Minimum-Rank Dynamic Output Consensus Design for Heterogeneous Nonlinear Multi-Agent Systems
aa r X i v : . [ c s . S Y ] M a y Minimum-Rank Dynamic Output Consensus Designfor Heterogeneous Nonlinear Multi-Agent Systems
Dinh Hoa Nguyen,
Member, IEEE
Abstract —In this paper, we propose a new and systematicdesign framework for output consensus in heterogeneous Multi-Input Multi-Output (MIMO) general nonlinear Multi-AgentSystems (MASs) subjected to directed communication topology.First, the input-output feedback linearization method is utilizedassuming that the internal dynamics is Input-to-State Stable(ISS) to obtain linearized subsystems of agents. Consequently,we propose local dynamic controllers for agents such that thelinearized subsystems have an identical closed-loop dynamicswhich has a single pole at the origin whereas other poles areon the open left half complex plane. This allows us to deal withdistinct agents having arbitrarily vector relative degrees and toderive rank- cooperative control inputs for those homogeneouslinearized dynamics which results in a minimum rank distributeddynamic consensus controller for the initial nonlinear MAS.Moreover, we prove that the coupling strength in the consensusprotocol can be arbitrarily small but positive and hence ourconsensus design is non-conservative. Next, our design approachis further strengthened by tackling the problem of randomlyswitching communication topologies among agents where werelax the assumption on the balance of each switched graph andderive a distributed rank- dynamic consensus controller. Lastly,a numerical example is introduced to illustrate the effectivenessof our proposed framework. I. I
NTRODUCTION
Cooperative control of multi-agent systems (MASs) hasgained much attention recently since there are a lot of practicalapplications, e.g., power grids, wireless sensor networks, trans-portation networks, systems biology, etc, can be formulated,analyzed and synthesized under the framework of MASs. Oneof the key features in MASs is the achievement of a globalobjective by performing local measurement and control at eachagent and simultaneously collaborating among agents usingthat local information.Employing the principle of relatively exchanged local in-formation, a very important and extensively studied subjectin MASs is the consensus problem where agents’ states oroutputs come to a non-zero agreement. A huge collection ofresults can be found for consensus of MASs, ranging from sin-gle integrator dynamics of agents with fixed and time-varyingcommunication topology [1]–[4] to general linear agents [5]and to nonlinear agents with disturbances, uncertainties, timedelays, etc [6], [7].For linear MASs, one way to develop a systematic consen-sus control design is to employ the LQR method, e.g. [8]–[11].The paper [8] designed optimal consensus laws for network
Dinh Hoa Nguyen is currently with Control System Laboratory, De-partment of Advanced Science and Technology, Toyota Technological In-stitute, 2-12-1 Hisakita, Tempaku-ku, Nagoya 468-8511, Japan. e-mail:[email protected], [email protected]. of integrators utilizing two LQR cost functions. LQR-basedconsensus designs for leader-follower MASs was presented in[9] in which only local LQR problems were solved and noglobal LQR problem was considered. Next, in our previousresearch [10], we introduced an LQR-based method to designa distributed consensus controller for general linear MASsbut the obtained controller is only sub-optimal. Recently, wehave proposed an approach in [11] to achieve a consensusdesign with a non-conservative coupling strength where analternative MAS model namely edge dynamics was presentedthat helps transforming the consensus design into an equivalentstability synthesis which can be derived by LQR method. Thisadvanced result will be employed subsequently in the currentarticle.In nonlinear MASs, many efforts have recently been con-ducted for consensus problem, of which most are based onpassivity theory and internal model principle, e.g. [6], [12]–[16], just to name a few. Consensus designs with linear andnonlinear output couplings were introduced in [6] for hetero-geneous SISO affine nonlinear MASs under the assumption ofpassive dynamics of agents. This work was then extended in[12] where the balanced condition of inter-agent communica-tion graph was relaxed to be strongly connected graph. Robuststatic output-feedback consensus controllers for heterogeneousSISO affine relative-degree-two passive sector-bounded MASsin presence of communication constraints were investigated in[14]. In [15], distributed output tracking consensus controllerswere proposed based on internal model principle and passivityfor heterogeneous MIMO affine nonlinear networks of agentswith relative degree one and two. In a recent work, [16] pro-posed a method to design distributed output tracking consensuscontrollers for heterogeneous SISO nonlinear MASs utilizinginternal model principle and nonlinear controller forms thatsatisfying global Lipschitz conditions. Although the passiveproperty can be found in a wide class of nonlinear systems,this approach requires the number of inputs and outputs ofa nonlinear agent to be the same and further assumptionsor conditions must be satisfied. Moreover, finding the energyfunction or the Lyapunov function to show the passivity ofnonlinear agents is not always easy.Some other researches on consensus of nonlinear MASsconsider specific problems, e.g. [17], [18]. A special class ofhomogeneous leader-follower MIMO affine MASs was studiedin [17] and sufficient conditions for static consensus con-trollers were given based on Lipschitz assumption for agentsand Lyapunov theory. Tracking consensus controller based oninternal model principle and a specific, complicated selectionof control input were introduced in [18] for a very special class of heterogeneous SISO relative-degree-one nonlinear MASs.These results are quite difficult to extend to more generalcontexts.On the other hand, our current article proposes a systematicframework to design linear dynamic output consensus con-trollers for leaderless heterogeneous MIMO general nonlinearMASs , following the idea of designing low-rank consensuscontroller with non-conservative coupling strength in our pre-vious work [11]. It is worth noting that there are essentialdifferences between the current paper and [11] as follows.First, the current article deals with heterogeneous nonlinear
MASs while [11] considered homogeneous linear
MASs.Second, this paper proposes consensus designs for MASs with directed and switching communication topologies, but [11]developed consensus controllers for MASs with undirectedand fixed structures. Third, the consensus controllers proposedin this paper can be freely designed to have rank , howeverthe ones in [11] could not.The remarkable features of our proposed framework areas follows. First, it gives us a distributed dynamic outputconsensus controller design for heterogeneous MIMO generalnonlinear MASs with arbitrary vector relative degree that: (i)has an arbitrarily small but positive coupling strength; (ii) has minimum rank , i.e., rank- . Second, the communication topol-ogy among agents is directed and can be randomly switching of which the component graphs need not to be balanced . Tothe best of our knowledge, there have not been similar resultsin the literature so far, and thus the aforementioned propertiesclearly show the contributions of this paper.II. P RELIMINARIES
A. Notations and Symbols
The following notations and symbols will be used in the pa-per. R , R − , and C stand for the sets of real, non-positive real,and complex number. Re( x ) denotes the real part of a complexnumber x . Moreover, n and n denote the n × vector withall elements equal to and , respectively; and I n denotesthe n × n identity matrix. Next, L f h ( x ) , ( ∂h ( x ) /∂x ) f ( x ) represents the notation for Lie derivative, and ⊗ stands forthe Kronecker product. On the other hand, λ ( A ) and λ min ( A ) denotes the eigenvalue set and the eigenvalue with smallest,non-zero real part of A , respectively. In addition, ≻ and (cid:23) denote the positive definiteness and positive semi-definitenessof a matrix. Lastly, K ∞ denotes the class of scalar function γ ( x ) : R + → R + which is continuous, strictly increasing,unbounded, and γ (0) = 0 ; and KL denotes the class of scalarfunction γ ( x, t ) : R + × R + → R + such that γ ( ., t ) ∈ K ∞ foreach t and γ ( x, t ) ց as t → ∞ . B. Graph Theory
Denote ( G , V , E ) the directed graph representing the infor-mation structure in a multi-agent system composing of N agents, where each node in G stands for an agent and each edgein G represents the interconnection between two agents; V and E represent the set of vertices and edges of G , respectively.There is an edge e ij ∈ E if agent i receives informationfrom agent j . The neighboring set of a vertex i is denoted by N i , { j : e ij ∈ E} . Moreover, let a ij be elements ofthe adjacency matrix A of G , i.e., a ij > if e ij ∈ E and a ij = 0 if e ij / ∈ E . The in-degree of a vertex i is denoted by deg in i , P Nj =1 a ij , then the in-degree matrix of G is denotedby D = diag { deg in i } i =1 ,...,N . Consequently, the Laplacianmatrix L associated to G is defined by L = D − A . The out-degree of a vertex i is denoted by deg out i , P Nj =1 a ji . Then G is said to be balanced if deg in i = deg out i ∀ i = 1 , . . . , N. A directed path connecting vertices i and j in G is a set ofconsecutive edges starting from i and stopping at j . Then G issaid to have a spanning tree if there exists a node called rootnode from which there are directed paths to every other node. Lemma 1: [19] The Laplacian matrix L always has a zeroeigenvalue with associated eigenvector N , and all non-zeroeigenvalues of L have positive real parts. Furthermore, L hasonly one zero eigenvalue if and only if G has a spanning tree.If the communication topology among agents is varied withtime then we will write the time-varying terms with the timeindex t , e.g., N i ( t ) , a ij ( t ) , G ( t ) , L ( t ) , etc. C. Consensus of Linear MASs
Let us consider an MAS composing of N identical linearagents whose dynamics is described by ˙ x i = Ax i + Bu i , i = 1 , . . . , N, (1)where x i ∈ R n , u i ∈ R m , A ∈ R n × n , B ∈ R n × m . A commonconsensus protocol for (1) is u i = − µK X j ∈N i a ij ( x i − x j ) , i = 1 , . . . , N, (2)where K ∈ R m × n is the consensus controller gain matrix, µ is the coupling strength. Lemma 2: [20], [21] A necessary and sufficient conditionfor the MAS with agent dynamics (1) to reach consensusdefined as follows, lim t →∞ k x i ( t ) − x j ( t ) k = 0 ∀ i, j = 1 , . . . , N, by the control law (2) is that its communication graph G has aspanning tree and A − µλ k BK are stable for all k = 2 , . . . , N, where λ k are non-zero eigenvalues of L .The following proposition shows a consensus design forlinear MASs with non-conservative coupling strength, whichserves as a basis for consensus design of nonlinear MASs withnon-conservative coupling strength in the next sections. Proposition 1:
Suppose that the following conditions aresatisfied: (i) the directed graph G representing the commu-nication structure in the MAS (1) has a spanning tree; (ii) ( A, B ) is controllable; (iii) λ ( A ) ∈ R − . Then this MASreaches consensus by the controller (2) for any µ > and K = RB T P , where R ∈ R m × m , R ≻ , and P ∈ R n × n , P ≻ is the unique solution of the following Riccati equation, P A + A T P + Q − P BRB T P = 0 , (3)in which Q ∈ R n × n , Q (cid:23) , and ( Q / , A ) is observable. Proof:
Based on the result of Lemma 2, the MAS(1) will reach consensus if condition (i) is satisfied and A − µλ k BRB T P are stable for all k = 2 , . . . , N, where λ k are non-zero eigenvalues of L . Note that K = RB T P is infact an LQR controller gain, and from optimal control theory[22], it is known that all eigenvalues of A − BRB T P areshifted to the left of the imaginary axis. On the other hand, Re( λ k ) > ∀ k = 2 , . . . , N since G has a spanning tree[20], [21]. Therefore, by scaling with a scalar parameter µλ k with positive real part for all k = 2 , . . . , N , the controller gain µλ k RB T P still shifts all eigenvalues of A to the left thoughit could be more or less depending on whether µλ k > or µλ k < . Since we have assumed that λ ( A ) ∈ R − , this meansall eigenvalues of A − µλ k BRB T P belong to the open left halfcomplex plane for all k = 2 , . . . , N , and thus the consensusis achieved in the MAS (1).III. O UTPUT C ONSENSUS OF H ETEROGENEOUS
SISON
ONLINEAR
MAS
S WITH F IXED D IRECTED T OPOLOGY
In this section, we present a novel approach to designdistributed controller for output consensus problem in hetero-geneous SISO nonlinear MASs with fixed topology. The SISOaffine nonlinear MASs will be investigated first in SectionIII-A then SISO general nonlinear MASs will consequentlybe studied in Section III-B based on the results obtained foraffine ones.
A. Heterogeneous SISO Affine Nonlinear MASs
Consider a network of N heterogeneous SISO affine non-linear agents whose models are described as follows, ˙ x i = f i ( x i ) + g i ( x i ) u i ,y i = h i ( x i ) , i = 1 , . . . , N, (4)where x i ∈ R n i , u i ∈ R , and y i ∈ R are the state vector,input, and output of the i th agent, respectively; f i , g i ∈ R n i and h i ∈ R are vector-valued and scalar-valued of continuous,differentiable nonlinear functions. Definition 1:
The affine nonlinear agent (4) is said to haverelative degree r i > if L g i L kf i h i ( x i ) = 0 as k = 0 , . . . , r i − ,L g i L kf i h i ( x i ) = 0 as k = r i − . (5) Definition 2:
A multi-agent system with dynamics of agentsdescribed by (4) is said to reach an output consensus if lim t →∞ | y i ( t ) − y j ( t ) | = 0 ∀ i, j = 1 , . . . , N. (6)The control design problem is to find a distributed controlstrategy for the agents (4) such that their outputs cooperativelyreach consensus while they unidirectionally exchange informa-tion through a directed graph G . Throughout this section, weutilize the following assumption of G . A1:
The directed graph G is time-invariant and has a spanningtree. Remark 1:
In some practical situations, the directed graph G could be time-varying due to the link failures, packetlosses, etc. This phenomenon of varied topology is usuallymodeled in the literature as deterministic switches (e.g., [23], [16]) or random switches (e.g., [24], [25]). For the clarity ofapproach representation, we first employ assumption A1 andwill investigate the scenario of switching topologies later in aseparated section.Consequently, we employ the input-output feedback lin-earization method [26] to derive linearized models of agentsand accordingly convert the output consensus problem ofinitial nonlinear MAS to a state consensus problem of a newlinearized MAS. More specifically, the nonlinear models ofagents are changed by a diffeomorphism Φ i ( x i ) = [ ξ Ti , η Ti ] T to normal forms ˙ ξ i, = ξ i, , ˙ ξ i, = ξ i, , ... ˙ ξ i,r i = α i ( ξ i , η i ) + β i ( ξ i , η i ) u i , ˙ η i = ϑ i ( ξ i , η i ) ,y i = ξ i, , i = 1 , . . . , N, (7)where ξ i,k , Φ i,k ( x i ) , L k − f i h i ( x ) , k = 1 , . . . , r i , ξ i =[ ξ i, , . . . , ξ i,r i ] T ∈ R r i , η i ∈ R n i − r i ; α i ( ξ i , η i ) , L r i f i h i ( x ) , β i ( ξ i , η i , d i ) , L g i L r i − f i h i ( x ) , ϑ i ( ξ i , η i ) ∈ R n i − r i . To avoidthe finite-escape-time (FET) phenomenon and guarantee theinternal stability of the closed-loop system, we employ thefollowing assumption [27], A2:
The internal dynamics ˙ η i = ϑ i ( ξ i , η i ) is input-to-statestable (ISS), i.e., there exist some functions γ i, ∈ KL and γ i, ∈ K ∞ such that k η i ( t ) k ≤ γ i, ( k η i (0) k , t ) + γ i, ( k ξ i ( t ) k ∞ ) . Then the design problem becomes finding a control law forthe linearized multi-agent system (7) such that linearized states ξ i, of agents are consensus. In light of assumption A2, weare able to set the control input for the i th agent as follows, u i = 1 β i ( ξ i , η i ) [ − α i ( ξ i , η i ) + ˆ u i ] , i = 1 , . . . , N, (8)where ˆ u i ∈ R is a new control input for the linearizedsubsystem. Since the dynamics of linearized subsystems aredifferent, a static consensus controller cannot be derived.Instead, we will propose a dynamic controller which is ableto make ξ i, , i = 1 , . . . , N converge to a common value, i.e.,output consensus for (4) is achieved. Define r = max i =1 ,...,N r i , i = 1 , . . . , N. (9)Consequently, each agent is equipped with the followingdynamic controller, ˙ φ i = D i ξ i + E i φ i + G i v i ,u i = H i φ i − α i ( ξ i , η i ) β i ( ξ i , η i ) , i = 1 , . . . , N, (10)where φ i ∈ R r − r i is the controller’s state vector in which φ i, = ˆ u i , v i ∈ R is a new control input, and G i = (cid:2) · · · (cid:3) T ∈ R r − r i ,H i = (cid:2) /β i ( ξ i , η i ) 0 · · · (cid:3) ∈ R × ( r − r i ) , D i = · · ·
00 0 0 · · · ... ... ... ... · · · − b − b · · · − b r i ∈ R ( r − r i ) × r i ,E i = · · ·
00 0 · · · ... ... ... · · · − b r i +1 − b r i +2 · · · − b r ∈ R ( r − r i ) × ( r − r i ) , of which b , . . . , b r are coefficients of the following charac-teristic equation whose poles are in R − , s r + b r s r − + · · · + b s = 0 . (11)Note that the free coefficient is chosen to be to ensure anon-zero consensus. Denote ˆ ξ i = [ ξ Ti , φ Ti ] T , then the overalllinearized dynamics of agents are made identical by thedynamic controllers (10), and has the following representation, ˙ˆ ξ i = A ˆ ξ i + Bv i ,u i = (cid:2) H i (cid:3) ˆ ξ i − α i ( ξ i , η i ) /β i ( ξ i , η i ) , i = 1 , . . . , N, (12)where A = · · ·
00 0 1 · · · ... ... ... ... · · · − b − b · · · − b r ∈ R r × r , B = · · · ∈ R r . We are now ready to state a foundation result of this paperin the following theorem where the coupling strength in theconsensus law for nonlinear MASs is non-conservative . Theorem 1:
The heterogeneous SISO affine nonlinear MAS(4) reaches an output consensus by the local dynamic con-trollers (10) and the cooperative controls v i = − µ (ˆ rB T P ) X j ∈N i a ij ( ˆ ξ i ( t ) − ˆ ξ j ( t )) , (13)for any µ > , where ˆ r > and P ∈ R r × r is the uniquepositive definite solution of the following Riccati equation P A + A T P + Q − ˆ rP BB T P = 0 , (14)of which Q ∈ R r × r , Q (cid:23) , and ( Q / , A ) is observable. Proof:
Since the incorporated models of linearized agentsin (12) are homogeneous, linear, and all of their poles are in R − , we can immediately apply the result of Proposition 1 inSection II-C for designing a distributed consensus controllerfor (12) under the form of (13). Note that in the currentsituation each linearized agent is SISO, so the weightingmatrix R becomes a scalar parameter that we denoted by ˆ r . Consequently, in combination with the local dynamic con-trollers (10), it gives us the output consensus of the initialnonlinear MAS (4).The control design for the whole system is demonstratedin Figure 1, where C ( s ) represents the transfer function ofidentical linearized systems (12). PSfrag replacements u i v i ξ i ˆ ξ i ˆ ξ j µ (ˆ rB T P ) X ( i,j ) ∈E a ij ˆ ξ j y i − α i β i Φ i ( x i ) Agent i th C ( s ) Cooperative controlLocal dynamic controller
Fig. 1. Block diagram of distributed output consensus design for SISO affinenonlinear MASs based on input-output feedback linearization.
Remark 2:
It can be seen in Theorem 1 that µ can bearbitrarily chosen as long as it is positive. On the other hand,in other researches, e.g., [5], [20], [21], [28], µ is lowerbounded by / Re( λ min ( L )) which can be extremely big as thenumber of agents increases and the inter-agent communicationtopology is sparse and hence is very conservative. Moreover, λ min ( L ) is a global information and therefore to make the con-sensus law fully distributed, other methods need to be furtherdeveloped to estimate this global term, e.g., adaptive designs[25]. This unexpectedly increases the complexity of the controldesign and implementation. Nevertheless, this conservatism isremoved in our work, which makes our consensus design non-conservative and more effective in design and implementation. Remark 3:
The output consensus design in Theorem 1 re-lies on the output y i , its first-order and higher-order derivativeswhich may not be available in some practical systems. In thatcases, local estimation techniques can be employed to obtainthe approximated values of those unmeasurable derivatives.Let us denote θ i = C ˆ ξ i the partial information that could beexchanged among agents with C ∈ R × r , then we can employa decentralized Luenberger observer [29] for each agent asfollows, ˙ˇ ξ i = A ˇ ξ i + Bv i + M ( θ i − ˇ θ i ) , ˇ θ i = C ˇ ξ i , i = 1 , . . . , N, (15)where M ∈ R r . Denote e i = ˆ ξ i − ˇ ξ i the error vector betweenthe real state ˆ ξ i and the estimated state ˇ ξ i . Then by subtracting(12) with (15), we obtain the following error model, ˙ e i = ( A − M C ) e i . (16)As a result, by selecting the observer gain M such that A − M C is stable, e ( t ) → as t → ∞ , i.e., ˇ ξ i → ˆ ξ i as t → ∞ .Lastly, the cooperative control input v i is modified by v i = − µ (ˆ rB T P ) X j ∈N i a ij ( ˇ ξ i ( t ) − ˇ ξ j ( t )) , (17)where ˇ ξ is obtained from the local observer (15).In Theorem 1, a local rank- r Riccati equation (14) needsto be solved to obtain the consensus controller, which wouldcost more computational time than expected for high relativedegree nonlinear agents. Hence, we next propose a method to derive a minimum rank distributed consensus controllerby solving a local rank- , i.e., scalar Riccati equation. Thecontroller is therefore fully analytical, which requires noadditional time for solving Riccati equation.First, we choose b , . . . , b r such that matrix A defined in(III-A) has only one eigenvalue at the origin while othereigenvalues belong to the open left half complex plane. Let ν T ∈ R × r be the left eigenvector of A associated with theeigenvalue . Second, select Q = νq ν T where q > .Suppose that ( Q / , A ) is still observable. Then, the rank- distributed dynamic consensus controller is derived as follows. Theorem 2:
The heterogeneous SISO affine nonlinear MAS(4) reaches an output consensus by the distributed dynamicconsensus controllers (10) with the rank- cooperative inputs v i , i = 1 , . . . , N , synthesized as follows, v i = − µ p q ˆ rν T X j ∈N i a ij ( ˆ ξ i ( t ) − ˆ ξ j ( t )) . (18)Furthermore, the consensus speed, i.e., the smallest non-zeroabsolute of real parts of closed-loop eigenvalues, is equal to min n µ p q ˆ rB T ν Re( λ min ( L )) , λ min ( − A ) o . (19) Proof:
First, we prove the rank- consensus controller’sformula. Let P = νp ν T , p > , then substituting Q and P back to the Riccati equation (14), we obtain ν (cid:0) p ν T A + A T νp + q − ˆ rν T BB T νp (cid:1) ν T = 0 , (20)which is equivalent to the vanishment of the expression insidethe bracket. Since ν T A = 0 , this implies q − ˆ rν T BB T νp =0 , which leads to p = √ q / √ ˆ rB T ν. Since (14) has a uniquepositive semidefinite solution, P = ν √ q / ( √ ˆ rB T ν ) ν T isindeed that unique one. Consequently, substituting this valueof P into the cooperative control input (13) gives us (18).Obviously, rank( v i ) = 1 , so together with (10) we derive arank- distributed dynamic consensus controller.Next, we reveal how to obtain the consensus speed. Em-ploying the same process in the proof of Theorem 3 in [11],we can easily show that the eigenvalue set of the closed-loopdynamics of the linearized MAS is given by [ γ ∈ λ ( L ) ,γ =0 − µγ p q ˆ rB T ν [ ( λ ( A ) \{ } ) . (21)Thus, the consensus speed is determined by (19). Remark 4:
It can be observed from Theorem 2 that µ , thepole of A closest to the imaginary axis, and q and ˆ r areparameters that affect to the consensus speed. Hence, we mayadjust them to obtain an expected consensus speed. B. Heterogeneous SISO General Nonlinear MASs
In this scenario, the models of agents are in the followinggeneral form ˙ x i = f i ( x i , u i ) ,y i = h i ( x i ) , i = 1 , . . . , N, (22)where x i ∈ R n i , u i ∈ R , and y i ∈ R are the state vector,input, and output of the i th agent, respectively; f i ∈ R n i and h i ∈ R are vector-valued and scalar-valued of continuous,differentiable nonlinear functions. Definition 3:
The general nonlinear agent (22) is said tohave relative degree r i if ∂∂u i L kf i h i ( x i ) = 0 as k = 0 , . . . , r i − ,∂∂u i L kf i h i ( x i ) = 0 as k = r i . (23)The dynamic distributed controller (10) for affine nonlinearMASs cannot be utilized in this scenario. However, it ispossible if we consider the following augmented models ofagents which are affine, ˙˜ x i = ˜ f i (˜ x i ) + ˜ g (˜ x i ) ˙ u i ,y i = ˜ h i (˜ x i ) , i = 1 , . . . , N, (24)where ˜ x i , [ x Ti , u i ] T , ˜ f i (˜ x i ) , [ f i ( x i , u i ) T , T , ˜ g i (˜ x i ) , [0 Tn i , T , ˜ h i (˜ x i ) , h i ( x i ) . Consequently, it can be easilychecked that the relative degree of the augmented affine non-linear agents (24), in the sense of Definition 1, are r i + 1 , i =1 , . . . , N . Similarly to the case of affine nonlinear MASs, weemploy the input-output linearization feedback approach withdiffeomorphisms ˜Φ i (˜ x i ) = [ ˜ ξ Ti , ˜ η Ti ] T assumed that the internaldynamics ˜ η i is ISS. Then the linearized subsystems of agentsare obtained by the following control inputs ˙ u i = 1˜ β i ( ˜ ξ i , ˜ η i ) [ − ˜ α i ( ˜ ξ i , ˜ η i ) + ˜ u i ] , i = 1 , . . . , N, (25)where ˜ β i ( ˜ ξ i , ˜ η i ) , L ˜ g i L r i ˜ f i ˜ h i (˜ x i ) ; ˜ α i ( ˜ ξ i , ˜ η i ) , L r i +1˜ f i ˜ h i (˜ x i ) ; ˜ u i ∈ R is a new control input for the linearized subsystem ofthe augmented nonlinear agents (24). Let r be defined as in(9). Then we propose the following dynamic controller ˙˜ φ i = ˜ D i ˜ ξ i + ˜ E i ˜ φ i + ˜ G i ˜ v i ,w i = ˜ H i ˜ φ i − ˜ α i ( ˜ ξ i , ˜ η i )˜ β i ( ˜ ξ i , ˜ η i ) , ˙ u i = w i , i = 1 , . . . , N, (26)where ˜ φ i ∈ R r − r i is a vector of controller’s states in which ˜ φ i, = ˜ u i ; ˜ v i ∈ R is a new control input; w i ∈ R is anadditional state of the controller; ˜ E i and ˜ G i are defined asfollows, ˜ D i = · · ·
00 0 0 · · · ... ... ... ... · · · − ˜ b − ˜ b · · · − ˜ b r i +1 ∈ R ( r − r i ) × ( r i +1) , ˜ E i = · · ·
00 0 · · · ... ... ... · · · − ˜ b r i +2 − ˜ b r i +2 · · · − ˜ b r +1 ∈ R ( r − r i ) × ( r − r i ) , ˜ G i = (cid:2) · · · (cid:3) T ∈ R r − r i , ˜ H i = (cid:2) / ˜ β i ( ˜ ξ i , ˜ η i ) 0 · · · (cid:3) ∈ R × ( r − r i ) , of which ˜ b , . . . , ˜ b r +1 are coefficients of the following char-acteristic equation whose poles are in R − , s r +1 + ˜ b r +1 s r + · · · + ˜ b s = 0 . (27)This controller can be viewed as a cascade of two controllersin which the first one composes of the first two equationsin (26) while the second one is an integrator corresponding tothe last equation (26). Subsequently, the closed-loop linearizeddynamics of agents are made homogeneous by the dynamiccontrollers (26), and has the following representation, ˙ ζ i = ˜ Aζ i + ˜ B ˜ v i ,w i = (cid:2) H i (cid:3) ˆ ξ i − ˜ α i ( ˜ ξ i , ˜ η i ) / ˜ β i ( ˜ ξ i , ˜ η i ) , i = 1 , . . . , N, (28)where ζ i = (cid:2) ˜ ξ Ti ˜ φ Ti (cid:3) T , ˜ A = · · ·
00 0 1 · · · ... ... ... ... · · · − ˜ b − ˜ b · · · − ˜ b r +1 ∈ R ( r +1) × ( r +1) , ˜ B = (cid:2) · · · (cid:3) T ∈ R r +1 . (29)The control design for the whole system is demonstratedin Figure 2, where ˜ C ( s ) represents the transfer function ofidentical linearized systems in (28) from the inputs ˜ v i to theoutputs w i , i = 1 , . . . , N . PSfrag replacements u i w i ˜ v i ˜ ξ i ζ i ζ j µ (˜ r ˜ B T ˜ P ) X ( i,j ) ∈E a ij ζ j y i − ˜ α i ˜ β i ˜Φ i (˜ x i ) Agent i th ˜ C ( s ) 1 s Cooperative controlLocal dynamic controller
Fig. 2. Block diagram of distributed output consensus design for SISOgeneral nonlinear MASs based on input-output feedback linearization.
Similarly to the scenario of SISO affine nonlinear MASs, wecan design rank- distributed dynamic consensus controllersfor the MAS (22). Let us select ˜ b , . . . , ˜ b r +1 such that matrix ˜ A defined in (29) has only one eigenvalue at the origin whileother eigenvalues belong to the open left half complex plane.Denote ˜ ν T ∈ R × ( r +1) the left eigenvector of ˜ A associatedwith the eigenvalue . Consequently, choose ˜ Q = ˜ ν ˜ q ˜ ν T , where ˜ q > . Assuming that ( ˜ Q / , ˜ A ) is still observable,then the rank- distributed dynamic consensus controller inthis case is introduced in the following theorem. Theorem 3:
The heterogeneous SISO general nonlinearMAS (22) reaches an output consensus by the distributed dy-namic consensus controllers (26) with the rank- cooperative inputs ˜ v i , i = 1 , . . . , N , synthesized as follows, ˜ v i = − µ p ˜ q ˜ r ˜ ν T X j ∈N i a ij ( ζ i ( t ) − ζ j ( t )) . (30)Furthermore, the consensus speed is equal to min n µ p ˜ q ˜ r ˜ B T ˜ ν Re( λ min ( L )) , λ min ( − ˜ A ) o . (31) Proof:
The proof of this theorem is the same as that ofTheorem 2, so we omit it for brevity.IV. O
UTPUT C ONSENSUS OF H ETEROGENEOUS
MIMON
ONLINEAR
MAS
S WITH F IXED D IRECTED T OPOLOGY
Consider a MAS composing of N heterogeneous MIMOaffine nonlinear agents whose models are described as follows, ˙ x i = f i ( x i ) + G i ( x i ) u i ,y i = h i ( x i ) , i = 1 , . . . , N, (32)where x i ∈ R n i , u i ∈ R m i , and y i ∈ R p are the state, input,and output vectors of the i th agent, respectively; f i ( x i ) ∈ R n i , h i ( x i ) ∈ R p , and G i ∈ R n i × m i are vector-valued andmatrix-valued of continuous, differentiable nonlinear func-tions; h i ( x i ) = [ h i, ( x i ) , . . . , h i,p ( x i )] T , h i,τ ( x i ) ∈ R ∀ τ =1 , . . . , p ; G i ( x i ) = [ g i, ( x i ) , . . . , g i,m i ( x i )] , g i,j ( x i ) ∈ R n i ∀ j = 1 , . . . , m i . The communication topology G amongagents is also assumed to satisfy assumption A1 in Section III.The output dimensions of all agents are equal to be meaningfulin the context of output consensus, which is defined as follows. Definition 4:
The outputs of agents whose models are de-scribed by (32) are said to reach a consensus if lim t →∞ k y i ( t ) − y j ( t ) k = 0 ∀ i, j = 1 , . . . , N. (33)In the use of input-output feedback linearization for non-linear systems [26], the dimensions of input and output areusually assumed to be equal, however we consider here a moregeneral context where those dimensions may be different.Accordingly, the definition of vector relative degree [26] canbe modified as follows. Definition 5:
The MIMO nonlinear system (32) is said tohave an extended vector relative degree { r i, , r i, , . . . , r i,p } ata point ˆ x i ∈ R n i if the following conditions hold.(i) L g i,j L kf i h i,l ( x i ) = 0 ∀ k = 0 , . . . , r i,l − j = 1 , . . . , m i ; ≤ l ≤ p ; and for all x i in a neighborhood of ˆ x i .(ii) rank(Π i ( x i )) = p at the point ˆ x i , where Π i ( x i ) is definedin (34).Note that condition (ii) above is satisfied if and only if p ≤ m i , i.e., the extended vector relative degree is only defined forthe MIMO affine nonlinear systems that have the number ofoutputs not more than the number of inputs. If p = m i then itreduces to the vector relative degree in [26] since condition (ii)means Π i ( x i ) is invertible at ˆ x i . Furthermore, this extendedvector relative degree allows us to treat a scenario that the vector relative degree in [26] cannot, where some agents in aMIMO affine nonlinear MASs have the same number of inputsand outputs but other agents do not.To design a distributed output consensus controller forthe MAS (32), we also try to obtain linearized models of Π i ( x i ) = L g i, L r i, − f i h i, ( x i ) L g i, L r i, − f i h i, ( x i ) · · · L g i,mi L r i, − f i h i, ( x i ) L g i, L r i, − f i h i, ( x i ) L g i, L r i, − f i h i, ( x i ) · · · L g i,mi L r i, − f i h i, ( x i ) ... ... . . . ... L g i, L r i,p − f i h i,p ( x i ) L g i, L r i,p − f i h i,p ( x i ) · · · L g i,mi L r i,p − f i h i,p ( x i ) . (34)agents and then design the consensus controller based on thosemodels. Similarly to the scenario of SISO nonlinear MASs,we utilize the input-output feedback linearization technique fornonlinear agents (32) that can be processed as follows. Denote κ i = P pl =1 r i,l , i = 1 , . . . , N. Consequently, the agents’models are changed to normal forms by a diffeomorphism Φ i ( x i ) = [ ξ Ti , η Ti ] T where Φ i ( x i ) , [Φ i, ( x i ) , . . . , Φ i,r i, ( x i ) , . . . , Φ pi, ( x i ) , . . . , Φ pi,r i,p ( x i ) , Φ i,κ i +1 ( x i ) , . . . , Φ i,n i ( x i )] T ,ξ li,j , Φ li,j ( x i ) , L j − f i h i,l ( x i ) ∀ j = 1 , . . . , r i,l ; ∀ l = 1 , . . . , p,ξ i , [ ξ i, , . . . , ξ i,r i, , . . . , ξ pi, , . . . , ξ pi,r i,p ] T ,η i , [Φ i,κ i +1 ( x i ) , . . . , Φ i,n i ( x i )] T . (35)The linearized model for agent i th is as follows, ˙ ξ li, = ξ li, , ˙ ξ li, = ξ li, , ... ˙ ξ li,r i,l = α li ( ξ i , η i ) + m i X k =1 β li,k ( ξ i , η i ) u i,k , ˙ η i = ϑ i ( ξ i , η i ) + m i X k =1 χ i,k ( ξ i , η i ) u i,k ,y li = ξ li, ∀ l = 1 , . . . , p ; i = 1 , . . . , N, (36)where u i,k is the k th control input of the i th agent, k =1 , . . . , m i , and α li ( ξ i , η i ) , L r i,l f i h i,l ( x i ) , β li,k ( ξ i , η i ) , L g i,k L r i,l − f i h i,l ( x i ) ,ϑ i ( ξ i , η i ) ∈ R n i − κ i ,χ i,k ( ξ i , η i ) , [ L g i,k Φ i,κ i +1 ( x i ) , . . . , L g i,k Φ i,n i ( x i )] T . The following assumption is employed to avoid the FETphenomenon.
A3:
The internal dynamics η i in (36) is ISS for all i =1 , . . . , N .Then the r i,l -th equations, l = 1 , . . . , p , in the linearizedmodel (36) of agents can be collected and written in thefollowing form ˙ˇ ξ i = ˇ α i ( ξ i , η i ) + Π i ( ξ i , η i ) u i , (37)where ˇ ξ i = [ ξ i,r i, , ξ i,r i, , . . . , ξ pi,r i,p ] T ∈ R p , ˇ α i ( ξ i , η i ) =[ α i ( ξ i , η i ) , . . . , α pi ( ξ i , η i )] T ∈ R p , Π i ( ξ i , η i ) ∈ R p × m i isdefined in (34). Since rank(Π i ( ξ i , η i )) = p , there exists aright inverse Π i ( x i ) † of Π i ( x i ) defined by Π i ( ξ i , η i ) † , Π i ( ξ i , η i ) T (Π i ( ξ i , η i )Π i ( ξ i , η i ) T ) − . Accordingly, the MIMO nonlinear system (agent) (32) can be input-output decoupledby the following controller u i = Π i ( ξ i , η i ) † ( − ˇ α i ( ξ i , η i ) + ˇ u i ) , (38)where ˇ u i = [ˇ u i, , . . . , ˇ u i,p ] T ∈ R p is a new control inputvector. As a result, the input-output decoupling is achievedfor each agent as follows, ˙ ξ li, = ξ li, , ˙ ξ li, = ξ li, , ... ˙ ξ li,r i,l = ˇ u i,l , ˙ η i = ϑ i ( ξ i , η i ) + m i X k =1 χ i,k ( ξ i , η i ) u i,k ,y li = ξ li, , ∀ l = 1 , . . . , p ; i = 1 , . . . , N. (39)At this point, we can see that the consensus design foroutput vectors of nonlinear agents (32) is decomposed intoindependent consensus designs of individual outputs of agentsthat is similar to SISO affine nonlinear MASs in Section III-A.Hence, all steps of designing consensus controller for SISOaffine nonlinear MASs can be adopted straightforwardly. Thus,to avoid the complexity and duplication in representing results,we skip the details here. Similar situation applies to MIMOgeneral nonlinear MASs.V. O UTPUT C ONSENSUS OF H ETEROGENEOUS N ONLINEAR
MAS
S UNDER S WITCHING T OPOLOGY
In this section, we aim at investigating the consensus designfor heterogeneous nonlinear MASs subjected to randomlyswitching topologies described by continuous-time Markovchains. Due to space limitation, only results for SISO affinenonlinear MASs are presented.Since the communication topology is randomly switching, G is time-varying and is denoted by G ( t ) . Suppose that G ( t ) switches among the elements of a finite set of ℓ topologies S G , {G , . . . , G ℓ } , where the switching process is repre-sented by a continuous-time Markov chain with a switchingsignal σ ( t ) ∈ { , . . . , ℓ } . Denote Q = [ q ij ] ∈ R ℓ × ℓ and π = [ π , . . . , π ℓ ] T the transition rate matrix and the stationarydistribution of this Markov process. Here, we assume that theMarkov process is ergodic, so π is unique, π k > ∀ k =1 , . . . , ℓ , and each state of the Markov chain can be reachedfrom any other state. Furthermore, we can also assume that theMarkov process starts from π [24]. As a result, the distributionof σ ( t ) is equal to π for all t ≥ .Let us denote the union of all possible topologies by G ∪ , [ k =1 ,...,ℓ G k . The following assumption is utilized. • A4: G ∪ has a spanning tree and is balanced. The consensus of agents in this context is defined as follows.
Definition 6:
The nonlinear MAS (4) with a randomlyswitching topology G ( t ) is said to reach a mean-squareconsensus for any initial condition of agents and any initialdistribution of the continuous-time Markov process if lim t →∞ E (cid:2) k x i ( t ) − x j ( t ) k (cid:3) = 0 , ∀ i, j = 1 , . . . , N, (40)where E [ · ] denotes the expectation taken with some chosenprobability measure. Remark 5:
Note that assumption A4 is milder than thefrequently used one in literature (see e.g., [24], [25]) whichassumes the balance of G k ∀ k = 1 , . . . , ℓ . This relaxation onthe switching topologies is an advantage that greatly broadenthe class of MAS topologies under random switches to achieveconsensus.Consequently, the following theorem shows that our pro-posed distributed rank- consensus controller design in Sec-tion III-A can be generalized to this scenario of switchingtopologies. Theorem 4:
Under assumption A4, the nonlinear MAS (4)reaches a mean-square consensus in the sense of Definition 6by the distributed dynamic consensus controller (10) with thefollowing rank- cooperative input v i ( t ) = − µ p q ˆ rν T X j ∈N i ( t ) a ij ( t )( ˆ ξ i ( t ) − ˆ ξ j ( t )) . (41)Moreover, the consensus speed is specified by π ∗ µ p q ˆ rB T νλ min (cid:0) L ∪ + L T ∪ (cid:1) , (42)where π ∗ = min k =1 ,...,ℓ π k , L ∪ is the Laplacian matrix associatedwith G ∪ . Proof:
Denote ˆ ξ ave = 1 N ( ˆ ξ (0) + · · · + ˆ ξ N (0)) ,δ i ( t ) = ˆ ξ i ( t ) − ˆ ξ ave , k = i, . . . , N,δ ( t ) = (cid:2) δ ( t ) T , . . . , δ N ( t ) T (cid:3) T . It is then followed from substituting the rank- cooperativeinput (41) to the homogeneous linearized systems (12) that ˙ δ ( t ) = [ I N ⊗ A − µ L ( t ) ⊗ ( B ˆ rB T P )] δ ( t ) . Let us define the following Lyapunov functions V ( t ) = E (cid:2) δ ( t ) T ( I N ⊗ P ) δ ( t ) (cid:3) ,V k ( t ) = E (cid:2) δ ( t ) T ( I N ⊗ P ) δ ( t )1 { σ ( t )= k } (cid:3) , k = 1 , . . . , ℓ, where · is the Dirac measure. Obviously, V ( t ) = P ℓk =1 V k ( t ) . Consequently, using Lemma 4.2 in [30], weobtain dV k ( t ) = E (cid:2) ( dδ ( t )) T ( I N ⊗ P ) δ ( t )1 { σ ( t )= k } + δ ( t ) T ( I N ⊗ P ) dδ ( t )1 { σ ( t )= k } (cid:3) + ℓ X j =1 q jk V j ( t ) dt + o ( dt ) , where o ( · ) stands for the Little-o notation. Accordingly, ˙ V k ( t ) = E (cid:2) δ ( t ) T [ I N ⊗ ( P A + AP ) − µ ( L ( t ) + L T ( t )) ⊗ ( P B ˆ rB T P )] δ ( t )1 { σ ( t )= k } (cid:3) = E (cid:2) − µδ ( t ) T [( L ( t ) + L T ( t )) ⊗ ( νq ν T )] δ ( t )1 { σ ( t )= k } (cid:3) , since P A = νp ν T A = 0 as shown in the proof of Theorem2. Hence, ˙ V ( t ) = E " − µδ ( t ) T ℓ X k =1 π k ( L k + L Tk ) ⊗ ( νq ν T ) ! δ ( t ) ≤ − π ∗ µ E (cid:2) δ ( t ) T [( L ∪ + L T ∪ ) ⊗ ( νq ν T )] δ ( t ) (cid:3) , since π k ≥ π ∗ ∀ k = 1 , . . . , ℓ , and P ℓk =1 L k = L ∪ . Onthe other hand, L ∪ + L T ∪ can be regarded as a Laplacianmatrix of an connected undirected graph due to assumptionA4. Therefore, there exists an orthogonal matrix U ∈ R N × N such that L ∪ + L T ∪ = U Λ U T where Λ = diag { λ i } i =1 ,...,N is a diagonal matrix whose diagonal elements are eigenvaluesof L ∪ + L T ∪ and U = h √ N N , U i with U ∈ R N × ( N − .Subsequently, δ ( t ) T [( L ∪ + L T ∪ ) ⊗ ( νq ν T )] δ ( t )= δ ( t ) T ( U ⊗ I n )[Λ ⊗ ( νq ν T )]( U T ⊗ I n ) δ ( t )= δ ( t ) T ( U ⊗ I n )diag { λ i νq ν T } i =2 ,...,N ( U T ⊗ I n ) δ ( t ) ≥ λ min (cid:0) L ∪ + L T ∪ (cid:1) p q ˆ rB T νδ ( t ) T ( U ⊗ I n )( I N − ⊗ P ) × ( U T ⊗ I n ) δ ( t )= λ min (cid:0) L ∪ + L T ∪ (cid:1) p q ˆ rB T νδ ( t ) T ( I N ⊗ P ) δ ( t ) . This leads to ˙ V ( t ) ≤ − π ∗ µ p q ˆ rB T νλ min (cid:0) L ∪ + L T ∪ (cid:1) V ( t ) . Thus, V ( t ) exponentially converges to with the speed π ∗ µ √ q ˆ rB T νλ min (cid:0) L ∪ + L T ∪ (cid:1) , i.e., the mean-square consen-sus is achieved with the speed specified in (42).VI. N UMERICAL E XAMPLE
To illustrate the proposed approach, let us consider a simpleMAS composing of distinct SISO affine nonlinear agentsdescribed by the following dynamics, • Agent : f ( x ) = [ − x , − x , + x , , x , , T , g ( x ) = [0 , , T , h ( x ) = x , . • Agent : f ( x ) = [ − x , − x , + x , , x , , T , g ( x ) = [0 , , T , h ( x ) = x , . • Agent : f ( x ) = [ x , x , + x , x , , x , + x , , x , + x , x , ] T , g ( x ) = [1 , , T , h ( x ) = x , . • Agent : f ( x ) = [ − x , − x , + x , , x , , T , g ( x ) = [0 , , T , h ( x ) = x , . • Agent : f ( x ) = [ − x , − x , + x , , x , , T , g ( x ) = [0 , , T , h ( x ) = x , .It can be verified that the relative degrees of agents , , , are and of agent is , then the maximum relative degreeof agents is and hence by following the consensus design inSection III-A, agents , , , will be equipped with dynamicconsensus controllers whereas agent will be incorporatedwith a static consensus controller. For agent i, i ∈ { , , , } : α i ( ξ i , η i ) = 0 , β i ( ξ i , η i ) = 1 .The matrices of local dynamic controller is: D i = (cid:2) − b (cid:3) ; E i = − b ; G i = 1 ; H i = 1 /β i ( ξ i , η i ) . For agent : α ( ξ , η ) = x , ( x , + x , ) + (6 x , +3 x , )( x , + x , ) + 3 x , ( x , + x , x , ) , β ( ξ , η ) = 1 . A. Fixed Directed Topology
The communication topology among agents and the associ-ated Laplacian matrix in this case are presented in Figure 3. L = − − − − − Fig. 3. Communication structure in the MAS.
Consequently, we would like to demonstrate the distributedminimum rank, i.e., rank- consensus controller in Theorem2 that shows the advanced features of our proposed approach.Let us choose b = 2 and b = 3 then matrix A ofhomogeneous linearized system has only one zero eigenvaluewith the associated left eigenvector ν T = (cid:2) (cid:3) .First, we attempt to verify the coupling strength µ to theconsensus of agents by selecting ˆ r = 1 , q = 1 and varying µ . The simulation results for the rank- consensus controllerin this case are displayed in Figure 4. It can be seen that evenwhen µ is very small, the consensus among agents is stillachieved. This confirms our claim on the non-conservativedesign of arbitrarily small but positive coupling strength.Furthermore, the consensus speed is slower as µ is smaller.That is because it solely depends on µ when µ is varied andsmall, which is deduced from the expression of consensusspeed (19). O u t pu t µ =0.1 Time [s] µ =0.05 µ =0.5 Fig. 4. Consensus of the given nonlinear MAS by a distributed rank- controller with non-conservative coupling strength µ . Next, we would like to check the effects of the parameters q and ˆ r to the consensus speed. Since the roles of q and ˆ r are similar in the consensus speed formula (19), let us choose ˆ r = 1 , µ = 1 , and change q . Then we can observe from simulation results in Figure 5 that the consensus speeds ofagents as q = 10 and q = 10 are similar and are fasterthan when q = 1 . This is explained by the consensus speeddetermined in (19) as follows. We have λ min ( − A ) = 2 while µ √ q ˆ rB T ν Re( λ min ( L )) = 1 . √ q , and hence when q = 1 the consensus speed is equal to . √ q = 1 . , but when q =10 or q = 100 the consensus speed is equal to λ min ( − A ) = 2 which is independent of q . q =100 O u t pu t q =10 Time [s] q =1 Fig. 5. Consensus of the given nonlinear MAS by a distributed rank- controller as q is changed. B. Switching Directed Topology
Here we assume that the communication topology amongagents is randomly switched between two directed graphs G and G shown in Figure 6, where the random processis described by a continuous-time Markov chain with gen-erator matrix Q = (cid:20) − − (cid:21) and the invariant distribution π = [ , ] . It can be seen that neither G nor G is balancedand none of them has a spanning tree, but their union graph G ∪ is balanced and has a spanning tree. Next, the parameters ofthe distributed rank- consensus controllers are ˆ r = 1 , µ = 1 , q = 1 . Then the simulation result is exhibited in Figure 7. Wecan observe that the outputs of nonlinear agents still reach aconsensus in spite of the switching topology. This confirmsour result in Theorem 4.
241 35 241 35 G G
241 35 G ∪ Fig. 6. Switching topologies of the given nonlinear MAS.
VII. C
ONCLUSIONS AND D ISCUSSIONS
This article has proposed a systematic framework to designdistributed dynamic rank- consensus controllers for a fairlygeneral class of heterogeneous MIMO nonlinear MASs sub-jected to fixed and randomly switching directed topologies.The framework has been developed based on the input-output Time [s] O u t pu t Fig. 7. Consensus of the given nonlinear MAS under a randomly switchingtopology by a distributed rank- controller. feedback linearization and LQR methods with the followingappealing properties. First, distributed dynamic consensus con-trollers are derived for heterogeneous MIMO nonlinear MASswith arbitrary vector relative degree . Second, the couplingstrength in the consensus controller can be arbitrarily smallbut positive which allows us to achieve consensus with anyspeed. Third, the dynamic consensus controller has minimumrank , i.e., rank- which is very computationally efficient. Andlast, the proposed design works well under randomly switchingtopologies where the switched graphs are unnecessary to bebalanced , which greatly relaxes the assumptions on switchingtopologies.The current results can be further developed in several direc-tions that are worth investigating. One issue is the robustnessand adaptability of the consensus controller in the presenceof time delays, unmeasured disturbances or noises, and modeluncertainties. Another direction is to design dynamic consen-sus controllers for heterogeneous nonlinear MASs under someconstraints for control inputs or state flows.A CKNOWLEDGMENT
The author would like to sincerely thank the anonymousreviewers for their valuable comments and suggestions thatsignificantly help to improving the quality of the paper.R
EFERENCES[1] A. Jadbabaie, J. Lin, and A. Morse, “Coordination of groups of mobileautonomous agents using nearest neighbor rules,”
IEEE Transaction onAutomatic Control , vol. 48(6), pp. 988–1001, 2003.[2] R. Olfati-Saber and R. M. Murray, “Consensus problems in networks ofagents with switching topology and time-delays,”
IEEE Transaction onAutomatic Control , vol. 49(9), pp. 1520–1533, 2004.[3] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooper-ation in networked multi-agent systems,”
Proceeding of the IEEE , vol.95(1), pp. 215–233, 2007.[4] W. Ren, R. W. Beard, and E. M. Atkins, “Information consensus inmultivehicle cooperative control,”
IEEE Control Systems Magazine , vol.27(2), pp. 71–82, 2007.[5] F. Xiao and L. Wang, “Consensus problems for high-dimensional multi-agent systems,”
IET Control Theory and Applications , vol. 1(3), pp.830–837, 2007.[6] N. Chopra and M. W. Spong, “Passivity-based control of multi-agentsystems,” in
Advances in Robot Control: From Everyday Physics toHuman-Like Movements . S. Kawamura and M. Svinin, Eds. New York:Springer-Verlag, 2006, pp. 107–134. [7] Y. Su and J. Huang, “Cooperative global robust output regulation fornonlinear uncertain multi-agent systems in lower triangular form,”
IEEETransactions on Automatic Control , vol. 60(9), pp. 2378–2389, 2015.[8] Y. Cao and W. Ren, “Optimal linear-consensus algorithms: An LQRperspective,”
IEEE Transactions on Systems, Man, and Cybernetics-PartB: Cybernetics , vol. 40(3), pp. 819–830, 2010.[9] H. Zhang, F. L. Lewis, and A. Das, “Optimal design for synchronizationof cooperative systems: State feedback, observer and output feedback,”
IEEE Transactions on Automatic Control , vol. 56(8), pp. 1948–1952,2011.[10] D. H. Nguyen and S. Hara, “Hierarchical decentralized stabilization fornetworked dynamical systems by LQR selective pole shift,” in
Proc. of19th IFAC World Congress , 2014, pp. 5778–5783.[11] D. H. Nguyen, “Reduced-order distributed consensus controller designvia edge dynamics,”
IEEE Transactions on Automatic Control , (Condi-tionally accepted). Available online at: http://arxiv.org/abs/1508.06346.[12] N. Chopra, “Output synchronization on strongly connected graphs,”
IEEE Transactions on Automatic Control , vol. 57(11), pp. 2896–2901,2012.[13] G.-B. Stan and R. Sepulchre, “Analysis of interconnected oscillatorsby dissipativity theory,”
IEEE Transactions on Automatic Control , vol.52(2), pp. 256–270, 2007.[14] U. Münz, A. Papachristodoulou, and F. Allgöwer, “Robust consensuscontroller design for nonlinear relative degree two multi-agent systemswith communication constraints,”
IEEE Transactions on AutomaticControl , vol. 56(1), pp. 145–161, 2011.[15] C. D. Persis and B. Jayawardhana, “On the internal model principle inthe coordination of nonlinear systems,”
IEEE Transactions on Controlof Network Systems , vol. 1(3), pp. 272–282, 2014.[16] F. D. Priscoli, A. Isidori, L. Marconi, and A. Pietrabissa, “Leader-following coordination of nonlinear agents under time-varying commu-nication topologies,”
IEEE Transactions on Control of Network Systems ,vol. 2(4), pp. 393–405, 2015.[17] W. Yu, G. Chen, and M. Cao, “Consensus in directed networks of agentswith nonlinear dynamics,”
IEEE Transactions on Automatic Control , vol.56(6), pp. 1436–1441, 2011.[18] Z. Ding, “Consensus output regulation of a class of heterogeneousnonlinear systems,”
IEEE Transactions on Automatic Control , vol.58(10), pp. 2648–2653, 2013.[19] W. Ren and R. W. Beard, “Consensus seeking in multi-agent systemsunder dynamically changing interaction topologies,”
IEEE Transactionon Automatic Control , vol. 50(5), pp. 655–661, 2005.[20] Z. Li, Z. Duan, G. Chen, and L. Huang, “Consensus of multiagentsystems and synchronization of complex networks: A unified viewpoint,”
IEEE Transaction on Circuits and Systems-I: Regular Papers , vol. 57(1),pp. 213–224, 2010.[21] C.-Q. Ma and J.-F. Zhang, “Necessary and sufficient conditions forconsensusability of linear multi-agent systems,”
IEEE Transactions onAutomatic Control , vol. 55(5), pp. 1263–1268, 2010.[22] B. D. O. Anderson and J. B. Moore,
Optimal Control: Linear QuadraticMethods . Englewood Cliffs, NJ: Prentice Hall, 1990.[23] H. Kim, H. Shim, J. Back, and J. H. Seo, “Consensus of output-coupledlinear multi-agent systems under fast switching network: Averagingapproach,”
Automatica , vol. 49, pp. 267–272, 2013.[24] K. You, Z. Li, and L. Xie, “Consensus condition for linear multi-agentsystems over randomly switching topologies,”
Automatica , vol. 49, pp.3125–3132, 2013.[25] Z. Li, G. Wen, Z. Duan, and W. Ren, “Designing fully distributedconsensus protocols for linear multi-agent systems with directed graphs,”
IEEE Transaction on Automatic Control , vol. 60(4), pp. 1152–1157,2015.[26] A. Isidori and A. J. Krener, “On feedback equivalence of nonlinearsystems,”
Systems and Control Letters , vol. 2, pp. 118–121, 1982.[27] E. D. Sontag, “Input to state stability: Basic concepts and results,” in
Nonlinear and Optimal Control Theory . Springer, 2006, pp. 163–220.[28] K. H. Movric and F. L. Lewis, “Cooperative optimal control for multi-agent systems on directed graph topologies,”
IEEE Transactions onAutomatic Control , vol. 59(3), pp. 769–774, 2014.[29] D. H. Nguyen, “A sub-optimal consensus design for multi-agent systemsbased on hierarchical LQR,”
Automatica , vol. 55, pp. 88–94, 2015.[30] M. D. Fragoso and O. L. V. Costa, “A unified approach for stochastic andmean square stability of continuous-time linear systems with markovianjumping parameters and additive disturbances,”