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

Authors: Yanrong Ge; Yangzhou Chen; Yaxiao Zhang; Zhonghe He
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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. Section2 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. Letℝn×n and ℂn×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. In presents the identity matrix of dimension n, and sometimes I is used for simplicity. 1N 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∈ℂn×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 withN homogeneous agents and assume they are described by
(1)xi+=Axi+Bui,i=1,…,N,
where A∈ℝn×n, B∈ℝn×m, (A,B) is assumed to be stabilizable, xi=xi(k)∈ℝn is the state of the current time k, xi+=xi(k+1) denotes the state at the next time k+1, and ui=ui(k)∈ℝm is the control input of the current time k.The control inputui will be constructed based on the available information of the agent i. Let 𝒩i denote the index set of the agents which can send their state information to the agent i. We call the set 𝒩={𝒩i: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∈𝒩i. Based on the directed edges, one can construct an adjacency matrix A=[aij]N×N, whose entries are defined as aij=0 for j=i, aij=1 for j∈𝒩i, and aij=0 for j∉𝒩i. The corresponding in-degree matrix and graph Laplacian are defined as D=diag{deg1,…,degN} and L=D-A, respectively, where degi=∑j=1Naij 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 matrixL∈ℝN×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 topology𝒩, we construct the following linear consensus protocol:
(2)ui=K1xi+K2∑j∈𝒩i‍wij(xj-xi),i=1,…,N,
where K1, K2∈ℝm×n are feedback gain matrices to be determined, which are relative to the consensus state and the convergence rate, respectively. W=:[wij]N×N is a weighted matrix associated with the information topology 𝒩. For the sake of expression, we also define the weighted adjacency matrix Aw=A∘W by using Hadamard product of matrices and the weighted Laplacian Lw=Dw-Aw with weights wij, where Dw=diag{deg1,…,degN} is the corresponding weighted in-degree matrix with weighted in-degrees degi=∑j∈𝒩iwij.Definition 2.
For the given information topology𝒩, the DLMASs (1) are said to achieve global state consensus via the protocol (2) if for any given initial state xi(0), i=1,…,N, there exists an n-dimensional vector function ξ(k) depending on the initial states such that limk→∞∥xi(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 topology𝒩, 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.Letx=[x1T,…,xNT]T. The dynamics of the DLMASs (1) with the protocol (2) is described by
(3)x+=Ψx,
where
(4)Ψ=IN⊗(A+BK1)-Lw⊗BK2.We propose a state linear transformation for the linear system (3) as follows:
(5)x¯=Tx,
where the block matrix T∈ℝnN×nN is defined as
(6)T=:[T~1NT⊗In],T~=[T1⋮TN-1].
And the matrix Ti=[Ti1,…,TiN],  i=1,…,N-1, is chosen such that the following two conditions are satisfied:(1)
the row vectors in each of the matricesTi are linearly independent, respectively;(2)
the identitiesTi(1N⊗In)=0, i=1,…,N-1, are held.Lemma 3.
The inverseT-1 of the matrix T admits the following form:
(7)T-1=[T¯11⋯T¯1,N-1N-1In⋮⋱⋮⋮T¯N-1,1⋯T¯N-1,N-1N-1InT¯N,1⋯T¯N,N-1N-1In]=:[T¯1⋯T¯N-1N-11N⊗In]=:[T^N-11N⊗In],
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:
(8)x¯+=TΨT-1x¯,
or the form of two equations
(9)y+=T~ΨT^y+N-1T~Ψ(1N⊗In)z,z+=(1NT⊗In)ΨT^y+N-1(1NT⊗In)Ψ(1N⊗In)z,
where x¯=[yT,zT]T, y=[x¯1T,…,x¯N-1T]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 limk→∞y(k)=0 for any bounded initial state x¯(0) of the system (8).Lemma 5.
Under the given information topology𝒩, 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∑i=1Nxi(k)=N-1x¯N(k).Lemma5 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:
(10)y+=A¯y,y∈ℝn(N-1),z+=C¯y+D¯z,z∈ℝn,
where A¯=T~(IN⊗(A+BK1)-Lw⊗BK2)T^, C¯=-(1NTLw⊗BK2)T^, and D¯=A+BK1.Combining Lemma5 with Lemma 6, we directly get the following theorem.Theorem 7.
Under the given information topology𝒩, 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
(11)ξ(k)=N-1((∑j=0k-1‍(A+BK1)jC¯A¯k-1-j)T~+1NT⊗(A+BK1)k(∑j=0k-1‍(A+BK1)jC¯A¯k-1-j))x(0).Subsequently, we give some interesting remarks and corollaries based on the result.Remark 8.
SinceT^T~=(IN-N-11N1NT)⊗In, 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:
(12)T=[1-10⋯001-1⋯0⋮⋮⋱⋮⋮0⋯01-1111⋯1]⊗In:=[T~01NT]⊗In.
The corresponding inverse matrix is
(13)T-1=1N[N-1N-2⋯11-1N-2⋯11⋮⋮⋱⋮⋮-1-2⋯11-1-2⋯-(N-1)1]⊗In:=[T^0N-11N]⊗In.
Thus, we can write A¯ and C¯ into
(14)A¯=IN-1⊗(A+BK1)-T~0LwT^0⊗BK2,C¯=-1NTLwT^0⊗BK2.Corollary 9.
Under the given information topology𝒩, the DLMASs (1) achieve global consensus via the protocol (2) if and only if all the matrices A+BK1-λiBK2 are Schur stable, where λi, i=1,…,N-1, are the eigenvalues of the matrix T~0LwT^0. Moreover, the state consensus function is expressed by
(15)ξ(k)=(ηT⊗(A+BK1)k)x(0),i=1,…,N,
where η satisfies ηTLw=0 and ηT1N=1.One can verify that ask→∞ the state consensus functions in formulas (11) and (15) are the same.Remark 10.
From Schur stability ofA¯ 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 -T~0LwT^0, a lack of directed spanning tree means that -T~0LwT^0 has a zero eigenvalue. In this case, we transform A¯ into its Jordan form via the matrix U⊗In, where U is the matrix such that U-1T~0LwT^0U=J is the Jordan form, and thus we have
(16)(U-1⊗In)A¯(U⊗In)=IN-1⊗(A+BK1)-J⊗BK2.
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.
IfA+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 T~0LwT^0 has N-1 eigenvalues with positive real part λi, i=1,…,N-1, and the matrices A+BK1-λiBK2, 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 (∑j=0k-1(A+BK1)jC¯A¯k-1-j)T~+1NT⊗(A+BK1)k=1NT⊗In.IfA+BK1=In, then the last condition in Corollary 13 becomes 1NTLw=0, or equivalently, the digraph G is either undirected connected or directed strong connected and balanced. More specially, if Lw is a symmetric matrix (equivalently, the digraph G becomes undirected connected), the condition 1NTLw=0 is satisfied and thus the average consensus is achieved.Remark 14.
WhenA=In, K1=0, and B=In, the DLMASs (1) are called a single-integrator one. In this case, A¯=I(N-1)n-T~0LwT^0⊗K2 and C¯=-1NTLwT^0⊗K2. We educe the following result.Corollary 15.
Under the given information topology𝒩, 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 -T~0LwT^0 is Hurwitz stable; that is, the digraph G admits a directed spanning tree; (2) the products λiμj, i=1,…,N-1, j=1,…,n, are in the open unit circle of the complex plane with the centre at (1,0), where λi, i=1,…,N-1, are the eigenvalues of the matrix T~0LwT^0 and μj, j=1,…,n, are the eigenvalues of the matrix K2. The corresponding consensus function (11) becomes a constant vector
(17)ξ=N-1(1NT⊗In-1NTLwT^0(T~0LwT^0)-1T~0⊗In)x(0).
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 1NTLw=0.Remark 16.
WhenA=[1101]⊗In, B=[01]⊗In, 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 ℝn.Corollary 17.
Under the given information topology𝒩, 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
(18)ξ(k)=N-1{(C¯(I2(N-1)n-A¯)-1+[0kIn00]C¯(I2(N-1)n-A¯)-1-[0In00]C¯(I2(N-1)n-A¯)-2)T~+1NT⊗[InkIn0In]}x(0).The consensus function above can be decomposed into the position consensus function(19)ξ1(k)=N-1{([In0]C¯(I2(N-1)n-A¯)-1+[0kIn]C¯(I2(N-1)n-A¯)-1-[0In]C¯(I2(N-1)n-A¯)-2)T~+1NT⊗[InkIn](I2(N-1)n-A¯)-1}x(0),
which is a linear function of discrete time k, and the constant velocity consensus function is as follows:
(20)ξ2(k)=N-1{([0In]C¯(I2(N-1)n-A¯)-1)T~+1NT⊗[0In]([0In]C¯(I2(N-1)n-A¯)-1)}x(0).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 topology𝒩, the double-integrator DLMASs (1) achieve global velocity average consensus via the protocol (2) if and only the matrix A¯ is Schur stable and [0In]C¯=0.It is obvious that if1NTLw=0, then C¯=0; that is, the last condition in Corollary 18 is satisfied, and thus, the state consensus function becomes
(21)ξ(k)=N-11NT⊗[InkIn0In]x(0).


## 4. Design of Gain Matrices

In this section, we discuss the third problem, that is, how to determine the weighted matrixW and the gain matrices K1 and K2, such that the DLMASs (1) achieve global state consensus via the protocol (2).Theorem7 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-λiBK2 are Schur stable, where λi, i=1,…,N-1, are the eigenvalues of the matrix T~0LwT^0. 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 matrixA^=A+BK1 is not Schur stable but (A^,B) is stabilizable, then there is a critical value δc∈(0,1], such that for any number δ with 0<δ<δc the modified Riccati inequality
(22)A^TPA^-P-(1-δ2)A^TPB(BTPB)-1BTPA^<0
admits a positive definite matrix solution P, where δc depends on the unstable eigenvalues of the matrix A^.We define functionsδi(ω)=1-ωλi and δ(ω)=maxi∈{1,…,N-1}|δi(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 -T~0LwT^0 is Hurwitz stable with N-1 eigenvalues -λi, 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(BTPB)-1BTP(A+BK1), where ω is an arbitrary constant satisfying δ=δ(ω)<δc, δc∈(0,1] is a critical value which depends on the unstable eigenvalues of the matrix A+BK1, and PT=P>0 is a solution of the algebraic Riccati inequality (22).Based on Theorem20, 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 matricesK1 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 Lw such that -T~0LwT^0 is Hurwitz stable with N-1 eigenvalues -λi, i=1,…,N-1.Step 2.
DesignK1 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 ofA^, which are composed of the stable eigenvalues λis(A^), i=1,…,ns and unstable ones λiu(A^), i=1,…,nu, ns+nu=n.Step 4.
Calculate the critical valueδc∈(0,1]. If B is invertible, then δc=(maxi={1,…,nu}|λiu(A^)|)-1. If B is of rank one, then δc=(∏i={1,…,nu}|λiu(A^)|)-1. Otherwise, apply Wonham decomposition to the unstable part (Au,Bu) 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 Au. Specifically, there is a nonsingular real matrix Q with a compatible dimension such that A~=Q-1AuQ and B~=Q-1Bu take the form
(23)A~=[A1***0A2⋯*⋮⋮⋱⋮00⋯Am],B~=[b1***0b2⋯*⋮⋮⋱⋮00⋯bm],
where the symbol * denotes possibly nonzero parts and (Aj,bj) with Aj∈ℝnj×nj and bj∈ℝnj for all j∈{1,…,m} is controllable and ∑j=1m‍nj=nu. In this case, δc is lower bounded by δc≥(∏i‍|λiu(Am*)|)-1)=δc′, where the index m* is defined by m*=argmaxj={1,…,m}(∏i‍|λiu(Aj)|) and Aj is the Jordan block of the unstable part of matrix A^.Step 5.
Calculate the valueω such that δ(ω)=maxi∈{1,…,N-1}|δi(ω)|.Step 6.
Solve (22) with δ=δ(ω) for a positive definite matrix P.Step 7.
Calculate the matrixK2=ω(BTPB)-1BTP(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=[h1Th2T⋯hNT]T∈ℝnN describe a constant formation of the agent network in a reference coordinate frame, where hi∈ℝn is the formation variable corresponding to the agent i. The variable hi-hj 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:
(24)ui=K1xi+K2∑j∈𝒩i‍wij(xj-xi-(hj-hi)),i=1,…,N.Definition 22.
Under the given information topology𝒩, the DLMASs (1) achieve the given formation h via the protocol (24) if ∥(xi(k)-xj(k))-(hi-hj)∥→0 as k→∞, for all i,j=1,…,N, that is, if there is a function ξ(k) such that ∥xi(k)-hi-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 topology𝒩, the DLMASs (1) achieve the formation h via the protocol (24) if and only if the matrix IN-1⊗(A+BK1)-T~0LwT^0⊗BK2 is Schur stable and (T~0⊗(A+BK1-In))h=0. Moreover, the reference state consensus function is
(25)ξ(k)=N-1((∑j=0k-1‍(A+BK1)jC¯A¯k-1-j)T~+1NT⊗(A+BK1)k(∑j=0k-1‍(A+BK1)jC¯A¯k-1-j))x(0).Similarly to Corollary11, we get the following corollary for the formation control.Corollary 24.
Under the given information topology𝒩, the DLMASs (1) achieve the formation h via the protocol (24) if and only if all the matrices A+BK1-λiBK2 are Schur stable and (T~0⊗(A+BK1-In))h=0, where λi, i=1,…,N-1, are the eigenvalues of the matrix T~0LwT^0.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 (T~0⊗(A+BK1-In))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:
(26)A=[222-22-1224324100],B=[1-121].
Supposed that we are given the information topology 𝒩={{4},{1},{2},{3}}, the weights wij=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 matrixA¯ 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(27)xi+=xi+vi,vi+=vi+ui,i=1,…,4,
where xi∈ℝ2, vi∈ℝ2, and ui∈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 T~0LwT^0 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 (T~0⊗(A+BK1-In))h=0. One can get δc=1 through Step 4 of the Algorithm 21. By Step 5, one can obtain ω=1.0, and the corresponding δ(ω)=2/2<δc. The modified Riccati inequality (22) admits a solution
(28)P=[88170017248000088170017248]×10-4.
The corresponding control gain is calculated as follows:
(29)K2=[000.06861.06860.06861.068600].
The eigenvalues of matrixA¯ 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 statesx1(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.



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*Source: 192351-2013-07-17.xml*