Robust Consensus of Linear Multi-Agent Systems under Input Constraints or Uncertainties
aa r X i v : . [ m a t h . O C ] M a y Robust Consensus of Linear Multi-Agent Systems underInput Constraints or Uncertainties ⋆ Dinh Hoa Nguyen, Tatsuo Narikiyo, Michihiro Kawanishi,
Control System Laboratory, Department of Advanced Science and Technology, Toyota Technological Institute, 2-12-1Hisakata, Tempaku-ku, Nagoya 468-8511, Japan.
Abstract
This paper proposes a new approach to analyze and synthesize robust consensus control laws for general linear leaderless multi-agent systems (MASs) subjected to input constraints or uncertainties. First, the MAS under input constraints or uncertainties isreformulated as a network of Lur’e systems. Next, two scenarios of communication topology are considered, namely undirectedand directed cyclic structures. In each case, a sufficient condition for consensus and the design of consensus controller gainare derived from solutions of a distributed LMI convex problem. Finally, a numerical example is introduced to illustrate theeffectiveness of the proposed theoretical approach.
Key words:
Multi-agent Systems; Consensus; Robustness; Input Constraints; Input Uncertainties; Lur’e Networks;Distributed LMI.
Multi-agent systems (MASs) and their cooperative con-trol problems have gained much attention since there area lot of practical applications, e.g., power grids, wirelesssensor networks, transportation networks, systems biol-ogy, etc., can be formulated, analyzed and synthesizedunder the framework of MASs. A key feature in MASsis the achievement of a global objective by performinglocal measurement and control at each agent and simul-taneously collaborating among agents using that localinformation. One of the most important and intensivelyinvestigated issues in MASs (and their applications) isthe consensus problem due to its attraction in both the-oretical and applied aspects [12–14].Since all real control systems are subjected to physicalconstraints on their inputs or states, MASs are not ex-ception. Therefore, the MAS consensus under input orstate constraints is a significant and realistic problem.Likewise, the MAS consensus in presence of input orstate uncertainties is realistic and worth studying. Prac-tical examples include consensus of vehicles with lim- ⋆ This paper was not presented at any IFAC meeting. Cor-responding author: D. H. Nguyen.
Email addresses: [email protected] (Dinh HoaNguyen), [email protected] (Tatsuo Narikiyo), [email protected] (Michihiro Kawanishi). ited speeds and limited working space, smart buildingsenergy control with temperature and humidity are re-quired in specific ranges, just to name a few. However,most of the early researches on MASs were not aware ofthose practical issues, and it has not been until recentlythat some studies have considered the cooperative con-trol of MASs in presence of input or state constraints oneach agent [4–6, 8, 9, 15, 16, 18–20].A constrained consensus problem was investigated in [9]where the states of agents are required to lie in individ-ual closed convex sets and the final consensus state mustbelong to the non-empty intersection of those sets. Ac-cordingly, a projected consensus algorithm was proposedand then applied to distributed optimization problems.Following this research line, [5] extended the result in [9]to the context where communication delays exist. In an-other work, [4] studied the state increment by utilizingthe model predictive control (MPC) method. In fact,using the MPC framework we can also incorporate in-put or state constraints, however the computational costcould be high. Therefore, distributed and fast MPC al-gorithms need to be developed to fit into the context oflarge-scale MASs. Another direction to deal with inputor state constraints is to employ the so-called discardedconsensus algorithms [6, 18]. Nevertheless, a disadvan-tage of these approaches as well as in [5, 9] is that theinitial states of agents must belong to some sets speci-fied by the constraints, or in other words the consensus
Preprint submitted to Automatica 7 November 2018 s only local. Moreover, only agents with single integra-tor dynamics were considered in [6, 18].To achieve the global or semi-global consensus in pres-ence of input or state constraints, some consensus lawswere presented in [8, 15], but they were only for leader-follower MASs. Another way to tackle the input or stateconstraints to derive global consensus is to reformu-late the constrained MAS as a network of Lur’e sys-tems [16,19,20]. The paper [16] considered linear agentswith bounded-constraint inputs and obtained a sufficientcondition for global consensus, but agents is limited tobe single-input and the network is undirected. Next, [20]and [19] investigated consensus problems where outputsof agents are incrementally bounded or passive, withdirected and undirected topologies. Consequently, suffi-cient conditions for global consensus were derived in theform of LMI convex problems.Following the ideas of achieving global consensus by re-formulating the considered MAS as a network of Lur’esystems in [16, 19, 20], this paper proposes a new ap-proach to design robust consensus controllers for gen-eral linear homogeneous leaderless MASs in presence ofinput constraints or uncertainties. The contributions ofthis paper are threefold. First, the proposed approach isapplicable for leaderless MASs with general linear dy-namics of agents, and the class of nonlinearities inducedby the input constraints or uncertainties is broader thanthose in [16, 19, 20]. Second, the consensus controllergain is computed from the solution of a distributed low-dimension convex problem with LMI constraints. Third,the proposed approach can be used for global consen-sus analysis and synthesis under both scenarios of undi-rected networks and a special class of directed networks.The following notation and symbols will be used in thepaper. R and C stand for the real and complex sets, and j denotes the complex unit. Moreover, n denotes the n × I n de-notes the n × n identity matrix. Next, ⊗ stands for theKronecker product, diag {} denotes diagonal or block-diagonal matrices, and sym( A ) denotes A + A T for anyreal matrix A . Lastly, ≻ and (cid:23) denote the positive defi-niteness and positive semi-definiteness of a matrix, andsimilar meanings are used for ≺ and (cid:22) . Denote ( G , V , E ) the graph representing the informationstructure in an MAS composing of N agents, where eachnode in G stands for an agent and each edge in G repre-sents 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 elementsof the adjacency matrix A of G , i.e., a ij > e ij ∈ E and a ij = 0 if e ij / ∈ E . The in-degree of a vertex i is de-noted by deg in i , P Nj =1 a ij , then the in-degree matrix of G is denoted by D = diag { deg in i } i =1 ,...,N . Consequently,the Laplacian matrix L associated to G is defined by L = D − A . The out-degree of a vertex i is denotedby deg out i , P Nj =1 a ji . Then G is said to be balanced ifdeg in i = deg out i ∀ i = 1 , . . . , N. A directed path connect-ing vertices i and j in G is a set of consecutive edgesstarting from i and stopping at j . Then G is said to havea spanning tree if there exists a node called root nodefrom which there are directed paths to every other node. G is undirected if and only if a ij = a ji ∀ i, j = 1 , . . . , N . Consider a MAS including of N identical agents withthe following linear dynamics˙ x i = Ax i + Bu i , i = 1 , . . . , N, (1)where x i ∈ R n is the state vector, u i ∈ R m is the controlinput, A ∈ R n × n , B ∈ R n × m . The following assump-tions will be employed. A1: ( A, B ) is stabilizable.
A2:
All eigenvalues of A is on the closed left half complexplane. A3: G is balanced and contains a spanning tree.Assumptions A1-A2 are necessary and sufficient suchthat the consensus can be achieved and stable (see e.g.[7]). And assumption A3 implies that G is connected ifit is undirected.Denote x = (cid:2) x T , . . . , x TN (cid:3) T , u = (cid:2) u T , . . . , u TN (cid:3) T . Thewhole MAS at the initial state is then described by˙ x = A x + B u. (2)It has recently been proved in our previous research [10]that without any further requirement on the control in-put or agents’ states, the MAS (2) can reach consensusin the sense of (5) by a control law in the following form, u = ( L ⊗ K ) x, (3)with a properly synthesized K . Nevertheless, in real ap-plications the inputs of agents are usually bounded insome certain ranges due to physical limitations of agents,and may contain some uncertainties because of uncer-tain communication links. As a result, the control law (3)can no longer guarantee the consensus of agents. There-fore, to take into account the aforementioned practicalissues, we will consider in this research the following con-trol scenario:2 Input constraint/uncertainty:
For all i ∈ [1 , N ], u i,k = f i,k ( z i,k ) ∀ k = 1 , . . . , m where z i,k ∈ R is theaggregated signal that the k th input of agent i re-ceived; f i,k : R → R is a continuous function thatsatisfies the following sector-bounded condition:( f i,k ( z i,k ) − δ k, z i,k )( f i,k ( z i,k ) − δ k, z i,k ) ≤ , ∀ k = 1 , . . . , m ; ∀ i = 1 , . . . , N, (4)where δ k, , δ k, ∈ R are known constants, δ k, < δ k, .Consequently, in presence of input constraints or uncer-tainties described above, each agent try to collaboratewith others to achieve a consensus defined as follows. Definition 1
The MAS with linear dynamics of agentsrepresented by (1) and the information exchange amongagents represented by G is said to reach a consensus if lim t →∞ k x i ( t ) − x j ( t ) k = 0 ∀ i, j = 1 , . . . , N. (5)Next, we introduce the control design problem investi-gated in this paper. • Design problem (Robust consensus under inputconstraints or uncertainties):
For the given linearMASs with dynamics of agents represented by (1) andthe information exchange among agents representedby G , find a control strategy to achieve consensus ofagents in the sense of (5) under the input constraints oruncertainties (4), for any initial conditions of agents. Under the input constraints or uncertainties (4), we pro-pose to use the following consensus control law u = f (( L ⊗ K ) x ) , (6)where f ( y ) , [ f T ( y ) , . . . , f TN ( y N )] T , f i ( y i ) , [ f Ti, ( y i, ), . . . , f Ti,N ( y i,N )] T , ∀ y = [ y T , . . . , y TN ] T . Then the MAS(2) with this control strategy can be rewritten in thefollowing form ˙ x = A x + B u,z = ( L ⊗ K ) x,u = f ( z ) , (7)which can be seen as a network of Lur’e systems. Notethat this Lur’e network is different from that in [19, 20]and the nonlinearity is more general.The following theorem presents a sufficient condition forrobust consensus under input constraints or uncertain-ties and how to design the consensus controller gain K . Theorem 1
The robust consensus is achieved for theMAS (2) with an undirected communication graph bythe control law (6) if there exist matrices X ∈ R n × n , Y ∈ R m × n and Z ∈ R m × m such that the following LMIproblem is feasible with a given ǫ > , " sym( AX + λ B ∆ Y ) + ǫX BZ + λ Y T (∆ − ∆ ) (cid:0) BZ + λ Y T (∆ − ∆ ) (cid:1) T − Z (cid:22) , " sym( AX + λ N B ∆ Y ) + ǫX BZ + λ N Y T (∆ − ∆ ) (cid:0) BZ + λ N Y T (∆ − ∆ ) (cid:1) T − Z (cid:22) ,X ≻ ,Z ≻ , Z is diagonal , (8) where ∆ = diag { δ k, } k =1 ,...,m , ∆ = diag { δ k, } k =1 ,...,m , λ ≤ λ ≤ · · · ≤ λ N are non-zero eigenvalues of L .Furthermore, the controller gain K is calculated by K = Y X − . (9) PROOF.
Consider a Lyapunov function V ( x ) = x T P x where P , P ⊗ P , P ∈ R N , P (cid:23) P ∈ R n , P ≻ V ( x ) gives us˙ V ( x ) = x T (cid:0) PA + A T P (cid:1) x + 2 x T PB u. Hence, for all ǫ > V ( x ) + ǫV ( x ) = x T (cid:0) PA + A T P + ǫ P (cid:1) x + 2 x T PB u. We now seek P such that ˙ V ( x ) + ǫV ( x ) ≤ P exists ifthere exist γ , , . . . , γ ,m , . . . , γ N, , . . . , γ N,m which arenon-negative such that˙ V ( x ) + ǫV ( x ) − N X i =1 m X k =1 γ i,k ( u i,k − δ k, z i,k )( u i,k − δ k, z i,k ) ≤ , (10)Let γ i,k = γ k > ∀ i = 1 , . . . , N and Γ = diag { γ k } k =1 ,...,m ,then (10) is satisfied if˙ V ( x ) + ǫV ( x ) − N X i =1 ( u i − ∆ z i ) T Γ( u i − ∆ z i ) ≤ , ⇔ " xu T " P P P T P xu (cid:22) ⇔ " P P P T P (cid:22) , (11)where P = PA + A T P + ǫ P − ( L T L ) ⊗ ( K T Γ∆ ∆ K ), P = PB + L T ⊗ ( K T Γ(∆ + ∆ )), P = − I N ⊗ Γ.Subsequently, employing Schur complement [1] to (11)3 sym( AX + B ˜∆ i Y ) + ǫX BZ + Y T ˆ∆ i − Y i Z T B T + ˆ∆ i Y − Z Y i AX + B ˜∆ i Y ) + ǫX BZ + Y T ˆ∆ i Z T B T + ˆ∆ i Y − Z (cid:22) , ∀ i = 2 , . . . , N. (12)results in P − P P − P T (cid:22)
0, which is equivalent to P ⊗ ( A T P + P A + ǫP ) + P ⊗ ( P B Γ − B T P )+ sym( 12 ( P L ) ⊗ [ P B (∆ + ∆ ) K ])+ 14 ( L T L ) ⊗ [ K T Γ(∆ − ∆ ) K ] (cid:22) . (13)Let us choose P = I N − N N TN then we can easily showthat P = P , L T P = L T , and P L = L for balancedgraphs. Therefore, (13) is equivalent to P ⊗ ( A T P + P A + ǫP + P B Γ − B T P )+ 14 ( L T L ) ⊗ [ K T Γ(∆ − ∆ ) K ]+ sym( 12 L ⊗ [ P B (∆ + ∆ ) K ]) (cid:22) . (14)Next, denote X , P − and multiply I N ⊗ X both to theleft and to the right of (14), we obtain P ⊗ ( XA T + AX + ǫX + B Γ − B T )+ 14 ( L T L ) ⊗ [ Y T Γ(∆ − ∆ ) Y ]+ sym( 12 L ⊗ [ B (∆ + ∆ ) Y ]) (cid:22) , (15)where Y , K X . For undirected graph G , L = L T , so letus denote U ∈ R N × N the orthogonal matrix that diago-nalizes L . Accordingly, applying a congruence transfor-mation with U ⊗ I n to (15) gives us XA T + AX + ǫX + B Γ − B T + 14 λ i Y T Γ(∆ − ∆ ) Y + sym( 12 λ i B (∆ + ∆ ) Y ) (cid:22) , (16)for all i = 2 , . . . , N , since U T P U = diag { , , . . . , } , U T L U = diag { , λ , . . . , λ N } , and U T L U = diag { , λ ,. . . , λ N } . By some simple mathematical manipulations,we can rewrite (16) as follows, XA T + AX + ǫX + λ i ( Y T ∆ B T + B ∆ Y )+ ( BZ + 12 λ i Y T (∆ − ∆ )) Z − ( ZB T + 12 λ i (∆ − ∆ ) Y ) (cid:22) , (17) where Z , Γ − . Then using Schur complement againwith (17) and noting that λ i , i = 3 , . . . , N − λ and λ N since λ ≤ λ , . . . , λ N − ≤ λ N , we obtain (8). Next, we haveseen that (16) implies (10) and hence˙ V ( x ) + ǫV ( x ) ≤ N X i =1 m X k =1 γ i,k ( u i,k − δ k, z i,k )( u i,k − δ k, z i,k ) ≤ , ⇒ ˙ V ( x ) ≤ − ǫV ( x ) ≤ . (18)Therefore, from Lasalle’s invariance principle wecan conclude that ξ globally exponentially con-verges to the largest invariance set contained in n x ∈ R nN (cid:12)(cid:12) ˙ V ( x ) = 0 o for any initial condition. Fur-thermore, it can be seen from (18) that ˙ V ( x ) = 0 if andonly if V ( x ) = 0 which is equivalent to x = N ⊗ ¯ x ,¯ x ∈ R n , i.e., the consensus is achieved. Remark 2
When the inputs of agents are subjected toboundedness, f i becomes the vector-valued saturationfunctions and hence δ k, = 0 , δ k, = 1 ∀ k = 1 , . . . , m .This particular case was investigated in [11] by a differ-ent control design. The method presented in this paperis more general and is applicable for more contexts thanthe one in [11]. Remark 3
Recently, there are several existing re-searches, e.g. [2], [17], which propose different distributedmethods to approximate the whole eigen-spectrum of theLaplacian matrix. These methods can be employed toestimate λ and λ N before solving the LMI problem (8).As a result, we can solve (8) in a distributed fashion. Remark 4
The results in [16] can be considered as aspecial case of our result in Theorem 1 with single-inputagents, input saturation, and Z = I m . Our result aremuch more general with the following properties: (i) itsrobustness to any constraint or uncertainty specified by(4); (ii) its applicability for leaderless MASs with generallinear dynamics of agents; (iii) an additional variable Z is introduced in the LMI problem (8), which makes theLMIs less restrictive (cf. identity matrix in LMI problems(8) and (12) in [16]); (iv) the term ǫX in the upper-left blocks of the matrices in the LMI problem (8) makesthe consensus speed faster since ˙ V ( x ) is exponentially onverged instead of being asymptotically converged asin [16]. Next, we present a design for directed networks in thefollowing theorem.
Theorem 5
The robust consensus is achieved for theMAS (2) with a directed cyclic unweighted communi-cation graph by the control law (6) if there exist matrices X ∈ R n × n , Y ∈ R m × n and Z ∈ R m × m such that theLMI problem (12) is feasible with a given ǫ > , wherefor all i = 2 , . . . , N , Y i , −
12 sin 2 π ( i − N sym( B (∆ + ∆ ) Y ) , ˆ∆ i , s (cid:18) − cos 2 π ( i − N (cid:19) (∆ − ∆ ) , ˜∆ i , (cid:18) − cos 2 π ( i − N (cid:19) (∆ + ∆ ) − ˆ∆ i . (19) Accordingly, the controller gain K is calculated by (9). PROOF.
Here, we employ the same Lyapunov functionas in the proof of Theorem 1, so all the steps until Eq.(15) are also applied for this scenario. Afterward, we notethat L is a circulant matrix since G is an unweighteddirected cyclic graph. Therefore, the sets of eigenvectorsof L T L , L , and L T are the same. Denote V ∈ R N × N the unitary matrix whose columns are eigenvectors of L and Λ , diag { , λ , . . . , λ N } . Consequently, we have L = V Λ V ∗ , L T = V Λ ∗ V ∗ , L T L = V Λ ∗ Λ V ∗ , and P = V diag { , , . . . , } V ∗ .Let λ i,r and λ i,ℓ be the real and imaginary parts of λ i , i = 2 , . . . , N , respectively. Then applying a congru-ence transformation with V ∗ ⊗ I n to (15) gives us XA T + AX + ǫX + 12 ( λ i,r + jλ i,ℓ ) B (∆ + ∆ ) Y + B Γ − B T + 12 ( λ i,r − jλ i,ℓ ) Y T (∆ + ∆ ) B T + 14 ( λ i,r + λ i,ℓ ) Y T Γ(∆ − ∆ ) Y (cid:22) . (20)Furthermore, we have λ i,r = 1 − cos π ( i − N and λ i,ℓ = − sin π ( i − N since L is a circulant matrix with the fol-lowing form L = − · · · − · · · · · · − − · · · . Denote X i , XA T + AX + ǫX + B Γ − B T + λ i,r sym( B (∆ +∆ ) Y ) + ( λ i,r + λ i,ℓ ) Y T Γ(∆ − ∆ ) Y . Subsequently,(20) is equivalent to " X i − Y i Y i X i (cid:22) . On the other hand,using Schur complement for X i , we obtain X i (cid:22) ⇔ " sym( AX + B ˜∆ Y ) + ǫX BZ + Y T ˆ∆ i Z T B T + ˆ∆ i Y − Z (cid:22) , (21)where Z , Γ − . As a result, we obtain (12). Consider a building temperature control problem wherethe target is to make the temperatures of different roomsbe identical by allowing the exchange of their tempera-tures through a communication network. For simplicity,the dynamics of each room can be described by a first-order transfer function aT s +1 , a > , T > , where thedelays of the heating or cooling processes are ignored.Consequently, each room is equipped with an integra-tor controller for the consensusability, i.e., the model ofeach agent is as ( T s +1) . Then we illustrate the input-constrained consensus de-sign in Theorem 1 and Theorem 5 with a simulation ofwhich a = 10 , T = 50, N = 3, and agents’ inputs arebounded in [ − . , . L are { , , } . Using ǫ = 0 . CVX [3], we obtain K = [ − . , − . ↔ ↔ { , , } . Resolving the LMI problem (8) using CVX [3] gives us K = [ − . , − .
50 100 150 200 250 300 350 400 450 50022232425262728
Time [s] T e m pe r a t u r e [ o C ] Time [s] C on t r o l i npu t s Fig. 1. Consensus of temperatures under input constraintwith all-to-all undirected communication structure.
Time [s] T e m pe r a t u r e [ o C ] Time [s] C on t r o l i npu t s Fig. 2. Consensus of temperatures under input constraintwith an undirected communication structure. graph. In this case, we obtain K = [ − . , − . CVX [3]. Thenthe simulation results are displayed in Figure 3. We canobserve that the temperature of all rooms reach consen-sus despite the presence of the bounded input constraintand the directed communication topology. Furthermore,the consensus value, consensus speed, and the transientresponses of rooms’ temperatures are different from theprevious cases of undirected communication structure.
In this section, we assume that the control inputs ofagents contain some uncertainties which may be due tothe uncertain communication links. More specifically, δ k, and δ k, are assumed to be 0 . .
2, respectively.This means the inputs of agents are multiplied with un-certain parameters K , K , K ∈ [0 . , . ↔ ↔ CVX [3] to obtain K = [ − . , − . Time [s] T e m pe r a t u r e [ o C ] Time [s] C on t r o l i npu t s Fig. 3. Consensus of temperatures under input constraintwith directed cyclic communication structure. agents’ inputs in the interval [0 . , . K = 1 . K = 1 . K = 0 . Time [s] T e m pe r a t u r e [ o C ] Time [s] C on t r o l i npu t s Fig. 4. Consensus of temperatures under uncertain inputwith an undirected communication structure.
Next, we investigate another context where the agentsare interconnected through a directed cyclic graph. Inthis situation, solving the LMI problem (12) using
CVX [3]gives us K = [ − . , − . . , . K , K , K as aboveare displayed in Figure 5 for comparison.Overall, we can conclude that the consensus of agentsunder input constraints or uncertainties depends on theinterconnection structure among agents. This is obvioussince the communication topology affects to the eigen-spectrum of the Laplacian matrix L which directly in-fluences the solutions of the LMI problems (8) and (12)and hence the consensus controller gain K . It is appar-ently different from the circumstance of consensus with-out any constraints or uncertainties since the consensusvalue is the average of initial conditions of agents inter-connected by a connected undirected graph regardlessof its structure.6
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Time [s] T e m pe r a t u r e [ o C ] Time [s] C on t r o l i npu t s Fig. 5. Consensus of temperatures under uncertain inputwith directed cyclic communication structure.
A new approach has been proposed in this paper to ana-lyze and synthesize robust consensus controllers for lin-ear leaderless MASs subjected to input constraints oruncertainties. The remarkable features of this approachare as follows. First, it is available for leaderless MASswith general linear dynamics of agents unlike the exist-ing results for special cases of single integrator or single-input agents. Second, the robust consensus design un-der sector-bounded input constraints or uncertainties isderived in the form of a distributed low-dimension LMIproblem which can be efficiently solved by off-the-shelfoptimization software. Third, the proposed approachcan deal with for both undirected and a special class ofdirected networks.The next researches would study more general classes ofdirected networks and take into account other practicalissues such as time delays, disturbances, etc., togetherwith the considered constraints and uncertainties.
ACKNOWLEDGMENT
This research is partially supported by Hitech ResearchCenter, projects for private universities, supplied fromthe Ministry of Education, Culture, Sports, Science andTechnology, Japan.
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