Balancing time-varying demand-supply in distribution networks: an internal model approach
aa r X i v : . [ m a t h . O C ] F e b Balancing time-varying demand-supply in distributionnetworks: an internal model approach
Claudio De Persis ∗ Abstract
The problem of load balancing in a distribution network under unknown time-varying demand and supply is studied. A set of distributed controllers which regulatethe amount of flow through the edges is designed to guarantee convergence of the solu-tion to the steady state solution. The results are then extended to a class of nonlinearsystems and compared with existing results. Incremental passivity and internal modelare the main analytical tools.
Cooperative control systems have been widely investigated in a variety of different contexts[16, 12, 15, 3]. Less attention has been devoted to cooperative control in the framework ofdynamical flow networks, with some interesting exceptions [9, 6, 8, 7, 17, 5]. The aim ofthis paper is to study a class of cooperative control algorithms in the context of distributionnetworks under exogenous inputs.
Main contribution.
We analyze and design distributed controllers at the edge which achieveload balancing in the presence of time-varying demand and supply (exogenous signals). Therole of internal model and incremental passivity is investigated for the problem at hand.Similar tools have been used for controlled synchronization and leader-follower formationcontrol in e.g. [18, 3, 15, 10] and references therein. We address a different problem and wetackle it in a novel way. The load distribution problem is then considered for a more generalclass of systems and this allows us to make a comparison with the results of [2] and [9].
Literature review.
The literature on the control of flow or distribution networks is wideand multi-disciplinary. Here we restrict ourselves to a very small portion of it, focusing on amodel which takes into account the amount of stored material at the nodes and mass balance.This class of systems has been used to model data networks [13] and supply chains [1] forinstance. Our paper focuses on the problem of stabilizing the flow network to a steady state ∗ C. De Persis is with ITM, Faculty of Mathematics and Natural Sciences (FWN), University of Groningen,9747 AG Groningen, The Netherlands, [email protected] and Department of Computer, Control andManagement Engineering, Sapienza Universit`a di Roma, Italy. This research is partially supported by anFWN starting grant.
Consider the system ˙ x = Bλ + P d (1)with x ∈ R n the state, λ ∈ R m the control vector and d ∈ R q , q ≤ n , a disturbance vector.The ( n × m ) matrix B is the incidence matrix of an undirected graph G = ( V, E ) where | V | = n , | E | = m . The ends of the edges of G are labeled with a ‘+’ and a ‘-’. Then b ik = +1 i is the positive end of k − i is the negative end of k x i ∈ R , i ∈ I := 1 , , . . . , n represents the quantity of material stored at the node i , λ k ∈ R , k = 1 , , . . . , m the flowthrough the edge k . The disturbance d j ∈ R represents the inflow or the outflow at somenode.The available measurements are the differences among the quantities stored at the nodesnamely, z = B T x .We assume that each disturbance d j is supposed to be generated by the exosystem˙ w j = S dj w j d j = Γ dj w j , j = 1 , . . . , q, where w j ∈ R p j is the state of the exosystem which describes the evolution of the in-flow/outflow j and Γ dj , S dj are suitable matrices. Considering more general classes of ex-osystems is left for future research. We give the system above the compact form˙ w = S d wd = Γ d w, (2)2here w = ( w T . . . w Tq ) T , d = ( d T . . . d Tq ) T , S d = block . diag( S d , . . . , S dq ), Γ d = block . diag(Γ d , . . . , Γ dq ).The model (1) and the overall exosystem (2) return the closed-loop system˙ w = S d w ˙ x = Bλ + P wz = B T x (3)where by a slight abuse of notation we renamed P Γ d simply as P .We are interested in the problem of distributing the cumulative imbalance of the networkdue to the in- and out-flow among the nodes. More formally the problem at hand is asfollows: Load balancing at the nodes
Find distributed dynamic feedback control laws˙ η k = Φ k η k + Λ k z k λ k = Ψ k η k + Γ k z k , k = 1 , . . . , m (4)such that, for each initial condition ( w , x , η ), the solution of the closed-loop system (3),(17) satisfies lim t → + ∞ z ( t ) = 0.In what follows we propose a solution to the problem. We focus on flow networks whose underlying graph satisfies the following standing assump-tion:
Assumption 1
The graph G is connected. The first result concerns the characterization of a “steady state” solution to the problem:
Lemma 1
Let Assumption 1 hold. For each w solution to ˙ w = S d w , if there exist a function λ w : R + → R m and a continuously differentiable function x w : R + → R n solution to ˙ x w = Bλ w + P w (5) and B T x w (6) then x w = n x w ∗ , ˙ x w ∗ = Tn P wn (7) and λ w = M w , for some matrix M. If the graph is a tree, then the matrix M is unique.Proof: From Assumption 1 and (6), one obtains that x w = n x w ∗ , for some function x w ∗ : R + → R . Replacing the expression of x w in (5) one has n ˙ x w ∗ = Bλ w + P w, (8)3s N ( B T ) = R ( n ), with N , R the null space and the range of a matrix, multiplying onthe left both sides of (8) by Tn yields ˙ x w ∗ = Tn P wn as claimed. Replace the latter in (8) toobtain
Y P w = Bλ w , Y = n Tn n − I n . By Assumption 1 and without loss of generality (up toa relabeling of the edges of the graph), Bλ w = B a λ wa + B b λ wb with B a full-column rank and λ wa ∈ R n − . If a solution λ wa to Y P w = Bλ w exists, then λ wa = ( B Ta B a ) − B Ta ( Y P w − B b λ wb ).Letting λ wb = 0 one obtains λ wa = M a w , with M a = ( B Ta B a ) − B Ta Y P .If G is a tree, then B is full-column rank and λ w = ( B T B ) − B T Y P w .In what follows, we assume that a solution to (5), (6) exists. Moreover, if m > n − n − B are linearlyindependent and we let the last m − n + 1 components of λ w be identically zero. Remark 1
From (7), by integration, one has x w ( t ) = n (cid:18) x w ∗ (0) + Z t Tn P w ( s ) n ds (cid:19) . Observe that x w depends on the initial condition and strictly speaking cannot be referredto as a steady state solution. Bearing in mind the interpretation of (1) as a flow networkand of P w the vector of the inflows and outflows of the network, the integral R t Tn P w ( s ) n ds can be seen as the cumulative imbalance of the network. In other words, if for any given w a solution to the load balancing problem exists, then the state at each node equals – up toa constant – the cumulative imbalance of the network.In the case of a network with no imbalance, i.e. Tn P w ( t ) = 0 for all t ≥ x w is a constant vector. Example 1
Consider the graph depicted in Fig. 1. The graph corresponds to system (1)with B = − − − , P = −
10 0 The solutions of (5)-(6) (with w = d ) are as follows˙ x w ∗ = d − d λ w = λ w + d + d λ w = λ w + d − d . A solution is obtained letting λ w = 0.We introduce now a system which generates the control signal λ w in Lemma 1. Considerthe input λ wk associated with the edge k , with k = 1 , , . . . , m . In general, such input maydepend on all the components of the disturbance vector w . Hence, to generate λ wk , thefollowing system is proposed: ˙ η k = S d η k u k = H k η k (9)The statement below is immediate. 4 − − − d d Figure 1: The distribution network considered in the Example 1.
Lemma 2
For any w solution to ˙ w = S d w , there exists a solution η wk to (9) such that H k η wk ( t ) = λ wk ( t ) for all t ≥ , where λ wk is the k th entry of λ w in Lemma 1.Proof: Choose w (0) as the initial condition of (9), then η k ( t ) = w ( t ) for all t ≥
0. As λ w = M w , then it suffice to choose H k as the k th row of M to have H k η wk ( t ) = H k w ( t ) = λ wk ( t ) for all t ≥ Remark 2
From the proof of Lemma 1 it turns out that m − n + 1 components of λ w canbe chosen identically zero. The matrices H k corresponding to these components are thenidentically zero as well. Hence, for k = n, n + 1 , . . . , m , the system (9) reduces trivially to u k = 0.The system (9) is completed by adding control inputs v k , v k to be designed for guaran-teeing that the response of the closed-loop system converges to the desired response for x .Hence, we set ˙ η k = S d η k + v k u k = H k η k + v k , k = 1 , , . . . , n − η k , v k ∈ R q , v k ∈ R , and u k = v k for k = n, n + 1 , . . . , m .We write (10) in the form ˙ η = Sη + v λ = Hη + v (11)where η = ( η T η T . . . η Tn − ) T , S = I n − ⊗ S d , where ⊗ denotes the Kronecker product, and H = (cid:18) H (cid:19) , H = block . diag( H , . . . , H n − ) . Observe that by Lemma 2, for any w and provided that v = , v = , there exists asolution η w to (11) which satisfies ˙ η w = Sη w λ w = Hη w (12) Theorem 1
Consider the system (1), where B is the incidence matrix of a graph G and d is a disturbance generated by the system (2). nder Assumption 1, provided that S dj is skew symmetric for each j = 1 , , . . . , q , thedynamic feedback controller (11) with v = − H T B T x and v = − B T x , namely ˙ η = Sη − H T B T xλ = Hη − B T x (13) guarantees boundedness of the state of the closed-loop system and asymptotic convergence of x ( t ) to n ( c ′ + R t Tn P w ( s ) n ds ) for some constant c ′ .Proof: Consider the overall closed-loop system˙ w = S d w ˙ x = B ( Hη + v ) + P w ˙ η = Sη + v z = B T x Introduce the new variables ˜ x = x − x w , ˜ η = η − η w . These satisfy˙˜ x = B ( Hη + v ) + P w − Bλ w − P w = BH ˜ η + B ( Hη w − λ w ) + Bv = BH ˜ η + Bv and ˙˜ η = Sη + v − Sη w = S ˜ η + v . Introduce the Lyapunov function V (˜ x, ˜ η ) = (cid:0) ˜ x T ˜ x + ˜ η T ˜ η (cid:1) . The function V computed alongthe solutions of system ˙˜ x = BH ˜ η + Bv ˙˜ η = S ˜ η + v (14)satisfies ˙ V (˜ x, ˜ η ) = ˜ x T ( BH ˜ η + Bv )+ ˜ η T ( S ˜ η + v ). Under the assumption of the skew-simmetryof S , one obtains ˙ V (˜ x, ˜ η ) = ˜ x T BH ˜ η + ˜ x T Bv + ˜ η T v . Set v = − H T B T ˜ x, v = − B T ˜ x. (15)Observe that by the connectivity of the graph and the definition of ˜ x , v = − H T B T x and v = − B T x . Then ˙ V (˜ x, ˜ η ) = −|| B T ˜ x || . Hence, (˜ x, ˜ η ) is bounded. By La Salle’s invarianceprinciple and connectivity of the graph, the solutions to (14) converge to the largest invariantset contained in { (˜ x, ˜ η ) : B T ˜ x = } = { (˜ x, ˜ η ) : ˜ x ∈ R ( n ) } .Observe that the system (14) with the inputs v as in (15) becomes˙˜ x = − BB T ˜ x + BH ˜ η ˙˜ η = S ˜ η − H T B T ˜ x (16)6n this invariant set the system (16) satisfies˙˜ x = BH ˜ η ˙˜ η = S ˜ η = B T ˜ x. Hence, ˜ x = n ˜ x ∗ . Replacing this expression in the equation for ˜ x and pre-multiplying bothsides by Tn , one obtains ˙˜ x ∗ = 0, that is ˜ x ∗ is a constant. Hence ˜ x = x − x w → n c for someconstant c . Bearing in mind the expression of x w obtained in Lemma 1, then one concludesthat x ( t ) → n ( c ′ + R t Tn P w ( s ) n ds ) for some constant c ′ . Remark 3
In the case of balanced demand/supply, the state x ( t ) converges to n c ′ forsome constant c ′ . Observe that T ˙ x = , that is T x ( t ) = T x (0). Hence, T x (0) =lim t →∞ T x ( t ) = nc ′ implies that x ( t ) converges to n T x (0) n . Hence under the effect of a time-varying but balanced demand/supply all the components of the state x ( t ) asymptoticallyconverge to the average of the initial distribution of material at the nodes.Bearing in mind the block diagonal nature of the matrices S , H and the definition z = B T x , the dynamic feedback controller (13) can be decomposed as the following set ofdynamic feedback controllers at the edges:˙ η k = S d η k − H Tk z k λ k = H k η k − z k , k = 1 , , . . . , n − k = S d , Λ k = − H Tk , Ψ k = H k , Γ k = −
1. By Remark 2, for k = n, n + 1 , . . . , m for which H k = the edge controllerbecomes a static one, i.e. λ k = − z k . Example 1 (Cont’d) Assume that d = α + β sin( ωt + ϕ ), with α > β > d = α .The supply is a periodic fluctuation around a constant value while the demand is a constant.Then the matrices S d and Γ d in (2) write as S d = ω − ω , Γ d = (cid:18) (cid:19) . Let λ w = 0. Then, for k = 1 ,
2, the matrices H k which allow to reproduce λ wk are H = (cid:0)
23 13 (cid:1) , H = (cid:0) − (cid:1) Then the controllers at the edges 1 and 2 are given by (17) with S d and H k as above and z = − x + x , z = − x + x . The controller at edge 3 is the static control law λ = − z = − ( x − x ).7 emark 4 (Passivity-based reinterpretation) The proof of Theorem 1 can be reinter-preted as follows. In view of Lemma 1, the system˙˜ x = B ˜ λz = B T ˜ x is the incremental model associated with system (1). Similarly, by Lemma 2, system˙˜ η = S ˜ η + H T ˜ v ˜ u = H ˜ η, where ˜ u = u − u w and u w := Hη w , is the incremental model associated with the internalmodel ˙ η = Sη + H T vu = Hη The systems are passive with respect to the storage functions V (˜ x ) = ˜ x T ˜ x and V (˜ η ) = ˜ η T ˜ η provided that S d is skew symmetric. The negative feedback interconnection of the twosystems, namely ˜ λ = λ ext − ˜ u ˜ v = u ext + z, is passive as well from the input ( λ ext , u ext ) to the output ( z, ˜ u ). The output feedback (cid:18) λ ext u ext (cid:19) = − (cid:18) K
00 0 (cid:19) (cid:18) z ˜ u (cid:19) gives asymptotic convergence of the closed-loop system to the largest invariant set where z = 0.We discuss briefly the difficulties related to the presence of possible state and inputconstraints. State constraints.
Consider a variation of the model (1) in which the positivity constrainton the amount of material stored at the nodes is enforced. The model becomes˙ x = ( Bλ + P w ) + x where ( b i λ + p i w ) + x i is the i th component of the vector ( Bλ + P w ) + x and( ζ i ) + x i = (cid:26) ζ i if ( x i >
0) or ( i = 0 and ζ i ≥ x i = 0 and ζ i < Tn P w = 0. As a conse-quence,
P w = − Bλ w and ˙ x = ˙˜ x = ( B ˜ λ ) + x . V (˜ x ) = ˜ x T ˜ x , with ˜ x = x − x w ∗ and x w ∗ >
0, satisfies˙ V (˜ x ) = ˜ x T ( B ˜ λ ) + x . Observe that ˜ x T ( B ˜ λ ) + x = P ni =1 ˜ x i ( b i ˜ λ ) + x i = ˜ x T ( B ˜ λ ). This shows that the system˙˜ x = ( B ˜ λ ) + x z = B T ˜ x is passive and the arguments of the previous remark can be used. A formal analysis requiresto take into account the discontinuity of the system. This is not pursued here for lack ofspace. Edge capacity constraints.
Constraints on the capacity of the edges can be modeled via asaturation function replacing λ in (1) with sat( λ ). Here, sat( λ ) = (sat( λ ) . . . sat( λ m )) T andsat( λ k ) = min {| λ k | , c } sign( λ k ). Following Lemma 1, let x w , λ w be such that˙ x w = B sat( λ w ) + P w and M such that sat( λ w ) = M w . For the problem to be feasible restrict the set of initialconditions w of the exosystem ˙ w = S d w in such a way that || M w ( t ) || ∞ < c for all t ≥ x = B [sat( λ ) − sat( λ w )]= B sat( λ ) − BM w
To tackle the problem, we assume the scenario in which at each edge a dynamic observerprovides ˆ w that converges to w asymptotically (or at each edge k there exists an estimatorwhich generates a local estimate ˆ w k of w ). Consider then the control input λ = − µ ( B T x ) + M ˆ w = − µ ( B T ˜ x ) + M ˆ w, where µ : R m → R m is a map such that each component is a monotonically increasingfunction which is zero at the origin. The incremental model writes as˙˜ x = B sat( − µ ( B T ˜ x ) + M ˆ w ) − BM w = B sat( − µ ( B T ˜ x ) + M w + M ( ˆ w − w )) − BM w.
The right-hand side is bounded and the solutions exists for all t ≥
0. Suppose that eachcomponent of µ is a function whose range is within [ − c , c ]. Then after a finite time,sat( − µ ( B T ˜ x ) + M w + M ( ˆ w − w )) = − µ ( B T ˜ x ) + M w + M ( ˆ w − w ) and the incrementalmodel evolves as ˙˜ x = − Bµ ( B T ˜ x ) + BM ( ˆ w − w ) . Consider the projected state y = Q ˜ x , where Q is an ( n − × n matrix such that Q n = , QQ T = I n − and Q T Q = I n − n Tn /n . It yields˙ y = − QBµ ( B T Q T y ) + QBM ( ˆ w − w ) . ; moreover the forcingterm is decaying to zero. Since the response of the system is bounded then the state y converges to the origin which implies that ˜ x converges to R ( n ). Then one can proceed asin the last part of the proof of Theorem 1. The proposed solution relies on the existence ofdistributed estimators for w , whose actual design is left as a topic for future research. In the previous section, the dynamics describing the evolution of the storage variable at eachnode was given by ˙ x i = b i λ + p i w, i = 1 , , . . . , n (18)where b i and p i are the i th row of the incidence matrix B and P respectively. Considernow a different case of a flow network in which the way material accumulates at the node isdescribed by a non-trivial dynamics, namely˙ x i = f i ( x i ) + b i λ + p i w, i = 1 , , . . . , n (19)with vector of measurements y i ∈ R m given by y i = b Ti x i . The nonlinear system (19) allows us to put the results of the paper in a broader contextand compare them with those in [2], [9] (see the end of the section). Observe that for k = 1 , , . . . , m , y ik is either x i , − x i or 0. The sum of the outputs y i over all the nodesreturns the vector of relative measurements z , z = B T x = n X i =1 y i . Each system ˙ x i = f i ( x i ) + b i λ + p i wy i = b Ti x i , i = 1 , , . . . , n (20)is assumed to be incrementally passive. Assumption 2
There exists a regular storage function V i : R × R × R + → R + such that ∂V i ∂t + ∂V i ∂x i ( f i ( x i ) + b i λ + p i w )+ ∂V i ∂x ′ i ( f i ( x ′ i ) + b i λ ′ + p i w ) ≤ ( y i − y ′ i ) T ( λ − λ ′ ) . Take V ( y ) = y T y ; then ˙ V ( y ) ≤ V ( y ) = 0 is identically zero if and only if B T Q T y = 0. Thisimplies that y = 0. In fact if this were not true, that is B T Q T y = 0 and y = 0, then QBB T Q T y = 0 as welland this would contradict that y = 0 since QBB T Q T is a non singular matrix. See [14] for a definition. emark 5 (A class of incrementally passive systems) Consider the linear dynamicsat the node (18) and the function V i = ( x i − x ′ i ) . Then the right-hand side of the inequalityabove becomes ( x i − x ′ i )( b i λ + p i w ) − ( x i − x ′ i )( b i λ ′ + p i w )= ( x i − x ′ i ) b i ( λ − λ ′ )= ( b Ti ( x i − x ′ i )) T ( λ − λ ′ )= ( y i − y ′ i ) T ( λ − λ ′ )which satisfies the dissipation inequality in Assumption 2.Suppose that the dynamics f i are equal to ∇ F i , with F i a twice continuously differentiableand concave function. Then the static nonlinearity − f i ( x i ) is incrementally passive, that is( x i − x ′ i )( f i ( x i ) − f i ( x ′ i )) ≤ . As a matter of fact f i ( x i ) − f i ( x ′ i ) = ∇ F i ( x i ) − ∇ F i ( x ′ i ) = ∇ F i ( ξ i )( x i − x ′ i ) for some ξ i lyingin the segment connecting x i , x ′ i . By concavity, ∇ F i ( ξ i ) ≤ x i − x ′ i )( f i ( x i ) − f i ( x ′ i )) ≤
0. Hence any system (20) with f i ( x i ) = ∇ F i ( x i ) and F i defined as before satisfiesAssumption 2.Lemma 1 is replaced by the following: Lemma 3
For each i = 1 , , . . . , n , for each w solution to ˙ w = S d w , there exist a function λ w : R + → R m and continuously differentiable bounded functions x wi : R + → R that satisfy ˙ x wi = f i ( x wi ) + b i λ w + p i w, i = 1 , , . . . , n n X i =1 b Ti x wi (21) only if there exists a solution x w ∗ : R + → R defined for all t ≥ to ˙ x w ∗ = Tn f ( x w ∗ ) n + Tn P wn , (22) where f ( x ) = ( f ( x ) . . . f n ( x )) T . If this is the case, then x wi = x w ∗ , i = 1 , , . . . , n,λ w = (cid:18) λ wa λ wb (cid:19) = (cid:18) M f ( x w ∗ ) + M w (cid:19) with λ wa ∈ R n − , λ wb ∈ R m − n +1 , and M , M suitable matrices.Proof: The second equality in (21) and Assumption 1 implies that x wi = x w ∗ for all i .Replacing the latter in the first equality implies that necessarily x w ∗ must be a solution ofthe inhomogeneous differential equation˙ x w ∗ = Tn f ( x w ∗ ) n + Tn P wn . x w ∗ exists for all t and let x wi = x w ∗ for each i . Then the secondequation in (21) is satisfied by the connectivity of the graph and the properties of theincidence matrix. Since x wi = x w ∗ for all i , it is seen that the first equation in (21) is satisfiedif and only if there exists λ w such that Y [ f ( x w ∗ ) − P w ] = Bλ w , with Y = n Tn n − I n . By connectivity of the graph, the rank of B is n −
1. If the graph has n − B is full-column rank and, provided that a solution λ w to the previous equationexists, it is given by λ w = ( B T B ) − B T Y ( f ( x w ∗ ) − P w ). If the graph has more than n − B as ( B a B b ) T with B a full-column rank. Then, provided that a solution to theprevious equation exists, it is given by λ wa = ( B Ta B a ) − B Ta [ Y ( f ( x w ∗ ) − P w ) − B b λ wb ]. Oneparticular solution is obtained for λ wb = 0 and λ wa = ( B Ta B a ) − B Ta Y ( f ( x w ∗ ) − P w ). Remark 6
If the inflow and outflow are balanced, i.e. Tn P w = 0, then the solution x w ∗ to(22) exists for all t and is bounded. In fact, consider the system˙ y = Tn f ( y ) n and the radially unbounded function V ( y ) = y . Then˙ V ( y ) = y Tn f ( y ) n = n X i =1 yf i ( y ) n . By the incremental passivity property of − f i , yf i ( y ) ≤ i and this implies ˙ V ( y ) ≤ x w ∗ . Remark 7
In the case the dynamics at the nodes are all the same, i.e. f i = f j for all i, j ,then the expression of λ w simplifies as λ w = (cid:18) λ wa λ wb (cid:19) = (cid:18) M (cid:19) w. This descends from the proof, since by definition of the matrix Y , Y f ( x w ∗ ) = .In the remaining of the section we assume that a solution to (21) exists.The parallel interconnection of the n subsystems (20) with input λ and output z = P ni =1 y i returns an incrementally passive systems. Formally Lemma 4
The parallel interconnection ˙ x = f ( x ) + b λ + p w. . . ˙ x n = f n ( x n ) + b n λ + p n wz = n X i =1 b Ti x i , enoted as ˙ x = f ( x ) + Bλ + P wz = B T x (23) is such that the storage function V ( x, x ′ ) = P ni =1 V i ( x i , x ′ i ) satisfies ∂V∂x ( f ( x ) + Bλ + P w ) + ∂V∂x ′ ( f ( x ′ ) + Bλ ′ + P w ) ≤ ( z − z ′ ) T ( λ − λ ′ ) . The proof is straightforward and is omitted. Consider now systems of the form˙ η k = φ k ( η k , v k ) u k = ψ k ( η k ) , k = 1 , , . . . , n − , (24)with the following two additional properties: Assumption 3
For each k = 1 , , . . . , n − , there exists regular functions W k ( η k , η ′ k ) suchthat ∂W k ∂η k φ ( η k , v k ) + ∂W k ∂η ′ k φ ( η ′ k , v ′ k ) ≤ ( u k − u ′ k )( v k − v ′ k ) . Assumption 4
For each k = 1 , , . . . , n − , for each w solution to ˙ w = S d w , there existsa bounded solution η wk to ˙ η k = φ k ( η k , such that λ wk = ψ k ( η wk ) . Assume that the system ˙ η wka = Tn f ( η wka ) n + Tn P η wkb n ˙ η wkb = S d η wkb is forward complete. Initialize the system as η wka (0) = x w ∗ (0) and η wkb (0) = w (0). Then η wka ( t ) = x w ∗ ( t ) and η wkb ( t ) = w ( t ) for all t ≥
0. Hence λ wk = M k f ( η ka ) + M k η kb , k =1 , , . . . , n −
1, where M k and M k are the k th rows of M and M respectively. On theother hand, λ wk = 0, k = n, n + 1 , . . . , m . An expression for φ k , ψ k , k = 1 , , . . . , n − φ k ( η k ,
0) = Tn f ( η ka ) n + Tn P η kb nS d η kb ,ψ k ( η k ) = M k f ( η ka ) + M k η kb .In the special case of nodes with the same dynamics ( f i = f j = ¯ f for all i, j ) ψ k ( η k ) simplifiesas M k η kb and a system that satisfies Assumptions 3 and 4 is˙ η k = S d η k + M T k v k u k = M k η k , W k ( η k ) = η Tk η k . Collect the systems (24) into a system with statevariable η = ( η T . . . η Tn − ) T , input v = ( v . . . v m ) T and output u = ( u . . . u m ) T , namely˙ η = Φ( η, v ) u = Ψ( η ) (25)with Φ( η, v ) = ( φ T . . . φ Tn − ) T , Ψ( η ) = ( ψ . . . ψ n − T ) T . The system is incrementally passivefrom v to u with storage function W ( η, η ′ ) = P n − k =1 W k ( η k , η ′ k ).The following holds: Theorem 2
Let Assumptions 1-4 hold. Suppose that a solution to (21) exists and x w ∗ isbounded. Consider the systems (23), with input λ and output z , and (25), with input v andoutput u , interconnected via the relations v = − z + v ext , λ = u + λ ext .The interconnected system is incrementally passive from the input ( λ Text v Text ) T to the output ( z T u T ) T . Moreover, the feedback ( λ Text v Text ) T = ( − Kz T T ) T , with K a positive definitediagonal matrix, guarantees lim t → + ∞ z ( t ) = .Proof: The feedback interconnection of incrementally passive systems is incrementallypassive ([14], Lemma 1). Hence ˙ x = f ( x ) + Bλ + P wz = B T x ˙ η = Φ( η, v ) u = Ψ( η ) λ = u + λ ext v = − z + v ext is incrementally passive from the input ( λ Text v Text ) T to the output ( z T u T ) T . The storagefunction U is given by the sum V + W where V, W are the functions defined above (inLemma 4 and after (25), respectively).Let λ ext = − Kz , v ext = . The system becomes˙ x = f ( x ) + B (Ψ( η ) − KB T x ) + P w ˙ η = Φ( η, − z ) z = B T x For a given solution w to ˙ w = S d w , let x w , λ w be as in Lemma 3 and η w as in Assumption 4.The functions x w and η w are a solution to the equations above with input ( λ Text v Text ) T = and output ( z T u T ) T = ( T λ wT ) T . In fact˙ x w = f ( x w ) + B Ψ( η w ) + P w = f ( x w ) + Bλ w + P w ˙ η w = Φ( η w , ) = B T x w .
14s in [14], by the incremental passivity of the feedback system and the existence of a solution( x w , η w ) of the feedback system such that z ( t ) = , any other solution ( x, η ) with input( λ Text v Text ) T = ( − Kz T T ) T satisfies˙ V (( x, η ) , ( x w , η w )) ≤ (( z T u T ) − ( T λ wT )) (cid:18) − Kz ( t ) (cid:19) = − z T Kz.
Bearing in mind the regularity of U and boundedness of x w , this yields boundedness of x .In view of the time-varying nature of the system, to infer convergence of z to zero, one canresort to Barbalat’s lemma. This guarantees convergence under the assumption that ˙ z isbounded. This in turn requires ˙ w bounded, which is the case here since S is skew symmetric. Corollary 1
If (i) f i = ¯ f for all i = 1 , , . . . , n , (ii) there exists a twice continuouslydifferentiable convex function F ( x ) such that ∇ F ( x ) = ¯ f ( x ) and (iii) Tn P w = 0 for all t ≥ , then the controllers ˙ η k = S d η k − M T k z k λ k = M k η k − z k , , k = 1 , , . . . , n − , and λ k = − z k , k = n, n + 1 , . . . , m , guarantee lim t → + ∞ z ( t ) = . The closed-loop system given in the corollary above takes the form˙ x = ∇ F ( x ) + Bλ + P w, z = B T x ˙ η = Sη − M T z, λ = M η − z , where we are assuming that m = n − x = ∇ F ( x ) + Bλ, z = B T x ˙ η = z, λ = − ψ ( η )with ψ a non-decreasing monotonic non-linearity (such as a saturation function), were stud-ied. The presence of the non-trivial dynamics S in our controller is due to the time varying-nature of the external input. In [2], ∇ F ( x ) has a unique equilibrium at the origin and thesystem ˙ x = ∇ F ( x ) + Bλ is strictly passive. In [9] it is shown that if the components ofthe vector field ∇ F ( x ) have different equilibria, ∇ F ( x ) is strongly concave and ψ introducessaturation constraints, then the system’s response exhibits state clustering. We have presented an internal model approach to the problem of balancing demand andsupply in a class of distribution networks. Extensions to nonlinear systems have also been15iscussed. Further research will focus on a detailed investigation of state and input con-straints and more complex models of demand and supply. The fulfillment of the internalmodel principle has to be understood for more general classes of nonlinear systems thanthose in Corollary 1. This will shed light on the relation between the results in this paperand the saddle-point perspective of [9]. Compared with other papers where the robustnessto time-varying inputs is studied using a frequency domain approach ([4]), our state spaceapproach allows us to consider more general classes of cooperative control systems.
References [1] A. Alessandri, M. Gaggero, and F. Tonelli. Min-max and predictive control for themanagement of distribution in supply chains.
IEEE Transactions on Control SystemsTechnology , 19(5):1075–1089, 2011.[2] M. Arcak. Passivity as a design tool for group coordination.
IEEE Transactions onAutomatic Control , 52(8):1380–1390, 2007.[3] H. Bai, M. Arcak, and J. Wen.
Cooperative Control Design: A Systematic, Passivity-Based Approach . Communications and Control Engineering. Springer, New York, 2011.[4] H. Bai, R. A. Freeman, and K. M. Lynch. Robust dynamic average consensus of time-varying inputs. In
Proceedings of the IEEE Conference on Decision and Control , pages3104–3109, 2010.[5] M. Bari´c and F. Borrelli. Distributed averaging with flow constraints. In
Proceedingsof the American Control Conference , pages 4834–4839, 2011.[6] D. Bauso, F. Blanchini, L. Giarr´e, and R. Pesenti. A decentralized solution for theconstrained minimum cost flow. In
Proceedings of the IEEE Conference on Decisionand Control , pages 661–666, 2010.[7] D. Bauso, F. Blanchini, and R. Pesenti. Average flow constraints and stabilizability inuncertain production-distribution systems.
Journal of Optimization Theory and Appli-cations , 144(1):12–28, 2009.[8] D. Bauso, F. Blanchini, and R. Pesenti. Optimization of long-run average-flow costin networks with time-varying unknown demand.
IEEE Transactions on AutomaticControl , 55(1):20–31, 2010.[9] M. B¨urger, D. Zelazo, and F. Allg¨ower. Network clustering: A dynamical systemsand saddle-point perspective. In
Proceedings of the IEEE Conference on Decision andControl , pages 7825–7830, 2011.[10] C. De Persis and B. Jayawardhana. On the internal model principle in formation controland in output synchronization of nonlinear systems. In
Proceedings of the 51th IEEEConference on Decision and Control , 2012.1611] R. De Santis and A. Isidori. On the output regulation for linear systems in the presenceof input saturation.
IEEE Transactions on Automatic Control , 46(1):156–160, 2001.[12] L. Moreau. Stability of multiagent systems with time-dependent communication links.
IEEE Transactions on Automatic Control , 50(2):169–182, 2005.[13] Franklin H. Moss and Adrian Segall. Optimal control approach to dynamic routing innetworks.
IEEE Transactions on Automatic Control , AC-27(2):329–339, 1982.[14] A. Pavlov and L. Marconi. Incremental passivity and output regulation.
Systems andControl Letters , 57(5):400–409, 2008.[15] G. Stan and R. Sepulchre. Analysis of interconnected oscillators by dissipativity theory.
IEEE Transactions on Automatic Control , 52(2):256–270, 2007.[16] J.N. Tsitsiklis, D.P. Bertsekas, and M. Athans. Distributed asynchronous deterministicand stochastic gradient optimization algorithms.
IEEE Transactions on AutomaticControl , AC-31(9):803–812, 1986.[17] A.J. van der Schaft and J. Wei. Distributed averaging with flow constraints. In
Pro-ceedings of the 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for NonLinear Control , pages 24–29, 2012.[18] P. Wieland, R. Sepulchre, and F. Allg¨ower. An internal model principle is necessaryand sufficient for linear output synchronization.