Decentralized design of leader-following consensus protocols for asymmetric matrix-weighted heterogeneous multiagent systems

Decen tralized design of leader-follo wing consensus proto cols for asymmetric matrix-w eigh ted heterogeneous m ultiagen t systems Lanhao Zhao 1 , 2 and Y angzhou Chen 1 , 2 (  ) 1 College of Articial Intelligence and A utomation, Beijing Universit y of T echnology . 2 Engineering Research Center of Digital Communit y , Ministry of Education. zhaolanhao@emails.bjut.edu.cn, yzchen@bjut.edu.cn (Corresponding author) Abstract. This pap er in v estigates a decentralized design approac h of leader-follo wing consensus protocols for heterogeneous multiagen t sys- tems under a xed comm unication topology with a directed spanning tree (DST) and asymmetric w eigh t matrix. First, a con trol proto col us- ing only the information of the neighbor on the DST of eac h agent is designed, which is called the consensus protocol with minimal communi- cation links. Particularly , the DST-based linear transformation method is used to transform the consensus problem into a partial v ariable stability problem of a corresp onding system, and a decentralized design method is proposed to nd the gain matrices in the proto cols. Next, the decen- tralized design approach is extended to the proto cols using all neigh b or information in the original communication top ology with the help of the matrix diagonally dominan t method. Some numerical simulations are giv en to illustrate the theoretical results. Keyw ords: Consensus, heterogeneous multi-agen t system, asymmetric matrix-w eighted, DST-based linear transformation, minimal communi- cation links, decen tralized design 1 In tro duction The consensus of multi-agen t systems (MASs) has received extensiv e attention in the control comm unit y in the past three decades (see, e.g., the review pa- p er [1]). How ev er, the follo wing c hallenging issues in consensus problems are still necessary to further inv estigate: decen tralized design approach of consen- sus proto cols for heterogeneous MASs and generic asymmetric matrix-w eigh ted comm unication top ology . In terms of consensus proto cols for heterogeneous MASs, tw o key problem need further attention: one is the decen tralized design approac h of the feedback gains based on the distributed structure of consensus proto cols. The other is ho w to use as little communication as p ossible from the neigh b ors of each agen t. F or heterogeneous MAS, the consensus problem of heterogeneous MAS with 2 Lanhao Zhao et al. lo w-order individuals was rst discussed and man y results w ere obtained (see, e.g., [2]). F or the heterogeneous MAS with high-order individuals, W ahrburg and A dam y [3] obtained sucient and necessary conditions for state consen- sus problems by comp ensating for parameter deviations to isomorphize agents. Ho wev er, isomorphism has strict premises and is not alwa ys feasible. Tian and Zhang [4] discussed the consensus problem of high-order heterogeneous MASs with unkno wn comm unication dela y based on a class of transfer function mo dels. Ho wev er, some severer assumptions are made in [4], for example, the MAS is required to b e semi-stable. Under more relaxed assumptions, our previous works for leaderless [5] and leader-following [12] prop osed a new metho d to deal with the consensus problem of heterogeneous MASs. Based on a linear transformation constructed according to the characteristics of MAS, the consensus problem of heterogeneous MAS is transformed into the partial v ariable stabilit y problem of the corresponding system [6], and some necessary and sucient conditions were obtained. It should be noted that the method dev elop ed in [12] allow ed that eac h agen t has its o wn exclusiv e feedbac k gain. How ev er, global information is still required when nding these feedbac k gains, which is dicult for large-scale heterogeneous MASs. T o address this problem, a fully distributed metho d was prop osed in [13] to deal with this problem, where although global information is not required, but the feedback gain of all agents w as set to b e the same. A natural extension is to allo w each individual to ha v e an exclusive feedbac k gain matrix. This will increase the degree of freedom in the design of the gain matrix, and it is esp ecially needed for heterogeneous MASs. F or the second problem men tioned ab ov e, researchers usually use in termittent communication metho ds suc h as sampled con trol [14] and ev en t-triggered con trol [15] to reduce comm uni- cation trac. On the other hand, using information from some neighbors rather than all neighbors is also a metho d of reducing comm unication [7]. Thus, this prompts us to consider a consensus proto col that only uses part of neighbor information and design feedbac k gain matrices decentrally . In addition, in practical applications, the weigh ts expressing communication relationships b et ween dieren t agents ma y ha ve dieren t v alues concerning dif- feren t comp onents. At this p oin t, weigh t in matrix form is needed to describ e the in teraction betw een t wo agen ts. The consensus problem with comm unication w eights in matrix form has b een addressed in our early researc h [8], although this topic w as not particularly emphasized there. The concept of the matrix-w eigh ted net work has also been prop osed b y T rinh et al.[9] when studying b earing-based formation control problems. Recently , the consensus problem with the matrix- w eighted netw ork has received further atten tion, and some signicant results ha ve b een obtained [10]. How ever, in most existing research, the weigh ted ma- trices are assumed to be symmetric [11], either p ositive denite or negativ e def- inite. Compared with a symmetric w eighted matrix, the asymmetric weigh ted matrix can describ e a wider range of netw orks. Asymmetric weigh ted matrices will cause dicult y in stability analysis in the work mentioned ab ov e, which is another key issue that inspires the research in this pap er. Therefore, a natural Decen tralized design of consensus proto cols 3 further consideration is to handle the consensus problem of heterogeneous MASs with asymmetric w eighted matrices. In this pap er, we attempt to prop ose a decen tralized design approach of leader-follo wing consensus proto cols for heterogeneous MASs with asymmetric w eighted matrices. The main con tributions are summarized b elo w. 1) F or the leader-following consensus problem of heterogeneous MASs with asymmetric weigh ted matrices, the DST-based linear transformation metho d is used in the analysis and design of the control protocol, and the partial v ariable stabilit y theory is adopted to obtain consensus criteria. Esp ecially , the results obtained relax the assumption of symmetric w eighted matrices in the existing w ork on matrix weigh ted MASs [9–11]. 2) A new consensus protocol is prop osed, where each agen t uses only the information of itself and its neighbors on the DST, and the gain matrices can b e designed via a decentralized procedure. This design approach is signican t in reducing comm unication load and is called the consensus proto col design with minimal comm unication links. 3) The decentralized design approach is extended to nd the gain matrices of the consensus proto col using all neigh b ors’ information by adopting a matrix diagonally dominant idea. Compared to previous w ork [12], the decen tralized design approach pro vides computational conv enience in designing feedback gains. As far as we know, this is the rst time to giv e the decentralized design metho d of consensus protocols for heterogeneous in terconnected MASs with asymmetric w eighted matrices. The rest of this pap er is organized as follo ws. Section I I presents preliminary kno wledge and the mo del of heterogeneous MASs with asymmetric weigh ted matrices. Section I II sho ws the decentralized design metho d of the consensus proto col for heterogeneous MASs with asymmetric weigh ted matrices. Section IV pro vides some numerical examples. The summary is made in Section V. 2 Heterogeneous MASs with asymmetric w eigh t matrix In this section, w e introduce some basic concepts and the model of heteroge- neous MAS with asymmetric weigh ted matrices. Let R n b e the n -dimensional Euclidean space and e 1 =  1 0 · · · 0  T ∈ R N +1 b e the unit v ector. The symbol ⊗ is used to express the Kronec ker product. Consider a linear parameter heterogeneous MAS comp osed of N + 1 agen ts, where agen t 0 is the leader and the other agents are the follow ers. The dynamic of eac h follow er is ˙ x i = A i x i + B i u i , i = 1 , · · · , N (1) and the dynamic of the leader is ˙ x 0 = A 0 x 0 + B 0 u 0 (2) where for i = 0 , 1 , · · · , N , A i ∈ R n × n , B i ∈ R n × m , x i ∈ R n , u i ∈ R m . Here u i , i = 1 , · · · , N are the con trol inputs of the follow ers to b e designed and u 0 is the external input of the leader. 4 Lanhao Zhao et al. The communication top ology of the MAS can b e represented by a directed graph G = ( V , E , W ) with w eight matrix W , where V = { 0 , 1 , 2 , · · · , N } , E ⊆ { ( j, i ) : i, j ∈ V } . The edge ( j, i ) ∈ E indicates that the information can b e transferred to the agent i from the agent j , and th us the agent j is called the neigh b or of the agen t i . The set N i is used to represen t the collection of all neigh b ors of agent i . There is no communication edge from the follow ers to the leader, which means that the leader’s state is not aected by the follow ers’ states, but only by external inputs. The weigh t matrix W is expressed by the blo c k adjacency matrix W = [ W ij ] N i,j =0 ∈ R n ( N +1) × n ( N +1) , where W ij ∈ R n × n denotes the comm unication weigh t matrix b et ween i and j and W ij  = 0 if and only if ( j, i ) ∈ E . W e emphasize that for the matrix-weigh ted MASs, the communication w eights are allo wed to be in the form of matrices with generic en tries. F or example, the comm unication weigh ts b etw een dierent comp onen ts of an individual and the corresp onding components of its neigh b ors ma y be dierent. F or the directed graph G , a sequence of end-to-end directed edges in the same direction is called a directed path (strong path). If there is a no de i that has a directed path to an y other no de j  = i , the G is called containing a directed spanning tree (DST) with the root no de i and thus quasi-strongly connected. F urthermore, if there exists a directed path from any no de to any other no de, then graph G is called strongly connected. F urthermore, to provide a clearer description of the connection b et w een the leader and the follo wers, w e introduce the blo c k matrix δ =  D T 1 , . . . , D T N  T , where D i = W i 0 ∈ R n × n is a non-zero matrix i the leader can communicate directly with the follo wer i , otherwise D i = 0 . In this pap er, it is assumed that there is a DST T = ( V , E T , W T ) with 0 as the ro ot no de. Each non-ro ot no de i ∈ { 1 , 2 , · · · , N } has only one paren t no de denoted b y k i and, without loss of generalit y , we assume that k i < i . 3 Decen tralized design of consensus proto col In this section, we rst discuss the decen tralized design of the consensus proto col, where eac h agent only uses the information of itself and its neigh bor on the DST and call it the consensus proto col with minimal communication links. Then we extend the approac h to the protocol that uses all neigh b or information b y means of the matrix diagonally dominan t idea. 3.1 Decen tralized design of consensus protocol with minimum comm unication links The w ell-known fact is that in consensus problems with xed communication top ologies, the existence of a DST is a necessary comm unication condition for the agents to reac h consensus. F rom the p erspective of graph connectivity re- quiremen ts, DST generally has the smallest num b er of edges compared to the o verall netw ork. Therefore, at least tw o b enets would b e gained if each agen t Decen tralized design of consensus proto cols 5 only used the information of its neighbor on the DST instead of using all neigh- b or information in the consensus protocol design. The rst one is to reduce the comm unication load by using as few communication connections as p ossible. The other one is that, as one sees later, one can directly design the gain matrices by a decen tralized approach. Therefore, we prop ose the consensus proto col as follows, where the follow er i , i = 1 , · · · , N , only uses its o wn state (if av ailable) and the relative state with its neigh b or on the DST u i = − G i x i + K i W i,k i ( x k i − x i ) (3) and G i , K i ∈ R m × n are the gain matrices that need to b e designed. Here W i,k i = D i ∈ R n × n if k i = 0 . The consensus proto col (3) consists of t wo parts. The rst part is the agent’s o wn state feedback, where the gain matrix G i is directly set to a zero matrix if the agent’s o wn state is not av ailable. The second part is the feedback of the relativ e state describing the eect of the neighbor information. R emark 1 . The proto col (3) is referred to as the protocol with minimal communi- cation links b ecause it only uses neighbor information on the DST. A t this p oin t, the communication top ology can b e replaced by the DST, which is a necessary condition for the agents to reac h consensus. If the n umber of communication connections used in the proto col is further reduced, it will inevitably damage the DST and ma y lead to the consensus not b eing reac hed. It follo ws from (1) and (3) that ˙ x i = ( A i − B i G i ) x i + B i K i W i,k i ( x k i − x i ) , i = 1 , · · · , N (4) By com bining (2) and (4), we rewrite the MAS as follo ws  ˙ x 0 ˙ x  =  A 0 0 B D K D δ M   x 0 x  + ˜ B u 0 (5) where x =  x T 1 , . . . , x T N  T , M = A D − B D ( G D + K D L W T + K D ∆ ) , A D = diag ( A 1 , · · · , A N ) B D = diag ( B 1 , · · · , B N ) , G D = diag ( G 1 , . . . , G N ) , K D = diag ( K 1 , . . . , K N ) ∆ = diag ( D 1 , . . . , D N ) , ˜ B =  B T 0 , 0 n × m , . . . , 0 n × m  T and L W T = [ L ij ] N i,j =1 is the blo ck Laplacian concerning the follow ers’ netw ork in the DST and dened as L ij =      W i,k i , j = i − W i,k i , j  = i, j = k i 0 , j  = i, j  = k i A ccording to the denition of L W T and the assumption k i < i , we easily conclude that matrix L W T is lo wer triangular. 6 Lanhao Zhao et al. Therefore, the leader-following state consensus problem can b e describ ed as follo ws: to nd the gain matrices G i and K i in the proto col (3) suc h that the closed-lo op system (5) meets lim t →∞ ∥ x i ( t ) − x 0 ( t ) ∥ = 0 , ∀ i ∈ { 1 , · · · , N } (6) In this case, it is said that the leader-following state consensus of MAS (1) and (2) with proto col (3) is ac hiev ed. The decentralized design refers to nding the gain matrices G i and K i with- out using global information suc h as Laplacian matrices. Next, we transform the consensus problem expressed by (5) into the partial v ariable stability problem of a corresp onding system based on the DST-based linear transformation metho d [19] and provide a decentralized design approach for the gain matrices. Consider the incidence matrix P 0 of the DST T = ( V , E T , W T ) with the leader as the ro ot no de. Let the edges of the DST be represen ted by ( k i , i ) ∈ E , i = 1 , 2 , · · · , N . Let e k i ,i ∈ R N +1 b e the incidence v ector of edge ( k i , i ) , where k i -th component is 1 , i -th comp onen t is − 1 , and the remaining components are 0 . Th us, we ha v e P 0 = [ e k 1 , 1 , e k 2 , 2 , · · · , e k N ,N ] . Construct the follo wing transformation matrix P =  P 0 e 1  T ⊗ I n =:  p ˜ P 0 1 0  ⊗ I n and the corresp onding linear transformation  y x 0  = P  x 0 x  (7) The linear transformation (7) turns system (5) in to  ˙ y ˙ x 0  =  p ˜ P 0 1 0  ⊗ I n  ·  A 0 0 B D K D δ M   0 1 ˜ P − 1 0 1 N  ⊗ I n   y x 0  +  ˆ B B 0  u 0 (8) and th us we get ˙ y = ¯ Ay + ˆ Ax 0 + ˆ B u 0 , ˙ x 0 = A 0 x 0 + B 0 u 0 (9) where ¯ A =  ˜ P 0 ⊗ I n  M  ˜ P − 1 0 ⊗ I n  ˆ A = h ( A 0 − A 1 + B 1 G 1 ) T , ( A 1 − B 1 G 1 − A 2 + B 2 G 2 ) T . . . , ( A N − 1 − B N − 1 G N − 1 − A N + B N G N ) T i T ˆ B =  B T 0 , 0 n × m , . . . , 0 n × m  T R emark 2 . Both matrices ¯ A and M hav e the same Hurwitz stabilit y b ecause ¯ A is obtained b y a similar transformation of M . Decen tralized design of consensus proto cols 7 In tro ducing the follo wing notations η =  x T 0 , u T 0  T , ¯ B =  ˆ A ˆ B  , ¯ D =  A 0 B 0 0 0  where ¯ B is N n × ( n + m ) matrix and ¯ D is ( n + m ) × ( n + m ) matrix, the equation (9) b ecomes ˙ y = ¯ Ay + ¯ B η ˙ η = ¯ D η (10) W e can get from (7) that y = 0 if and only if x 0 = x 1 = . . . = x N . Therefore, w e hav e the follo wing lemma. Lemma 1. The le ader-fol lowing state c onsensus of MAS (5) c an b e achieve d i the zer o e quilibrium p oint of system (10) is glob al ly asymptotic al ly y -stable. No w, the consensus problem has b een transformed into a partial v ariable stabilit y problem of the corresponding system (10), and then we can giv e the y -stabilit y condition of (10) based on the partial stability theory [6]. First, construct the observ abilit y matrix of ( ¯ B , ¯ D ) as follo ws V k = h ¯ B T ( ¯ B ¯ D ) T · · ·  ¯ B ¯ D k  T i T , k = 0 , . . . , ( n + m − 1) Next, matrices L 1 ∈ R h × ( n + m ) and L 3 ∈ R ( n + m ) × h are obtained via the follo wing steps. 1) Let s = min { k : rank V k = rank V k +1 } and h = rank V s . 2) F or matrix V s , remain the rst row if it is not zero, otherwise it will b e remo ved. Then starting from the second ro w, remov e all rows linearly related to the previously remained rows. All the remained rows form a new matrix, denoted as L 1 . 3) The set of linearly indep enden t columns of L 1 , say , the columns i 1 , · · · , i h of L 1 , constitutes a new rev ersible matrix L 2 . 4) The ro w i j of L 3 is the ro w j of the in v erse matrix of L 2 , j = 1 , . . . , h , and the rest ro ws of L 3 are set to b e zero. Lemma 2. The zer o e quilibrium state of the system (10) is glob al ly asymptoti- c al ly y -stable i the fol lowing auxiliary system (11) is asymptotic al ly stable ˙ ξ = ¯ M ξ , ¯ M =  ¯ A ¯ B L 3 0 L 1 ¯ D L 3  (11) Lemma 1 rev eals the relationship b etw een the consensus problem of the MAS and the partial v ariable stability problem of a corresp onding system. Lemma 2 pro vides a criterion of partial v ariable stability based on the auxiliary system. Therefore, it follows from the tw o lemmas that the consensus of MAS (1) and (2) under con trol proto col (3) is reached if and only if ¯ M is Hurwitz. W e note that ¯ M is an upper triangular block matrix, and th us its Hurwitz stabilit y is equiv alen t to that of ¯ A and L 1 ¯ D L 3 . It implies that the state consensus of MAS 8 Lanhao Zhao et al. (1) and (2) under con trol protocol (3) is ac hieved if and only if the matrix M in (5) and L 1 ¯ D L 3 in (11) are b oth Hurwitz stable. Let A i ∗ = A i − B i G i and B i ∗ = B i K i ( W ik i + D i ) . Then according to the c haracteristic that matrix L W T is a low er triangular matrix, the matrix M can b e written as a lo w er triangular blo c k matrix M =     A 1 ∗ − B 1 ∗ 0 · · · 0 ∗ A 2 ∗ − B 2 ∗ · · · 0 · · · · · · · · · · · · ∗ ∗ · · · A N ∗ − B N ∗     and th us we get the follo wing theorem. Theorem 1. The le ader-fol lowing state c onsensus of MAS (1) and (2) under c ontr ol pr oto c ol (3) is achieve d i matric es A i ∗ − B i ∗ , i = 1 , · · · , N and L 1 ¯ D L 3 ar e Hurwitz stable. F urthermore, w e p oin t out that the gain matrices G i and K i to b e designed in protocol (3) hav e dieren t roles. First, L 1 ¯ D L 3 expresses the heterogeneous c haracteristics of MAS (1) and (2) and only dep ends on the c hoice of gain matrix G i , i = 1 , · · · , N . So one can rst choose G i , i = 1 , · · · , N (if state x i a v ailable) suc h that L 1 ¯ D L 3 is Hurwitz stable. Moreov er, as shown in Example 2 in Section V, the gain matrix G i is related to the con vergence rate of the MAS, so one can rst select them according to the actual demand for the conv ergence rate. Next, after G i , i = 1 , · · · , N being selected, one further determines K i suc h that A i ∗ − B i ∗ for i = 1 , · · · , N are Hurwitz stable. R emark 3 . It is ob vious that the design of the gain matrix K i according to The- orem 1 only dep ends on the parameters of agen t i and thus the design procedure is decen tralized. 3.2 Decen tralized design of consensus proto col that uses all neigh b or information In this subsection, we extend the decentralized design approac h to the case of the proto col that uses all neigh b or information using the matrix diagonally dominan t idea. Consider the follo wing proto col that uses all the neigh b ors’ information u i = − G i x i + K i X j ∈ N i W i,j ( x j − x i ) (12) In this case, the block Laplacian L W = [ L ij ] N i,j =1 concerning the follow ers’ net work in the original comm unication top ology is dened as L ij =    P N k =1 ,k  = i W ik , j = i − W ij , j  = i, j ∈ N i 0 , j  = i, j / ∈ N i Decen tralized design of consensus proto cols 9 Similarly , we use the DST-based linear transformation to transform the con- sensus problem of MAS (1) and (2) with the proto col (12) into the partial v ariable stabilit y problem of the following system ˙ y = ¯ A ′ y + ˆ Ax 0 + ˆ B u 0 ˙ x 0 = A 0 x 0 + B 0 u 0 (13) where ¯ A ′ =  ˜ P 0 ⊗ I n  M ′  ˜ P − 1 0 ⊗ I n  , M ′ = A D − B D ( G D + K D L W + K D ∆ ) W e hav e the follo wing similar result with the same matrices ¯ B , ¯ D , L 1 , and L 3 as b efore. Lemma 3. The zer o e quilibrium state of the system (13) is glob al ly asymptoti- c al ly y -stable i the fol lowing auxiliary system (14) is asymptotic al ly stable: ˙ ξ = ¯ M ′ ξ , ¯ M ′ =  ¯ A ′ ¯ B L 3 0 L 1 ¯ D L 3  (14) A ccording to Lemma 3, the leader-following state consensus of MAS (1) and (2) under control proto col (12) is achiev ed i ¯ M ′ is Hurwitz stable, or equiv a- len tly , if and only if ¯ A ′ and L 1 ¯ D L 3 are Hurwitz stable. F urthermore, the tw o matrices ¯ A ′ and M ′ ha ve the same Hurwitz stability b ecause ¯ A ′ is obtained b y a similar transformation of M ′ . Let A ′ i = A i − B i G i , B ′ i = B i K i  P N k =1 ,k  = i W ik + D i  . Then the matrix M ′ can b e written in the follo wing form     A ′ 1 − B ′ 1 − B 1 K 1 W 12 · · · − B 1 K 1 W 1 N − B 2 K 2 W 21 A ′ 2 − B ′ 2 · · · − B 2 K 2 W 2 N · · · · · · · · · · · · − B N K N W N 1 − B N K N W N 2 · · · A ′ N − B ′ N     Therefore, w e ha ve the following result using the Gerschgorlin circle theorem of blo c k matrices [18], where the matrix norms are dened as ∥ H ∥ = max 1 ≤ i ≤ m n X j =1 | h ij | ,    H − 1    − 1 = min 1 ≤ i ≤ m n X j =1 | h ij | for an y matrix H = [ h ij ] ∈ C m × n . Theorem 2. The le ader-fol lowing state c onsensus of MAS (1) and (2) with c ontr ol pr oto c ol (12) is achieve d if matrix L 1 ¯ D L 3 is Hurwitz and the gain matric es K i , i = 1 , ..., N in (12) ar e sele cte d such that the fol lowing Gerschgorlin cir cles     ( A ′ i − B ′ i − λI n ) − 1     − 1 ≦ X j  = i,j ∈ N i ∥− B i K i W i,j ∥ (15) ar e al l lo c ate d in the op en left half c omplex plane. 10 Lanhao Zhao et al. A ccording to Lemma 3, we need to ensure that M ′ and L 1 ¯ D L 3 are Hurwitz stable. The stability of matrix M ′ is closely related to the eigenv alues of the matrix M ′ . A ccording to the Gerschgorlin circle theorem of blo c k matrices [18], all eigenv alues of the matrix are also lo cated in the left half complex plane as long as all Gersc hgorlin circles (15) are placed in the left half complex plane. Therefore, for each agen t i , w e can design K i to adjust the p osition of the cor- resp onding Gerschgorlin circle so that it is lo cated in the op en left half complex plane. That is, if the control gain matrices K i .i = 1 , 2 , · · · , N are selected to mak e suc h that Gerschgorlin circles (15) are all located in the open left half complex plane, matrix M ′ is Hurwitz. If matrix L 1 ¯ D L 3 is also Hurwitz at this time, then the leader-follo wing state consensus of MAS (1) and (2) under con trol proto col (12) is ac hiev ed. R emark 4 . The information used in designing the feedback gain matrix K i in- cludes the co ecien t matrix information of agen t i and the neigh bor information it communicates with, but without using other agen t information that is not di- rectly communicates with it in the netw ork. F rom this p ersp ectiv e, the design metho d prop osed in Theorem 2 is still decentralized. This result improv es the conclusion giv en in [12] by solving matrix inequalities that con tain global net- w ork information. As sho wn in the pro of of Theorems 1 and 2, the conclusion giv en through the linear transformation metho d do es not require whether the weigh t on the edge is a symmetric matrix, which is dierent from the assumption that the w eigh ted matrix on the edge is p ositiv e denite or negativ e denite commonly used in most existing researc h on matrix weigh ted MASs [10, 11]. So to our knowledge, Theorem 1 and Theorem 2 pro vide a decentralized design metho d for consensus proto cols in heterogeneous MASs under the condition that the weigh t matrix is asymmetric for the rst time. The sp ecic selection pro cess of the gain matrix K i will b e further sho wn in Example 3 in Section IV. 4 Num b erical examples In this section, we provide sev eral n umerical examples to verify the eectiveness of the prop osed metho d and the obtained results. Firstly , the follo wing example is used to v erify the decentralized design approac h in Theorem 1. 0 2 4 3 1 Fig. 1. Communication top ology in Example 1 Decen tralized design of consensus proto cols 11 Example 1 . Consider the MAS with the comm unication topology sho wn in Fig- ure 1, where no de 0 is the leader and nodes from 1 to 4 are the follow ers. The parameter matrices for their dynamics are as follo ws A 0 =  2 0 0 4  , B 0 =  1 1  A 1 =  1 0 3 4  , B 1 =  2 1  , A 2 =  1 2 1 4  , B 2 =  2 3  A 3 =  1 1 0 4  , B 3 =  1 4  , A 4 =  2 2 3 5  , B 4 =  2 4  The w eight matrices with resp ect to the communication in Figure 1 are as follows W 10 =  1 2 3 4  , W 20 =  4 3 3 4  , W 41 =  1 2 5 4  , W 42 =  2 2 3 4  , W 32 =  1 2 0 4  , W 34 =  1 0 3 4  and D 1 = W 10 , D 2 = W 20 . Consider the DST with edges (0 , 1)(0 , 2)(2 , 3)(2 , 4) . No des 1 and 2 directly receiv e the leader’s information, and nodes 3 and 4 ha ve a common neighbor no de 2 on the DST. F or the design of the gain matrices, rst, select G i as follo ws G 1 =  1 1  , G 2 =  2 2  , G 3 =  3 3  , G 4 =  4 4  suc h that L 1 ¯ D L 3 is Hurwitz stable. Then select K i according to Theorem 1 K 1 =  0 . 7 0 . 1  , K 2 =  − 2 . 786 2 . 464  , K 3 =  2 − 1 . 625  , K 4 =  − 12 . 25 6  suc h that A 1 − B 1 G 1 − B 1 K 1 D 1 , A 2 − B 2 G 2 − B 2 K 2 D 2 A 3 − B 3 G 3 − B 3 K 3 W 32 , A 4 − B 4 G 4 − B 4 K 4 W 42 are Hurwitz stable. Figure 2 sho ws the state errors, b oth for the initial states x 0 = (2 . 5 , 2 . 5) , x 1 = (0 . 5 , 0) , x 2 = (1 , 0) , x 3 = (1 . 5 , 0) , x 4 = (2 , 0) . 0 2 4 6 8 10 12 14 16 18 20 t/s -2.5 -2 -1.5 -1 -0.5 0 x1 1 -x0 1 x1 2 -x0 2 x2 1 -x0 1 x2 2 -x0 2 x3 1 -x0 1 x3 2 -x0 2 x4 1 -x0 1 x4 2 -x0 2 Fig. 2. State errors in Example 1 12 Lanhao Zhao et al. R emark 5 . When solving the feedback gain matrix K i , only the information of agen t i and the communication weigh t betw een itself and its paren t node are used. Therefore, the solving pro cess is decen tralized. As men tioned ab o ve, the gain matrices G i and K i pla y dierent roles in the consensus process. Below, w e will use an example to demonstrate the impact of the gain matrix G i on the consensus pro cess. Example 2 . Consider the same MAS in Example 1. Now we set G i to b e zero matrices. One can v erify L 1 ¯ D L 3 is Hurwitz stable. Next, w e select K i according to Theorem 1 K 1 =  0 . 2222 0 . 6111  , K 2 =  − 2 . 5 2 . 75  , K 3 =  2 . 5 − 1 . 10313  , K 4 =  − 10 . 25 6  suc h that A 1 − B 1 K 1 D 1 , A 2 − B 2 K 2 D 2 , A 3 − B 3 K 3 W 32 , A 4 − B 4 K 4 W 42 are Hurwitz stable. 0 2 4 6 8 10 12 14 16 18 20 t/s -2.5 -2 -1.5 -1 -0.5 0 x1 1 -x0 1 x1 2 -x0 2 x2 1 -x0 1 x2 2 -x0 2 x3 1 -x0 1 x3 2 -x0 2 x4 1 -x0 1 x4 2 -x0 2 Fig. 3. State errors in Example 2 Figure 3 shows the state errors with initial states x 0 = (2 . 5 , 2 . 5) , x 1 = (0 . 5 , 0) , x 2 = (1 , 0) , x 3 = (1 . 5 , 0) , x 4 = (2 , 0) . By comparing Example 1 and Example 2, one can observ e that the gain matrices G i ha ve an impact on the con vergence rate of the consensus pro cess. Next, we will demonstrate the decentralized design metho d under proto cols that use all the neigh b ors’ information. Example 3 . Consider the same MAS as in Example 1. W e rst select G i G 1 =  1 1  , G 2 =  2 2  , G 3 =  3 3  , G 4 =  4 4  suc h that L 1 ¯ D L 3 is Hurwitz stable. A ccording to Theorem 2, w e design matrices K i making the follo wing Gerschgorlin circles ( i = 1 , 2 , 3 , 4 )     ( A ′ i − B ′ i − λI n ) − 1     − 1 ≦ X j  =1 ,j ∈ N i ∥− B i K i W i,j ∥ Decen tralized design of consensus proto cols 13 to b e all lo cated in the op en left half complex plane, where λ are the eigen v alues of matrix M ′ . Th us K i can b e selected as K 1 =  0 . 7 0 . 1  , K 2 =  − 2 . 786 2 . 464  K 3 =  2 − 1 . 625  , K 4 =  − 12 . 25 5  Figure 4 shows the state errors with initial states x 0 = (2 . 5 , 2 . 5) , x 1 = (0 . 5 , 0) , x 2 = 0 2 4 6 8 10 12 14 16 18 20 t/s -2.5 -2 -1.5 -1 -0.5 0 x1 1 -x0 1 x1 2 -x0 2 x2 1 -x0 1 x2 2 -x0 2 x3 1 -x0 1 x3 2 -x0 2 x4 1 -x0 1 x4 2 -x0 2 Fig. 4. State errors in Example 3 (1 , 0) , x 3 = (1 . 5 , 0) , x 4 = (2 , 0) . R emark 6 . When solving the gain matrix K i , only the information of agent i and the communication w eight b et ween itself and its neigh b or no de are used. Therefore, the solving pro cess is decen tralized. 5 CONCLUSION This paper studies the decen tralized design of consensus proto cols for heteroge- neous MASs with asymmetric weigh t matrices. A consensus proto col with min- imal communication links is designed, and a decentralized design approach for the gain matrices in the proto col is obtained. 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