# RobustH∞ Consensus Control for Linear Discrete-Time Swarm Systems with Parameter Uncertainties and Time-Varying Delays

Authors: Xiao Jia; Laihong Hu; Fujun Feng; Jun Xu
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

RobustH∞ consensus control problems of linear swarm systems with parameter uncertainties and time-varying delays are investigated. In this literature, a linear consensus protocol for high-order discrete-time swarm systems is proposed. Firstly, the robust H∞ consensus control problem of discrete-time swarm systems is transformed into a robust H∞ control problem of a set of independent uncertain systems. Secondly, sufficient linear matrix inequality conditions for robust H∞ consensus analysis of discrete-time swarm systems are given by the stability theory, and a H∞ performance level γ is determined meanwhile. Thirdly, the convergence result is derived as a final consensus value of swarm systems. Finally, numerical examples are presented to demonstrate theoretical results.


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## Body

## 1. Introduction

As a hot topic of swarm systems, distributed cooperative control for swarm systems has gained an increasing attention from many communities because of its wide applications in various areas including rendezvous [1–3], filtering [4–6], formation control [7–9], synchronization of power-driven machine [10–12], and other areas [13–16]. It should be also pointed out that some relevant novel nearest results in aerospace have been reported in [17–19]. Furthermore, as an essential research aspect of distributed cooperative control for swarm systems, the consensus problem, which is referred to an agreement on some common values by all agents, has received extensive research in recent years [20–22].According to the difference of the system dynamics, the research objective of the consensus problem for swarm systems can be classified into three types: first-order swarm systems, second-order swarm systems, and high-order swarm systems. For first-order swarm systems, the consensus problem with input and communication delays was studied by the frequency analysis in [23]. The consensus of second-order agents with multiple time-varying delays was investigated based on the graph theory in [24]. In combination with realistic applications, high-order swarm systems are more valuable and general than low-order swarm systems due to the practicality and complexity of the former. Some significant theoretical achievements about the consensus of continuous-time high-order swarm systems have been discovered in [25]. However, digital signals are processed rather than analog signals under engineering background, so discrete-time swarm systems should be paid more attention. The distributed output feedback consensus problem of general linear model discrete-time swarm systems with fixed topology was presented in [26, 27]. Based upon the leader-following model, the consensus problem of discrete-time swarm systems with directed graphs and external disturbances was investigated in [28]. Furthermore, the consensus problem of discrete-time swarm systems with switching topologies based on the matrix inequality theory was presented in [29–31].As the industrial technology continues to develop, the research on the control theory becomes more mature. Due to operating state changes, external disturbances and model errors of swarm systems in the real world, robustH∞ consensus control problems of swarm systems were wildly explored by researchers, as shown in [32–34]. Robust H∞ consensus control problems of first-order swarm systems with external disturbances and model uncertainties were presented in [32]. The second-order robust H∞ consensus control problem of swarm systems with measurement noises and asymmetric delays was studied in [33]. Distributed H2 and H∞ consensus control problems of high-order linear swarm systems with x,y,z‐dependent noise were investigated in [34]. However, due to the general existence of parameter uncertainties and time delays of swarm systems in practical application, it is more meaningful and important to take these constraints into consideration in the theoretical analysis of robust H∞ consensus control problems of high-order linear discrete-time swarm systems, which can been deemed as inadequately available in current literature.As is well known, precise dynamics models of swarm systems can be hardly established in practice, and many kinds of faults in systems can also contribute to uncertainties of the dynamics model. Besides, components of systems will be out of order or aged with the increase of working hours. Therefore, researches on the robustH∞ consensus problem of high-order linear discrete-time swarm systems with parameter uncertainties are very valuable and significant. Moreover, time delays cannot be avoided in nature and they take place so commonly in biological, physical, and electrical systems. The acquisition of parameters becomes more difficult when time delays exist in systems. If time delays are ignored when the consensus problem of swarm systems with parameter uncertainties is discussed, the stability of the system is hard to be guaranteed. Motivated by the above analysis, for high-order linear discrete-time swarm systems with parameter uncertainties to achieve consensus by the robust H∞ control method, time delays are necessary to be considered and the problem remains the most open as we know.This paper studies robustH∞ control consensus problems of high-order linear discrete-time swarm systems with parameter uncertainties and time-varying delays, and the innovations of this paper are threefold: (i) a sufficient linear matrix inequality (LMI) condition such that swarm systems can achieve the robust H∞ consensus subject to parameter uncertainties and time-varying delays is derived; (ii) a H∞ performance index γ of swarm systems is obtained; and (iii) the final consensus value of swarm systems is developed. Compared with the previous work on the robust H∞ consensus of high-order swarm systems, the novelties of this literature are also threefold. Firstly, only the robust H∞ consensus of high-order linear swarm systems was investigated in [35–39], where the H∞ performance index was not obtained. However, in order to show the performance of the robust consensus of high-order linear discrete-time swarm systems, the H∞ performance index, which is necessary and important, was involved in this paper. Secondly, the final consensus value was obtained by LMI theory in this paper, which could better show the advantages of the proposed consensus protocol, but most existing works for the robust H∞ consensus of high-order linear swarm systems subject to parameter uncertainties and time delays, including [35–38], did not consider this useful point. Thirdly, to better correspond with the practical application, influences of time delays on the robust consensus of swarm systems were sufficiently considered in this paper, which did not appear in [37–39].The remainder of the paper is organized as follows. Section2 states several important conclusions of the graph theory and the problem description in which the robust H∞ consensus protocol is proposed. Section 4 shows a sufficient LMI condition that can guarantee the robust H∞consensus of swarm systems and a final consensus value of swarm systems. The effectiveness of theoretical results is illustrated via simulation examples in Section 5. Finally, Section 5 draws the conclusion.Notations.
A matrix can be defined to be positive if each entry of the matrix is positive. A square matrix can be known as Schur stable if each eigenvalue of the matrix lies in the open unit diskΘ=λ∈ℂ:λ<1. diaga11,a22,⋯,ann denotes the diagonal matrix, where a11,a22,⋯,ann are diagonal entries. E refers to the expectation. The symbol ⊗ represents the Kronecker product. The transpose of a matrix M is denoted as MT. I stands for the identity matrix with appropriate dimension. Let 1N=1,⋯,1T be the vector of N ones with all components 1.


## 2. Preliminaries and Problem Description

### 2.1. Graph Theory

In general, exchanges of information amongN agents in swarm systems can be shown as a directed graph G=V,ℰ,A, where V=v1,⋯,vN denotes the node set, ℰ⊆vi,vj:vi,vj∈V represents the edge set, and A=aij refers to the weighted adjacency matrix with the element aij>0 if vj,vi∈ℰ while aij=0 if vj,vi∉ℰ. Define the node index of graph G as a finite index set ℐ=1,2,⋯,N. Let Ni=vj∈V:vj,vi∈ℰ stand for the set of all neighbours of node vi, where the information flows from node vj to node vi. Let subset J⊆V of the nodes in graph G be a cluster. Suppose that NJ=∪vi∈JNi=vj∈V:vi∈J,vi,vj∈ℰ represents the neighbour set of a cluster NJ. It can be defined that the in-degree matrix of node vi is deginvi=∑j=1Naji, and the out-degree matrix of node vi is degoutvi=∑j=1Naij. Note that a diagonal matrix Δ=Δij can be provided to describe the degree matrix of digraph G, which is defined as follows:
(1)Δij=0,i≠j,degoutvi,i=j.The definition of the Laplacian matrix in connection with digraphG can be shown in matrix form as ℒG=L=Δ−A.


### 2.2. Problem Description

Consider a linear swarm system composed ofN homogeneous agents marked as 1,2,⋯,N, in which the ith agent’s dynamics can be expressed as a state update equation
(2)xik+1=A+ΔAxik+B+ΔBuik+Bωωik,zik=Cxik,where xik∈Rd indicates the state vector; uik=ui,1k,ui,2k,⋯,ui,m1kT∈Rm1 and ωik=ωi,1k,ωi,2k,⋯,ωi,m2kT∈Rm2, respectively, represent the input matrix and the external disturbance; A∈Rd×d, B∈Rd×m1, Bω∈Rd×m2, and C∈Rl×d are the given constant matrices; zik∈Rl is the control output; ΔA and ΔB stand for uncertainties in the system matrix and input channel, respectively, such that
(3)ΔAΔB=DFE1E2,where D denotes a positive constant matrix and F refers to an unknown matrix function which meets constraints as follows:
(4)FTF≤I.The parameter uncertaintiesΔA and ΔB are considered permissible when equations (3) and (4) hold meanwhile.Remark 1.
When there is the external disturbance in system (2), external disturbance ωik=ωi,1k,ωi,2k,⋯,ωi,m2kT∈Rm2 belongs to l2 signal shown in [33, 40], which satisfies ∑k=1nωi,jk2<∞i=1,⋯,N,j=1,⋯,m2, that is, the energy of the external disturbance in system (2) is bounded on any finite interval. In this case, the robust H∞ control problem can be analysed in an appropriate and meaningful way.To achieve consensus for linear swarm system (2), the following control protocols are adopted:
(5)uik=K∑j∈Niaijxjk−τk−xik,where 0≤τk≤τmax, K is a constant gain matrix with appropriate dimensions, and τk∈ℤ∗ denotes time-varying delays of swarm systems. Then, to show the consensus analysis and design for swarm system (2), the definition is presented as follows:Definition 1.
For the given gain matrixK and any given bounded initial states, swarm system (2) with consensus protocol (5) are assured to arrive consensus if there exists a vector-valued ck depending on the initial states such that limk→∞xk−1N⊗ck=0, where ck denotes the final consensus value.
Letxk=x1Tk,⋯,xNTkT, ωxk=ω1T,⋯,ωNTT, and zk=z1Tk,⋯,zNTkT, and we can write swarm system (2) with equation (5) into a compact form as follows:
(6)xk+1=IN⊗A+ΔA−Δ⊗BΔBKxk+Δ−L⊗BΔBKxk−τk+IN⊗Bωωxk,xk=ϕk,k∈−τmax,−τmax+1,⋯,0,zk=IN⊗Cxk,where BΔBK=B+ΔBK and ϕk is an initial value at k.The suboptimal robustH∞ consensus control problem of system (6) is stated to find a distributed protocol (5) such that
(i)
withωxk=0, system (6) is asymptotically stable for all acceptable uncertain matrices F
(ii)
withωxk interpreted as deterministic l2 signal, the transfer function from ωxk to zk of system (6), which is denoted by Tωz, satisfies Tωz∞<γ for all acceptable uncertain matrices F and a given allowable scalar γ>0, where Tωz∞, is the H∞ norm of Tωz, defined by Tωz∞=supωx∈RNdσ¯TωzjωIn order to make an in-depth analysis for the robustH∞ consensus control problem of system (6), we presume henceforth that the communication graph G is connected and give the following lemma about the graph theory.Lemma 1 (see [41]).
For the Laplacian matrixL associated with an undirected graph G, the following description holds: (i) 0 is an eigenvalue of L, and its corresponding eigenvector is 1N, and (ii) 0 is a simple eigenvalue of L, and all the other eigenvalues are positive and real if G is connected.Assume thatλii=1,2,⋯,N denotes eigenvalues of matrix L∈RN of an undirected graph G. Moreover, λ1=0, which satisfies λ1≤λ2≤⋯≤λN, is related to its corresponding eigenvector u¯1=1/N−1/21N. Thus, an orthogonal matrix can be found in the form
(7)U=1N1N−1TN1N−1TNU¯,such that UTLU=diagλ1,λ2,⋯,λN.Theorem 1.
For a givenγ>0, system (6) is asymptotically stable and Tωz∞<γ, if and only if the following N systems are simultaneously and asymptotically stable and all of the H∞ norms of their transfer function matrices are less than γ:
(8)x~ik+1=AΔiiBKx~ik+BΔλiKx~ik−τk+Bωω~i,xk,x~k=ϕ~k,k∈−τmax,−τmax+1,⋯,0,z~ik=Cx~ik,i=1,2,⋯,N,where AΔiiBK=A+ΔA−ΔiiBΔBK and BΔλiK=Δii−λiB+ΔBK.Proof.
Letλii=1,2,⋯,N be the eigenvalues of matrix L∈RN for an undirected topology G, where λ1=0 with the corresponding eigenvector u¯1=1/N−1/21N, and λ1≤λ2≤⋯≤λN. An orthogonal matrix can be found in the form
(9)U=1N1N−1TN1N−1TNU¯,such that UTLU=diagλ1,λ2,⋯,λN=Λ. Assume that
(10)x~k=UT⊗Idxk=x~cTk,x~rTkT=x~1Tk,x~2Tk,⋯,x~NTkT.
Then, system (6) can be rewritten in terms of x~k as
(11)x~k+1=IN⊗A+ΔA−Δ⊗BΔBKx~kk−Δ−Λ⊗BΔBKx~k−τk+UT⊗Bωωxk,x~k=ϕ~k,k∈−τmax,−τmax+1,⋯0,zk=U⊗Cx~k.
Furthermore, reformulate the disturbance variableωxk and the performance variable zk via
(12)ω~xk=UT⊗Im2ωxk=ω~1,xTk,ω~2,xTk,⋯,ω~N,xTkT,(13)z~k=UT⊗Ilzk=z~1Tk,z~2Tk,⋯z~NTkT.
Subsequently, substituting (12) and (13) into (11) gives
(14)x~k+1=IN⊗A+ΔA−Δ⊗BΔBKx~k−Δ−Λ⊗B+ΔBKx~k−τk+IN⊗Bωω~xk,x~k=ϕ~k,k∈−τmax,−τmax+1,⋯,0,z~k=IN⊗Cx~k.
In fact, system (14) consists of N individual systems in (6). Denote by Tω~z~∞ and Tω~iz~i∞the transfer function matrices of system (14) and (6), respectively. Afterwards, we can deduce from equations (6), (12), (13), and (14) that
(15)Tω~z~=diagTω~1z~1,Tω~2z~2,⋯,Tω~Nz~N=UT⊗IlTωzU⊗Im2,which implies that
(16)Tω~z~∞=maxi=2,3,⋯,NTω~iz~i∞=Tωz∞.
Therefore, the proof of Theorem1 is completed.In addition, it is worth mentioning that, let(17)x~k=x~cTk,x~rTkT=x~cTk,x~r,2Tk,⋯,x~r,NTkT,ω~xk=ω~c,xTk,ω~r,xTkT=ω~c,xTk,ω~r,2xTk,⋯,ω~r,NxTkT,z~=z~cTk,z~rTkT=z~cTk,z~r,2Tk,⋯,z~r,NTkT.By Lemma1, we can convert system (14) into N subsystems as follows:
(18)x~ck+1=A+ΔA−Δ11BΔBKx~ck+Δ11B+ΔBKx~ck−τk+Bωω~c,xk,x~ck=ϕ~ck,k∈−τmax,−τmax+1,⋯,0,z~ck=Cx~ck,(19)x~r,ik+1=AΔiiBKx~r,ik−BΔλiKx~r,ik−τk+Bωω~i,xk,x~r,ik=ϕ~r,ik,k∈−τmax,−τmax+1,⋯,0,z~r,ik=Cx~r,ik,i=2,3,⋯,N.Obviously, swarm system (6) can achieve consensus if subsystem (19) is asymptotically stable. Note that subsystem (18) determines the final consensus value of swarm system (6), and the details of it will be discussed below.Remark 2.
By Theorem1, the distributed H∞ consensus control problem of swarm system (6) is transformed into the robust H∞ control problems of N subsystems (8), so that a set of independent systems which have the same dimensions as a single agent in (2) is obtained. Therefore, the computational complexity is decreased evidently and the stability theory of swarm systems can be used. The essential tools producing this result by right of the state space decomposition approach.


## 2.1. Graph Theory

In general, exchanges of information amongN agents in swarm systems can be shown as a directed graph G=V,ℰ,A, where V=v1,⋯,vN denotes the node set, ℰ⊆vi,vj:vi,vj∈V represents the edge set, and A=aij refers to the weighted adjacency matrix with the element aij>0 if vj,vi∈ℰ while aij=0 if vj,vi∉ℰ. Define the node index of graph G as a finite index set ℐ=1,2,⋯,N. Let Ni=vj∈V:vj,vi∈ℰ stand for the set of all neighbours of node vi, where the information flows from node vj to node vi. Let subset J⊆V of the nodes in graph G be a cluster. Suppose that NJ=∪vi∈JNi=vj∈V:vi∈J,vi,vj∈ℰ represents the neighbour set of a cluster NJ. It can be defined that the in-degree matrix of node vi is deginvi=∑j=1Naji, and the out-degree matrix of node vi is degoutvi=∑j=1Naij. Note that a diagonal matrix Δ=Δij can be provided to describe the degree matrix of digraph G, which is defined as follows:
(1)Δij=0,i≠j,degoutvi,i=j.The definition of the Laplacian matrix in connection with digraphG can be shown in matrix form as ℒG=L=Δ−A.


## 2.2. Problem Description

Consider a linear swarm system composed ofN homogeneous agents marked as 1,2,⋯,N, in which the ith agent’s dynamics can be expressed as a state update equation
(2)xik+1=A+ΔAxik+B+ΔBuik+Bωωik,zik=Cxik,where xik∈Rd indicates the state vector; uik=ui,1k,ui,2k,⋯,ui,m1kT∈Rm1 and ωik=ωi,1k,ωi,2k,⋯,ωi,m2kT∈Rm2, respectively, represent the input matrix and the external disturbance; A∈Rd×d, B∈Rd×m1, Bω∈Rd×m2, and C∈Rl×d are the given constant matrices; zik∈Rl is the control output; ΔA and ΔB stand for uncertainties in the system matrix and input channel, respectively, such that
(3)ΔAΔB=DFE1E2,where D denotes a positive constant matrix and F refers to an unknown matrix function which meets constraints as follows:
(4)FTF≤I.The parameter uncertaintiesΔA and ΔB are considered permissible when equations (3) and (4) hold meanwhile.Remark 1.
When there is the external disturbance in system (2), external disturbance ωik=ωi,1k,ωi,2k,⋯,ωi,m2kT∈Rm2 belongs to l2 signal shown in [33, 40], which satisfies ∑k=1nωi,jk2<∞i=1,⋯,N,j=1,⋯,m2, that is, the energy of the external disturbance in system (2) is bounded on any finite interval. In this case, the robust H∞ control problem can be analysed in an appropriate and meaningful way.To achieve consensus for linear swarm system (2), the following control protocols are adopted:
(5)uik=K∑j∈Niaijxjk−τk−xik,where 0≤τk≤τmax, K is a constant gain matrix with appropriate dimensions, and τk∈ℤ∗ denotes time-varying delays of swarm systems. Then, to show the consensus analysis and design for swarm system (2), the definition is presented as follows:Definition 1.
For the given gain matrixK and any given bounded initial states, swarm system (2) with consensus protocol (5) are assured to arrive consensus if there exists a vector-valued ck depending on the initial states such that limk→∞xk−1N⊗ck=0, where ck denotes the final consensus value.
Letxk=x1Tk,⋯,xNTkT, ωxk=ω1T,⋯,ωNTT, and zk=z1Tk,⋯,zNTkT, and we can write swarm system (2) with equation (5) into a compact form as follows:
(6)xk+1=IN⊗A+ΔA−Δ⊗BΔBKxk+Δ−L⊗BΔBKxk−τk+IN⊗Bωωxk,xk=ϕk,k∈−τmax,−τmax+1,⋯,0,zk=IN⊗Cxk,where BΔBK=B+ΔBK and ϕk is an initial value at k.The suboptimal robustH∞ consensus control problem of system (6) is stated to find a distributed protocol (5) such that
(i)
withωxk=0, system (6) is asymptotically stable for all acceptable uncertain matrices F
(ii)
withωxk interpreted as deterministic l2 signal, the transfer function from ωxk to zk of system (6), which is denoted by Tωz, satisfies Tωz∞<γ for all acceptable uncertain matrices F and a given allowable scalar γ>0, where Tωz∞, is the H∞ norm of Tωz, defined by Tωz∞=supωx∈RNdσ¯TωzjωIn order to make an in-depth analysis for the robustH∞ consensus control problem of system (6), we presume henceforth that the communication graph G is connected and give the following lemma about the graph theory.Lemma 1 (see [41]).
For the Laplacian matrixL associated with an undirected graph G, the following description holds: (i) 0 is an eigenvalue of L, and its corresponding eigenvector is 1N, and (ii) 0 is a simple eigenvalue of L, and all the other eigenvalues are positive and real if G is connected.Assume thatλii=1,2,⋯,N denotes eigenvalues of matrix L∈RN of an undirected graph G. Moreover, λ1=0, which satisfies λ1≤λ2≤⋯≤λN, is related to its corresponding eigenvector u¯1=1/N−1/21N. Thus, an orthogonal matrix can be found in the form
(7)U=1N1N−1TN1N−1TNU¯,such that UTLU=diagλ1,λ2,⋯,λN.Theorem 1.
For a givenγ>0, system (6) is asymptotically stable and Tωz∞<γ, if and only if the following N systems are simultaneously and asymptotically stable and all of the H∞ norms of their transfer function matrices are less than γ:
(8)x~ik+1=AΔiiBKx~ik+BΔλiKx~ik−τk+Bωω~i,xk,x~k=ϕ~k,k∈−τmax,−τmax+1,⋯,0,z~ik=Cx~ik,i=1,2,⋯,N,where AΔiiBK=A+ΔA−ΔiiBΔBK and BΔλiK=Δii−λiB+ΔBK.Proof.
Letλii=1,2,⋯,N be the eigenvalues of matrix L∈RN for an undirected topology G, where λ1=0 with the corresponding eigenvector u¯1=1/N−1/21N, and λ1≤λ2≤⋯≤λN. An orthogonal matrix can be found in the form
(9)U=1N1N−1TN1N−1TNU¯,such that UTLU=diagλ1,λ2,⋯,λN=Λ. Assume that
(10)x~k=UT⊗Idxk=x~cTk,x~rTkT=x~1Tk,x~2Tk,⋯,x~NTkT.
Then, system (6) can be rewritten in terms of x~k as
(11)x~k+1=IN⊗A+ΔA−Δ⊗BΔBKx~kk−Δ−Λ⊗BΔBKx~k−τk+UT⊗Bωωxk,x~k=ϕ~k,k∈−τmax,−τmax+1,⋯0,zk=U⊗Cx~k.
Furthermore, reformulate the disturbance variableωxk and the performance variable zk via
(12)ω~xk=UT⊗Im2ωxk=ω~1,xTk,ω~2,xTk,⋯,ω~N,xTkT,(13)z~k=UT⊗Ilzk=z~1Tk,z~2Tk,⋯z~NTkT.
Subsequently, substituting (12) and (13) into (11) gives
(14)x~k+1=IN⊗A+ΔA−Δ⊗BΔBKx~k−Δ−Λ⊗B+ΔBKx~k−τk+IN⊗Bωω~xk,x~k=ϕ~k,k∈−τmax,−τmax+1,⋯,0,z~k=IN⊗Cx~k.
In fact, system (14) consists of N individual systems in (6). Denote by Tω~z~∞ and Tω~iz~i∞the transfer function matrices of system (14) and (6), respectively. Afterwards, we can deduce from equations (6), (12), (13), and (14) that
(15)Tω~z~=diagTω~1z~1,Tω~2z~2,⋯,Tω~Nz~N=UT⊗IlTωzU⊗Im2,which implies that
(16)Tω~z~∞=maxi=2,3,⋯,NTω~iz~i∞=Tωz∞.
Therefore, the proof of Theorem1 is completed.In addition, it is worth mentioning that, let(17)x~k=x~cTk,x~rTkT=x~cTk,x~r,2Tk,⋯,x~r,NTkT,ω~xk=ω~c,xTk,ω~r,xTkT=ω~c,xTk,ω~r,2xTk,⋯,ω~r,NxTkT,z~=z~cTk,z~rTkT=z~cTk,z~r,2Tk,⋯,z~r,NTkT.By Lemma1, we can convert system (14) into N subsystems as follows:
(18)x~ck+1=A+ΔA−Δ11BΔBKx~ck+Δ11B+ΔBKx~ck−τk+Bωω~c,xk,x~ck=ϕ~ck,k∈−τmax,−τmax+1,⋯,0,z~ck=Cx~ck,(19)x~r,ik+1=AΔiiBKx~r,ik−BΔλiKx~r,ik−τk+Bωω~i,xk,x~r,ik=ϕ~r,ik,k∈−τmax,−τmax+1,⋯,0,z~r,ik=Cx~r,ik,i=2,3,⋯,N.Obviously, swarm system (6) can achieve consensus if subsystem (19) is asymptotically stable. Note that subsystem (18) determines the final consensus value of swarm system (6), and the details of it will be discussed below.Remark 2.
By Theorem1, the distributed H∞ consensus control problem of swarm system (6) is transformed into the robust H∞ control problems of N subsystems (8), so that a set of independent systems which have the same dimensions as a single agent in (2) is obtained. Therefore, the computational complexity is decreased evidently and the stability theory of swarm systems can be used. The essential tools producing this result by right of the state space decomposition approach.


## 3. Main Results

In this section, the robust stability and existence conditions of theH∞ consensus control problem for swarm system (2) with protocol (5) is provided by using a basis-dependent Lyapunov function and the LMI approach. Above all, we should introduce the following two lemmas.Lemma 2 (Jensen inequality [42]).
For any constant matrixM≥0, M∈Rn×n, and ψi∈Rn, positive integers α1 and α2 satisfying α2≥α1≥1, an inequality holds as follows:
(20)−α2−α1+1∑i=α1α2ψTiMψi≤−∑i=α1α2ψiTM∑i=α1α2ψi.Lemma 3.
Suppose that equal positive integer time-varying delaysτk exist in whole communication network of swarm system (2). Meanwhile, the undirected graph G is fixed and connected. Then, by use of control protocol (5), the robust consensus of swarm system (2) with H∞-norm consensus performance bound γ can be achieved globally asymptotically if we can prove that, there are a feedback matrix K with suitable dimensions and four positive-definite matrices P, R, Q1, and Q2 with d×d dimensions satisfying
(21)Ω~i=Ω~i11Ω~i12Ω~i13Ω~i14∗Ω~i220Ω~i24∗∗Ω~i330∗∗∗Ω~i44<0,i=2,⋯,N,where
(22)Ω~i11=AΔiiBKTPAΔiiBK−P+τmaxQ1+Q2+τmaxAΔiiBK−ITRAΔiiBK−I+CTC−Rτmax,Ω~i12=τmaxAΔiiBK−ITRBΔλiK+AΔiiBKTPBΔλiK,Ω~i13=Rτmax,Ω~i14=AΔiiBKTPBω+τmaxAΔiiBK−ITRBω,Ω~i22=BΔλiKTPBΔλiK+τmaxBΔλiKTRBΔλiK−Q1,Ω~i24=BΔλiKTPBω+τmaxBΔλiKTRBω,Ω~i33=−Q2−Rτmax,Ω~i44=BωTPBω+τmaxBωTRBω−γ2I.Proof.
For purpose of consensus analysis, we construct a Lyapunov function as follows:(23)Vk=∑i=15Vik,where
(24)V1k=∑i=2Nx~r,iTkPx~r,ik,V2k=∑i=2N∑s=k−τkk−1x~r,iTsQ1x~r,is,V3k=∑i=2N∑s=k−τmaxk−1x~r,iTsQ2x~r,is,V4k=∑i=2N∑s=−τmax+1−1∑l=k+sk−1x~r,iTlQ1x~r,il,V5k=∑i=2N∑s=−τmax−1∑l=k+sk−1η~r,iTlRη~r,il,for η~r,il=x~r,il+1−x~r,il.
Then, we can define the derivative ofVk as
(25)ΔVk=Vk+1−Vk=∑i=15ΔVik=∑i=15Vik+1−Vik.
From the solution of (8), ΔV1k can be rewritten as follows:
(26)ΔV1k=V1k+1−V1k=∑i=2Nx~r,iTk+1Px~r,ik+1−x~r,iTkPx~r,ik=∑i=2Nx~r,iTkAΔiiBKTPAΔiiBK−Px~r,ik−2∑i=2Nx~r,iTkAΔiiBKTPBΔλiKx~r,ik−τk+2∑i=2Nx~r,iTkAΔiiBKTPBωω~i,xk−2∑i=2Nx~r,iTk−τkBΔλiKTPBωω~i,xk+∑i=2Nx~r,iTk−τkBΔλiKTPBΔλiKx~r,ik−τk+∑i=2Nω~i,xTkBωTPBωω~i,xk.
ΔV2k can be reformulated as
(27)ΔV2k=V2k+1−V2k=∑i=2Nx~r,iTkQ1x~r,ik−∑i=2Nx~r,iTk−τkQ1x~r,ik−τk+∑i=2N∑s=k+1−τk+1k−1x~r,iTsQ1x~r,is−∑i=2N∑s=k+1−τkk−1x~r,iTsQ1x~r,is≤∑i=2Nx~r,iTkQ1x~r,ik−∑i=2Nx~r,iTk−τkQ1x~r,ik−τk+∑i=2N∑s=k+1−τmaxk−1x~r,iTsQ1x~r,is.
ΔV3k can be rewritten as
(28)ΔV3k=V3k+1−V3k=∑i=2Nx~r,iTkQ2x~r,ik−x~r,iTk−τmaxQ2x~r,ik−τmax.
ΔV4k can be reformulated as
(29)ΔV4k=V4k+1−V4k=∑i=2N∑s=−τmax+1−1∑l=k+1+sk−1x~r,iTlQ1x~r,il−∑i=2N∑s=−τmax+1−1∑l=k+sk−1x~r,iTlQ1x~r,il=∑i=2N∑s=−τmax+1−1x~r,iTkQ1x~r,ik−x~r,iTk+sQ1x~r,ik+s=∑i=2Nτmax−1x~r,iTkQ1x~r,ik−∑l=k−τmax+1k−1x~r,iTlQ1x~r,il.
ΔV5k can be rewritten as
(30)ΔV5k=V5k+1−V5k=∑i=2N∑s=−τmax−1η~r,iTkRη~r,ik−η~r,iTk+sRη~r,ik+s=∑i=2Nτmaxη~r,iTkRη~r,ik−∑l=k−τmaxk−1η~r,iTlRη~r,il.
Furthermore, the termτmaxη~r,iTkRη~r,ik can be shown that
(31)τmaxη~r,iTkRη~r,ik=τmaxx~r,ik+1−x~r,ikTRx~r,ik+1−x~r,ikτmaxAΔiiBK−Ix~r,ik+BΔλiKx~r,ik−τk+Bωω~i,xkTR×AΔiiBK−Ix~r,ik+BΔλiKKx~r,ik−τk+Bωω~i,xk=τmaxx~r,iTk×AΔiiBK−ITRAΔiiBK−Ix~r,ik+2x~r,iTkAΔiiBK−ITRBΔλiKx~r,ik−τk+2τmaxx~r,iTkAΔiiBK−ITRBωω~i,xk+2τmaxx~r,iTk−τkBΔλiKTRBωω~i,xk+τmaxx~r,iTk−τkBΔλiKTRBΔλiKx~r,i×k−τk+τmaxω~i,xTkBωTRBωω~i,xk.
At the same time, according to Lemma2, the following inequality holds:
(32)−∑l=k−τmaxk−1η~r,iTlRη~r,il≤−∑l=k−τmaxk−1η~r,ilTRτmax∑l=k−τmaxk−1η~r,il=−x~r,ik−x~r,ik−τmaxTRτmaxx~r,ik−x~r,ik−τmax.
Letξ~ik=x~r,iTk,x~r,iTk−τk,x~r,iTk−τmax,ω~i,xTkT. Then, it follows from (8) and (25)–(32) that
(33)∑i=2Nz~r,iTkz~r,ik−γ2ω~i,xTkω~i,xTk+ΔVk=∑i=2Nx~r,iTkCTCx~r,ik−γ2ω~i,xTkω~i,xTk+∑i=15ΔVik≤∑i=2Nξ~iTkΩ~iξ~ik<0,if system (8) is asymptotically stable. This implies that the robust consensus of swarm system (2) with H∞-norm consensus performance bound γ can be achieved globally asymptotically by the use of control protocol (5) if matrix inequality (21) holds.
Therefore, the conclusion of Lemma3 is drawn.Remark 3.
In Lemma3, for making sure that system (8) can achieve robust consensus with a H∞ performance index γ, a sufficient condition is obtained. Nevertheless, it is not difficult to find that (21) is a nonlinear matrix inequality (NMI) and therein lies parameter uncertainties.
To cope with the uncertain matricesF and the nonlinear terms of (21), the following two lemmas are given.Lemma 4 (see [43]).
For given matricesY, D, and E of appropriate dimensions where Y is symmetric, a matrix inequality holds as follows:
(34)Y+DFE+ETFTDT<0,for all F subject to FTF≤I, if and only if there exists a scalar ε>0 such that
(35)Y+εDDT+ε−1ETE<0.Lemma 5 (Schur complement [44]).
The linear matrix inequality(36)QxSxSxTRx>0,where Qx=QxT, Rx=RxT, and Sx are dependent on x, is the identical meaning as anyone of the following two descriptions:
(1)
Qx>0, Rx−SxTQx−1Sx>0
(2)
Rx>0, Qx−SxRx−1SxT>0Theorem 2.
Consider swarm system (2) with a fixed, undirected, and connected communication topologyG. The robust consensus of swarm system (2) with H∞-norm consensus performance bound γ can be achieved globally asymptotically by use of control protocol (5) if we can find two scaling factors ε1>0 and ε2>0, a feedback matrix K with suitable dimensions and six positive-definite matrices P, S, R, M, Q1, and Q2 with d×d dimensions which are a settlement of the minimization problem as follows:
(37)mintrPS+RM,subject to
(38)−S+ε1DDT0AiBKBλiK0Bω000−1τmaxM+ε2DDTAiBK−IBλiK0Bω00ATA−IT−P+τmaxQ1+Q2+CTC−Rτmax0Rτmax0E1TE1TBλiKTBλiKT0−Q100Δii−λiE2KTΔii−λiE2KT00RTτmax0−Q2−Rτmax00000E1Δii−λiE2K0−γ2I0000E1Δii−λiE2K00−ε1I0000K000−ε2I<0,i=2,3,⋯,N,(39)PIIS>0,P>0,S>0,(40)RIIM>0,R>0,M>0,where AiBK=A−ΔiiBK and BλiK=Δii−λiBK. It can also be said if the minimization problem (37)–(40) has an available settlement P, S, R, M, Q1, Q2, and K, the robust consensus of swarm system (2) with H∞-norm consensus performance bound γ can be achieved globally asymptotically by the use of control protocol (5).Proof.
Inequality (21) can be expressed as
(41)AΔiiBKTBΔλiKT0BωTPAΔiiBKBΔλiK0Bω+τmaxAΔiiBK−ITBΔλiKT0BωTRAΔiiBK−IBΔλiK0Bω+−P+τmaxQ1+Q2+CTC−Rτmax0Rτmax0∗−Q100∗∗−Q2−Rτmax0∗∗∗−γ2I≤0.
According to Lemma5, one can obtain that inequality (41) can be rewritten as follows:
(42)−P−10AΔiiBKBΔλiK0Bω∗−1τmaxR−1AΔiiBK−IBΔλiK0Bω∗∗−P+τmaxQ1+Q2+CTC−Rτmax0Rτmax0∗∗∗−Q100∗∗∗∗−Q2−Rτmax0∗∗∗∗∗−γ2I<0.
By replacingΔA=DFE1 and ΔB=DFE2, inequality (42) is equivalently written as follows:
(43)−P−10AiBKBλiK0Bω∗−1τmaxR−1AiBK−IBλiK0Bω∗∗−P+τmaxQ1+Q2+CTC−Rτmax0Rτmax0∗∗∗−Q100∗∗∗∗−Q2−Rτmax0∗∗∗∗∗−γ2I+D00000F00E1Δii−λiE2K00+0D0000F00E1Δii−λiE2K00+00E1Δii−λiE2K00TFTD00000T+00E1Δii−λiE2K00TFT0D0000T≤0.
In virtue of Lemma4 and Lemma 5, inequality (43) comes into existence for all acceptable uncertain matrices F if and only if we can prove that there are two scaling factors ε1>0 and ε2>0 satisfying
(44)−P−1+ε1DDT0AiBKBλiK0Bω000−1τmaxR−1+ε2DDTAiBK−IBλiK0Bω00ATA−IT−P+τmaxQ1+Q2+CTC−Rτmax0Rτmax0E1TE1TBλiKTBλiKT0−Q100Δii−λiE2KTΔii−λiE2KT00RTτmax0−Q2−Rτmax00000E1Δii−λiE2K0−γ2I0000E1Δii−λiE2K00−ε1I0000K000−ε2I<0,i=2,3,⋯,N.
By the Cone Complementarity Linearization (CCL) algorithm in [45], we set S=P−1 and M=R−1. Then, the optimization problem (37)–(40) can be reasonably settled by transforming the nonlinear matrix inequality (NMI) (21).
Thus, the proof of Theorem2 is completed.Remark 4.
According to Theorem2, notice that the optimization problem (40)–(45) transformed from the NMI (21) can be reasonably solved based upon Algorithm 1. Hereafter, the linear swarm system (2) with the control protocol (5) achieves robust H∞consensus.Algorithm 1: A possible way to the optimization problem (37)–(40).
Step 1: Seek out a set of available values ofP0, S0, R0, and M0 for LMI (38)–(40), if there does not exist, then exit. Set k=0.
Step 2: SetPk=P0, Sk=S0, Rk=R0, and Mk=M0 and seek out Pk, Sk, Rk, and Mk which are a settlement for the LMI problem mintrPkS+SkP+RkM+MkR subject to LMI (38)–(40).
Step 3: IftrPkS+SkP+RkM+MkR<4d+ε, then exit. Otherwise, set k=k+1 and switch to Step 2.It should be pointed out that the final outcome of first “exit” in Step 1 is that available initial values ofP0, S0, R0, and M0 cannot be obtained. In this case, we set k=0 and repeat Step 1. As long as the available values are obtained, we switch to Step 2. Then, second “exit” in Step 3 means that suitable values of Pk, Sk, Rk, and Mk are obtained.Theorem 3.
Withωxk interpreted as deterministic l2 signal, when swarm system (6) achieves robust H∞ consensus with parameter uncertainties and time-varying delays, the final consensus value ck satisfies
(45)limk→∞ck−xCk=0,where xCk can be the following form:
(46)xCk=1N⊗1NAΔ11BKk∑i=1Nxi0+∑i=0k−1∑l=0l=k−i−1∑j=1j=N1NAΔ11BKiBωωx,jl,k≤τk,xCk+1=AΔBKxCk+B+ΔBKxCk−τk+1NT⊗1N∑i=1NBωωx,ik,k>τk,for AΔ11BK=A+ΔA−Δ11BΔBK.Proof.
In the following, we consider two cases for the proof.
Case 1. (k≤τk).
In this case, system (18) can be equivalently written as follows:
(47)x~ck+1=AΔ11BKx~ck+Bωω~c,xk,z~ck=Cx~ck.
Assume thatxCk=U⊗Idx~cTk,0T and xC¯k=U⊗Id0,x~rTkT, then from (10), we can uniquely decompose xk into xk=xCk+xC¯k. Considering the mentioned above, we can also conclude that if the robust H∞ consensus of system (6) is achieved, the subsystem (19) should be Schur stable, which implies that the response of system (19) due to xC¯0 complies with limk→∞xC¯k=0. Therefore, the final consensus value ck depends totally on xCk. Since x~cTk,0T=e1⊗x~k, we have xC0=u¯1⊗x~c0=u¯1⊗e1T⊗Idx~0, and because x~0=UT⊗Idx0, then we can obtain xC0=u¯1⊗x~c0=u¯1⊗e1T⊗Idx~0, that is to say
(48)xC0=u¯1⊗e1T⊗Id∗UT⊗Idx0=u¯1⊗e1TUT⊗Idx0=u¯1⊗1N1NT⊗Idx0=1NT⊗1N∑i=1Nxi0.
Likewise, letωC,xk=U⊗Idω~c,xTk,0T and ω~C¯,xk=U⊗Id0,ω~r,xTkT, then by (12), xk can be uniquely decomposed as ωxk=ωC,xk+ωC¯,xk. If system (6) achieves the robust H∞ consensus, the response of system (19) due to ωC¯,x0 also should comply with limk→∞ωC¯,xk=0. Since ω~c,xTk,0T=e1⊗ω~xk, we have ω~C,xk=u¯1⊗ω~c,xk=u¯1⊗e1T⊗Idω~xk, and because ω~xk=UT⊗Idωxk, then we can obtain ωC,xk=u¯1⊗ω~c,xk=u¯1⊗e1T⊗Idω~xk, that is
(49)ωC,xk=u¯1⊗e1T⊗Id∗UT⊗Idωxk=u¯1⊗e1TUT⊗Idωxk=u¯1⊗1N1NT⊗Idωxk=1NT⊗1N∑i=1Nωx,ik.
Hence, we have(50)xCk=AΔ11BKkxC0+1N⊗∑i=0k−1AΔ11BKiBωω~c,xk−i−1=1N⊗1NAΔ11BKk∑i=1Nxi0+∑i=0k−1∑l=0l=k−i−1AΔ11BKiBωωC,xl=1N⊗1NAΔ11BKk∑i=1Nxi0+∑i=0k−1∑l=0l=k−i−1∑j=1j=N1NAΔ11BKiBωωx,jl.
Case 2. (k>τk).
In this case, we also definexCk=U⊗Idx~cTk,0T and xC¯k=U⊗Id0,x~rTkT. Similarly, we can have xCk=1NT⊗1/N∑i=1Nxik and ωC,xk=1NT⊗1/N∑i=1Nωx,ik. By (18), we can obtain
(51)xCk+1=AΔ11BKxCk+B+ΔBK∗xCk−τk+1NT⊗1N∑i=1NBωωx,ik,k>τk.
Thus, the final consensus valueck complies with limk→∞ck−xCk=0,k=0,1,2,⋯.
The conclusion of Theorem3 is obtained.Remark 5.
Withωxk interpreted as deterministic l2 signal, the final consensus value ck of system (6) is presented by Theorem 3. In the k≤τk case, the final consensus value ck has two divisions, one is 1/NAΔ11BKk∑i=1Nxi0, which is associated with the system matrix A+ΔA, the neighbouring feedback matrix K, and the initial state x0; and the other is ∑i=0k−1∑l=0l=k−i−1∑j=1j=N1/NAΔ11BKiBωωx,jl, which is related to the external disturbance ωxk. This implies that the external disturbance ωxk has an effect on the final consensus value, which is also related to the system matrix A+ΔA, the neighbouring feedback matrix K, and the initial state x0. The result is different from that of the swarm system without external disturbances and parameter uncertainties, which is discussed in [29]. In the k>τk case, we get a linear recursion formula (51) for xCk, which is the limit of the final consensus value ck as k→∞. This implies that the final consensus value ck is associated with the states xCk, xCk−τk, and the external disturbance ωxk when k>τk.


## 4. Results and Discussion

In this section, we firstly use a simulation example to validate the effectiveness of the theoretical results, and then apply the control protocol (5) to achieve state consensus for a swarm system which consists of 8 agents. The dynamics of these agents are represented by (2), and relevant system matrices is given as follows:
(52)A=22220−22220001,B=0.2−0.41,D=0.10000.20000.3,E1=0.10.300.20.40001,E2=0.20.10.3,F=r1000r2000r3,where r1, r2, and r3 are uncertain parameters satisfying −1≤r1≤1, −1≤r2≤1, and −1≤r3≤1, respectively. Then, swarm system (2) can be reformulated as
(53)xik+1=A+DFE1xik+B+DFE2uik.We analyze the robustH∞ consensus among the above 8 agents with an undirected topology G as shown in Figure 1.Figure 1
Communication topologyG.Assume that the initial state of each agent is randomly given asx10=1,5,−2T, x20=2,4,3T, x30=1,1,2T, x40=3,2,1T, x50=5,6,−2T, x60=−3,3,4T, x70=−2,−4,−3T, and x80=−5,−2,−1T and set r1=0.15, r2=0.25, r3=0.15, and τmax=2. The dynamics of each agent is combined with protocol (5). Let γ=2 and suppose that the exogenous disturbance inputs are selected as ωi,xk=0.1ie−0.5ksink. By Theorem 2, we can get that
(54)K=0.0408−0.04700.1564,P=2.24690.12280.07500.12282.55840.05130.07500.05132.1872,S=1.11400.00020.02160.00021.13360.03150.02160.05131.3031,R=0.52060.0090−0.01600.00900.48210.0249−0.01600.02491.1160,M=1.9259−0.03840.0426−0.03842.0822−0.07060.0426−0.07061.3476,Q1=0.25270.00380.08640.00380.20920.00990.08640.00990.3334,Q1=0.9301−0.00030.0061−0.00030.92220.00230.00610.00230.9131.The simulation results are produced from Figures2–4. The state trajectories of swarm system (2) are shown. In addition, the final consensus value ck deduced from Theorem 3 is marked with the red asterisk.Figure 2
State 1 trajectories of swarm system (2).Figure 3
State 2 trajectories of swarm system (2).Figure 4
State 3 trajectories of swarm system (2).In view of Figures2–4, we can see easily that the state trajectories of swarm system (2) can be asymptotically convergent to the common value ck, associated with rjj=1,2,3. From Definition 1, we can also find clearly that the robust H∞ consensus of swarm system (2) with control protocol (5) is achieved. Therefore, the effectiveness of the theoretical results is demonstrated.


## 5. Conclusions

RobustH∞ consensus control problems of linear high-order discrete-time swarm systems with parameter uncertainties and time-varying delays were investigated. A robust H∞ consensus control protocol was proposed while the impact of time-varying delays is involved. LMI conditions for robust H∞ consensus analysis were given, and a final consensus value was obtained from the derivation process. If swarm systems satisfy the sufficient LMI conditions, then we can ensure that swarm systems with parameter uncertainties and time-varying delays could achieve robust H∞ consensus, and the convergence result could be determined as a final consensus value. Finally, the effectiveness of the theoretical results was verified by relevant simulation examples. Further studies will focus on the impact of switching topologies and directed graphs on the consensus problem of linear discrete-time swarm systems.



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*Source: 7278531-2019-07-24.xml*