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

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.


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][5][6], rendezvous, and the formation of various moving objects [7][8][9][10][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][13][14][15][16][17][18].This paper focuses on the study of highorder 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 selfpropelled 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 firstorder systems.The extension to second-order systems was done for the systems with time-varying delays and timevarying 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][35][36][37][38][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.

Problem Description
Before stating the consensus problem, we give some basic concepts and notations.Let R × and C × be the sets of  ×  real matrices and complex matrices, respectively.Matrices, if not explicitly stated, have appropriate dimensions in all settings.The superscript "" means transpose for real matrices, and the superscript "" means conjugate transpose for complex matrices.  presents the identity matrix of dimension , and sometimes  is used for simplicity. 1  denotes the vector of dimension  with all entries equal to one.0 is applied to denote zero matrices/vectors of any size, with zero components.A matrix  ∈ C × 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: We consider DLMASs with  homogeneous agents and assume they are described by where  ∈ R × ,  ∈ R × , (, ) is assumed to be stabilizable,   =   () ∈ R  is the state of the current time ,  +  =   ( + 1) denotes the state at the next time  + 1, and   =   () ∈ R  is the control input of the current time .
The control input   will be constructed based on the available information of the agent .Let N  denote the index set of the agents which can send their state information to the agent .We call the set N = {N  :  = 1, . . ., } the information topology of the DLMASs (1).It is well known that one can use a digraph  = (, ) to express the information topology N, where  = {1, . . ., } is the index set of  agents,  ⊆  ×  is the set of directed edges to describe the information interaction between agents; that is, (, ) ∈  ⇔  ∈ N  .Based on the directed edges, one can construct an adjacency matrix  = [  ] × , whose entries are defined as   = 0 for  = ,   = 1 for  ∈ N  , and   = 0 for  ∉ N  .The corresponding in-degree matrix and graph Laplacian are defined as  = diag{deg 1 , . . ., deg  } and  =  − , respectively, where deg  = ∑  =1   is the in-degree of the vertex .A directed spanning tree of the digraph  is a tree covering all the vertices of the digraph.The following results are well known.
Lemma 1 (see [22]).The Laplacian matrix  ∈ R × has the following properties: (1) all of the eigenvalues of  are either in the open right half complex plan or equal to 0; (2) 0 is a simple eigenvalue of  if and only if the digraph  contains a directed spanning tree.
Given the information topology N, we construct the following linear consensus protocol: where  1 ,  2 ∈ R × are feedback gain matrices to be determined, which are relative to the consensus state and the convergence rate, respectively. =: [  ] × is a weighted matrix associated with the information topology N.For the sake of expression, we also define the weighted adjacency matrix   =  ∘  by using Hadamard product of matrices and the weighted Laplacian   =   −   with weights   , where   = diag{deg 1 , . . ., deg  } is the corresponding weighted in-degree matrix with weighted in-degrees deg  = ∑ ∈N    .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   (0),  = 1, . . ., , there exists an -dimensional vector function () depending on the initial states such that lim  → ∞ ‖  () − ()‖ = 0.The function () 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  and feedback gain matrices  1 and  2 to find the conditions of the DLMASs (1) achieving global state consensus via the protocol (2); (ii) to calculate the state consensus function () if the DLMASs (1) achieve global state consensus via the protocol (2); (iii) to determine the matrices  1 and  2 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.

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.
Using the linear transformation (5), we transform the linear system (3) into the following system: or the form of two equations where  = [  ,   ]  ,  = [  1 , . . .,   −1 ]  , and  =   .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  (or asymptotically stable in short) if lim  → ∞ () = 0 for any bounded initial state (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  = 0 of the linear system (8) is asymptotically -stable.Moreover, the state consensus function of the agents is Lemma 5 builds a bridge between the consensus problem and the partial stability problem.Now we focus on the asymptotical -stability of the linear system (8).We can verify the following lemma.Lemma 6.The system (9) is of the following form: where  = T(  ⊗ ( +  1 ) −   ⊗  2 ) T,  = −(1 T    ⊗  2 ) T, and  =  +  1 .
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  in (10) is Schur stable.Moreover, the state consensus function is Subsequently, we give some interesting remarks and corollaries based on the result.Remark 8. Since T T = (  −  −1 1  1   ) ⊗   , the result of Theorem 7 is in fact independent of the choice of the matrix  although both  and formula (11) in Theorem 7 contain T and T. Hence, for simplicity, we take it in the following form: The corresponding inverse matrix is Thus, we can write  and  into where  satisfies     = 0 and   1  = 1.
One can verify that as  → ∞ the state consensus functions in formulas (11) and ( 15) are the same.
Remark 10.From Schur stability of  in the formula (14), we can conclude that if  +  1 is not Schur stable, it is a necessary condition of the consensus that the digraph  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 − T0   T0 , a lack of directed spanning tree means that − T0   T0 has a zero eigenvalue.In this case, we transform  into its Jordan form via the matrix  ⊗   , where  is the matrix such that  −1 T0   T0  =  is the Jordan form, and thus we have One can verify that the eigenvalues of  +  1 are the members of the eigenvalues of , and thus  is not Schur stable if  +  1 is not Schur stable.On the other hand, if  +  1 is Schur stable, one can take  2 = 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) 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 (∑
If  +  1 =   , then the last condition in Corollary 13 becomes 1      = 0, or equivalently, the digraph  is either undirected connected or directed strong connected and balanced.More specially, if   is a symmetric matrix (equivalently, the digraph  becomes undirected connected), the condition 1      = 0 is satisfied and thus the average consensus is achieved.Remark 14.When  =   ,  1 = 0, and  =   , the DLMASs (1) are called a single-integrator one.In this case,  =  (−1) − T0   T0 ⊗  2 and  = −1     T0 ⊗  2 .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 − T0   T0 is Hurwitz stable; that is, the digraph  admits a directed spanning tree; (2) the products     ,  = 1, . . .,  − 1,  = 1, . . ., , are in the open unit circle of the complex plane with the centre at (1, 0), where   ,  = 1, . . .,  − 1, are the eigenvalues of the matrix T0   T0 and   ,  = 1, . . ., , are the eigenvalues of the matrix  2 .The corresponding consensus function (11) becomes a constant vector 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     = 0.
Remark 16.When  = [ 1 1 0 1 ]⊗  ,  = [ 0 1 ]⊗  , and  1 = 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  dimensional space R  .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  is Schur stable.Moreover, the consensus function is The consensus function above can be decomposed into the position consensus function which is a linear function of discrete time , and the constant velocity consensus function is as follows: 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.

It is obvious that if 1 𝑇
= 0, then  = 0; that is, the last condition in Corollary 18 is satisfied, and thus, the state consensus function becomes

Design of Gain Matrices
In this section, we discuss the third problem, that is, how to determine the weighted matrix  and the gain matrices  1 and  2 , such that the DLMASs (1) achieve global state consensus via the protocol (2).Theorem 7 shows that the matrices ,  1 , and  2 should be taken to ensure that the matrix  is Schur stable.Furthermore, from Corollary 11 we see that if the matrix  with respect to the information topology N has been given, we need only to design the gain matrices  1 and  2 to ensure that the matrices  +  1 −    2 are Schur stable, where   ,  = 1, . . .,  − 1, are the eigenvalues of the matrix T0   T0 .The matrix  1 is often taken to obtain an expected consensus dynamics.The matrix  2 is designed to achieve state consensus and expected convergence rate.Its design needs the following lemma.
Theorem 20.Supposing that the matrix (, ) is stabilizable, the gain matrix  1 has been taken such that the expected consensus dynamic matrix  +  1 is not Schur stable, and the weighted matrix  with respect to the information topology N is given such that − T0   T0 is Hurwitz stable with  − 1 eigenvalues −  ,  = 1, . . .,  − 1; then, for the DLMASs (1) to achieve state consensus via the protocol (2), the matrix  2 can be designed as  2 = (  ) −1   ( +  1 ), where  is an arbitrary constant satisfying  = () <   ,   ∈ (0, 1] is a critical value which depends on the unstable eigenvalues of the matrix  +  1 , and   =  > 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  1 and  2 in the protocol (2).Algorithm 21.Design procedure of the gain matrices  1 and  2 .
Step 1. Verify the stabilizability condition of (, ) and the spanning tree condition of the information topology N. If neither of them is satisfied, then stop.Otherwise, design the weighted Laplacian   such that − T0   T0 is Hurwitz stable with  − 1 eigenvalues −  ,  = 1, . . .,  − 1.
Step 2. Design  1 such that Â =  +  1 is the matrix of the expected consensus dynamics of the DLMASs (1) and is not Schur stable.
Step 4. Calculate the critical value Otherwise, apply Wonham decomposition to the unstable part (  ,   ) of ( Â, ) to convert the multiple input system to  single input subsystems, where  is the number of the Jordan blocks of matrix   .Specifically, there is a nonsingular real matrix  with a compatible dimension such that Ã =  −1    and B =  −1   take the form where the symbol * denotes possibly nonzero parts and (  ,   ) with   ∈ R   ×  and   ∈ R   for all  ∈ {1, . . ., } is controllable and , where the index  * is defined by  * = arg max ={1,...,} (∏  |   (  )|) and   is the Jordan block of the unstable part of matrix Â.

Application to Formation Control
In this section, the consensus approach is modified to solve the formation control problem of the DLMASs (1).
∈ R  describe a constant formation of the agent network in a reference coordinate frame, where ℎ  ∈ R  is the formation variable corresponding to the agent .
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  −1 ⊗ ( +  1 ) − T0   T0 ⊗  2 is Schur stable and ( T0 ⊗ ( +  1 −   ))ℎ = 0.Moreover, the reference state consensus function is Similarly to Corollary 11, we get the following corollary for the formation control.

Numerical Examples
In this section, we give some illustrative examples.

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 highorder 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.

Figure 2 :
Figure 2: Position and velocity trajectories of mobile robots.