Robust Consensus Analysis and Design under Relative State Constraints or Uncertainties
aa r X i v : . [ m a t h . O C ] M a y Robust Consensus Analysis and Design underRelative State Constraints or Uncertainties
Dinh Hoa Nguyen,
Member, IEEE,
Tatsuo Narikiyo,and Michihiro Kawanishi,
Member, IEEE
Abstract —This paper proposes a new approach to analyzeand design distributed robust consensus control protocols forgeneral linear leaderless multi-agent systems (MASs) in presenceof relative-state constraints or uncertainties. First, we show thatthe MAS robust consensus under relative-state constraints oruncertainties is equivalent to the robust stability under stateconstraints or uncertainties of a transformed MAS. Next, thetransformed MAS under state constraints or uncertainties isreformulated as a network of Lur’e systems. By employing S-procedure, Lyapunov theory, and Lasalle’s invariance principle,a sufficient condition for robust consensus and the design ofrobust consensus controller gain are derived from solutions of adistributed LMI convex problem. Finally, numerical examplesare introduced to illustrate the effectiveness of the proposedtheoretical approach.
I. I
NTRODUCTION
Multi-agent systems (MASs) and their cooperative controlproblems have been extensively studied and applied to manypractical systems, e.g., power grids, wireless sensor networks,transportation networks, systems biology, etc., because of theirkey advantage of achieving global objectives by performinglocal measurements and controls at each agent and simultane-ously collaborating among agents using that local information.Among many interesting problems, consensus is one of themost important and intensively investigated issues in MASsdue to its attraction in both theory and applications [1]–[3].In practical MASs, agents’ inputs or states and the ex-changed information among agents are subjected to con-straints or uncertainties due to physical limitations of agentsor uncertain communication channels. Realistic examples areconsensus of vehicles with limited speeds and working space,smart buildings energy control with temperature and humidityare required in specific ranges, just to name a few. Therefore,the MAS consensus under constraints and uncertainties onthe inputs, states, or relative states of agents is a significant,realistic problem and is worth studying. However, this problemwas not investigated in the early researches on MASs and itjust has been considered in some recent studies [4]–[14].A constrained consensus problem was investigated in [4]where the states of agents are required to lie in individualclosed convex sets and the final consensus state must belongto the non-empty intersection of those sets. Accordingly, aprojected consensus algorithm was proposed and then appliedto distributed optimization problems. Following this researchline, [5] extended the result in [4] to the context wherecommunication delays exist. In another work, [6] studiedthe state increment by utilizing the model predictive control(MPC) method. However, distributed and fast MPC algorithms
The authors are with Control System Laboratory, Department ofAdvanced Science and Technology, Toyota Technological Institute,2-12-1 Hisakata, Tempaku-ku, Nagoya 468-8511, Japan. Emails:[email protected], [email protected], [email protected] need to be further developed in order to use in large-scaleMASs. Another direction is to employ the discarded consen-sus algorithms [7], [8]. Nevertheless, a requirement of theseapproaches as well as in [4], [5] is that the initial states ofagents must belong to some sets specified by the constraints,i.e., the consensus is only local. Moreover, only agents withsingle integrator dynamics were considered in [7], [8].To achieve the global or semi-global consensus in presenceof input or state constraints, some consensus laws werepresented in [9], [10], but they were only for leader-followerMASs. In other researches, [11]–[13] derived global consensusunder input or state constraints by reformulating the con-strained MAS as a network of Lur’e systems and employingLyapunov theory. The paper [11] considered linear agents withinput saturation but agents’ dynamics is limited to be single-input. Next, [12] and [13] investigated consensus problems forgeneral linear MASs where outputs of agents are incrementallybounded or passive and obtained sufficient conditions forglobal consensus in the form of LMI convex problems.On the other hand, the MAS consensus under relative-stateconstraints has been recently studied in [14] within a veryspecial context where the input matrices of agents are identitymatrices and the consensus controller gain is a diagonal matrix.Then sufficient conditions were proposed for the cases of -norm and ∞ -norm bounded constraints on relative statesof agents. Nevertheless, the consensus is only local and noconsensus controller design was given in [14].This paper proposes a new approach to analyze and design distributed robust consensus controllers for general linearhomogeneous leaderless MASs to achieve global consensus under relative-state constraints or uncertainties which arein the form of a sector-bounded condition . Our approachcovers broader systems and scenarios than those in the ex-isting researches, and hence constitutes our first contribution.Consequently, we further develop the edge dynamics proposedin [15] to achieve that the currently considered problem is equivalent to a distributed robust stabilization problem understate constraints or uncertainties for a transformed MAS .This serves as our second contribution. Next, the transformedMAS is rewritten as a network of Lur’e systems and therobust stabilization problem is formulated as a distributedconvex LMI problem. In comparison with the one in [11]for a similar type of Lur’e networks, our LMI problem isless conservative that: (i) employs a more general methodnamely the S-procedure; (ii) gives an exponential convergenceto consensus instead of asymptotic convergence. Furthermore,our consensus controller gain is much more general than thediagonal one in [14]. Those advantages clearly show our thirdcontribution.The following notation and symbols will be used in thepaper. R and C stand for the real and complex sets. Moreover, n denotes the n × vector with all elements equal to , and I n denotes the n × n identity matrix. Next, ⊗ stands for theKronecker product, diag {} denotes diagonal or block-diagonalmatrices, and sym( A ) denotes A + A T for any real matrix A .Lastly, ≻ and (cid:23) denote the positive definiteness and positivesemi-definiteness of a matrix, and similar meanings are usedfor ≺ and (cid:22) . II. P
ROBLEM D ESCRIPTION
Consider a MAS consisting of N identical agents with thefollowing 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 control input, A ∈ R n × n , B ∈ R n × m . The whole MAS is then describedby ˙ x = ( I N ⊗ A ) x + ( I N ⊗ B ) u, (2)where x = (cid:2) x T , . . . , x TN (cid:3) T , u = (cid:2) u T , . . . , u TN (cid:3) T . Let G be anundirected graph representing the information structure in theMAS, in which each node in G represents an agent and eachedge in G represents the interconnection between two agents.Denote L ∈ R N × N and E ∈ R N × M the Laplacian matrix andthe incidence matrix associated with G . Then L = EE T and E T N = 0 .The following assumptions will be employed. A1: ( A, B ) is stabilizable. A2:
All eigenvalues of A is on the closed left half complexplane. A3: G is undirected and connected.Assumptions A1–A2 are necessary and sufficient such that theconsensus can be achieved and stable (see e.g. [16]). Next, theconsensus of agents is defined as follows. Definition 1:
The MAS with linear dynamics of agentsrepresented by (1) and the information exchange among agentsrepresented by G is said to reach a consensus if lim t →∞ k x i ( t ) − x j ( t ) k = 0 ∀ i, j = 1 , . . . , N. (3)Due to physical limitations on the communication range andbandwidth of agents or uncertain information channels, theexchanged relative states among agents could be bounded orcontain some uncertainties. To take into account those practicalissues in the control analysis and design, we define in thefollowing a new state vector and a new control input, z , ( E T ⊗ I n ) x, w , ( E T ⊗ I m ) u. Let L † be the generalized pseudoinverse of the Laplacianmatrix L [17]. Then multiplying both sides of (2) with E T ⊗ I n gives us the following edge dynamics [15], ˙ z = [( E T L † E ) ⊗ A ] z + ( I M ⊗ B ) w. (4)Since z composes of all relative states of connected agents, (2)is consensus if this edge dynamics is stabilized. Moreover, allrelative-state constraints and uncertainties are now representedin term of z . Accordingly, the following control scenario shallbe investigated. • Relative-State Constraints/Uncertainties:
For all j ∈ [1 , M ] , y j,k = φ j,k ( z j,k ) ∀ k = 1 , . . . , n where y j,k ∈ R is the k th component of the signal exchanged throughthe edge j ; φ j,k : R → R is a continuous function thatsatisfies the following sector-bounded condition: ( φ j,k ( z j,k ) − σ k, z j,k )( φ j,k ( z j,k ) − σ k, z j,k ) ≤ ∀ k = 1 , . . . , n ; ∀ j = 1 , . . . , M, (5)where σ k, , σ k, ∈ R are known constants, σ k, < σ k, . Consequently, we present the control analysis and designproblem considered in this paper. • Global robust consensus under relative-state con-straints or uncertainties:
For the given linear MAS withdynamics of agents represented by (1) and the informa-tion exchange among agents represented by G , find acondition and a control strategy to achieve consensus ofagents in the sense of (3) subjected to the relative-stateconstraints or uncertainties (5), for any initial conditionsof agents.III. C ONSENSUS A NALYSIS AND D ESIGN UNDER R ELATIVE -S TATE C ONSTRAINTS OR U NCERTAINTIES
A. Equivalence to Robust Stabilization in Presence of StateConstraints or Uncertainties
Denote ¯ L = E T L † E and L e = E T E . Lemma 1: [15] The following statements hold.(i) L e has exactly N − non-zero eigenvalues, which areequal to positive eigenvalues of L while all other eigen-values of L e if exist are .(ii) ¯ L has exactly N − non-zero eigenvalues, which are allequal to , and other eigenvalues of ¯ L if exists are .Let U ∈ R M × M be an orthogonal matrix that diagonalizes ¯ L , and ˜ z , ( U T ⊗ I n ) z, ˜ w , ( U T ⊗ I m ) w. Subsequently, we obtain from (4) that ˙˜ z = (cid:0) ¯Γ ⊗ A (cid:1) ˜ z + ( I M ⊗ B ) ˜ w, (6)where ¯Γ = diag { , I N − } includes all eigenvalues of ¯ L in itsdiagonal (due to Lemma 1). Now, let us partition U , the stateand input vectors in (6) as follows, U = (cid:2) U U (cid:3) , ˜ z = (cid:20) ˜ z ˜ z (cid:21) , ˜ w = (cid:20) ˜ w ˜ w (cid:21) , (7)where U ∈ R M × ( M − N +1) , U ∈ R M × ( N − , ˜ z ∈ R n ( M − N +1) , ˜ z ∈ R n ( N − , ˜ w ∈ R n ( M − N +1) , ˜ w ∈ R n ( N − . Then (6) is equivalent to ˙˜ z = ( I M − N +1 ⊗ B ) ˜ w , ˙˜ z = ( I N − ⊗ A )˜ z + ( I N − ⊗ B ) ˜ w , (8)and ˜ z = ( U T ⊗ I n ) z , ˜ z = ( U T ⊗ I n ) z .Let Γ ∈ R ( N − × ( N − be the diagonal matrix includingall non-zero eigenvalues of L in its diagonal, and V ∈ R N × N is an orthogonal matrix such that V T LV = (cid:20) (cid:21) . (9)Partitioning V into [ V , V ] where V ∈ R N , V ∈ R N × ( N − .Then LV = V Γ ⇔ V T LV = Γ , (10)since V T V = I N − .Denote Φ( z ) , [ φ T ( z ) , . . . , φ TM ( z M )] T , φ j ( z j ) , [ φ Tj, ( z j, ) , . . . , φ Tj,N ( z j,M )] T , ∀ z = [ z T , . . . , z TM ] T . Theorem 1:
Let U be chosen as E T V Γ − / , then thefollowing distributed robust stabilizing controller in presenceof state constraints or uncertainties ˜ w = F (cid:0) U T ⊗ I n (cid:1) Φ( z ) , (11)for the transformed edge dynamics (8) with F = F ⊗ K , K ∈ R m × n and F = (cid:20) (cid:21) , (12)is equivalent to the following distributed robust consensuscontroller under relative-state constraints or uncertainties (5), u = ( E ⊗ K )Φ( z ) , (13)for the initial MAS (2). Furthermore, ˜ z ( t ) = 0 ∀ t ≥ . Proof:
First, we show that the orthogonality of U issatisfied with U chosen to be E T V Γ − / . Indeed, U T U =Γ − / V T EU = 0 since EU = 0 due to a fact that E ¯ L = E . Moreover, U T U = Γ − / V T EE T V Γ − / =Γ − / ΓΓ − / = I N − .Next, multiplying to the left of (10) with E T gives us L e E T V = E T V Γ , ⇔ L e E T V Γ − / = E T V Γ / , ⇔ Γ − / V T EL e E T V Γ − / = Γ − / V T EE T V Γ / , ⇔ U T L e U = Γ . (14)On the other hand, ¯ LU = 0 , which leads to L e U = 0 since L e ¯ L = L e . Therefore, we obtain F = U T L e U ⇔ U F U T = L e . (15)Consequently, ˜ w = F (cid:0) U T ⊗ I n (cid:1) Φ( z ) , ⇔ [( U T E T ) ⊗ I m ] u = (cid:2)(cid:0) F U T (cid:1) ⊗ K (cid:3) Φ( z ) . (16)Since U T E T = [ EU , EU ] T = [0 , V Γ / ] T and F U T =[0 , E T V Γ / ] T , (16) is equivalent to h(cid:16) Γ / V T (cid:17) ⊗ I m i u = h(cid:16) Γ / V T E (cid:17) ⊗ K i Φ( z ) , ⇔ [( V V T ) ⊗ I m ] u = [( V V T E ) ⊗ K ]Φ( z ) , (17)by multiplying both to the left and to the right of (17) with ( V Γ − / ) ⊗ I m . Note that V = √ N N , then V V T = I N − V V T = I N − N N TN . Hence, we obtain [( I N − N N TN ) ⊗ I m ] u = [([ I N − N N TN ] E ) ⊗ K ]Φ( z ) = ( E ⊗ K )Φ( z ) , since TN E = 0 . This is equivalent to u = ( E ⊗ K )Φ( z ) + ( N ⊗ I m ) u for any u ∈ R m . Since we are not interested in self-feedback inputs for agents, u = 0 or equivalently u = ( E ⊗ K )Φ( z ) . On the other hand, we have ˜ z = U T z = [( U T E T ) ⊗ I n ] x = 0 ∀ t ≥ , since EU = 0 .Employing the result of Theorem 1 to the transformed edgedynamics (8), it can be deduced that we only need to designa distributed robust stabilizing controller for the subsystem ˙˜ z = ( I N − ⊗ A )˜ z + ( I N − ⊗ B ) ˜ w , (18) having the following form ˜ w = (cid:2)(cid:0) Γ U T (cid:1) ⊗ K (cid:3) Φ( z ) , (19)which is directly calculated from (11). Thus, the interestingresult of Theorem 1 is that the distributed robust consensusdesign (13) under relative-state constraints or uncertainties forthe initial MAS (2) is equivalent to a simpler problem of syn-thesizing a distributed robust stabilizing controller (19) understate constraints or uncertainties for a new MAS (18) whichhas lower dimension. In the next section, we will presentan approach to design such a distributed robust stabilizingcontroller. Remark 1: If G is a spanning tree then M = N − andhence ¯ L = I N − . Then we do not need the additional trans-formation (6). Therefore, all results here and in subsequentsections are derived with ˜ w and ˜ z replaced by w and z ,respectively. B. Distributed Robust Stabilizing Controller Synthesis
The transformed edge dynamics (18) together with therobust stabilizing controller (19) can be rewritten in thefollowing form of a network of Lur’e systems, ˙˜ z = A ˜ z + B v,z = ( U ⊗ I n )˜ z ,v = Φ( z ) , (20)where A = I N − ⊗ A , B = (cid:0) Γ U T (cid:1) ⊗ ( BK ) .The following theorem presents a sufficient condition forachieving the robust stabilization of (18) and equivalently therobust consensus of the initial MAS (2), and then how todesign the consensus controller gain K . Theorem 2:
When Σ and Σ are not multipliers of identitymatrices, the MAS (18) is robustly stabilized by the distributedstabilizing control law (19) and equivalently the robust consen-sus under relative-state constraints or uncertainties is achievedfor the initial MAS (2) by the distributed controller (13) ifthere exist matrices X ∈ R n × n , Y ∈ R m × n and Z ∈ R m × m such that the following LMI problem is feasible with ǫ > , (cid:20) sym( AX + λ BY Σ ) + ǫX λ BY + (Σ − Σ ) Z ( λ BY + (Σ − Σ ) Z ) T − Z (cid:21) (cid:22) , (cid:20) sym( AX + λ N BY Σ ) + ǫX λ N BY + (Σ − Σ ) Z ( λ N BY + (Σ − Σ ) Z ) T − Z (cid:21) (cid:22) ,X ≻ , X is diagonal , (cid:20) Z XX Ψ − (cid:21) (cid:23) , Ψ ≻ , Ψ is diagonal . (21)Moreover, the controller gain K is calculated by K = Y X − . Proof:
Consider a Lyapunov function V (˜ z ) = ˜ z T P ˜ z where P , I N − ⊗ P , P ∈ R n , P ≻ . Taking the derivativeof V (˜ z ) gives us ˙ V (˜ z ) = ˜ z T (cid:0) PA + A T P (cid:1) ˜ z + 2˜ z T PB v. Hence, for all ǫ > we have ˙ V (˜ z ) + ǫV (˜ z ) = ˜ z T (cid:0) PA + A T P + ǫ P (cid:1) ˜ z + 2˜ z T PB v. We now seek P such that ˙ V (˜ z ) + ǫV (˜ z ) ≤ as longas (5) holds. Using the S-procedure [18], such P exists ifthere exist ψ , , . . . , ψ ,n , . . . , ψ M, , . . . , ψ M,n which are non-negative such that ˙ V (˜ z ) + ǫV (˜ z ) − M X j =1 n X k =1 ψ j,k ( v j,k − σ k, z j,k )( v j,k − σ k, z j,k ) ≤ . (22)Let ψ j,k = ψ k > ∀ j = 1 , . . . , M and Ψ =diag { ψ k } k =1 ,...,n , then (22) is satisfied if ˙ V (˜ z ) + ǫV (˜ z ) − M X j =1 ( v j − Σ z j ) T Ψ( v j − Σ z j ) ≤ , ⇔ (cid:20) ˜ z v (cid:21) T (cid:20) P P P T P (cid:21) (cid:20) ˜ z v (cid:21) (cid:22) ⇔ (cid:20) P P P T P (cid:21) (cid:22) , (23)where P = PA + A T P + ǫ P − I N − ⊗ (ΨΣ Σ ) , P = PB + U T ⊗ (Ψ(Σ + Σ )) , P = − I M ⊗ Ψ .Subsequently, employing Schur complement [18] to (23)results in P − P P − P T (cid:22) , which is equivalent to I N − ⊗ ( A T P + P A + ǫP − ΨΣ Σ )+ 12 Γ ⊗ sym( P BK (Σ + Σ )) + Γ ⊗ ( P BK Ψ − K T B T P )+ 14 I N − ⊗ (cid:2) Ψ(Σ + Σ ) (cid:3) (cid:22) . (24)Since Γ is diagonal, (24) is equivalent to A T P + P A + ǫP − ΨΣ Σ + λ k P BK Ψ − K T B T P + 12 λ k sym( P BK (Σ + Σ )) + 14 Ψ(Σ + Σ ) (cid:22) , ⇔ A T P + P A + ǫP + λ k P BK Ψ − K T B T P + 12 λ k sym( P BK (Σ + Σ )) + 14 Ψ(Σ − Σ ) (cid:22) . Next, denote X , P − and multiply X both to the left andto the right of the equation above, we obtain XA T + AX + ǫX + λ k BK Ψ − K T B T + 12 λ k sym( BK (Σ + Σ ) X ) + 14 X Ψ(Σ − Σ ) X (cid:22) . (25)If X is diagonal then (25) is equivalent to sym( AX + λ k BY Σ ) + ǫX + 12 [ λ k BY + (Σ − Σ ) Z ] × Z − [ λ k BY + (Σ − Σ ) Z ] T (cid:22) , (26)where Y , KX , Z = X Ψ . Then using Schur complementagain with (26) leads to (cid:20) sym( AX + λ k BY Σ ) + ǫX λ k BY + (Σ − Σ ) Z ( λ k BY + (Σ − Σ ) Z ) T − Z (cid:21) (cid:22) , (27)for all k = 2 , . . . , N . Since λ ≤ λ , . . . , λ N − ≤ λ N , wecan represent λ i , i = 3 , . . . , N − as convex combinations of λ and λ N . Thus, we derive (21). Theorem 3:
Suppose that Σ = σ I n and Σ = σ I n thenthe MAS (18) is robustly stabilized by the distributed stabi-lizing control law (19) and equivalently the robust consensus under relative-state constraints or uncertainties is achieved forthe initial MAS (2) by the distributed controller (13) if thereexist matrices X ∈ R n × n , Y ∈ R m × n and Z ∈ R m × m suchthat the following LMI problem is feasible with ǫ > , (cid:20) sym( AX + σ λ BY ) + ǫX λ BY + ( σ − σ ) Z ( λ BY + ( σ − σ ) Z ) T − Z (cid:21) (cid:22) , (cid:20) sym( AX + σ λ N BY ) + ǫX λ N BY + ( σ − σ ) Z ( λ N BY + ( σ − σ ) Z ) T − Z (cid:21) (cid:22) ,X ≻ , (cid:20) Z XX Ψ − (cid:21) (cid:23) , Ψ ≻ , Ψ is diagonal . (28)Moreover, the controller gain K is calculated by K = Y X − . Proof:
Consider the same Lyapunov function as in theproof of Theorem 2. Then all steps until (25) are also true inthis scenario. Accordingly, substituting Σ = σ I n and Σ = σ I n into (25) gives us sym( AX + σ λ k BY ) + ǫX + 12 [ λ k BY + ( σ − σ ) Z ] × Z − [ λ k BY + ( σ − σ ) Z ] T (cid:22) , (29)where Y , KX , Z = X Ψ X . Then using Schur complementagain with (29) and notes that λ i , i = 3 , . . . , N − can berepresented as convex combinations of λ and λ N , we obtain(28). Remark 2:
Recently, there are several existing researches,e.g. [19], [20], which propose different distributed methodsto approximate the whole eigen-spectrum of the Laplacianmatrix. These methods can be employed to estimate λ and λ N before solving the LMI problems (21), (28). As a result,we can solve (21) and (28) in a distributed fashion. Remark 3:
The difference between Theorem 3 and The-orem 2 is that the variable X in (28) is not required tobe diagonal while that in (21) is. Therefore, if Σ and Σ are multipliers of identity matrices, i.e., the upper and lowersector slopes for relative state constraints or uncertainties ofall agents are the same then the associated LMI problem isless conservative and hence its feasibility is increased. Remark 4:
As stated in the introduction, our method toderive LMI problems (21) and (28) for the Lur’e network (20)is more general than the method for a similar Lur’e network in[11]. On the other hand, the problem setting in this paper is ina different form of Lur’e networks with that in [13]. Therefore,the obtained results are not similar. More specifically, [13] usesa linear cooperative input and another nonlinear input with adifferent input matrix E satisfying the incrementally passive orincrementally sector-bounded condition, which is less generalthan our sector-bounded condition (5).IV. N UMERICAL E XAMPLES
A. Practical Consensus of Mobile Robots
Consider a group of N identical -wheel robots with front-wheel steering. The variables and parameters of each robotare illustrated in Figure 1 where the center of mass is denotedby M i whose position in a given coordinate ( O, x, y ) is represented by ( x Mi , y Mi ) . The rotation and steering anglesare denoted by θ i and ϕ i , respectively. Accordingly, ω i and v i represent the angular and longitudinal velocity. To takeinto account practical factors such as robots’ dimensions andcollision avoidance, we shall investigate the consensus ofthe robots’ heading points C i instead of M i . This practicalconsensus concept is demonstrated in Figure 1. (cid:1829) (cid:1841) (cid:1876) (cid:3014) (cid:3) (cid:1876) (cid:3004) (cid:3) (cid:1877) (cid:3014) (cid:3) (cid:1877) (cid:3004) (cid:3) (cid:1876) (cid:3) (cid:1877) (cid:3) (cid:2016) (cid:2033) (cid:1874) (cid:1839) (cid:2030) (cid:1841) (cid:1877) (cid:3) (cid:1876) (cid:3) Fig. 1. Demonstration of variables related to a -wheel robot and the conceptof practical robot consensus. Next, the robot’s model in term of the coordinates of C i isas follows, ˙ x i = M i [ v i , ω i ] T , i = 1 , . . . , N, (30)where x i = [ x Ci , y Ci ] T ; M i = (cid:20) cos θ i − r sin θ i sin θ i r cos θ i (cid:21) . Denote u i , M i [ v i , ω i ] T then each robot can be represented by a setof two integrators ˙ x i = u i , i = 1 , . . . , N. We have (cid:20) v i ω i (cid:21) = M − i u i = (cid:20) cos θ i sin θ i − sin θ i /r cos θ i /r (cid:21) u i , (cid:20) ˜ u i, ˜ u i, (cid:21) . Therefore, the real control inputs v i and ϕ i to each robot arecomputed by v i = ˜ u i, ; ϕ i = arctan ω i v i = arctan ˜ u i, ˜ u i, . (31)Consequently, we consider the constraint k x i − x j k ∞ ≤ α on relative states of connected robots, which implies thatthe communication range between robots is limited to √ α .This is indeed a robust consensus problem under relative-stateconstraints within our framework.Employing Theorem 3, we solve the LMI problem (28)with A = 0 , B = 1 and obtain − ǫλ σ < K < . In thesimulation, we set r = 2 [dm], α = 3 [dm], ǫ = 0 . , and G is a full graph, then choose K = − . since λ = 3 . Thesimulation result in Figure 2 then confirms that the consensusamong robots is achieved even though there is a constraint onrelative state exchange of robots, where the arrows representthe vectors −−−→ M i C i of robots. Moreover, Figure 3 shows thatthe exchanged relative states of robots always satisfy the givenbounded constraint. B. Consensus of Oscillator Networks
To further illustrate the proposed approach, we consider aconsensus problem in a network of identical linear oscillatorswith the following model, ˙ x k = Ax k + Bu k , k = 1 , , , (32) Fig. 2. Trajectories of -wheel robots reaching consensus. R e l a t i v e x − c oo r d i na t e x C1 −x C2 x C2 −x C3 x C3 −x C1 Time [s] R e l a t i v e y − c oo r d i na t e y C1 −y C2 y C2 −y C3 y C3 −y C1 Fig. 3. Bounded relative coordinates of heading points of -wheel robots. where A = (cid:20) − (cid:21) , B = (cid:20) (cid:21) , (33)and the initial conditions of the oscillators are (1 , − ; ( − , ; (4 , − , respectively. We then assume that G is a full graphand the exchanged relative states among agents are boundedin [ − , . With ǫ = 0 . , solving the LMI problem (28) givesus K = [ − . , − . . Consequently, Figure 4 revealsthat the oscillators exhibit synchronized oscillations whereasFigure 5 shows that the relative states of oscillators satisfy thebounded constraints, i.e., the robust consensus is achieved.Next, we demonstrate the effectiveness of our approach inthe scenario that the exchanged relative states among agentscontain some uncertainties that results in Σ = 0 . I and Σ = 1 . I . Then we solve the LMI problem (28) to obtain K = [ − . , − . . In the simulation, we randomlygenerate those uncertainties in the interval [0 . , . . We thenobserve that the synchronization of oscillators are achieved forany uncertainties in the given range. Particularly, Figure 6–7display the oscillator network’s responses for a specific casewhere the uncertainties on two relative states of oscillators are . and . . V. C ONCLUSION
An approach has been proposed in this paper to analyze andsynthesize distributed global robust consensus controllers forgeneral linear leaderless MASs under relative state constraintsor uncertainties with the following appealing features. First, itis available for a broader class of MASs and for constraints oruncertainties described by a sector-bounded condition which
Time [s] F i r s t s t a t e s Time [s] S e c ond s t a t e s Fig. 4. Synchronization of oscillators under relative state constraints.
Time [s] s t r e l a t i v e s t a t e s x −x x −x x −x Time [s] r e l a t i v e s t a t e s x −x x −x x −x Fig. 5. Bounded relative states of oscillators.
Time [s] F i r s t s t a t e s Time [s] S e c ond s t a t e s Fig. 6. Synchronization of oscillators under relative state uncertainties.
Time [s] s t r e l a t i v e s t a t e s x −x x −x x −x Time [s] r e l a t i v e s t a t e s x −x x −x x −x Fig. 7. Relative states of oscillators. is more general than that in the existing researches. Second, itshows that the global robust consensus design with relativestate constraints or uncertainties is equivalent to a robuststabilizing design with state constraints or uncertainties ofa transformed MAS. Third, a sufficient condition for global robust consensus and the global robust consensus controllergain are derived from the solutions of a distributed convexLMI problem which is less conservative than in other studies.A
CKNOWLEDGMENT
The author would like to thank Toyota Technological Insti-tute for its supports. R
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