Optimal regulation of flow networks with transient constraints{}^{\star}

# Optimal regulation of flow networks with transient constraints⋆

[    [    [ ENTEG, Faculty of Science and Engineering, University of Groningen, Nijenborgh 4, 9747 AG Groningen, the Netherlands. (e-mail: s.trip@rug.nl; t.w.scholten@rug.nl; c.de.persis@rug.nl). Institute of Engineering, Hanze University of Applied Sciences, Zernikeplein 11, 9747 AS Groningen, the Netherlands.
###### Abstract

This paper investigates the control of flow networks, where the control objective is to regulate the measured output (e.g storage levels) towards a desired value. We present a distributed controller that dynamically adjusts the inputs and flows, to achieve output regulation in the presence of unknown disturbances, while satisfying given input and flow constraints. Optimal coordination among the inputs, minimizing a suitable cost function, is achieved by exchanging information over a communication network. Exploiting an incremental passivity property, the desired steady state is proven to be globally asymptotically attractive under the closed loop dynamics. Two case studies (a district heating system and a multi-terminal HVDC network) show the effectiveness of the proposed solution.

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thanks: [

footnoteinfo]This work is partially supported by the Danish Council for Strategic Research (contract no. 11-116843) within the ‘Programme Sustainable Energy and Environment’, under the ‘EDGE’ (Efficient Distribution of Green Energy) research project and by the research grant ‘Flexiheat’ (Ministerie van Economische Zaken, Landbouw en Innovatie). Preliminary results have appeared in Trip et al. [2017].
Both authors contributed equally.

First]Sebastian Trip First,Second]Tjardo Scholten First]Claudio De Persis

ontrol of networks, Optimization, Passivity, Distributed control.

## 1 Introduction

Flow networks (also known as distribution or transportation networks) consist of edges that are used to model the exchange of material (flow) between the nodes. The design and regulation of these networks received significant attention due to its many applications, including supply chains (Alessandri et al. [2011]), heating, ventilation and air conditioning (HVAC) systems (Gupta et al. [2015]), data networks (Moss and Segall [1982]), traffic networks (Iftar [1999], Coogan and Arcak [2015]) and compartmental systems (Blanchini et al. [2016], Como [2017]). If the considered objective is static, the study of flow networks has a long history within the field of network optimization (Bertsekas [1998], Rockafellar [1984]). Many practical networks must on the other hand react dynamically on changes in the external conditions such as a change in the demand. In these cases continuous feedback controllers are required, that dynamically adjust inputs at the nodes and the flows along the edges, and the design of such controllers is the subject of this work.

Since flow networks are ubiquitous in engineering systems, many solutions have been proposed to coordinate them, exploiting methodologies from e.g. passivity (Arcak [2007]) and model predictive control (Koeln and Alleyne [2017]). We focus on flow networks where the nodes can store the considered material (Kotnyek [2003]). A common objective in such networks is that the stored material needs to be regulated towards desired setpoints, despite the presence of an unknown demand. This is commonly achieved by actively controlling the flows on the edges (Wei and van der Schaft [2013], Bürger and De Persis [2015], Xiang et al. [2017]) using dynamic flow controllers. These controllers on the edges generally provide a form of integral action, that shows some benefits over networks lacking these dynamics. For example, the presence of an integral action permits the achievement of output regulation, in contrast to approximate regulation (Giordano [2016]). This inability to achieve output regulation in the presence of unknown disturbances can be observed in systems where the flow on an edge proportionally depends on the potential difference of its adjacent nodes. This is found in e.g. compartmental systems (Blanchini et al. [2016] Como [2017]). Furthermore, in most cases, the capacity of the edges is constrained, requiring careful design of the flow controllers. Naturally, the control of flows only permits to distribute the material within the network. In case there is no possibility to adjust the input to the network, a necessary requirement for stability is that all uncontrollable inflows and outflows sum to zero (Wei [2016]). Since this is generally not the case, additional controllable inputs are required that might have their own capacity constraints.

### 1.1 Main contributions

In this work we focus on flow networks, where at various nodes, an unknown amount of material (disturbance) is supplied to, or extracted from, the network. Despite these disturbances, we require the various storage levels at the nodes (or an ‘output function’ thereof) to be regulated towards desired values. We aim at achieving this so-called output regulation, by optimally allocating the required inputs among the nodes that possess a controllable external input. Here, only a subset of the nodes is assumed to have a controllable input, where a cost function relates the provided input to associated costs. We particularly propose a distributed control solution to enhance robustness to failures and to improve the scalability. Furthermore, the proposed solution respects capacity constraints that the inputs and flows might have.

Although various of these aspects have been addressed before, the way how we incorporate them within a coherent approach is new. Furthermore, the proposed controllers are shown to achieve the overall objective outlined above globally, i.e. independent of the initialization of the system. We elaborate on some specific contributions below.

(i) In flow networks it is desirable to meet certain optimality criteria, prescribing e.g. the optimal flows within the network and the optimal inputs to the network. Examples of the former include a ‘maximum flow’, ‘quickest flow’ or ‘minimum cost flow’, and achieving them received a considerable amount of attention in the past (see Kotnyek [2003], Skutella [2009] and references therein). On the other hand, when optimal inputs are considered, costs are often associated to the amount of generated input (materials), and optimization thereof has been studied thoroughly within the setting of smart (electricity) grids (Trip et al. [2016], Dörfler et al. [2016]). In this paper we apply this idea to general flow networks (Scholten et al. [2016]), where only a subset of the nodes can generate an input. A communication network then connects the various nodes, where relevant information on the costs is exchanged.

(ii) The distributed controllers are designed to enjoy certain passivity properties. That passivity plays an outstanding role in the coordination of systems is well recognized (Arcak [2007]). Particularly, incremental passivity (Pavlov and Marconi [2008]) has been exploited to analyze the stability of flow networks (see. e.g. Bürger et al. [2015] and Bürger and De Persis [2015]), but also of virtual networks in the setting of distributed optimization (Tang et al. [2016]) and game theory (Gadjov and Pavel [2017]). To prove asymptotic convergence to the desired state, generally, some form of strict output passivity (e.g. as a result of damping) is required. The considered flow networks in this work do not enjoy this property, due to the preservation of the material, making the controller design more challenging. We propose a ‘dynamic extension’ of previously considered integral-type controllers, to ensure convergence to a point, preventing the network to converge to a limit cycle, exhibiting oscillations. Although the approach is tailored to the system at hand, the design offers new perspectives on similar systems lacking dissipation. In case physical considerations forbid this dynamic extension, global convergence to the desired output can be achieved by carefully selecting nodes that have a controllable input. This selection is related to the zero forcing set of the underlying graph of the network (Monshizadeh et al. [2014], Trefois and Delvenne [2015]), and this work provides an interesting link between zero forcing sets and the application of an invariance principle for dynamical systems.

(iii) The proposed distributed controllers are applied, besides flow networks, to compartmental systems, studied in e.g. Blanchini et al. [2016] and Como [2017], and we show that additional control on some inputs and flows is sufficient to achieve regulation. Although setpoint regulation for (linear) compartmental systems has been studied before in Lee and Ahn [2015] and Ahn et al. [2017], our approach is different. In the aforementioned works, the flows are adjusted by properly altering the system parameters of the network, whereas we consider here the parameters constant and dynamically adjust the flows on some edges that are independent of the state of the network.

(iv) We provide two case studies that exemplify the use of flow networks to describe interconnected physical systems. In the first case study, we consider district heating systems (Scholten et al. [2015]) and improve upon existing results by guaranteeing asymptotic convergence to a desired setpoint, where only a subset of the nodes are required to have a controllable input. In the second case study, we consider voltage regulation and current sharing in multi-terminal high voltage direct current (HVDC) networks (Zonetti et al. [2015], Andreasson et al. [2016]). Despite the fact that these networks have already been studied extensively, the proposed control solution is noteworthy in that it provides means to limit current injections during transients and does not require all terminals to be controlled.

### 1.2 Outline

The paper is structured as follows. In Section 2 we introduce the considered flow network model. Next, in Section 3, we state our control objective of optimal output regulation and discuss various constraints under which the control objective should be achieved. In Section 4 we propose a distributed controller and study the feasibility of the control problem in more detail. Exploiting incremental passivity properties of the network and the controllers, the stability analysis of the closed loop system is carried out in Section 5. In Section 6, we study two modifications to the controlled flow network, widening the scope of this work. Two case studies are presented in Section 7. Finally, the conclusions and future directions are given in Section 8.

### 1.3 Notation

Let be the vector of all zeros of suitable dimension and let be the vector containing all ones of length . The -th element of vector is denoted by or, if it enhances the readability, by . We define to be the range of function . A steady state solution to system , is denoted by , i.e. . In case the argument of a function is clear from the context, we occasionally write as . Let be a matrix, then is the image of and is the kernel of . In case is a positive definite (positive semi-definite) matrix, we write (). Lastly, we denote the cardinality of a set as .

For convenience we provide, in Table 1, an overview of some important symbols appearing in this work.

## 2 Flow networks

In this paper we consider a network of physically interconnected undamped dynamical systems. The topology of the system is described by an undirected graph , where is the set of nodes and is the set of edges connecting the nodes. We represent the topology by its corresponding incidence matrix , where the entries of are defined by arbitrarily labelling the ends of the edges in with a ‘+’ and a ‘–’, and letting

 bik=⎧⎨⎩+1if node i is the positive end of edge k−1if node i is the negative end of edge k0otherwise.

Let be the set of actuated nodes that are controlled by an external input and let . We define

 ei={1i∈Ve0otherwise. (1)

The dynamics of node are given by

 Txi˙xi(t)= −∑k∈EiBikλk(t)+eiui(t)−di (2a) yi(t)= hi(xi(t)), (2b)

where is the storage (inventory) level, the control input, a constant111Usually we have in the classical flow networks, where a material is transported. See however Subsection 7.2 for an example where ., is a constant unknown disturbance and the measured output with a continuously differentiable and strictly increasing function. Moreover, is the set of edges connected to node and is the flow on edge . We can represent the complete network compactly as222For the sake of simplicity, the dependence of the variables on time is omitted in most of the remainder this paper.

 Tx˙x =−Bλ+Eu−d (3a) y = h(x), (3b)

where , , , and . Without loss of generality we assume that only the first nodes have a controllable external input, i.e. , and consequently is of the form

 E=[Ip×p0(n−p)×p]. (4)

Furthermore, and of which the -th component is given by . Throughout this work we will study the control of the inputs to the nodes and the control of the flows on the edges. We make two basic assumptions on the network that allows us to formulate the control objectives explicitly in the next section. First, in order to guarantee that each node can be reached from anywhere in the graph we make the following assumption on the topology:

###### Assumption 1 (Connectedness)

The graph is connected.

We recall (see e.g. [Bapat, 2010, Lemma 2.2]) the following useful lemma:

###### Lemma 1 (Rank of B)

Let be a graph with nodes and let be the incidence matrix of . Then the rank of is if and only if is connected.

Second, to compensate for the disturbances to the network, the following assumption is required:

###### Assumption 2 (Controllable inputs)

There is at least one node that has a controllable external input, i.e. .

An immediate consequence of Assumption 1 and its related Lemma 1 is the following result:

###### Lemma 2 (Rank of [BE])

If Assumption 1 is satisfied, then Assumption 2 is equivalent to being full row rank, i.e. .

Particularly, we will use the fact that the pseudoinverse of constitutes a right inverse, which has been exploited within a similar context in e.g. Blanchini et al. [2016].

## 3 Optimal regulation with input and flow constraints

In this section we discuss two control objectives and the various input and flow constraints under which the objectives should be reached. We start with discussing the two objectives. The first objective is concerned with the output in (3), at steady state.

###### Objective 1 (Output regulation)

Let be a desired constant setpoint, then the output of (3) asymptotically converges to , i.e.

 limt→∞∥h(x(t))−¯¯¯y∥ =0. (5)
###### Remark 1 (Tracking of a ramp)

In case that , Objective 1 can immediately be extended to the possibility of tracking a linear transition from the current setpoint to a new setpoint with . To do so, the desired reference signal is modelled as a ramp, i.e.

 ¯¯¯y(t)=¯¯¯y(t1)+t−t1t2−t1(¯¯¯y(t2)−¯¯¯y(t1)). (6)

After a coordinate transformation , we obtain a system of the same form as (3a), where the evolution of is described by

 Tx˙~x=−Bλ+Eu−~d. (7)

The corresponding constant disturbance is now given by

 ~d=d+1t2−t1(¯¯¯y(t2)−¯¯¯y(t1)). (8)

Note that boundedness of does not imply boundedness of as increases or decreases constantly over time. Therefore, the used invariance principle in the later sections is not immediately applicable if we consider the original variables of the system. Nevertheless, the subsequent analysis can be applied to the incremental system (7) if we consider as the state.

To ensure feasibility of Objective 1, the following assumption is made:

###### Assumption 3 (Feasible setpoint)

The desired setpoint satisfies

 ¯¯¯yi∈R(hi) for all i∈V. (9)

At a state where is constant and satisfies system (3a) necessarily satisfies

 0=−Bλ+Eu−d. (10)

Premultiplying (10) with results in

 0=\mathds1TnEu−\mathds1Tnd, (11)

such that at a steady state the total input to the network needs to be equal to the total disturbance. If there are two or more inputs to the network (i.e. ), it is natural to wonder if the total input can be coordinated optimally among the nodes. To this end, we assign a strictly convex linear-quadratic cost function to each input of the form

 Ci(ui)=12qiu2i+riui+si, (12)

with and . The total cost can be expressed as

 C(u)= ∑i∈VeCi(ui)=12uTQu+rTu+s, (13)

where , and . Minimizing (13), while satisfying the equilibrium condition (10), gives rise to the following optimization problem:

 minimizeu,λ C(u) (14) subject to 0=−Bλ+Eu−d.

It is possible to explicitly characterize the solution to (14) and we do so in the following lemma:

###### Lemma 3 (Solution to optimization problem (14))

The solution to (14) is given by

 ¯¯¯u=Q−1(κ−r), (15)

where

 κ=ET\mathds1n\mathds1Tn\mathds1TpQ−1\mathds1p(d+EQ−1r). (16)

Proof. The proof follows standard arguments from convex optimization and from realizing ([Trip et al., 2016, Lemma 4]) that the constraint in (14) can be equivalently replaced by

 \mathds1Tn(Eu−d)=0. (17)

###### Remark 2 (Identical marginal costs)

Note that we can rewrite (15) as

 κ= Q¯¯¯u+r, (18)

and that . It follows that, when evaluated at the solution to (14), the so-called marginal costs are identical for all (Hoy et al. [2011]).

We are now ready to state the second control objective.

###### Objective 2 (Optimal feedforward input)

The input at the nodes asymptotically converge to the solution to (14), i.e.

 limt→∞∥u(t)−¯¯¯u∥= 0, (19)

with as in (15).

We now turn our attention to possible constraints on the control inputs and under which the objectives should be reached. First, in physical systems the input is generally constrained by a minimum value (often zero, preventing a negative input) and a maximum value, representing e.g. a production capacity.

###### Constraint 1 (Input limitations)

The inputs at the nodes satisfy

 u−i< ui(t)

with being suitable constants.

Second, the flows on the edges are often constrained to be unidirectional and to be within the capacity of the edges.

###### Constraint 2 (Flow capacity)

The flows on the edges satisfy

 λ−k< (21)

with being suitable constants.

Note that physical limitations and safety requirements demand that the constraints should be satisfied for all time and not only at steady state.

###### Remark 3 (Special cases)

The unconstrained case can be regarded as a particular example of the considered setting. This is obtained by taking as a lower and as an upper bound for both and . Moreover, if we take or , the flow on edge is constrained to be unidirectional.

In many applications it is desirable to have a distributed control architecture where controllers rely only on local information to decrease communications, to increase robustness and to improve the scalability of the control scheme. We therefore require that the controllers to be designed, only depend on information available from adjacent nodes in the physical flow network or adjacent nodes in a digital communication network that is deployed to ensure optimality (see the next section).

For convenience, we summarize the objectives and constraints yielding the following controller design problem.

###### Problem 1 (Controller design problem)

Design distributed controllers that regulate the external inputs at the nodes and the flows on the edges, such that

 limt→∞∥h(x(t))−¯¯¯y∥=0limt→∞∥u(t)−¯¯¯u∥=0, (22)

where is the desired setpoint and is as in (15). Furthermore,

 λ−k< λk(t)<λ+ku−i< ui(t)

for all , and .

###### Remark 4 (Positive systems)

A common requirement is that, additionally to Objective 1 and Objective 2, the state has to be nonnegative, i.e. for all . Although, achieving output regulation, with , is in practical cases sufficient to ensure that for all , when the system is suitably initialized (see also the case studies in Section 7), a theoretical guarantee is difficult to obtain, due to the presence of an unknown and constant disturbance . An interesting future endeavor is to study the design of controllers achieving Objective 1 and Objective 2 within the setting of so-called positive systems (Benvenuti and Farina [2002], Valcher and Misra [2014], Arneson et al. [2016], Ebihara et al. [2017]).

## 4 Controller design

In this section we propose distributed input and flow controllers that achieve the various objectives under the constraints discussed in the previous section. The controllers will be designed to enjoy a passivity property and asymptotic stability of the closed loop system will derive from a suitable power preserving interconnection of the flow network and the controllers. Both the passivity property as well as the stability of the closed loop system will be discussed in the next section.

Before introducing the controllers, we make two observation. First, by premultiplying both sides of (3a) with , we obtain that

 \mathds1TnTx˙x=\mathds1Tn(Eu−d), (24)

which shows that the aggregated storage level are independent of the flows , that distribute the material within the network. Second, at steady state, (24) becomes

 0=\mathds1Tn(E¯¯¯u−d), (25)

which implies that a balance between the total input and disturbance is required to obtain a steady state. The first observation motivates the design of a flow controller, aiming at distributing the deviation from the desired output, , equally among the nodes, i.e. , for all . The controllers at the nodes, regulating the external input to the network, are then designed to steer the deviation from the desired output to zero, by optimally allocating the external inputs to the network, such that the total input is identical to the total disturbance. We start with discussing the flow controller in more detail.

### 4.1 Flow controller

We design a controller that regulates the flows on the edges, aiming at consensus in the error (balancing), while obtaining a useful passivity property of the resulting closed loop system when interconnected with (3). Consider the following controller:

 Tμ˙μ= BT(h(x)−¯¯¯y)−(f(μ)−ξ)Tξ˙ξ= f(μ)−ξλ= f(μ), (26)

where are diagonal matrices with strictly positive entries, and the mapping , with , has suitable properties discussed in Assumptions 5 and 6 below. Moreover, is the incidence matrix reflecting the topology of the physical network, which implies that the flow controller on edge only requires information from its adjacent nodes (see also Figure 1). Note that the term determines the difference in the output error of the two adjacent nodes to edge . As will be discussed in Remark 9 and Subsection 6.2, the state is introduced to prove convergence to a constant flow, preventing oscillations. The passivity property, mentioned before, is derived in Lemma 6 in the next section.

### 4.2 Controller at the nodes

Next, we design an input controller at each node that adjusts the external input to the network. Inspired by the result in Trip and De Persis [2017], where a similar control problem is considered in the setting of power networks, we propose the controller

 Tθ˙θ=−ET(h(x)−¯¯¯y)−(g(θ)−ϕ)Tϕ˙ϕ= g(θ)−ϕ−QLcom(Qϕ+r)u= g(θ), (27)

where are diagonal matrices with strictly positive entries, and the mapping , with , has suitable properties discussed in Assumptions 5 and 6 below. Moreover, is the Laplacian matrix reflecting the communication topology (see also Figure 2). This communication ensures that, at steady state, a consensus is obtained in the marginal costs, i.e. . In order to guarantee that all marginal costs converge to the same value we make the following assumption on the communication network.

###### Assumption 4 (Communication network)

The graph reflecting the communication topology is balanced333A directed graph is balanced if the (weighted) in-degree is equal to the (weighted) out-degree of every node. and strongly connected.

###### Lemma 4 (Consequence of Assumption 4)

If Assumption 4 is satisfied, then is a positive semi-definite matrix and

 ϕTLcomϕ=0, (28)

if and only if .

Proof. The proof follows immediately from [Olfati-Saber and Murray, 2004, Theorem 7]. Specifically, since the communication graph is balanced, is positive semi-definite and (28) satisfies

 ϕTLcomϕ=12ϕT(Lcom+(Lcom)T)ϕ=ϕT^Lcomϕ, (29)

where is a Laplacian matrix corresponding to the communication network with undirected edges. Furthermore, , due to the connectendness of the communication network.

Again, we introduced an additional state , to ensure convergence to a constant point, whereas the term provides an integral action to reduce the output error at the node .

###### Remark 5 (Local and exchanged information)

According to (27), every controller at node , measures and compares it with the desired set point . Information on the marginal costs () is exchanged among neighbours over a communication network with a topology described by . Controller (27) is therefore fully distributed. The output is chosen to satisfy Constraint 1, and is discussed in more detail in the next subsection.

### 4.3 Feasibility of the control problem

To ensure feasibility of the controller design problem, we impose two assumptions on the controllers (26) and (27). The first assumption guarantees that the controllers are able to generate a (feedforward) control signal, that is required to attain a steady state of the system.

###### Assumption 5 (Attainability of the steady state)

Consider functions and , in respectively (26) and (27). Let be as in (15). There exists444If has any solution , then all solutions are given by , for an arbitrary vector , where denotes the Moore-Penrose pseudoinverse of . The existence of a solution is shown in the proof of Lemma 5. a , such that for all . Furthermore, for all .

Moreover, the controllers (26) and (27) can be designed to satisfy constraints (20) and (21), by properly selecting and . Since and , the following assumption is sufficient to ensure that the inputs and flows do not exceed their limitations.

###### Assumption 6 (Controller outputs)

Functions and , in respectively (26) and (27), are continuously differentiable, strictly increasing and satisfy

 R(fk)=(λ−k,λ+k)R(gi)=(u−i,u+i), (30)

for all and all .

The property of and being continuously differentiable and strictly increasing functions, is exploited within the various proofs to establish the global convergence properties, and ensures e.g. the existence of an inverse function. Possible choices for and , that satisfy Assumption 6, include e.g. the function in absence of any constraints, and also, upon proper scaling, the constraint enforcing functions , (see also the case studies in Section 7).

Before we analyse the stability of the system we investigate the properties of the steady state. To do so, we write system (3) in closed loop with controllers (26) and (27), obtaining

 Tx˙x=−Bf(μ)+Eg(θ)−dTμ˙μ= BT(h(x)−¯¯¯y)−(f(μ)−ξ)Tξ˙ξ= f(μ)−ξ (31a) Tθ˙θ=−ET(h(x)−¯¯¯y)−(g(θ)−ϕ)Tϕ˙ϕ= g(θ)−ϕ−QLcom(Qϕ+r). (31b)

Any equilibrium of system (31) satisfies

 0 =−Bf(¯¯¯μ)+Eg(¯¯¯θ)−d (32a) 0 =BT(h(¯¯¯x)−¯¯¯y)−(f(¯¯¯μ)−¯ξ) (32b) 0 =f(¯¯¯μ)−¯¯¯ξ (32c) 0 =−ET(h(¯¯¯x)−¯¯¯y)−(g(¯¯¯θ)−¯¯¯ϕ) (32d) 0 =(g(¯¯¯θ)−¯¯¯ϕ)−QLcom(Q¯¯¯ϕ+r). (32e)

We will now show that under Assumptions 16 there exists at least one solution to (32) and all solutions (32) satisfy the control objectives.

###### Lemma 5 (Equilibria)

Let Assumptions 15 hold. Then, there exists an equilibrium of system (31). Moreover, any equilibrium is such that and , where is the optimal control input given by (15).

Proof. To prove the statement, we first show that at least one equilibrium of system (31) exists. By Assumption 5, , and we set . Also, we set . Bearing in mind that , we have that (32e) holds. Furthermore, by definition, satisfies Since the graph is connected (Assumption 1) and , we have that . For this reason, there exists a satisfying , and any solution is given by for an arbitrary vector . By Assumption 5, there exists at least one such that . Taking such a , setting and , shows that (32a), (32c) hold. Since (Assumption 3), setting shows (32b) and (32d). Hence, there exists a state that satisfies the equations (32) and is therefore an equilibrium of (31).

Next, we show that any equilibrium necessarily satisfies and , where is the optimal control input given by (15). From (32c), holds and we will show that this implies that necessarily . By (32e), bearing in mind that is the Laplacian of a balanced and strongly connected graph (Assumption 4), we have according to Lemma 4 that . This, together with (32d), implies that . By (32b) and , we also have . Hence,

 [\mathds1TpQ−1ETBT](h(¯¯¯x)−¯¯¯y)=0. (33)

We now prove that necessarily . Suppose, ad absurdum, that there exists such that

 [\mathds1TpQ−1ETBT]v=0. (34)

By Assumption 1, it follows that with a scalar. Then , which is, by definition of in (4) and Assumption 2, equivalent to . This implies that , contradicting that . Hence, necessarily and by strict monotonicity of , we must have that .

Since , it follows from (32d) that , and by strict monotonicity of , that . Moreover, from (32e) we obtain that , and since the communication graph is strongly connected due to Assumption 4, we have that . Since , we obtain from (32a) that . Bearing in mind that satisfies and , we have consequently that , with as in (15).

As a consequence of Lemma 5 we have that if Assumptions 15 hold, system (31) is equivalent to

 Tx˙x=−B(f(μ)−f(¯¯¯μ))+E(g(θ)−g(¯¯¯θ))Tμ˙μ= BT(h(x)−h(¯¯¯x))−((f(μ)−f(¯¯¯μ))−(ξ−¯¯¯ξ))Tξ˙ξ= (f(μ)−f(¯¯¯μ))−(ξ−¯¯¯ξ) (35a) Tθ˙θ=−ET(h(x)−h(¯¯¯x))−(g(θ)−g(¯¯¯θ))+(ϕ−¯¯¯ϕ)Tϕ˙ϕ= (g(θ)−g(¯¯¯θ))−(ϕ−¯¯¯ϕ)−QLcomQ(ϕ−¯¯¯ϕ), (35b)

a form that will be exploited in the stability analysis.

## 5 Stability analysis

In this section we analyze the stability of the closed-loop system (31). The analysis is foremost based on LaSalle’s invariance principle and exploits useful properties of interconnected incrementally passive systems. To facilitate the discussion, we first recall the following definition:

###### Definition 1 (Incremental passivity)

System

 ˙x= f(x,u)y= h(x), (36)

, the state space, , is incrementally passive555With some abuse of terminology, we state the incremental passivity property with respect to a steady state solution. This is in contrast to the ‘usual’ definition where the incremental passivity property holds with respect to any solution (Pavlov and Marconi [2008]). with respect to a constant triplet satisfying

 0= f(¯¯¯x,¯¯¯u)¯¯¯y= h(¯¯¯x), (37)

if there exists a continuously differentiable and radially unbounded function , such that for all , and ,

 ˙V(⋅)=∂V∂xf(x,u)+∂V∂¯¯¯xf(¯¯¯x,¯¯¯u)≤(y−¯¯¯y)T(u−¯¯¯u). (38)

We now proceed with establishing the incremental passivity property of (31a), that is the proposed flow controller (26) renders the network dynamics (3) incrementally passive with respect to the input and output .

###### Lemma 6 (Incremental passivity of (31a))

Let Assumptions 15 hold. System (31a) with input and output is incrementally passive with respect to the constant satisfying (32a)-(32c). Namely, the radially unbounded storage function satisfies

 ˙V1(⋅)= (h(x)−h(¯¯¯x))TE(g(θ)−g(¯¯¯θ)) (39) −(f(μ)−ξ)T(f(μ)−ξ),

along the solutions to (31a).

Proof. Consider the storage function

 V1(x,¯¯¯x,μ,¯¯¯μ,ξ,¯¯¯ξ)=∑i∈VTxi∫xi¯¯¯xihi(y)−hi(¯¯¯xi)dy+∑k∈ETμk∫μk¯¯¯μkfk(y)−fk(¯¯¯μk)dy+12(ξ−¯¯¯ξ)TTξ(ξ−¯¯¯ξ). (40)

Since and are strictly increasing functions, the incremental storage function is radially unbounded. Furthermore, satisfies along the solutions to (31a), or equivalently along the solutions to (35a),

 (41)

Since , indeed satisfies along the solutions to (31a).

We now prove a similar result for (31b), that is the controller (27) is incrementally passive with respect to the input and output .

###### Lemma 7 (Incremental passivity of (31b))

Let Assumptions 15 hold. System (31b) with input and output is incrementally passive with respect to satisfying (32d)-(32e). Namely, the radially unbounded storage function satisfies

 ˙V2(⋅)=−(g(θ)−ϕ)T(g(θ)−ϕ)−(ϕ−¯¯¯ϕ)TQLcomQ(ϕ−¯¯¯ϕ)−(g(θ)−g(¯¯¯θ))TET(h(x)−h(¯¯¯x)), (42)

along the solutions to (31b).

Proof. Consider the storage function

 V2(θ,¯¯¯θ,ϕ,¯¯¯ϕ)=∑i∈VTθi∫θi¯θigi(y)−gi(¯¯¯θi)dy+12(ϕ−¯¯¯ϕ