝜁(𝑡)−^𝑡_{𝑡−ℎ𝑚}𝜉^𝑇(𝑠)𝑒^{−𝛼ℎ𝑚}𝒮_1𝜎(𝑡)+𝒵_1𝜎(𝑡)𝜉(𝑠)−𝑑𝑠_{𝑡−ℎ𝑚(𝑡)𝑡−ℎ}𝜉^𝑇(𝑠)𝑒^{−𝛼ℎ𝑀}𝒮_2𝜎(𝑡)+𝒵_2𝜎(𝑡)𝜉(𝑠)−𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}𝜉^𝑇(𝑠)𝑒^{−𝛼ℎ𝑀}𝒮_2𝜎(𝑡)+𝒵_3𝜎(𝑡)𝜉(𝑠)𝑑𝑠. (35)
Thus, if inequalities (17)–(19) hold, then, an upper bound of 𝑉̇_4𝜎(𝑡) can be bounded as 𝑉̇_4𝜎(𝑡)+𝛼𝑉_4𝜎(𝑡)≤𝜁^𝑇(𝑡)Ξ_4𝜎(𝑡)+Θ_𝜎(𝑡)𝜁(𝑡). (36) Also, the system (11) with the augmented vector 𝜁(𝑡) can be rewritten as Υ_𝜎(𝑡)𝜁(𝑡)=0_𝑁𝑛⋅1. (37) Then, from (26)–(36), a leader-following stability condition for the system (11) is 𝜁^𝑇(𝑡)Ξ_𝜎(𝑡)𝜁(𝑡)<0 (38) subject to (37). If inequality (38) holds, then 𝑉̇_𝜎(𝑡)(𝑥(𝑡))+𝛼𝑉_𝜎(𝑡)(𝑥(𝑡))<0. (39) From (37), the vector 𝜁(𝑡) can be parameterized as 𝜁(𝑡)=Υ^⟂_𝜎(𝑡)𝑦(𝑡), (40) where 𝑟=rank(Υ_𝜎(𝑡))<𝑁𝑛 and 𝑦(𝑡)∈ℝ^𝑟.
By utilizing Lemma 11 (ii) with (40), condition (38) is equivalent to the following inequality: 𝜁^𝑇(𝑡)Ξ_𝜎(𝑡)𝜁(𝑡)=Υ^⟂_𝜎(𝑡)𝑦(𝑡)^𝑇Ξ_𝜎(𝑡)Υ^⟂_𝜎(𝑡)𝑦(𝑡)=𝑦^𝑇(𝑡)Υ^⟂_𝜎(𝑡)^𝑇Ξ_𝜎(𝑡)Υ^⟂_𝜎(𝑡)𝑦(𝑡)<0. (41) From inequality (41), if the LMIs (15)–(19) hold, then the presented leader-following stability condition (38) is satisfied.
Furthermore, according to the works [7–9], for an arbitrary piecewise constant switching signal 𝜎(𝑡) and any 𝑡>0, assume that 𝑡_𝑙(𝑙=1,2,…,0<𝑡₁<𝑡₂<⋯) are the switching points of 𝜎(𝑡) over the interval (0,𝑡). As mentioned above, the 𝑖_𝑙th subsystem is activated when 𝑡∈[𝑡_𝑙,𝑡_𝑙+1). Integrating (39) from 𝑡_𝑙 to 𝑡 gives 𝑉_𝜎(𝑡)(𝑥(𝑡))≤𝑒^{−𝛼(𝑡−𝑡𝑙)}𝑉_{𝜎(𝑡𝑙)}𝑥𝑡_𝑙. (42) Using (21) and (24), at the switching instant 𝑡_𝑙, we get 𝑉_{𝜎(𝑡𝑙)}𝑥𝑡_𝑙≤𝜇𝑉_𝜎(𝑡⁻_𝑙)𝑥𝑡⁻_𝑙. (43) Therefore, from (42), (43), and the relation 𝜅=𝑁_𝜎(0,𝑡)≤(𝑡−0)/𝜏_𝑎, it follows that 𝑉_𝜎(𝑡)(𝑥(𝑡))≤𝑒^{−𝛼(𝑡−𝑡𝑙)}𝜇𝑉_𝜎(𝑡⁻_𝑙)𝑥𝑡⁻_𝑙≤⋯≤𝑒^{−𝛼(𝑡−0)}𝜇^𝜅×𝑉_𝜎(0)(𝑥(0))≤𝑒^{−(𝛼−ln𝜇/𝜏𝑎)𝑡}𝑉_𝜎(0)(𝑥(0)). (44) Furthermore, from (24), the following inequalities hold: 𝑎‖𝑥(𝑡)‖²≤𝑉_𝜎(𝑡)(𝑥(𝑡,𝑉))_𝜎(0)(𝑥(0))≤𝑏‖𝑥(0)‖²_𝑐1, (45) where 𝑎 and 𝑏 are defined in (22). Combining (44) and (45) leads to 𝑎‖𝑥(𝑡)‖²≤𝑉_𝜎(𝑡)(𝑥(𝑡))≤𝑏𝑒^{−(𝛼−ln𝜇/𝜏𝑎)𝑡}‖𝑥(0)‖²_𝑐1, (46) which implies (22). Thus, by Definition 9 with 𝑡₀=0, the system (11) is exponentially stable. This completes our proof.

Remark 13. Since the proposed method is satisfied with condition (21), we have 𝛼−ln𝜇𝜏_𝑎>0, (47) which implies that switching signal can guarantee that the switched system is stable; in other words, condition (47) corresponds to the exponential stability condition 𝛾>0 in Definition 9.

Remark 14. When 𝜇=1 and 𝛼>0, we have 𝜏_𝑎>ln𝜇/𝛼=0, which means that the switching signal can be arbitrary. At this time, with 𝛼=0, we get the asymptotically stability condition for the concerned system. If 𝜇>1 and 𝛼→0; that is, 𝜏_𝑎→∞, then there is no switching. Also, the following condition means that a mode-independent Lyapunov-Krasovskii functional is required for all subsystems: 𝑃_𝑖=𝑃_𝑗=𝑃,𝑄_𝑝𝑖=𝑄_𝑝𝑗=𝑄_𝑝,𝑅_𝑞𝑖=𝑅_𝑞𝑗=𝑅_𝑞,𝑆_𝑞1𝑖𝑆_𝑞2𝑖𝑆^𝑇_𝑞2𝑖𝑆_𝑞3𝑖=𝑆_𝑞1𝑗𝑆_𝑞2𝑗𝑆^𝑇_𝑞2𝑗𝑆_𝑞3𝑗=𝑆_𝑞1𝑆_𝑞2𝑆^𝑇_𝑞2𝑆_𝑞3𝑝=1,2,3,𝑞=1,2. (48) Moreover, in this paper, switching signals are generated with the occurrence probability of self-delay, Pr{𝜌(𝑡)=1}=𝜌₀=Pr{𝜎(𝑡)=1}.

Theorem 12 provides the leader-following criterion for system (11) in the framework of LMIs when the leader-following protocol gain is known. Based on the results of Theorem 12, we will propose a leader-following protocol design method for system (11) which will be introduced as follows.

Theorem 15. For given scalars 0<ℎ_𝑚≤ℎ_𝑀, ℎ_𝐷<1, and 𝛼>0, the agents in the system (11) converge to the position of leader exponentially, if there exist positive definite matrices 𝑃_𝑙∈ℝ^𝑛×𝑛, 𝑄_𝑖𝑙∈ℝ^𝑛×𝑛(𝑖=1,2,3), 𝑅_𝑖𝑙∈ℝ^𝑛×𝑛(𝑖=1,2), and 𝑆_𝑖𝑗𝑙∈ℝ^𝑛×𝑛(𝑖=1,2,𝑗=1,3), any matrices 𝑆_𝑖2𝑙∈ℝ^𝑛×𝑛(𝑖=1,2), 𝑊_𝑙∈ℝ^𝑛×𝑛, and any symmetric matrices 𝑍_𝑖𝑙∈ℝ^𝑛×𝑛(𝑖=1,2,3) satisfying LMIs (16)–(19) and Ξ_𝑙+Γ_𝑙+Γ^𝑇_𝑙<0(𝑙=1,2), (49) where Γ_𝑙=[𝜗₁+𝜗₅][(𝐹_𝑙⊗𝑌_𝑙)𝜗^𝑇₁+(𝐺_𝑙⊗𝑌_𝑙)𝜗^𝑇₃−(𝐼_𝑁⊗𝑋_𝑙)𝜗^𝑇₅].

Then, the MASs (11) under the leader-following protocol gain 𝐾_𝑙=𝑋_𝑙⁻¹𝑌_𝑙 are exponentially stable.

Proof. Design the gains 𝐾₁ and 𝐾₂, with the same Lyapunov-Krasovskii functional candidate in (24), by using similar method in (26)–(36) and considering the following zero equalities with any matrices 𝑋₁ and 𝑋₂: 0=2𝜁^𝑇(𝑡)𝜗₁𝐼_𝑁⊗𝑋_𝑙+𝜗₅𝐼_𝑁⊗𝑋_𝑙Υ_𝑙𝜁(𝑡)(𝑙=1,2). (50) Then, by defining 𝑌₁=𝑋₁𝐾₁ and 𝑌₂=𝑋₂𝐾₂, equalities (50) lead to 0=2𝜁^𝑇(𝑡)𝜗₁+𝜗₅𝐹_𝑙⊗𝑌_𝑙𝜗^𝑇₁+𝐺_𝑙⊗𝑌_𝑙𝜗^𝑇₃−𝐼_𝑁⊗𝑋_𝑙𝜗^𝑇₅𝜁(𝑡)(𝑙=1,2). (51) Therefore, from (26)–(36), by consideration of zero equalities (51) and application of the S-procedure [30], a sufficient condition guaranteeing stability for the system (11) can be 𝜁^𝑇(𝑡)⁴_𝑖=1Ξ_𝑖𝑙+Θ_𝑙+Γ_𝑙+Γ^𝑇_𝑙𝜁(𝑡)<0(𝑙=1,2). (52) In conclusion, condition (52) is equivalent to LMIs (49).

Remark 16. When information about the upper bound of  ℎ̇(𝑡) is unknown, then by setting 𝑄_2𝜎(𝑡)=0 in the Lyapunov-Krasovskii functional (24), the results with only the condition 0<ℎ_𝑚≤ℎ(𝑡)≤ℎ_𝑀 can be obtained, and their proofs are similar to the proof of Theorems 12 and 15, so it is omitted.

4. An Illustrative Example

In this section, one numerical example will be shown to illustrate the effectiveness of the proposed Theorem 15. In an example, MATLAB LMI Control Toolbox [39] is used to solve LMI problems.

Consider MASs (11) with 4 missiles (agents) in 2-dimensional plan (e.g., the latitude and longitude coordinate), that is, 𝑁=4, 𝑛=2, and the graph 𝒢 of topology form described in Figure 3.

Figure 3 

The interconnection topology.

From Figure 1, the matrices 𝒜, ℬ, and 𝒟 related to graph 𝒢 are represented as follow: ,{}{}.𝒜=01.5000001.700.7001.300.60ℬ=diag1.5,1.7,0.7,1.9,𝒟=diag0,2.2,0,0 (53) Thus, in view of switched system (11), the parameters of each subsystem are(i)subsystem 1 when 𝜎(𝑡)=1 (for 𝜌(𝑡)=1) 𝐹₁=−ℬ,𝐺₁=𝒜−𝒟=−1.51.5000−1.701.700.7−0.701.300.6−1.9, (54)(ii)subsystem 2 when 𝜎(𝑡)=2 (for 𝜌(𝑡)=0) 𝐹₂(){},𝐺=−ℬ+𝒟=−diag1.5,3.9,0.7,1.9₂=𝒜. (55)For the system mentioned above, by applying Theorem 15, the leader-following protocol gains 𝐾₁ and 𝐾₂ and the range of average dwell time 𝜏_𝑎 with fixed 𝜇=1.05, 𝛼=0.1, ℎ_𝐷=0.2, and ℎ_𝑚=0.1 are 𝐾₁{},𝐾=diag0.2074,0.2074₂=diag{0.2521,0.2521}, (56) and 𝜏_𝑎>ln𝜇/𝛼=ln1.05/0.1=0.4879. At this time, the maximum delay bound, ℎ_𝑀, is 0.76. In order to confirm this result, we set the condition on time-delay as ℎ(𝑡)=0.1+0.66sin²(0.3𝑡). In addition, let us choose the occurrence probability of self-delay, 𝜌₀, as 0.7. Then, the operation modes of switching signal are shown in Figure 4.

Figure 4 

The curve of the operation modes.

It is assumed that the position of leader is set by 𝑝₀=[5,−3]^𝑇. Then, the simulation results are given in Figure 5. This figure shows that each agent with the responses converges to the position of leader under the protocol gains in (56) and the switching signal 𝜎(𝑡).

Figure 5 

Positions of the agents and leader.

Also, taking 𝜏_𝑎=1, and from (22), we can further obtain the estimate of state decay ‖𝑥(𝑡)‖≤16.2698𝑒^{−0.0256𝑡}‖𝑥(0)‖_𝑐1=𝜒(0), (57) and its result is shown in Figure 6.

Figure 6 

The estimate of state decay (57).

5. Conclusions

In this paper, the delay-dependent leader-following problem for the MASs with both communication delay and randomly occurring self-delay was studied. In Theorem 12, the leader-following stability criterion for the MASs was proposed by introducing the new piecewise Lyapunov-Krasovskii functional and constructing the new model of switched MASs (11). In Theorem 15, based on the result of Theorem 12, the leader-following protocol gains for the switched MASs were designed. One illustrative example has been given to show the effectiveness and usefulness of the designed protocol gains.

Acknowledgment

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0009273 and 2008-0062611).