# Output Consensus of Heterogeneous Multiagent Systems with Physical and Communication Graphs

Authors: Junwei Wang; Kairui Chen; Qiuli Liu
Journal: Complexity
 (2018)

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## Abstract

This paper deals with the output consensus problem of heterogeneous networks in a leader-follower manner that are interconnected by a physical coupling graph. The network under consideration consists of linear agents with different dynamics/dimensions. Both the state-feedback and output-feedback control protocols based upon information flow prescribed by a separate communication graph are developed, using the internal model principle and relative outputs of neighboring agents. With the small-gain theorem, we convert the consensus problem into aH∞ control problem of decoupled linear systems having the same dimensions as a single agent, where the disturbance attenuation constraints depend on the largest singular value related to the global information of physical and communication graphs. Then, we provide local synthesis procedures for control gains in terms of feasible solutions of algebraic Riccati equations. Finally, simulation examples are presented to verify the performance of the theoretical results.


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## Body

## 1. Introduction

In the past decade, consensus (including leaderless consensus and leader-follower consensus) has become a hot topic due to broad applications of cooperative control of multiagent systems (MASs) [1–4]. It is concerned with designing control protocols for each agent in the presence of limited information exchanges such that eventually the group reaches an agreement on certain quantities of interest. Early consensus works focus on MASs with very primitive individual agent dynamics (such as single integrators or double integrators) [5–8], where only the information exchange among agents determines time evolution of the networks. Starting from them, the MASs with finite-dimensional linear dynamics have enticed intensive attention from the control community, which are more general and include integrator dynamics as a special case [9–13].In the past literature of consensus, it is commonly assumed that all agents in a MAS are modeled byidentical linear equations, which is said to be homogeneous networks. In such MASs, control protocols are designed upon either relative states, relative observer states, or relative output measurements of neighboring agents to produce state consensus (see [11, 14, 15] and references therein). However, in practice, the dynamics of individual agents are hardly exactly identical. For these MASs (often formulated as heterogeneous networks), previous control protocols for homogeneous networks not only fail to guarantee the stability of state consensus but also make state consensus meaningless in the case that the state-space dimensions of individual agents are different. Therefore, recent interest has shifted to output consensus of heterogeneous MASs, where all agents reach an agreement on their partial-state outputs through relative output information exchange with neighboring agents [16–26].Within the literature addressing the output consensus problem of heterogeneous MASs, two approaches are of particular interest: the feedforward approach and the internal model approach. In the feedforward approach, a state or output feedback protocol is designed based on the solution of the output regulator equations [19]. More recently, the internal model principle has been extended to the study of output consensus control [20–23] and output containment control [27] of heterogeneous networks.However, most of previous works are restricted to MASs interconnected bya single communication topology. However, coordination problems encountered in the cyber-physical systems (e.g., power generators and robot teams) can often be rephrased as consensus problems in which two interaction topologies coexist, i.e., the physical coupling topology and the communication topology [28, 29]. When that is the case, the first physical coupling topology connects the dynamics of the agents [30, 31] and then control protocols are computed using the information flow prescribed by the second communication topology [32, 33]. To our best knowledge, the very few works that explicitly address the consensus problems of linear MASs with more than one interaction topologies are reported in [34, 35]. In [34], only the homogeneous dynamics of agents are considered. Although Lombana and di Bernardo [35] has investigated the heterogeneous case, the agents are required to have identical dimension. Furthermore, the input of each agent should have the order as many as that of the agent itself.To fill up the abovementioned gaps, this paper considers the distributed control problem of linear heterogeneous MASs coupled by a physical coupling graphGp, which is in general more difficult than that of homogeneous MASs. The control protocols are designed upon the allowed information flow prescribed by a separate communication graph Gc. Compared to existing methods, the salient features of this paper are summarized as follows.
(1)
We study linear MASs in a generalized environment in the sense that they have not only physical couplings between their dynamics (captured by the physical coupling graphGp) but also a second communication graph Gc allowing information flow for the design of control protocols. Furthermore, the agents have their own individual dynamics. Thus, the resulting multiagent systems are heterogeneous and coupled by multiplex interaction topologies
(2)
Relying on the relative outputs between neighboring agents, both the state-feedback and output-feedback protocols are proposed. In each protocol, a dynamic compensator is constructed by the internal model principle
(3)
The key element of our approach for the analysis of leader-follower output consensus is the small-gain theorem, which allows us to recast the original consensus problem as theH∞ control problem of a set of decoupled linear systems with the same dimensions as a single agent
(4)
The decomposedH∞ control problem consists of both the state-feedback and output-feedback cases, with a disturbance attenuation constraint depending on the largest singular value related to the global information of physical and communication graphs. Accordingly, the control gains can be determined by solving the algebraic Riccati equationsThe rest of this paper is organized as follows. In Section2, the notation used throughout the paper is given and the output consensus problem is formulated. Sections 3 and 4 are devoted to the main results for the state-feedback control and output-feedback control, respectively, where the global sufficient condition and local design procedures are presented. In Section 5, numerical simulations are presented to validate the main results of the paper. Section 6 concludes the paper.


## 2. Preliminaries and Problem Formulation

Letℝ and ℝn×n stand for the sets of real numbers and n×n real matrices, respectively. jℝ denotes the imaginary axis. In and 1n represent an identity matrix of dimension n and a vector in ℝn with elements being all ones, respectively. ⋅ denotes the Euclidean vector norm or the corresponding induced matrix 2-norm. For symmetric matrices X and Y, X≥Y (respectively, X>Y) means that X−Y is positive semidefinite (respectively, positive definite). diag… defines a block-diagonal matrix. For two matrices X and Y, X⊗Y denotes the Kronecker product and σ¯X is the maximum singular value of X. If not explicitly stated, matrices are assumed to have compatible dimensions.


### 2.1. Graph Theory

A weighted directed graph (digraph) is denoted byG=V,ℰ, where V=1,…,N denotes the set of nodes and ℰ⊆V×V denotes the set of edges. A directed edge in ℰ is denoted by the ordered pair of nodes j,i, meaning that node i can receive information from node j. The set of neighbors of node i is denoted by Ni=j∈V:j,i∈ℰ. No self-loop is allowed; hence, aii=0, ∀i∈V. The adjacency matrix G=gij∈ℝN×N associated with the digraph G is defined by gij>0 if j,i∈ℰ and gij=0 otherwise. The Laplacian matrix L=lij∈ℝN×N is defined as lii=∑k∈Nigik and lij=−gij for all j≠i. A path on the digraph is an ordered set of distinct nodes i1,…,ik such that iq−1,iq∈ℰ. A digraph is said to have a spanning tree if there exists a node, called the root, which has a directed path to every other node in the graph.


### 2.2. Problem Formulation

We consider a heterogeneous group withN followers and a leader, in which the followers are labeled as agents 1,…,N and the leader is labeled as agent 0. Motivated by the applications of distributed control in cyber-physical systems, we model the interaction topology between the N followers by two digraphs: one is the physical coupling digraph Gp=V,ℰp with the adjacency matrix Gp=gijp∈ℝN×N and Laplacian matrix Lp=lijp∈ℝN×N and the other is the communication digraph Gc=V,ℰc with the adjacency matrix Gc=gijc∈ℝN×N and Laplacian matrix Lc=lijc∈ℝN×N.The dynamics ofith follower is given by
(1)ẋi=Aixi+Biui+Di∑j∈Nipgijpyj−yi,yi=Cixi,i∈V,where xi∈ℝni is the state, yi∈ℝq the measured output, ui∈ℝmi is the control input which will be designed based on the communication graph Gc, gijp is the element of Gp, and Nip is the set of neighbors of node i corresponding to the physical coupling graph Gp. Ai, Bi, Ci, and Di are the known real constant matrices of appropriate dimensions describing the nominal dynamics of follower i. To avoid redundancy of measurement, it is assumed that Ci has full row rank, i.e., rankCi=q.The leader0 which acts as an exosystem that generates a reference trajectory for the group obeys the dynamic model.
(2)ẋ0=A0x0,y0=C0x0,where x0∈ℝn0 and y0∈ℝq denote the state and measured output, respectively. We assume that the leader does not receive any information from followers and can be observed only from a small subset of followers. To depict whether the followers are connected to the leader, we define a leader adjacency matrix G0=diagg10,…,gN0∈ℝN×N, where gi0>0 if the ith follower obtains information from the leader 0 and gi0=0 otherwise.Remark 1.
Note that MAS (1) and (2) is called heterogeneous because each agent has its own distinct dynamics and their dimensions are also nonidentical. Attempts to address the distributed control problems for such systems have been addressed [19–23]. Unfortunately, the aforementioned schemes ignore the coupling effect of the originally existed physical coupling graph, thus cannot be straightforwardly applied to the considered control problem. This renders the contribution of this paper with respect to previous works considerable.Throughout this paper, we shall use the following concept of output consensus for the heterogeneous MAS (1) and (2).Definition 1.
The heterogeneous MAS (1) and (2) coupled by the physical coupling graph Gp is said to reach output consensus if for all i∈V, there exist distributed control laws ui relying on the communication graph Gc such that the following local output error:
(3)eit≜yit−y0t→0ast→∞.The aim of this work is to solve the output consensus problem of the MAS (1) and (2) under the physical coupling graph Gp by designing the control protocols ui. More specifically, we consider two cases: (1) the state-feedback case in which the full state information of each follower can be measured and (2) the output-feedback case in which only partial state information of each follower is measured. To solve the aforementioned control problem, we need to introduce the concept of the internal model [36], which will help us define the dynamic compensator in the process of distributed control design.Definition 2.
A pair of matricesS1,S2 is said to incorporate a q-copy internal model of matrix A0 if
(4)S1=diagα1,…,αq⏟q−tuple,S2=diagθ1,…,θq⏟q−tuple,where αi,θii=1,…,q are controllable and the minimal polynomial of A0 divides the characteristics polynomial of αi.We conclude this section by introducing some assumptions that will be required to derive the main results of this paper.Assumption 1.
The pairAi,Bi is stabilizable, and the pair Ai,Ci is detectable.Assumption 2.
A0 has no eigenvalues with negative real parts.Assumption 3.
The communication graphGc contains a spanning tree, and at least one root node in Gc can observe information from the leader.Assumption 4.
For allλ that are the eigenvalues of A0:
(5)rankAi−λIBiCi0=ni+q.


## 2.1. Graph Theory

A weighted directed graph (digraph) is denoted byG=V,ℰ, where V=1,…,N denotes the set of nodes and ℰ⊆V×V denotes the set of edges. A directed edge in ℰ is denoted by the ordered pair of nodes j,i, meaning that node i can receive information from node j. The set of neighbors of node i is denoted by Ni=j∈V:j,i∈ℰ. No self-loop is allowed; hence, aii=0, ∀i∈V. The adjacency matrix G=gij∈ℝN×N associated with the digraph G is defined by gij>0 if j,i∈ℰ and gij=0 otherwise. The Laplacian matrix L=lij∈ℝN×N is defined as lii=∑k∈Nigik and lij=−gij for all j≠i. A path on the digraph is an ordered set of distinct nodes i1,…,ik such that iq−1,iq∈ℰ. A digraph is said to have a spanning tree if there exists a node, called the root, which has a directed path to every other node in the graph.


## 2.2. Problem Formulation

We consider a heterogeneous group withN followers and a leader, in which the followers are labeled as agents 1,…,N and the leader is labeled as agent 0. Motivated by the applications of distributed control in cyber-physical systems, we model the interaction topology between the N followers by two digraphs: one is the physical coupling digraph Gp=V,ℰp with the adjacency matrix Gp=gijp∈ℝN×N and Laplacian matrix Lp=lijp∈ℝN×N and the other is the communication digraph Gc=V,ℰc with the adjacency matrix Gc=gijc∈ℝN×N and Laplacian matrix Lc=lijc∈ℝN×N.The dynamics ofith follower is given by
(1)ẋi=Aixi+Biui+Di∑j∈Nipgijpyj−yi,yi=Cixi,i∈V,where xi∈ℝni is the state, yi∈ℝq the measured output, ui∈ℝmi is the control input which will be designed based on the communication graph Gc, gijp is the element of Gp, and Nip is the set of neighbors of node i corresponding to the physical coupling graph Gp. Ai, Bi, Ci, and Di are the known real constant matrices of appropriate dimensions describing the nominal dynamics of follower i. To avoid redundancy of measurement, it is assumed that Ci has full row rank, i.e., rankCi=q.The leader0 which acts as an exosystem that generates a reference trajectory for the group obeys the dynamic model.
(2)ẋ0=A0x0,y0=C0x0,where x0∈ℝn0 and y0∈ℝq denote the state and measured output, respectively. We assume that the leader does not receive any information from followers and can be observed only from a small subset of followers. To depict whether the followers are connected to the leader, we define a leader adjacency matrix G0=diagg10,…,gN0∈ℝN×N, where gi0>0 if the ith follower obtains information from the leader 0 and gi0=0 otherwise.Remark 1.
Note that MAS (1) and (2) is called heterogeneous because each agent has its own distinct dynamics and their dimensions are also nonidentical. Attempts to address the distributed control problems for such systems have been addressed [19–23]. Unfortunately, the aforementioned schemes ignore the coupling effect of the originally existed physical coupling graph, thus cannot be straightforwardly applied to the considered control problem. This renders the contribution of this paper with respect to previous works considerable.Throughout this paper, we shall use the following concept of output consensus for the heterogeneous MAS (1) and (2).Definition 1.
The heterogeneous MAS (1) and (2) coupled by the physical coupling graph Gp is said to reach output consensus if for all i∈V, there exist distributed control laws ui relying on the communication graph Gc such that the following local output error:
(3)eit≜yit−y0t→0ast→∞.The aim of this work is to solve the output consensus problem of the MAS (1) and (2) under the physical coupling graph Gp by designing the control protocols ui. More specifically, we consider two cases: (1) the state-feedback case in which the full state information of each follower can be measured and (2) the output-feedback case in which only partial state information of each follower is measured. To solve the aforementioned control problem, we need to introduce the concept of the internal model [36], which will help us define the dynamic compensator in the process of distributed control design.Definition 2.
A pair of matricesS1,S2 is said to incorporate a q-copy internal model of matrix A0 if
(4)S1=diagα1,…,αq⏟q−tuple,S2=diagθ1,…,θq⏟q−tuple,where αi,θii=1,…,q are controllable and the minimal polynomial of A0 divides the characteristics polynomial of αi.We conclude this section by introducing some assumptions that will be required to derive the main results of this paper.Assumption 1.
The pairAi,Bi is stabilizable, and the pair Ai,Ci is detectable.Assumption 2.
A0 has no eigenvalues with negative real parts.Assumption 3.
The communication graphGc contains a spanning tree, and at least one root node in Gc can observe information from the leader.Assumption 4.
For allλ that are the eigenvalues of A0:
(5)rankAi−λIBiCi0=ni+q.


## 3. The Case of State-Feedback Design

In this section, we consider the output consensus problem of MAS (1) and (2) using the state-feedback control of the form:
(6)ui=Kxixi+Kzizi,żi=S1izi+S2ieiv,where zi∈ℝsi represents the internal controller state and Kxi and Kzi are the state-feedback gain and feedforward gain to be designed, respectively. S1i and S2i are constant matrices such that the pair S1i,S2i contains a q-copy internal model of A0 as defined in Definition 2. eiv is the neighborhood output error of follower i defined by
(7)eiv=1∑j∈Nicgijc+gi0∑j∈Nicgijcyi−yj+gi0yi−y0,where gijc are elements of Gc and Nic is the set of neighbors of node i corresponding to the communication graph Gc.The output consensus problem will be solved using the state-feedback control law (6) in two steps. First, a global sufficient condition for output consensus is provided by applying the output regulation technique [36], and we next present detailed local design procedures for control gains Kxi and Kzi based on the small-gain theorem [37] and the solution of H∞ control problems [38].


### 3.1. Global Condition for Output Consensus

Denoting(8)β=β1T,…,βNTTβ=x,e,z,xl=xT,zTT,x~0=1N⊗x0,Φ~=diagΦiΦ=A,B,C,D,Kx,Ky,Kz,S1,S2,A~0=IN⊗A0,C~0=IN⊗C0,H~p=Lp⊗Iq,H~c=diag1∑j∈Nicgijc+gi0Lc+G0⊗Iq,and substituting the protocol (6) into MAS (1) and (2), we can write the closed-loop dynamics of the overall network in the following compact form:
(9)ẋl=Alxl+Blx~0,x~̇0=A~0x~0,e=Clxl−C~0x~0,where
(10)Al=A~+B~K~x−D~H~pC~B~K~zS~2H~cC~S~1,Bl=0−S~2H~cC~0,Cl=C~0.It can be seen from (9) that the problem of output consensus will be solved if the global output error et=Clxlt−C~0x~0t→0 as t→∞. The following theorem gives a global sufficient condition on the system matrix Al for the problem.Theorem 1.
Consider the heterogeneous MAS (1) and (2) and let Assumptions 2-3 be satisfied. Then, the output consensus problem is solvable by the state-feedback protocol (6) if the system matrix Al in (10) is Hurwitz.Proof 1.
Assume thatAl in (10) is Hurwitz. According to the output regulation theory (Huang [36]), the global output error et=Clxl−C~0x~0 in (9) will converge to zero if the linear matrix equation:
(11)XlA~0=AlXl+Bl,0=ClXl−C~0,has a unique solution Xl.
It is now required to show that under the hypotheses of this theorem, the unique solutionXl to (11) indeed exists. Recall that since Al is exponentially stable and A~0 is completely unstable under Assumption 2, σAl∩σA~0=∅. Then, the first equation of (11) which is a Sylvester equation has a unique solution Xl. By decomposing Xl as Xl=Xl1T,Xl2TT, the first equation of (11) can be expanded as
(12)Xl1A~0=A~−D~H~pC~+B~K~xXl1+B~K~zXl2,Xl2A~0=S~1Xl2+S~2H~cC~Xl1−C~0.
Since the pairS1i,S2i incorporates a q-copy internal model of A0, the pair S~1,S~2 also incorporates a q-copy internal model of A~0. So the solution Xl=Xl1T,Xl2TT can guarantee that 0=ClXl−C~0 with the help of Assumption 3. Therefore, Xl satisfies both equations of (11). This completes the proof.


### 3.2. Local Design for Control Gains

Based on Theorem1, what remains is to design gain matrices Kxi and Kzi to make Al in (10) Hurwitz, which is equivalent to stabilizing the following linear systems coupled through graphs Gp and Gc:
(13)x¯i=A¯ix¯i−Di0∑j∈Niplijpy¯j−0S2i1∑j∈Nicgijc+gi0∑j∈Nicgijcy¯jy¯i=C¯ix¯i,i∈V,,where x¯i=xiT,ziTT and
(14)A¯i=Ai+BiKxiBiKziS2iCiS1i,C¯i=Ci0.By lettingx¯=x¯1T,…,x¯NTT, y¯=y¯1T,…,y¯NTT, and
(15)B¯i=−Di0,−0S2i,Θ¯=diagΘ¯iΘ=A,B,C,Ψ=Lp⊗Iq0+G¯c⊗0Iq,G¯c=diag1∑j∈Nicgijc+gi0Gc,we can express the global form of system (13) as
(16)x¯=A¯x¯+B¯Ψy¯,y¯=C¯x¯.Lemma 1.
Suppose the transfer functionTis=C¯isIn−A¯i−1B¯i∀i∈V is stable. Then, Al in (10) is Hurwitz if
(17)maxi∈VTis∞<1σ¯Ψ,where Tis∞ denotes the H∞ norm defined by Tis∞=supω∈Rσ¯Tijω.Proof 2.
Notice that stability of matrixAl in (10) is equivalent to stability of system (16). According to the small-gain theorem (Khalil [37]), system (16) is stable if
(18)Ts∞Ψ∞<1,where Ts=diagT1s,…,TNs.
It follows from(19)Ts∞Ψ∞=diagT1s,…,TNs∞σ¯Ψ≤maxi∈NTis∞σ¯Ψthat (17) is a sufficient condition for (18). This completes the proof.Remark 2.
Very recently, the cooperative output regulation problem was studied for heterogeneous MASs in a similarH∞ control framework [23], where both state feedback and dynamic output feedback were considered. Interestingly, Li. et al. [23] proposed a less conservative small-gain-type solvability condition than the ones in [21, 22]. The main characteristic of the small-gain-type condition of [23] is that the communication graph is not separated from the H∞-type index of all agents, which is directly used for the graph weight optimization and H∞-control-based protocol gain design. Particularly, the square structure of the communication graph-related matrix in [23] allows that the solvability condition can be preexplicitly ensured for three special cases of the graph, the agent dynamcis and the exo-system dynamics. However, it can be seen from our graph-related matrix Ψ in (15) that the introduction of the physical graph Gp not only destroys the square structure of Ψ but also may lead to σ¯Ψ>1. Therefore, the developed graph weight optimization method and the feasibility discussion for special MASs in [23] are not applicable to our considered MAS (1) and (2) with both physical graph Gp and communication graph Gc.By Theorem1 and Lemma 1, the output consensus is linked to the H∞ control theory. In order to obtain the state-feedback protocol (6) that makes MAS (1) and (2) achieves output consensus, it suffices to find gains Kxi and Kzi such that a state-feedback controller internally stabilizes the linear system Tis with disturbance attenuation bound 1/σ¯Ψ. For this purpose, we should consider the linear system:
(20)x^̇i=A^ix^i+B^iu^i+B¯iw^i,z^i=C¯ix^i,under the state-feedback control:
(21)u^i=KxiKzix^i.Here,x^i∈ℝni+si is the state, u^i∈ℝmi is the control input, w^i∈ℝ2q is the disturbance input, z^i∈ℝq is the controlled output, C¯i and B¯i are as in (14) and (15), and
(22)A^i=Ai0S2iCiS1i,B^i=Bi0.The following result explains how to obtain control gainsKxi and Kzi such that (21) solves the H∞ problem of (20) with a prescribed disturbance attenuation constraint γ>0.Lemma 2.
Letγ>0 be a prescribed level of disturbance attenuation and δi>0 be a given constant. Then, the linear system (20) is stabilizable with an H∞-norm bound γ via the state-feedback controller (21) with gains given by
(23)KxiKzi=−B^iTP^i,if there exists a positive definite solution P^i>0 for the Riccati equation:
(24)A^iTP^i+P^iA^i+P^i1γ2B¯iB¯iT−2B^iB^iTP^i+C¯iTC¯i+δiI=0.Proof 3.
When we apply the control (21) to the linear system (20), the closed-loop system from w^i to z^i is given by
(25)x^̇i=A¯ix^i+B¯iw^i,z^i=C¯ix^i,with A¯i in (14) expressed as
(26)A¯i=A^i−B^iB^iTP^i.
We first prove the asymptotic stability of the closed-loop system (25) without external disturbance (i.e., w^i=0), by considering the Lyapunov function (Xie and de Souza Carlos [38])
(27)Vt=x^iTP^ix^i,where P^i>0 is a solution of (24).
DifferentiatingVt along the trajectory of the closed-loop system (25) yields
(28)V̇t=x^iTA¯iTP^i+P^iA¯ix^i=x^iTA^iTP^i+P^iA^i−2P^iB^iB^iTP^ix^i=−x^iT1γi2P^iB¯iB¯iTP^i+C¯iTC¯i+δiIx^i<0,then the closed-loop system (25) is asymptotically stable with w^i=0.
Next, we establish the upper boundγw^i2 for the L20,∞ norm of w^i by introducing
(29)J=∫0∞z^iTz^i−γ2w^iTw^idt.
Assumingx^i0=0, we have that for any nonzero w^i∈L20,∞:
(30)J=∫0∞z^iTz^i−γ2w^iTw^idt=∫0∞z^iTz^i−γ2w^iTw^i+ddtx^iTP^ix^idt−x^iT∞P^ix^i∞=∫0∞x^iTA^iTP^i+P^iA^i+P^i1γ2B¯iB¯iT−2B^iB^iTP^i+C¯iTC¯ix^idt−∫0∞γ2w^i−1γ2B¯iTP^ix^iTw^i−1γ2B¯iTP^ix^idt−x^iT∞P^ix^i∞.
Considering (24), one can show that J<0, i.e., z^i2≤γw^i2 for any nonzero w^i∈L20,∞. This ends the proof.Based on Theorem1 and Lemmas 1-2, the following theorem yields a state-feedback protocol for output consensus of heterogeneous networks.Theorem 2.
For any given0<γ≤1/σ¯Ψ, consider the heterogeneous MAS (1) and (2) under Assumptions 1–4. The output consensus problem is solved by the state-feedback protocol (6) if we design control gains Kxi and Kzi as in (23) where P^i>0 is the solution of the Riccati equation (24).Remark 3.
From Theorem2, the state-feedback protocol (6) can be designed along the following steps (Algorithm 1).Algorithm 1
Step 1. Choose the pair S1i,S2i to contain a q-copy internal model of A0.
Step 2. For the given physical coupling graph Gp and communication graph Gc, compute σ¯Ψ for Ψ in (15).
Step 3. Choose the disturbance attenuation constraint γ such that 0<γ≤1/σ¯Ψ and δi>0.
Step 4. Solve the Riccati equation (24) to get gain matrices Kxi and Kzi according to (23).


## 3.1. Global Condition for Output Consensus

Denoting(8)β=β1T,…,βNTTβ=x,e,z,xl=xT,zTT,x~0=1N⊗x0,Φ~=diagΦiΦ=A,B,C,D,Kx,Ky,Kz,S1,S2,A~0=IN⊗A0,C~0=IN⊗C0,H~p=Lp⊗Iq,H~c=diag1∑j∈Nicgijc+gi0Lc+G0⊗Iq,and substituting the protocol (6) into MAS (1) and (2), we can write the closed-loop dynamics of the overall network in the following compact form:
(9)ẋl=Alxl+Blx~0,x~̇0=A~0x~0,e=Clxl−C~0x~0,where
(10)Al=A~+B~K~x−D~H~pC~B~K~zS~2H~cC~S~1,Bl=0−S~2H~cC~0,Cl=C~0.It can be seen from (9) that the problem of output consensus will be solved if the global output error et=Clxlt−C~0x~0t→0 as t→∞. The following theorem gives a global sufficient condition on the system matrix Al for the problem.Theorem 1.
Consider the heterogeneous MAS (1) and (2) and let Assumptions 2-3 be satisfied. Then, the output consensus problem is solvable by the state-feedback protocol (6) if the system matrix Al in (10) is Hurwitz.Proof 1.
Assume thatAl in (10) is Hurwitz. According to the output regulation theory (Huang [36]), the global output error et=Clxl−C~0x~0 in (9) will converge to zero if the linear matrix equation:
(11)XlA~0=AlXl+Bl,0=ClXl−C~0,has a unique solution Xl.
It is now required to show that under the hypotheses of this theorem, the unique solutionXl to (11) indeed exists. Recall that since Al is exponentially stable and A~0 is completely unstable under Assumption 2, σAl∩σA~0=∅. Then, the first equation of (11) which is a Sylvester equation has a unique solution Xl. By decomposing Xl as Xl=Xl1T,Xl2TT, the first equation of (11) can be expanded as
(12)Xl1A~0=A~−D~H~pC~+B~K~xXl1+B~K~zXl2,Xl2A~0=S~1Xl2+S~2H~cC~Xl1−C~0.
Since the pairS1i,S2i incorporates a q-copy internal model of A0, the pair S~1,S~2 also incorporates a q-copy internal model of A~0. So the solution Xl=Xl1T,Xl2TT can guarantee that 0=ClXl−C~0 with the help of Assumption 3. Therefore, Xl satisfies both equations of (11). This completes the proof.


## 3.2. Local Design for Control Gains

Based on Theorem1, what remains is to design gain matrices Kxi and Kzi to make Al in (10) Hurwitz, which is equivalent to stabilizing the following linear systems coupled through graphs Gp and Gc:
(13)x¯i=A¯ix¯i−Di0∑j∈Niplijpy¯j−0S2i1∑j∈Nicgijc+gi0∑j∈Nicgijcy¯jy¯i=C¯ix¯i,i∈V,,where x¯i=xiT,ziTT and
(14)A¯i=Ai+BiKxiBiKziS2iCiS1i,C¯i=Ci0.By lettingx¯=x¯1T,…,x¯NTT, y¯=y¯1T,…,y¯NTT, and
(15)B¯i=−Di0,−0S2i,Θ¯=diagΘ¯iΘ=A,B,C,Ψ=Lp⊗Iq0+G¯c⊗0Iq,G¯c=diag1∑j∈Nicgijc+gi0Gc,we can express the global form of system (13) as
(16)x¯=A¯x¯+B¯Ψy¯,y¯=C¯x¯.Lemma 1.
Suppose the transfer functionTis=C¯isIn−A¯i−1B¯i∀i∈V is stable. Then, Al in (10) is Hurwitz if
(17)maxi∈VTis∞<1σ¯Ψ,where Tis∞ denotes the H∞ norm defined by Tis∞=supω∈Rσ¯Tijω.Proof 2.
Notice that stability of matrixAl in (10) is equivalent to stability of system (16). According to the small-gain theorem (Khalil [37]), system (16) is stable if
(18)Ts∞Ψ∞<1,where Ts=diagT1s,…,TNs.
It follows from(19)Ts∞Ψ∞=diagT1s,…,TNs∞σ¯Ψ≤maxi∈NTis∞σ¯Ψthat (17) is a sufficient condition for (18). This completes the proof.Remark 2.
Very recently, the cooperative output regulation problem was studied for heterogeneous MASs in a similarH∞ control framework [23], where both state feedback and dynamic output feedback were considered. Interestingly, Li. et al. [23] proposed a less conservative small-gain-type solvability condition than the ones in [21, 22]. The main characteristic of the small-gain-type condition of [23] is that the communication graph is not separated from the H∞-type index of all agents, which is directly used for the graph weight optimization and H∞-control-based protocol gain design. Particularly, the square structure of the communication graph-related matrix in [23] allows that the solvability condition can be preexplicitly ensured for three special cases of the graph, the agent dynamcis and the exo-system dynamics. However, it can be seen from our graph-related matrix Ψ in (15) that the introduction of the physical graph Gp not only destroys the square structure of Ψ but also may lead to σ¯Ψ>1. Therefore, the developed graph weight optimization method and the feasibility discussion for special MASs in [23] are not applicable to our considered MAS (1) and (2) with both physical graph Gp and communication graph Gc.By Theorem1 and Lemma 1, the output consensus is linked to the H∞ control theory. In order to obtain the state-feedback protocol (6) that makes MAS (1) and (2) achieves output consensus, it suffices to find gains Kxi and Kzi such that a state-feedback controller internally stabilizes the linear system Tis with disturbance attenuation bound 1/σ¯Ψ. For this purpose, we should consider the linear system:
(20)x^̇i=A^ix^i+B^iu^i+B¯iw^i,z^i=C¯ix^i,under the state-feedback control:
(21)u^i=KxiKzix^i.Here,x^i∈ℝni+si is the state, u^i∈ℝmi is the control input, w^i∈ℝ2q is the disturbance input, z^i∈ℝq is the controlled output, C¯i and B¯i are as in (14) and (15), and
(22)A^i=Ai0S2iCiS1i,B^i=Bi0.The following result explains how to obtain control gainsKxi and Kzi such that (21) solves the H∞ problem of (20) with a prescribed disturbance attenuation constraint γ>0.Lemma 2.
Letγ>0 be a prescribed level of disturbance attenuation and δi>0 be a given constant. Then, the linear system (20) is stabilizable with an H∞-norm bound γ via the state-feedback controller (21) with gains given by
(23)KxiKzi=−B^iTP^i,if there exists a positive definite solution P^i>0 for the Riccati equation:
(24)A^iTP^i+P^iA^i+P^i1γ2B¯iB¯iT−2B^iB^iTP^i+C¯iTC¯i+δiI=0.Proof 3.
When we apply the control (21) to the linear system (20), the closed-loop system from w^i to z^i is given by
(25)x^̇i=A¯ix^i+B¯iw^i,z^i=C¯ix^i,with A¯i in (14) expressed as
(26)A¯i=A^i−B^iB^iTP^i.
We first prove the asymptotic stability of the closed-loop system (25) without external disturbance (i.e., w^i=0), by considering the Lyapunov function (Xie and de Souza Carlos [38])
(27)Vt=x^iTP^ix^i,where P^i>0 is a solution of (24).
DifferentiatingVt along the trajectory of the closed-loop system (25) yields
(28)V̇t=x^iTA¯iTP^i+P^iA¯ix^i=x^iTA^iTP^i+P^iA^i−2P^iB^iB^iTP^ix^i=−x^iT1γi2P^iB¯iB¯iTP^i+C¯iTC¯i+δiIx^i<0,then the closed-loop system (25) is asymptotically stable with w^i=0.
Next, we establish the upper boundγw^i2 for the L20,∞ norm of w^i by introducing
(29)J=∫0∞z^iTz^i−γ2w^iTw^idt.
Assumingx^i0=0, we have that for any nonzero w^i∈L20,∞:
(30)J=∫0∞z^iTz^i−γ2w^iTw^idt=∫0∞z^iTz^i−γ2w^iTw^i+ddtx^iTP^ix^idt−x^iT∞P^ix^i∞=∫0∞x^iTA^iTP^i+P^iA^i+P^i1γ2B¯iB¯iT−2B^iB^iTP^i+C¯iTC¯ix^idt−∫0∞γ2w^i−1γ2B¯iTP^ix^iTw^i−1γ2B¯iTP^ix^idt−x^iT∞P^ix^i∞.
Considering (24), one can show that J<0, i.e., z^i2≤γw^i2 for any nonzero w^i∈L20,∞. This ends the proof.Based on Theorem1 and Lemmas 1-2, the following theorem yields a state-feedback protocol for output consensus of heterogeneous networks.Theorem 2.
For any given0<γ≤1/σ¯Ψ, consider the heterogeneous MAS (1) and (2) under Assumptions 1–4. The output consensus problem is solved by the state-feedback protocol (6) if we design control gains Kxi and Kzi as in (23) where P^i>0 is the solution of the Riccati equation (24).Remark 3.
From Theorem2, the state-feedback protocol (6) can be designed along the following steps (Algorithm 1).Algorithm 1
Step 1. Choose the pair S1i,S2i to contain a q-copy internal model of A0.
Step 2. For the given physical coupling graph Gp and communication graph Gc, compute σ¯Ψ for Ψ in (15).
Step 3. Choose the disturbance attenuation constraint γ such that 0<γ≤1/σ¯Ψ and δi>0.
Step 4. Solve the Riccati equation (24) to get gain matrices Kxi and Kzi according to (23).


## 4. The Case of Output-Feedback Design

We now deal with the output-feedback design for achieving output consensus of system (1) and (2), which is of the form:
(31)ui=Kyiyi+Kzizi,żi=S1izi+S2ieiv,where zi∈ℝsi represents the internal controller state and Kyi and Kzi are the output-feedback gain and feedforward gain to be determined, respectively. S1i and S2i are constant matrices such that the pair S1i,S2i contains a q-copy internal model as defined in Definition 2 of A0.Interconnecting the MAS (1) and (2) using the output-feedback protocol (31), the closed-loop system can be still obtained as (9) but with only Al in (10) redefined by
(32)Al=A~+B~K~yC~−D~H~pC~B~K~zS~2H~cC~S~1.Therefore, Theorem1 holds to solve the output consensus problem by the output-feedback protocol (31) with the corresponding system matrix Al in (32). Similarly, Lemma 1 holds with A¯i in (14) redefined by
(33)A¯i=Ai+BiKyiCiBiKziS2iCiS1i.As in Section3 for the case of state-feedback design, we now consider the linear system:
(34)x^̇i=A^ix^i+B^iu^i+B¯iw^i,y^i=C^ix^i,z^i=C¯ix^i,under the output-feedback control:
(35)u^i=KyiKziy^i.Here,x^i∈ℝni+si is the state, u^i∈ℝmi is the control input, w^i∈ℝ2q is the disturbance input, y^i∈ℝq+si is the measured output, z^i∈ℝq is the controlled output, C¯i and B¯i are as in (14) and (15), A^i and B^i are as in (22), and
(36)C^i=Ci00I.Motivated by the static output-feedback design for the single agents [39], we establish the following Lemma on how to compute output-feedback control for the H∞ control problem of linear systems.Lemma 3.
Letγ>0 be a prescribed level of disturbance attenuation and δi>0 be a given constant. The linear system (34) is stabilizable with an H∞-norm bound γ via the output-feedback controller (35) if there exist matrices Kyi, Kzi, and Li such that
(37)KyiKziC^i=−B^iTP^i+Li,where P^i>0 is a solution of the Riccati equation:
(38)A^iTP^i+P^iA^i+P^i1γ2B¯iB¯iT−2B^iB^iTP^i+C¯iTC¯i−LiTB^iTP^i−P^iB^iLi+δiI=0.Proof 4.
The proof is similar to Lemma2 and is thus omitted here.Based on Theorem1 and Lemmas 1 and 3, the main result on output consensus using the output-feedback design is established below.Theorem 3.
For any given0<γ<1/σ¯Ψ, consider the heterogeneous MAS (1) and (2) under Assumptions 1–4. The output consensus problem is solved by the output-feedback protocol (31) if we design control gains Kyi and Kzi as in (37) where P^i>0 is the solution of the Riccati equation (38).Remark 4.
From Theorem3, the output-feedback protocol (31) should be designed by solving the coupled matrix (37) and (38). These equations can be efficiently solved by the following steps (Algorithm 2) without requiring the initial stabilizing gains [39].Algorithm 2
Step 1. Choose the pair S1i,S2i to contain a q-copy internal model of A0.
Step 2. For the given physical coupling graph Gp and communication graph Gc, compute σ¯Ψ
forΨ in (15).
Step 3. Choose the disturbance attenuation constraint γ such that 0<γ≤1/σ¯Ψ and δi>0.
Step 4. Set n=0 and Li0=0.
Step 5. Find P^in through solving
A^iTP^in+P^inA^i+P^in1γ2B¯iB¯iT−2B^iB^iTP^in+C¯iTC¯i−LiTnB^iTP^in−P^inB^iLin+δiI=0.
Update the control gainKyiKzi and the matrix Li using
Kyin+1Kzin+1=−B^iTP^in+LinC^iTC^iC^iT−1,Lin+1=−Kyin+1Kzin+1C^i−B^iTPin.
Step 6. If Kyin+1Kzin+1 and KyinKzin are close enough to each other,
stop and setKyiKzi=Kyin+1Kzin+1. Otherwise, set n=n+1 and go
to Step 5.


## 5. Simulation Results

Consider a group of one leader and three followers with the communication network topologyGc and the physical network topology Gp as shown in Figure 1. The corresponding Laplacian matrices Lc and Lp are given by
(39)Lc=0.10−0.1000−0.02−0.040.06,Lp=0.06−0.06000.02−0.02000,with the leader adjacency matrix G0=diag0,1,1.Figure 1
The physical graphGp and communication graph Gc for simulation examples.The dynamics of the leader and each follower is described as below.(40)ẋ0=12−3−1x0,y0=10x0,(41)ẋ1=x1+u1+∑j∈N1pg1jpyj−y1,y1=x1,(42)ẋ2=0110x2+01u2+11∑j∈N2pg2jpyj−y2,y2=10x2,(43)ẋ3=0−2000−3001x3+201040,u3+011∑j∈N3pg3jpyj−y3,y3=110x3.It can be easily verified that Assumptions1–4 are satisfied by designing the internal model S1i,S2i for the ith follower as
(44)S1i=12−3−1,S2i=01,i=1,2,3.With the Laplacian matricesLc and Lp in (39), we have σ¯Ψ=0.1022 and 1/σ¯Ψ=9.7857.


### 5.1. The State-Feedback Case

In this case, we use the state-feedback protocol (6). Based on Algorithm 1, we set γ=6.0<1/σ¯Ψ and σ1=σ2=σ3=0.6. Then, by solving the Riccati equation (24), the gain matrix Kxi and Kzi in (23) can be designed as
(45)Kx1Kz1=−1.82120.3001−0.6437,Kx2Kz2=−2.4747−1.73952.10510.5652,Kx3Kz3=−0.5999−0.0816−1.0550−0.1554−0.43001.0873−1.1951−2.5010−0.8602−0.3173.Thus, according to Theorem2, MAS (40), (41), (42), and (43) has achieved the output consensus by using the state-feedback control (6) with control gains Kxi and Kzi in (45). The evolutions of the agent outputs and tracking errors are shown in Figures 2 and 3, respectively.Figure 2
The follower outputyii=1,2,3 and leader output y0 by state-feedback control (6).Figure 3
The tracking errorei=yi−y0i=1,2,3 by state-feedback control (6).


### 5.2. The Output-Feedback Case

In this case, we use the output-feedback protocol (31) with the output matrices C^i in (34) as
(46)C^1=I3,C^2=100000100001,C^3=110000001000001,and other parameters are kept the same as the state-feedback case.Based on Algorithm2 and setting γ=8.0<1/σ¯Ψ and σ1=σ2=σ3=0.8, we select the gain matrix Kyi and Kzi in (37) as follows:
(47)Ky1Kz1=−1.88490.2938−0.7361,L1=000,Ky2Kz2=−2.58622.38170.4205,L2=0−2.738400,Ky3Kz3=−0.2330−0.5079−0.5760−0.0517−0.9173−0.2407,L3=−0.27470.2747−2.6205002.0712−2.0712−4.124500.Thus, according to Theorem3, MAS (40), (41), (42), and (43) has achieved output consensus by using the output-feedback control (31) with control gains Kyi and Kzi in (47). The evolutions of the agent outputs and tracking errors are shown in Figures 4 and 5, respectively.Figure 4
The follower outputyii=1,2,3 and leader output y0 by output-feedback control (31).Figure 5
The tracking errorei=yi−y0i=1,2,3 by output-feedback control (31).


## 5.1. The State-Feedback Case

In this case, we use the state-feedback protocol (6). Based on Algorithm 1, we set γ=6.0<1/σ¯Ψ and σ1=σ2=σ3=0.6. Then, by solving the Riccati equation (24), the gain matrix Kxi and Kzi in (23) can be designed as
(45)Kx1Kz1=−1.82120.3001−0.6437,Kx2Kz2=−2.4747−1.73952.10510.5652,Kx3Kz3=−0.5999−0.0816−1.0550−0.1554−0.43001.0873−1.1951−2.5010−0.8602−0.3173.Thus, according to Theorem2, MAS (40), (41), (42), and (43) has achieved the output consensus by using the state-feedback control (6) with control gains Kxi and Kzi in (45). The evolutions of the agent outputs and tracking errors are shown in Figures 2 and 3, respectively.Figure 2
The follower outputyii=1,2,3 and leader output y0 by state-feedback control (6).Figure 3
The tracking errorei=yi−y0i=1,2,3 by state-feedback control (6).


## 5.2. The Output-Feedback Case

In this case, we use the output-feedback protocol (31) with the output matrices C^i in (34) as
(46)C^1=I3,C^2=100000100001,C^3=110000001000001,and other parameters are kept the same as the state-feedback case.Based on Algorithm2 and setting γ=8.0<1/σ¯Ψ and σ1=σ2=σ3=0.8, we select the gain matrix Kyi and Kzi in (37) as follows:
(47)Ky1Kz1=−1.88490.2938−0.7361,L1=000,Ky2Kz2=−2.58622.38170.4205,L2=0−2.738400,Ky3Kz3=−0.2330−0.5079−0.5760−0.0517−0.9173−0.2407,L3=−0.27470.2747−2.6205002.0712−2.0712−4.124500.Thus, according to Theorem3, MAS (40), (41), (42), and (43) has achieved output consensus by using the output-feedback control (31) with control gains Kyi and Kzi in (47). The evolutions of the agent outputs and tracking errors are shown in Figures 4 and 5, respectively.Figure 4
The follower outputyii=1,2,3 and leader output y0 by output-feedback control (31).Figure 5
The tracking errorei=yi−y0i=1,2,3 by output-feedback control (31).


## 6. Conclusion

The output consensus problem has been solved for a group of heterogeneous linear agents whose dynamics are coupled by a physical coupling graph. Distributed control protocols relying on the relative output information have been proposed for both state-feedback and output-feedback cases. By bringing together output regulation theory andH∞ control technique, we have demonstrated that the developed protocols guarantee that the output of each follower synchronizes to that of the leader. The computational algorithms for computing control gains were provided by solving the algebraic Riccati equations. It should be noted that the recent work of Li. et al. [23] introduces an interesting topic of discussing the preexplicit issue for output consensus of special MASs. Unfortunately, until now, we have not established the similar results for MASs with multiplex interaction graphs (e.g., the communication graph Gc and the physical graph Gp considered here), which leaves an open problem for our future study. Also, it is worth extending our results to solve the similar problem of uncertain MASs with complicated agent dynamics.



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*Source: 9091380-2018-11-01.xml*