Recommended by Sophie Tarbouriech Accepted Manuscript Solvability conditions and design for H & H2 almost state synchronization of homogeneous multi-agent systems Anton A. Stoorvogel, Ali Saberi, Meirong Zhang, Zhenwei Liu PII: DOI: Reference: S0947-3580(17)30260-1 https://doi.org/10.1016/j.ejcon.2018.08.001 EJCON 281 To appear in: European Journal of Control Received date: Revised date: Accepted date: 17 July 2017 25 July 2018 6 August 2018 Please cite this article as: Anton A. Stoorvogel, Ali Saberi, Meirong Zhang, Zhenwei Liu, Solvability conditions and design for H & H2 almost state synchronization of homogeneous multi-agent systems, European Journal of Control (2018), doi: https://doi.org/10.1016/j.ejcon.2018.08.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT Solvability conditions and design for H? & H2 almost state synchronization of homogeneous multi-agent systems a School CR IP T Anton A. Stoorvogelb,?, Ali Saberia , Meirong Zhangc , Zhenwei Liua of Electrical Engineering and Computer Science, Washington State University, Pullman, WA, USA of Electrical Engineering, Mathematics and Computer Science, University of Twente, P.O. Box 217, Enschede, The Netherlands c School of Engineering and Applied Science, Gonzaga University, Spokane, WA, USA b Department AN US Abstract This paper studies the H? and H2 almost state synchronization problem for homogeneous multi-agent systems with general linear agents affected by external disturbances and with a directed communication topology. Agents are connected via diffusive fullstate coupling or diffusive partial-state coupling. A necessary and sufficient condition is developed for the solvability of the H? and H2 almost state synchronization problem. M Moreover, a family of protocols based on either an algebraic Riccati equation (ARE) method or a directed eigenstructure assignment method are developed such that the impact of disturbances on the network disagreement dynamics, expressed in terms of ED the H? and H2 norm of the corresponding closed-loop transfer function, is reduced to any arbitrarily small value. The protocol for full-state coupling is static, while for PT partial-state coupling it is dynamic. Keywords: Multi-agent systems, H? and H2 almost state synchronization, CE Distributed control AC 1. Introduction Over the past decade, the synchronization problem of multi-agent system (MAS) has received substantial attention because of its potential applications in cooperative control of autonomous vehicles, distributed sensor network, swarming and flocking and ? Corresponding author Email addresses: A.A.Stoorvogel@utwente.nl (Anton A. Stoorvogel), saberi@eecs.wsu.edu (Ali Saberi), zhangm@gonzaga.edu (Meirong Zhang), zhenwei.liu@wsu.edu (Zhenwei Liu) Preprint submitted to Elsevier August 14, 2018 ACCEPTED MANUSCRIPT others. The objective of synchronization is to secure an asymptotic agreement on a common state or output trajectory through decentralized protocols (see [1, 2, 3, 4] and references therein). CR IP T State synchronization inherently requires homogeneous MAS (i.e. agents have identical dynamics). Most works have focused on state synchronization based on diffu- sive full-state coupling, where the agent dynamics progress from single- and double- integrator dynamics (e.g. [5], [6], [7], [8], [9]) to more general dynamics (e.g. [10], [11], [12], [13]). State synchronization based on diffusive partial-state coupling has also been considered (e.g. [14], [15], [10], [16], [17], [18], [19]). AN US Most research has focused on the idealized case where the agents are not affected by external disturbances. In the literature where external disturbances are considered, ?-suboptimal H? design is developed for MAS to achieve H? norm from an external disturbance to the synchronization error among agents less than an, a priori given, bound ?. In particular, [15], [20] considered the H? norm from an external disturbance to the output error among agents. [21] considered the H? norm from an external disturbance M to the state error among agents. These papers do not present an explicit methodology for designing protocols. The papers [22] and [23] try to obtain an H? norm from a ED disturbance to the average of the states in a network of single or double integrators. By contrast, [24] introduced the notion of H? almost synchronization for homoge- neous MAS, where the goal is to reduce the H? norm from an external disturbance to PT the synchronization error, to any arbitrary desired level. But it requires an additional layer of communication among distributed controllers, which is completely dispensed in this paper. This work is extended later in [25], [26], and [27]. The paper [27], where CE heterogeneous MAS are considered, provides a solution for the case of right-invertible agents with the addional objective beyond output synchronization that the agents track AC a regulated signal given to some or all of the agents. Although homogeneous MAS, which are considered in this paper, are a subset of heterogeneous MAS, the results of [27] cannot be directly applied to the case of full-state coupling since the agents are not right-invertible. Secondly, the results for synchronization without regulation cannot be obtained from results obtained for regulated synchronization. Thirdly, we consider state synchronization instead of output synchronization in both full- and partial-state 2 ACCEPTED MANUSCRIPT coupling. Finally, by restricting to homogeneous networks more explicit designs can be obtained under weaker conditions. In this paper, we will study H? almost state synchronization for a MAS with full- CR IP T state coupling or partial-state coupling. We will also study H2 almost state synchronization, since it is closely related to the problems of H? almost state synchronization. In H? we look at the worst case disturbance with the only constraints being the power, while in H2 we only consider white noise disturbances which is a more restrictive class. In both cases, disturbances or noises are restricted in the process, not in the measurement. Our contribution in this paper is three-fold. AN US ? We obtain necessary and sufficient conditions for H? and H2 almost state synchronization for a MAS in the presence of external disturbances, ? We develop a protocol design for H? and H2 almost state synchronization based on an algebraic Riccati equation (ARE) method, ? We develop a protocol design for H? and H2 almost state synchronization based M on an asymptotic time-scale eigenstructure assignment (ATEA) method for the full-state coupling case, and on the direct eigenstructure assignment method for ED the partial-state coupling case. It is worth noting that our solvability conditions and protocol designs are developed for a MAS associated with a set of network graphs. Specifically, only rough information PT of a network graph is utilized. 1.1. Notations and definitions CE Given a matrix A ? CmОn , A0 denotes its conjugate transpose, k Ak is the induced 2-norm. A square matrix A is said to be Hurwitz stable if all its eigenvalues are in AC the open left half complex plane. A ? B depicts the Kronecker product between A and B. In denotes the n-dimensional identity matrix and 0n denotes n О n zero matrix; sometimes we drop the subscript if the dimension is clear from the context. Given a complex number ?, Re(?) is the real part of ? and Im(?) is the imaginary part of ?. A weighted directed graph G is defined by a triple (V, E, A) where V = {1, . . . , N } is a node set, E is a set of pairs of nodes indicating connections among nodes, and 3 ACCEPTED MANUSCRIPT A = [ai j ] ? R N ОN is the weighting matrix, and ai j > 0 iff (i, j) ? E which denotes an edge from node j to node i. In our case, we have aii = 0. A path from node i 1 to i k is a sequence of nodes {i 1, . . . , i k } such that (i j+1, i j ) ? E for j = 1, . . . , k ? 1. A CR IP T directed tree is a subgraph (subset of nodes and edges) in which every node has exactly one parent node except for one node, called the root, which has no parent node. In this case, the root has a directed path to every other node in the tree. A directed spanning tree is a directed tree containing all the nodes of the graph. For a weighted graph G, a matrix L = [` i j ] with AN US PN ? ? aik , i = j, ? k=1 `i j = ? ? ? ?ai j , i , j, ? is called the Laplacian matrix associated with the graph G. The Laplacian L has all its eigenvalues in the closed right half plane and at least one eigenvalue at zero associated with right eigenvector 1. A specific class of graphs needed in this paper is presented below: N denote the set of directed graphs Definition 1. For any given ? ? ? > 0, let G?, ? M with N nodes that contain a directed spanning tree and for which the corresponding Laplacian matrix L satisfies kLk < ? while its nonzero eigenvalues have a real part ED larger than or equal to ?. 2. Problem formulation PT Consider a MAS composed of N identical linear time-invariant agents of the form, CE x? i = Ax i + Bui + E?i , yi = C x i , (i = 1, . . . , N ) (1) where x i ? Rn , ui ? Rm , yi ? R p are respectively the state, input, and output vectors AC of agent i, and ?i ? R? is the external disturbances. The communication network provides each agent with a linear combination of its own outputs relative to that of other neighboring agents. In particular, each agent i ? {1, . . . , N } has access to the quantity, ?i = N X j=1 ai j (yi ? y j ), 4 (2) ACCEPTED MANUSCRIPT where ai j ? 0 and aii = 0 indicate the communication among agents. This communication topology of the network can be described by a weighted and directed graph G with nodes corresponding to the agents in the network and the weight of edges given rewritten as ?i = N X CR IP T by the coefficient ai j . In terms of the coefficients of the Laplacian matrix L, ? i can be (3) `i j y j . j=1 We refer to this case as partial-state coupling. Note that if C has full column rank then, without loss of generality, we can assume that C = I, and the quantity ? i becomes N X j=1 ai j (x i ? x j ) = N X `i j x j . AN US ?i = We refer to this case as full-state coupling. (4) j=1 If the graph G describing the communication topology of the network contains a directed spanning tree, then it follows from [9, Lemma 3.3] that the Laplacian matrix L has a simple eigenvalue at the origin, with the corresponding right eigenvector 1 and all M the other eigenvalues are in the open right-half complex plane. Let ? 1, . . . , ? N denote the eigenvalues of L such that ? 1 = 0 and Re(? i ) > 0, i = 2, . . . , N. PT ED Let N be any agent and define x? i = x N ? x i and *. x? 1 +/ . x? = ... .. /// . / x? N ?1 , - and *. ?1 +/ . ? = ... .. /// . . / ,? N - CE Obviously, synchronization is achieved if x? = 0. That is lim (x i (t) ? x N (t)) = 0, t?? ?i, ? {1, . . . , N ? 1}. (5) We denote by T? x? , the transfer function from ? to x? AC Remark 1. Agent N is not necessarily a root agent. Obviously, (5) is equivalent to the condition that lim (x i (t) ? x j (t)) = 0, t?? ?i, j ? {1, . . . , N }. We formulate below four almost state synchronization problems for a network with either H2 or H? almost synchronization. 5 ACCEPTED MANUSCRIPT Problem 1. Consider a MAS described by (1) and (4). Let G be a given set of graphs such that G ? G N . The H? almost state synchronization problem via full-state coupling (in short H? -ASSFS) with a set of network graphs G is to find, if possible, a ui = F (?)? i , CR IP T linear static protocol parameterized in terms of a parameter ?, of the form, (6) such that, for any given real number r > 0, there exists an ? ? such that for any ? ? (0, ? ? ] and for any graph G ? G, (5) is satisfied for all initial conditions in the absence of AN US disturbances and the closed loop transfer matrix T? x? satisfies kT? x? k? < r. (7) Problem 2. Consider a MAS described by (1) and (3). Let G be a given set of graphs such that G ? G N . The H? almost state synchronization problem via partial-state coupling (in short H? -ASSPS) with a set of network graphs G is to find, if possible, a linear time-invariant dynamic protocol parameterized in terms of a parameter ?, of the M form, ?? i = Ac (?) ? i + Bc (?)? i , (8) ED ui = Cc (?) ? i + Dc (?)? i , where ? i ? Rn c , such that, for any given real number r > 0, there exists an ? ? such that for any ? ? (0, ? ? ] and for any graph G ? G, (5) is satisfied for all initial conditions in PT the absence of disturbances and the closed loop transfer matrix T? x? satisfies (7). Problem 3. Consider a MAS described by (1) and (4). Let G be a given set of graphs CE such that G ? G N . The H2 almost state synchronization problem via full-state coupling (in short H2 -ASSFS) with a set of network graphs G is to find, if possible, a linear static protocol parameterized in terms of a parameter ?, of the form (6) such AC that, for any given real number r > 0, there exists an ? ? such that for any ? ? (0, ? ? ] and for any graph G ? G, (5) is satisfied for all initial conditions in the absence of disturbances and the closed loop transfer matrix T? x? satisfies kT? x? k2 < r. 6 (9) ACCEPTED MANUSCRIPT Problem 4. Consider a MAS described by (1) and (3). Let G be a given set of graphs such that G ? G N . The H2 almost state synchronization problem via partial-state coupling (in short H2 -ASSPS) with a set of network graphs G is to find, if possible, CR IP T a linear time-invariant dynamic protocol parameterized in terms of a parameter ?, of the form (8) such that, for any given real number r > 0, there exists an ? ? such that for any ? ? (0, ? ? ] and for any graph G ? G, (5) is satisfied for all initial conditions in the absence of disturbances and the closed loop transfer matrix T? x? satisfies (9). Note that the problems of H? almost state synchronization and H2 almost state synchronization are closely related. Roughly speaking, H2 almost synchronization is AN US easier to achieve than H? almost synchronization. This is related to the fact that in H? we look at the worst case disturbance with the only constraints being the power: Z T 1 lim sup ?i0 (t)?i (t)dt < ?. T ?? 2T ?T while in H2 we only consider white noise disturbances which is a more restrictive class. M 3. MAS with full-state coupling In this section, we establish a connection between the almost state synchronization ED among agents in the network and a robust H? or H2 almost disturbance decoupling problem via state feedback with internal stability (in short H? or H2 -ADDPSS) (see PT [28]). Then, we use this connection to derive the necessary and sufficient condition and design appropriate protocols. CE 3.1. Necessary and sufficient condition for H? -ASSFS The MAS system described by (1) and (4) after implementing the linear static AC protocol (6) is described by x? i = Ax i + BF (?)? i + E?i , for i = 1, . . . , N. Let *. x 1 +/ . x = ... .. /// , . / ,x N - *. ?1 +/ . ? = ... .. /// . . / ,? N 7 ACCEPTED MANUSCRIPT Then, the overall dynamics of the N agents can be written as x? = (I N ? A + L ? BF (?))x + (I N ? E)?. (10) CR IP T We define the robust H? -ADDPSS with bounded input as follows. Given ? ? C, there should exist M > 0 such that for any given real number r > 0, we can find a parameterized controller u = F (?)x for the following subsystem, AN US x? = Ax + ? Bu + B?, (11) (12) such that for any ? ? ? the following hold: 1. The interconnection of the systems (12) and (11) is internally stable; 2. The resulting closed-loop transfer function T? x from ? to x has an H? norm less than r. M 3. The resulting closed-loop transfer function T?u from ? to u has an H? norm less than M. ED In the above, ? denotes all possible locations for the nonzero eigenvalues of the Laplacian matrix L when the graph varies over the set G. It is also important to note that M is independent of the choice for r. PT In the following lemma we give a necessary condition for the H? -ASSFS. Moreover, for sufficiency, we connect the H? -ASSFS problem to the robust H? -ADDPSS with CE bounded input problem which we will address later. Lemma 1. Let G be a set of graphs such that the associated Laplacian matrices are AC uniformly bounded and let ? consist of all possible nonzero eigenvalues of Laplacian matrices associated with graphs in G. (Necessity) The H? -ASSFS for the MAS described by (1) and (4) given G is solvable by a parameterized protocol ui = F (?)? i only if im E ? im B. 8 (13) ACCEPTED MANUSCRIPT (Sufficiency) The H? -ASSFS for the MAS described by (1) and (4) given G is solved by a parameterized protocol ui = F (?)? i if the robust H? -ADDPSS with bounded input for the system (12) with ? ? ? is solved by the parameterized controller u = F (?)x. CR IP T Proof: Note that L has eigenvalue 0 with associated right eigenvector 1. Let (14) L = T SL T ?1, with T unitary and SL the upper-triangular Schur form associated to the Laplacian matrix L such that SL (1, 1) = 0. Let *. ? 1 +/ . ? In )x = ... .. /// , . / ,? N - *. ??1 +/ . ? I)? = ... .. /// . / ,?? N - AN US ? := (T ?1 ?? = (T ?1 where ? i ? Cn and ??i ? Cq . In the new coordinates, the dynamics of ? can be written as which is rewritten as N X s1 j BF (?)? j + E ??1, ED ?? 1 = A? 1 + (15) M ??(t) = (I N ? A + SL ? BF (?))? + (T ?1 ? E)?, j=2 ?? i = ( A + ? i BF (?))? i + N X s i j BF (?)? j + E ??i , (16) PT j=i+1 ?? N = ( A + ? N BF (?))? N + E ?? N , CE for i ? {2, . . . , N ? 1} where SL = [s i j ]. The first column of T is an eigenvector of ? L associated to eigenvalue 0 with length 1, i.e. it is equal to ▒1/ N. Using this we AC obtain: *. x N ? x 1 +/ *.*.?1 0 и и и 0 1+/ +/ .. .. / .. .. // .... // x N ? x 2 / .. 0 ?1 . . . // // (T ? I )? / . . / x? = ... = ? I n n .. .. / // .... .. // .. .. .. / . . . . 0 . // .... // // . x ? x 0 и и и 0 ?1 1 N ?1 , N ,, = 0 V ? In ?, 9 ACCEPTED MANUSCRIPT for some suitably chosen matrix V . Therefore we have (17) CR IP T *. ? 2 +/ . x? = (V ? In ) ... .. /// , . / ,? N - Note that since T is unitary, also the matrix T ?1 is unitary and the matrix V is uniformly bounded. Therefore the H? norm of the transfer matrix from ? to x? can be made arbitrarily small if and only if the H? norm of the transfer matrix from ?? to ? can be made arbitrarily small. AN US In order for the H? norm from ?? to ? to be arbitrarily small we need the H? norm from ?? N to ? N to be arbitrarily small. From classical results (see [29, 30]) on H? almost disturbance decoupling we find that this is only possible if (13) is satisfied. Conversely, suppose u = F (?)x solves the robust H? -ADDPSS with bounded input for (12) and assume (13) is satisfied. We show next that ui = F (?)? i solves the H? -ASSFS for the MAS described by (1) and (4). Let X be such that E = BX. M The fact that u = F (?)x solves the robust H? -ADDPSS with bounded input for (12) implies that for small ? we have that A + ? BF (?) is asymptotically stable for all ? ? ?. ED In particular, A + ? i BF (?) is asymptotically stable for i = 2, . . . , N which guarantees that ? i ? 0 for i = 2, . . . , N for zero disturbances and all initial conditions. Therefore we have state synchronization. PT Next, we are going to show that for any r? > 0, we can choose ? sufficiently small such that the transfer matrix from ?? to ? i is less than r? for i = 2, . . . , N. This guarantees AC CE that we can achieve (7) for any r > 0. We have that T?? x (s) = (sI ? A ? ? BF (?)) ?1 B, ? T?u (s) = F (?)(sI ? A ? ? BF (?)) ?1 B. For a given M and parameter ?, the following is satisfied kT?? x k? < r? ? , ? kT?u k? < M for all ? ? ? where r? ? is a parameter depending on ? with the property that lim??0 r? ? = 0. Denote ?i = F (?)? i . 10 ACCEPTED MANUSCRIPT When i = N, it is easy to find that, иии 0 and hence ?N T??? N = T?u 0 X , kT??? N k? < r?, kT??? N k? < M?N provided k X k r? ? < r?, k X k M < M?N . иии 0 X CR IP T T??? N = T?? xN 0 (18) Recall that we can make r? ? arbitrarily small by reducing ? without affecting the bound AN US M. Assume kT??? j k? < r?, holds for j = i + 1, . . . , N. We have: kT??? j k? < M? j M ?? ?? N X s i j T??? j (s) ??? T??? i (s) = T?? ix (s) ???ei ? X + ?? ?? j=i+1 ?? ?? N X ? ?i T??? i (s) = T?u (s) ??ei ? X + s i j T??? j (s) ??? ?? ?? j=i+1 where ei is a row vector of dimension N with elements equal to zero except for the ith ED component which is equal to 1. Since PT N N X X e ? X + < k X k + s T |s i j | M? j i j ??? j i j=i+1 j=i+1 ? CE we find: kT??? i k? < r?, kT??? i k? < M?i (19) AC provided: *. k X k + , N X |s i j | M? j +/ r? ? < r?, *. k X k + j=i+1 , N X |s i j | M? j +/ M? < M?i . j=i+1 - (20) Note that s i j depends on the graph in G but since the Laplacian matrices associated to graphs in G are uniformly bounded we find that also the s i j are uniformly bounded. In this way for any arbitrary r?, we can recursively obtain the bounds in (19) for i = 2, . . . , N provided we choose ? sufficiently small such that the corresponding r? ? satisfies (18) 11 ACCEPTED MANUSCRIPT and (20) for i = 2, . . . , N ? 1. Hence, we can choose ? sufficiently small such that the transfer matrix from ?? to ? i is less than r? for i = 2, . . . , N. As noted before this guarantees that we can achieve (7) for any r > 0. CR IP T N for some given ?, ? > 0, we For the case where the set of graphs G equals G?, ? develop necessary and sufficient conditions for the solvability of the H? -ASSFS for MAS as follows: Theorem 1. Consider a MAS described by (1) and (4) with an associated graph from N for some ?, ? > 0. the set G = G?, ? Then, the H? -ASSFS is solvable if and only if (13) is satisfied and ( A, B) is AN US stabilizable. Proof: From Lemma 1, we note that (13) is actually a necessary condition for H? - ASSFS. Clearly, also ( A, B) stabilizable is a necessary condition. Sufficiency is a direct result of Theorems 3 or Theorem 5 for H? -ASSFS. M 3.2. Necessary and sufficient conditions for H2 -ASSFS We define the robust H2 -ADDPSS with bounded input as follows. Given ? ? C, ED there should exist M > 0 such that for any given real number r > 0, we can find a parameterized controller (11) for the system, (12) such that the following holds for any ? ? ?: PT 1. The interconnection of the systems (11) and (12) is internally stable; 2. The resulting closed-loop transfer function T? x from ? to x has an H2 norm less CE than r. 3. The resulting closed-loop transfer function T?u from ? to u has an H? norm less AC than M. In the above, ? denotes all possible locations for the nonzero eigenvalues of the Laplacian matrix L when the graph varies over the set G. It is also important to note that M is independent of the choice for r. Note that we need to consider two aspects in our controller H2 disturbance rejection and robust stabilization (because of a set of network N ). The latter translates in the H norm constraint from ? to u. graphs G?, ? ? 12 ACCEPTED MANUSCRIPT Lemma 2. Let G be a set of graphs such that the associated Laplacian matrices are uniformly bounded and let ? consist of all possible nonzero eigenvalues of Laplacian matrices associated with graphs in G. CR IP T (Necessity) The H2 -ASSFS for the MAS described by (1) and (4) given G is solvable by a parameterized protocol ui = F (?)? i only if (13) is satisfied. (Sufficiency)The H2 -ASSFS for the MAS described by (1) and (4) given G is solvable by a parameterized protocol ui = F (?)? i if the robust H2 -ADDPSS with bounded input for the system (12) with ? ? ? is solved by the parameterized controller u = F (?)x. Proof: The proof is similar to the proof of Lemma 1. This time we need the H2 norm AN US from ?? N to ? N to be arbitrarily small and also H2 almost disturbance decoupling then immediately yields that we need that (13) is satisfied. The rest of the proof follows the same lines except that we require the H2 norm from ?? to ? j arbitrarily small while we keep the H? norm from ?? to ? j bounded. Recall that for any two stable, strictly proper transfer matrices T1 and T2 we have: M kT1T2 k2 ? kT1 k2 kT2 k? which we need in the modifications of the proof of Lemma 1. ED N (with given ?, ? > 0), we develop For the case with a set of graph G = G?, ? necessary and sufficient conditions for the solvability of the H2 -ASSFS for MAS as follows: PT Theorem 2. Consider a MAS described by (1) and (4) with an associated graph from N for some ?, ? > 0. the set G = G?, ? CE Then, the H2 -ASSFS is solvable if and only if (13) is satisfied and ( A, B) is stabilizable. Proof: We have already noted before that (13) is actually a necessary condition for AC H2 -ASSFS. Clearly, also ( A, B) being stabilizable is a necessary condition. Sufficiency for H2 -ASSFS, is a direct result of either Theorem 4 or Theorem 6. 3.3. Protocol design for H? -ASSFS and H2 -ASSFS We present below two protocol design methods for both H? -ASSFS and H2 -ASSFS problems. One relies on an algebraic Riccati equation (ARE), and the other is based on 13 ACCEPTED MANUSCRIPT an asymptotic time-scale eigenstructure assignment (ATEA) method. 3.3.1. ARE-based method ( A, B) is stabilizable. We consider the protocol, ui = ?F? i , where ? = 1 ? CR IP T Using an algebraic Riccati equation, we can design a suitable protocol provided (21) and F = ?B 0 P with P being the unique solution of the continuous-time algebraic Riccati equation AN US A0 P + P A ? 2 ?PBB 0 P + I = 0, (22) where ? is a lower bound for the real part of the non-zero eigenavlues of all Laplacian N . matrices associated with a graph in G = G?, ? The main result regarding H? -ASSFS is stated as follows. M Theorem 3. Consider a MAS described by (1) and (4) such that (13) is satisfied. Let any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network N be defined. graphs G?, ? ED N is If ( A, B) is stabilizable then the H? -ASSFS stated in Problem 1 with G = G?, ? solvable. In particular, for any given real number r > 0, there exists an ? ? , such that for any ? ? (0, ? ? ), the protocol (21) achieves state synchronization and the resulting PT system from ? to x i ? x j has an H? norm less than r for any i, j ? 1, . . . , N and for CE N . any graph G ? G?, ? Proof: Using Lemma 1, we know that we only need to verify that u = ?F x solves the robust H? -ADDPSS with bounded input for the system (12) with ? ? ?. Given AC N , we know that ? ? ? implies Re ? ? ?. Clearly, the Laplacian matrices are G ? G?, ? uniformly bounded since kLk ? ?. Consider the interconnection of (12) and u = ?F x. We define V (x) = x 0 Px 14 ACCEPTED MANUSCRIPT and we obtain: V? = x 0 ( A ? ?? BB 0 P) 0 Px + ? 0 B 0 Px + x 0 P( A ? ?? BB 0 P)x + x 0 PB? CR IP T = x 0 PBB 0 Px ? x 0 x ? 2? ?x 0 PBB 0 Px + 2x 0 PB? ? (1 ? ?? )x 0 PBB 0 Px ? x 0 x + ?? ? 0? ? ? ?2 ?u 0u ? x 0 x + ?? ? 0? which implies that the system is asymptotically stable and the H? norm of the transfer function from ? to x is less that ?/ ? while the H? norm of the transfer function from AN US ? to u is less that 2/ ? 2 . Therefore, u = ?F x solves the robust H? -ADDPSS with bounded input for the system (12) as required. For H2 -ASSFS we have the following classical result: Lemma 3. Consider an asymptotically stable system: M p? = A1 p + B1 ? The H2 norm from ? to p is less than ? if there exists a matrix Q such that: ED A1 Q + Q A10 + B1 B10 ? 0, Q < ?I The main result regarding H2 -ASSFS is stated as follows. PT Theorem 4. Consider a MAS described by (1) and (4) such that (13) is satisfied. Let any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network CE N be defined. graphs G?, ? N is If ( A, B) is stabilizable then the H2 -ASSFS stated in Problem 3 with G = G?, ? solvable. In particular, for any given real number r > 0, there exists an ? ? , such that AC for any ? ? (0, ? ? ), the protocol (21) achieves state synchronization and the resulting system from ? to x i ? x j has an H? norm less than r for any i, j ? 1, . . . , N and for N . any graph G ? G?, ? Proof: Using Lemma 2, we know that we only need to verify that u = ?F x solves the robust H2 -ADDPSS with bounded input for the system (12) with ? ? ?. We use the 15 ACCEPTED MANUSCRIPT same feedback as in the proof of Theorem 3. In the proof of Theorem 3 it is already shown that the closed loop system is asymptotically stable and the H? norm of the transfer function from ? to u is bounded. The only remaining part of the proof is to CR IP T show that the H2 norm from ? to x can be made arbitrarily small. Using the algebraic Riccati equation it is easy to see that we have: ( A ? ?? BB 0 P) 0 P + P( A ? ?? BB 0 P) + ? ?PBB 0 P ? 0 for large ?. But then we have: AN US Q? ( A ? ?? BB 0 P) 0 + ( A ? ?? BB 0 P)Q? + BB 0 ? 0 for Q? = ? ? ?1 P?1 . Then Lemma 3 immediately yields that we can make the H2 norm from ? to x arbitrarily small by choosing a sufficiently small ?. 3.3.2. ATEA-based method The ATEA-based design is basically a method of time-scale structure assignment M in linear multivariable systems by high-gain feedback [31]. In the current case, we do not need the full structure presented in the above method. It is sufficient to note that ED there exists non-singular transformation matrix Tx ? RnОn (See [32, Theorem 1]) such PT that x? 1 x? = *. +/ = Tx x, , x? 2 - (23) CE and the dynamics of x? is represented as x?? 1 = A?11 x? 1 + A?12 x? 2, x?? 2 = A?21 x? 1 + A?22 x? 2 + ? B?u + B??, (24) AC with B? invertible, and that ( A, B) is stabilizable implies that ( A?11, A?12 ) is stabilizable. Choose F1 such that A?11 + A?12 F1 is asymptotically stable. In that case a suitable protocol for (1) is (25) ui = F? ? i , where F? is designed as F? = 1 ?1 B? F1 ? 16 ?I Tx (26) ACCEPTED MANUSCRIPT The main result regarding H? -ASSFS is stated as follows. The result is basically the same as Theorem 3 except for a different design protocol. Theorem 5. Consider a MAS described by (1) and (4) such that (13) is satisfied. Let CR IP T any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network N be defined. graphs G?, ? N is If ( A, B) is stabilizable then the H? -ASSFS stated in Problem 1 with G = G?, ? solvable. In particular, for any given real number r > 0, there exists an ? ? , such that for any ? ? (0, ? ? ), the protocol (25) achieves state synchronization and the resulting system from ? to x i ? x j has an H? norm less than r for any i, j ? 1, . . . , N and for AN US N . any graph G ? G?, ? Proof: Similarly to the proof of Theorem 3, we only need to establish that u = F? x solves the robust H? -ADDPSS with bounded input for the system (12) with ? ? ?. N , we know that ? ? ? implies Re ? ? ?. Given G ? G?, ? After a basis transformation, the interconnection of (12) and u = F? x is equal to M the interconnection of (24) and (25). We obtain: x?? 1 = A?11 x? 1 + A?12 x? 2, Define ED ? x?? 2 = (? A?21 + ?F1 ) x? 1 + (? A?22 ? ?I) x? 2 + ? B??. PT x? 1 = x? 1, (27) x? 2 = x? 2 ? F1 x? 1 . CE Then we can write this system (27) in the form: x?? 1 = A?11 x? 1 + A?12 x? 2, ? x?? 2 = ? A?21 x? 1 + (? A?22 ? ?I) x? 2 + ? B??, (28) AC where A?11 = A?11 + A?12 F1, A?12 = A?12, A?21 = A?21 ? F1 A?11 + A?22 ? F1 A?12, A?22 = A?22 ? F1 A?12 . In the absence of the external disturbances, the above system (28) is asymptotically stable for small enough ?. 17 ACCEPTED MANUSCRIPT Since A?11 = A?11 + A?12 F1 is Hurwitz stable, there exists P > 0 such that the Lyapunov 0 P = ?I holds. For the dynamics x? , we define a Lyapunov function equation P A?11 + A?11 1 V1 = x? 10 P x? 1 . Then the derivative of V1 can be bounded CR IP T 0 V?1 ? ?k x? 1 k 2 + x? 20 A?12 P x? 1 + x? 10 P A?12 x? 2 ? ?k x? 1 k 2 + 2 Re( x? 10 P A?12 x? 2 ) ? ?k x? 1 k 2 + r 1 k x? 1 kk x? 2 k, where 2kP A?12 k ? r 1 . Now define a Lyapunov function V2 = ? x? 20 x? 2 for the dynamics AN US x? 2 . The derivative of V2 can then also be bounded. V?2 ? ?2 Re(?)k x? 2 k 2 + 2? Re( x? 20 A?21 x? 1 ) + 2? x? 20 A?22 x? 2 + 2? Re( x? 20 B??) ? ?2 Re(?)k x? 2 k 2 + ?r 2 k x? 1 kk x? 2 k + ?r 3 k x? 2 k 2 + ?r 4 k?kk x? 2 k ? ? ?k x? 2 k 2 + ?r 2 k x? 1 k k x? 2 k + ?r 4 k?k k x? 2 k for a small enough ?, where we choose r 2, r 3, r 4 such that 2k A?22 k ? r 3, M 2k A?21 k ? r 2, and 2k B?k ? r 4 . ED Let V = V1 + ?V2 for some ? > 0. Then, we have V? ? ?k x? 1 k 2 + r 1 k x? 1 kk x? 2 k ? ? ?k x? 2 k 2 + ??r 2 k x? 1 kk x? 2 k + ??r 4 k?kk x? 2 k. CE PT We have that r 1 k x? 1 k k x? 2 k ? r 12 k x? 2 k 2 + 41 k x? 1 k 2, ??r 2 k x? 1 k k x? 2 k ? ? 2 ? 2 r 22 k x? 1 k 2 + 14 k x? 2 k 2, ??r 4 k?k k x? 2 k ? ? 2 ? 2 r 42 k?k 2 + 14 k x? 2 k 2 . AC Now we choose ? such that ? ? = 1 + r 12 and r 5 = ?r 4 . Then, we obtain V? ? ? 12 k x? 1 k 2 ? 12 k x? 2 k 2 + ? 2 r 52 k?k 2 ? ? 21 k x?k 2 + ? 2 r 52 k?k 2, for a small enough ?. From the above, we have that kT? x? k? < 2?r 5 , which immediately leads to kT? x k? < r for any real number r > 0 as long as we choose ? small enough. 18 ACCEPTED MANUSCRIPT and hence: T?u (s) = ? ?1 0 B??1 T? x? (s) kT?u k? ? k B??1 kr 5 . CR IP T On the other hand: Therefore, u = F? x solves the robust H? -ADDPSS with bounded input for the system (12) as required. The main result regarding H2 -ASSFS is stated as follows. Theorem 6. Consider a MAS described by (1) and (4) such that (13) is satisfied. Let AN US any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network N be defined. graphs G?, ? N is If ( A, B) is stabilizable then the H2 -ASSFS stated in Problem 1 with G = G?, ? solvable. In particular, for any given real number r > 0, there exists an ? ? , such that for any ? ? (0, ? ? ), the protocol (25) achieves state synchronization and the resulting N . graph G ? G?, ? M system from ? to x i ? x j has an H2 norm less than r for any i, j ? 1, . . . , N and for any ED Proof: Using Lemma 2, we know that we only need to verify that the feedback solves the robust H2 -ADDPSS with bounded input for the system (12) with ? ? ?. We use the same feedback as in the proof of Theorem 5. In the proof of Theorem 5 it is already PT shown that the closed loop system is asymptotically stable and the H? norm of the transfer function from ? to u is bounded. The only remaining part of the proof is CE to show that the H2 norm from ? to x can be made arbitrarily small. This clearly is equivalent to showing that the system (28) has an arbitrary small H2 norm from ? to AC x? 1 and x? 2 for sufficiently small ?. Choose Q such that 0 Q A?11 + A?11 Q = ?I 19 ACCEPTED MANUSCRIPT In that case we have: ? 0 + * ?Q ? /+. ?I - , 0 0 + 0 *0 ? / Acl + . ?I ,0 ? ? *.? 0 , ?( A?12 + A?21 Q) for sufficiently small ? where: A?11 Acl = *. , A?21 A?12 A?22 ? +/ ? ? I- ? 0 ) ?( A?12 + Q A?21 +/ ? ??? I - AN US and we used that 0 + / B? B? 0? CR IP T ? ?Q * Acl . , 0 ? + ? 0 ? 2 ?. We then obtain for sufficiently small ? that: ? 0 + * ?Q ? /+. ?I - , 0 M ? ?Q * Acl . , 0 0 + 0 *0 ? / Acl + . ?I ,0 0 + / ?0 B? B? 0- Then Lemma 3 immediately yields that we can make the H2 norm from ? to x arbitrarily ED small by choosing a sufficiently small ?. 4. MAS with partial-state coupling PT In this section, similar to the approach of the previous section, we show first that the almost state synchronization among agents in the network with partial-state CE coupling can be solved by equivalently solving a robust H? or H2 almost disturbance decoupling problem via measurement feedback with internal stability (in short H? or H2 -ADDPMS). Then, we design a controller for such a robust H? or H2 -ADDPMS AC with bounded input. 20 ACCEPTED MANUSCRIPT 4.1. Necessary and sufficient condition for H? -ASSPS The MAS system described by (1) and (3) after implementing the linear dynamical protocol (8) is described by for i = 1, . . . , N, where (29) AN US xi x? i = *. +/ . , ?i - CR IP T ? ? A BCc (?) + BDc (?) + E ? ? ? ? / x? i + *. /? i + *. +/ ?i , x?? i = *. ? ? ? ? Ac (?) ? ? , 0 , Bc (?) ,0 ? ? ? y = C 0 x? i , i ? ? ? ? ? N ? X ? ? ? ? ?i = `i j y j, ? ? ? j=1 ? Define M *. x? 1 +/ . x? = ... .. /// , . / , x? N - and *. ?1 +/ . ? = ... .. /// , . / ,? N - ED A BCc (?) + BDc (?) + E / , B? = *. / , E? = *. +/ , A? = *. , 0 Ac (?) , Bc (?) ,0Then, the overall dynamics of the N agents can be written as 0 . (30) PT x?? = (I N ? A? + L ? B?C?) x? + (I N ? E?)?. C? = C We define a robust H? -ADDPMS with bounded input as follows. Given ? ? C, CE there should exist M > 0 such that for any given real number r > 0, we can find a AC parameterized controller ?? = Ac (?) ? + Bc (?)y, u = Cc (?) ? + Dc (?)y, (31) where ? ? Rn c , for the following system, x? = Ax + ? Bu + B?, y = Cx such that the following holds for any ? ? ?: 21 (32) ACCEPTED MANUSCRIPT 1. The closed-loop system of (31) and (32) is internally stable 2. The resulting closed-loop transfer function T? x from ? to x has an H? norm less than r. CR IP T 3. The resulting closed-loop transfer function T?u from ? to u has an H? norm less than M. In the above, ? denotes all possible locations for the nonzero eigenvalues of the Laplacian matrix L when the graph varies over the set G. It is also important to note that M is independent of the choice for r. Lemma 4. Consider the system: AN US In order to obtain our main result, we will need the following lemma: x? = Ax + Bu + E?, y = Cx z=x M with ( A, B) stabilizable and (C, A) detectable. The H? -ADDPMS for the above system is defined as the problem to find for any r > 0 a controller of the form (31) such that the closed loop system is internally stable while the H? norm from ? to z is less than ED r. The H? -ADDPMS is solvable if and only if: 1. im E ? im B, PT 2. ( A, E, C, 0) is left-invertible, 3. ( A, E, C, 0) is minimum-phase. CE Proof: From [29] we immediately obtain that the H? -ADDPMS is solvable if and only AC if: 1. im E ? im B 2. ( A, E, C, 0) is at most weakly non-minimum-phase and left-invertible. 3. For any ? > 0 and every invariant zero s0 of ( A, E, C, 0), there exists a matrix K such that sI ? A ? BKC is invertible and k(s0 I ? A ? BKC) ?1 E k? < ? 22 (33) ACCEPTED MANUSCRIPT Choose a suitable basis such that: A11 A = *. , A21 A12 + /, A22 - B1 B = *. +/ , , B2 - E1 E = *. +/ , , E2 - C= I 0 *. sI ? A11 .. .. ?A21 I , ?A12 sI ? A22 0 E1 + // E2 // / 0- CR IP T Assume s0 is an imaginary axis zero of ( A, E, C, 0). In that case the rank of the matrix: AN US drops for s = s0 . This implies the existence of p , 0 and q , 0 such that *. ?A12 +/ p = *. E1 +/ q. , s0 I ? A22 , E2 - The final condition for H? almost disturbance decoupling requires for any ? > 0 the (s0 I ? A ? BKC) ?1 Eq s0 I ? A11 ? B1 K = *. , ?A21 ? B2 K PT 0 = *. +/ , , p- ?1 ?A12 + / s0 I ? A22 - ED s0 I ? A11 ? B1 K = *. , ?A21 ? B2 K M existence of a K such that (33) is satisfied. However: ?1 ?A12 + / s0 I ? A22 - *. ?A12 +/ p , s0 I ? A22 - *. s0 I ? A11 ? B1 K , ?A21 ? B2 K ?A12 + *0+ /. / s0 I ? A22 - , p- kpk > ?kqk. AC CE which yields a contradiction if ? is such that Therefore we cannot have any invariant zeros in the imaginary axis. In other words, the system ( A, E, C, 0) needs to be minimum-phase instead of weakly minimum-phase. Conversely, if ( A, E, C, 0) is minimum-phase it is easy to verify that for any ? > 0 there exists K such that (33) is satisfied. 23 ACCEPTED MANUSCRIPT Theorem 7. Consider the MAS described by (1) and (3) with ( A, B) stabilizable and (C, A) detectable. N be defined. Then, (Part I) Let ?, ? > 0 be given such that a set of graphs G?, ? protocol (8) for any ? > ? > 0 if and only if im E ? im B CR IP T N is solvable by a parameterized the H? -ASSPS for the MAS with any graph G ? G?, ? (34) while ( A, E, C, 0) is minimum phase and left-invertible. (Part II) Let G be a set of graphs such that the associated Laplacian matrices are AN US uniformly bounded and let ? consist of all possible nonzero eigenvalues of Laplacian matrices associated with graphs in G. Then, the H? -ASSPS for the MAS with any graph G ? G is solved by a parameterized protocol (8) if the robust H? -ADDPMS with bounded input for the system (32) with ? ? ? is solved by the parameterized controller (31). *. ? 1 +/ . ? In ) x? = ... .. /// , . / ,? N - ?? = (T ?1 ED ? := (T ?1 M Proof: By using L = T SL T ?1 , we define *. ??1 +/ . ? I)? = ... .. /// . / ,?? N - where ? i ? Cn+n c and ??i ? Cq . In the new coordinates, the dynamics of ? can be PT written as ??(t) = (I N ? A? + SL ? B?C?? + (T ?1 ? E)?, (35) AC CE which is rewritten as ?? 1 = A?? 1 + N X s1 j B?C?? j + E? ??1, j=2 ?? i = ( A? + ? i B?C?)? i + N X s i j B?C?? j + E? ??i , j=i+1 ?? N = ( A? + ? N B?C?)? N + E? ?? N , with i ? {2, . . . , N ? 1} where 0 E? = *. +/ , ,E - SL = [s i j ]. 24 (36) ACCEPTED MANUSCRIPT As in the case of full-state coupling, we can show that: (37) CR IP T *. ? 2 +/ . x? = (V ? In ) ... .. /// , . / ,? N - for some suitably chosen matrix V which is uniformly bounded. Therefore the H? norm of the transfer matrix from ? to x? can be made arbitrarily small if and only if the H? norm of the transfer matrix from ?? to ? can be made arbitrarily small. In order for the H? norm from ?? to ? to be arbitrarily small we need the H? norm AN US from ?? N to ? N to be arbitrarily small. In other words, the robust H? -ADDPMS with bounded input has to be solvable for the system x? = Ax + ? Bu + E?, y = Cx From the results of Lemma 4, we find that this is only possible if (34) is satisfied and ( A, E, C, 0) is left-invertible and minimum phase. M On the other hand, suppose (31) solves the robust H? -ADDPMS with bounded input of (32) and assume (34) is satisfied. We need to show that (8) solves the H? -ASSFS for ED the MAS described by (1) and (3). This follows directly from arguments very similar to the approach used in the proof of Lemma 1. PT 4.2. Necessary and sufficient condition for H2 -ASSPS The MAS system described by (1) and (3) after implementing the linear dynamical CE protocol (8) is described by (29) for i = 1, . . . , N, and, as before, the overall dynamics AC of the N agents can be written as x?? = (I N ? A? + L ? B?C?) x? + (I N ? E?)?. (38) We define a robust H2 -ADDPMS with bounded input as follows. Given ? ? C, there should exist M > 0 such that for any given real number r > 0, we can find a parameterized controller ?? = Ac (?) ? + Bc (?)y, u = Cc (?) ? + Dc (?)y, 25 (39) ACCEPTED MANUSCRIPT where ? ? Rn c , for the following system, x? = Ax + ? Bu + B?, (40) such that the following holds for any ? ? ?: CR IP T y = Cx 1. The closed-loop system of (39) and (40) is internally stable 2. The resulting closed-loop transfer function T? x from ? to x has an H2 norm less than r. than M. AN US 3. The resulting closed-loop transfer function T?u from ? to u has an H? norm less In the above, ? denotes all possible locations for the nonzero eigenvalues of the Laplacian matrix L when the graph varies over the set G. It is also important to note that M is independent of the choice for r. The following lemma, provides a necessary condition for the H2 -ADDPMS: M Lemma 5. Consider the system: x? = Ax + Bu + E?, ED y = Cx z=x PT with ( A, B) stabilizable and (C, A) detectable. The H2 -ADDPMS for the above system is defined as the problem to find for any r > 0 a controller of the form (39) such that the closed loop system is internally stable while the H? norm from ? to z is less than CE r. The H2 -ADDPMS is solvable only if: AC 1. im E ? im B 2. ( A, E, C, 0) is at most weakly non-minimum-phase and left-invertible. Proof: This follows directly from [29]. Theorem 8. Consider the MAS described by (1) and (3) with ( A, B) stabilizable and (C, A) detectable. 26 ACCEPTED MANUSCRIPT N be defined. Then, (Part I) Let ?, ? > 0 be given such that a set of graphs G?, ? N is solvable by a parameterized the H2 -ASSPS for the MAS with any graph G ? G?, ? protocol (8) for any ? > ? > 0 only if (41) CR IP T im E ? im B while ( A, E, C, 0) is at most weakly non-minimum phase and left-invertible . (Part II) Let G be a set of graphs such that the associated Laplacian matrices are uniformly bounded and let ? consist of all possible nonzero eigenvalues of Laplacian matrices associated with graphs in G. Then, the H2 -ASSPS for the MAS with any AN US graph G ? G is solved by a parameterized protocol (8) if the robust H2 -ADDPMS with bounded input for the system (40) with ? ? ? is solved by the parameterized controller (39). Proof: Similar, to the proof of Theorem 7, the dynamics can be written in the form (36). Using (37), we note the H2 norm of the transfer matrix from ? to x? can be made made arbitrarily small. M arbitrarily small if and only if the H2 norm of the transfer matrix from ?? to ? can be In order for the H2 norm from ?? to ? to be arbitrarily small we need the H2 norm ED from ?? N to ? N to be arbitrarily small. In other words, the robust H2 -ADDPMS with bounded input has to be solvable for the system PT x? = Ax + ? Bu + E?, y = Cx From the results of Lemma 5, we find that this is only possible if (41) is satisfied, CE ( A, E, C, 0) is left-invertible and at most weakly non-minimum phase. On the other hand, suppose (39) solves the robust H2 -ADDPMS with bounded input of (40) and assume (41) is satisfied. We need to show that (8) solves the H2 -ASSFS for AC the MAS described by (1) and (3). This follows directly from arguments very similar to the approach used in the proof of Lemma 1. 4.3. Protocol design for H? -ASSPS We present below two protocol design methods based on robust stabilization for the case E = B and therefore the case where ( A, B, C, 0) is minimum-phase. One relies 27 ACCEPTED MANUSCRIPT on an algebraic Riccati equation (ARE) method, and the other is based on the direct eigenstructure assignment method. 4.3.1. ARE-based method CR IP T Using an algebraic Riccati equation, we can design a suitable protocol. As in the full-state coupling case, we choose F = ?B 0 P with P = P 0 > 0 being the unique solution of the continuous-time algebraic Riccati equation A0 P + P A ? 2 ?PBB 0 P + I = 0, (42) where ? is a lower bound for the real part of the non-zero eigenvalues of all Laplacian AN US N . matrices associated with a graph in G?, ? Since ( A, B, C, 0) is minimum-phase then for any ? there exists ? small enough such that AQ + Q A0 + BB 0 + ? ?4 Q2 ? ??2 QC 0CQ = 0 (43) has a solution Q > 0. We then consider the following protocol: M ?? i = ( A + K? C) ? i ? K? ? i , ui = F? ? i , where F? = ? ?1 B 0 P, (44) ED K? = ? ?12 QC 0 The main result in this section is stated as follows. PT Theorem 9. Consider a MAS described by (1) and (3) with ( A, B) stabilizable and (C, A) detectable. Let any real numbers ?, ? > 0 and a positive integer N be given, CE N be defined. and hence a set of network graphs G?, ? N is solvable. In particular, for The H? -ASSPS stated in Problem 2 with G = G?, ? any given real number r > 0, there exists an ? ? , such that for any ? ? (0, ? ? ), the AC protocol (44) achieves state synchronization and the resulting system from ? to x i ? x j N . has an H? norm less than r for any i, j ? 1, . . . , N and for any graph G ? G?, ? Proof: Using Theorem 7, we know that we only need to verify that ?? = ( A + K? C) ? ? K? y, u = F? ?, 28 (45) ACCEPTED MANUSCRIPT solves the robust H? -ADDPMS with bounded input for the system (32) with ? ? ?. N , we know that ? ? ? implies Re ? ? ?. Obviously A + BF and Given G ? G?, ? ? A + K? C are both asymptotically stable by construction and hence the intersection of equal to: T? x = I where: I * 0 . ,T?3 ?1 ?T2 + / I - *. T1 +/ ,?T?4 - AN US T1 (s) = (sI ? A ? ? BF? ) ?1 B CR IP T (32) and (45) is asymptotically stable. The closed loop transfer function from ? to x is (46a) T2 (s) = ?(sI ? A ? ? BF? ) ?1 BF? (46b) T?3 (s) = ?(sI ? A ? K? C + ? BF? ) ?1 BF? (46c) T?4 (s) = (sI ? A ? K? C + ? BF? ) ?1 B (46d) As argued in the proof of Theorem 3, we have: ? , ? kT2 k? < M kT1 k? < 2|?| 2? ? 2. 2 ? ? PT where ED On the other hand, (43) implies according to the bounded real lemma: kT3 k? < ? 2 T3 (s) = (sI ? A ? K? C) ?1 B CE Note that: T?3 = ?(I + ?T3 F? ) ?1T3 F? AC which yields, using (47), that kT?3 k? < (1 ? ?M1 ) ?1 ?M1 < 2?M1 for small ? where M1 is such that: |?|kB 0 Pk < ?kB 0 Pk = M1 29 (47) ACCEPTED MANUSCRIPT The above yields: kT? x k? < ?M2 to: I * F? . ,T?3 T?u = F? ?T2 + / I - which yields using similar arguments as above that: ?1 *. T1 +/ ,?T?4 - kT?u k? < M3 CR IP T for some suitable constant M2 . The closed loop transfer function from ? to u is equal AN US for some suitable constant M3 independent of ?. Therefore the H? norm of the transfer matrix T? x becomes arbitrarily small for sufficiently small ? while the H? norm of the transfer matrix T?u remains bounded. 4.3.2. Direct method For ease of presentation, we only consider the case q = 1, i.e. the case where we M have a scalar measurement. We consider the state feedback gain F? given in (26), that is ED F? = where Tx is defined in (23). 1 ?1 B? F1 ? ?I Tx , Next, we consider the observer design. Note that the system ( A, B, C, 0) is minimum- PT phase and left-invertible. In that case there is a nonsingular matrix ?x such that, by AC CE defining x? = ?x x, we obtain the system x? a = Aa x a + L ad y, x? d = Ad x d + Bd (u + ? + Eda x a + Edd x d ), y = Cd x d . where xa x? = ?x x = *. +/ , ,xd - 30 (48) ACCEPTED MANUSCRIPT with x a ? Rn?? and x d ? R ? and where the matrices Ad ? R ?О? , Bd ? R ?О1 , and Cd ? R1О? have the special form 1 .. . иии иии иии .. . 0 0 0+ .. /// ./ // , 1// / 0- *.0+/ .. .. // . Bd = ... /// , ..0// . / ,1- Cd = 1 0 0 . CR IP T *.0 .. .. . Ad = ... ..0 . ,0 иии (49) Furthermore, the eigenvalues of Aa are the invariant zeros of ( A, B, C) and hence Aa is asymptotically stable. The transformation ?x can be calculated using available AN US software, either numerically [33] or symbolically [34]. Next, define a high-gain scaling matrix S? := diag(1, ? 2, . . . , ? 2??2 ), and define the output injection matrix M 0 +/ . K? = ?x *. ?2 ?1 ,? S? K - (50) (51) protocol: ED where K is such that Ad + Bd K is asymptotically stable. We then consider the following ?? i = ( A + K? C) ? i ? K? ? i , (52) PT ui = F? ? i , The main result in this section is stated as follows. CE Theorem 10. Consider a MAS described by a SISO system (1) and (3). Let any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network graphs AC N be defined. G?, ? N is If ( A, B) is stabilizable then the H? -ASSPS stated in Problem 2 with G = G?, ? solvable. In particular, for any given real number r > 0, there exists an ? ? , such that for any ? ? (0, ? ? ), the protocol (52) achieves state synchronization and the resulting system from ? to x i ? x j has an H? norm less than r for any i, j ? 1, . . . , N and for N . any graph G ? G?, ? 31 ACCEPTED MANUSCRIPT Proof: We use a similar argument as in the proof of Theorem 10. We know that we only need to verify that ?? = ( A + K? C) ? ? K? y, (53) CR IP T u = F? ?, solves the robust H? -ADDPMS with bounded input for the system (32) with ? ? ?. N , we know that ? ? ? implies Re ? ? ?. Obviously A + BF and Given G ? G?, ? ? A + K? C are both asymptotically stable by construction and hence the intersection of (32) and (45) is asymptotically stable. As in the proof of Theorem 9, the closed loop transfer function from ? to x is equal to: T? x = I I * 0 . ,T?3 ?1 ?T2 + / I - *. T1 +/ ,?T?4 - AN US (54) where, as before, we use the definitions in (46) but with our modified F? and K? . As argued in the proof of Theorem 5, we have: M kT1 k? < M1 ?, kT2 k? < M2 . for suitable constants M1, M2 > 0. Finally ED sI ? Aa T3 (s) = ?x *. ,?Bd Eda PT where ?1 L ad Cd + / Z1 - *. 0 +/ , Bd - Z1 = sI ? Ad ? ? ?2 S??1 KCd ? Bd Edd AC CE We obtain: with I T3 (s) = ? ?x *. ,0 2n 0 + * sI ? Aa /. S??1 - ,? 2n Bd Eda ?1 L ad Cd + / Z2 - *. 0 +/ , Bd - Z2 = sI ? ? ?2 Ad ? ? ?2 KCd + ? 2n Bd Edd S??1, using that ? ?2 Ad = S? Ad S??1, S? Bd = ? 2n Bd 32 and Cd S??1 = Cd . ACCEPTED MANUSCRIPT Note that Ed? is bounded for ? < 1. Next, we note that: L ad Cd +/ ?2 sI ? ? ( Ad + KCd ) - ?1 *. 0 +/ , Bd - CR IP T sI ? Aa X? (s) = *. , 0 ?(sI ? Aa ) ?1 L ad + 2 / (? sI ? Ad ? KCd ) ?1 Bd = ? 2 *. I , - From the above we can easily conclude that there exists M such that k X? k? < M? 2 . We have: where 0 + / (I + ? 2 X? Ed? ) ?1 X? S??1 - AN US I T3 = ? 2n ?x *. ,0 Ed? = ? 2n?2 Eda ? 2n?2 Edd S??1 which is clearly bounded for ? < 1. This clearly implies, using our bounds for X? and Ed? , that there exists M3 > 0 such that: for small ? since M kT3 k ? ? 2 M3 PT ED I ? 2n *. ,0 Our bound for T3 guarantees that 0 + / < ? 2 I. S??1 - kT?3 k? < ?M4, kT?4 k? < ?M5, CE for suitable M4 and M5 . Moreover kF? k < ? ?1 M0 AC Given our bounds, we immediately obtain from (54) that there exists M6 such that kT? x k? < M6 ?. The closed loop transfer function from ? to u is equal to: T?u = F? I * F? . ,T?3 33 ?T2 + / I - ?1 *. T1 +/ ,?T?4 - ACCEPTED MANUSCRIPT which yields, using similar arguments as above, that: kT?u k? < M7 CR IP T for some suitable constant M7 independent of ?. In other words, the transfer function from ? to x is arbitrarily small for sufficiently small ? while the transfer function from ? to u is bounded which completes the proof. 4.4. Protocol design for H2 -ASSFS We present below two protocol design methods based on robust stabilization for the AN US case E = B. The necessary condition provided earlier shows that ( A, B, C, 0) need only be at most weakly non-minimum-phase. The following designs are provided under the stronger assumption that ( A, B, C, 0) is minimum-phase. 4.4.1. ARE-based method We consider the protocol (44) already used in the case of H? -ASSFS. It is easy to M verify using a similar proof that this protocol also solves the robust H2 -ADDPMS with bounded input and therefore solves H2 -ASSPS for the MAS. Using the same notation ED as before, this relies on the fact that we have N1 such that kT1 k2 < ?N1 PT which follows directly from the full-state coupling case. On the other hand we have N2 CE such that kT3 k2 < ? 2 N2 AC since Q ? 0 for ? ? 0 and ( A ? KC)Q + Q( A ? KC) 0 + BB 0 ? 0. It is then easily shown that kT?4 k2 < ? 2 N3 for some N3 > 0. The rest of the proof is then as before in the case of H? -ASSPS. 34 Figure 1: The communication topology 4.4.2. Direct method CR IP T ACCEPTED MANUSCRIPT We consider the protocol (52) already used in the case of H? -ASSFS. It is easy to AN US verify using a similar proof that this protocol also solves the robust H2 -ADDPMS with bounded input and therefore solves H2 -ASSPS for the MAS. Using the same notation as before, this relies on the fact that we have N1 such that kT1 k2 < ?N1 M which follows directly from the full-state coupling case. On the other hand we have N2 ED such that kT?4 k2 < ?N2 using that X? has an H2 norm of order ?. The rest of the proof is then as before in the PT case of H? -ASSPS. 5. Example CE In this section, we illustrate our results on a homogeneous MAS of N = 6 agents. We consider the H? almost state synchronization problem via partial-state coupling. AC The agent model is given by: *.?2 A = ... 2 . ,5 0 0+ // 2 0// , / 4 2- *.0+/ B = ...3/// , . / ,2- 35 *.0+/ C 0 = ...1/// , . / ,1- *.0+/ E = ...3/// , . / ,2- ACCEPTED MANUSCRIPT with disturbances ?1 = sin(3t), ?2 = cos(t), ?3 = 0.5, CR IP T ?4 = sin(2t) + 1, ?5 = sin(t), ?6 = cos(2t). The communication topology is shown in Figure 1 with the Laplacian matrix ?1 2 ?1 0 ?2 0 0 0 ?2 0 4 0 ?3 2 0 0+ // 0 0// / 0 0// // ?2 0// // 2 0// / ?1 1- AN US *.1 .. ..0 ..0 L = .. ..0 .. ..0 . ,0 0 0 0 0 We design a controller of the form (44) based on an ARE-based method. The feedback gain F? = ? ?1 B 0 P with P given by the algebraic Riccati equation (22) and K? = ? ?12 QC 0 given by the algebraic Riccati equation (43). When choosing ? = 0.3 and ? = 0.01, we M get the controller PT ED 0 0 *.?2 +/ *. 0 +/ . / ?? i = .. 2 ?299.214 ?301.214// ? i + ...301.214/// ? i , . / . / 203.194, 5 ?199.194 ?201.194, ui = ?34.068 ?30.5702 ?27.0943 ? i ; AC CE while when choosing ? = 0.01 and ? = 0.0001, the controller is 0 0 + *.?2 *. 0 +/ // . ?? i = .. 2 ?29999 ?30001// ? i + ...30001/// ? i , . / . / 5 ?19999 ?20001 20003 , - , ui = ?1022 ?917.1 ?812.8 ? i . The results are shown in Figure 2. It is clear that when ? goes smaller, the H? norm from the disturbance to the relative error between the states of the different agents gets smaller. The controller inputs for all agents are shown in Figure 3. 36 =0.3 and =0.01 10 5 0 -5 -10 0 5 10 15 20 25 30 =0.01 and =0.0001 10 5 0 -5 -10 0 5 10 15 35 AN US State errors among 6 agents Time CR IP T State errors among 6 agents ACCEPTED MANUSCRIPT 20 25 30 40 35 40 30 35 40 30 35 40 Time M Figure 2: State errors among N = 6 agents =0.3 and =0.01 ED 500 0 -500 -1000 PT Control inputs of 6 agents 1000 -1500 0 10 15 20 25 Time =0.01 and =0.0001 10 4 1 AC CE Control inputs of 6 agents 5 0.5 0 -0.5 -1 -1.5 -2 0 5 10 15 20 25 Time Figure 3: The controller inputs of N = 6 agents 37 ACCEPTED MANUSCRIPT 6. Conclusion In this paper, we have studied H? and H2 almost state synchronization for MAS with identical linear agents affected by external disturbances. The communication network CR IP T is directed and coupled through agents? states or outputs. We have first developed the necessary and sufficient conditions on agents? dynamics for the solvability of H? and H2 almost state synchronization problems. Then, we have designed protocols to achieve H? and H2 almost state synchronization among agents based on two methods. One is ARE-based method and the other is ATEA-baed method. The future work could be to extend the results of this paper to nonlinear agents, that is, H? and H2 almost AN US state synchronization for MAS with identical nonlinear agents affected by external disturbances. References [1] H. Bai, M. Arcak, J. Wen, Cooperative control design: a systematic, passivity- M based approach, Communications and Control Engineering, Springer Verlag, 2011. ED [2] M. Mesbahi, M. Egerstedt, Graph theoretic methods in multiagent networks, Princeton University Press, Princeton, 2010. [3] W. Ren, Y. Cao, Distributed Coordination of Multi-agent Networks, Communica- PT tions and Control Engineering, Springer-Verlag, London, 2011. [4] C. Wu, Synchronization in complex networks of nonlinear dynamical systems, CE World Scientific Publishing Company, Singapore, 2007. AC [5] R. Olfati-Saber, J. Fax, R. Murray, Consensus and cooperation in networked multi-agent systems, Proc. of the IEEE 95 (1) (2007) 215?233. [6] R. Olfati-Saber, R. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Aut. Contr. 49 (9) (2004) 1520?1533. [7] W. Ren, On consensus algorithms for double-integrator dynamics, IEEE Trans. Aut. Contr. 53 (6) (2008) 1503?1509. 38 ACCEPTED MANUSCRIPT [8] W. Ren, E. Atkins, Distributed multi-vehicle coordinate control via local information, Int. J. Robust & Nonlinear Control 17 (10-11) (2007) 1002?1033. [9] W. Ren, R. Beard, Consensus seeking in multiagent systems under dynamically CR IP T changing interaction topologies, IEEE Trans. Aut. Contr. 50 (5) (2005) 655?661. [10] L. Scardovi, R. Sepulchre, Synchronization in networks of identical linear systems, Automatica 45 (11) (2009) 2557?2562. [11] S. Tuna, LQR-based coupling gain for synchronization of linear systems, available: AN US arXiv:0801.3390v1 (2008). [12] P. Wieland, J. Kim, F. AllgШwer, On topology and dynamics of consensus among linear high-order agents, International Journal of Systems Science 42 (10) (2011) 1831?1842. [13] T. Yang, S. Roy, Y. Wan, A. Saberi, Constructing consensus controllers for networks with identical general linear agents, Int. J. Robust & Nonlinear Control M 21 (11) (2011) 1237?1256. [14] H. Kim, H. Shim, J. Back, J. Seo, Consensus of output-coupled linear multi-agent ED systems under fast switching network: averaging approach, Automatica 49 (1) (2013) 267?272. PT [15] Z. Li, Z. Duan, G. Chen, L. Huang, Consensus of multi-agent systems and synchronization of complex networks: A unified viewpoint, IEEE Trans. Circ. & CE Syst.-I Regular papers 57 (1) (2010) 213?224. [16] J. Seo, J. Back, H. Kim, H. Shim, Output feedback consensus for high-order linear AC systems having uniform ranks under switching topology, IET Control Theory and Applications 6 (8) (2012) 1118?1124. [17] J. Seo, H. Shim, J. Back, Consensus of high-order linear systems using dynamic output feedback compensator: Low gain approach, Automatica 45 (11) (2009) 2659?2664. 39 ACCEPTED MANUSCRIPT [18] Y. Su, J. Huang, Stability of a class of linear switching systems with applications to two consensus problem, IEEE Trans. Aut. Contr. 57 (6) (2012) 1420?1430. IEEE Trans. Aut. Contr. 55 (10) (2009) 2416?2420. CR IP T [19] S. Tuna, Conditions for synchronizability in arrays of coupled linear systems, [20] Y. Zhao, Z. Duan, G. Wen, G. Chen, Distributed H? consensus of multi-agent systems: a performance region-based approach, Int. J. Contr. 85 (3) (2015) 332? 341. [21] I. Saboori, K. Khorasani, H? consensus achievement of multi-agent systems with AN US directed and switching topology networks, IEEE Trans. Aut. Contr. 59 (11) (2014) 3104?3109. [22] P. Lin, Y. Jia, Robust H? consensus analysis of a class of second-order multiagent systems with uncertainty, IET Control Theory and Applications 4 (3) (2010) 487?498. M [23] P. Lin, Y. Jia, L. Li, Distributed robust H? consensus control in directed networks of agents with time-delay, Syst. & Contr. Letters 57 (8) (2008) 643?653. ED [24] E. Peymani, H. Grip, A. Saberi, Homogeneous networks of non-introspective agents under external disturbances - H? almost synchronization, Automatica 52 PT (2015) 363?372. [25] E. Peymani, H. Grip, A. Saberi, X. Wang, T. Fossen, H? almost ouput syn- CE chronization for heterogeneous networks of introspective agents under external disturbances, Automatica 50 (4) (2014) 1026?1036. AC [26] M. Zhang, A. Saberi, H. F. Grip, A. A. Stoorvogel, H? almost output synchronization for heterogeneous networks without exchange of controller states, IEEE Trans. Control of Network Systems 2 (4) (2015) 348?357. [27] M. Zhang, A. Saberi, A. A. Stoorvogel, P. Sannuti, Almost regulated output synchronization for heterogeneous time-varying networks of non-introspective 40 ACCEPTED MANUSCRIPT agents and without exchange of controller states, in: American Control Conference, Chicago, IL, 2015, pp. 2735?2740. [28] A. Saberi, P. Sannuti, B. M. Chen, H2 Optimal Control, Prentice Hall, Englewood CR IP T Cliffs, NJ, 1995. [29] A. Saberi, Z. Lin, A. Stoorvogel, H2 and H? almost disturbance decoupling problem with internal stability, Int. J. Robust & Nonlinear Control 6 (8) (1996) 789?803. CWI Tracts, Amsterdam, 1986. AN US [30] H. Trentelman, Almost invariant subspaces and high gain feedback, Vol. 29 of [31] A. Saberi, P. Sannuti, Time-scale structure assignment in linear multivariable systems using high-gain feedback, Int. J. Contr. 49 (6) (1989) 2191?2213. [32] A. Saberi, Decentralization of large-scale systems: a new canonical form for linear M multivariable systems, IEEE Trans. Aut. Contr. 30 (11) (1985) 1120?1123. [33] X. Liu, B. Chen, Z. Lin, Linear systems toolkit in matlab: structural decompositions and their applications, Journal of Control Theory and Applications 3 (3) ED (2005) 287?294. [34] H. Grip, A. Saberi, Structural decomposition of linear multivariable systems using AC CE PT symbolic computations, Int. J. Contr. 83 (7) (2010) 1414?1426. 41 structure assignment method are developed such that the impact of disturbances on the network disagreement dynamics, expressed in terms of ED the H? and H2 norm of the corresponding closed-loop transfer function, is reduced to any arbitrarily small value. The protocol for full-state coupling is static, while for PT partial-state coupling it is dynamic. Keywords: Multi-agent systems, H? and H2 almost state synchronization, CE Distributed control AC 1. Introduction Over the past decade, the synchronization problem of multi-agent system (MAS) has received substantial attention because of its potential applications in cooperative control of autonomous vehicles, distributed sensor network, swarming and flocking and ? Corresponding author Email addresses: A.A.Stoorvogel@utwente.nl (Anton A. Stoorvogel), saberi@eecs.wsu.edu (Ali Saberi), zhangm@gonzaga.edu (Meirong Zhang), zhenwei.liu@wsu.edu (Zhenwei Liu) Preprint submitted to Elsevier August 14, 2018 ACCEPTED MANUSCRIPT others. The objective of synchronization is to secure an asymptotic agreement on a common state or output trajectory through decentralized protocols (see [1, 2, 3, 4] and references therein). CR IP T State synchronization inherently requires homogeneous MAS (i.e. agents have identical dynamics). Most works have focused on state synchronization based on diffu- sive full-state coupling, where the agent dynamics progress from single- and double- integrator dynamics (e.g. [5], [6], [7], [8], [9]) to more general dynamics (e.g. [10], [11], [12], [13]). State synchronization based on diffusive partial-state coupling has also been considered (e.g. [14], [15], [10], [16], [17], [18], [19]). AN US Most research has focused on the idealized case where the agents are not affected by external disturbances. In the literature where external disturbances are considered, ?-suboptimal H? design is developed for MAS to achieve H? norm from an external disturbance to the synchronization error among agents less than an, a priori given, bound ?. In particular, [15], [20] considered the H? norm from an external disturbance to the output error among agents. [21] considered the H? norm from an external disturbance M to the state error among agents. These papers do not present an explicit methodology for designing protocols. The papers [22] and [23] try to obtain an H? norm from a ED disturbance to the average of the states in a network of single or double integrators. By contrast, [24] introduced the notion of H? almost synchronization for homoge- neous MAS, where the goal is to reduce the H? norm from an external disturbance to PT the synchronization error, to any arbitrary desired level. But it requires an additional layer of communication among distributed controllers, which is completely dispensed in this paper. This work is extended later in [25], [26], and [27]. The paper [27], where CE heterogeneous MAS are considered, provides a solution for the case of right-invertible agents with the addional objective beyond output synchronization that the agents track AC a regulated signal given to some or all of the agents. Although homogeneous MAS, which are considered in this paper, are a subset of heterogeneous MAS, the results of [27] cannot be directly applied to the case of full-state coupling since the agents are not right-invertible. Secondly, the results for synchronization without regulation cannot be obtained from results obtained for regulated synchronization. Thirdly, we consider state synchronization instead of output synchronization in both full- and partial-state 2 ACCEPTED MANUSCRIPT coupling. Finally, by restricting to homogeneous networks more explicit designs can be obtained under weaker conditions. In this paper, we will study H? almost state synchronization for a MAS with full- CR IP T state coupling or partial-state coupling. We will also study H2 almost state synchronization, since it is closely related to the problems of H? almost state synchronization. In H? we look at the worst case disturbance with the only constraints being the power, while in H2 we only consider white noise disturbances which is a more restrictive class. In both cases, disturbances or noises are restricted in the process, not in the measurement. Our contribution in this paper is three-fold. AN US ? We obtain necessary and sufficient conditions for H? and H2 almost state synchronization for a MAS in the presence of external disturbances, ? We develop a protocol design for H? and H2 almost state synchronization based on an algebraic Riccati equation (ARE) method, ? We develop a protocol design for H? and H2 almost state synchronization based M on an asymptotic time-scale eigenstructure assignment (ATEA) method for the full-state coupling case, and on the direct eigenstructure assignment method for ED the partial-state coupling case. It is worth noting that our solvability conditions and protocol designs are developed for a MAS associated with a set of network graphs. Specifically, only rough information PT of a network graph is utilized. 1.1. Notations and definitions CE Given a matrix A ? CmОn , A0 denotes its conjugate transpose, k Ak is the induced 2-norm. A square matrix A is said to be Hurwitz stable if all its eigenvalues are in AC the open left half complex plane. A ? B depicts the Kronecker product between A and B. In denotes the n-dimensional identity matrix and 0n denotes n О n zero matrix; sometimes we drop the subscript if the dimension is clear from the context. Given a complex number ?, Re(?) is the real part of ? and Im(?) is the imaginary part of ?. A weighted directed graph G is defined by a triple (V, E, A) where V = {1, . . . , N } is a node set, E is a set of pairs of nodes indicating connections among nodes, and 3 ACCEPTED MANUSCRIPT A = [ai j ] ? R N ОN is the weighting matrix, and ai j > 0 iff (i, j) ? E which denotes an edge from node j to node i. In our case, we have aii = 0. A path from node i 1 to i k is a sequence of nodes {i 1, . . . , i k } such that (i j+1, i j ) ? E for j = 1, . . . , k ? 1. A CR IP T directed tree is a subgraph (subset of nodes and edges) in which every node has exactly one parent node except for one node, called the root, which has no parent node. In this case, the root has a directed path to every other node in the tree. A directed spanning tree is a directed tree containing all the nodes of the graph. For a weighted graph G, a matrix L = [` i j ] with AN US PN ? ? aik , i = j, ? k=1 `i j = ? ? ? ?ai j , i , j, ? is called the Laplacian matrix associated with the graph G. The Laplacian L has all its eigenvalues in the closed right half plane and at least one eigenvalue at zero associated with right eigenvector 1. A specific class of graphs needed in this paper is presented below: N denote the set of directed graphs Definition 1. For any given ? ? ? > 0, let G?, ? M with N nodes that contain a directed spanning tree and for which the corresponding Laplacian matrix L satisfies kLk < ? while its nonzero eigenvalues have a real part ED larger than or equal to ?. 2. Problem formulation PT Consider a MAS composed of N identical linear time-invariant agents of the form, CE x? i = Ax i + Bui + E?i , yi = C x i , (i = 1, . . . , N ) (1) where x i ? Rn , ui ? Rm , yi ? R p are respectively the state, input, and output vectors AC of agent i, and ?i ? R? is the external disturbances. The communication network provides each agent with a linear combination of its own outputs relative to that of other neighboring agents. In particular, each agent i ? {1, . . . , N } has access to the quantity, ?i = N X j=1 ai j (yi ? y j ), 4 (2) ACCEPTED MANUSCRIPT where ai j ? 0 and aii = 0 indicate the communication among agents. This communication topology of the network can be described by a weighted and directed graph G with nodes corresponding to the agents in the network and the weight of edges given rewritten as ?i = N X CR IP T by the coefficient ai j . In terms of the coefficients of the Laplacian matrix L, ? i can be (3) `i j y j . j=1 We refer to this case as partial-state coupling. Note that if C has full column rank then, without loss of generality, we can assume that C = I, and the quantity ? i becomes N X j=1 ai j (x i ? x j ) = N X `i j x j . AN US ?i = We refer to this case as full-state coupling. (4) j=1 If the graph G describing the communication topology of the network contains a directed spanning tree, then it follows from [9, Lemma 3.3] that the Laplacian matrix L has a simple eigenvalue at the origin, with the corresponding right eigenvector 1 and all M the other eigenvalues are in the open right-half complex plane. Let ? 1, . . . , ? N denote the eigenvalues of L such that ? 1 = 0 and Re(? i ) > 0, i = 2, . . . , N. PT ED Let N be any agent and define x? i = x N ? x i and *. x? 1 +/ . x? = ... .. /// . / x? N ?1 , - and *. ?1 +/ . ? = ... .. /// . . / ,? N - CE Obviously, synchronization is achieved if x? = 0. That is lim (x i (t) ? x N (t)) = 0, t?? ?i, ? {1, . . . , N ? 1}. (5) We denote by T? x? , the transfer function from ? to x? AC Remark 1. Agent N is not necessarily a root agent. Obviously, (5) is equivalent to the condition that lim (x i (t) ? x j (t)) = 0, t?? ?i, j ? {1, . . . , N }. We formulate below four almost state synchronization problems for a network with either H2 or H? almost synchronization. 5 ACCEPTED MANUSCRIPT Problem 1. Consider a MAS described by (1) and (4). Let G be a given set of graphs such that G ? G N . The H? almost state synchronization problem via full-state coupling (in short H? -ASSFS) with a set of network graphs G is to find, if possible, a ui = F (?)? i , CR IP T linear static protocol parameterized in terms of a parameter ?, of the form, (6) such that, for any given real number r > 0, there exists an ? ? such that for any ? ? (0, ? ? ] and for any graph G ? G, (5) is satisfied for all initial conditions in the absence of AN US disturbances and the closed loop transfer matrix T? x? satisfies kT? x? k? < r. (7) Problem 2. Consider a MAS described by (1) and (3). Let G be a given set of graphs such that G ? G N . The H? almost state synchronization problem via partial-state coupling (in short H? -ASSPS) with a set of network graphs G is to find, if possible, a linear time-invariant dynamic protocol parameterized in terms of a parameter ?, of the M form, ?? i = Ac (?) ? i + Bc (?)? i , (8) ED ui = Cc (?) ? i + Dc (?)? i , where ? i ? Rn c , such that, for any given real number r > 0, there exists an ? ? such that for any ? ? (0, ? ? ] and for any graph G ? G, (5) is satisfied for all initial conditions in PT the absence of disturbances and the closed loop transfer matrix T? x? satisfies (7). Problem 3. Consider a MAS described by (1) and (4). Let G be a given set of graphs CE such that G ? G N . The H2 almost state synchronization problem via full-state coupling (in short H2 -ASSFS) with a set of network graphs G is to find, if possible, a linear static protocol parameterized in terms of a parameter ?, of the form (6) such AC that, for any given real number r > 0, there exists an ? ? such that for any ? ? (0, ? ? ] and for any graph G ? G, (5) is satisfied for all initial conditions in the absence of disturbances and the closed loop transfer matrix T? x? satisfies kT? x? k2 < r. 6 (9) ACCEPTED MANUSCRIPT Problem 4. Consider a MAS described by (1) and (3). Let G be a given set of graphs such that G ? G N . The H2 almost state synchronization problem via partial-state coupling (in short H2 -ASSPS) with a set of network graphs G is to find, if possible, CR IP T a linear time-invariant dynamic protocol parameterized in terms of a parameter ?, of the form (8) such that, for any given real number r > 0, there exists an ? ? such that for any ? ? (0, ? ? ] and for any graph G ? G, (5) is satisfied for all initial conditions in the absence of disturbances and the closed loop transfer matrix T? x? satisfies (9). Note that the problems of H? almost state synchronization and H2 almost state synchronization are closely related. Roughly speaking, H2 almost synchronization is AN US easier to achieve than H? almost synchronization. This is related to the fact that in H? we look at the worst case disturbance with the only constraints being the power: Z T 1 lim sup ?i0 (t)?i (t)dt < ?. T ?? 2T ?T while in H2 we only consider white noise disturbances which is a more restrictive class. M 3. MAS with full-state coupling In this section, we establish a connection between the almost state synchronization ED among agents in the network and a robust H? or H2 almost disturbance decoupling problem via state feedback with internal stability (in short H? or H2 -ADDPSS) (see PT [28]). Then, we use this connection to derive the necessary and sufficient condition and design appropriate protocols. CE 3.1. Necessary and sufficient condition for H? -ASSFS The MAS system described by (1) and (4) after implementing the linear static AC protocol (6) is described by x? i = Ax i + BF (?)? i + E?i , for i = 1, . . . , N. Let *. x 1 +/ . x = ... .. /// , . / ,x N - *. ?1 +/ . ? = ... .. /// . . / ,? N 7 ACCEPTED MANUSCRIPT Then, the overall dynamics of the N agents can be written as x? = (I N ? A + L ? BF (?))x + (I N ? E)?. (10) CR IP T We define the robust H? -ADDPSS with bounded input as follows. Given ? ? C, there should exist M > 0 such that for any given real number r > 0, we can find a parameterized controller u = F (?)x for the following subsystem, AN US x? = Ax + ? Bu + B?, (11) (12) such that for any ? ? ? the following hold: 1. The interconnection of the systems (12) and (11) is internally stable; 2. The resulting closed-loop transfer function T? x from ? to x has an H? norm less than r. M 3. The resulting closed-loop transfer function T?u from ? to u has an H? norm less than M. ED In the above, ? denotes all possible locations for the nonzero eigenvalues of the Laplacian matrix L when the graph varies over the set G. It is also important to note that M is independent of the choice for r. PT In the following lemma we give a necessary condition for the H? -ASSFS. Moreover, for sufficiency, we connect the H? -ASSFS problem to the robust H? -ADDPSS with CE bounded input problem which we will address later. Lemma 1. Let G be a set of graphs such that the associated Laplacian matrices are AC uniformly bounded and let ? consist of all possible nonzero eigenvalues of Laplacian matrices associated with graphs in G. (Necessity) The H? -ASSFS for the MAS described by (1) and (4) given G is solvable by a parameterized protocol ui = F (?)? i only if im E ? im B. 8 (13) ACCEPTED MANUSCRIPT (Sufficiency) The H? -ASSFS for the MAS described by (1) and (4) given G is solved by a parameterized protocol ui = F (?)? i if the robust H? -ADDPSS with bounded input for the system (12) with ? ? ? is solved by the parameterized controller u = F (?)x. CR IP T Proof: Note that L has eigenvalue 0 with associated right eigenvector 1. Let (14) L = T SL T ?1, with T unitary and SL the upper-triangular Schur form associated to the Laplacian matrix L such that SL (1, 1) = 0. Let *. ? 1 +/ . ? In )x = ... .. /// , . / ,? N - *. ??1 +/ . ? I)? = ... .. /// . / ,?? N - AN US ? := (T ?1 ?? = (T ?1 where ? i ? Cn and ??i ? Cq . In the new coordinates, the dynamics of ? can be written as which is rewritten as N X s1 j BF (?)? j + E ??1, ED ?? 1 = A? 1 + (15) M ??(t) = (I N ? A + SL ? BF (?))? + (T ?1 ? E)?, j=2 ?? i = ( A + ? i BF (?))? i + N X s i j BF (?)? j + E ??i , (16) PT j=i+1 ?? N = ( A + ? N BF (?))? N + E ?? N , CE for i ? {2, . . . , N ? 1} where SL = [s i j ]. The first column of T is an eigenvector of ? L associated to eigenvalue 0 with length 1, i.e. it is equal to ▒1/ N. Using this we AC obtain: *. x N ? x 1 +/ *.*.?1 0 и и и 0 1+/ +/ .. .. / .. .. // .... // x N ? x 2 / .. 0 ?1 . . . // // (T ? I )? / . . / x? = ... = ? I n n .. .. / // .... .. // .. .. .. / . . . . 0 . // .... // // . x ? x 0 и и и 0 ?1 1 N ?1 , N ,, = 0 V ? In ?, 9 ACCEPTED MANUSCRIPT for some suitably chosen matrix V . Therefore we have (17) CR IP T *. ? 2 +/ . x? = (V ? In ) ... .. /// , . / ,? N - Note that since T is unitary, also the matrix T ?1 is unitary and the matrix V is uniformly bounded. Therefore the H? norm of the transfer matrix from ? to x? can be made arbitrarily small if and only if the H? norm of the transfer matrix from ?? to ? can be made arbitrarily small. AN US In order for the H? norm from ?? to ? to be arbitrarily small we need the H? norm from ?? N to ? N to be arbitrarily small. From classical results (see [29, 30]) on H? almost disturbance decoupling we find that this is only possible if (13) is satisfied. Conversely, suppose u = F (?)x solves the robust H? -ADDPSS with bounded input for (12) and assume (13) is satisfied. We show next that ui = F (?)? i solves the H? -ASSFS for the MAS described by (1) and (4). Let X be such that E = BX. M The fact that u = F (?)x solves the robust H? -ADDPSS with bounded input for (12) implies that for small ? we have that A + ? BF (?) is asymptotically stable for all ? ? ?. ED In particular, A + ? i BF (?) is asymptotically stable for i = 2, . . . , N which guarantees that ? i ? 0 for i = 2, . . . , N for zero disturbances and all initial conditions. Therefore we have state synchronization. PT Next, we are going to show that for any r? > 0, we can choose ? sufficiently small such that the transfer matrix from ?? to ? i is less than r? for i = 2, . . . , N. This guarantees AC CE that we can achieve (7) for any r > 0. We have that T?? x (s) = (sI ? A ? ? BF (?)) ?1 B, ? T?u (s) = F (?)(sI ? A ? ? BF (?)) ?1 B. For a given M and parameter ?, the following is satisfied kT?? x k? < r? ? , ? kT?u k? < M for all ? ? ? where r? ? is a parameter depending on ? with the property that lim??0 r? ? = 0. Denote ?i = F (?)? i . 10 ACCEPTED MANUSCRIPT When i = N, it is easy to find that, иии 0 and hence ?N T??? N = T?u 0 X , kT??? N k? < r?, kT??? N k? < M?N provided k X k r? ? < r?, k X k M < M?N . иии 0 X CR IP T T??? N = T?? xN 0 (18) Recall that we can make r? ? arbitrarily small by reducing ? without affecting the bound AN US M. Assume kT??? j k? < r?, holds for j = i + 1, . . . , N. We have: kT??? j k? < M? j M ?? ?? N X s i j T??? j (s) ??? T??? i (s) = T?? ix (s) ???ei ? X + ?? ?? j=i+1 ?? ?? N X ? ?i T??? i (s) = T?u (s) ??ei ? X + s i j T??? j (s) ??? ?? ?? j=i+1 where ei is a row vector of dimension N with elements equal to zero except for the ith ED component which is equal to 1. Since PT N N X X e ? X + < k X k + s T |s i j | M? j i j ??? j i j=i+1 j=i+1 ? CE we find: kT??? i k? < r?, kT??? i k? < M?i (19) AC provided: *. k X k + , N X |s i j | M? j +/ r? ? < r?, *. k X k + j=i+1 , N X |s i j | M? j +/ M? < M?i . j=i+1 - (20) Note that s i j depends on the graph in G but since the Laplacian matrices associated to graphs in G are uniformly bounded we find that also the s i j are uniformly bounded. In this way for any arbitrary r?, we can recursively obtain the bounds in (19) for i = 2, . . . , N provided we choose ? sufficiently small such that the corresponding r? ? satisfies (18) 11 ACCEPTED MANUSCRIPT and (20) for i = 2, . . . , N ? 1. Hence, we can choose ? sufficiently small such that the transfer matrix from ?? to ? i is less than r? for i = 2, . . . , N. As noted before this guarantees that we can achieve (7) for any r > 0. CR IP T N for some given ?, ? > 0, we For the case where the set of graphs G equals G?, ? develop necessary and sufficient conditions for the solvability of the H? -ASSFS for MAS as follows: Theorem 1. Consider a MAS described by (1) and (4) with an associated graph from N for some ?, ? > 0. the set G = G?, ? Then, the H? -ASSFS is solvable if and only if (13) is satisfied and ( A, B) is AN US stabilizable. Proof: From Lemma 1, we note that (13) is actually a necessary condition for H? - ASSFS. Clearly, also ( A, B) stabilizable is a necessary condition. Sufficiency is a direct result of Theorems 3 or Theorem 5 for H? -ASSFS. M 3.2. Necessary and sufficient conditions for H2 -ASSFS We define the robust H2 -ADDPSS with bounded input as follows. Given ? ? C, ED there should exist M > 0 such that for any given real number r > 0, we can find a parameterized controller (11) for the system, (12) such that the following holds for any ? ? ?: PT 1. The interconnection of the systems (11) and (12) is internally stable; 2. The resulting closed-loop transfer function T? x from ? to x has an H2 norm less CE than r. 3. The resulting closed-loop transfer function T?u from ? to u has an H? norm less AC than M. In the above, ? denotes all possible locations for the nonzero eigenvalues of the Laplacian matrix L when the graph varies over the set G. It is also important to note that M is independent of the choice for r. Note that we need to consider two aspects in our controller H2 disturbance rejection and robust stabilization (because of a set of network N ). The latter translates in the H norm constraint from ? to u. graphs G?, ? ? 12 ACCEPTED MANUSCRIPT Lemma 2. Let G be a set of graphs such that the associated Laplacian matrices are uniformly bounded and let ? consist of all possible nonzero eigenvalues of Laplacian matrices associated with graphs in G. CR IP T (Necessity) The H2 -ASSFS for the MAS described by (1) and (4) given G is solvable by a parameterized protocol ui = F (?)? i only if (13) is satisfied. (Sufficiency)The H2 -ASSFS for the MAS described by (1) and (4) given G is solvable by a parameterized protocol ui = F (?)? i if the robust H2 -ADDPSS with bounded input for the system (12) with ? ? ? is solved by the parameterized controller u = F (?)x. Proof: The proof is similar to the proof of Lemma 1. This time we need the H2 norm AN US from ?? N to ? N to be arbitrarily small and also H2 almost disturbance decoupling then immediately yields that we need that (13) is satisfied. The rest of the proof follows the same lines except that we require the H2 norm from ?? to ? j arbitrarily small while we keep the H? norm from ?? to ? j bounded. Recall that for any two stable, strictly proper transfer matrices T1 and T2 we have: M kT1T2 k2 ? kT1 k2 kT2 k? which we need in the modifications of the proof of Lemma 1. ED N (with given ?, ? > 0), we develop For the case with a set of graph G = G?, ? necessary and sufficient conditions for the solvability of the H2 -ASSFS for MAS as follows: PT Theorem 2. Consider a MAS described by (1) and (4) with an associated graph from N for some ?, ? > 0. the set G = G?, ? CE Then, the H2 -ASSFS is solvable if and only if (13) is satisfied and ( A, B) is stabilizable. Proof: We have already noted before that (13) is actually a necessary condition for AC H2 -ASSFS. Clearly, also ( A, B) being stabilizable is a necessary condition. Sufficiency for H2 -ASSFS, is a direct result of either Theorem 4 or Theorem 6. 3.3. Protocol design for H? -ASSFS and H2 -ASSFS We present below two protocol design methods for both H? -ASSFS and H2 -ASSFS problems. One relies on an algebraic Riccati equation (ARE), and the other is based on 13 ACCEPTED MANUSCRIPT an asymptotic time-scale eigenstructure assignment (ATEA) method. 3.3.1. ARE-based method ( A, B) is stabilizable. We consider the protocol, ui = ?F? i , where ? = 1 ? CR IP T Using an algebraic Riccati equation, we can design a suitable protocol provided (21) and F = ?B 0 P with P being the unique solution of the continuous-time algebraic Riccati equation AN US A0 P + P A ? 2 ?PBB 0 P + I = 0, (22) where ? is a lower bound for the real part of the non-zero eigenavlues of all Laplacian N . matrices associated with a graph in G = G?, ? The main result regarding H? -ASSFS is stated as follows. M Theorem 3. Consider a MAS described by (1) and (4) such that (13) is satisfied. Let any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network N be defined. graphs G?, ? ED N is If ( A, B) is stabilizable then the H? -ASSFS stated in Problem 1 with G = G?, ? solvable. In particular, for any given real number r > 0, there exists an ? ? , such that for any ? ? (0, ? ? ), the protocol (21) achieves state synchronization and the resulting PT system from ? to x i ? x j has an H? norm less than r for any i, j ? 1, . . . , N and for CE N . any graph G ? G?, ? Proof: Using Lemma 1, we know that we only need to verify that u = ?F x solves the robust H? -ADDPSS with bounded input for the system (12) with ? ? ?. Given AC N , we know that ? ? ? implies Re ? ? ?. Clearly, the Laplacian matrices are G ? G?, ? uniformly bounded since kLk ? ?. Consider the interconnection of (12) and u = ?F x. We define V (x) = x 0 Px 14 ACCEPTED MANUSCRIPT and we obtain: V? = x 0 ( A ? ?? BB 0 P) 0 Px + ? 0 B 0 Px + x 0 P( A ? ?? BB 0 P)x + x 0 PB? CR IP T = x 0 PBB 0 Px ? x 0 x ? 2? ?x 0 PBB 0 Px + 2x 0 PB? ? (1 ? ?? )x 0 PBB 0 Px ? x 0 x + ?? ? 0? ? ? ?2 ?u 0u ? x 0 x + ?? ? 0? which implies that the system is asymptotically stable and the H? norm of the transfer function from ? to x is less that ?/ ? while the H? norm of the transfer function from AN US ? to u is less that 2/ ? 2 . Therefore, u = ?F x solves the robust H? -ADDPSS with bounded input for the system (12) as required. For H2 -ASSFS we have the following classical result: Lemma 3. Consider an asymptotically stable system: M p? = A1 p + B1 ? The H2 norm from ? to p is less than ? if there exists a matrix Q such that: ED A1 Q + Q A10 + B1 B10 ? 0, Q < ?I The main result regarding H2 -ASSFS is stated as follows. PT Theorem 4. Consider a MAS described by (1) and (4) such that (13) is satisfied. Let any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network CE N be defined. graphs G?, ? N is If ( A, B) is stabilizable then the H2 -ASSFS stated in Problem 3 with G = G?, ? solvable. In particular, for any given real number r > 0, there exists an ? ? , such that AC for any ? ? (0, ? ? ), the protocol (21) achieves state synchronization and the resulting system from ? to x i ? x j has an H? norm less than r for any i, j ? 1, . . . , N and for N . any graph G ? G?, ? Proof: Using Lemma 2, we know that we only need to verify that u = ?F x solves the robust H2 -ADDPSS with bounded input for the system (12) with ? ? ?. We use the 15 ACCEPTED MANUSCRIPT same feedback as in the proof of Theorem 3. In the proof of Theorem 3 it is already shown that the closed loop system is asymptotically stable and the H? norm of the transfer function from ? to u is bounded. The only remaining part of the proof is to CR IP T show that the H2 norm from ? to x can be made arbitrarily small. Using the algebraic Riccati equation it is easy to see that we have: ( A ? ?? BB 0 P) 0 P + P( A ? ?? BB 0 P) + ? ?PBB 0 P ? 0 for large ?. But then we have: AN US Q? ( A ? ?? BB 0 P) 0 + ( A ? ?? BB 0 P)Q? + BB 0 ? 0 for Q? = ? ? ?1 P?1 . Then Lemma 3 immediately yields that we can make the H2 norm from ? to x arbitrarily small by choosing a sufficiently small ?. 3.3.2. ATEA-based method The ATEA-based design is basically a method of time-scale structure assignment M in linear multivariable systems by high-gain feedback [31]. In the current case, we do not need the full structure presented in the above method. It is sufficient to note that ED there exists non-singular transformation matrix Tx ? RnОn (See [32, Theorem 1]) such PT that x? 1 x? = *. +/ = Tx x, , x? 2 - (23) CE and the dynamics of x? is represented as x?? 1 = A?11 x? 1 + A?12 x? 2, x?? 2 = A?21 x? 1 + A?22 x? 2 + ? B?u + B??, (24) AC with B? invertible, and that ( A, B) is stabilizable implies that ( A?11, A?12 ) is stabilizable. Choose F1 such that A?11 + A?12 F1 is asymptotically stable. In that case a suitable protocol for (1) is (25) ui = F? ? i , where F? is designed as F? = 1 ?1 B? F1 ? 16 ?I Tx (26) ACCEPTED MANUSCRIPT The main result regarding H? -ASSFS is stated as follows. The result is basically the same as Theorem 3 except for a different design protocol. Theorem 5. Consider a MAS described by (1) and (4) such that (13) is satisfied. Let CR IP T any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network N be defined. graphs G?, ? N is If ( A, B) is stabilizable then the H? -ASSFS stated in Problem 1 with G = G?, ? solvable. In particular, for any given real number r > 0, there exists an ? ? , such that for any ? ? (0, ? ? ), the protocol (25) achieves state synchronization and the resulting system from ? to x i ? x j has an H? norm less than r for any i, j ? 1, . . . , N and for AN US N . any graph G ? G?, ? Proof: Similarly to the proof of Theorem 3, we only need to establish that u = F? x solves the robust H? -ADDPSS with bounded input for the system (12) with ? ? ?. N , we know that ? ? ? implies Re ? ? ?. Given G ? G?, ? After a basis transformation, the interconnection of (12) and u = F? x is equal to M the interconnection of (24) and (25). We obtain: x?? 1 = A?11 x? 1 + A?12 x? 2, Define ED ? x?? 2 = (? A?21 + ?F1 ) x? 1 + (? A?22 ? ?I) x? 2 + ? B??. PT x? 1 = x? 1, (27) x? 2 = x? 2 ? F1 x? 1 . CE Then we can write this system (27) in the form: x?? 1 = A?11 x? 1 + A?12 x? 2, ? x?? 2 = ? A?21 x? 1 + (? A?22 ? ?I) x? 2 + ? B??, (28) AC where A?11 = A?11 + A?12 F1, A?12 = A?12, A?21 = A?21 ? F1 A?11 + A?22 ? F1 A?12, A?22 = A?22 ? F1 A?12 . In the absence of the external disturbances, the above system (28) is asymptotically stable for small enough ?. 17 ACCEPTED MANUSCRIPT Since A?11 = A?11 + A?12 F1 is Hurwitz stable, there exists P > 0 such that the Lyapunov 0 P = ?I holds. For the dynamics x? , we define a Lyapunov function equation P A?11 + A?11 1 V1 = x? 10 P x? 1 . Then the derivative of V1 can be bounded CR IP T 0 V?1 ? ?k x? 1 k 2 + x? 20 A?12 P x? 1 + x? 10 P A?12 x? 2 ? ?k x? 1 k 2 + 2 Re( x? 10 P A?12 x? 2 ) ? ?k x? 1 k 2 + r 1 k x? 1 kk x? 2 k, where 2kP A?12 k ? r 1 . Now define a Lyapunov function V2 = ? x? 20 x? 2 for the dynamics AN US x? 2 . The derivative of V2 can then also be bounded. V?2 ? ?2 Re(?)k x? 2 k 2 + 2? Re( x? 20 A?21 x? 1 ) + 2? x? 20 A?22 x? 2 + 2? Re( x? 20 B??) ? ?2 Re(?)k x? 2 k 2 + ?r 2 k x? 1 kk x? 2 k + ?r 3 k x? 2 k 2 + ?r 4 k?kk x? 2 k ? ? ?k x? 2 k 2 + ?r 2 k x? 1 k k x? 2 k + ?r 4 k?k k x? 2 k for a small enough ?, where we choose r 2, r 3, r 4 such that 2k A?22 k ? r 3, M 2k A?21 k ? r 2, and 2k B?k ? r 4 . ED Let V = V1 + ?V2 for some ? > 0. Then, we have V? ? ?k x? 1 k 2 + r 1 k x? 1 kk x? 2 k ? ? ?k x? 2 k 2 + ??r 2 k x? 1 kk x? 2 k + ??r 4 k?kk x? 2 k. CE PT We have that r 1 k x? 1 k k x? 2 k ? r 12 k x? 2 k 2 + 41 k x? 1 k 2, ??r 2 k x? 1 k k x? 2 k ? ? 2 ? 2 r 22 k x? 1 k 2 + 14 k x? 2 k 2, ??r 4 k?k k x? 2 k ? ? 2 ? 2 r 42 k?k 2 + 14 k x? 2 k 2 . AC Now we choose ? such that ? ? = 1 + r 12 and r 5 = ?r 4 . Then, we obtain V? ? ? 12 k x? 1 k 2 ? 12 k x? 2 k 2 + ? 2 r 52 k?k 2 ? ? 21 k x?k 2 + ? 2 r 52 k?k 2, for a small enough ?. From the above, we have that kT? x? k? < 2?r 5 , which immediately leads to kT? x k? < r for any real number r > 0 as long as we choose ? small enough. 18 ACCEPTED MANUSCRIPT and hence: T?u (s) = ? ?1 0 B??1 T? x? (s) kT?u k? ? k B??1 kr 5 . CR IP T On the other hand: Therefore, u = F? x solves the robust H? -ADDPSS with bounded input for the system (12) as required. The main result regarding H2 -ASSFS is stated as follows. Theorem 6. Consider a MAS described by (1) and (4) such that (13) is satisfied. Let AN US any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network N be defined. graphs G?, ? N is If ( A, B) is stabilizable then the H2 -ASSFS stated in Problem 1 with G = G?, ? solvable. In particular, for any given real number r > 0, there exists an ? ? , such that for any ? ? (0, ? ? ), the protocol (25) achieves state synchronization and the resulting N . graph G ? G?, ? M system from ? to x i ? x j has an H2 norm less than r for any i, j ? 1, . . . , N and for any ED Proof: Using Lemma 2, we know that we only need to verify that the feedback solves the robust H2 -ADDPSS with bounded input for the system (12) with ? ? ?. We use the same feedback as in the proof of Theorem 5. In the proof of Theorem 5 it is already PT shown that the closed loop system is asymptotically stable and the H? norm of the transfer function from ? to u is bounded. The only remaining part of the proof is CE to show that the H2 norm from ? to x can be made arbitrarily small. This clearly is equivalent to showing that the system (28) has an arbitrary small H2 norm from ? to AC x? 1 and x? 2 for sufficiently small ?. Choose Q such that 0 Q A?11 + A?11 Q = ?I 19 ACCEPTED MANUSCRIPT In that case we have: ? 0 + * ?Q ? /+. ?I - , 0 0 + 0 *0 ? / Acl + . ?I ,0 ? ? *.? 0 , ?( A?12 + A?21 Q) for sufficiently small ? where: A?11 Acl = *. , A?21 A?12 A?22 ? +/ ? ? I- ? 0 ) ?( A?12 + Q A?21 +/ ? ??? I - AN US and we used that 0 + / B? B? 0? CR IP T ? ?Q * Acl . , 0 ? + ? 0 ? 2 ?. We then obtain for sufficiently small ? that: ? 0 + * ?Q ? /+. ?I - , 0 M ? ?Q * Acl . , 0 0 + 0 *0 ? / Acl + . ?I ,0 0 + / ?0 B? B? 0- Then Lemma 3 immediately yields that we can make the H2 norm from ? to x arbitrarily ED small by choosing a sufficiently small ?. 4. MAS with partial-state coupling PT In this section, similar to the approach of the previous section, we show first that the almost state synchronization among agents in the network with partial-state CE coupling can be solved by equivalently solving a robust H? or H2 almost disturbance decoupling problem via measurement feedback with internal stability (in short H? or H2 -ADDPMS). Then, we design a controller for such a robust H? or H2 -ADDPMS AC with bounded input. 20 ACCEPTED MANUSCRIPT 4.1. Necessary and sufficient condition for H? -ASSPS The MAS system described by (1) and (3) after implementing the linear dynamical protocol (8) is described by for i = 1, . . . , N, where (29) AN US xi x? i = *. +/ . , ?i - CR IP T ? ? A BCc (?) + BDc (?) + E ? ? ? ? / x? i + *. /? i + *. +/ ?i , x?? i = *. ? ? ? ? Ac (?) ? ? , 0 , Bc (?) ,0 ? ? ? y = C 0 x? i , i ? ? ? ? ? N ? X ? ? ? ? ?i = `i j y j, ? ? ? j=1 ? Define M *. x? 1 +/ . x? = ... .. /// , . / , x? N - and *. ?1 +/ . ? = ... .. /// , . / ,? N - ED A BCc (?) + BDc (?) + E / , B? = *. / , E? = *. +/ , A? = *. , 0 Ac (?) , Bc (?) ,0Then, the overall dynamics of the N agents can be written as 0 . (30) PT x?? = (I N ? A? + L ? B?C?) x? + (I N ? E?)?. C? = C We define a robust H? -ADDPMS with bounded input as follows. Given ? ? C, CE there should exist M > 0 such that for any given real number r > 0, we can find a AC parameterized controller ?? = Ac (?) ? + Bc (?)y, u = Cc (?) ? + Dc (?)y, (31) where ? ? Rn c , for the following system, x? = Ax + ? Bu + B?, y = Cx such that the following holds for any ? ? ?: 21 (32) ACCEPTED MANUSCRIPT 1. The closed-loop system of (31) and (32) is internally stable 2. The resulting closed-loop transfer function T? x from ? to x has an H? norm less than r. CR IP T 3. The resulting closed-loop transfer function T?u from ? to u has an H? norm less than M. In the above, ? denotes all possible locations for the nonzero eigenvalues of the Laplacian matrix L when the graph varies over the set G. It is also important to note that M is independent of the choice for r. Lemma 4. Consider the system: AN US In order to obtain our main result, we will need the following lemma: x? = Ax + Bu + E?, y = Cx z=x M with ( A, B) stabilizable and (C, A) detectable. The H? -ADDPMS for the above system is defined as the problem to find for any r > 0 a controller of the form (31) such that the closed loop system is internally stable while the H? norm from ? to z is less than ED r. The H? -ADDPMS is solvable if and only if: 1. im E ? im B, PT 2. ( A, E, C, 0) is left-invertible, 3. ( A, E, C, 0) is minimum-phase. CE Proof: From [29] we immediately obtain that the H? -ADDPMS is solvable if and only AC if: 1. im E ? im B 2. ( A, E, C, 0) is at most weakly non-minimum-phase and left-invertible. 3. For any ? > 0 and every invariant zero s0 of ( A, E, C, 0), there exists a matrix K such that sI ? A ? BKC is invertible and k(s0 I ? A ? BKC) ?1 E k? < ? 22 (33) ACCEPTED MANUSCRIPT Choose a suitable basis such that: A11 A = *. , A21 A12 + /, A22 - B1 B = *. +/ , , B2 - E1 E = *. +/ , , E2 - C= I 0 *. sI ? A11 .. .. ?A21 I , ?A12 sI ? A22 0 E1 + // E2 // / 0- CR IP T Assume s0 is an imaginary axis zero of ( A, E, C, 0). In that case the rank of the matrix: AN US drops for s = s0 . This implies the existence of p , 0 and q , 0 such that *. ?A12 +/ p = *. E1 +/ q. , s0 I ? A22 , E2 - The final condition for H? almost disturbance decoupling requires for any ? > 0 the (s0 I ? A ? BKC) ?1 Eq s0 I ? A11 ? B1 K = *. , ?A21 ? B2 K PT 0 = *. +/ , , p- ?1 ?A12 + / s0 I ? A22 - ED s0 I ? A11 ? B1 K = *. , ?A21 ? B2 K M existence of a K such that (33) is satisfied. However: ?1 ?A12 + / s0 I ? A22 - *. ?A12 +/ p , s0 I ? A22 - *. s0 I ? A11 ? B1 K , ?A21 ? B2 K ?A12 + *0+ /. / s0 I ? A22 - , p- kpk > ?kqk. AC CE which yields a contradiction if ? is such that Therefore we cannot have any invariant zeros in the imaginary axis. In other words, the system ( A, E, C, 0) needs to be minimum-phase instead of weakly minimum-phase. Conversely, if ( A, E, C, 0) is minimum-phase it is easy to verify that for any ? > 0 there exists K such that (33) is satisfied. 23 ACCEPTED MANUSCRIPT Theorem 7. Consider the MAS described by (1) and (3) with ( A, B) stabilizable and (C, A) detectable. N be defined. Then, (Part I) Let ?, ? > 0 be given such that a set of graphs G?, ? protocol (8) for any ? > ? > 0 if and only if im E ? im B CR IP T N is solvable by a parameterized the H? -ASSPS for the MAS with any graph G ? G?, ? (34) while ( A, E, C, 0) is minimum phase and left-invertible. (Part II) Let G be a set of graphs such that the associated Laplacian matrices are AN US uniformly bounded and let ? consist of all possible nonzero eigenvalues of Laplacian matrices associated with graphs in G. Then, the H? -ASSPS for the MAS with any graph G ? G is solved by a parameterized protocol (8) if the robust H? -ADDPMS with bounded input for the system (32) with ? ? ? is solved by the parameterized controller (31). *. ? 1 +/ . ? In ) x? = ... .. /// , . / ,? N - ?? = (T ?1 ED ? := (T ?1 M Proof: By using L = T SL T ?1 , we define *. ??1 +/ . ? I)? = ... .. /// . / ,?? N - where ? i ? Cn+n c and ??i ? Cq . In the new coordinates, the dynamics of ? can be PT written as ??(t) = (I N ? A? + SL ? B?C?? + (T ?1 ? E)?, (35) AC CE which is rewritten as ?? 1 = A?? 1 + N X s1 j B?C?? j + E? ??1, j=2 ?? i = ( A? + ? i B?C?)? i + N X s i j B?C?? j + E? ??i , j=i+1 ?? N = ( A? + ? N B?C?)? N + E? ?? N , with i ? {2, . . . , N ? 1} where 0 E? = *. +/ , ,E - SL = [s i j ]. 24 (36) ACCEPTED MANUSCRIPT As in the case of full-state coupling, we can show that: (37) CR IP T *. ? 2 +/ . x? = (V ? In ) ... .. /// , . / ,? N - for some suitably chosen matrix V which is uniformly bounded. Therefore the H? norm of the transfer matrix from ? to x? can be made arbitrarily small if and only if the H? norm of the transfer matrix from ?? to ? can be made arbitrarily small. In order for the H? norm from ?? to ? to be arbitrarily small we need the H? norm AN US from ?? N to ? N to be arbitrarily small. In other words, the robust H? -ADDPMS with bounded input has to be solvable for the system x? = Ax + ? Bu + E?, y = Cx From the results of Lemma 4, we find that this is only possible if (34) is satisfied and ( A, E, C, 0) is left-invertible and minimum phase. M On the other hand, suppose (31) solves the robust H? -ADDPMS with bounded input of (32) and assume (34) is satisfied. We need to show that (8) solves the H? -ASSFS for ED the MAS described by (1) and (3). This follows directly from arguments very similar to the approach used in the proof of Lemma 1. PT 4.2. Necessary and sufficient condition for H2 -ASSPS The MAS system described by (1) and (3) after implementing the linear dynamical CE protocol (8) is described by (29) for i = 1, . . . , N, and, as before, the overall dynamics AC of the N agents can be written as x?? = (I N ? A? + L ? B?C?) x? + (I N ? E?)?. (38) We define a robust H2 -ADDPMS with bounded input as follows. Given ? ? C, there should exist M > 0 such that for any given real number r > 0, we can find a parameterized controller ?? = Ac (?) ? + Bc (?)y, u = Cc (?) ? + Dc (?)y, 25 (39) ACCEPTED MANUSCRIPT where ? ? Rn c , for the following system, x? = Ax + ? Bu + B?, (40) such that the following holds for any ? ? ?: CR IP T y = Cx 1. The closed-loop system of (39) and (40) is internally stable 2. The resulting closed-loop transfer function T? x from ? to x has an H2 norm less than r. than M. AN US 3. The resulting closed-loop transfer function T?u from ? to u has an H? norm less In the above, ? denotes all possible locations for the nonzero eigenvalues of the Laplacian matrix L when the graph varies over the set G. It is also important to note that M is independent of the choice for r. The following lemma, provides a necessary condition for the H2 -ADDPMS: M Lemma 5. Consider the system: x? = Ax + Bu + E?, ED y = Cx z=x PT with ( A, B) stabilizable and (C, A) detectable. The H2 -ADDPMS for the above system is defined as the problem to find for any r > 0 a controller of the form (39) such that the closed loop system is internally stable while the H? norm from ? to z is less than CE r. The H2 -ADDPMS is solvable only if: AC 1. im E ? im B 2. ( A, E, C, 0) is at most weakly non-minimum-phase and left-invertible. Proof: This follows directly from [29]. Theorem 8. Consider the MAS described by (1) and (3) with ( A, B) stabilizable and (C, A) detectable. 26 ACCEPTED MANUSCRIPT N be defined. Then, (Part I) Let ?, ? > 0 be given such that a set of graphs G?, ? N is solvable by a parameterized the H2 -ASSPS for the MAS with any graph G ? G?, ? protocol (8) for any ? > ? > 0 only if (41) CR IP T im E ? im B while ( A, E, C, 0) is at most weakly non-minimum phase and left-invertible . (Part II) Let G be a set of graphs such that the associated Laplacian matrices are uniformly bounded and let ? consist of all possible nonzero eigenvalues of Laplacian matrices associated with graphs in G. Then, the H2 -ASSPS for the MAS with any AN US graph G ? G is solved by a parameterized protocol (8) if the robust H2 -ADDPMS with bounded input for the system (40) with ? ? ? is solved by the parameterized controller (39). Proof: Similar, to the proof of Theorem 7, the dynamics can be written in the form (36). Using (37), we note the H2 norm of the transfer matrix from ? to x? can be made made arbitrarily small. M arbitrarily small if and only if the H2 norm of the transfer matrix from ?? to ? can be In order for the H2 norm from ?? to ? to be arbitrarily small we need the H2 norm ED from ?? N to ? N to be arbitrarily small. In other words, the robust H2 -ADDPMS with bounded input has to be solvable for the system PT x? = Ax + ? Bu + E?, y = Cx From the results of Lemma 5, we find that this is only possible if (41) is satisfied, CE ( A, E, C, 0) is left-invertible and at most weakly non-minimum phase. On the other hand, suppose (39) solves the robust H2 -ADDPMS with bounded input of (40) and assume (41) is satisfied. We need to show that (8) solves the H2 -ASSFS for AC the MAS described by (1) and (3). This follows directly from arguments very similar to the approach used in the proof of Lemma 1. 4.3. Protocol design for H? -ASSPS We present below two protocol design methods based on robust stabilization for the case E = B and therefore the case where ( A, B, C, 0) is minimum-phase. One relies 27 ACCEPTED MANUSCRIPT on an algebraic Riccati equation (ARE) method, and the other is based on the direct eigenstructure assignment method. 4.3.1. ARE-based method CR IP T Using an algebraic Riccati equation, we can design a suitable protocol. As in the full-state coupling case, we choose F = ?B 0 P with P = P 0 > 0 being the unique solution of the continuous-time algebraic Riccati equation A0 P + P A ? 2 ?PBB 0 P + I = 0, (42) where ? is a lower bound for the real part of the non-zero eigenvalues of all Laplacian AN US N . matrices associated with a graph in G?, ? Since ( A, B, C, 0) is minimum-phase then for any ? there exists ? small enough such that AQ + Q A0 + BB 0 + ? ?4 Q2 ? ??2 QC 0CQ = 0 (43) has a solution Q > 0. We then consider the following protocol: M ?? i = ( A + K? C) ? i ? K? ? i , ui = F? ? i , where F? = ? ?1 B 0 P, (44) ED K? = ? ?12 QC 0 The main result in this section is stated as follows. PT Theorem 9. Consider a MAS described by (1) and (3) with ( A, B) stabilizable and (C, A) detectable. Let any real numbers ?, ? > 0 and a positive integer N be given, CE N be defined. and hence a set of network graphs G?, ? N is solvable. In particular, for The H? -ASSPS stated in Problem 2 with G = G?, ? any given real number r > 0, there exists an ? ? , such that for any ? ? (0, ? ? ), the AC protocol (44) achieves state synchronization and the resulting system from ? to x i ? x j N . has an H? norm less than r for any i, j ? 1, . . . , N and for any graph G ? G?, ? Proof: Using Theorem 7, we know that we only need to verify that ?? = ( A + K? C) ? ? K? y, u = F? ?, 28 (45) ACCEPTED MANUSCRIPT solves the robust H? -ADDPMS with bounded input for the system (32) with ? ? ?. N , we know that ? ? ? implies Re ? ? ?. Obviously A + BF and Given G ? G?, ? ? A + K? C are both asymptotically stable by construction and hence the intersection of equal to: T? x = I where: I * 0 . ,T?3 ?1 ?T2 + / I - *. T1 +/ ,?T?4 - AN US T1 (s) = (sI ? A ? ? BF? ) ?1 B CR IP T (32) and (45) is asymptotically stable. The closed loop transfer function from ? to x is (46a) T2 (s) = ?(sI ? A ? ? BF? ) ?1 BF? (46b) T?3 (s) = ?(sI ? A ? K? C + ? BF? ) ?1 BF? (46c) T?4 (s) = (sI ? A ? K? C + ? BF? ) ?1 B (46d) As argued in the proof of Theorem 3, we have: ? , ? kT2 k? < M kT1 k? < 2|?| 2? ? 2. 2 ? ? PT where ED On the other hand, (43) implies according to the bounded real lemma: kT3 k? < ? 2 T3 (s) = (sI ? A ? K? C) ?1 B CE Note that: T?3 = ?(I + ?T3 F? ) ?1T3 F? AC which yields, using (47), that kT?3 k? < (1 ? ?M1 ) ?1 ?M1 < 2?M1 for small ? where M1 is such that: |?|kB 0 Pk < ?kB 0 Pk = M1 29 (47) ACCEPTED MANUSCRIPT The above yields: kT? x k? < ?M2 to: I * F? . ,T?3 T?u = F? ?T2 + / I - which yields using similar arguments as above that: ?1 *. T1 +/ ,?T?4 - kT?u k? < M3 CR IP T for some suitable constant M2 . The closed loop transfer function from ? to u is equal AN US for some suitable constant M3 independent of ?. Therefore the H? norm of the transfer matrix T? x becomes arbitrarily small for sufficiently small ? while the H? norm of the transfer matrix T?u remains bounded. 4.3.2. Direct method For ease of presentation, we only consider the case q = 1, i.e. the case where we M have a scalar measurement. We consider the state feedback gain F? given in (26), that is ED F? = where Tx is defined in (23). 1 ?1 B? F1 ? ?I Tx , Next, we consider the observer design. Note that the system ( A, B, C, 0) is minimum- PT phase and left-invertible. In that case there is a nonsingular matrix ?x such that, by AC CE defining x? = ?x x, we obtain the system x? a = Aa x a + L ad y, x? d = Ad x d + Bd (u + ? + Eda x a + Edd x d ), y = Cd x d . where xa x? = ?x x = *. +/ , ,xd - 30 (48) ACCEPTED MANUSCRIPT with x a ? Rn?? and x d ? R ? and where the matrices Ad ? R ?О? , Bd ? R ?О1 , and Cd ? R1О? have the special form 1 .. . иии иии иии .. . 0 0 0+ .. /// ./ // , 1// / 0- *.0+/ .. .. // . Bd = ... /// , ..0// . / ,1- Cd = 1 0 0 . CR IP T *.0 .. .. . Ad = ... ..0 . ,0 иии (49) Furthermore, the eigenvalues of Aa are the invariant zeros of ( A, B, C) and hence Aa is asymptotically stable. The transformation ?x can be calculated using available AN US software, either numerically [33] or symbolically [34]. Next, define a high-gain scaling matrix S? := diag(1, ? 2, . . . , ? 2??2 ), and define the output injection matrix M 0 +/ . K? = ?x *. ?2 ?1 ,? S? K - (50) (51) protocol: ED where K is such that Ad + Bd K is asymptotically stable. We then consider the following ?? i = ( A + K? C) ? i ? K? ? i , (52) PT ui = F? ? i , The main result in this section is stated as follows. CE Theorem 10. Consider a MAS described by a SISO system (1) and (3). Let any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network graphs AC N be defined. G?, ? N is If ( A, B) is stabilizable then the H? -ASSPS stated in Problem 2 with G = G?, ? solvable. In particular, for any given real number r > 0, there exists an ? ? , such that for any ? ? (0, ? ? ), the protocol (52) achieves state synchronization and the resulting system from ? to x i ? x j has an H? norm less than r for any i, j ? 1, . . . , N and for N . any graph G ? G?, ? 31 ACCEPTED MANUSCRIPT Proof: We use a similar argument as in the proof of Theorem 10. We know that we only need to verify that ?? = ( A + K? C) ? ? K? y, (53) CR IP T u = F? ?, solves the robust H? -ADDPMS with bounded input for the system (32) with ? ? ?. N , we know that ? ? ? implies Re ? ? ?. Obviously A + BF and Given G ? G?, ? ? A + K? C are both asymptotically stable by construction and hence the intersection of (32) and (45) is asymptotically stable. As in the proof of Theorem 9, the closed loop transfer function from ? to x is equal to: T? x = I I * 0 . ,T?3 ?1 ?T2 + / I - *. T1 +/ ,?T?4 - AN US (54) where, as before, we use the definitions in (46) but with our modified F? and K? . As argued in the proof of Theorem 5, we have: M kT1 k? < M1 ?, kT2 k? < M2 . for suitable constants M1, M2 > 0. Finally ED sI ? Aa T3 (s) = ?x *. ,?Bd Eda PT where ?1 L ad Cd + / Z1 - *. 0 +/ , Bd - Z1 = sI ? Ad ? ? ?2 S??1 KCd ? Bd Edd AC CE We obtain: with I T3 (s) = ? ?x *. ,0 2n 0 + * sI ? Aa /. S??1 - ,? 2n Bd Eda ?1 L ad Cd + / Z2 - *. 0 +/ , Bd - Z2 = sI ? ? ?2 Ad ? ? ?2 KCd + ? 2n Bd Edd S??1, using that ? ?2 Ad = S? Ad S??1, S? Bd = ? 2n Bd 32 and Cd S??1 = Cd . ACCEPTED MANUSCRIPT Note that Ed? is bounded for ? < 1. Next, we note that: L ad Cd +/ ?2 sI ? ? ( Ad + KCd ) - ?1 *. 0 +/ , Bd - CR IP T sI ? Aa X? (s) = *. , 0 ?(sI ? Aa ) ?1 L ad + 2 / (? sI ? Ad ? KCd ) ?1 Bd = ? 2 *. I , - From the above we can easily conclude that there exists M such that k X? k? < M? 2 . We have: where 0 + / (I + ? 2 X? Ed? ) ?1 X? S??1 - AN US I T3 = ? 2n ?x *. ,0 Ed? = ? 2n?2 Eda ? 2n?2 Edd S??1 which is clearly bounded for ? < 1. This clearly implies, using our bounds for X? and Ed? , that there exists M3 > 0 such that: for small ? since M kT3 k ? ? 2 M3 PT ED I ? 2n *. ,0 Our bound for T3 guarantees that 0 + / < ? 2 I. S??1 - kT?3 k? < ?M4, kT?4 k? < ?M5, CE for suitable M4 and M5 . Moreover kF? k < ? ?1 M0 AC Given our bounds, we immediately obtain from (54) that there exists M6 such that kT? x k? < M6 ?. The closed loop transfer function from ? to u is equal to: T?u = F? I * F? . ,T?3 33 ?T2 + / I - ?1 *. T1 +/ ,?T?4 - ACCEPTED MANUSCRIPT which yields, using similar arguments as above, that: kT?u k? < M7 CR IP T for some suitable constant M7 independent of ?. In other words, the transfer function from ? to x is arbitrarily small for sufficiently small ? while the transfer function from ? to u is bounded which completes the proof. 4.4. Protocol design for H2 -ASSFS We present below two protocol design methods based on robust stabilization for the AN US case E = B. The necessary condition provided earlier shows that ( A, B, C, 0) need only be at most weakly non-minimum-phase. The following designs are provided under the stronger assumption that ( A, B, C, 0) is minimum-phase. 4.4.1. ARE-based method We consider the protocol (44) already used in the case of H? -ASSFS. It is easy to M verify using a similar proof that this protocol also solves the robust H2 -ADDPMS with bounded input and therefore solves H2 -ASSPS for the MAS. Using the same notation ED as before, this relies on the fact that we have N1 such that kT1 k2 < ?N1 PT which follows directly from the full-state coupling case. On the other hand we have N2 CE such that kT3 k2 < ? 2 N2 AC since Q ? 0 for ? ? 0 and ( A ? KC)Q + Q( A ? KC) 0 + BB 0 ? 0. It is then easily shown that kT?4 k2 < ? 2 N3 for some N3 > 0. The rest of the proof is then as before in the case of H? -ASSPS. 34 Figure 1: The communication topology 4.4.2. Direct method CR IP T ACCEPTED MANUSCRIPT We consider the protocol (52) already used in the case of H? -ASSFS. It is easy to AN US verify using a similar proof that this protocol also solves the robust H2 -ADDPMS with bounded input and therefore solves H2 -ASSPS for the MAS. Using the same notation as before, this relies on the fact that we have N1 such that kT1 k2 < ?N1 M which follows directly from the full-state coupling case. On the other hand we have N2 ED such that kT?4 k2 < ?N2 using that X? has an H2 norm of order ?. The rest of the proof is then as before in the PT case of H? -ASSPS. 5. Example CE In this section, we illustrate our results on a homogeneous MAS of N = 6 agents. We consider the H? almost state synchronization problem via partial-state coupling. AC The agent model is given by: *.?2 A = ... 2 . ,5 0 0+ // 2 0// , / 4 2- *.0+/ B = ...3/// , . / ,2- 35 *.0+/ C 0 = ...1/// , . / ,1- *.0+/ E = ...3/// , . / ,2- ACCEPTED MANUSCRIPT with disturbances ?1 = sin(3t), ?2 = cos(t), ?3 = 0.5, CR IP T ?4 = sin(2t) + 1, ?5 = sin(t), ?6 = cos(2t). The communication topology is shown in Figure 1 with the Laplacian matrix ?1 2 ?1 0 ?2 0 0 0 ?2 0 4 0 ?3 2 0 0+ // 0 0// / 0 0// // ?2 0// // 2 0// / ?1 1- AN US *.1 .. ..0 ..0 L = .. ..0 .. ..0 . ,0 0 0 0 0 We design a controller of the form (44) based on an ARE-based method. The feedback gain F? = ? ?1 B 0 P with P given by the algebraic Riccati equation (22) and K? = ? ?12 QC 0 given by the algebraic Riccati equation (43). When choosing ? = 0.3 and ? = 0.01, we M get the controller PT ED 0 0 *.?2 +/ *. 0 +/ . / ?? i = .. 2 ?299.214 ?301.214// ? i + ...301.214/// ? i , . / . / 203.194, 5 ?199.194 ?201.194, ui = ?34.068 ?30.5702 ?27.0943 ? i ; AC CE while when choosing ? = 0.01 and ? = 0.0001, the controller is 0 0 + *.?2 *. 0 +/ // . ?? i = .. 2 ?29999 ?30001// ? i + ...30001/// ? i , . / . / 5 ?19999 ?20001 20003 , - , ui = ?1022 ?917.1 ?812.8 ? i . The results are shown in Figure 2. It is clear that when ? goes smaller, the H? norm from the disturbance to the relative error between the states of the different agents gets smaller. The controller inputs for all agents are shown in Figure 3. 36 =0.3 and =0.01 10 5 0 -5 -10 0 5 10 15 20 25 30 =0.01 and =0.0001 10 5 0 -5 -10 0 5 10 15 35 AN US State errors among 6 agents Time CR IP T State errors among 6 agents ACCEPTED MANUSCRIPT 20 25 30 40 35 40 30 35 40 30 35 40 Time M Figure 2: State errors among N = 6 agents =0.3 and =0.01 ED 500 0 -500 -1000 PT Control inputs of 6 agents 1000 -1500 0 10 15 20 25 Time =0.01 and =0.0001 10 4 1 AC CE Control inputs of 6 agents 5 0.5 0 -0.5 -1 -1.5 -2 0 5 10 15 20 25 Time Figure 3: The controller inputs of N = 6 agents 37 ACCEPTED MANUSCRIPT 6. Conclusion In this paper, we have studied H? and H2 almost state synchronization for MAS with identical linear agents affected by external disturbances. The communication network CR IP T is directed and coupled through agents? states or outputs. We have first developed the necessary and sufficient conditions on agents? dynamics for the solvability of H? and H2 almost state synchronization problems. Then, we have designed protocols to achieve H? and H2 almost state synchronization among agents based on two methods. One is ARE-based method and the other is ATEA-baed method. The future work could be to extend the results of this paper to nonlinear agents, that is, H? and H2 almost AN US state synchronization for MAS with identical nonlinear agents affected by external disturbances. References [1] H. Bai, M. Arcak, J. Wen, Cooperative control design: a systematic, passivity- M based approach, Communications and Control Engineering, Springer Verlag, 2011. ED [2] M. Mesbahi, M. Egerstedt, Graph theoretic methods in multiagent networks, Princeton University Press, Princeton, 2010. [3] W. Ren, Y. Cao, Distributed Coordination of Multi-agent Networks, Communica- PT tions and Control Engineering, Springer-Verlag, London, 2011. [4] C. Wu, Synchronization in complex networks of nonlinear dynamical systems, CE World Scientific Publishing Company, Singapore, 2007. AC [5] R. Olfati-Saber, J. Fax, R. Murray, Consensus and cooperation in networked multi-agent systems, Proc. of the IEEE 95 (1) (2007) 215?233. [6] R. Olfati-Saber, R. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Aut. Contr. 49 (9) (2004) 1520?1533. [7] W. Ren, On consensus algorithms for double-integrator dynamics, IEEE Trans. Aut. Contr. 53 (6) (2008) 1503?1509. 38 ACCEPTED MANUSCRIPT [8] W. Ren, E. Atkins, Distributed multi-vehicle coordinate control via local information, Int. J. Robust & Nonlinear Control 17 (10-11) (2007) 1002?1033. [9] W. Ren, R. Beard, Consensus seeking in multiagent systems under dynamically CR IP T changing interaction topologies, IEEE Trans. Aut. Contr. 50 (5) (2005) 655?661. [10] L. Scardovi, R. Sepulchre, Synchronization in networks of identical linear systems, Automatica 45 (11) (2009) 2557?2562. [11] S. Tuna, LQR-based coupling gain for synchronization of linear systems, available: AN US arXiv:0801.3390v1 (2008). [12] P. Wieland, J. Kim, F. AllgШwer, On topology and dynamics of consensus among linear high-order agents, International Journal of Systems Science 42 (10) (2011) 1831?1842. [13] T. Yang, S. Roy, Y. Wan, A. Saberi, Constructing consensus controllers for networks with identical general linear agents, Int. J. Robust & Nonlinear Control M 21 (11) (2011) 1237?1256. [14] H. Kim, H. Shim, J. Back, J. Seo, Consensus of output-coupled linear multi-agent ED systems under fast switching network: averaging approach, Automatica 49 (1) (2013) 267?272. PT [15] Z. Li, Z. Duan, G. Chen, L. Huang, Consensus of multi-agent systems and synchronization of complex networks: A unified viewpoint, IEEE Trans. Circ. & CE Syst.-I Regular papers 57 (1) (2010) 213?224. [16] J. Seo, J. Back, H. Kim, H. Shim, Output feedback consensus for high-order linear AC systems having uniform ranks under switching topology, IET Control Theory and Applications 6 (8) (2012) 1118?1124. [17] J. Seo, H. Shim, J. Back, Consensus of high-order linear systems using dynamic output feedback compensator: Low gain approach, Automatica 45 (11) (2009) 2659?2664. 39 ACCEPTED MANUSCRIPT [18] Y. Su, J. Huang, Stability of a class of linear switching systems with applications to two consensus problem, IEEE Trans. Aut. Contr. 57 (6) (2012) 1420?1430. IEEE Trans. Aut. Contr. 55 (10) (2009) 2416?2420. CR IP T [19] S. Tuna, Conditions for synchronizability in arrays of coupled linear systems, [20] Y. Zhao, Z. Duan, G. Wen, G. Chen, Distributed H? consensus of multi-agent systems: a performance region-based approach, Int. J. Contr. 85 (3) (2015) 332? 341. [21] I. Saboori, K. Khorasani, H? consensus achievement of multi-agent systems with AN US directed and switching topology networks, IEEE Trans. Aut. Contr. 59 (11) (2014) 3104?3109. [22] P. Lin, Y. Jia, Robust H? consensus analysis of a class of se

1/--страниц