Robust Connectivity Analysis for Multi-Agent Systems
RROBUST CONNECTIVITY ANALYSIS FOR MULTI-AGENT SYSTEMS
D. BOSKOS AND D. V. DIMAROGONAS
Abstract.
In this report we provide a decentralized robust control approach, which guar-antees that connectivity of a multi-agent network is maintained when certain bounded inputterms are added to the control strategy. Our main motivation for this framework is to de-termine abstractions for multi-agent systems under coupled constraints which are furtherexploited for high level plan generation. Introduction
Cooperative control of multi-agent systems constitutes a highly active area of research duringthe last two decades. Typical objectives are the consensus problem, which is concerned withfinding a protocol that achieves convergence to a common value [9], reference tracking [1] andformation control [6]. A common feature in the approach to these problems is the design ofdecentralized control laws in order to achieve a global goal.In the case of mobile robot networks with limited sensing and communication ranges, con-nectivity maintenance plays a fundamental role [14]. In particular, it is required to constrainthe control input in such a way that the network topology remains connected during the evo-lution of the system. For instance, in [6] the rendezvous and formation control problems arestudied while preserving connectivity, whereas in [3] swarm aggregation is achieved by meansof a control scheme that guarantees both connectivity and collision avoidance.In our approach we provide a control law for each agent comprising of a decentralized feedbackcomponent and a free input term, which ensures connectivity maintenance, for all possiblefree input signals up to a certain bound of magnitude. The motivation for this approachcomes from distributed control and coordination of multi-agent systems with locally assignedLinear Temporal Logic (LTL) specifications. In particular, by virtue of the invariance androbust connectivity maintenance properties, it is possible to define well posed decentralizedabstractions for the multi-agent system which can be exploited for motion planning. The latterproblem has been studied in our recent work [2] for the single integrator dynamics case.In this work, we design a bounded control law which results in network connectivity ofthe system for all future times provided that the initial relative distances of interconnectedagents and the free input terms satisfy appropriate bounds. Furthermore, in the case of aspherical domain, it is shown that adding an extra repulsive vector field near the boundary of thedomain can also guarantee invariance of the solutions and simultaneously maintain the robustconnectivity property. The latter framework enables the construction of finite abstractions forthe single integrator case.The rest of the report is organized as follows. Section 2 introduces basic notation and pre-liminaries. In Section 3, results on robust connectivity maintenance are provided and explicitcontrollers which establish this property are designed. In Section 4, the corresponding con-trollers are appropriately modified, in order to additionally guarantee invariance of the solution a r X i v : . [ c s . S Y ] M a r D. BOSKOS AND D. V. DIMAROGONAS for the case of a spherical domain. We summarize the results and discuss possible extensionsin Section 5. 2.
Preliminaries and Notation
Notation.
We use the notation | x | for the Euclidean norm of a vector x ∈ R n . For a matrix A ∈ R m × n we use the notation | A | := max {| Ax | : x ∈ R n } for the induced Euclidean matrixnorm and A T for its transpose. For two vectors x, y ∈ R n (= R n × ) we denote their inner productby (cid:104) x, y (cid:105) := x T y . Given a subset S of R n , we denote by cl( S ), int( S ) and ∂S its closure, interiorand boundary, respectively, where ∂S := cl( S ) \ int( S ). For R >
0, we denote by B ( R ) theclosed ball with center 0 ∈ R n and radius R . Given a vector x = ( x , . . . , x n ) ∈ R n we define thecomponent operators c l ( x ) := x l , l = 1 , . . . , n . Likewise, for a vector x = ( x , . . . , x N ) ∈ R Nn we define the component operators c l ( x ) := ( c l ( x ) , . . . , c l ( x N )) ∈ R N , l = 1 , . . . , n .Consider a multi-agent system with N agents. For each agent i ∈ { , . . . , N } := N we usethe notation N i for the set of its neighbors and |N i | for its cardinality. We also consider anordering of the agent’s neighbors which we denote by j , . . . , j |N i | . E stands for the undirectednetwork’s edge set and { i, j } ∈ E iff j ∈ N i . The network graph G := ( N , E ) is connected if foreach i, j ∈ N there exists a finite sequence i , . . . , i l ∈ N with i = i , i l = j and { i k , i k +1 } ∈ E ,for all k = 1 , . . . , l −
1. Consider an arbitrary orientation of the network graph G , which assignsto each edge { i, j } ∈ E precisely one of the ordered pairs ( i, j ) or ( j, i ). When selecting thepair ( i, j ) we say that i is the tail and j is the head of edge { i, j } . By considering a numbering l = 1 , . . . , M of the graph’s edge set we define the N × M incidence matrix D ( G ) correspondingto the particular orientation as follows: D ( G ) kl := , if vertex k is the head of edge l − , if vertex k is the tail of edge l , otherwiseThe graph Laplacian L ( G ) is the N × N positive semidefinite symmetric matrix L ( G ) := D ( G ) × D ( G ) T . If we denote by the vector (1 , . . . , ∈ R N , then L ( G ) = D ( G ) T = 0. Let0 = λ ( G ) ≤ λ ( G ) ≤ · · · ≤ λ N ( G ) be the ordered eigenvalues of L ( G ). Then each correspondingset of eigenvectors is orthogonal and λ ( G ) > G is connected.2.2. Problem Statement.
We focus on single integrator multi-agent systems with dynamics˙ x i = u i , x i ∈ R n , i = 1 , . . . , N (2.1)We aim at designing decentralized control laws of the form u i := k i ( x i , x j , . . . , x j |N i | ) + v i (2.2)which ensure that appropriate apriori bounds on the initial relative distances of interconnectedagents guarantee network connectivity for all future times, for all free inputs v i bounded bycertain constant. In particular, we assume that the network graph is connected as long asthe maximum distance between two interconnected agents does not exceed a given positiveconstant R . In addition, we make the following connectivity hypothesis for the initial states ofthe agents. (ICH) We assume that the agents’ communication graph is initially connected and thatmax {| x i (0) − x j (0) | : { i, j } ∈ E} ≤ ˜ R for certain constant ˜ R ∈ (0 , R ) (2.3) OBUST CONNECTIVITY ANALYSIS FOR MULTI-AGENT SYSTEMS 3
Potential Functions.
We proceed by defining certain mappings which we exploit inorder to design the control law (2.2) and prove that network connectivity is maintained. Let r : R ≥ → R ≥ be a continuous function satisfying the following property. (P) r ( · ) is increasing and r (0) > P ( ρ ) = (cid:90) ρ r ( s ) sds, ρ ∈ R ≥ (2.4)For each pair ( i, j ) ∈ N × N with { i, j } ∈ E we define the potential function V ij : R Nn → R ≥ as V ij ( x ) = P ( | x i − x j | ) , ∀ x = ( x , . . . , x N ) ∈ R Nn (2.5)Notice that V ij ( · ) = V ji ( · ). Furthermore, it can be shown that V ij ( · ) is continuously differen-tiable and that ∂∂x i V ij ( x ) = r ( | x i − x j | )( x i − x j ) T (2.6)where ∂∂x i stands for the derivative with respect to the x i -coordinates. Remark 2.1.
Notice that we are only interested in the values of the mappings r ( · ) and P ( · ) inthe interval [0 , R ], which stands for the maximum distance that two interconnected agents mayachieve before losing connectivity. Yet, defining them on the whole positive line provides uscertain technical flexibilities for the analysis employed in the proof of connectivity maintenance.3. Connectivity Analysis
In the following proposition we provide a control law (2.2) and an upper bound on the magnitudeof the input terms v i ( · ) which guarantee connectivity of the multi-agent network. Proposition 3.1.
For the multi-agent system (2.1) , assume that (ICH) is fulfilled and definethe control law u i = − (cid:88) j ∈N i r ( | x i − x j | )( x i − x j ) + v i (3.1) for certain continuous r ( · ) satisfying Property (P). Also, consider a constant δ > and define K := 2 (cid:112) N ( N − | D ( G ) T | λ ( G ) (3.2) where D ( G ) is the incidence matrix of the systems’ graph and λ ( G ) the second eigenvalue ofthe graph Laplacian. We assume that the positive constant δ , the maximum initial distance ˜ R and the function r ( · ) satisfy the restrictions δ ≤ K r (0) sr ( s ) , s ≥ ˜ R (3.3) with K as given in (3.2) and M P ( ˜ R ) ≤ P ( R ) (3.4) where P ( · ) is given in (2.4) , and M = |E| is the cardinality of the system’s graph edge set.Then, the system remains connected for all positive times, provided that the input terms v i ( · ) , i = 1 , . . . , N satisfy | v i ( t ) | ≤ δ, ∀ t ≥ D. BOSKOS AND D. V. DIMAROGONAS
Proof.
For the proof we follow parts of the analysis in [6] (see also [8, Section 7.2]). Considerthe energy function V := 12 N (cid:88) i =1 (cid:88) j ∈N i V ij (3.6)where the mappings V ij , { i, j } ∈ E are given in (2.5). Then it follows from (2.6) that ∂∂x i V ( x ) = (cid:88) j ∈N i r ( | x i − x j | )( x i − x j ) T (3.7)Also, in accordance with [8, Section 7.2] we have for l = 1 , . . . , n that c l (cid:88) j ∈N i r ( | x i − x j | )( x i − x j ) = L w ( x ) c l ( x ) (3.8)The weighted Laplacian matrix L w ( x ) is given as L w ( x ) = D ( G ) W ( x ) D ( G ) T (3.9)where D ( G ) is the incidence matrix of the communication graph (see Notation) and W ( x ) := diag { w ( x ) , . . . , w M ( x ) } := diag { r ( | x i − x j | ) , { i, j } ∈ E} (3.10)(recall that M = |E| ). Then, by evaluating the time derivative of V along the trajectories of(2.1)-(3.1) and taking into account (3.6), (3.7) and (3.8) we get˙ V = − n (cid:88) l =1 c l (cid:18) ∂∂x V ( x ) (cid:19) c l ( ˙ x )= − n (cid:88) l =1 c l ( x ) T L w ( x )( L w ( x ) c l ( x ) − c l ( v ))= − n (cid:88) l =1 c l ( x ) T L w ( x ) c l ( x ) + n (cid:88) l =1 c l ( x ) T L w ( x ) c l ( v ) ≤ − n (cid:88) l =1 c l ( x ) T L w ( x ) c l ( x ) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) l =1 c l ( x ) T L w ( x ) c l ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.11)We want to provide appropriate bounds for the right hand side of (3.11) which can guaranteethat the sign of ˙ V is negative whenever the maximum distance between two agents exceeds thebound ˜ R on the maximum initial distance as given in (2.3). First, we provide certain usefulinequalities between the eigenvalues of the weighted Laplacian L w ( x ) and the Laplacian matrixof the graph L ( G ). Notice, that due to (3.10), for each i = 1 , . . . , M we have w i ( x ) = r ( | x k − x (cid:96) | )for certain { k, (cid:96) } ∈ E and hence, by virtue of Property (P), it holds0 < r (0) ≤ w i ( x ) ≤ max { k,(cid:96) }∈E r ( | x k − x (cid:96) | ) (3.12)From (3.12), it follows that L w ( x ) has precisely the same properties with those provided for L ( G ) in the Notation subsection. Furthermore, it holds λ ( x ) ≥ λ ( G ) r (0) (3.13) OBUST CONNECTIVITY ANALYSIS FOR MULTI-AGENT SYSTEMS 5 where 0 = λ ( x ) < λ ( x ) ≤ · · · ≤ λ N ( x ) and 0 = λ ( G ) < λ ( G ) ≤ · · · ≤ λ N ( G ) are theeigenvalues of L w ( x ) and the Laplacian matrix of the graph L ( G ), respectively. Indeed, in orderto show (3.13), notice that L w ( x ) = D ( G )diag { w ( x ) , . . . , w M ( x ) } D ( G ) T = D ( G )diag { r (0) , . . . , r (0) } D ( G ) T + D ( G )diag { w ( x ) − r (0) , . . . , w M ( x ) − r (0) } D ( G ) T = r (0) L ( G ) + B where (3.12) implies that B := D ( G )diag { w ( x ) − r (0) , . . . , w M ( x ) − r (0) } D ( G ) T is positivesemidefinite. Hence, it holds L w ( x ) (cid:23) r (0) L ( G ), with (cid:23) being the partial order on the set ofsymmetric N × N matrices and thus, we deduce from Corollary 7.7.4(c) in [4, page 495] that(3.13) is fulfilled.In the sequel we introduce some additional notation. Let H be the subspace H := { x ∈ R Nn : x = x = · · · = x N } For a vector x ∈ R Nn we denote by ¯ x its projection to the subspace H , and x ⊥ its orthogonalcomplement with respect to that subspace, namely x ⊥ := x − ¯ x . By taking into account thatfor all y ∈ H we have D ( G ) T c l ( y ) = 0 and hence, due to (3.9), that c l ( y ) ∈ ker( L w ( x )), itfollows that for every vector x ∈ R Nn with x = ¯ x + x ⊥ it holds L w ( x ) c l (¯ x ) = 0 ⇒ L w ( x ) c l ( x ) = L w ( x ) c l (¯ x + x ⊥ ) = L w ( x ) c l ( x ⊥ ) (3.14)We also denote by ∆ x ∈ R Mn the stack column vector of the vectors x i − x j , { i, j } ∈ E with theedges ordered as in the case of the incidence matrix. It is thus straightforward to check thatfor all x ∈ R Nn D ( G ) T x = ∆ x (3.15)and furthermore, due to (3.10) and (3.12), that | W ( x ) | ≤ r ( | ∆ x | ∞ ) (3.16)where | ∆ x | ∞ := max {| ∆ x i | , i = 1 , . . . , M } (3.17)Before proceeding we state the following elementary facts, whose proofs can be found in theAppendix. In particular, for the vectors x = ( x , . . . , x N ) , y = ( y , . . . , y N ) ∈ R Nn the followingproperties hold. Fact I. | L w ( x ) c l ( x ⊥ ) | ≥ λ ( x ) | c l ( x ⊥ ) | , ∀ l = 1 , . . . , n (3.18) Fact II. n (cid:88) l =1 | c l ( x ) || c l ( y ) | ≤ | x || y | (3.19) Fact III. | x ⊥ | ≥ (cid:112) N − | ∆ x | (3.20) Fact IV. √ | x ⊥ | ≥ | ∆ x | ∞ (3.21)We are now in position to bound the derivative of the energy function V and exploit the resultin order to prove the desired connectivity maintenance property. We break the subsequentproof in two main steps. D. BOSKOS AND D. V. DIMAROGONAS
Step 1: Bound estimation for the rhs of (3.11) .Bound for the first term in (3.11) . By taking into account (3.14), it follows that n (cid:88) l =1 c l ( x ) T L w ( x ) c l ( x ) = n (cid:88) l =1 | L w ( x ) c l ( x ) | = n (cid:88) l =1 (cid:12)(cid:12) L w ( x ) c l ( x ⊥ ) (cid:12)(cid:12) (3.22)and by exploiting Fact I and (3.13), we get n (cid:88) l =1 (cid:12)(cid:12) L w ( x ) c l ( x ⊥ ) (cid:12)(cid:12) ≥ n (cid:88) l =1 λ ( x ) | c l ( x ⊥ ) | ≥ n (cid:88) l =1 [ λ ( G ) r (0)] | c l ( x ⊥ ) | = [ λ ( G ) r (0)] n (cid:88) l =1 (cid:12)(cid:12) c l ( x ⊥ ) (cid:12)(cid:12) = [ λ ( G ) r (0)] | x ⊥ | (3.23)Thus, it follows from (3.22) and (3.23) that n (cid:88) l =1 c l ( x ) T L w ( x ) c l ( x ) ≥ [ λ ( G ) r (0)] | x ⊥ | (3.24) Bound for the second term in (3.11) . For this term, we have from (3.9) and (3.15) that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) l =1 c l ( x ) T L w ( x ) c l ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n (cid:88) l =1 | c l ( x ) T L w ( x ) c l ( v ) | = n (cid:88) l =1 | c l ( x ) T D ( G ) W ( x ) D ( G ) T c l ( v ) | = n (cid:88) l =1 | c l (∆ x ) T W ( x ) D ( G ) T c l ( v ) |≤ n (cid:88) l =1 | c l (∆ x ) || W ( x ) || D ( G ) T || c l ( v ) | (3.25)By taking into account (3.16), we obtain n (cid:88) l =1 | c l (∆ x ) || W ( x ) || D ( G ) T || c l ( v ) | ≤ n (cid:88) l =1 | c l (∆ x ) | r ( | ∆ x | ∞ ) | D ( G ) T || c l ( v ) | (3.26)Also, by exploiting Fact II, we get that n (cid:88) l =1 | c l (∆ x ) | r ( | ∆ x | ∞ ) | D ( G ) T || c l ( v ) | = r ( | ∆ x | ∞ ) | D ( G ) T | n (cid:88) l =1 | c l (∆ x ) || c l ( v ) |≤ r ( | ∆ x | ∞ ) | D ( G ) T || ∆ x || c l ( v ) |≤ r ( | ∆ x | ∞ ) | D ( G ) T || ∆ x |√ N | v | ∞ (3.27)where | v | ∞ := max {| v i | , i = 1 , . . . , N } OBUST CONNECTIVITY ANALYSIS FOR MULTI-AGENT SYSTEMS 7
Hence, it follows from (3.25)-(3.27) that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) l =1 c l ( x ) T L w ( x ) c l ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ √ N | D ( G ) T || ∆ x | r ( | ∆ x | ∞ ) | v | ∞ (3.28)Thus, we get from (3.11), (3.24) and (3.28) that˙ V ≤ − [ λ ( G ) r (0)] | x ⊥ | + √ N | D ( G ) T || ∆ x | r ( | ∆ x | ∞ ) | v | ∞ and by exploiting Facts III and IV, that˙ V ≤ − [ λ ( G ) r (0)] (cid:112) N − | ∆ x | √ | ∆ x | ∞ + √ N | D ( G ) T || ∆ x | r ( | ∆ x | ∞ ) | v | ∞ = | ∆ x | (cid:18) − √ N − λ ( G ) r (0)] | ∆ x | ∞ + √ N | D ( G ) T | r ( | ∆ x | ∞ ) | v | ∞ (cid:19) By using the notation | ∆ x | ∞ := s , in order to guarantee that the above rhs is negative for s ≥ ˜ R , it should hold − λ ( G ) (cid:112) ( N − r (0) s + √ N | D ( G ) T | r ( s ) | v | ∞ ≤ , ∀ s ≥ ˜ R ⇐⇒ (cid:112) N ( N − | D ( G ) T | λ ( G ) | v | ∞ ≤ r (0) sr ( s ) , ∀ s ≥ ˜ R or equivalently | v | ∞ ≤ K r (0) sr ( s ) , ∀ s ≥ ˜ R (3.29)with K as given in (3.2). Hence, we have shown that for v satisfying (3.29) the followingimplication holds | ∆ x | ∞ ≥ ˜ R ⇒ ˙ V ≤ Step 2: Proof of connectivity.
By assuming that conditions (3.5), (3.3) and (3.4) in the statement of the proposition arefulfilled and recalling that according to (ICH) (2.3) holds, we can show that the system willremain connected for all future times. Indeed, let x ( · ) be the solution of the closed loop system(2.1)-(3.1) with initial condition satisfying (2.3), defined on the maximal right interval [0 , T max ).We claim that the system remains connected on [0 , T max ), namely, that max {| x i ( t ) − x j ( t ) | : { i, j } ∈ E} ≤ R for all t ∈ [0 , T max ), which by boundedness of the dynamics on the set F := { x ∈ R Nn : | x i − x j | ≤ R, ∀{ i, j } ∈ E} implies that T max = ∞ . In order to prove thelast assertion, assume on the contrary that T max < ∞ . Then, by taking into account that x ( t )remains in F for all t ∈ [0 , T max ) and that the dynamics are bounded on F , it follows that x ( t )remains in a compact subset of R Nn for all t ∈ [0 , T max ) and hence, that it can be extended,contradicting maximality of [0 , T max ). We proceed with the proof of connectivity. First, noticethat V ( x (0)) ≤ P ( R ) (3.31) D. BOSKOS AND D. V. DIMAROGONAS
Indeed, by exploiting (2.3) and (3.4) we get that V ( x (0)) = 12 N (cid:88) i =1 (cid:88) j ∈N i P ( | x i (0) − x j (0) | ) ≤ N (cid:88) i =1 (cid:88) j ∈N i P ( ˜ R ) = M P ( ˜ R ) ≤ P ( R ) (3.32)In order to prove our claim, it suffices to show that V ( x ( t )) ≤ P ( R ) , ∀ t ∈ [0 , T max ) (3.33)because if | x i ( t ) − x j ( t ) | > R for certain t ∈ [0 , T max ) and { i, j } ∈ E , then V ( x ( t )) ≥ P ( | x i ( t ) − x j ( t ) | ) > P ( R ). We prove (3.33) by contradiction. Indeed, suppose on the contrary that thereexists T ∈ (0 , T max ) (due to (3.31)) such that V ( x ( T )) > P ( R ) (3.34)and define τ := min { t ∈ [0 , T ] : V ( x (¯ t )) > P ( R ) , ∀ ¯ t ∈ ( t, T ] } (3.35)which due to (3.34) and continuity of V ( x ( · )) is well defined. Then it follows from (3.31) and(3.35) that V ( x ( τ )) = 12 P ( R ) , V ( x ( t )) > P ( R ) , ∀ t ∈ ( τ, T ] (3.36)hence, there exists ¯ τ ∈ ( τ, T ) such that˙ V ( x (¯ τ )) = V ( x ( T )) − V ( x ( τ )) T − τ > V ( x (¯ τ )) > P ( R ) (3.38)which implies that there exists { i, j } ∈ E with | x i (¯ τ ) − x j (¯ τ ) | > ˜ R (3.39)Indeed, if (3.39) does not hold, then we can show as in (3.32) that V ( x (¯ τ )) ≤ P ( R ) whichcontradicts (3.38). Notice that by virtue of (3.5) and (3.3), (3.29) is fulfilled. Hence, we get from(3.39) that | ∆ x (¯ τ ) | ∞ > ˜ R and thus from (3.30) it follows that ˙ V ( x (¯ τ )) ≤
0, which contradicts(3.37). We conclude that (3.33) holds and the proof is complete. (cid:3)
In the following corollary, we apply the result of Proposition 3.1 in order to provide twoexplicit feedback laws of the form (3.1), a linear and a nonlinear one and compare their perfor-mance in the subsequent remark.
Corollary 3.2.
For the multi agent system (2.1) , assume that (ICH) is fulfilled and considerthe control law (2.2) as given by (3.1) . By imposing the additional requirement r (0) = r ( ˜ R ) = 1 and defining δ := ˜ RK (3.40) OBUST CONNECTIVITY ANALYSIS FOR MULTI-AGENT SYSTEMS 9 with ˜ R and K as given in (2.3) and (3.2) , respectively, the system remains connected for allpositive times, provided that the function r ( · ) and the constant ˜ R are selected as in the followingtwo cases (L) and (NL) (providing a linear and a nonlinear feedback, respectively). Case (L).
We select r ( s ) := 1 , s ≥ and ˜ R ≤ √ M R (3.42) where M is the cardinality of the system’s graph edge set. Case (NL).
We select r ( s ) := , s ∈ [0 , ˜ R ] s ˜ R , s ∈ ( ˜ R, R ] R ˜ R , s ∈ ( R, ∞ ) (3.43) and ˜ R ≤ (cid:18) M − (cid:19) R (3.44) Proof.
For the proof we apply the result of Proposition (3.1). In particular, it suffices to showthat for both cases (L) and (NL) the selection of the function r ( · ) and the initial maximumdistance ˜ R satisfy (3.3) and (3.4), with δ as given by (3.40). Case (L).
Indeed, it follows from (3.40) and (3.41) that δ = ˜ RK = 1 K ˜ Rr ( s ) r (0) ≤ K sr ( s ) r (0) , ∀ s ≥ ˜ R hence, (3.3) is fulfilled. Furthermore, it follows from (3.41) and (2.4) that M P ( ˜ R ) ≤ P ( R ) ⇐⇒ M (cid:90) ˜ R r ( s ) sds ≤ (cid:90) R r ( s ) sds ⇐⇒ M (cid:90) ˜ R sds ≤ (cid:90) R ˜ R sds ⇐⇒ M ˜ R ≤ R ⇐⇒ ˜ R ≤ √ M R
Case (NL).
Also in this case, it follows from (3.40) and (3.43) that δ = ˜ RK = ˜ RK ss r (0) ≤ K s s ˜ R r (0) = 1 K sr ( s ) r (0) , ∀ s ∈ ( ˜ R, R ]and that δ = ˜ RK ≤ ˜ RK sR r (0) = 1 K s s ˜ R r (0) ≤ K s R ˜ R r (0) = 1 K sr ( s ) r (0) , ∀ s ≥ R hence, (3.3) is again fulfilled. In addition, it follows from (3.43) and (2.4) that M P ( ˜ R ) ≤ P ( R ) ⇐⇒ M (cid:90) ˜ R r ( s ) sds ≤ (cid:90) R r ( s ) sds ⇐⇒ M (cid:90) ˜ R r ( s ) sds ≤ (cid:90) ˜ R r ( s ) sds + (cid:90) R ˜ R r ( s ) sds ⇐⇒ ( M − (cid:90) ˜ R sds ≤ R (cid:90) R ˜ R s ds ⇐⇒ ( M −
1) ˜ R ≤ R s (cid:12)(cid:12)(cid:12)(cid:12) R ˜ R ⇐⇒ M −
12 ˜ R ≤ R −
13 ˜ R ⇐⇒ M −
36 ˜ R + 26 ˜ R ≤ R ⇐⇒ M −
12 ˜ R ≤ R ⇐⇒ ˜ R ≤ (cid:18) M − (cid:19) R (cid:3) Remark 3.3.
At this point we derive the advantage of using the nonlinear controller overthe linear one by comparing the ratio of the maximal initial relative distance that maintainsconnectivity for these two cases. In both cases we have the same bound on the free input termsand the same feedback law up to some distance between neighboring agents, which allows usto compare their performance under the criterion of maximizing the largest initial distancebetween two interconnected agents. In particular, this ratio, which depends on the number ofedges in the systems’ graph, is given by
Rat ( M ) := √ M (cid:16) M − (cid:17) (3.45)It is evident from the plot of Rat ( M ) in Figure 1 that it is a decreasing function of M withvalues less than 1 for M ≥
1. The latter property follows quite intuitively if we take a look atFigure 2. Indeed, as sown in the proof of Proposition (3.1), both the maximal initial distance˜ R L for the linear and ˜ R NL for the nonlinear case are expressed by virtue of (3.4) as the solutionsof the equation P ( ˜ R ) P ( R ) − P ( ˜ R ) = M − . In Figure 2, both the quotient of the area inside the orangeframe over the area in the red frame and the quotient of the violet area over the blue area arethe same and equal to P ( ˜ R ) P ( R ) − P ( ˜ R ) = M − , with P ( · ) as defined for both the cases (L) and (NL)through (2.4), (3.41) and (3.43). It is thus straightforward that the quotient of ˜ R L over ˜ R NL ,namely Rat ( M ) should be less than 1. Yet, we also provide a formal proof of this argument bystudying the monotonicity of Rat ( · ).For convenience we consider the 6-th power of Rat ( M ), namely the function Rat ( M ) := √ M (cid:16) M − (cid:17) = (3 M − M Hence, by evaluating the derivative of
Rat ( M ) we obtain ddM Rat ( M ) = ddM (3 M − M = 2(3 M − · M − (3 M − M M
6= 3 · M (3 M − M − (3 M − M = 32 · M − M (1 − M ) < , for M > Figure 1.
This figure shows the ratio √ M / (cid:16) M − (cid:17) for the number of edgesranging from 2 to 150 sr ( s ) s ˜ R L ˜ R NL R Figure 2. ˜ R L and ˜ R NL and norms of the respective feedback control laws for M = 12 Invariance Analysis
In what follows, we assume that the agents’ initial states belong to a given domain D ⊂ R n .In order to simplify the subsequent analysis, we assume that D = int( B ( R )), namely theinterior of the ball with center 0 ∈ R n and radius R >
0. We aim at designing an appropriatemodification of the feedback law (3.1) which guarantees that the trajectories of the agentsremain in D for all future times.For each ε ∈ (0 , R ), let N ε be the region with distance ε from the boundary of ∂D towards theinterior of D , namely N ε := { x ∈ R n : R − ε ≤ | x | < R } (4.1)and D ε := D \ N ε (4.2)We proceed by defining a repulsive from the boundary of D vector field, which when addedto the dynamics of each agent in (3.1), will ensure the desired invariance of the closed loopsystem and simultaneously guarantee the same robust connectivity result established above.Let h : [0 , → [0 ,
1] be a Lipschitz continuous function that satisfies h (0) = 0; h (1) = 1; h ( · ) strictly increasing (4.3)We define the vector field g : D → R n as g ( x ) := (cid:40) − cδh (cid:16) ε + | x |− Rε (cid:17) x | x | , if x ∈ N ε , if x ∈ D ε (4.4)with h ( · ) as given above and appropriate positive constants c , δ which serve as design parame-ters. Then, it follows from (4.3), (4.4) and the Lipschitz property for h ( · ) that the vector field g ( · ) is Lipschitz continuous on D .Having defined the mappings for the extra term in the dynamics of the modified controllerwhich will guarantee the desired invariance property, we now state our main result. Proposition 4.1.
For the multi-agent system (2.1) , assume that D = int( B ( R )) , for certain R > and that (ICH) is fulfilled. Furthermore, let ε ∈ (0 , R ) , N ε and D ε as defined by (4.1) and (4.2) , respectively and assume that the initial states of all agents lie in D ε . Then, thereexists a control law (2.2) (with free inputs v i ) which guarantees both connectivity and invarianceof D for the solution of the system for all future times and is defined as u i = g ( x i ) − (cid:88) j ∈N i r ( | x i − x j | )( x i − x j ) + v i (4.5) with g ( · ) given in (4.4) and certain r ( · ) satisfying Property (P). We choose the same positiveconstant δ in both (3.5) and (4.4) and select the constant c in (4.4) greater that 1. Then theconnectivity-invariance result is valid provided that the parameters δ , ˜ R and the function r ( · ) satisfy the restrictions (3.3) , (3.4) and the input terms v i ( · ) , i = 1 , . . . , N satisfy (3.5) .Proof. We break the proof in two steps. In the first step, we show that as long as the invarianceassumption is satisfied, namely, the solution of the closed loop system (2.1)-(4.5) is defined andremains in D , network connectivity is maintained. In the second step, we show that for alltimes where the solution is defined, it remains inside a compact subset of D , which impliesthat the solution is defined and remains in D for all future times, thus providing the desiredinvariance property. OBUST CONNECTIVITY ANALYSIS FOR MULTI-AGENT SYSTEMS 13
Step 1: Proof of network connectivity.
The proof of this step is based on an appropriate modification of the corresponding proofof Proposition 3.1. In particular, we exploit the energy function V as given by (3.6) and showthat when | ∆ x | ∞ ≥ ˜ R , namely, when the maximum distance between two agents exceeds ˜ R then its derivative along the solutions of the closed loop system is negative. Thus by using thesame arguments with those in proof of Proposition 3.1 we can deduce that the system remainsconnected. Indeed, by evaluating the derivative of V along the solutions of (2.1)-(4.5) we obtain˙ V = N (cid:88) i =1 ∂∂x i V ( x ) ˙ x i = N (cid:88) i =1 ∂∂x i V ( x ) g ( x i ) − n (cid:88) l =1 c l ( x ) T L w ( x ) c l ( x ) + n (cid:88) l =1 c l ( x ) T L w ( x ) c l ( v ) ≤ N (cid:88) i =1 ∂∂x i V ( x ) g ( x i ) − n (cid:88) l =1 c l ( x ) T L w ( x ) c l ( x ) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) l =1 c l ( x ) T L w ( x ) c l ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4.6)By taking into account (3.11) and using precisely the same arguments with those in proof ofSteps 1 and 2 of Proposition 3.1 it suffices to show that the first term of inequality (4.6), whichby virtue of (3.7) is equal to N (cid:88) i =1 ∂∂x i V ( x ) g ( x i ) = N (cid:88) i =1 (cid:88) j ∈N i r ( | x i − x j | ) (cid:104) ( x i − x j ) , g ( x i ) (cid:105) is nonpositive for all x ∈ D . Given the partition D ε , N ε of D , we consider for each agent i ∈ N the partition N D ε i , N N ε i of its neighbors’ set, corresponding to its neighbors that belong to D ε and N ε , respectively. Also, we denote by E N ε the set of edges { i, j } with both x i , x j ∈ N ε .Then, by taking into account that due to (4.4), g ( x i ) = 0 for x i ∈ D ε , it follows that N (cid:88) i =1 (cid:88) j ∈N i r ( | x i − x j | ) (cid:104) ( x i − x j ) , g ( x i ) (cid:105) = (cid:88) { i ∈N : x i ∈ N ε } (cid:88) j ∈N i r ( | x i − x j | ) (cid:104) ( x i − x j ) , g ( x i ) (cid:105) = (cid:88) { i ∈N : x i ∈ N ε } (cid:88) j ∈N Dεi ∪N Nεi r ( | x i − x j | ) (cid:104) ( x i − x j ) , g ( x i ) (cid:105) = (cid:88) { i ∈N : x i ∈ N ε } (cid:88) j ∈N Dεi r ( | x i − x j | ) (cid:104) ( x i − x j ) , g ( x i ) (cid:105) + (cid:88) { i,j }∈E Nε r ( | x i − x j | )[ (cid:104) ( x i − x j ) , g ( x i ) (cid:105) + (cid:104) ( x j − x i ) , g ( x j ) (cid:105) ] (4.7)In order to prove that both terms in (4.7) are less than or equal to zero and hence derive ourdesired result on the sign of ˙ V , we exploit the following facts. Fact V.
Consider the vectors α, β, γ ∈ R n with the following properties: | α | = 1 , | β | = 1 (4.8) (cid:104) α, γ (cid:105) ≥ , (cid:104) β, γ (cid:105) ≤ λ α , λ β , µ α , µ β ∈ R ≥ satisfying λ α ≥ λ β , µ α ≥ µ β (4.10)it holds (cid:104) ( µ α α − µ β β ) , ˜ δ (cid:105) ≥ δ := λ α α + γ − λ β β (4.12)We provide the proof of Fact V in the Appendix. Fact VI.
For any x, ˜ x ∈ N ε with x = λ ˜ x , λ > y ∈ cl( D ε ) it holds (cid:104) (˜ x − y ) , x (cid:105) ≥ y ∈ cl( D ε ) ⇒ | y | ≤ R − ε and x, ˜ x ∈ N ε ⇒ R − ε ≤ | x | and R − ε ≤ | ˜ x | . Hence we have that (cid:104) (˜ x − y ) , x (cid:105) ≥ | x || ˜ x | − | x || y | ≥ Proof of the fact that the first term in (4.7) is nonpositive.
For each i, j in the firstterm in (4.7) we get by applying Fact VI with x, ˜ x = x i ∈ N ε and y = x j ∈ D ε that r ( | x i − x j | ) (cid:104) ( x i − x j ) , g ( x i ) (cid:105) = r ( | x i − x j | ) − cδh (cid:16) ε + | x i |− R ε (cid:17) | x i | (cid:104) ( x i − x j ) , x i (cid:105) ≤ Proof of the fact that the second term in (4.7) is nonpositive.
We exploit Fact V inorder to prove that for each { i, j } ∈ E N ε the quantity (cid:104) ( x i − x j ) , g ( x i ) (cid:105) + (cid:104) ( x j − x i ) , g ( x j ) (cid:105) (4.13)in the second term of (4.7) is nonpositive as well. Notice that both x i , x j ∈ N ε and withoutloss of generality we may assume that | x i | ≥ | x j | (4.14)namely, that x i is farther from the boundary of D ε than x j . Then by setting α := x i | x i | (4.15) β := x j | x j | (4.16) γ :=˜ x i − ˜ x j (4.17) OBUST CONNECTIVITY ANALYSIS FOR MULTI-AGENT SYSTEMS 15 with ˜ x i := x i − ( | x i | + ε − R ) x i | x i | (4.18)˜ x j := x j − ( | x j | + ε − R ) x j | x j | (4.19)and λ α := | x i | + ε − R (4.20) λ β := | x j | + ε − R (4.21) µ α := cδh ( | x i | + ε − R ) (4.22) µ β := cδh ( | x j | + ε − R ) (4.23)it follows from (4.15) and (4.16) that | α | = | β | = 1and from (4.3), (4.14), (4.20), (4.21), (4.22) and (4.23) that λ α ≥ λ β ≥ , µ α ≥ µ β ≥ | ˜ x i | = | ˜ x j | = R − ε ⇒ ˜ x i , ˜ x j ∈ ∂D ε . Thus, itfollows from (4.15), (4.16), (4.17) and application of Fact VI with x = x i , ˜ x = ˜ x i and y = ˜ x j that (cid:104) α, γ (cid:105) ≥ (cid:104) β, γ (cid:105) ≤ δ = λ α α + γ − λ β β = x i − x j (4.24)Thus we establish by virtue of (4.4), (4.11), (4.12), (4.15), (4.16), (4.22), (4.23) and (4.24) that (cid:104) ( µ α α − µ β β ) , ˜ δ (cid:105) = −(cid:104) ( g ( x i ) − g ( x j )) , ( x i − x j ) (cid:105) ≥ ⇐⇒(cid:104) ( x i − x j ) , g ( x i ) (cid:105) + (cid:104) ( x j − x i ) , g ( x j ) (cid:105) ≤ Step 2: Proof of forward invariance of D with respect to the solution of (2.1) - (4.5) . We proceed by proving that the control law (4.5) also guarantees the desired invarianceproperty for the solutions of system (2.1)-(4.5), provided that the input terms v i ( · ), i = 1 , . . . , N satisfy (3.5). Let [0 , T max ) be the maximal forward interval for which the solution x ( · ) of (2.1)-(4.5) with x (0) ∈ D ε exists and remains inside D . We claim that for all t ∈ [0 , T max ) thesolution remains inside cl( D ˜ c − c ε ) with˜ c = 1 h − (cid:0) c (cid:1) ⇐⇒ h (cid:18) c (cid:19) = 1 c ; ˜ c > c > h ( · ) are given in the statement of the proposition and (4.3), respectively.Then, it also follows from the fact that x ( t ) remains in the compact subset cl( D ˜ c − c ε ) of D for all t ∈ [0 , T max ), that T max = ∞ , which provides the desired result. In order to prove our claim,we need to define certain auxiliary mappings. For each i ∈ { , . . . , N } we define the functions m i ( t ) := (cid:26) ε + | x i ( t ) | − R , if x i ( t ) ∈ N ε , if x i ( t ) ∈ D ε , t ∈ [0 , T max ) (4.26)and m ( t ) := max { m i ( t ) : i = 1 , . . . , N } , t ∈ [0 , T max ) (4.27)where m i ( t ) denotes the distance of agent i from D ε at time t and m ( t ) is the maximum overthose distances for all agents. Hence, for all t ∈ [0 , T max ) and all ¯ ε ∈ (0 , ε ] we have the followingequivalences x i ( t ) ∈ N ¯ ε ⇐⇒ m i ( t ) ∈ [ ε − ¯ ε, ε ) (4.28) x i ( t ) ∈ ∂D ¯ ε ⇐⇒ m i ( t ) = ε − ¯ ε (4.29)and for all i = 1 , . . . , N that x i ( t ) ∈ cl( D ¯ ε ) , ∀ i = 1 , . . . , N ⇐⇒ m ( t ) ∈ [0 , ε − ¯ ε ] (4.30)Notice that the functions m i ( · ), i = 1 , . . . , N and m ( · ) are continuous and due to our hypothesisthat x (0) ∈ D ε , satisfy m (0) = 0 (4.31)We claim that m ( t ) ≤ ε ˜ c , ∀ t ∈ [0 , T max ) (4.32)with ˜ c as given in (4.25). Indeed, suppose on the contrary that there exists T ∈ (0 , T max ) suchthat m ( T ) = ˜ ε ∈ (cid:16) ε ˜ c , ε (cid:17) (4.33)and define τ := min (cid:26) ˜ τ ∈ [0 , T ] : m ( t ) ≥ (cid:16) ˜ ε + ε ˜ c (cid:17) , ∀ t ∈ [˜ τ , T ] (cid:27) (4.34)Then it follows from (4.33) that τ is well defined and from (4.31), (4.34) and the continuity of m ( · ) that m ( τ ) = 12 (cid:16) ˜ ε + ε ˜ c (cid:17) (4.35)and that there exists a sequence ( t ν ) ν ∈ N with t ν (cid:38) τ and m ( t ν ) ≥ (cid:16) ˜ ε + ε ˜ c (cid:17) , ∀ ν ∈ N (4.36)From (4.27), (4.35), (4.36) and the infinite pigeonhole principle we deduce that there exists i ∈ { , . . . , N } and a subsequence ( t ν k ) k ∈ N of ( t ν ) ν ∈ N such that m i ( t ν k ) ≥ (cid:16) ˜ ε + ε ˜ c (cid:17) , ∀ k ∈ N ; m i ( τ ) = 12 (cid:16) ˜ ε + ε ˜ c (cid:17) (4.37)Thus, it follows by virtue of (4.28) and (4.29) that x i ( t ν k ) ∈ N ε − ( ˜ ε + ε ˜ c ) , ∀ k ∈ N ; x i ( τ ) ∈ ∂D ε − ( ˜ ε + ε ˜ c ) (4.38) OBUST CONNECTIVITY ANALYSIS FOR MULTI-AGENT SYSTEMS 17
Set W ( x ) = | x | and notice that due to (4.38) it holds | x i ( t ν k ) | ≥ | x i ( τ ) | , ∀ k ∈ N . The latterimplies that ddt W ( x i ( t )) (cid:12)(cid:12)(cid:12)(cid:12) t = τ = lim k →∞ W ( x i ( t ν k )) − W ( x i ( τ )) t ν k − τ ≥ ddt W ( x i ( t )) (cid:12)(cid:12)(cid:12)(cid:12) t = τ = ∇ W ( x i ( τ )) ˙ x i ( τ )= x i ( τ ) T g ( x i ( τ )) + v i ( τ ) − (cid:88) j ∈N i r ( | x i ( τ ) − x j ( τ ) | )( x i ( τ ) − x j ( τ )) (4.40)By taking into account (4.33) and (4.38) we get that | x i ( τ ) | = R − ε + 12 (cid:16) ˜ ε + ε ˜ c (cid:17) > R − ε + ε ˜ c = R − ˜ c − c ε and hence from (4.3), (4.4) and (4.25) that | g ( x i ( τ )) | > cδh (cid:32) ε + R − ˜ c − c ε − R ε (cid:33) = cδh (cid:18) − ˜ c − c (cid:19) = cδh (cid:18) c (cid:19) = δ (4.41)Also, due to (4.4) it holds x i ( τ ) = − ag ( x i ( τ )) (4.42)for certain a >
0. Then we get from (4.41), (4.42) and the fact that | v i ( τ ) | ≤ δ that x i ( τ ) T [ g ( x i ( τ )) + v i ( τ )] ≤ x i ( τ ) T g ( x i ( τ )) + | x i ( τ ) || v i ( τ ) | = −| x i ( τ ) || g ( x i ( τ )) | + | x i ( τ ) || v i ( τ ) | = −| x i ( τ ) | ( | g ( x i ( τ )) | − | v i ( τ ) | ) < x j ( τ ) ∈ cl( D ε − ( ˜ ε + ε ˜ c )) for all j ∈ N i andfrom (4.38) that x i ( τ ) ∈ N ε − ( ˜ ε + ε ˜ c ). Thus, it follows from Fact VI that x i ( τ ) T ( x i ( τ ) − x j ( τ )) = (cid:104) x i ( τ ) , ( x i ( τ ) − x j ( τ )) (cid:105) ≤ x ( t ) remains in the compact subset cl( D ˜ c − c ε ) of D for all t ∈ [0 , T max ). Thus, T max = ∞ and we conclude that the solution x ( · ) of the system remains in D for all t ≥ (cid:3) Conclusions
We have provided a distributed control scheme which guarantees connectivity of a multi-agent network governed by single integrator dynamics. The corresponding control law is robustwith respect to additional free input terms which can further be exploited for motion planning.For the case of a spherical domain, adding a repulsive vector field near the boundary ensuresthat the agents remain inside the domain for all future times. The latter framework is motivatedby the fact that it allows us to abstract the behaviour of the system through a finite transitionsystem and exploit formal method tools for high level planning.
Further research directions include the generalization of the invariance result of Section 4 forthe case where the domain is convex and has smooth boundary and the improvement of thebound on the free input terms, by allowing the bound to be state dependent.6.
Appendix
In the Appendix, we provide the proofs of Facts I, II, III and IV which were used in proofof Proposition 3.1 and of Fact V, in proof of Proposition 4.1. For convenience we state theelementary inequality 2( | w | + | z | ) ≥ | w − z | , ∀ w, z ∈ R n (6.1) Proof of Fact I.
Let { e k } k =1 ,...,N be an orthonormal basis of eigenvectors corresponding tothe ordered eigenvalues of L w ( x ). Then, for each l = 1 , . . . , n we have that c l ( x ⊥ ) = N (cid:88) k =2 µ k e k ; µ k ∈ R , k = 2 , . . . , N and hence, that | c l ( x ⊥ ) | = (cid:32) N (cid:88) k =2 µ k (cid:33) Thus, we get that | L w ( x ) c l ( x ⊥ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) k =2 µ k L w ( x ) e k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) k =2 µ k λ k ( x ) e k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:32) N (cid:88) k =2 ( µ k λ k ( x )) (cid:33) ≥ λ ( x ) (cid:32) N (cid:88) k =2 µ k (cid:33) = λ ( x ) | c l ( x ⊥ ) | which establishes (3.18). Proof of Fact II.
By taking into account the Cauchy Schwartz inequality we obtain n (cid:88) l =1 | c l ( x ) || c l ( y ) | ≤ (cid:32) n (cid:88) l =1 | c l ( x ) | (cid:33) (cid:32) n (cid:88) l =1 | c l ( y ) | (cid:33) = (cid:32) n (cid:88) l =1 N (cid:88) i =1 c l ( x i ) (cid:33) (cid:32) n (cid:88) l =1 N (cid:88) i =1 c l ( y i ) (cid:33) = (cid:32) N (cid:88) i =1 n (cid:88) l =1 c l ( x i ) (cid:33) (cid:32) N (cid:88) i =1 n (cid:88) l =1 c l ( y i ) (cid:33) = (cid:32) N (cid:88) i =1 | x i | (cid:33) (cid:32) N (cid:88) l =1 | y i | (cid:33) = | x || y | and hence (3.19) holds. OBUST CONNECTIVITY ANALYSIS FOR MULTI-AGENT SYSTEMS 19
Proof of Fact III.
By the definition of x ⊥ and ¯ x , it follows that there exists ˜ x ∈ R n such that x − x ⊥ = ¯ x = (˜ x, . . . , ˜ x ) ∈ R Nn . Hence, we have that | x ⊥ | = | x − ¯ x | = | ( x , . . . , x N ) − (˜ x, . . . , ˜ x ) | = | ( x − ˜ x, . . . , x N − ˜ x ) | = (cid:32) N (cid:88) i =1 | x i − ˜ x | (cid:33) ⇒ (cid:112) N − | x ⊥ | = (cid:32) N (cid:88) i =1 N − | x i − ˜ x | (cid:33) = (cid:88) { i,j }∈E ( K ( N )) | x i − ˜ x | + | x j − ˜ x | ) where E ( K ( N )) stands for the edge set of the complete graph with vertex set N . Then, itfollows from (6.1) that (cid:88) { i,j }∈E ( K ( N )) | x i − ˜ x | + | x j − ˜ x | ) ≥ (cid:88) { i,j }∈E ( K ( N )) | x i − x j | ≥ (cid:88) { i,j }∈E | x i − x j | = | ∆ x | which provides the desired result. Proof of Fact IV.
Notice that (3.21) is equivalently written as √ | x ⊥ | ≥ max { i,j }∈E | x i − x j | ⇐⇒ | x ⊥ | ≥ max { i,j }∈E | x i − x j | ⇐⇒ (cid:32) N (cid:88) i =1 | x i − ˜ x | (cid:33) ≥ max { i,j }∈E | x i − x j | with ˜ x ∈ R n as in proof of Fact III. Let { ˆ i, ˆ j } ∈ E such that | x ˆ i − x ˆ j | = max { i,j }∈E | x i − x j | .Then, by taking into account (6.1) we have2 (cid:32) N (cid:88) i =1 | x i − ˜ x | (cid:33) ≥ | x ˆ i − ˜ x | + | x ˆ j − ˜ x | ) ≥ | x ˆ i − x ˆ j | = max { i,j }∈E | x i − x j | and thus (3.21) is fulfilled. Proof of Fact V.
By taking into account (4.8)-(4.10) and (4.12) we evaluate (cid:104) ( µ α α − µ β β ) , ˜ δ (cid:105) = (cid:104) ( µ α α − µ β β ) , ( λ α α − λ β β ) + γ (cid:105) = (cid:104) ( µ α α − µ β β ) , ( λ α α − λ β β ) (cid:105) + µ α (cid:104) α, γ (cid:105) − µ β (cid:104) β, γ (cid:105)≥ (cid:104) ( µ α α − µ β β ) , ( λ α α − λ β β ) (cid:105) = µ α λ α | α | − ( µ α λ β + µ β λ α ) (cid:104) α, β (cid:105) + µ β λ β | β | ≥ µ α λ α | α | − ( µ α λ β + µ β λ α ) | α || β | + µ β λ β | β | = µ α λ α − µ α λ β − µ β λ α + µ β λ β = µ α ( λ α − λ β ) − µ β ( λ α − λ β ) = ( µ α − µ β )( λ α − λ β ) ≥ Acknowledgements
This work was supported by the EU STREP RECONFIG: FP7-ICT-2011-9-600825, theH2020 ERC Starting Grant BUCOPHSYS and the Swedish Research Council (VR).
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Department of Automatic Control, School of Electrical Engineering, KTH Royal Institute ofTechnology, Osquldas v¨ag 10, 10044, Stockholm, Sweden
E-mail address : [email protected] Department of Automatic Control, School of Electrical Engineering, KTH Royal Institute ofTechnology, Osquldas v¨ag 10, 10044, Stockholm, Sweden
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