Output Consensus for Networks of Non-Identical Introspective Agents

Output Consensus for Networks of Non-identical Introspective Agents

Tao Yang1 Ali Saberi1 Anton A. Stoorvogel2 Havard˚ Fjær Grip1

Abstract— In this paper, we consider three problems, namely, In a network of non-identical agents, the agents’ states the output consensus problem, the model-reference output may have different dimensions. In this case, the state con- consensus problem, and the regulation of output consensus sensus is not even properly defined, and it is more natural problem, for a network of non-identical right-invertible linear agents. The network provides each agent with a linear to aim for output consensus—that is, asymptotic agreement combination of multiple agents’ outputs. We assume that all on the agents’ partial-state outputs. Chopra and Spong [1] the agents are introspective, meaning that they have access to studied the output consensus problem for weakly minimum- their own local measurements. Under this assumption, we then phase nonlinear systems of relative degree one, using a pre- propose a distributed linear protocol to solve each problem for a feedback to create a single-integrator system with decoupled broad class of network topologies, including not only Laplacian topologies for a directed graph which contains a directed zero dynamics. Kim, Shim, and Seo [5] considered the output spanning tree, but a wide family of asymmetric topologies. consensus problem for uncertain single-input single-output, minimum-phase linear systems, by embedding an identical I. INTRODUCTION model within each agent, the output of which is tracked by The problem of achieving consensus among agents in the actual agent output. a network—that is, asymptotic agreement on the agents’ The design mentioned in Section I-A generally rely on state or output trajectories—has been extensively studied in some sort of self-knowledge that is separate from the in- recent years and has yielded some advances in e.g., sensor formation transmitted over the network. More specifically, networking [8], [9], [7], [15] and autonomous vehicle control the agents may know their own state, their own output, applications [14], [12], [11], [13]. or their own state/output relative to that of the reference Much of the attention has been devoted to achieving state trajectory. We shall refer to agents that possess this type consensus for a network of identical agents, where each of self-knowledge as introspective agents. In this paper, we agent has access to a linear combination of its own state assume that all the agents in the network are introspective. relative to that of neighboring agents (e.g., [8], [9], [7], [12], [11], [21]). Roy, Saberi, and Herlugson [15] and Yang, Roy, B. Contributions of this paper Wan, and Saberi [29] considered the state consensus prob- In this paper we consider networks of non-identical, intro- lem for more general network topologies. A more realistic spective, right-invertible linear agents, where each agent has scenario—that is, each agent receives a linear combination of access to its own local measurement, and receives a linear its own partial-state output and that of neighboring agents— combination of multiple agents’ outputs. We propose a dis- has been considered in [10], [22], [23], [6]. The results of [6] tributed linear protocol to solve each of the three problems— was expanded by [30] to more general network topologies. that is, output consensus problem, model-reference output Many of the results on the consensus problem are rooted consensus problem, and regulation of output consensus prob- in the seminal work of Wu and Chua [26], [27] in the circuit lem—under a set of straightforward assumptions about the community, which gives conditions on a network topology agents and the network topology. The assumptions about the for synchronization of coupled nonlinear oscillators. agents are much more general than that made in [1], [5], and the assumption about the network topology includes not only A. Non-identical agents and output consensus Laplacian topologies for a directed graph which contains Recent activities in the consensus literature have been a directed spanning tree, but a wide family of asymmetric focused on achieving consensus for a network of non- topologies. identical agents. This problem is challenging and only some partial results are available, see for instance [4], [28], [1], C. Preliminaries and notations [5], [25]. m n T Given a matrix A R × , A denotes its transpose, and ∈ n n !i(A) is its i’th eigenvalue. A R × is said to be Hurwitz The work of Ali Saberi and Tao Yang is partially supported by ∈ National Science Foundation grant NSF-0901137, NAVY grants ONR stable if all its eigenvalues are in the open left-half plane. KKK777SB001 and ONR KKK760SB0012. The work of Havard˚ Fjær Grip denotes the Kronecker product between two matrices of is supported by the Research Council of Norway. ⊗ p q 1School of Electrical Engineering and Computer Science, appropriate dimensions. A and a matrix B R × is defined mp nq ∈ Washington State University, Pullman, WA 99164-2752, U.S.A. as the R × matrix E-mail: [email protected], [email protected], [email protected]. a11B ... a1nB 2 . . . Department of Electrical Engineering, Mathematics and Computer Sci- A B = . .. . , ence, University of Twente, P.O. Box 217, 7500AE Enschede, The Nether- ⊗  . .  lands. E-mail: [email protected]. a B ... a B  m1 mn    where aij denotes element (i, j) of A. In denotes the identity will be seen explicitly in Section III-B. On the other hand, in matrix of dimension n, similarly, 0n denotes the square the regulation of output consensus problem, we have hardly matrix of dimension n with all zero elements; we sometimes any restrictions on (Cr,Ar). drop the subscript if the dimension is clear in the context. When clear form the context, 1 denotes the column vector A. Available information with all entries equal to one. For a set of vectors x1,...,xn, The agents are introspective, meaning that the agents have we denote by col(x1,...,xn) the column vector obtained by access to their own local information. Specifically, each agent stacking the elements of x1,...,xn. has access to In this paper, we will make use of the classical result m zi = Ci xi. (4) on stabilizing a matrix by scaling. Let us recall Fisher and Fuller’s result from [2]. Thus without any communications among agents provided by Lemma 1: (Fisher and Fuller) Consider an n n matrix the network, we might still be able to simply asymptotically G. If there exists a permutation matrix P such that× all the stabilize each individual agent and obtain output consensus 1 leading principal minors of PGP− are nonzero, then there with zero consensus trajectory, that is limt " yi(t)=0. But → exists a diagonal matrix D such that the eigenvalues of DG to achieve nontrivial consensus behavior, each agent needs are all in the open right-half complex plane (or, alternatively, to have some information provided by the network. In in the open left-half complex plane). particular, we assume that each agent i observes a linear combination of the outputs of multiple agents, that is, Note that the proof of Lemma 1 given in [2] is constructive. n II. PROBLEM STATEMENT #i = $ gijy j, (5) We consider a network of n multiple-input multiple-output j=1 agents of the form where gij R are scalars, referred as observation weights. ∈ Note that the observation weight gij represents the influence x˙i = Aixi + Biui, (1) (through network communication) of each agent j’s output yi = Cixi, ' on agent i’s observation. We find it natural to assemble the ni mi p for i 1,...,n , where xi R , ui R and yi R . weights gij into an n n network topology G =[gij]. ∈ { } ∈ ∈ ∈ × For the output consensus problem, our goal is to achieve In the model-reference output consensus and the regulation agreement asymptotically on the agents’ outputs; that is to of output consensus problems, it is obvious that a non-empty ensure that limt "(yi(t) y j(t)) = 0 for all i, j 1,...,n . subset of agents observes their outputs relative to that of → − ∈ { } Output consensus by itself does not impose any conditions the reference model given by (2), (3), respectively to ensure on asymptotic behavior of the output of an individual agent that the consensus trajectory tracks the reference trajectory. beyond the fact that the outputs of all the agents have to be Specifically, let I 1,...,n be such a subset. Then, each the same asymptotically. agent i 1,...,n ⊂has{ access} to the quantity In this paper we are, however, also interested in the model- ∈ { } reference output consensus problem. In this problem, our 1, i I , %i = ei(yi yr), ei = ∈ (6) − 0, i / I . goal is not only to achieve output consensus, but to make ' ∈ the consensus trajectory follow the output of the reference In the model-reference output consensus problem, all the system/virtual leader given by agents need to know the input ur of the reference model. x˙r = Arxr + Brur, (2) B. Linear protocols yr = Crxr, ' p Having the data available for solving each problem, in this where yr R . That is, we wish to ensure that limt "(yi(t) → paper, we restrict our attention to linear protocols. y (t)) = 0∈ for all i 1,...,n . − r More specifically, for solving the output consensus prob- Finally, we consider∈ { the regulation} of output consensus lem, we consider the linear protocol of the form problem, where the output of each agent has to track the same c ¯ c ¯ trajectory, generated by an arbitrary autonomous exosystem x˙i = Ac,ixi + Bc,i col(zi,#i), ¯ c ¯ (7) x˙ = A x , ui = Cc,ixi + Dc,i col(zi,#i), r r r (3) ' yr = Crxr, c qi ' for i 1,...,n , where xi R is the state of agent i’th r p ∈ { } ∈ where xr R and yr R . That is we wish to ensure that protocol. ∈ ∈ limt "(yi(t) yr(t)) = 0 for all i 1,...,n . For solving the model-reference output consensus prob- Notice→ that− the model-reference∈ output{ consensus} problem lem, we consider the linear protocols of the form is more general than the regulation of output consensus c ¯ c ¯ x˙i = Ac,ixi + Bc,i col(zi,#i,%i,ur), problem, since we would get (3) if we choose u r = 0 in c (8) ui = C¯c,ix + D¯ c,i col(zi,#i,%i,ur), (2). However, in order to solve the model-reference output ' i consensus problem, we require stronger assumptions on for i 1,...,n , where xc Rqi is the state of agent i’th ∈ { } i ∈ (Cr,Ar,Br) and that all the agents have to know ur, which protocol. p For solving the regulation of output consensus problem, where u¯i R is a new input, such that the resulting system we consider the linear protocol of the form (1) and (10)∈ can be written as the following form c ¯ c ¯ ˙ x˙i = Ac,ixi + Bc,i col(zi,#i,%i), x¯i = Ax¯i + B(u¯i + 'i), c (9) (11) ui = C¯c,ix + D¯ c,i col(zi,#i,%i), yi = Cx¯i, ' i ' p for i 1,...,n , where xc Rqi is the state of agent i’th where 'i R is given by dynamical equations ∈ { } i ∈ ∈ protocol. x˜˙ = H x˜ , i i i (12) 'i = Rix˜i, C. Assumptions ' for some Hurwitz stable matrix H . We make the following assumption about the agents. i Assumption 1: For each i 1,...,n , assume that The proof of Lemma 2 is given in Appendix A. ∈ { } 1) (Ai,Bi) is stabilizable; Remark 2: Note that matrices A, B, and C in Lemma 2 2) (Ci,Ai) is detectable; can be chosen arbitrarily as long as the system (C,A,B) has 3) (Ci,Ai,Bi) is right-invertible; and no invariant zeros, has uniform rank n q, is invertible, and all m the eigenvalues of A are in the closed left-half plane. They 4) (Ci ,Ai) is detectable. play a role as design parameters. In this paper, we shall use In the output consensus problem, we make the following this property in many locations for various purposes. assumption about the network topology. Assumption 2: We assume that the network topology G Lemma 2 shows that we can design a protocol based on the has only one zero eigenvalue, with right eigenvector 1, and local measurement (4), to transform each non-identical agent there exists a permutation matrix P such that all the leading model given by (1) into a new model given by (11). The new 1 models are almost identical; that is, they are the same for principal minors of PGP− , of size less than n are nonzero. each agent except for an exponentially decaying signal ' i. In the model-reference output consensus problem and regu- In the following three sections we shall first solve each of lation of output consensus problem, we make the following our three problems with respect to the new, almost-identical assumption about the network topology. models given by (11), and then combine each result with Assumption 3: We assume that the network topology G Lemma 2 to solve each problem with respect to the agent has only one zero eigenvalue, with right eigenvector 1, and models given by (1). there exist a permutation matrix P and a constant & > 0 such 1 A. The output consensus problem that all the leading principal minors of P(G+&E)P− , where E = diag(e1,...,en), are nonzero. The following lemma gives conditions under which the output consensus problem for a network of agents (11) can Remark 1: As a special case, if G is a Laplacian matrix be solved by using #i given by (5). of a digraph which contains a directed spanning tree, then it Lemma 3: Consider a network n agents of the form (11). is easy to check that Assumption 2 is always satisfied, and Let Assumption 2 be satisfied. Then there exists a protocol Assumption 3 is always satisfied by choosing eK = 1 for the of the form root agent K, and arbitrary & > 0. q˙ = A q + B # , i c,i i c,i i (13) u¯i = Cc,iqi, III. PROTOCOL DESIGN ' such that limt "(yi(t) y j(t)) = 0 for all i, j 1,...,n . In this section, we design a distributed protocol of the form → − ∈ { } (7), (8), and (9) to solve each of three problems formulated The proof of Lemma 3 is given in Appendix B. in Section II, respectively. The first stage of our design Combining the results of Lemma 2 and Lemma 3, we procedure of all three protocols is to design a protocol based obtain the following result. on local measurement (4) to make all the agents substantially Theorem 1: Consider n agents of the form (1). Let As- the same except for different exponentially decaying signals. sumptions 1 and 2 be satisfied. Then there exists a protocol This is shown in the following lemma. of the form (7) that solves the output consensus problem, Lemma 2: Consider a network of agents (1) with the local that is, limt "(yi(t) y j(t)) = 0 for all i, j 1,...,n . → − ∈ { } measurement zi given by (4). Let Assumption 1 be satisfied. Let us make several comments regarding Theorem 1. Let n be the integer that is equal to the largest degree of q The network topology conditions given in Assumption all infinite zeros of (Ai,Bi,Ci), i 1,...,n . Given arbitrary • ∈ { } 2 are satisfied for a broad class of matrices, including matrices A pnq pnq , B pnq p, and C p pnq such R × R × R × a Laplacian topology for a network which contains a that the system∈ (C,A,B) has∈ no invariant zeros,∈ has uniform directed spanning tree, and a class of matrices known as rank n , is invertible, and all the eigenvalues of A are in q D-semistable matrices, which have a single eigenvalue the closed left-half plane, then for each i 1,...,n , there at the origin with the corresponding right eigenvector exists a local output feedback of the following∈ { form} 1. For the definition of D-semistability, please see [15], p˙ = A p + B z + E u¯ , [3]. It is clear that D-semistable matrices includes a wide i c,i i c,i i c,i i (10) ui = Cc,i pi + Dc,iu¯i, family of matrices with more general entry sign pattern ' than the Laplacian matrix, and hence admits consensus such as step or sinusoidal signals, generated by an arbitrary control for a wider set of observation capabilities. exosystem system given by (3). In this case we use ideas The consensus trajectory is a linear combination of from output regulation [18]. • modes of the open loop system (11). Note that from To begin with, we make following classical assumption Remark 2, we know that the eigenvalues of A can be about the reference model given by (3). assigned arbitrarily as long as all the eigenvalues of A Assumption 5: All the eigenvalues of Ar are on imaginary are in the closed left-half plane. Therefore, the output axis, and the pair (Cr,Ar) is detectable. consensus trajectory captures various types of consensus The following lemma gives conditions under which the behavior. In order to avoid the trivial consensus, at least regulation of output consensus problem for a network of some of the eigenvalues of A need to be located on the agents (11) and the reference model given by (3) can be imaginary axis. solved by using #i given by (5) provided by the network, B. The model-reference output consensus problem and %i given by (6). Our focus so far has been on achieving output consensus, Lemma 5: Consider n agents of the form (11) and the without regard to the particular consensus trajectory. In this reference model (3). Let Assumptions 3 and 5 be satisfied. section, we consider the model-reference output consensus Then there exists a protocol of the form problem. q˙ = A q + B col(# ,% ), i c,i i c,i i i (15) To begin with, we make following assumption about the u¯i = Cc,iqi, reference model/virtual leader given by (2). ' such that limt "(yi(t) yr(t)) = 0 for all i 1,...,n . Assumption 4: (Cr,Ar,Br) has no invariant zeros, has uni- → − ∈ { } form rank nq, is invertible, and all the eigenvalues of Ar are The proof of Lemma 5 is given in Appendix D. in the closed left-half plane. In order to prove Lemma 5, we need following properties which can be guaranteed by our design. The conditions on (Cr,Ar,Br) given in Assumption 4 are not 1) Since the system (C,A,B) in Lemma 2 has no invariant as restrictive as it might appear. In fact, any trajectory y r p ∈ zeros and is invertible, from [18], we know that the follow- R , which is nq times differentiable can be generated by (2) ing equations with unknowns ( pnq r and ) p r, with (Cr,Ar,Br), which has no invariant zeros, has uniform R × i R × commonly know as the regulator∈ equations ∈ rank nq, is invertible, and all the eigenvalues of Ar are in the closed left-half plane. The most natural choice of reference (Ar = A( + B), (16a) model is p chains of integrators with each integrator of length Cr = C(, (16b) nq. The following lemma gives conditions under which the have a unique solution in terms of ( and ). model-reference output consensus problem for a network of 2) Since the matrix A in Lemma 2 is arbitrarily assignable agents (11) and the reference model given by (2) can be as long as all the eigenvalues of A are in the closed left-half solved by using #i given by (5) provided by the network, % i plane, we choose A such that A and Ar have no common given by (6) and the input to the reference model u r. eigenvalues. Lemma 4: Consider n agents of the form (11) and the Combining the results of Lemma 2 and Lemma 5, we reference model (2). Let Assumptions 3 and 4 be satisfied. obtain the following result. Then there exists a protocol of the form Theorem 3: Consider n agents of the form (1) and the reference model (3). Let Assumptions 3 and 5 be satis- q˙i = Ac,iqi + Bc,i col(#i,%i,ur), (14) fied. 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Using the above, we can find the following state space representation Next let us choose a particular protocol of the form (13) as of the resulting system with inputw ˜ i and output yi T q˙i =(A KC BB P(+))qi + diK#i, −T − (24) x˙i,q = Axˆ i,q + Bˆ(w˜i + Di,axi,a), u¯i = B P(+)qi, ˆ (21) ' − yi = Cxi,q, where K is a matrix such that A KC is Hurwitz stable, and ' − where + is sufficiently small. 0 I Let us define relative-state variables as p¯i = x¯i x¯n, ˆ p(nq 1) ˆ 0 ˆ − A = − , B = , C = Ip 0 , and q¯i = qi qn for all i 1,...,n 1 . We then de- Di Ip − ∈ { − } +, -. + . fine p¯ = col(p¯1,...,p¯n 1), q¯ = col(q¯1,...,q¯n 1), and , = p pnq p ,ni,a - − − for some matrices Di R × and Di,a R × . col(p¯,q¯,x˜1,...,xñ 1,xñ). ∈ ∈ − Note that the informationz ˜i is available for agent i since With just a little bit algebra, from (5), (11), (12) and (24), the local information zi is available for agent i and the we obtain the dynamical equations of the state variable , as 1 2 internal states xi and xi are always available for agent i. T In 1 A In 1 (BB P(+)) Q1 Q2 Moreover, sincez ˜i is detectable which follows directly from − ⊗ − − ⊗ T ˙ (DG) (KC) In 1 (A KC BB P(+)) 00 m , = ⊗ − ⊗ − − , , the detectability of the pair (Ci ,Ai), we can always design an  00Q3 0  local observer such that the observer errorx ˜i = col(xi,a,xi,q) 000Hn −   col(xî,a,xî,q) satisfies x˜˙i = Hix˜i, for some Hurwitz stable Hi.  (25) Now, let us choose the following pre-feedback where Q1 = blkdiag(BR1,...,BRn 1),Q2 = 1n 1 (BRn), − − − ⊗ and Q3 = blkdiag(H1,...,Hn 1). w˜i = Mu¯i Di,axî,a + A¯xî,q Dixî,q, (22) − − − It is then easy to see that the output consensus problem p is solved if the relative-state system (25) is asymptotically wherex î,a andx î,q are observer estimates, and u¯i R is a 1 ∈ stable, that is, all the eigenvalues of the matrix new input. Defining Ri = M− Di,a Di A¯ , 'i = Rix˜i, and − T xi,q = x¯i, we then obtain (11) by applying the pre-feedback In 1 A In 1 (BB P(+)) , - − ⊗ − − ⊗ T (22) to (21). (DG) (KC) In 1 (A KC BB P(+)) Notice that the above three stages can be represented by + ⊗ − ⊗ − − . 1 2 are in the open left-half plane due to the upper block- (10) by defining pi = col(xi ,xi ,xî,a,xî,q). The above three stages can be captured in Fig. 1. triangular form of the system matrix of (25) and the fact that H1,...,Hn are Hurwitz stable. With just a little bit algebra, we can show that the above matrix is similar to In 1 A˜ +(DG) (B˜C˜), where − u¯i w˜i wi rank vi squaring ui yi ⊗ ⊗ M Γi2 equalizing down agent i A BBTP(+) 0 + precompensator precompensator A˜ = , B˜ = , C˜ = C 0 . (26) 0 A −KC BBTP(+) K + − − . + . Following the methodology of [27], we define,U such- that 2 1 xi J = U − DGU, where J is the Jordan canonical form of 1 x DG. Then, using the matrix U In 1 to perform a similarity Pre-feedback Observer i ⊗ − transform of the matrix In 1 A˜ +(DG) (B˜C˜), we obtain zi − ⊗ ⊗ 1 (U − I)[I A˜ +(DG) (B˜C˜)](U I)=I A˜ + J (B˜C˜). ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ Fig. 1. First three stages of the design The resulting matrix is upper block-triangular, we see that its eigenvalues are the union of the eigenvalues of A˜ + !iB˜C˜ for each !i of the matrix DG. B. Proof of Lemma 3 From [20, Theorem 4], we see that all the eigenvalues of ˜ ˜ ˜ Since G satisfies Assumption 2, according to the proof A + !iBC are in the open left-half plane, and thus relative- of classical result of Fisher and Fuller (quoted as Lemma state dynamics (25) is asymptotically stable. Hence, the 1 above) given in [2], we can design a diagonal matrix output consensus is achieved. D = diag(d1,...,dn) such that all the eigenvalues of DG C. Proof of Lemma 4 are in the closed right-half plane, and that DG has only Let us first note that from Lemma 2, we know that the one eigenvalue at origin, with right eigenvector 1. Next, system (C,A,B) is arbitrarily assignable as long as the system we define the matrix DG as the (n 1) (n 1) matrix (C,A,B) has no invariant zeros, has uniform rank n , is obtained by removing the last row and− last× column− from the q T T invertible, and all the eigenvalues of A are in the closed left- matrix DG dn1gn, where gn is the last row of G, and we − half plane. Therefore, we choose (C,A,B)=(Cr,Ar,Br). define * = mini=1,...,n 1 Re(!i(DG)). It is easy to see that all − Given the matrix E, let be such that Assumption 3 is the eigenvalues of DG are the nonzero eigenvalues of DG, & satisfied. According to the proof of classical result of Fisher therefore, * > 0. Let P(+)=PT(+) > 0 be the unique solution and Fuller (quoted as Lemma 1 above) given in [2], we of the algebraic Riccati equation can design a matrix D = diag(d1,...,dn) such that all the T T A P(+)+P(+)A *P(+)BB P(+)++Ipn = 0. (23) eigenvalues of D(G + &E) are in the open right-half plane, − q and we define * = mini=1,...,n Re(!i(D(G + &E))) > 0. Let Notice that (Cf ,A f ) is detectable since A and Ar have T P(+)=P (+) > 0 be the unique solution of (23). Let us no common eigenvalues while (C,A) and (C(,Ar) are de- choose a particular protocol of the form (14) as tectable. This follows from the fact that I 0 I 0 A 0 I 0 T r C( C q˙i =(A KC BB P(+))qi + diK#i + &diK%i, ( I A f ( I = 0 A , Cf ( I =[ ]. −T − (27) − u¯i = B P(+)qi + ur, ' − Given1 the2 matrix1 E,2 let1& be2 such that1 Assumption2 3 is where K is a matrix such that A KC is Hurwitz stable, and satisfied. According to the proof of classical result of Fisher − + is sufficiently small. and Fuller (quoted as Lemma 1 above) given in [2], we Note that the protocol state qi is an estimate of x¯i can design a diagonal matrix D = diag(d ,...,dn) such that − 1 xr. Let us define ri = x¯i xr, r = col(r ,...,rn), q = all the eigenvalues of D(G + &E) are in the open right- − 1 col(q1,...,qn),˜x = col(x˜1,...,xñ) and , = col(r,q,x˜). Note half plane. Define * = mini=1,...,n Re!i(D(G + &E)) > 0. Let n n T that $ j=1 gijCx¯j = $ j=1 gijC(x¯j xr) since G1 = 0, we then P(+)=P (+) > 0 be the unique solution of the algebraic obtain − Riccati equation

T In A In (BB P(+)) Q4 A P( )+P( )AT P( )BT B P( )+ I = 0. (32) ⊗ − ⊗ T f + + f * + f f + + pnp+r ,˙ = D(G + &E) (KC) In (A KC BB P(+)) 0 , , − ⊗ ⊗ − − / 00Q50 Notice that we need that the unique solution P(+) of (32) (28) satisfies lim+ 0 P(+)=0. In order to achieve this, it is where Q4 = blkdiag(BR1,...,BRn), and Q5 = → required that all the eigenvalues of A f are in the closed left- blkdiag(H1,...,Hn). half plane from [24, Theorem 3, Lemma 4]. This is satisfied Following the same lines as the proof of Lemma 3, due to the lower block-triangular form of A f , and the facts we can easily show that the closed-loop system (28) is that all the eigenvalues of Ar are on imaginary axis and all asymptotically stable, and thus the model-reference output the eigenvalues of A are in the closed left-half plane. For consensus is achieved. each i 1,...,n , let us choose the following protocol ∈ { } T D. Proof of Lemma 5 q˙i =(A f Kf Cf B f B f P(+))qi + diKf #i + &diKf %i, T− − (33) u f,i = B P(+)qi, Let us first expand the exosystem (3) as ' − f where K is a matrix such that A K C is Hurwitz stable f f − f f x˙r = Arxr, and + is sufficiently small. -˙ = A- + B)xr,  Note that the protocol state qi is an estimate of xe,i = x f ,i yr(t)=Crxr(t), −  x f ,r. Let us define xe = col(xe,1,...,xe,n), q = col(q1,...,qn), x˜ = col(x˜1,...,xñ), and , = col(xe, p,x˜). with -(0)=(xr(0), where ( and ) are the unique solution of regulator equation (16). It is then easy to verify that -(t)= With some algebra, from (30), (31), and (33), we obtain the closed loop system (xr(t) for all t and thus yr(t)=Crxr(t)=C(xr(t)=C-(t). T The reason behind this expansion is that the exosystem also In A f In (B f B f P(+)) Q6 ⊗ − ⊗ T ˙ = (D(G + &E)) (Kf Cf ) In (A f Kf Cf B f B P(+)) 0 , contains a target for the state of the individual agents. , ⊗ ⊗ − − f , Next, we expand each individual agent (11) as / 00Q50 (34) 0 x˙r,i = Arxr,i + ui,1, where Q6 =(In )blkdiag(R1,...,Rn). x¯˙ = Ax¯ + B)x + Bu + B' , ⊗ B  i i r,i i,2 i Following the+ same. lines as the proof of Lemma 3, y (t)=Cx¯ (t),  i i we can easily show that the closed loop system (34) is where we have used u¯i = )xr,i + ui,2. Note that the first state asymptotically stable. The above argument shows that there equation for xr,i is basically part of our consensus protocol. exists a protocol of the form (15) that solves the regulation Let us define x f ,i = col(xr,i,x¯i), x f ,r = col(xr,-), u f ,i = of output consensus for multi-agent system (11). col(ui,1,ui,2), u f ,r = 0, and A 0 I 0 A = r , B = , C = 0 C . (29) f B) A f 0 B f + . + . , - We then obtain the expanded reference model x˙ = A x + B u , f ,r f f ,r f f ,r (30) yr = C x , ' f f ,r where u f ,r = 0 and n individual agents 0 x˙f ,i = A f x f ,i + B f (u f ,i + ), 'i (31)  + .  yi = Cf x f ,i. 