State Consensus Analysis and Design for High-Order Discrete-Time Linear Multiagent Systems

Abstract

The paper deals with the state consensus problem of high-order discrete-time linear multiagent systems (DLMASs) with fixed information topologies. We consider three aspects of the consensus analysis and design problem: (1) the convergence criteria of global state consensus, (2) the calculation of the state consensus function, and (3) the determination of the weighted matrix and the feedback gain matrix in the consensus protocol. We solve the consensus problem by proposing a linear transformation to translate it into a partial stability problem. Based on the approach, we obtain necessary and sufficient criteria in terms of Schur stability of matrices and present an analytical expression of the state consensus function. We also propose a design process to determine the feedback gain matrix in the consensus protocol. Finally, we extend the state consensus to the formation control. The results are explained by several numerical examples.

1. Introduction

In recent years, the consensus problem of multiagent systems (MASs) has been becoming a significant research topic because of its broad practical applications, including the work load balance in a network of parallel computers [1], the clock synchronization [2], distributed decision [3], consensus filtering and estimation in sensor networks [4–6], rendezvous, and the formation of various moving objects [7–11] such as underwater vehicles, aircrafts, satellites, mobile robots, and intelligent vehicles in automated highway systems, to name only a few. Hence, its study has captured attention of the researchers from different disciplines.

MASs are comprised of locally interacting agents equipped with dedicated sensing, computing, and communication devices. The consensus problem of MASs is to design a distributed control law for each agent, using only information from itself and its neighbors, such that all agents achieve an agreement on some quantities of interest. To design and analyse this class of systems, one needs to consider three essential elements: (1) a dynamic model describing the states of the agents, which can be either continuous time or discrete time, linear or nonlinear, homogeneous or heterogeneous, time varying or time invariant, low order or high order; (2) an information topology describing communication network between the agents, which can be either undirected or directed, fixed or switched; (3) a protocol (control input) for each of the agents describing how the agents interact on each other according to the given information topology, which can be synchronous or asynchronous, with or without time delay.

Up to now, numerous researches have been done for continuous-time MASs in different settings from the above cases [10, 12–18]. This paper focuses on the study of high-order discrete-time linear multiagent systems (DLMASs) by proposing a linear transformation to translate the consensus problem into a partial stability problem. Although this approach can be extended to any setting from the above cases, we pay our attention only to the case of fixed information topology and in the absence of time delay for giving prominence to the trait of the approach. Here we give an overview mainly to the DLMASs.

Reference [19] first proposed an interesting model for self-propelled particle systems, where all agents move in a plane with the same speed but different headings, and showed that in the model all agents might eventually move in the same direction despite the absence of centralized coordination. Reference [20] further gave a mathematically rigorous qualitative analysis. Then, a theoretical explanation was given for the consensus behavior of Vicsek model on the basis of graph theory in [21, 22]. A necessary and sufficient condition was given for the average consensus criterion in [23]. Reference [24] further considered the case of switching network topologies for the average consensus. The average consensus was investigated for the systems with uncertain communication environments and time-varying topologies in [25] and with communication constraints in [26]. Reference [27] presented convergence results for the time-varying protocol in the absence or presence of communication delays. Reference [28] proposed an asynchronous time-varying consensus protocol. Reference [29] further discussed nonlinear systems with time-dependent communication links. References [30, 31] addressed the case with both time-varying delays and switching information topologies and provided a class of effective consensus protocols by repeatedly using the same state information at two time steps.

The researches mentioned above were limited to first-order systems. The extension to second-order systems was done for the systems with time-varying delays and time-varying interaction topology in [32], and for the systems with nonuniform time-delays and dynamically changing topologies in [33].

Recent researches were turned to the high-order DLMASs in [34–39]. Reference [34] studied a class of dynamic average consensus algorithms that allow a group of agents to track the average of their reference inputs. Reference [35] proposed an observer-type protocol based on the relative outputs of neighboring agents. Reference [36] studied the convergence speed for the high-order systems with random networks and arbitrary weights. Reference [37] addressed the high-order systems with or without delays. These researches were focused on the consensus convergence criteria for the proposed protocols. Another significant topic is the design of the gain matrices of the protocols in [38, 39].

This paper deals with both analysis and design problems of the state consensus for general high-order DLMASs. Compared with the existing works, the contributions of the paper are summarized as follows. Firstly, motivated by [12], we improve the protocol by adding a self-feedback of the agent to achieve the expected consensus dynamics, whereas [13] introduced the internal model to change the given dynamic to achieve the expected consensus dynamics and [14, 15] introduced the virtual leader to guide the multiagent systems to achieve the expected consensus dynamics. Secondly, we propose a state linear transformation to translate the consensus problem into a partial stability problem. The approach is motivated by the error variable method or the state space decomposition method in [12, 16]. However, our improvement can more spontaneously and conveniently deal with various settings of the consensus problems. Based on the partial stability theory, we educe new necessary and sufficient consensus convergence criteria in terms of stability of matrices and moreover give an explicit analytical expression of the state consensus function based on the different contributions of the initial states of the agents and the protocols. Thirdly, based on stability theorem, we give a design procedure to determine the gain matrices in the protocol on the basis of algebraic Riccati inequality similarly to [38, 39]. Fourthly, we extend the state consensus results to the formation control problem.

The remainder of the paper is organized as follows. Section 2 introduces some basic concepts and notations, and formulates the problem under investigation. Section 3 firstly introduces a linear transformation which translates the consensus problem of the multiagent systems into a partial stability problem of the corresponding transformed system, and then educes a new necessary and sufficient condition for the multiagent system to achieve global state consensus and presents an analytical expression of the state consensus function. Section 4 shows a design procedure to determine the gain matrices in the state consensus protocol. Section 5 extends the approach for the analysis and design of the state consensus to the formation control problem. Section 6 gives numerical examples to explain the theoretical results. Section 7 concludes the paper. All the proofs of the results are deposited in the appendix for the sake of reading.

2. Problem Description

Before stating the consensus problem, we give some basic concepts and notations. Let ℝ^𝑛×𝑛 and ℂ^𝑛×𝑛 be the sets of 𝑛×𝑛 real matrices and complex matrices, respectively. Matrices, if not explicitly stated, have appropriate dimensions in all settings. The superscript “𝑇” means transpose for real matrices, and the superscript “𝐻” means conjugate transpose for complex matrices. 𝐼_𝑛 presents the identity matrix of dimension 𝑛, and sometimes 𝐼 is used for simplicity. 𝟏_𝑁 denotes the vector of dimension 𝑁 with all entries equal to one. 0 is applied to denote zero matrices/vectors of any size, with zero components. A matrix 𝐴∈ℂ^𝑛×𝑛 is said to be Schur stable if all of its eigenvalues have magnitude less than 1. The Kronecker product is denoted by ⊗ and the Hadamard product by ∘ in [40]. The following properties of the Kronecker product will be used: (1)(𝐴⊗𝐵)(𝐶⊗𝐷)=𝐴𝐶⊗𝐵𝐷; (2)(𝐴+𝐵)⊗𝐶=𝐴⊗𝐶+𝐵⊗𝐶; (3)(𝐴⊗𝐵)⁻¹=𝐴⁻¹⊗𝐵⁻¹.

We consider DLMASs with 𝑁 homogeneous agents and assume they are described by 𝑥⁺_𝑖=𝐴𝑥_𝑖+𝐵𝑢_𝑖,𝑖=1,…,𝑁, (1) where 𝐴∈ℝ^𝑛×𝑛, 𝐵∈ℝ^𝑛×𝑚, (𝐴,𝐵) is assumed to be stabilizable, 𝑥_𝑖=𝑥_𝑖(𝑘)∈ℝ^𝑛 is the state of the current time 𝑘, 𝑥⁺_𝑖=𝑥_𝑖(𝑘+1) denotes the state at the next time 𝑘+1, and 𝑢_𝑖=𝑢_𝑖(𝑘)∈ℝ^𝑚 is the control input of the current time 𝑘.

The control input 𝑢_𝑖 will be constructed based on the available information of the agent 𝑖. Let 𝒩_𝑖 denote the index set of the agents which can send their state information to the agent 𝑖. We call the set 𝒩={𝒩_𝑖∶𝑖=1,…,𝑁} the information topology of the DLMASs (1). It is well known that one can use a digraph 𝐺=(𝑉,𝐸) to express the information topology 𝒩, where 𝑉={1,…,𝑁} is the index set of 𝑁 agents, 𝐸⊆𝑉×𝑉 is the set of directed edges to describe the information interaction between agents; that is, (𝑗,𝑖)∈𝐸⇔𝑗∈𝒩_𝑖. Based on the directed edges, one can construct an adjacency matrix 𝐴=[𝑎_𝑖𝑗]_𝑁×𝑁, whose entries are defined as 𝑎_𝑖𝑗=0 for 𝑗=𝑖, 𝑎_𝑖𝑗=1 for 𝑗∈𝒩_𝑖, and 𝑎_𝑖𝑗=0 for 𝑗∉𝒩_𝑖. The corresponding in-degree matrix and graph Laplacian are defined as 𝐷=diag{deg₁,…,deg_𝑁} and 𝐿=𝐷−𝐴, respectively, where deg_𝑖=∑^𝑁_𝑗=1𝑎_𝑖𝑗 is the in-degree of the vertex 𝑖. A directed spanning tree of the digraph 𝐺 is a tree covering all the vertices of the digraph. The following results are well known.

Lemma 1 (see [22]). The Laplacian matrix 𝐿∈ℝ^𝑁×𝑁 has the following properties: (1) all of the eigenvalues of 𝐿 are either in the open right half complex plan or equal to 0; (2) 0 is a simple eigenvalue of 𝐿 if and only if the digraph 𝐺 contains a directed spanning tree.

Given the information topology 𝒩, we construct the following linear consensus protocol: 𝑢_𝑖=𝐾₁𝑥_𝑖+𝐾_{2𝑗∈𝒩𝑖}𝑤_𝑖𝑗𝑥_𝑗−𝑥_𝑖,𝑖=1,…,𝑁, (2) where 𝐾₁, 𝐾₂∈ℝ^𝑚×𝑛 are feedback gain matrices to be determined, which are relative to the consensus state and the convergence rate, respectively. 𝑊=∶[𝑤_𝑖𝑗]_𝑁×𝑁 is a weighted matrix associated with the information topology 𝒩. For the sake of expression, we also define the weighted adjacency matrix 𝐴_𝑤=𝐴∘𝑊 by using Hadamard product of matrices and the weighted Laplacian 𝐿_𝑤=𝐷_𝑤−𝐴_𝑤 with weights 𝑤_𝑖𝑗, where 𝐷_𝑤=diag{deg₁,…,deg_𝑁} is the corresponding weighted in-degree matrix with weighted in-degrees deg_𝑖=∑_{𝑗∈𝒩𝑖}𝑤_𝑖𝑗.

Definition 2. For the given information topology 𝒩, the DLMASs (1) are said to achieve global state consensus via the protocol (2) if for any given initial state 𝑥_𝑖(0), 𝑖=1,…,𝑁, there exists an 𝑛-dimensional vector function 𝜉(𝑘) depending on the initial states such that lim_𝑘→∞‖𝑥_𝑖(𝑘)−𝜉(𝑘)‖=0. The function 𝜉(𝑘) is called a state consensus function.

In this paper, we will address the following three aspects of the state consensus problem: (i) to give criteria of global state consensus, that is, for any given information topology 𝒩, weighted matrix 𝑊 and feedback gain matrices 𝐾₁ and 𝐾₂ to find the conditions of the DLMASs (1) achieving global state consensus via the protocol (2); (ii) to calculate the state consensus function 𝜉(𝑘) if the DLMASs (1) achieve global state consensus via the protocol (2); (iii) to determine the matrices 𝐾₁ and 𝐾₂ such that the DLMASs (1) achieve global state consensus via the protocol (2).

First of all, we transform the state consensus problem to the partial stability problem. Then, based on the partial stability theorem framework, we educe new necessary and sufficient consensus convergence criteria and state a procedure to determine the gain matrices in the protocol on the basis of algebraic Riccati inequality. We also give an explicit analytical expression of the state consensus function based on the respective contributions of the initial states and the protocols. Finally, we extend the results to formation control.

3. State Consensus Analysis

In this section, we first introduce a linear transformation which translates the consensus problem of the multiagent systems into a partial stability problem of the corresponding transformed system. Then, we educe a necessary and sufficient condition for the DLMASs (1) to achieve global state consensus via the protocol (2), and present an analysis expression of the state consensus function. Finally, we discuss some interesting remarks and corollaries based on the result.

Let 𝑥=[𝑥^𝑇₁,…,𝑥^𝑇_𝑁]^𝑇. The dynamics of the DLMASs (1) with the protocol (2) is described by 𝑥⁺=Ψ𝑥, (3) where Ψ=𝐼_𝑁⊗𝐴+𝐵𝐾₁−𝐿_𝑤⊗𝐵𝐾₂. (4)

We propose a state linear transformation for the linear system (3) as follows: 𝑥=𝑇𝑥, (5) where the block matrix 𝑇∈ℝ^{𝑛𝑁×𝑛𝑁} is defined as 𝑇𝟏𝑇=∶_𝑇𝑁⊗𝐼_𝑛,𝑇𝑇=⋮¹𝑇_𝑁−1. (6) And the matrix 𝑇_𝑖=[𝑇_𝑖1,…,𝑇_𝑖𝑁],  𝑖=1,…,𝑁−1, is chosen such that the following two conditions are satisfied:(1)the row vectors in each of the matrices 𝑇_𝑖 are linearly independent, respectively;(2)the identities 𝑇_𝑖(𝟏_𝑁⊗𝐼_𝑛)=0, 𝑖=1,…,𝑁−1, are held.

Lemma 3. The inverse 𝑇⁻¹ of the matrix 𝑇 admits the following form: 𝑇⁻¹=𝑇₁₁⋯𝑇_1,𝑁−1𝑁⁻¹𝐼_𝑛⋮⋱⋮⋮𝑇_𝑁−1,1⋯𝑇_{𝑁−1,𝑁−1}𝑁⁻¹𝐼_𝑛𝑇_𝑁,1⋯𝑇_{𝑁,𝑁−1}𝑁⁻¹𝐼_𝑛=∶𝑇₁⋯𝑇_𝑁−1𝑁⁻¹𝟏_𝑁⊗𝐼_𝑛=∶𝑇𝑁⁻¹𝟏_𝑁⊗𝐼_𝑛, (7) where 𝑇_𝑖𝑗, 𝑖=1,…,𝑁, 𝑗=1,…,𝑁−1, are the 𝑛×𝑛 blocks indefinitely described.

Using the linear transformation (5), we transform the linear system (3) into the following system: 𝑥⁺=𝑇Ψ𝑇⁻¹𝑥, (8) or the form of two equations 𝑦⁺=𝑇Ψ𝑇𝑦+𝑁⁻¹𝑇Ψ𝟏_𝑁⊗𝐼_𝑛𝑧𝑧,₊=𝟏^𝑇_𝑁⊗𝐼_𝑛Ψ𝑇𝑦+𝑁⁻¹𝟏^𝑇_𝑁⊗𝐼_𝑛Ψ𝟏_𝑁⊗𝐼_𝑛𝑧, (9) where 𝑥=[𝑦^𝑇,𝑧^𝑇]^𝑇, 𝑦=[𝑥^𝑇₁,…,𝑥^𝑇_𝑁−1]^𝑇, and 𝑧=𝑥_𝑁.

We show that the state consensus problem of the DLMASs (1) with the protocol (2) can be transformed into a partial stability problem.

Definition 4 (see [41]). The linear system (8) is said to be asymptotically stable with respect to 𝑦 (or asymptotically 𝑦-stable in short) if lim_𝑘→∞𝑦(𝑘)=0 for any bounded initial state 𝑥(0) of the system (8).

Lemma 5. Under the given information topology 𝒩, the DLMASs (1) achieve global state consensus via the protocol (2) if and only if the equilibrium point 𝑥=0 of the linear system (8) is asymptotically 𝑦-stable. Moreover, the state consensus function of the agents is 𝜉(𝑘)=𝑁⁻¹∑^𝑁_𝑖=1𝑥_𝑖(𝑘)=𝑁⁻¹𝑥_𝑁(𝑘).

Lemma 5 builds a bridge between the consensus problem and the partial stability problem. Now we focus on the asymptotical 𝑦-stability of the linear system (8). We can verify the following lemma.

Lemma 6. The system (9) is of the following form: 𝑦⁺=𝐴𝑦,𝑦∈ℝ^{𝑛(𝑁−1)},𝑧⁺=𝐶𝑦+𝐷𝑧,𝑧∈ℝ^𝑛, (10) where 𝐴=𝑇(𝐼_𝑁⊗(𝐴+𝐵𝐾₁)−𝐿_𝑤⊗𝐵𝐾₂)𝑇, 𝐶=−(𝟏^T_𝑁𝐿_𝑤⊗𝐵𝐾₂)𝑇, and 𝐷=𝐴+𝐵𝐾₁.

Combining Lemma 5 with Lemma 6, we directly get the following theorem.

Theorem 7. Under the given information topology 𝒩, the DLMASs (1) achieve global state consensus via the protocol (2) if and only if matrix 𝐴 in (10) is Schur stable. Moreover, the state consensus function is 𝜉(𝑘)=𝑁^−1𝑘−1_𝑗=0𝐴+𝐵𝐾₁^𝑗𝐶𝐴^{𝑘−1−𝑗}𝑇+𝟏^𝑇_𝑁⊗𝐴+𝐵𝐾₁^𝑘𝑥(0). (11)

Subsequently, we give some interesting remarks and corollaries based on the result.

Remark 8. Since 𝑇𝑇=(𝐼_𝑁−𝑁⁻¹𝟏_𝑁𝟏^𝑇_𝑁)⊗𝐼_𝑛, the result of Theorem 7 is in fact independent of the choice of the matrix 𝑇 although both 𝐴 and formula (11) in Theorem 7 contain 𝑇 and 𝑇. Hence, for simplicity, we take it in the following form: 𝑇=1−10⋯001−1⋯0⋮⋮⋱⋮⋮0⋯01−1111⋯1⊗𝐼_𝑛𝑇∶=⁰𝟏^𝑇_𝑁⊗𝐼_𝑛. (12) The corresponding inverse matrix is 𝑇⁻¹=1𝑁()1𝑁−1𝑁−2⋯11−1𝑁−2⋯11⋮⋮⋱⋮⋮−1−2⋯11−1−2⋯−𝑁−1⊗𝐼_𝑛∶=𝑇₀𝑁⁻¹𝟏_𝑁⊗𝐼_𝑛. (13) Thus, we can write 𝐴 and 𝐶 into 𝐴=𝐼_𝑁−1⊗𝐴+𝐵𝐾₁−𝑇₀𝐿_𝑤𝑇₀⊗𝐵𝐾₂,𝐶=−𝟏^T_𝑁𝐿_𝑤𝑇₀⊗𝐵𝐾₂. (14)

Corollary 9. Under the given information topology 𝒩, the DLMASs (1) achieve global consensus via the protocol (2) if and only if all the matrices 𝐴+𝐵𝐾₁−𝜆_𝑖𝐵𝐾₂ are Schur stable, where 𝜆_𝑖, 𝑖=1,…,𝑁−1, are the eigenvalues of the matrix 𝑇₀𝐿_𝑤𝑇₀. Moreover, the state consensus function is expressed by 𝜉(𝑘)=𝜂^𝑇⊗𝐴+𝐵𝐾₁^𝑘𝑥(0),𝑖=1,…,𝑁, (15) where 𝜂 satisfies 𝜂^𝑇𝐿_𝑤=0 and 𝜂^𝑇𝟏_𝑁=1.

One can verify that as 𝑘→∞ the state consensus functions in formulas (11) and (15) are the same.

Remark 10. From Schur stability of 𝐴 in the formula (14), we can conclude that if 𝐴+𝐵𝐾₁ is not Schur stable, it is a necessary condition of the consensus that the digraph 𝐺 expressing the information topology 𝒩 has a directed spanning tree. In fact, since the condition of directed spanning tree is equivalent to Hurwitz stability of −𝑇₀𝐿_𝑤𝑇₀, a lack of directed spanning tree means that −𝑇₀𝐿_𝑤𝑇₀ has a zero eigenvalue. In this case, we transform 𝐴 into its Jordan form via the matrix 𝑈⊗𝐼_𝑛, where 𝑈 is the matrix such that 𝑈⁻¹𝑇₀𝐿_𝑤𝑇₀𝑈=𝐽 is the Jordan form, and thus we have 𝑈⁻¹⊗𝐼_𝑛𝐴𝑈⊗𝐼_𝑛=𝐼_𝑁−1⊗𝐴+𝐵𝐾₁−𝐽⊗𝐵𝐾₂. (16) One can verify that the eigenvalues of 𝐴+𝐵𝐾₁ are the members of the eigenvalues of 𝐴, and thus 𝐴 is not Schur stable if 𝐴+𝐵𝐾₁ is not Schur stable. On the other hand, if 𝐴+𝐵𝐾₁ is Schur stable, one can take 𝐾₂=0 to make the DLMASs (1) achieve global consensus, which implies that for any initial states all the agents always converge to the equilibrium point 0.

Hence, from the formula (16) we can educe another global consensus criterion.

Corollary 11. If 𝐴+𝐵𝐾₁ is not Schur stable, then under the given information topology 𝒩, the DLMASs (1) achieve global state consensus via the protocol (2) if and only if 𝑇₀𝐿_𝑤𝑇₀ has 𝑁−1 eigenvalues with positive real part 𝜆_𝑖, 𝑖=1,…,𝑁−1, and the matrices 𝐴+𝐵𝐾₁−𝜆_𝑖𝐵𝐾₂, 𝑖=1,…,𝑁−1, are Schur stable.

Remark 12. If the state consensus function in (11) is a constant vector equal to the average of the initial states of all the agents, the consensus is called the average consensus. From the formula (11) we educe the following result on the average consensus.

Corollary 13. The DLMASs (1) achieve global average consensus via the protocol (2) if and only the matrix 𝐴 is Schur stable and (∑^𝑘−1_𝑗=0(𝐴+𝐵𝐾₁)^𝑗𝐶𝐴^{𝑘−1−𝑗})𝑇+𝟏^𝑇_𝑁⊗(𝐴+𝐵𝐾₁)^𝑘=𝟏^𝑇_𝑁⊗𝐼_𝑛.

If 𝐴+𝐵𝐾₁=𝐼_𝑛, then the last condition in Corollary 13 becomes 𝟏^𝑇_𝑁𝐿_𝑤=0, or equivalently, the digraph 𝐺 is either undirected connected or directed strong connected and balanced. More specially, if 𝐿_𝑤 is a symmetric matrix (equivalently, the digraph 𝐺 becomes undirected connected), the condition 𝟏^𝑇_𝑁𝐿_𝑤=0 is satisfied and thus the average consensus is achieved.

Remark 14. When 𝐴=𝐼_𝑛, 𝐾₁=0, and 𝐵=𝐼_𝑛, the DLMASs (1) are called a single-integrator one. In this case, 𝐴=𝐼_{(𝑁−1)𝑛}−𝑇₀𝐿_𝑤𝑇₀⊗𝐾₂ and 𝐶=−𝟏^𝑇_𝑁𝐿_𝑤𝑇₀⊗𝐾₂. We educe the following result.

Corollary 15. Under the given information topology 𝒩, the single-integrator DLMASs (1) achieve global state consensus via the protocol (2) if and only if the following two conditions are held simultaneously: (1) the matrix −𝑇₀𝐿_𝑤𝑇₀ is Hurwitz stable; that is, the digraph 𝐺 admits a directed spanning tree; (2) the products 𝜆_𝑖𝜇_𝑗, 𝑖=1,…,𝑁−1, 𝑗=1,…,𝑛, are in the open unit circle of the complex plane with the centre at (1,0), where 𝜆_𝑖, 𝑖=1,…,𝑁−1, are the eigenvalues of the matrix 𝑇₀𝐿_𝑤𝑇₀ and 𝜇_𝑗, 𝑗=1,…,𝑛, are the eigenvalues of the matrix 𝐾₂. The corresponding consensus function (11) becomes a constant vector 𝜉=𝑁⁻¹𝟏^𝑇_𝑁⊗𝐼_𝑛−𝟏^𝑇_𝑁𝐿_𝑤𝑇₀𝑇₀𝐿_𝑤𝑇₀⁻¹𝑇₀⊗𝐼_𝑛𝑥(0). (17) Moreover, the single-integrator DLMASs (1) achieve global average consensus via the protocol (2) if and only both of the above conditions are satisfied and in addition 𝟏^𝑇_𝑁𝐿_𝑤=0.

Remark 16. When 𝐴=¹¹₀₁⊗𝐼_𝑛, 𝐵=⁰₁⊗𝐼_𝑛, and 𝐾₁=0, the DLMASs (1) are called a double-integrator one, whose state vector can be seen as consisting of the position and velocity in the 𝑛 dimensional space ℝ^𝑛.

Corollary 17. Under the given information topology 𝒩, the double-integrator DLMASs (1) achieve global state consensus via the protocol (2) if and only if 𝐴 is Schur stable. Moreover, the consensus function is 𝜉(𝑘)=𝑁⁻¹𝐶𝐼_{2(𝑁−1)𝑛}−𝐴⁻¹+0𝑘𝐼_𝑛00𝐶𝐼_{2(𝑁−1)𝑛}−𝐴⁻¹−0𝐼_𝑛00𝐶𝐼_{2(𝑁−1)𝑛}−𝐴⁻²𝑇+𝟏^𝑇_𝑁⊗𝐼_𝑛𝑘𝐼_𝑛0𝐼_𝑛𝑥(0). (18)

The consensus function above can be decomposed into the position consensus function 𝜉₁(𝑘)=𝑁⁻¹𝐼_𝑛0𝐶𝐼_{2(𝑁−1)𝑛}−𝐴⁻¹+0𝑘𝐼_𝑛𝐶𝐼_{2(𝑁−1)𝑛}−𝐴⁻¹−0𝐼_𝑛𝐶𝐼_{2(𝑁−1)𝑛}−𝐴⁻²𝑇+𝟏^𝑇_𝑁⊗𝐼_𝑛𝑘𝐼_𝑛𝑥(0), (19) which is a linear function of discrete time 𝑘, and the constant velocity consensus function is as follows: 𝜉₂(𝑘)=𝑁⁻¹0𝐼_𝑛𝐶𝐼_{2(𝑁−1)𝑛}−𝐴⁻¹𝑇+𝟏^𝑇_𝑁⊗0𝐼_𝑛𝑥(0). (20)

Similarly, we can define the velocity average consensus, that is, if the DLMASs (1) achieve global consensus via the protocol (2), and the velocity consensus function is a constant vector equal to the average of the initial velocities of all the agents.

Corollary 18. Under the given information topology 𝒩, the double-integrator DLMASs (1) achieve global velocity average consensus via the protocol (2) if and only the matrix 𝐴 is Schur stable and 0𝐼_𝑛𝐶=0.

It is obvious that if 𝟏^𝑇_𝑁𝐿_𝑤=0, then 𝐶=0; that is, the last condition in Corollary 18 is satisfied, and thus, the state consensus function becomes 𝜉(𝑘)=𝑁⁻¹𝟏^𝑇_𝑁⊗𝐼_𝑛𝑘𝐼_𝑛0𝐼_𝑛𝑥(0). (21)

4. Design of Gain Matrices

In this section, we discuss the third problem, that is, how to determine the weighted matrix 𝑊 and the gain matrices 𝐾₁ and 𝐾₂, such that the DLMASs (1) achieve global state consensus via the protocol (2).

Theorem 7 shows that the matrices 𝑊, 𝐾₁, and 𝐾₂ should be taken to ensure that the matrix 𝐴 is Schur stable. Furthermore, from Corollary 11 we see that if the matrix 𝑊 with respect to the information topology 𝒩 has been given, we need only to design the gain matrices 𝐾₁ and 𝐾₂ to ensure that the matrices 𝐴+𝐵𝐾₁−𝜆_𝑖𝐵𝐾₂ are Schur stable, where 𝜆_𝑖, 𝑖=1,…,𝑁−1, are the eigenvalues of the matrix 𝑇₀𝐿_𝑤𝑇₀. The matrix 𝐾₁ is often taken to obtain an expected consensus dynamics. The matrix 𝐾₂ is designed to achieve state consensus and expected convergence rate. Its design needs the following lemma.

Lemma 19 (see [38]). Supposing that the matrix 𝐴=𝐴+𝐵𝐾₁ is not Schur stable but (𝐴,𝐵) is stabilizable, then there is a critical value 𝛿_𝑐∈(0,1], such that for any number 𝛿 with 0<𝛿<𝛿_𝑐 the modified Riccati inequality 𝐴^𝑇𝑃𝐴−𝑃−1−𝛿²𝐴^𝑇𝑃𝐵𝐵^𝑇𝑃𝐵⁻¹𝐵^𝑇𝑃𝐴<0 (22) admits a positive definite matrix solution 𝑃, where 𝛿_𝑐 depends on the unstable eigenvalues of the matrix 𝐴.

We define functions 𝛿_𝑖(𝜔)=1−𝜔𝜆_𝑖 and 𝛿(𝜔)=max_{𝑖∈{1,…,𝑁−1}}|𝛿_𝑖(𝜔)|. Motivated by [38], we get the following theorem.

Theorem 20. Supposing that the matrix (𝐴,𝐵) is stabilizable, the gain matrix 𝐾₁ has been taken such that the expected consensus dynamic matrix 𝐴+𝐵𝐾₁ is not Schur stable, and the weighted matrix 𝑊 with respect to the information topology 𝒩 is given such that −𝑇₀𝐿_𝑤𝑇₀ is Hurwitz stable with 𝑁−1 eigenvalues −𝜆_𝑖, 𝑖=1,…,𝑁−1; then, for the DLMASs (1) to achieve state consensus via the protocol (2), the matrix 𝐾₂ can be designed as 𝐾₂=𝜔(𝐵^𝑇𝑃𝐵)⁻¹𝐵^𝑇𝑃(𝐴+𝐵𝐾₁), where 𝜔 is an arbitrary constant satisfying 𝛿=𝛿(𝜔)<𝛿_𝑐, 𝛿_𝑐∈(0,1] is a critical value which depends on the unstable eigenvalues of the matrix 𝐴+𝐵𝐾₁, and 𝑃^𝑇=𝑃>0 is a solution of the algebraic Riccati inequality (22).

Based on Theorem 20, we give the following algorithm of determining the feedback gain matrices 𝐾₁ and 𝐾₂ in the protocol (2).

Algorithm 21. Design procedure of the gain matrices 𝐾₁ and 𝐾₂.

Step 1. Verify the stabilizability condition of (𝐴,𝐵) and the spanning tree condition of the information topology 𝒩. If neither of them is satisfied, then stop. Otherwise, design the weighted Laplacian 𝐿_𝑤 such that −𝑇₀𝐿_𝑤𝑇₀ is Hurwitz stable with 𝑁−1 eigenvalues −𝜆_𝑖, 𝑖=1,…,𝑁−1.

Step 2. Design 𝐾₁ such that 𝐴=𝐴+𝐵𝐾₁ is the matrix of the expected consensus dynamics of the DLMASs (1) and is not Schur stable.

Step 3. Calculate all the eigenvalues of 𝐴, which are composed of the stable eigenvalues 𝜆^𝑠_𝑖(𝐴), 𝑖=1,…,𝑛_𝑠 and unstable ones 𝜆^𝑢_𝑖(𝐴), 𝑖=1,…,𝑛_𝑢, 𝑛_𝑠+𝑛_𝑢=𝑛.

Step 4. Calculate the critical value 𝛿_𝑐∈(0,1]. If 𝐵 is invertible, then 𝛿_𝑐=(max_{𝑖={1,…,𝑛𝑢}}|𝜆^𝑢_𝑖(𝐴)|)⁻¹. If 𝐵 is of rank one, then 𝛿_𝑐=(∏_{𝑖={1,…,𝑛𝑢}}|𝜆^𝑢_𝑖(𝐴)|)⁻¹. Otherwise, apply Wonham decomposition to the unstable part (𝐴_𝑢,𝐵_𝑢) of (𝐴,𝐵) to convert the multiple input system to 𝑚 single input subsystems, where 𝑚 is the number of the Jordan blocks of matrix 𝐴_𝑢. Specifically, there is a nonsingular real matrix 𝑄 with a compatible dimension such that 𝐴=𝑄⁻¹𝐴_𝑢𝑄 and 𝐵=𝑄⁻¹𝐵_𝑢 take the form 𝐴𝐴=¹∗∗∗0𝐴₂⋯∗⋮⋮⋱⋮00⋯𝐴_𝑚,𝑏𝐵=¹∗∗∗0𝑏₂⋯∗⋮⋮⋱⋮00⋯𝑏_𝑚, (23) where the symbol ∗ denotes possibly nonzero parts and (𝐴_𝑗,𝑏_𝑗) with 𝐴_𝑗∈ℝ^{𝑛𝑗×𝑛𝑗} and 𝑏_𝑗∈ℝ^𝑛𝑗 for all 𝑗∈{1,…,𝑚} is controllable and ∑^𝑚_𝑗=1𝑛_𝑗=𝑛_𝑢. In this case, 𝛿_𝑐 is lower bounded by 𝛿_𝑐≥(∏_𝑖|𝜆^𝑢_𝑖(𝐴_𝑚∗)|)⁻¹)=𝛿^′_𝑐, where the index 𝑚^∗ is defined by 𝑚^∗=argmax_{𝑗={1,…,𝑚}}(∏_𝑖|𝜆^𝑢_𝑖(𝐴_𝑗)|) and 𝐴_𝑗 is the Jordan block of the unstable part of matrix 𝐴.

Step 5. Calculate the value 𝜔 such that 𝛿(𝜔)=max_{𝑖∈{1,…,𝑁−1}}|𝛿_𝑖(𝜔)|.

Step 6. Solve (22) with 𝛿=𝛿(𝜔) for a positive definite matrix 𝑃.

Step 7. Calculate the matrix 𝐾₂=𝜔(𝐵^𝑇𝑃𝐵)⁻¹𝐵^𝑇𝑃(𝐴+𝐵𝐾₁).

5. Application to Formation Control

In this section, the consensus approach is modified to solve the formation control problem of the DLMASs (1). Let ℎ=ℎ^𝑇₁ℎ^𝑇₂⋯ℎ^𝑇_𝑁^𝑇∈ℝ^𝑛𝑁 describe a constant formation of the agent network in a reference coordinate frame, where ℎ_𝑖∈ℝ^𝑛 is the formation variable corresponding to the agent 𝑖. The variable ℎ_𝑖−ℎ_𝑗 denotes the relative formation vector between the agents 𝑖 and 𝑗, which is assumed to be independent of the reference coordinate.

We modify the consensus protocol (2) and propose a distributed formation protocol as follows: 𝑢_𝑖=𝐾₁𝑥_𝑖+𝐾_{2𝑗∈𝒩𝑖}𝑤_𝑖𝑗𝑥_𝑗−𝑥_𝑖−ℎ_𝑗−ℎ_𝑖,𝑖=1,…,𝑁. (24)

Definition 22. Under the given information topology 𝒩, the DLMASs (1) achieve the given formation ℎ via the protocol (24) if ‖(𝑥_𝑖(𝑘)−𝑥_𝑗(𝑘))−(ℎ_𝑖−ℎ_𝑗)‖→0 as 𝑘→∞, for all 𝑖,𝑗=1,…,𝑁, that is, if there is a function 𝜉(𝑘) such that ‖𝑥_𝑖(𝑘)−ℎ_𝑖−𝜉(𝑘)‖→0 as 𝑘→∞, for all 𝑖=1,…,𝑁, where 𝜉(𝑘) is called reference state consensus function.

Theorem 23. Under the given information topology 𝒩, the DLMASs (1) achieve the formation ℎ via the protocol (24) if and only if the matrix 𝐼_𝑁−1⊗(𝐴+𝐵𝐾₁)−𝑇₀𝐿_𝑤𝑇₀⊗𝐵𝐾₂ is Schur stable and (𝑇₀⊗(𝐴+𝐵𝐾₁−𝐼_𝑛))ℎ=0. Moreover, the reference state consensus function is 𝜉(𝑘)=𝑁^−1𝑘−1_𝑗=0𝐴+𝐵𝐾₁^𝑗𝐶𝐴^{𝑘−1−𝑗}𝑇+𝟏^𝑇_𝑁⊗𝐴+𝐵𝐾₁^𝑘𝑥(0). (25)

Similarly to Corollary 11, we get the following corollary for the formation control.

Corollary 24. Under the given information topology 𝒩, the DLMASs (1) achieve the formation ℎ via the protocol (24) if and only if all the matrices 𝐴+𝐵𝐾₁−𝜆_𝑖𝐵𝐾₂ are Schur stable and (𝑇₀⊗(𝐴+𝐵𝐾₁−𝐼_𝑛))ℎ=0, where 𝜆_𝑖, 𝑖=1,…,𝑁−1, are the eigenvalues of the matrix 𝑇₀𝐿_𝑤𝑇₀.

Remark 25. Note that not all kinds of formation structure can be achieved for the DLMASs (1) by using the protocol (24). The achievable formation structures have to satisfy the constraints (𝑇₀⊗(𝐴+𝐵𝐾₁−𝐼_𝑛))ℎ=0. The formation protocol (24) for a given achievable formation ℎ can be constructed analogously by using the Algorithm 21 in Section 4.

6. Numerical Examples

In this section, we give some illustrative examples.

Example 26 (state consensus analysis). We consider the DLMASs (1) consisting of four agents described by the following matrices: 2√𝐴=22−√22−12√243√241100,𝐵=−121. (26) Supposed that we are given the information topology 𝒩={{4},{1},{2},{3}}, the weights 𝑤_𝑖𝑗=0.5, and the gain matrices 𝐾₁=[−1,−√2/2,√2/2] and 𝐾₂=[0.4444,−0.4714,0.1571], we are required to verify the consensus convergence.
The matrix 𝐴 is Schur stable since its eigenvalues are 0.3482±0.7182𝑖, 0.8746±0.2786𝑖,0.7885±0.4643𝑖,0.7728±0.5782𝑖, and 0.0001. Therefore, according to Theorem 7, the DLMASs (1) with the matrices in (26) achieve global consensus via the protocol (2) under the given information topology 𝒩 and the gain matrix 𝐾₁, 𝐾₂. For the initial states 𝑥₁(0)=[0,0.6,0.6]^𝑇, 𝑥₂(0)=[1.2,1.5,1.8]^𝑇, 𝑥₃(0)=[1.8,2.4,2.1]^𝑇, and 𝑥₄(0)=[3.0,2.7,3.6]^𝑇, the corresponding state consensus function in (11) is 𝜉(𝑘)=[1.5,2.7094cos(0.7854𝑘−1.6296),2.7094cos(0.7854𝑘−0.0588)]^𝑇. Figure 1 shows the state trajectories of the agents and the trajectory of the state consensus function marked by circles.

Figure 1 

State trajectories of DLMASs.

Example 27 (gain matrices design for a formation control). The consensus problem of multiagent systems has many practical applications, such as formation control of mobile robots and cooperative control of unmanned airborne vehicles. In this example, we consider that DLMASs consisting of four mobile robots are described as 𝑥⁺_𝑖=𝑥_𝑖+𝑣_𝑖,𝑣⁺_𝑖=𝑣_𝑖+𝑢_𝑖,𝑖=1,…,4, (27) where 𝑥_𝑖∈ℝ², 𝑣_𝑖∈ℝ², and 𝑢_𝑖∈ℝ² are the position, the velocity, and the acceleration input of the robot 𝑖, respectively. The information topology and the weights are the same as in Example 26. The eigenvalues of the matrix 𝑇₀𝐿_𝑤𝑇₀ are 𝜆₁=0.5+0.5𝑖, 𝜆₂=0.5−0.5𝑖, and 𝜆₃=1. We choose ℎ₁=[6,6,0,0]^𝑇, ℎ₂=[−6,6,0,0]^𝑇, ℎ₃=[−6,−6,0,0]^𝑇, and ℎ₄=[6,−6,0,0]^𝑇 for the formation ℎ of the mobile robots. Note that (𝐴,𝐵) in (27) is stabilizable. The eigenvalues of matrix 𝐴 are 1 (4 multiples). We assume that the expected consensus dynamics of the agents is not changed, and thus, we have 𝐾₁=0. It is easy to verify that the formation ℎ satisfies the constraints (𝑇₀⊗(𝐴+𝐵𝐾₁−𝐼_𝑛))ℎ=0. One can get 𝛿_𝑐=1 through Step 4 of the Algorithm 21. By Step 5, one can obtain 𝜔=1.0, and the corresponding 𝛿(𝜔)=√2/2<𝛿_𝑐. The modified Riccati inequality (22) admits a solution 𝑃=88170017248000088170017248×10⁻⁴. (28) The corresponding control gain is calculated as follows: 𝐾₂=000.06861.06860.06861.068600. (29)
The eigenvalues of matrix 𝐴 are 0.5336±0.5392𝑖 (2 multiples), 0.9321±0.0049𝑖 (2 multiples), 0.9314 (2 multiples), and 0 (2 multiples), that is, the matrix 𝐴 is Schur stable. By Theorem 23, the system (27) via the protocol (24) achieves the formation ℎ.
For the initial states 𝑥₁(0)=[−9,0.6,−18,1.8]^𝑇, 𝑥₂(0)=[12,1.5,27,0.9]^𝑇, 𝑥₃(0)=[−21,1.8,24,2.4]^𝑇, and 𝑥₄(0)=[21,3,27,3.6]^𝑇, we get the reference state consensus function 𝜉(𝑘)=[0.75+1.725𝑘,1.725,15+2.175𝑘,2.175]^𝑇 according to (25). Figure 2 shows the position and velocity trajectories of the mobile robots and the trajectory of the reference state consensus function marked by dots, and Figure 3 shows the formation evolution trajectories. From Figure 3 it is clear to see that the mobile robots achieve the expected formation, where the symbol diamond denotes the initial position, and the symbol circle denotes the position at the moment 𝑘=80 of the mobile robots, respectively.

Figure 2 

Position and velocity trajectories of mobile robots.

Figure 3 

Formation of four mobile robots.

7. Conclusions

We considered the state consensus problem of high-order discrete-time linear multiagent systems with fixed directed information topology. A linear transformation approach was proposed to translate the consensus problem of multiagent systems into a partial stability problem of the corresponding transformed systems. We have shown that the approach is powerful in dealing with the three aspects of the consensus problem: (1) the criteria of global state consensus, (2) the calculation of the state consensus function, and (3) the determination the weighted matrix and the feedback gain matrix. Precisely, we have educed new necessary and sufficient consensus criteria in terms of Schur stability of a matrix related to the weighted Laplacian matrix and presented an analytical expression of the state consensus function. In addition, we have stated a design process of determining the feedback gain matrix under the condition of each agent being stabilizable. The consensus algorithm has been further applied to solve the formation control problem of multiagent systems.

Though the work in this paper focuses on the high-order discrete-time linear multiagent systems with fixed information topology and without time delay, it is undoubted that the approach can be easily extended to more complex cases, which will be dealt with in the future works.

Appendix

Proof of Lemma 3. Assuming the inverse to be 𝑇⁻¹=𝑇₁⋯𝑇_𝑁−1𝑇_𝑁^𝑇 with columns 𝑇_𝑖,  𝑖=1,…,𝑁, we prove that the equality 𝑇_𝑁=𝑁⁻¹𝟏_𝑁⊗𝐼_𝑛 is correct. Since the matrix 𝑇^𝑇 is invertible and thus each column of 𝑇_𝑁 can be linearly represented by the columns of the matrix 𝑇^𝑇, so the matrix 𝑇_𝑁 can be represented by 𝑞₁+𝑞₂, where each column of the matrix 𝑞₁ can be linearly represented by matrices 𝑇^𝑇_𝑖, 𝑖=1,…,𝑁−1, and 𝑞₂ by those of the matrix 𝟏_𝑁⊗𝐼_𝑛. Left multiplying 𝑇_𝑁=𝑞₁+𝑞₂ by 𝑞^𝑇₁ gets 𝑞^𝑇₁𝑇_𝑁=𝑞^𝑇₁𝑞₁+𝑞^𝑇₁𝑞₂. Because of 𝑇𝑇⁻¹=𝐼_𝑁𝑛, one has 𝑇_𝑖𝑇_𝑁=0, 𝑖=1,…,𝑁−1, which implies 𝑞^𝑇₁𝑇_𝑁=0. On the other hand, from the equalities 𝑇_𝑖(𝟏_𝑁⊗𝐼_𝑛)=0, 𝑖=1,…,𝑁−1, one gets 𝑇_𝑖𝑞₂=0, 𝑖=1,…,𝑁−1, which means 𝑞^𝑇₁𝑞₂=0. Hence, from the equality 𝑞^𝑇₁𝑇_𝑁=𝑞^𝑇₁𝑞₁+𝑞^𝑇₁𝑞₂, one deduces 𝑞^𝑇₁𝑞₁=0, that is, 𝑞₁=0 and thus 𝑇_𝑁=𝑞₂. In other words, one can write 𝑇_𝑁 into 𝑇_𝑁=(𝟏_𝑁⊗𝐼_𝑛)𝛼, where 𝛼 is a matrix of the order 𝑛×𝑛. From the identity 𝑇𝑇=𝐼_𝑁𝑛, one has (𝟏_𝑁⊗𝐼_𝑛)^𝑇=𝐼_𝑛, and thus, from 𝑇_𝑁=(𝟏_𝑁⊗𝐼_𝑛)𝛼 one can get 𝐼_𝑛=𝑁𝛼, that is, 𝛼=𝑁⁻¹𝐼_𝑛. Finally, one has the expression 𝑇_𝑁=𝑁⁻¹𝟏_𝑁⊗𝐼_𝑛.

Proof of Lemma 5. In fact, if there is 𝜉(𝑘;𝑥(0)) such that lim_𝑘→∞‖𝑥_𝑖(𝑘;𝑥_𝑖(0))−𝜉(𝑘;𝑥(0))‖=0, 𝑖=1,…,𝑁, it follows that lim_𝑘→∞‖𝑥_𝑖(𝑘)‖=0, 𝑖=1,…,𝑁−1, in virtue of 𝑥_𝑖=𝑇_𝑖(𝑥−𝟏_𝑁⊗𝜉), 𝑖=1,…,𝑁, and therefore the necessary has been proved. Conversely, by virtue of Lemma 3, one can verify 𝑥_𝑖=∑^𝑁−1_𝑗=1𝑇_𝑖𝑗𝑥_𝑗+𝑁⁻¹𝑥_𝑁, 𝑖=1,…,𝑁−1. So from lim_𝑘→∞‖𝑥_𝑖(𝑘)‖=0, 𝑖=1,…,𝑁−1, it follows that lim_𝑘→∞‖𝑥_𝑖(𝑘)−𝜉(𝑘)‖=0, 𝑖=1,…,𝑁, where 𝜉(𝑘)=𝑁⁻¹∑^𝑁_𝑖=1𝑥_𝑖(𝑘)=𝑁⁻¹𝑥_𝑁(𝑘), and thus, the sufficiency has been verified.

Proof of Lemma 6. By observation, we only need to show that 𝑇Ψ(𝟏_𝑁⊗𝐼_𝑛)=0, (𝟏^𝑇_𝑁⊗𝐼_𝑛)Ψ𝑇=−(𝟏^𝑇_𝑁𝐿_𝑤⊗𝐵𝐾₂)𝑇, and (𝟏^𝑇_𝑁⊗𝐼_𝑛)Ψ(𝟏_𝑁⊗𝐼_𝑛)=𝑁(𝐴+𝐵𝐾₁). In fact, since 𝐿_𝑤𝟏_𝑁=0, (𝟏^𝑇_𝑁⊗𝐼_𝑛)𝑇=0, and 𝑇(𝟏_𝑁⊗𝐼_𝑛)=0, we have that 𝑇Ψ𝟏_𝑁⊗𝐼_𝑛=𝑇𝐼_𝑁⊗𝐴+𝐵𝐾₁−𝐿_𝑤⊗𝐵𝐾₂𝟏_𝑁⊗𝐼_𝑛=𝑇𝐼_𝑁⊗𝐴+𝐵𝐾₁𝟏_𝑁⊗𝐼_𝑛=𝑇𝟏_𝑁⊗𝐴+𝐵𝐾₁𝟏=0,_𝑇𝑁⊗𝐼_𝑛Ψ𝑇=𝟏^𝑇_𝑁⊗𝐼_𝑛𝐼_𝑁⊗𝐴+𝐵𝐾₁−𝐿_𝑤⊗𝐵𝐾₂𝑇=𝟏^𝑇_𝑁⊗𝐴+𝐵𝐾₁−𝟏^𝑇_𝑁𝐿_𝑤⊗𝐵𝐾₂𝑇=−𝟏^𝑇_𝑁𝐿_𝑤⊗𝐵𝐾₂𝟏𝑇,_𝑇𝑁⊗𝐼_𝑛Ψ𝟏_𝑁⊗𝐼_𝑛=𝟏^𝑇_𝑁⊗𝐼_𝑛𝐼_𝑁⊗𝐴+𝐵𝐾₁−𝐿_𝑤⊗𝐵𝐾₂𝟏_𝑁⊗𝐼_𝑛=𝟏^𝑇_𝑁⊗𝐼_𝑛𝐼_𝑁⊗𝐴+𝐵𝐾₁𝟏_𝑁⊗𝐼_𝑛=𝟏^𝑇_𝑁𝟏_𝑁⊗𝐴+𝐵𝐾₁=𝑁𝐴+𝐵𝐾₁. (A.1) Thus, system (9) becomes 𝑦⁺=𝑧𝑇Ψ𝑇𝑦,₊=−𝟏^𝑇_𝑁𝐿_𝑤⊗𝐵𝐾₂𝑇𝑦+𝐴+𝐵𝐾₁𝑧. (A.2) Denoting 𝐴=𝑇Ψ𝑇, 𝐶=−(𝟏^𝑇_𝑁𝐿_𝑤⊗𝐵𝐾₂)𝑇, and 𝐷=𝐴+𝐵𝐾₁, where Ψ=𝐼_𝑁⊗(𝐴+𝐵𝐾₁)−𝐿_𝑤⊗𝐵𝐾₂, we get (10).

Proof of Theorem 7. The necessary and sufficient condition is verified directly by using the Lemmas 5 and 6. Now we focus on the calculation of the state consensus function. The first equation in (10) gives 𝑦(𝑘)=𝐴^𝑘𝑦(0). From the second equation in (10) we have 𝑧(𝑘)=𝐶𝑦(𝑘−1)+()=𝐷𝑧𝑘−1𝐶𝑦(𝑘−1)+𝐷()+𝐶𝑦𝑘−2+⋯𝐷^𝑘−1𝐶𝑦(0)+𝐷^𝑘𝑧(0). (A.3) Substituting 𝑦(𝑘)=𝐴^𝑘𝑦(0), 𝑦(0)=𝑇𝑥(0), 𝐷=𝐴+𝐵𝐾₁ and 𝑧(0)=(𝟏^𝑇_𝑁⊗𝐼_𝑛)𝑥(0) into the above formula, one gets 𝑧(𝑘)=((∑^𝑘−1_𝑗=0(𝐴+𝐵𝐾₁)^𝑗𝐶𝐴^{𝑘−1−𝑗})𝑇+𝟏^𝑇_𝑁⊗(𝐴+𝐵𝐾₁)^𝑘)𝑥(0). Thus, by Lemma 5 one has the state consensus function in (11).

Proof of Corollary 9. First of all, one easily verifies that the Schur stability of 𝐴 is equivalent to Schur stability of all the matrices 𝐴+𝐵𝐾₁−𝜆_𝑖𝐵𝐾₂ by transforming 𝐴 into its Jordan form. We focus on the calculation of the state consensus function. Rewrite the system (3) as 𝑥⁺=(𝐼_𝑁⊗(𝐴+𝐵𝐾₁)−𝐿_𝑤⊗𝐵𝐾₂)𝑥. So for the left eigenvector 𝜂 with the property 𝜂^𝑇𝟏_𝑁=1 of the Laplacian 𝐿_𝑤 with respect to the zero eigenvalue, we obtain (𝜂^𝑇⊗𝐼_𝑛)𝑥⁺=(𝐴+𝐵𝐾₁)(𝜂^𝑇⊗𝐼_𝑛)𝑥. When the consensus is achieved, we have that 𝜉(𝑘)=(𝜂^𝑇⊗𝐼_𝑛)𝑥(𝑘) and thus the state consensus function is 𝜉(𝑘)=(𝐴+𝐵𝐾₁)^𝑘(𝜂^𝑇⊗𝐼_𝑛)𝑥(0), which can be written into the form (15).

Proof of Corollary 15. In this case, the two conditions in Corollary 11 become: (1) the matrix −𝑇₀𝐿_𝑤𝑇₀ is Hurwitz stable, that is, the digraph 𝐺 admits a directed spanning tree; (2) for all the eigenvalues 𝜆_𝑖, 𝑖=1,…,𝑁−1, of the matrix 𝑇₀𝐿_𝑤𝑇₀, all the matrix 𝐼_𝑛−𝜆_𝑖𝐾₂ are Schur stable. Let 𝜇_𝑗,𝑗=1,…,𝑛, be the eigenvalues of the matrix 𝐾₂. Hence, the Schur stability of the matrix 𝐼_𝑛−𝜆_𝑖𝐾₂ is equivalent to that the products 𝜆_𝑖𝜇_𝑗, 𝑖=1,…,𝑁−1, 𝑗=1,…,𝑛, are in the open unit circle of the complex plane with the centre at (1,0).
Now we calculate the state consensus function. In this case, the state consensus function becomes 𝜉(𝑘)=𝑁⁻¹{𝐶(𝐼_{(𝑁−1)𝑛}−𝐴^𝑘)(𝐼_{(𝑁−1)𝑛}−𝐴)⁻¹𝑇+𝟏^𝑇_𝑁⊗𝐼_𝑛}𝑥(0). Since 𝐴 is Schur stable, instead of the previous state consensus function 𝜉(𝑘), we take the following state consensus value 𝜉=𝑁⁻¹{𝐶(𝐼_{(𝑁−1)𝑛}−𝐴)⁻¹𝑇+𝟏^𝑇_𝑁⊗𝐼_𝑛}𝑥(0). Since 𝑇=𝑇₀⊗𝐼_𝑛, 𝐴=𝐼_{(𝑁−1)𝑛}−𝑇₀𝐿_𝑤𝑇₀⊗𝐾₂, and 𝐶=−𝟏^𝑇_𝑁𝐿_𝑤𝑇₀⊗𝐾₂, and noticing that (𝑇₀𝐿_𝑤𝑇₀⊗𝐾₂)⁻¹=(𝑇₀𝐿_𝑤𝑇₀)⁻¹⊗𝐾₂⁻¹, the previous state consensus value can be written into (17).

Proof of Corollary 17. Since 𝐴^𝑗=1𝑗01⊗𝐼_𝑛, (A.4) Equation (11) becomes 𝜉(𝑘)=𝑁^−1𝑘−1_𝑗=0𝐼_𝑛𝑗𝐼_𝑛0𝐼_𝑛𝐶𝐴^{𝑘−1−𝑗}𝑇+𝟏^𝑇_𝑁⊗𝐼_𝑛𝑘𝐼_𝑛0𝐼_𝑛𝑥(0)=𝑁⁻¹𝐶^𝑘−1_𝑗=0𝐴^{𝑘−1−𝑗}+^𝑘−1_𝑗=00𝑗𝐼_𝑛00𝐶𝐴^{𝑘−1−𝑗}𝑇+𝟏^𝑇_𝑁⊗𝐼_𝑛𝑘𝐼_𝑛0𝐼_𝑛𝑥(0). (A.5) Letting 𝑋=^𝑘−1_𝑗=00𝑗𝐼_𝑛00𝐶𝐴^{𝑘−1−𝑗} (A.6) and 𝑗^′=𝑗+1, we get 𝑋=^𝑘_𝑗′=10𝑗^′−1𝐼_𝑛00𝐶𝐴^{𝑘−1−𝑗′+1}=^𝑘_𝑗′=10𝑗^′𝐼_𝑛00𝐶𝐴^{𝑘−1−𝑗′+1}−0𝐼_𝑛00𝐶𝐼_{2(𝑁−1)𝑛}−𝐴^𝑘=^𝑘−1_𝑗′=00𝑗^′𝐼_𝑛00𝐶𝐴^{𝑘−1−𝑗′+1}+0𝑘𝐼_𝑛00𝐶−0𝐼_𝑛00𝐶𝐼_{2(𝑁−1)𝑛}−𝐴^𝑘𝐼_{2(𝑁−1)𝑛}−𝐴⁻¹=𝑋𝐴+0𝑘𝐼_𝑛00𝐶−0𝐼_𝑛00𝐶𝐼_{2(𝑁−1)𝑛}−𝐴^𝑘×𝐼_{2(𝑁−1)𝑛}−𝐴⁻¹. (A.7) Thus, we have the following: 𝑋=0𝑘𝐼_𝑛00𝐶𝐼_{2(𝑁−1)𝑛}−𝐴⁻¹−0𝐼_𝑛00𝐶𝐼_{2(𝑁−1)𝑛}−𝐴^𝑘𝐼_{2(𝑁−1)𝑛}−𝐴⁻². (A.8) Substituting it into (A.5), we get 𝜉(𝑘)=𝑁⁻¹𝐶𝐼_{2(𝑁−1)𝑛}−𝐴^𝑘𝐼_{2(𝑁−1)𝑛}−𝐴⁻¹+0𝑘𝐼_𝑛00𝐶𝐼_{2(𝑁−1)𝑛}−𝐴⁻¹−0𝐼_𝑛00𝐶𝐼_{2(𝑁−1)𝑛}−𝐴^𝑘𝐼_{2(𝑁−1)𝑛}−𝐴⁻²𝑇+𝟏^𝑇_𝑁⊗𝐼_𝑛𝑘𝐼_𝑛0𝐼_𝑛𝑥(0). (A.9) If 𝐴 is Schur stable, the consensus function above can be replaced by 𝜉(𝑘)=𝑁⁻¹𝐶𝐼_{2(𝑁−1)𝑛}−𝐴⁻¹+0𝑘𝐼_𝑛00𝐶𝐼_{2(𝑁−1)𝑛}−𝐴⁻¹−0𝐼_𝑛00𝐶𝐼_{2(𝑁−1)𝑛}−𝐴⁻²𝑇+𝟏^𝑇_𝑁⊗𝐼_𝑛𝑘𝐼_𝑛0𝐼_𝑛𝑥(0). (A.10)

Proof of Theorem 20. Supposed that the matrices 𝐿_𝑤 and 𝐾₁ have been given, by Corollary 11, we need only to verify that the matrix 𝐾₂ ensures that all the matrices 𝐴+𝐵𝐾₁−𝜆_𝑖𝐵𝐾₂ are Schur stable, where 𝜆_𝑖, 𝑖=1,…,𝑁−1, are the eigenvalues of the matrix 𝑇₀𝐿_𝑤𝑇₀.
It is clear that there exists 𝜔>0 such that |𝛿_𝑖(𝜔)|≤𝛿(𝜔)<𝛿_𝑐, 𝑖=1,…,𝑁−1. We can verify that the following systems 𝜉_𝑖(𝑘+1)=(𝐴+𝐵𝐾₁−𝜆_𝑖𝐵𝐾₂)𝜉_𝑖(𝑘), 𝑖=1,…,𝑁−1, admit a common Lyapunov function 𝑉(𝜉_𝑖)=𝜉^𝐻_𝑖𝑃𝜉_𝑖. In fact, let 𝐾₂=𝜔(𝐵^𝑇𝑃𝐵)⁻¹𝐵^𝑇𝑃(𝐴+𝐵𝐾₁), then we have that Δ𝑉𝜉_𝑖=𝑉𝜉_𝑖(𝑘+1)−𝑉𝜉_𝑖(𝑘)=𝜉^𝐻_𝑖(𝑘)𝐴+𝐵𝐾₁−𝜆_𝑖𝜔𝐵𝐵^𝑇𝑃𝐵⁻¹𝐵^𝑇𝑃𝐴+𝐵𝐾₁^𝐻×𝑃𝐴+𝐵𝐾₁−𝜆_𝑖𝜔𝐵(𝐵^𝑇𝑃𝐵)⁻¹𝐵^𝑇𝑃×𝐴+𝐵𝐾₁−𝑃𝜉_𝑖(𝑘)=𝜉^𝐻_𝑖(𝑘)𝐴+𝐵𝐾₁^𝑇𝑃𝐴+𝐵𝐾₁−𝜆−𝑃_𝐻𝑖+𝜆_𝑖𝜔+𝜆^𝐻_𝑖𝜆_𝑖𝜔²×𝐴+𝐵𝐾₁^𝑇𝑃𝐵𝐵^𝑇𝑃𝐵⁻¹𝐵^𝑇𝑃𝐴+𝐵𝐾₁𝜉_𝑖(𝑘)=𝜉^𝐻_𝑖(𝑘)𝐴+𝐵𝐾₁^𝑇𝑃𝐴+𝐵𝐾₁−𝛿−𝑃1−_𝑖(𝜔)²𝐴+𝐵𝐾₁^𝑇𝑃𝐵𝐵^𝑇𝑃𝐵⁻¹×𝐵^𝑇𝑃𝐴+𝐵𝐾₁𝜉_𝑖(𝑘)≤𝜉^𝐻_𝑖(𝑘)𝐴+𝐵𝐾₁^𝑇𝑃𝐴+𝐵𝐾₁−−𝑃1−𝛿_𝑖(𝜔)𝐴+𝐵𝐾₁^𝑇𝑃𝐵𝐵^𝑇𝑃𝐵⁻¹×𝐵^𝑇𝑃𝐴+𝐵𝐾₁𝜉_𝑖(𝑘)<0. (A.11) That is, all the matrices 𝐴+𝐵𝐾₁−𝜆_𝑖𝐵𝐾₂, 𝑖=1,…,𝑁−1, are Schur stable.

Proof of Theorem 23. Let 𝑥_𝑖=𝑥_𝑖−ℎ_𝑖, 𝑖=1,…,𝑁. Then the DLMASs (1) reach the formation ℎ if and only if ‖(𝑥_𝑖(𝑘)−𝑥_𝑗(𝑘)‖→∞ as 𝑘→∞, for all 𝑖,𝑗=1,…,𝑁. So the formation problem on the variables 𝑥_𝑖 is transformed to the consensus problem on the variables 𝑥_𝑖.
From (1), one gets 𝑥⁺_𝑖=𝑥⁺_𝑖−ℎ_𝑖=𝐴(𝑥_𝑖+ℎ_𝑖)+𝐵𝑢_𝑖−ℎ_𝑖, 𝑖=1,…,𝑁. Substituting the protocol (24) and writing it into the vector form, we have that 𝑥⁺=𝐼_𝑁⊗𝐴+𝐵𝐾₁−𝐿_𝑤⊗𝐵𝐾₂𝑥+𝐼_𝑁⊗𝐴+𝐵𝐾₁−𝐼_𝑛ℎ. (A.12) Introducing the state linear transformation 𝑥=𝑇𝑥 for the linear system (A.12), one obtains 𝑥⁺=𝑇𝐼_𝑁⊗𝐴+𝐵𝐾₁−𝐿_𝑤⊗𝐵𝐾₂𝑇⁻¹𝑥+𝑇𝐼_𝑁⊗𝐴+𝐵𝐾₁−𝐼_𝑛ℎ. (A.13) Let 𝑦=[𝑥^𝑇₁,…,𝑥^𝑇_𝑁−1]^𝑇, then 𝑦⁺=𝐼_𝑁−1⊗𝐴+𝐵𝐾₁−𝑇₀𝐿_𝑤𝑇₀⊗𝐵𝐾₂𝑦+𝑇₀⊗𝐼_𝑁𝐼_𝑁⊗𝐴+𝐵𝐾₁−𝐼_𝑛ℎ (A.14) is held. Since 𝑦=0 must be the equilibrium point if the DLMASs (1) reach the formation ℎ, one has (𝑇₀⊗(𝐴+𝐵𝐾₁−𝐼_𝑛))ℎ=0. The residuary proof is similar to that of Theorem 7.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (61079001 and 61273006), National High Technology Research and Development Program of China (863 Program) (2011AA110301), Specialized Research Fund for the Doctoral Program of Higher Education of China (20111103110017), Hebei Province Science and Technology Research and Development Planning Project (10203548D), and Hebei Province Science and Technology Conditions Building Program (11963546D).