Optimization of the H ∞ -norm of Dynamic Flow Networks
Alexander Johansson, Jieqiang Wei, Henrik Sandberg, Karl H. Johansson, Jie Chen
OOptimization of the H ∞ -norm of Dynamic Flow Networks Alexander Johansson, Jieqiang Wei, Henrik Sandberg, Karl H. Johansson and Jie Chen
Abstract — In this paper, we study the H ∞ - norm of linearsystems over graphs, which is used to model distributionnetworks. In particular, we aim to minimize the H ∞ - normsubject to allocation of the weights on the edges. The op-timization problem is formulated with LMI (Linear-Matrix-Inequality) constraints. For distribution networks with one port,i.e., SISO systems, we show that the H ∞ - norm coincides withthe effective resistance between the nodes in the port. Moreover,we derive an upper bound of the H ∞ - norm, which is interms of the algebraic connectivity of the graph on which thedistribution network is defined. I. I
NTRODUCTION
In this paper we study robustness of a basic model for thedynamics of a distribution network. Identifying the networkwith a undirected graph we associate with every vertex ofthe graph a state variable corresponding to storage , and withevery edge a control input variable corresponding to flow .Furthermore, some of the vertices serve as terminals wherean unknown flow may enter or leave the network in such away that the total sum of inflows and outflows is equal tozero. Many control protocols are designed for a distributedcontrol structure (the control input corresponding to a givenedge only depending on the difference of the state variablesof the adjacent vertices) which will ensure that the statevariables associated to all vertices will converge to the samevalue, i.e., reach consensus, [5],[12].In this paper, we consider the distribution network con-trolled by proportional controllers on the edges and studythe robustness property with respect to the controller gain,i.e., the edges weights. In particular, we are interested inminimizing the H ∞ - norm by allocating the edge weights.The distribution networks can be seen as linear time-invariant port-Hamiltonian systems [2], [1], but also residesin the category of state-space symmetric systems [21], [11],[16], [22], [15]. One important property of the state-spacesymmetric system is that its H ∞ - norm is attained at thezero frequency [20], which is employed to solve the currentproblem.The contributions of this paper are: The problem of mini-mizing the H ∞ - norm of the distribution networks subject tothe allocation of the edge weights is formulated and writtenwith LMIs as constraints. Moreover, we give an interpretation *This work is supported by Knut and Alice Wallenberg Foundation,Swedish Research Council, Swedish Foundation for Strategic Research andby Hong Kong Research Grants Council (CityU 11200415).A. Johansson, J. Wei, H. Sandberg and K.H. Johansson are withthe ACCESS Linnaeus Centre, School of Electrical Engineering. KTHRoyal Institute of Technology, SE-100 44 Stockholm, Sweden. Emails: { jieqiang, kallej } @kth.se Jie Chen is with the Department of Electronic Engineering, City Univer-sity of Hong Kong, Hong Kong, China. of the Riccati inequality which regards definitness of aLaplacian to a graph containing both positive and negativeweights on the edges. As a consequence of the interpretation,it is shown for distribution networks with one port, that the H ∞ - norm (or induced L - gain) is equal to the effectiveresistance between the nodes in the port. Then, an upperbound of the H ∞ - norm is derived, which relates to thealgebraic connectivity of the graph on which the distributionnetwork is defined. The results in this paper can be relevantwhen designing robust multi-agent systems. In particularwhen considering a malicious attacker, e.g., [17].The structure of the paper is as follows. Some prelim-inaries will be given in Section II. The considered classof dynamic flow networks is given and the optimizationproblem is formulated in Section III. The main results ispresented in Section IV. In Section VI there is a numericalexample which demonstrates some results from this paper.Conclusions and future work are given in Section VII andVIII, respectively. Notation.
A positive semi-definite (symmetric) matrix M is denoted as M (cid:60) . A positive definite (symmetric) matrix M is denoted as M (cid:31) . The i th row of a matrix M isgiven by M i . The element on the i th row and j th columnof a matrix M is denoted M ij . The vectors e , e , . . . , e n denote the canonical basis of R n , whereas the vectors n and n represent a n -dimensional column vector with eachentry being and , respectively. We will omit the subscript n when no confusion arises. The euclidean norm is denotedas | · | , for a vector x ∈ R n , | x | = ( x + · · · + x n ) .II. P RELIMINARIES
In this section, we briefly review some essentials aboutgraph theory [6], and give some definitions for robust anal-ysis [24].
A. Graph Theory
An undirected graph G = ( W , V , E ) consists of a finite setof nodes V = { v , ..., v n } , a set of edges E = {E , ..., E m } which contains unordered pairs of elements of V , and a setof corresponding edge weights W = { w , ..., w m } . Graphswith unit weights, i.e., w i = 1 , for i = 1 , ..., m , are denotedas G = ( V , E ) . The set of neighbours to node i is N i = { v j | ( v i , v j ) ∈ E} . The graph Laplacian L ∈ R n × n is defined component-wiselyas L ij = (cid:80) j ∈ N i w ij if i = j, − w ij if j ∈ N i \ { i } , if j / ∈ N i . a r X i v : . [ m a t h . O C ] O c t iven an orientation for each edge, the incidence matrix B ∈ R n × m is defined as B ij = if E j starts in node v i , − if E j ends in node v i , else . These two matrices are related by L = BW B T , where W = diag( w , ..., w m ) . If W (cid:62) then the eigenvaluesof L w can be structured as λ (cid:54) λ (cid:54) ... (cid:54) λ n , where the eigenvector corresponding to λ = 0 is (cid:62) = [1 , ..., T . The second smallest eigenvalue, i.e., λ , iscommonly referred to as the algebraic connectivity [9] andis a measure of how connected a graph is. Furthermore, if G is connected, then λ > .If some weights are negative, the Laplacian can be decom-posed as L = L + + L − = B + W + B T + + B − W − B T − , where B + and B − are incidence matrices correspondingto the positive and negative sub-graphs, respectively. Thematrices W + and W − are the weights of the positive andnegative sub graphs, respectively. This decomposition is alsoused in e.g., [7].A measure of the connectivity between two nodes in G = ( W , V , E ) is the effective resistance [8]. The effectiveresistance between the nodes v i and v j is defined as R ij = ( e i − e j ) T L † ( e i − e j ) , where L † is the Moore-Penrose pseudo inverse of L . Lemma 1 ([23],Theorem III.3):
Assume G = ( W , V , E ) has one edge with negative weight and the negative edgeis E − = ( u, v ) . Let G + be the positive sub-graph of G = ( W , V , E ) and assume it is connected. Then L ( G ) is positive semi-definite if and only if | W ( E − ) | (cid:54) R − uv ( G + ) , where W ( E − ) is the negative weight and R uv denotes theeffective resistance between node u and v . B. L -Norm and induced L -Gain In this subsection, we recall some definitions from robustcontrol. The notations used in this paper are fairly standardand are consistent with [24], [18]. The space of square-integrable signals f : [0 , ∞ ) → R n is denoted by L [0 , ∞ ) .For the linear time-invariant system ˙ x = Ax + Bu, (1) y = Cx + Du, the transfer matrix is G ( s ) = C ( sI − A ) − B + D , whichhas the impulse response g ( t ) = L − { G ( s ) } = Ce At B + ( t ) + Dδ ( t ) , where δ ( t ) is the unit impulse and + ( t ) is the unit stepdefined as + ( t ) = (cid:40) , t (cid:62) , , t < . If x (0) = 0 , then we have y ( t ) = (cid:82) t g ( t − τ ) u ( τ ) dτ. Thenthe induced L - gain is defined as (cid:107) g (cid:107) − ind = sup u ∈ L [0 , ∞ ) (cid:107) y (cid:107) (cid:107) u (cid:107) = sup u ∈ L [0 , ∞ ) (cid:107) g ∗ u (cid:107) (cid:107) u (cid:107) , where (cid:107) u ( t ) (cid:107) = (cid:16) (cid:82) ∞ | u ( t ) | dt (cid:17) . This induced L - gain, i.e., (cid:107) g (cid:107) − ind or (cid:107) G (cid:107) − ind , isoften called the H ∞ - norm, denoted as (cid:107) G (cid:107) ∞ . It is well-know that (cid:107) G (cid:107) ∞ = sup ω ∈ R ¯ σ { G ( jω ) } , where ¯ σ ( A ) denotethe largest singular value of the matrix A .For the system (1) with D = 0 , the bounded real lemma[24] implies that (cid:107) G (cid:107) ∞ (cid:54) γ if and only if there exists P = P (cid:62) (cid:31) such that P A + A (cid:62) P + C (cid:62) C + 1 γ P BB (cid:62) P (cid:52) . (2)III. P ROBLEM F ORMULATION
We consider the dynamical distribution network definedon a graph G = {V , E} with |V| = n and |E| = m . On thevertices, we consider integrators, given as ˙ x = u, x, u ∈ R n , (3) z = x, z ∈ R n . Here the i th element of x and u , i.e. x i and u i , are the stateand input variables associated with the i th vertex of the graph.System (3) defines a port-Hamiltonian system [1], satisfyingthe energy-balance ddt | x | u T z. As a next step we will extend the dynamical system (3)with an external input d of inflows and outflows ˙ x = u + Ed, d ∈ R k ,z = x, where E is a n × k matrix whose columns consist of oneelement which is (inflow) and one element − (outflow),while the rest of the elements are zero. A port is a set ofnodes(terminals) to where the external flow which enter andleave the network sums to zero. Thus, E specifies k ports.To achieve a state consensus, many controllers whichprovide the flows on the edges of G have been proposed,with the following general form ˙ η k = f k ( η k , ζ k ) ,µ k = g k ( η k , ζ k ) , k = 1 , , . . . , m (4)where η k , ζ k , µ k are respectively the states, input and outputof the controller on the k th edge of G . Denote the stacked vec-tors of η k , ζ k , µ k as η, ζ, µ respectively. With the controller(4), the state variables x i , i = 1 , , . . . , n, are controlled by2he controller output µ k , k = 1 , , . . . , m, in the followingmanner u + BW µ = 0 , where B ∈ R n × m is the incidence matrix of the digraph G , and W is the diagonal matrix corresponding to the gainof the controller to the edges. In addition, the controller isdriven by the relative output of the systems (3) on vertices,i.e ζ = B T z. It is known that, if d = 0 , the state agreement of thesystem (3) can be achieved by P-control and PI-control. Forthe P-control, the closed-loop is, ˙ x = − L w x + Ed,y = E T x, (5)where y is a vector with the components being the statedifference at each port. Example 1:
One physical interpretation of the system (5)is a basic model of a dynamic flow network, where there arewater reservoirs on the nodes and pipes on the edges. Thereservoirs are identical cylinders and the pipes are horizontal.The state x is constituted by the water levels in the reservoirsand the pressures are proportional to the water levels. Theflow in the pipes are passively driven by pressure differencebetween the reservoirs. The weights W are representing thecapacities of the pipes, in terms of diameter and friction.The passive flow from reservoir i to reservoir j is then q ij = w ij ( x i − x j ) . The external input d can e.g. beinterpreted as flow in pumps which are distributing waterinside the network. The output y is then the differencebetween water levels of the reservoirs which the pumps arepumping to and the reservoirs which the pumps are pumpingfrom.Another physical interpretation of the system (5) is a mass-damper system, where there are masses on the nodes anddampers on the edges. The damping force is proportional tothe relative velocity of the connected masses. The state x isconstituted by the momentum of the masses. The weights W are representing the damping constants. The input d is representing external forces, which are exposing somemasses to push and some masses to pull. The total pushis equal to the total pull. The output y is the difference inmomentum between the masses which are exposed to pushand the masses which are exposed to pull.There are many other interpretations and applications ofthe system (5). Others are e.g., chemical reaction networks[3] and consensus protocols [19].In this paper we are interested in the following problem:For a given topology, how to achieve the best robust per-formance of the system (5) by arranging the weights on theedges, i.e., min W (cid:107) G (cid:107) ∞ (6) s.t., (cid:88) w i = c, w i (cid:62) , where G is the transfer function of the system (5), W = diag( w , ..., w m ) and w i , for i = 1 , ..., m , are theweights on the edges. The constant c is the constraint on thesum of all edge weights. Example 2 (flow network continued):
For the flow net-work interpretation of the system (5), the optimization prob-lem above is to allocate capacities of the water pipes such the H ∞ - norm of the flow network is minimized. The constant c represents the total capacity of the pipes.IV. H ∞ - NORM OF THE DISTRIBUTION NETWORK
A. Optimization problem reformulated with LMI constraints
We start this subsection by reformulating problem (6) asan equivalent optimization problem with LMIs as constraints,which can then be efficiently solved numerically using, e.g.,with Yalmip [13].
Theorem 2:
Consider the system (5). If the H ∞ - norm isless than or equal to γ , then the following LMI is satisfied, (cid:20) L w EE (cid:62) γI k (cid:21) (cid:60) . (7) Proof:
Denote U (cid:62) = [ n , u (cid:62) , . . . , u (cid:62) n ] and U (cid:62) = [ u (cid:62) , . . . , u (cid:62) n ] , for which U L w U (cid:62) = diag(0 , λ , . . . , λ n ) =: Λ . Denote ˆΛ = diag( λ , . . . , λ n ) . Then the system (5) has equal H ∞ - norm as the system ˙˜ x = − Λ˜ x + U Ed,z = E (cid:62) U (cid:62) ˜ x. Notice that the first row of
U E is zero, thus the H ∞ - normof the system (5) equals the H ∞ - norm of the system ˙ˆ x = − ˆΛˆ x + U Ed,z = E (cid:62) U (cid:62) ˆ x. (8)Due to symmetry of the system and by Theorem 6 in [20],the H ∞ - norm of the system (8) is (cid:107) E (cid:62) U (cid:62) ˆΛ − U E (cid:107) . The H ∞ - norm of the system (5) is then less or equal to γ if andonly if (cid:107) E (cid:62) U (cid:62) ˆΛ − U E (cid:107) (cid:52) γ. By the property of real symmetric matrix, we can furtherrewrite the previous constrain as E (cid:62) U (cid:62) ˆΛ − U E (cid:52) γI k .By Schur complement, we have (cid:20) ˆΛ U EE (cid:62) U (cid:62) γI k (cid:21) (cid:60) , which is equivalent to (cid:20) Λ U EE (cid:62) U (cid:62) γI k (cid:21) (cid:60) . By pre and post multiplication of matrix diag( U (cid:62) , I k ) and diag( U, I k ) , respectively, the previous inequality is trans-formed to (cid:20) L w EE (cid:62) γI k (cid:21) (cid:60) . Then the conclusion follows.3 emark 1:
By Theorem 2, the optimization problem (6)is equivalent to min W γs.t., (cid:20) L w EE (cid:62) γI k (cid:21) (cid:60) , (cid:88) w i = c, w i (cid:62) . (9)Since the constraints are LMIs, this optimization problem canefficiently be solved with Yalmip. The set up above is usedlater in Section VI, there the optimal edge weight allocationis determined for the system which is illustrated in Figure 1and the optimal H ∞ - norm is verified in a simulation.In Theorem 2, we proved that the inequality (7) is satisfiedif the H ∞ - norm is less than or equal to γ . Moreover, bythe bounded real lemma we have that if (cid:107) G (cid:107) ∞ (cid:54) γ , thereexists P = P (cid:62) (cid:31) such that − P L w − L Tw P + EE T + 1 γ P EE T P (cid:52) . (10)In the next result, we provide one explicit solution to (10). Theorem 3:
Consider the system (5). If the H ∞ - norm isless then or equal to γ , then P = γI is a solution to theRiccati inequality (10). Proof: If (cid:107) G (cid:107) ∞ (cid:54) γ , then by the Schur complement,the LMI (7) is equivalent to − L w + EE T γ (cid:52) . (11)Furthermore, notice that by choosing P = γI , the Riccatiinequality (10) is equivalent to (11). Hence the conclusionfollows. B. Graphical interpretation of the Riccati inequality fordynamic flow networks
In this subsection, we give a graphical interpretation of theRiccati inequality (7) which is equivalent to (11) (by Schurcomplement), for a special type of dynamic flow networks.More precisely, we assume that each column of E has exactlytwo non-zero elements, one is and the other is − . By thisrestriction of E , it has the structure of an incidence matrixand EE T is therefore a Laplacian. For γ > , let us define L γ = − γ EE T , and denote the corresponding graph as G γ = ( W γ , V γ , E γ ) ,where W γ = {− γ , ..., − γ } and V γ = V . The set of edges E γ is determined by E . Recall that G = ( W , V , E ) is thegraph on which system (5) is defined. Moreover, we define ˜ L = L w + L γ , which is a Laplacian with both positiveand negative weights on the edges. The inequality (11) thenequals to ˜ L (cid:60) . Hence the H ∞ - norm of system (5)coincides with the largest magnitude of the negative weights γ , which yields a positive definite Laplacian ˜ L . Notice thatit is possible for ˜ L to have negative weights. Example 3:
The connection between G = ( W , V , E ) and G γ = ( W γ , V γ , E γ ) is illustrated in this example. Consider asystem as in (5), where w w w w d d d d Fig. 1. The graph on which the system (12) is defined. The external inputsto the system, i.e., d and d , is also marked. The output from the systemis y = x − x and y = x − x . The ports of this system are ( d , y ) and ( d , y ) . L w = w + w − w − w − w w + w − w − w w + w − w − w − w w + w ,E T = (cid:34) − − (cid:35) . (12)This dynamic flow system is defined on the graph G , whichis illustrated in Figure 1. For this system, the graph G γ whichcorresponds to L γ is illustrated in Figure 2. The induced L - gain from d = [ d , d ] T to y = [ x − x , x − x ] T isupper bounded by the largest magnitude of the weights − γ which yields a positive definite ˜ L . C. Connection between H ∞ - norm and effective resistance In the previous subsection, we reinterpret the H ∞ - normof system (5) in the scenario of the Laplacian ˜ L withnegative weights. Further conclusions can be drawn fromthe reasoning above about the definiteness of ˜ L if we set therestriction to SISO case, i.e., E = e i − e j . The H ∞ - norm isthen shown to coincide with the effective resistance betweennode i and j . Theorem 4:
Consider system (5) defined on G = ( W , V , E ) , which is undirected and only containspositive edge weights. Moreover, assume that there is oneport, i.e., d ∈ R and E = e i − e j . Then the induced L -gainfrom d to y is γ = R ij ( L w ) , where R ij denotes the effective resistance between the nodesin the port. Namely, nodes the i and j .4 − γ − γ Fig. 2. For the system (12), which is illustrated in Figure 1. The graphwhich corresponds to L γ = − γ EE T is illustrated in this figure. Theinduced L - gain from [ d , d ] T to [ y , y ] T is upper bounded by thelargest magnitude of − γ , which yields a positive definite ˜ L = L w + L γ . Proof:
First note that (11) is composed by a positive anda negative graph Laplacian. The negative graph Laplacian hasweights − γ . By Lemma 1, the inequality (11) is satisfied ifand only if γ (cid:54) R − ij ( L w ) ⇐⇒ γ (cid:62) R ij ( L w ) . V. H ∞ - NORM B OUNDED BY A LGEBRAIC C ONNECTIVITY
In Sections IV-B and IV-C, we showed that the H ∞ - normhas explicit graphical interpretation for a special matrix E .In this section, we focus on general matrices E . Here weprovide one preliminary result which relates the H ∞ - normof system (5) to the algebraic connectivity of the underlyinggraph. This result can be used if the location of the ports isunknown. Lemma 5:
Consider system (5). Then, the H ∞ - norm isbounded by γ = ¯ λ EE T λ , where λ is the second smallest eigenvalue of the weightedLaplacian L w and ¯ λ EE T is the largest eigenvalue of EE T . Proof:
The result is shown by using that L = 0 and E T = 0 and by applying the Courant-Fischer principle(e.g., [14] and [4]) on inequality (11). Remark 2:
By the previous lemma, maximizing the alge-braic connectivity of the graph G (with respect to the edgeweights) is suboptimal to minimizing the H ∞ - norm, forgiven a E . This result can be relevant for design of robustsystems when E is unknown. This is e.g. the scenario if amalicious attacker is considered and the attacked nodes areunknown. . . d d d d Fig. 3. Flow network (12) with the optimal allocated pipe capacities. I.e., w = 0 . , w = 0 . , w = w = 0 . The H ∞ - norm correspondingto this allocation is γ ∗ = 5 . Example 4:
Consider the dynamic flow network (5) andthe capacity of the pipes is to be allocated in order tominimize the H ∞ - norm of the system, i.e., the optimizationproblem (6). However, the only information about E which isavailable is the largest eigenvalue of EE T , i.e., ¯ λ EE T . Sincefull information about E is not available, it is not possibleto minimize the H ∞ - norm. Instead, by Lemma 5, we canminimize an upper bound by max W λ ( L w ) s.t., (cid:88) ω i = c. This problem of maximizing algebraic connectivity withrespect to the edge weights is well-studied, e.g., [9] and [10].VI. N
UMERICAL E XAMPLE
In this section we will demonstrate the results from SectionIV-A. For this purpose the dynamic flow network in Example3 (Figure 1) is used. We aim to allocate capacities of thepipes in order to minimize the H ∞ - norm. The optimalallocation of the pipe capacity and the optimal H ∞ - norm isdetermined numerically in Yalmip and the optimization setup (9).The total pipe capacity is set to c = 1 . The optimalallocation of the pipe capacity is w ∗ = 0 . , w ∗ = 0 . , w ∗ = w ∗ = 0 and the optimal H ∞ - norm is γ ∗ = 5 . Theflow network with optimally allocated pipe capacities is seenin Figure 3.Next, we induce an input to the system in order to verifythe H ∞ - norm. The input is d ( t ) = [1 , T if (cid:54) t < , [1 , T if (cid:54) t < , [0 , T if (cid:54) t. ig. 4. The L -norm of the output, i.e., || y ( t ) || , is seen together with the L -norm of the input scaled with the induced L - gain, i.e., γ ∗ || d ( t ) || . In Figure 4, the L -norm of the output, i.e. || y ( t ) || , isseen together with the L -norm of the input, scaled with theoptimal H ∞ - norm, i.e. γ ∗ || d ( t ) || . In the figure it is seenthat || y ( t ) || (cid:54) γ ∗ || d ( t ) || , hence the H ∞ - norm is verified.VII. C ONCLUSIONS
For the dynamic flow networks which we have considered,we have derived an optimization set up with LMIs asconstraints, which minimizes the H ∞ - norm with respect tothe allocation of the capacity of the pipes. Moreover, for theflow networks, we have interpreted the Riccati inequality asa definiteness criterion of a Laplacian to a graph containingboth positive and negative weights on the edges. For flownetworks which are SISO, more precisely, E = e i − e j , wehave shown that the H ∞ -norm coincides with the effectiveresistance between node i and node j . Moreover, we havederived an upper bound of the induced H ∞ -norm of theflow networks. This upper bound relates to the algebraicconnectivity on which the flow network is defined. Thisupper bound can be relevant when full information aboutthe input matrix, i.e E , is not available. Then, the capacitiesof the pipes can be allocated to get a suboptimal solutionwhich bounds the H ∞ - norm.VIII. F UTURE WORK
A related future topic is the problem of minimizingthe H ∞ - norm of dynamic flow networks with respect totopology, more precisely, a limited amount of edges is to beallocated in a graph with fixed vertices. Another future topicis to consider a fixed graph (both topology and weights), butconsider saturation of the flow on the edges. The problemis then to minimize the induced L -gain with respect toallocation of the saturation limits.R EFERENCES[1] A.J. van der Schaft. L -Gain and Passivity Techniques in NonlinearControl , volume 218, 2nd edition, Springer, London, 2000 of LectureNotes in Control and Information Sciences . Springer-Verlag, Berlin,1996.[2] A.J van der Schaft and B.M. Maschke. The Hamiltonian formulationof energy conserving physical systems with external ports.
Archiv f¨urElektronik und ¨Ubertragungstechnik , 49:362–371, 1995.[3] A.J. van der Schaft and B.M. Maschke. Port-Hamiltonian systems ongraphs.
SIAM J. Control and Optimization , 51(2):906–937, 2013. [4] R. Bellman.
Introduction to Matrix Analysis . MacGraw-Hill series inmatrix theory. McGraw-Hill Interamericana, 1960.[5] F. Blanchini, S. Miani, and W. Ukovich. Control of production-distribution systems with unknown inputs and system failures.
IEEETransactions on Automatic Control , 45(6):1072–1081, 2000.[6] B. Bollobas.
Modern Graph Theory , volume 184 of
Graduate Textsin Mathematics . Springer, New York, 1998.[7] Wei Chen, Dan Wang, Ji Liu, Tamer Baar, Karl H. Johansson, andLi Qiu. On semidefiniteness of signed laplacians with application tomicrogrids. , 49(22):97 – 102, 2016.[8] F. D¨orfler and F. Bullo. Kron reduction of graphs with applicationsto electrical networks.
IEEE Transactions on Circuits and Systems I:Regular Papers , 60(1):150–163, 2013.[9] Miroslav Fiedler. Algebraic connectivity of graphs.
CzechoslovakMathematical Journal , 23(2):298–305, 1973.[10] Arpita Ghosh, Stephen Boyd, and Amin Saberi. Minimizing effectiveresistance of a graph.
SIAM Review , 50(1):37–66, 2008.[11] M. Ikeda. Symmetric controllers for symmetric plants. In
Proceedingsof the 3rd European control conference , pages 989–994, 1995.[12] J. Wei and A.J. van der Schaft. Load balancing of dynamicaldistribution networks with flow constraints and unknown in/outflows.
Systems & Control Letters , 62(11):1001–1008, 2013.[13] J. Lofberg. Yalmip : a toolbox for modeling and optimization inmatlab. In , pages 284–289, 2004.[14] Bojan Mohar. The laplacian spectrum of graphs. In
Graph Theory,Combinatorics, and Applications , pages 871–898. Wiley, 1991.[15] T. Nagashio and T. Kida. Symmetric controller design for symmetricplant using matrix inequality conditions. In
Proceedings of the 44thIEEE Conference on Decision and Control , pages 7704–7707, 2005.[16] L. Qiu. On the robustness of symmetric systems.
Systems & ControlLetters , 27(3):187 – 190, 1996.[17] A. Rai, D. Ward, S. Roy, and S. Warnick. Vulnerable links and securearchitectures in the stabilization of networks of controlled dynamicalsystems. In , pages 1248–1253, 2012.[18] Anders Rantzer. Scalable control of positive systems.
EuropeanJournal of Control , 24:72 – 80, 2015.[19] R.O. Saber and R.M. Murray. Consensus protocols for networks ofdynamic agents. In
American Control Conference, 2003. Proceedingsof the 2003 , volume 2, pages 951–956, 2003.[20] K. Tan and K. M. Grigoriadis. Stabilization and H ∞ control ofsymmetric systems: an explicit solution. Systems & Control Letters ,44(1):57 – 72, 2001.[21] J. C. Willems. Realization of systems with internal passivity andsymmetry constraints.
Journal of the Franklin Institute , 301(6):605 –621, 1976.[22] G. Yang, J. Wang, and Y. Soh. Decentralized control of symmetricsystems.
Systems & Control Letters , 42(2):145 – 149, 2001.[23] D. Zelazo and M. Brger. On the definiteness of the weighted laplacianand its connection to effective resistance. In , pages 2895–2900, 2014.[24] K. Zhou and J.C. Doyle.
Essentials of Robust Control . Prentice HallModular Series f. Prentice Hall, 1998.. Prentice HallModular Series f. Prentice Hall, 1998.