Dynamic coupling design for nonlinear output agreement and

time-varying flow control

Mathias Burger a, Claudio De Persis b

aInstitute for Systems Theory and Automatic Control, University of Stuttgart, Pfaffenwaldring 9, 70550 Stuttgart, Germany

bITM, Faculty of Mathematics and Natural Sciences, University of Groningen, Nijenborgh 4, 9747 AG Groningen, theNetherlands

Abstract

This paper studies the problem of output agreement in networks of nonlinear dynamical systems under time-varying distur-bances, using dynamic diffusive couplings. Necessary conditions are derived for general networks of nonlinear systems, andthese conditions are explicitly interpreted as conditions relating the node dynamics and the network topology. For the class ofincrementally passive systems, necessary and sufficient conditions for output agreement are derived. The approach proposed inthe paper lends itself to solve flow control problems in distribution networks. As a first case study, the internal model approachis used for designing a controller that achieves an optimal routing and inventory balancing in a dynamic transportation networkwith storage and time-varying supply and demand. It is in particular shown that the time-varying optimal routing problemcan be solved by applying an internal model controller to the dual variables of a certain convex network optimization problem.As a second case study, we show that droop-controllers in microgrids have also an interpretation as internal model controllers.

1 Introduction

Output agreement has evolved as one of the most impor-tant control objectives in cooperative control. It appearsin various contexts, ranging from distributed optimiza-tion ([TBA86]), formation control ([OSFM07]) up tooscillator synchronization ([SS07]). Over the last years,it has become evident that the internal model principletakes a central role in output agreement problems, seee.g. [WSA11], [BAW11], [DJ14], [De 13]. Independentlyand in parallel, it was shown that passivity and relatedsystems theoretic concepts have outstanding relevance

? A preliminary version of this paper has been presented in[BP13].??The work of M. Burger is supported by the German Re-search Foundation (DFG) within the Cluster of Excellencein Simulation Technology (EXC 310/2) at the University ofStuttgart. The work of C. De Persis is supported by the re-search grants Quantized Information and Control for Forma-tion Keeping (The Netherlands Organisation for ScientificResearch), Efficient Distribution of Green Energy (DanishResearch Council of Strategic Research), Flexiheat (Minis-terie van Economische Zaken, Landbouw en Innovatie), andby a starting grant of the Faculty of Mathematics and Nat-ural Sciences, University of Groningen.

Email addresses:mathias.buerger@ist.uni-stuttgart.de (MathiasBurger), c.de.persis@rug.nl (Claudio De Persis).

in the analysis and synthesis of synchronizing net-works and output agreement problems, see e.g, [Arc07],[BAW11], [vdSM13], [DJ12], [BZA14], [SS07], [SAS10].The present paper studies output agreement in networksof heterogeneous nonlinear dynamical systems affectedby external disturbances and presents an approach thatcombines elements from internal model control withthose known in passivity-based cooperative control. Wefollow here the trail opened in [PM08] for centralizedoutput regulation and provide necessary and sufficientconditions for the solution of the output agreementproblem for the class of incrementally passive systems.Our results provide a bridge connecting the two com-plementary approaches for output agreement problems,namely the internal model approach on the one hand,and the passivity-based approach on the other hand.The proposed approach is inherently different fromother internal model approaches such as [WSA11],[WWA13] and [IMC13]. In [WSA11] each node is aug-mented with a local controller that contains a referencesystem, identical for all nodes, and the local controllerstrack the reference system. The local (“virtual”) copiesof the reference system are then synchronized with staticdiffusive couplings. The approach considered in thepresent paper differs in central points. Most obviously,dynamic couplings, rather than local controllers, areinvestigated. Furthermore, external signals are assumedto affect the node dynamics, a case that is not covered

Preprint submitted to Automatica 27 November 2014

in [WSA11]. Incrementally passive systems and distur-bance rejection are also dealt with in [DJ14]. However,the framework we propose here, inspired by [PM08], iscompletely different and leads to a family of new dis-tinct results that have not been considered in [DJ14].The main contribution of this paper is the developmentof a control framework for network systems that inte-grates the ideas of internal model and passivity-basedcooperative control. The proposed framework leads toboth, constructive methods for the design of distributedcontrollers and a novel understanding of existing con-trol approaches. In particular, we consider networks ofheterogeneous nonlinear systems, interacting accordingto an undirected network topology. The objective is todesign dynamic controllers placed on the edges of thenetwork such that output agreement is achieved. Nec-essary conditions for the feasibility of the problem arepresented for the general case of heterogeneous non-linear systems with external disturbances. Sufficientconditions for convergence to output agreement are de-rived for incrementally passive systems. Following this,we present a relevant class of heterogeneous nonlinearsystems for which all assumptions are met and the cou-pling controllers can be found following a constructivedesign procedure. The constructiveness of the result isfurther demonstrated via the design of optimal routingcontrollers for distribution systems with time-varyingdemand. To explain the relation to existing approachesthe special situations where output agreement can bereached with static diffusive couplings or where thedisturbances are constant are discussed. Based on thisresults, it is shown that droop-controllers in microgrids,as, e.g., studied in [SPDB13], are designed exactly inaccordance to the internal model control approach.The remainder of the paper is organized as follows.The problem formulation and necessary conditions foroutput agreement are presented in Section 2. Sufficientconditions for output agreement in networks of incre-mentally passive systems are discussed in Section 3. Aconstructive procedure for the design of such controllersfor a class of nonlinear systems is presented in Section 4.The design procedure is applied to a time-varying opti-mal distribution problem in Section 5. In Section 6, therelation to known methods in the literature is formallydiscussed, and an interpretation of droop-controllers ainternal model controllers is provided in Section 7.

Notation: The set of (positive) real numbers is denotedby R (R≥). Given two matrixes A and B, the Kroneckerproduct is denoted by A ⊗ B. The Moore-Penrose in-verse (or pseudo-inverse) of a non-invertible matrix A isdenoted by A†. The range-space and null-space of a ma-trix B are denoted by R(B) and N (B), respectively. Agraph G = (V,E) is an object consisting of a finite setof nodes, |V | = n, and edges, |E| = m. The incidencematrix B ∈ Rn×m of the graph G with arbitrary orien-tation, is a 0,±1 matrix with [B]ik having value ‘+1’if node i is the initial node of edge k, ‘-1’ if it is the ter-minal node, and ‘0’ otherwise.

2 Problem formulation and necessary condi-tions

We consider a network of dynamical systems defined ona connected, undirected graph G = (V,E). Each noderepresents a nonlinear system

xi = fi(xi, ui, wi)

yi = hi(xi, wi), i = 1, 2, . . . , n,(1)

where xi ∈ Rri is the state, and ui, yi ∈ Rp are the inputand output, respectively. Each system (1) is driven bythe time-varying signal wi ∈ Rqi , representing, e.g., adisturbance or reference. We assume that the exogenoussignals wi are generated by systems of the form

wi = si(wi), wi(0) ∈ Wi, (2)

whereWi is a set whose properties are specified below. Acommon assumption in nonlinear output regulation the-ory is neutral stability 1 of the exo-systems [IB08]. Forthe purpose of this paper, it is advantageous to restrictthe discussion to a slightly smaller class of exo-systems.

Assumption 1 The vector field si(wi) satisfies for allwi, w

′i the inequality

(wi − w′i)T (si(wi)− si(w′i)) ≤ 0. (3)

Remarkably, Assumption 1 includes in particular neu-trally stable linear exo-systems, i.e., a linear functionsi(wi) = Siwi with skew-symmetric matrix Si, i.e. STi +Si = 0, satisfies the requirements. These linear exosys-tems can generate signals that are combinations of con-stant and periodic modes.

In addition, our assumption includes various nonlineardynamical systems. For example, it has been shown in[DdG09, Sec. 4.3] that nonlinear Chua’s oscillators sat-isfy Assumption 1. We stack together the signals wi, fori = 1, 2, . . . , n, and obtain the vector w ∈ Rq, whichsatisfies the equation w = s(w). In what follows, when-ever we refer to the solutions of w = s(w), we assumethat the initial condition is chosen in a compact setW = W1 × . . . ×Wn. The set W is assumed to be for-ward invariant for the system w = s(w). Similarly, let x,u, and y be the stacked vectors of xi, ui, and yi, respec-tively. Using this notation, the totality of all systems is

w = s(w)

x = f(x, u, w)

y = h(x,w)

(4)

1 The dynamics (2) is said to be neutrally stable if it admitsa Lyapunov stable equilibrium point for w = 0 and thereexists a neighborhood of Poisson stable points around w = 0.

2

with state space W × X and X a compact subset ofRr1 × . . .× Rrn .The control objective is to reach output agreement of allnodes in the network, independent of the exact represen-tation of the time-varying external signals. We aim toachieve this control objective by a suitable design of dy-namic couplings between the systems. This means, be-tween any pair of neighboring nodes, i.e., on any edgeof G, a dynamical system (in the following called “con-troller”) is placed, taking the form

ξk = Fk(ξk, vk)

λk = Hk(ξk, vk), k = 1, 2, . . . ,m,(5)

with state ξk ∈ Rlk , input vk ∈ Rp and output λk ∈ Rp.Using the same notational convention as before, we de-fine ξ and λ as the stacked state and output vector. To-gether, the controllers (5) give raise to the overall con-troller

ξ = F (ξ, v)

λ = H(ξ, v),(6)

where ξ ∈ Ξ, a compact subset of Rl1×. . .×Rlm . The col-lection (6) of dynamical systems (5) generate the over-all output λ that determines the control input u appliedto the network system (4) via the interconnection (9)below. This motivates the choice of referring to the dy-namical systems (5) as controllers.

Throughout the paper the following interconnectionstructure between the plants, placed on the nodes of G,and the controllers, placed on the edges of G, is consid-ered. A controller (5), associated with edge k connectingnodes i, j, has access to the relative outputs yi − yj . Invector notation, the relative outputs of the systems are

z = (BT ⊗ Ip)y, (7)

where B is the (n ×m) signed incidence matrix of thegraph G. The controllers are then driven by the systemsvia the interconnection condition

v = −z, (8)

where v are the stacked inputs of the controllers. Addi-tionally, the output of the controllers influence the inci-dent systems via the interconnection 2

u = (B ⊗ Ip)λ. (9)

2 The interconnection structure (7), (9) naturally representsa canonical structure for distributed control laws. This struc-ture is often considered in the context of passivity-basedcooperative control, see e.g., [Arc07], [BAW11], [vdSM13],[DJ12], [BZA14]. In fact, it can also model a physical cou-pling of the dynamical systems in the network, as for thecase studies in Section 5 and 7.

xi = fi(xi, ui, wi)

yi = hi(xi, wi)

i = 1, 2, . . . , n

wi = si(wi)

B ⊗ Ip (BT ⊗ Ip)

−

ξk = Fk(ξk, vk)

λk = Hk(ξk, vk),

k = 1, 2, . . . ,m,

y(t)

z(t)

v(t)λ(t)

u(t)

Fig. 1. Structure of the internal model control scheme.

Due to this interconnection structure the dynamics onthe network can be represented as a closed-loop dynam-ics as illustrated in Figure 1. We are now ready to for-mally introduce the output agreement problem.

Definition 1 (Output Agreement Problem) Theoutput agreement problem is solvable for the process(4) under the interconnection relations (7), (8), (9),if there exists controllers (6), such that every solution(w(t), x(t), ξ(t)) originating fromW×X ×Ξ is boundedand satisfies limt→∞ (BT ⊗ Ip) y(t) = 0.

2.1 Necessary Conditions

We first investigate the necessary conditions for theoutput agreement problem to be solvable. To this pur-pose, we strengthen the requirement on the convergenceof the regulation error to the origin, requiring thatlimt→∞ (BT ⊗ Ip) y(t) = 0 uniformly in the initialcondition ([IB08]). The closed-loop system (4), (6), (7),(8), (9) can be written as

w = s(w)

x = f(x, (B ⊗ Ip)H(ξ,−(BT ⊗ Ip)h(x,w)), w)

ξ = F (ξ,−(BT ⊗ Ip)h(x,w)).

(10)

Definition 2 (ω-limit set) The ω-limit set Ω(W×X ×Ξ) is the set of points (w, x, ξ) for which there exists asequence of pairs (tk, (wk, xk, ξk)) with tk → +∞ and(wk, xk, ξk) ∈ W×X ×Ξ such that ϕ(tk, (wk, xk, ξk))→(w, x, ξ) as k → +∞, where ϕ(·, ·) is the flow of (10).

If the output agreement problem is solvable, then the ω-limit set Ω(W×X ×Ξ) is nonempty, compact, invariantand uniformly attractsW×X×Ξ under the flow of (10).Furthermore, the ω-limit set must satisfy

Ω(W ×X × Ξ) ⊆(w, x, ξ) ∈ W ×X × Ξ : (BT ⊗ Ip)h(x,w) = 0

.

3

This set is the graph of a map defined on the whole Wand is invariant for the closed-loop system. By the in-variance, for any solution w of the exosystem originatingfrom W, there exists (xw, uw, ξw) such that

xw = f(xw, uw, w)

0 = (BT ⊗ Ip)h(xw, w)(11)

and

ξw = F (ξw,0)

uw = (B ⊗ Ip)H(ξw,0).(12)

Proposition 1 If the output agreement problem is solv-able, then, for every w solution to w = s(w) originatingin W, there must exist solutions (xw, uw, ξw) such thatthe equations (11), (12) are satisfied.

In a controller-independent form, the constraints (11)and (12) require that there exists (xw, uw) satisfying

xw = f(xw, uw, w), yw = h(xw, w)

uw ∈ R(B ⊗ Ip), yw ∈ N (BT ⊗ Ip),(13)

where uw ∈ R(B ⊗ Ip) denotes that at every time t thevector uw(t) is contained in the respective vector space.Let in the following uw be a solution to (13), and λwpbe a trajectory satisfying uw = (B ⊗ Ip)λ

wp . The tra-

jectory λwp is uniquely defined if and only if the graphG has no cycles. Otherwise, the matrix B has a non-trivial nullspace, see [GR01]. In the most general form,the existence of a feedforward controller is equivalent tothe constraint that there exists an integer d and mapsτ :W → Rd, φ : Rd → Rd and ψ : Rd → Rmp satisfying

∂τ

∂ws(w) = φ(τ(w))

λwp + λw0 = ψ(τ(w)), λw0 ∈ N(B ⊗ Ip

).

(14)

Note that there might be an infinite number of possiblecontrollers that can generate the desired steady stateinput uw. If the constraint (14) holds, the system

η = φ(η), η ∈ Rd

λ = ψ(η)(15)

has the property that if η0 = τ(w(0)), then the solutionη(t) to (15) starting from η0 is such that (B⊗ Ip)λ(t) =uw(t) for all t ≥ 0. We denote by ηw such a solution to(15) such that (B ⊗ Ip)ψ(ηw(t)) = uw(t) for all t ≥ 0.We then let λw(t) := ψ(ηw(t)). Here λw(t) is one of theinfinite many realizations of the map λwp (t)+λw0 (t), with

λw0 ∈ N(B ⊗ Ip

).

To design a controller that decomposes into controllerson the edges of G, we introduce a vector ηk ∈ Rd for eachedge k = 1, . . . ,m, and denote with ψk the entries of thevector valued function ψ corresponding to the edge k.Each edge is now assigned a controller of the form

ηk = φ(ηk), λk = ψk(ηk), k = 1, 2, . . . ,m. (16)

With the stacked vector η = [ηT1 , . . . , ηTm]T , and vector

valued functions φ(η) = [φ(η1)T , . . . , φ(ηm)T ]T , ψ(η) =[ψ1(η1)T , . . . , ψm(ηm)T ]T , the overall controller (16) is

η = φ(η)

λ = ψ(η).(17)

If the initial condition is chosen as η0 = 1m ⊗ τ(w(0))then the solution η(t) to (17) starting from η0 is suchthat λ(t) = λw(t) for all t ≥ 0.

Remark 1 Necessary conditions for output agreementof nonlinear systems without disturbances and node con-trollers were presented in [WWA13]. We formalize herethe necessary conditions in the presence of external dis-turbances and dynamic couplings. The main differencebetween the two formulations, is that in the present for-mulation the synchronous solution depends on the dis-turbance representation and we allow for non-zero in-puts from the network in agreement, i.e., in general wewill have uw 6= 0 in (13). As it will become evident lateron, this formulation allows us to solve problems wherethe approach of [WWA13] is not applicable.

2.2 Discussion: The Regulator Equations

The necessary conditions (11) and (12) are a weaker formof the regulator equations of [IB90]. We discuss nextthat in the considered networked systems the classicalregulator equations might fail to admit a solution whilethe output agreement problem is still feasible. This mo-tivates the presentation of the more general condition(11). However, analyzing the regulator equations stillleads to valuable insights, as it reveals the outstandingrole of passivity in the context of networked systems.

If the systems (1) are such that for each given exogeneousinput w(t) there exists a unique steady state response,and the ω-limit set can be expressed as Ω(W×X ×Ξ) =(w, x, ξ) : x = π(w), ξ = πc(w), then xw = π(w) andthe regulator equations (11) express the existence of aninvariant manifold where the “regulation error” (BT ⊗Ip)y is identically zero provided that the control inputuw is applied. Furthermore, (12) express the existenceof a controller able to provide uw. In this case, (11), (12)

4

take the familiar expressions, see e.g. [IB90]:

∂π

∂ws(w) = f(π(w), (B ⊗ Ip)H(πc(w)), w)

0 = (BT ⊗ Ip)h(π(w), w)(18)

and

∂πc∂w

s(w) = F (πc(w),0) (19)

However, there is a substantial structural difference be-tween the output agreement problem considered hereand output regulation problems, that can be best seenfor linear dynamical systems. Suppose each system (1)is of the form

xi = Aixi +Giui + Piwiyi = Cixi,

(20)

with a linear exosystem wi = Siwi. Let in the followingA = block.diag(A1, . . . , An), G = block.diag(G1, . . . ,Gn), P = block.diag(P1, . . . , Pn) and C = block.diag(C1,. . . , Cn). The exosystems are stacked into the dynamicsw = Sw, with S = block.diag(S1, . . . , Sn). The classicalresult of [Fra76] states that one can take xw = Πw andλw = Γw such that the regulator equations (18) takethe form of Sylvester equations

ΠS = AΠ + G(B ⊗ Ip)Γ + P

(BT ⊗ Ip)CΠ = 0.(21)

Under controllability and observability assumptions,feasibility of (21) is necessary and sufficient for outputregulation of linear systems. We will see next, that dueto the networked structure of the considered problemsthe assumptions might fail to hold, although the outputagreement problem is solvable (as we show in the nextsections). First note that the regulator equations (21)have a solution if and only if

rank([ A− sIr G(B ⊗ Ip)

(BT ⊗ Ip)C 0mp×mp

])= #rows, (22)

for all s ∈ σ(S), where r =∑ni=1 ri and σ(S) is the

spectrum of S. The condition states that no pole of thestacked exosystem is a transmission zero of the systemfrom input λ to output z = (B⊗Ip)T y. To focus the dis-cussion on the impact of the constraints resulting fromthe network, we impose the following assumption:

Assumption 2 For each system i ∈ 1, . . . , n

rank([Ai − sIri Gi

Ci 0p×p

])= ri + p, ∀ s ∈ σ(S).

The main observation is that the rank condition can beviolated due to the networked structure of the problem.We summarize this in the result below.

Proposition 2 Suppose Assumption 2 holds. The rankcondition (22) is violated if either of the following holds:

(1) G contains a cycle;

(2) R(H(s)(B⊗ Ip)

)∩N

((BT ⊗ Ip)

)6= 0 for some

s ∈ σ(S), where H(s) = C(sI − A)−1G.

Moroever, the conditions are necessary provided that forall s ∈ σ(S), s 6∈ σ(A).

The proof is presented in the appendix. The first con-ditions shows that the regulator equations (21) have nosolution if graph contains cycles or if the transfer func-tions of the dynamical systems “rotate” R(B ⊗ Ip) insuch a way that it intersects nontrivially its orthogonalspaceN (BT ⊗Ip). The previous result gives an intuitionabout a class of systems for which the output agreementproblem is feasible.

Corollary 1 Assume Assumption 2 holds and G con-tains no cycles. Suppose furthermore that all eigenvaluesof S have zero real part. Then the equations (21) are fea-sible if H(s) is strictly positive real. 3

The proof is presented in the appendix. The result sug-gests that passivity takes an outstanding role in outputagreement problems. In fact, we show next, that a cer-tain type of passivity is sufficient for the feasibility ofthe output agreement problem.

3 Incremental passivity as sufficient conditionfor output agreement

In this section we highlight that the output agreementproblem is solvable for a special class of nonlinear sys-tems, namely incrementally passive systems. Our ap-proach follows the line of [PM08], where the notion of aregular storage function was introduced.

Definition 3 ([PM08]) A storage function V (t, x, x′)is called regular if for any sequence (tk, xk, x

′k), k =

1, 2, . . ., such that x′k is bounded, tk tends to infinity, and|xk| → ∞, it holds that V (tk, xk, x

′k)→∞, as k →∞.

The dissipativity characterization of incremental passiv-ity provided in [PM08] is as follows.

3 We refer to [Kha02, Def. 6.4] for the definition of a strictlypositive real transfer function.

5

Definition 4 The system (1) is said to be incrementallypassive if there exists a C1 regular storage function Vi :R≥0 × Rri × Rri → R≥0 such that for any two inputsui, u

′i and any two solutions xi,x

′i, corresponding to these

inputs, the respective outputs yi, y′i satisfy

∂Vi∂t

+∂Vi∂xi

fi(xi, ui, wi) +∂Vi∂x′i

fi(x′i, u′i, wi)

≤ (yi − y′i)T (ui − u′i).(23)

Incremental passivity is an input-output property thatcompares two arbitrary trajectories of the dynamicalsystem. While it is a more restrictive than classical pas-sivity it has been a relevant concept in modern controltheory, see e.g., [DV74], [Sas99], [PM08]. Related con-cepts have proven to be very useful in the study of net-work systems ([SS07], [SAS10] – more details on the com-parison between our approach and the one in those pa-pers are found in the first paragraph of Subsection 6.1).In [PM08, Theorem 2], a procedure is proposed to ren-der a class of nonlinear systems incrementally passive bythe design of a suitable local feedback controller. In thecontext of this paper, the following two classes of incre-mentally passive systems take an outstanding role.

Example 1 Linear systems of the form (20) that arepassive from the input ui to the output yi are also incre-mentally passive, with Vi = 1

2 (xi − x′i)TQi(xi − x′i) and

Qi = QTi > 0 the matrix such that ATi Qi + QiAi ≤ 0and QiGi = CTi . Remarkably, system (20) is incremen-tally passive if its transfer function is positive real. Thus,in view of Corollary 1, incremental passivity takes anoutstanding role for the feasibility of output agreementproblems.

Example 2 Nonlinear systems of the form

xi = fi(xi) +Giui + Piwi

yi = Cixi(24)

with fi(xi) = ∇Fi(xi), Fi(xi) twice continuously dif-ferentiable and concave, and Gi = CTi are incremen-tally passive. In fact, by concavity of Fi(xi), (xi −x′i)

T (fi(xi)− fi(x′i)) ≤ 0, and Vi = 12 (xi−x′i)T (xi−x′i)

is the incremental storage function.

For the sake of brevity, we will in the followingsometimes write Vi for the directional derivative∂Vi

∂t + ∂Vi

∂xifi(xi, ui, wi) + ∂Vi

∂x′ifi(x

′i, u′i, wi).

In the previous section, it was shown that the controllersat the edges have to take the form (16). Now, they mustbe completed by considering additional control inputsthat guarantee the achievement of the steady state.While we require the internal model to be identical for

all edges, i.e., φ(ηk), the augmented systems might bedifferent. Then, the controllers (16) modify as

ηk = φk(ηk, vk)

λk = ψk(ηk), k = 1, 2, . . . ,m,(25)

where all controllers reduce to the common inter-nal model if no external forcing is applied, i.e.,φk(ηk, 0) = φ(ηk). The controller is then said to havethe internal model property. The following is the mainstanding assumption that the controllers must satisfy.

Assumption 3 For each k = 1, 2, . . . ,m, there existsregular storage functions Wk(ηk, η

′k), with Wk : Rqk ×

Rqk → R≥0 such that

∂Wk

∂ηkφk(ηk, vk) +

∂Wk

∂η′kφk(η′k, v

′k)

≤ (λk − λ′k)T (vk − v′k).

(26)

The existence of an incrementally passive controller is ingeneral hard to verify. In fact, it depends on the struc-ture of the exo-system (2) as well as on the properties ofthe node dynamics, i.e., in particular on uw as definedby (13). However, there is an important class of prob-lems, where an incrementally passive controller can bedesigned in a constructive way.

Example 3 The important example when the design ispossible is when the feedforward control input is linear,that is (14) is satisfied with τ = Id, φ = s and ψ is alinear function of its argument. In this case, we let

φk(ηk, 0) = s(ηk), ψk(ηk) = Mkηk

and define

φk(ηk, vk) = s(ηk) +MTk vk. (27)

Then, by definition of s as the gradient of a concave func-tion, the storage functionWk(ηk, η

′k) = 1

2 (ηk−η′k)T (ηk−η′k) satisfies

∂Wk

∂ηkφk(ηk, vk) +

∂Wk

∂η′kφk(η′k, v

′k) =

(ηk − η′k)T (s(ηk)− s(η′k))

+(ηk − η′k)TMTk (vk − v′k)

≤ (ψk(ηk)− ψk(η′k))T (vk − v′k),

(28)

that is (26).

We state below the main result of the section that, whileextending to networked systems the results of [PM08],provides a solution to the output agreement problem inthe presence of time-varying disturbances.

6

Theorem 1 Consider the network G with dynamics onthe nodes (4). Suppose all exosystems satisfy (3), theregulator equations (11) hold, and all node dynamics areincrementally passive. Consider the controllers

η = φ(η, v)

λ = ψ(η) + ν(29)

where φ and ψ are the stacked functions of φk(ηk, vk)and ψk(ηk), and ν is an additional input to be designed.Suppose the controllers have the internal model propertyand satisfy Assumption 3. Then, the controller (29) withthe interconnection structure

u = (B ⊗ Ip)λ, v = −(BT ⊗ Ip)y. (30)

and ν := v = −(BT ⊗ Ip)y solves the output agreementproblem, that is every solution starting fromW ×X ×Ξis bounded and

limt→+∞

(BT ⊗ Ip)y(t) = 0.

Proof: By the incremental passivity property of thex subsystem in (4) and (11), it is true that

∂V

∂t+∂V

∂xf(x, u, w) +

∂V

∂xwf(xw, uw, w)

≤ (y − yw)T (u− uw),

where V =∑i Vi. Similarly by Assumption 3, the sys-

tem (29) satisfies

∂W

∂ηφ(η, v) +

∂W

∂ηwφ(ηw) ≤ (λ− λw)T v − (ν − νw)T v,

with W =∑kWk and φ(ηw) = 1m ⊗ φ(ηw). Bearing

in mind the interconnection constraints u = (B ⊗ Ip)λ,uw = (B ⊗ Ip)λw, and v = −(BT ⊗ Ip)y, and lettingU((x, xw), (η, ηw)) = V (x, xw) +W (η, ηw) we obtain

U((x, xw), (η, ηw)) := V (x, xw) + W (η, ηw)

≤ (y − yw)T (u− uw) + (λ− λw)T v − (ν − νw)T v

= (y − yw)T (B ⊗ Ip)(λ− λw)

− (λ− λw)T (BT ⊗ Ip)y + (ν − νw)T (BT ⊗ Ip)y.

By definition of output agreeement, (BT ⊗ Ip)yw = 0and the previous equality becomes

U((x, xw), (η, ηw)) ≤ νT (BT ⊗ Ip)y= −||(BT ⊗ Ip)y||2 = −zT z,

(31)

by definition of ν = −z and νw = 0. Since U is non-negative and non-increasing, then U(t) is bounded. As

xw, ηw are bounded 4 and U is regular, then x, η arebounded as well. Hence the solutions exist for all t. In-tegrating the latter inequality we obtain

∫ +∞

0

zT (s)z(s)ds ≤ U(0).

By Barbalat’s lemma, if one proves that ddtz

T (t)z(t) is

bounded then one can conclude that zT (t)z(t) → 0.Now, z(t) = (BT ⊗ Ip)y = (BT ⊗ Ip)h(x,w) is boundedbecause x,w are bounded. If h is continuously differen-tiable and x, w are bounded, then z is bounded and onecan infer that d

dtzT (t)z(t) is bounded. By assumption, w

is the solution of w = s(w) starting from a forward in-variant compact set. Hence, both w and w are bounded.On the other hand, x satisfies

x = f(x, (B ⊗ Ip)ψ(η)− z, w)

which proves that it is bounded because x, η, z wereproven to be bounded, while w is bounded by assump-tion. Therefore, x, w are bounded and this implies thatddtz

T (t)z(t) is bounded. Then by Barbalat’s Lemma wehave limt→+∞ z(t) = 0 as claimed.

Remarkably, the assumptions on the interconnectionscan be weakened, if stronger assumptions on the nodedynamics are imposed.

Corollary 2 Let all assumptions of Theorem 1 hold, butassume furthermore that all node dynamics are outputstrictly incrementally passive, that is, there exists a C1

regular storage function Vi, and a positive definite func-tion ρi : Rp → R, such that for any two inputs ui, u

′i and

corresponding outputs yi, y′i

V ≤ −ρi(yi − y′i) + (yi − y′i)T (ui − u′i). (32)

Then the output agreement problem is feasible with theinterconnection (30) and ν = 0.

Proof: Consider the storage function usedin proof of Theorem 1, i.e., U((x, xw), (η, ηw)) =V (x, xw) + W (η, ηw). After repeating the steps of theproof of Theorem 1, but using the output strict pas-sivity property and setting ν = 0, (31) is now replaced

by U((x, xw), (η, ηw)) ≤ −∑ni=1 ρi(yi − ywi ), where

ywi = ywj for all i 6= j. Now, we can proceed as in theproof of Theorem 1.

4 By definition, (w, xw, ηw) belongs to the ω-limit set, whichis compact. Hence, xw, ηw are bounded.

7

4 Constructive controller design for a class ofnonlinear systems

We present now a fairly large class of heterogeneous non-linear systems for which all assumptions of Theorem 1are satisfied and the controller design follows a construc-tive procedure. Consider the systems introduced in Ex-ample 2, namely

wi = si(wi)

xi = f0(xi) +Gui + Piwi

yi = Cxi i = 1, 2, . . . , n

(33)

where, compared with (24), we have chosen the systemsto have the same dynamics, i.e. fi(xi) = f0(xi) for alli = 1, 2, . . . , n, and we have set Gi = CTi = G = CT .Assuming that the dynamics of the systems are the samefacilitate the design of incrementally passive distributedcontrollers, as we see in the proof below.

Proposition 3 Consider systems (33), where f0 = ∇Fand F is a twice continuously differentiable and concavemap,G = CT and full column rank matrix, and the mapssi satisfy (3). Moreover, assume that R(Pi) ⊆ R(G) fori = 1, 2, . . . , n. Given the vector of disturbances w =(w1, . . . , wn), assume there exists a bounded solution x∗to the system

x = f0(x) +

∑ni=1 Piwin

. (34)

Then:

(1) there exists a bounded solution xw, uw to the regu-lator equations (11);

(2) there exists controllers at the edges of the form

ηk = s(ηk) +Hkvk

λk = HTk ηk + νk, k = 1, 2, . . . ,m

(35)

such that output agreement problem is solved forthe systems (33), interconnected with the controllers(35) via the conditions (30).

Proof: Take any solution xw∗ to (34). By definition

xw∗ = f0(xw∗ ) +

∑ni=1 Piwin

. (36)

Observe that such a solution xw∗ is necessarily bounded.As a matter of fact, in view of the assumptions onf0, the incremental dissipation inequality (23) holdand the incremental storage function V (xw∗ , x∗) =12 (xw∗ − x∗)T (xw∗ − x∗) satisfies V ≤ 0 (in system (34)

inputs are absent). Hence V (xw∗ , x∗) is bounded and byregularity of V and boundedness of x∗, x

w∗ is bounded.

Define now

Guwi = −(Piwi −

∑ni=1 Piwin

). (37)

Observe that since∑ni=1Gu

wi = 0 by construction, and

G is full-column rank, then uw ∈ R(B ⊗ Ip), i.e., therequirement imposed by the interconnection condition(30) is fulfilled. An explicit expression for uw can begiven. Let

(In ⊗G)uw = ((1n1Tnn− In)⊗ Ir)Pw =: (Y ⊗ Ir)Pw

where r is the dimension of the state space of each sys-tem. Hence (37) can be rewritten as

Guwi =

n∑j=1

YijPjwj , with Yij = [Y ]ij .

There exists a solution uwi to the latter equation if andonly if GG†bi = bi with bi =

∑nj=1 YijPjwj , where

G† is the Moore-Penrose pseudo inverse. Recalling thatR(Pj) ⊆ R(G), we can assume the existence of matricesΓj such that

Pjwj = GΓjwj .

As a result

bi =

n∑j=1

YijPjwj =

n∑j=1

YijGΓjwj = G

n∑j=1

YijΓjwj

that is bi ∈ R(G), and GG†bi = GG†G

n∑j=1

YijΓjwj =

G

n∑j=1

YijΓjwj =

n∑j=1

YijGΓjwj =

n∑j=1

YijPjwj = bi.

Then the unique solution to (37) is uwi = G†n∑j=1

YijPjwj .

Replacing (37) into (36), the latter becomes

xw∗ = f0(xw∗ ) +Guwi + Piwi.

The latter holds true for all i = 1, 2, . . . , n thus show-ing that (xw, uw) = ((1n ⊗ Ir)xw∗ , ((uw)T , . . . , (uwn )T )T )solves the regulator equations.Bearing in mind that (B ⊗ Ip)λw = uw, we have

λw = −(B† ⊗ Ip)(In ⊗G†)(Y ⊗ Ir)Pw= −(In ⊗G†)(B†Y ⊗ Ir)Pw= −(B†Y ⊗G†)Pw,

(38)

8

that is λw = Hw. Using the embedding (14) with τ =Id, φ = s and ψ(η) = Hη, and an analogous decompo-sition as in (16), the internal model controller takes theform

ηk = s(ηk)

λk = HTk ηk, k = 1, 2, . . . ,m.

The addition of the control term Hkvk

ηk = s(ηk) +Hkvk

λk = HTk ηk, k = 1, 2, . . . ,m.

renders the system incrementally passive, in view of theincrementally passive nature of the map s(·). Recall (seeExample 2) that the condition on F that defines thedynamics of the systems according to the identity f0 =∇F guarantees incremental passivity of systems (33).Hence, we are under the conditions of Theorem 1 andone concludes that the controllers

ηk = s(ηk)−Hkzk

λk = HTk ηk − zk, k = 1, 2, . . . ,m.

(39)

with z = (BT ⊗ Ip)y guarantee that the output agree-ment problem is solved.

The proof is constructive. The controllers, designed asin (39), provide an explicit solution to the output agree-ment problem for systems (33). In (39), the vector fields is given by the exosystems generating the disturbancesaffecting (33), while the input matrix Hk is obtainedfrom the matrix H defined in (38).The controllers (39) can be stacked together to the dy-namics

η = s(η)− Hvλ = Hη − ν,

(40)

where η = [ηT1 , . . . , ηTm]T ∈ Rmr, s(η) = [s(η1)T , . . . ,

s(ηm)T ]T , and H = block.diagH1, . . . ,Hm. Bearingin mind the controllers (40), one conclusion that fol-lows immediately from the proof of the result is thatsteady state solution of the controllers ηw can be takenas ηw = 1m ⊗ w. That is, one possible steady statesolution of the output agreement problem is that eachcontroller dynamics reproduces exactly the disturbancesignal. This observation can be used to redesign thecontrollers. In particular, additional communication be-tween the different (distributed) controllers can be usedto improve the convergence of the controllers.

4.1 Adding communication between controllers

Consider an additional communication network Gcomm,having one node for each controller, and one edge if

the controllers can exchange data. 5 For simplicity,we assume that Gcomm is an undirected connectedgraph with Laplacian matrix Lcomm ∈ Rm×m. As weshall see below, the additional communication termallows us to add a diffusive coupling between thevarious controllers that explicitly enforces the conver-gence of all the controllers states ηk to the same sig-nal. This in turn guarantees that the stacked vectorHη = block.diagHT

1 , . . . ,HTm(ηT1 . . . ηTm)T converges

to Hη∗, for some η∗. We recall that under the conditionsthat the convergence to the solution of the output agree-ment problem is uniform in the initial conditions, such asignal η∗ must satisfy (B ⊗ Ip)Hη∗ = uw. If in additionthe graph is acyclic and the system η = s(η), λ = Hηis incrementally observable 6 , then necessarily, η∗ = w,i.e. the internal model controllers asymptotically syn-chronize to the disturbance w.

By revisiting now the proof of Theorem 1, one can di-rectly see that the assumption of incremental passivityof the controllers, i.e., Assumption 3, is stricter thannecessary. In particular, one can require the incremen-tal passivity property (26) not to hold with respect toany two trajectories, but only with respect to the realand the steady state trajectory, i.e., with η′k = ηwk , v

′k =

0, λ′k = λw. Thus, one can replace Assumption 3 withthe following weaker assumption.

Assumption 3a Let ηw = τ(w) and λw = ψ(τ(w))be a solution to (14), and let vw = 0. For eachk = 1, 2, . . . ,m, there exists regular storage functionsWk(ηk, η

wk ), with Wk : Rqk × Rqk → R≥0 such that

∂Wk

∂ηkφk(ηk, vk) +

∂Wk

∂ηwkφk(ηwk ,0) ≤ (λk − λwk )T vk.

It can be readily seen that the proof of Theorem 1 re-mains valid if Assumption 3 is replaced by Assumption3a. In particular, the following result holds.

Proposition 4 Let all assumptions of Proposition 3hold and let Lcomm ∈ Rm×m be the Laplacian matrix ofcommunication graph. Then the distributed controllerwith communication of the form

η = s(η)− (Lcomm ⊗ Ir)η − Hvλ = HT η

(41)

interconnected with the node dynamics (33) according to(30), solves the output agreement problem Furthermore,limt→∞ ||ηk(t)− ηj(t)|| = 0 for all k 6= j.

5 For instance, two controllers can exchange data if theircorresponding edges are incident to the same node in theoriginal graph G.6 The system η = s(η), λ = Hη is incrementally observableif any two solutions η, η′ to η = s(η) which yield the sameoutput necessarily coincide, i.e. η = η′.

9

Proof: Under the assumptions of Proposition 3 itholds that ηw = w. Consequently (Lcomm ⊗ Ir)ηw = 0.The controller (41) satisfies Assumption 3a, since thedirectional derivative of W = 1

2 (η − ηw)T (η − ηw) is

W ≤ −(η − ηw)T (Lcomm ⊗ Ir)(η − ηw) + (λ− λw)T v.

Mirroring the proof of Theorem 1, the derivative of thestorage function U((x, xw), (η, ηw)) satisfies

U ≤ −‖(BT ⊗ Ip)y‖2 − ηT (Lcomm ⊗ Ir)η.

Thus, with the same arguments as in the proof of Theo-rem 1, convergence can be concluded. Additionally, thisproves that limt→∞ ‖ηk(t)− ηj(t)‖ = 0 for all k 6= j.

5 Time-varying Optimal Distribution Control

We illustrate next the applicability of the proposed con-troller design method to an (optimal) distribution con-trol problem in networks with storage. This is a simpleyet meaningful class of systems for which the theory de-veloped so far provides distributed flow controllers thatcan be proven to be optimal at steady state.Consider an inventory system with n inventories and mtransportation lines, and let B be the incidence matrixof the transportation network. The dynamics of the in-ventory system is given as

x = Bλ+ Pw, (42)

where x ∈ Rn represents the storage level, λ ∈ Rm theflow along one line, andPw an external in-/outflow of theinventories, i.e., the supply or demand. This basic modelis studied, e.g., in [BBP06], [vW12] or in a discrete-timeform in [BB12], [DBO+13].We assume here that the exact realization of the sup-ply/demand w is unknown, while it is known that it isgenerated by a dynamics of the form (2) The distribu-tion and balancing problem is to design controllers on theedges of the network, using only measurements of thestorage levels of the incident inventories and regulatingthe flows λk such that instantaneously all possible sup-ply/demand is satisfied and all inventory levels evolvesynchronously. By choosing u = Bλ, the problem canbe readily formulated as an output agreement problemwith time varying disturbance.The regulator equations (13) for the distribution prob-lem are

xw(t) = uw(t) + Pw(t),

uw(t) ∈ R(B), xw(t) ∈ N (BT ).(43)

The solution to the regulator equation (43) is

xw(t) = 1n

(xw0 +

∫ t

0

1TnPw(s)

nds)

(44)

where xw0 belongs to the projection of ω(W × X × Ξ)onto Rr1+...+rn . To see (44), note that uw(t) ∈ R(B)⇔1Tnu

w(t) = 0. Let now xw(t) = 1nxw∗ (t), for some

xw∗ (t) ∈ R. Then, multiplying (43) from the left withthe all ones vector gives nxw = 1TPw(t), leading to thedesired expression. The following observation is now adirect consequence.

Proposition 5 The output agreement problem isfeasible only if the accumulated imbalance w(t) =∫ t

01TnPw(s)n ds is bounded for all t ≥ 0.

Otherwise the inventory levels (i.e., xw) will grow un-bounded. The corresponding input is naturally given asuw(t) = −∆nPw(t), with ∆n = (In − 1

n1n1Tn ), namelythe projection of the supply/demand vector to the spaceorthogonal to span1n. Next, we verify that the nec-essary conditions for the output agreement problem aresatisfied by showing feasibility of (14). Note that the con-troller output must satisfyBλw = −(In− 1

n1n1Tn )Pw(t).

Proposition 6 If the network contains a spanning treethen the condition (14) is feasible.

Proof: Let T ⊆ G be a spanning tree. Assumewithout loss of generality that the edges are labeledin such a way that the flow vector can be written asλw = [λwTT , λwTT ]T , where λwT are the flows on the edgesin T and λwT are the flows in all other edges. Similarly, theincidence matrix can be represented as B = [BT , BT ].A feasible flow solution λwp can now be chosen as λwT = 0

and λwT = −(BTT BT )−1BTT Pw(t). Note that λwT routesexactly the balanced component of the supply/demandthrough the network, since λwT = −(BTT BT )−1BTT (In −1n1n1Tn )Pw(t) since BTT 1n = 0n. Define now

H =

[−(BTT BT )−1BTT P

0

],

and note that λwp = Hw. Thus, τ = Id, φ(·) = s(·) andψ being the linear function defined by H, solve (14).

After augmenting the controller with external outputs,a possible routing controller is

η = s(η) +HT v

λ = Hη + ν.(45)

Note that if s(·) satisfies the standing assumption (3),then this controller is incrementally passive.

5.1 Optimal Distribution Control

We enlarge our control objective and aim to design afeedback controller that achieves an optimal routing.

10

That is, we want to regulate the flows such that theyminimize the quadratic cost function

P(λ) =1

2λTQλ, (46)

with Q = diag(q1, . . . , qm) and qk > 0.We exploit therefore that the internal model controllerachieving balancing is not unique. In particular, we re-design the controller (45) in such a way that it routesthe balanced component of the flow through the networkin such a way that at each time instant the cost (46) isminimized. That is, asymptotically the routing shouldbe such that at each time instant t the following staticoptimization problem is solved

minλP(λ), s.t. 0 = Bλ+ ∆nPw, (47)

where w = w(t) is the supply at the respective time. Letnow ζ ∈ Rn be the multiplier for the equality constraint.The Lagrangian function of (47) is

L(λ, ζ) =1

2λTQλ+ ζT (Bλ+ ∆nPw).

One can express the optimality conditions in terms ofthe dual solution as

Qλ+BT ζ = 0, Bλ+ ∆nPw = 0,

from which B(Q−1BT ζ)+∆nPw = 0, with the optimalrouting being λ = −Q−1BT ζ. Thus the optimal rout-ing/supply pairs are defined as the set

Γ = (λ,w) : Qλ ∈ R(BT ), Bλ+ ∆nPw = 0.

We formalize the optimal distribution problem as fol-lows:

Definition 5 The time-varying optimal distributionproblem is solvable for the system (42), if there existsa controller (6) such that any solution originating fromW × X × Ξ satisfies (i) limt→∞BTx(t) = 0 and (ii)limt→∞ distΓ(λ(t), w(t)) = 0.

To solve the problem, we proceed in this way. Instead ofdesigning the controller directly for the flows, we designthe controllers for the multipliers. We take τ = Id andφ(·) = s(·) and design a controller of the form

η = s(η) +Hvv

ζ = Hζη,

where Hζ and Hv are suitable input and output matri-ces to be designed next. The routing will then be de-fined as λ(t) = Q−1BT ζ(t). For designingHζ , note that,

provided that v = 0 and the initial condition η(0) isproperly chosen, the system above generates the solutionηw(t) = w(t). Then Hζ must be design in such a waythat ζw(t) = Hζη

w(t) satisfies the optimality condition

BQ−1BTHζηw(t) + ∆nPw(t) = 0.

The matrix LQ = BQ−1BT is a weighted Laplacian ma-trix. As LQ has one eigenvalue at zero, with the cor-responding eigenvector 1, it is not invertible. However,since ηw(t) = w(t), one possible solution is

Hζ = −L†QP, (48)

where L†Q is the Moore-Penrose-inverse of LQ, see

e.g., [GX04]. From the properties of L†Q follows that

BQ−1BTHζηw + ∆nPw = −BQ−1BTL†QPη

w +∆nPw = −∆nPη

w+∆nPw = 0 as desired. Now, as thecontroller should be incrementally passive with input vand output λ, we can design it in the form (45) taking

H = Q−1BTHζ . (49)

Then, to have incremental passivity, we simply chooseHv = HT . This choice of the input and output matrix forthe controller (45) ensures that the optimal distributionproblem is solved.

Proposition 7 Consider the inventory system (42) withthe supply generated by the linear dynamics w = s(w),satisfying (3). Consider the controller

η = s(η)−HT z

λ = Hη − z.

with the interconnection condition z = BTx. Then, ev-ery solution of the closed-loop system is bounded and (i)limt→+∞BTx = 0, and (ii) limt→+∞ distΓ(λ(t), w(t)) =0, that is the time-varying optimal distribution problemis solvable.

Proof: First note that the optimal routing λw(t) =Q−1BT ζw(t) satisfies the identity

Bλw(t) + Pw(t) = −BQ−1BTL†QPηw(t) + Pw(t) =

−∆nPηw(t) + Pw(t) = 1

1TPw(t)

n.

Since 11TPw(t)n = xw (see (44)), the optimal routing is

such that xw(t) = Bλw(t) + Pw(t).

Now, consider the storage function U(x − xw, η, ηw) =12‖x− x

w‖2 + 12‖η− η

w‖2 along the solutions of the au-

tonomous system ddt (x−x

w) = B(λ−λw) = −BBT (x−

11

xw) +BHη−BHηw, η = s(η)−HTBT (x− xw), ηw =s(ηw). It satisfies

U =− (x− xw)TBBTx+ (x− xw)TBH(η − ηw)

+ (η − ηw)T (s(η)− s(ηw))− (η − ηw)THTBTx

≤− ‖BTx‖2,

due to the incremental passivity of the exosystem, i.e.,(η−ηw)T (s(η)−s(ηw)) ≤ 0. Since U is positive semidef-inite and ηw is bounded (again by the incremental pas-sivity property of the exosystem), we have that x−xw, η,ηw are all bounded. Then, by LaSalle’s invariance prin-ciple, the trajectories converge to the largest invariantset such that BT (x−xw) = BTx = 0. Thus, there existsx∗ such that on this set x−xw = x∗1 and the dynamicsevolves as

x∗1 = BHη −BHηw, η = s(η), ηw = s(ηw). (50)

After multiplying by 1n1T from the left, it follows x∗ = 0,

proving that x must approach xw modulo a constant.This proves the claim (i) of the statement.To prove claim (ii), note that inserting x∗ = 0 into (50)and bearing in mind that ηw = w gives the necessarycondition that in the set where BT (x− xw) = 0 it musthold that

0 = BHη −BHηw = BHη −BHw =

−BQ−1BTL†QP (η − w) = −∆nP (η − w).

Hence, ∆nPη = ∆nPw. The flow on the invariant set

is λ = Q−1BTL†QPη, while the optimal flow is λw =

Q−1BTL†QPw. Together with the previous condition,

this implies that there is a vector ν ∈ N (B) such thatλ = λw + ν. We will show next that ν must be identicalto zero. Note that ν = λ−λw, and must therefore satisfy

ν = Q−1BTL†QP (η − w).

Multiplying the previous equation from the left by νTQleads to

νTQν = 0

since νTBT = 0. AsQ is by assumption positive definite,the only solution is ν = 0. This proves that in the setwhere BTx = 0 it must hold that λ = λw, completingthe proof.

If additionally communication between the controllersis allowed, the controller can be augmented with a con-sensus term of the form (41).

5.2 Simulation Example

We illustrate the performance of the controller on a de-sign example. Consider a network with four inventories

x1 x2

x3 x4

λ1

λ2

λ3λ4

λ5

w1 w2

w3 w4

Fig. 2. Inventory network with four inventories and five trans-portation lines.

Time

Inve

ntor

yL

evel

s1

2

3

4

5

6

7

8

9

0 50 100 150 200 250

Fig. 3. Inventory levels xi(t), i ∈ 1, 2, 3, 4, under time–varying supply/demand.

and five transportation lines as illustrated in Figure 2.The supply/demand at each inventory is generated bythe linear dynamics

wi =

[0 −sisi 0

]wi

with s1 = 0.1, s2 = 0.7, s3 = −0.4 and s4 = −0.2 andinitial conditions wi(0) = [1, 1]T . The flow cost functionis of the form (46) with Q = diag1, 2, 3, 4, 5. The con-troller is implemented in the distributed form with com-munication (41), where H is chosen to satisfy (49), andGcomm is chosen such that two controllers communicateif they are incident to the same inventory.The simulation results for the inventory levels are shownin Figure 3. Note that the supply/demand is not bal-anced, but the accumulated imbalance is bounded. Thecontroller achieves a balancing of the inventory levels. Asan example, the flow λ1(t) is shown in Figure 4. The flowapproaches fairly quickly the time-varying optimal flow.The simulations illustrate that the controller achievesboth objectives, the balancing of the inventory levels andthe optimal routing of the flow through the network.

To illustrate the relevance of the dynamic couplingdesign, we compare in Figure 5 the long-time behav-ior of our controller to an integral routing controllerthat is designed assuming an unknown but static sup-

12

Flow

Time

3

2

1

0

0

-1

-2

20 40 60 80 100 120

Fig. 4. Flow λ1(t) (solid) and optimal flow on the corre-sponding edge (dashed).

ply and demand. It was shown in [Wv13] that integralcontrollers can solve distribution problems with staticinflows, and in [BZA13] that they can in this case alsosolve an optimal distribution problem. Following thedesign of [Wv13] (and combining it with the optimaldesign proposed in [BZA13]) the integral routing con-troller takes the form (see [Wv13] or [BZA13]) )

ηk = vk, λk = −q−1k ηk − vk. (51)

Figure 5 compares the long-term behavior of the inte-gral controller (51) (dashed, red) to the internal modelcontroller (black, solid). One can readily observe thatthe integral controllers do not achieve a balancing ofthe inventory levels, while the internal model controllerachieves an exact balancing. Note that the integral con-troller (51) would solve the optimal distribution prob-lem (47) asymptotically exactly if the supply was con-stant and balanced. However, for the time-varying sup-ply the resulting steady state flow violates the equalityconstraint of (47). Thus, the integral controller cannotachieve asymptotically an optimal routing.

6 Relation to known results

The presented controller design methodology can be seenas an extension of several existing control approaches fornetworked systems. We clarify next the relation of theproposed approach to existing ones.

6.1 Static Couplings

Significant work has been dedicated to the study ofsynchronizing oscillatory systems without disturbancesusing static diffusive couplings. In the early work[Hal97], Hale studied the synchronization of nonlin-ear oscillatory systems with infinitely strong couplings.Subsequent works studied synchronization of identi-cal nonlinear systems that have a property known as

Inve

ntor

yL

evel

s

Time250 300 350 400 450

Fig. 5. Long-term inventory levels with the internal modelcontroller (solid, black) and the integral controller (dashed,red).

relaxed cocoerciveness ([SS07], [SAS10], [DdR11]) orsemi-passivity ([PN01]). We show next that, while syn-chronization of heterogeneous systems with externaldisturbances require in general dynamic couplings, alsoin our framework static diffusive couplings are sufficientif all systems already share a common internal model.

Proposition 8 (Static Coupling) Consider the sys-tem (4) and suppose all node dynamics are incremen-tally passive. If there exists a solution to the regulatorequations (11) with uw(t) = 0, then, the static controllerλ = ν with the interconnection u = (B ⊗ Ip)λ, andν = −(BT ⊗ Ip)y solves the output agreement problem.

Proof: By the incremental passivity property of thesubsystems it is true that V ≤ (y−yw)T (u−uw), whereV =

∑ni=1 Vi. Now, since uw = 0 and yw ∈ N ((BT⊗Ip))

the coupling u = −Ly = −(B⊗ Ip)(BT ⊗ Ip)y = −(B⊗Ip)(B

T ⊗ Ip)(y − yw) gives V ≤ −(y − yw)TL(y − yw).Convergence and boundedness can now be shown as inthe proof of Theorem 1.

Please note that the input to the systems computes inthis case as u = −(L ⊗ Ip)y, where L = BBT is theLaplacian matrix of the (undirected) graph. Thus, forhomogeneous systems, our controller design method re-duces to the well-known Laplacian coupling, as studied,e.g., in [SS07], [SAS10], [DdR11].

6.2 Static Disturbances

For the special case of output agreement problemswith constant disturbances, our approach providesan interpretation of integral controllers that havebeen intensively studied in various different contexts([vW12],[Wv13], [BZA14], [BZA13], [SPDB13]). Con-trol of passive system with constant disturbances isstudied, e.g., in [JOGC07] or [HAP11]. The stability

13

of passive networks with static disturbance signals hasbeen discussed in [vW12]. We derive here slightly moregeneral controllers (or dynamic couplings) as [BZA14],[BZA13], using the internal model control approach.

Proposition 9 Consider the network G with dynamicson the nodes (4). Supposewi is some constant signal, i.e.,si(wi) = 0, the regulator equations (11) hold and (4) areincrementally passive. Then, any controller of the form

ηk = vkλk = ψk(ηk) + νk, k ∈ 1, . . . ,m

(52)

with ψk(·) satisfying the strong monotonicity condition

(ψk(η)− ψk(η′))T (η − η′) ≥ c‖η − η′‖2, ∀η, η′ (53)

for some positive constant c, and interconnection con-straints (30) solves the output agreement problem.

Proof: Let xw and uw be solutions to the regulatorequations (11). By the structure of (11) follows imme-diately that vw = −(BT ⊗ Ip)h(xw, w) = 0. Since thedisturbance is static, i.e., w = 0, the conditions (14) aresolved with φ(·) = 0 and τ(w) such that

λwp + λw0 = ψ(τ(w)) (54)

for some λwp satisfying uw = (B⊗ Ip)λwp and some λw0 ∈N (B ⊗ Ip) and constant. Thus, there is not a unique so-lution to (14), but rather for any λw0 ∈ N (B ⊗ Ip) thereexists exactly one τ(w) solving (14) (the existence ofmore than one solution τ(w) would contradict the strongmonotonicity condition (53)). Select now ηw = τ(w) asa solution to (54) an arbitrary λw0 ∈ N (B ⊗ Ip), and letλw = λwp + λw0 .One can construct now for each controller a storage func-tion Wk that satisfies Assumption 3a, i.e., that showspassivity with respect to the constant signals λwk andvwk = 0. Let in the following Ψk : Rq → R be a twice con-tinuously differentiable function such that ∇Ψk(ηk) =ψk(ηk). Since, by assumption, ψk satisfy the monotonic-ity condition (53) all Ψk are strongly convex. Considernow the storage function ([JOGC07], [BZA14]):

Wk(ηk, ηwk ) = Ψk(ηk)−Ψk(ηwk )−∇ΨT

k (ηwk )(ηk − ηwk ).(55)

Since Ψk is convex and, by the global under-estimatorproperty of the gradient, we have Ψk(ηk) ≥ Ψk(ηwk ) +∇ΨT

k (ηwk )(ηk − ηwk ) for each ηk, ηwk . Since Ψk is strongly

convex, then it is in particular strictly convex and theprevious inequality holds if and only if ηk = ηwk . ThenWk

is regular ([JOGC07]). Hence, Wk is a positive regular

storage function. Furthermore,

∂Wk

∂ηkφk(ηk, vk) = (ψk(ηk)− ψk(ηwk ))T vk

= (λk − λwk )T vk.

In the case of constant disturbances Assumption 3a isalways fulfilled by controllers of the form (52). Mim-icking now the proof of Theorem 1, using the storagefunction U((x, xw), (η, ηw)) = V (x, xw) + W (η, ηw),with W (η, ηw) =

∑mk=1Wk(ηk, η

wk ), one obtains

U ≤ −‖(BT ⊗Ip)y‖2. With the same arguments as usedin the proof of Theorem 1, it follows that the controller(52) solves the output agreement problem in the case ofstatic disturbances, that is limt→∞(BT ⊗ Ip)y(t) = 0.

7 Power Systems Droop-Control as InternalModel Control

The previous discussion suggests, that the incremen-tal passivity approach can provide novel insights intoexisting results. In the same direction, we show nextthat a dynamic oscillatory model of microgrids withfrequency-droop controllers studied in [SPDB13] can bere-interpreted in the internal model framework. This re-interpretation has several merits. It unveils the frame-work underlying some recent synchronization results inoscillatory networks, thus opening a new perspective onthe analysis (and control) of these networks. Moreover,it provides an interesting case study that has attracteda great deal of attention and to which our approach con-veniently applies.The model of [SPDB13] for the frequency-droop con-troller is

Diθi = P ∗i − Pe,i, i ∈ 1, . . . , n, (56)

where Di is the inverse of the controller gain, P ∗i is theinverters nominal power, and Pe,i is the active electricpower. The active electric power is given by

Pe,i =

n∑j=1

αij sin(θi − θj), (57)

where αij are constants depending on the node voltagesand the line admittance. The coefficients are symmetricαij = αji and only non-zero if the two nodes i and j areconnected by a line. We refer to [SPDB13] for a detaileddiscussion of the model. As in [SPDB13], we restrict thediscussion in the following to acyclic networks.In the model, the dynamics in the nodes (56) representsthe controllers, while the couplings between the nodes,i.e., (57), are physical laws. Although the situation is re-versed to the basic setup of this paper, we can still inter-pret the droop-controller as an internal model controller.Consider the node dynamics (1) as (56), with node state

14

xi = θi, constant external signal wi = P ∗i , satisfying

wi = 0, input ui = −Pe,i and output yi = θi, i.e.,

Dixi = P ∗i + ui, yi = xi. (58)

The node dynamics is output strictly incrementally pas-sive since for any to inputs ui, u

′i and the two corre-

sponding outputs yi, y′i, it holds that

(yi − y′i)(ui − u′i) = (yi − y′i)(Diyi − P ∗i −Diy′i + P ∗i )

= Di‖yi − y′i‖2.

From the interpretation of the node dynamics (1) as(56), one notices that u = −Pe = −BA sin(BTx), wheresin(z) = [sin(z1), . . . , sin(zm)]T , A = diaga1, . . . , am,ak = αij = αji, and k is the label of the edge connectingnodes i, j. One can then interpret the latter equationas the first one of the interconnection conditions (30),provided that λ = A sin(BTx). Set now η = BTx. Then

η = −BT yλ = A sin(η).

(59)

This can be understood as the stacked controllers (25),where, for all k, φk(ηk, vk) = vk, v = BT y, ψk(ηk) =ak sin(ηk). Hence, rewriting the model (56)-(57) in thisway leads directly to an interpretation as an internalmodel control loop of the form (52), where the feed-through term can be omitted, i.e., ν = 0, since the nodedynamics is output strictly incrementally passive (seeCorollary 2). We can now restate the result of [SPDB13]in the context of internal model control.

Proposition 10 Consider the droop-controller dynam-ics in the form (58) and (59) and let the underlying net-work G be acyclic. Then

(1) the regulator equations (11) are solved by xwi =∑n

i=1P∗i∑n

i=1Di

=: yw and uwi = Di

∑n

i=1P∗i∑n

i=1Di− P ∗i for all

i = 1, 2, . . . , n;(2) the embedding condition (14) is feasible if and only

if ‖A−1(BTB)−1BTuw‖∞ < 1;(3) if the necessary conditions (11) and (14) hold,

the solutions to the closed loop dynamics (58)and (59) with interconnection u = Bλ and thatoriginate sufficiently close to xw and ηw :=sin−1(A−1(BTB)−1BTuw) satisfy limt→∞ ‖yi −yw‖ → 0.

The proof follows completely along the lines of the inter-nal model control approach (except the local nature ofthe stability result) and exploits in particular the resultsfor static disturbances of Section 6.2. For completeness,we provide the proof in the appendix.

8 Conclusions

The paper has investigated output agreement problemsin the presence of time-varying disturbances, with aparticular focus on the role of dynamic internal-model-based controllers. The proposed methodology is appli-cable to a variety of problems, including time-varyingdistribution control and frequency droop control, andlends itself to several possible extensions. In many otherdistribution networks, similar to the inventory system,the constraints imposed by the network induces a non-unique solution such that is becomes meaningful todesign controllers with optimal features. Our approachnaturally lends itself to providing such solutions. More-over, the potentials of our approach in the context ofthe two case studies have not been fully explored yet.Phenomena to be studied are for instance the presenceof constraints on the input and state variables. For thecase of power systems, other classes of controllers couldbe considered, dealing for instance with the presence oftime-varying exogenous inputs.

A Appendix

Proof of Proposition 2

Proof: (Necessity) If the rank condition in (22) isviolated then there exists a nonzero vector (x0, λ0) suchthat [

A− sIr G(B ⊗ Ip)(B ⊗ Ip)T C 0np×np

][x0

λ0

]= 0. (A.1)

This equality can be made explicit as

(A− sIr)x0 + G(B ⊗ Ip)λ0 = 0

(B ⊗ Ip)T Cx0 = 0.(A.2)

As s 6∈ σ(A), then necessarily λ0 6= 0.The graph G may or may not contain a cycle. If it does,then statement (1) of the thesis holds. If not, then (B⊗Ip)λ0 6= 0 (recall that λ0 6= 0). Since s is not an eigen-value of any Ai, then from (A.2)

x0 = (sIr − A)−1G(B ⊗ Ip)λ0

(B ⊗ Ip)T C(sIr − A)−1G(B ⊗ Ip)λ0 = 0

The latter shows thatR(H(s)(B⊗Ip))∩N ((B⊗Ip)T ) 6=0, that is statement (2) of the thesis.(Sufficiency) If G contains cycles, then there exists λ0 6=0, such that (B ⊗ Ip)λ0 = 0. Consider now the nonzero

15

vector (x0, λ0) = (0, λ0). The product[A− sIr G(B ⊗ Ip)

(B ⊗ Ip)T C 0mp×mp

][0

λ0

]

returns a zero vector. This shows that the null space ofthe matrix on the left-hand side is non-trivial, that iscondition (22) is violated.If R(H(s)(B ⊗ Ip)) ∩ N ((BT ⊗ Ip)) 6= 0, then thereexists a vector λ0 6= 0 such that

(BT ⊗ Ip)H(s)(B ⊗ Ip)λ0 = 0.

By definition of H(s) = C(sIr−A)−1G, the latter equal-ity becomes

(BT ⊗ Ip)C(sIr − A)−1G(B ⊗ Ip)λ0 = 0.

Define x0 = (sIr − A)−1G(B ⊗ Ip)λ0. Then

(B ⊗ Ip)T Cx0 = 0

(A− sIr)x0 + G(B ⊗ Ip)λ0 = 0,

which is the same as (A.1). But this shows that condition(22) is violated.

Proof of Corollary 1

Proof: Under the given assumptions, the equations(21) are feasible if and only if Condition 2 in Proposition2 does not hold. Suppose, by contradiction, that H(s)is strictly positive real and Condition 2 in Proposition2 holds. The condition can equivalently be expressed asfollows: there exist vectors v ∈ Rnp and 0 6= β ∈ Rpsuch that

H(s)(B ⊗ Ip)v = 1n ⊗ β, ∀s ∈ σ(S).

Multiplying the previous condition from the left byvT (BT ⊗ Ip) leads to

vT (BT ⊗ Ip)H(s)(B ⊗ Ip)v = 0, ∀s ∈ σ(S).

Since all s ∈ σ(S) have zero real part, is equivalent tovT(H(jω) + H(−jω)

)v = 0 for all v = (B ⊗ Ip)v and

for some ω ∈ R.This is a contradiction since H(s) being strictly positivereal implies that

(H(jω) + H(−jω)

)is positive definite

for all ω ∈ R. This proves the statement.

Proof of Proposition 10

Proof: The first statement follows directly af-ter summing all equations (58) and noting that there

must be a scalar valued function y∗ ∈ R such thatyi = xwi = y∗ for all i ∈ 1, . . . , n.To prove the second statement, we choose φ(τ(w)) = 0since w = 0. Now, note that uw = Bλw, andsince the network is acyclic, λw is uniquely definedas λw = (BTB)−1BTuw. Since for an acyclic net-work N (B) = 0, the second condition in (14) be-comes (BTB)−1BTuw = ψ(τ(w)), where we can takeψ(τ(w)) = A sin(τ(w)). Thus, we have

τ(w) = sin−1(A−1(BTB)−1BTuw),

which exists if and only if ‖A−1(BTB)−1BTuw‖∞ < 1.To prove local stability we use the standard stor-age function (55). Note that it can be defined withΨk(ηk) = ak(1 − cos(ηk)). If the conditions (14) hold,choose ηw = τ(w). Stability follows now with the stor-age function U((x, xw), (η, ηw) =

∑ni=1 Vi(xi, x

wi ) +∑m

k=1Wk(ηk, ηwk ), with Vi(xi, x

wi ) = 0 and Wk de-

fined as (55). Note that U is positive semidefinitein a neighborhood around xw and ηw and such thatη, ηw ∈ (−π2 ,

π2 )m, and satisfies

U ≤ −∑ni=1Di‖yi(t)− ywi (t)‖2

= −∑ni=1D

−1i ‖

∑mk=1 bikak(sin(ηk(t))− sin(ηwk (t)))‖2,

(A.3)due to output strict incremental passivity of (58). Notethat the latter inequality involves only the variablesη, ηw. Hence, it shows that the trajectories of the closed-loop system η = −BT y = −BTD−1(P ∗ + BAsin(η))are bounded and converge to the set of points whereBAsin(η) = BAsin(ηw) (i.e., to the set of points wheresin(η) = sin(ηw), since the graph has no cycles andA is a diagonal matrix) or, equivalently, to the set ofpoints where yi = yw = xw∗ for all i. Thus, any trajec-tory originating sufficiently close to xw and ηw satisfieslimt→∞ ‖yi − yw‖ → 0.

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