representationofnetworksandsystemswithdelay: ddes,ddfs,ode

Representation ofNetworks andSystemswithDelay:

DDEs,DDFs,ODE-PDEs andPIEs

MatthewM. Peet a,

aSchool for the Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ, 85298USA.

Abstract

Delay-Differential Equations (DDEs) are the most common representation for systems with delay. However, the DDE repre-sentation is limited. In network models with delay, the delayed channels are low-dimensional and accounting for this hetero-geneity is not possible in the DDE framework. In addition, DDEs cannot be used to model difference equations. Furthermore,estimation and control of systems in DDE format has proven challenging, despite decades of study. In this paper, we exam-ine alternative representations for systems with delay and provide formulae for conversion between representations. First, weexamine the Differential-Difference (DDF) formulation which allows us to represent the low-dimensional nature of delayedinformation. Next, we examine the coupled ODE-PDE formulation, for which backstepping methods have recently becomeavailable. Finally, we consider the algebraic Partial-Integral Equation (PIE) representation, which allows the optimal estima-tion and control problems to be solved efficiently through the use of recent software packages such as PIETOOLS. In eachcase, we consider a very general class of delay systems, specifically accounting for all four possible sources of delay - statedelay, input delay, output delay, and process delay. We then apply these representations to 3 archetypical network models.

Key words: Delay, PDEs, Networked Control

1 Introduction

Delay-Differential Equations (DDEs) are a convenientshorthand notation used to represent what is perhapsthe simplest form of spatially-distributed phenomenon- transport. Because of their notational simplicity, it iscommon to use DDEs to model very complex systemswith multiple sources of delay - including almost all mod-els of control over and of “networks”.

To illustrate the ways in which delays can complicatean otherwise straightforward control problem, considercontrol of a swarm of N UAVs over a wireless network.In this case, each UAV, i, has a state, xi(t) ∈ Rni whichmay represent, e.g. displacement (the concatenation ofall such states is denoted x). Each UAV has local sensorswhich measure yi and this information is transmitted toa centralized control authority. There is also a central-ized vector of inputs, u, a regulated vector of outputs, z,and a vector of disturbances, w - including both process

Email address: [email protected] (Matthew M. Peet).1 This work was supported by the National Science Foun-dation under grant No. 1739990.

and sensor noise. We model this system as follows.

xi(t) = aixi(t) +∑N

j=1aijxj(t− τij)

+ b1iw(t− τi) + b2iu(t− hi)z(t) = C1x(t) +D12u(t)

yi(t) = c2ixi(t− τi) + d21iw(t− τi) (1)

• ai is the internal dynamics of the UAV i• aij is the effect of UAV j on the state of UAV i.• b1i is the disturbance to the motion of UAV i• b2i is the effect of the central command on UAV i• c2i is the measurement of the state of UAV i• d21i is the disturbance to the sensor on UAV i• C1 gives the weight on states of the fleet of UAVs to

minimize in the optimal control problem• D12 gives the weight on actuator commands to mini-

mize in the optimal control problem• τij is the time taken for changes in state of UAV j to

affect UAV i• hi is the time taken for a command from the central

authority to reach UAV i• τi is the time it takes the process disturbance (wind,

tracking signal, et c.) to reach UAV i• τi is the time taken for measurements collected at UAVi to reach the central authority

Preprint submitted to Automatica 18 December 2020

This relatively simple model shows that delayed channels

are often low dimensional (Rni vs. R∑

ni) and specifiesfour separate yet individually significant sources of delay.Specifically, we have: state delay (τij); input delay (hi);process delay (τi); and output delay (τi).

This UAV network is modeled as a DDE - a structureformulated in Eqn. (2) in Sec. 2. If we consider controlof such a network, however, we find that while there arealgorithms for control of DDEs (See [15]), these algo-rithms are complex and are memory-limited to a rela-tively small number of UAVs (perhaps 4-5). The premiseof this paper, however, is that the limitations of thesealgorithms are not caused by inefficiency of the algo-rithms, but rather by the failure to account for the lowdimensional nature of the delayed channels. Specifically,we note that in the UAV model, while the concatenatedstate, x(t), is high-dimensional, the individual delayedchannels, xi(t), are of much lower dimension. If we rep-resent the network as a DDE using the formulation inSubsec. 6.1, then the low-dimensional nature of the de-layed channels is lost. Furthermore, DDEs cannot repre-sent some important system designs - including a modelof feedback described in Subsection 7.

For these reasons, in Sec. 3, we consider the use of Dif-ferential Difference Equations (DDFs). The DDF canbe used to model both DDEs and neutral-type sys-tems, while also allowing for the assignment of delayedinformation to heterogeneous low-dimensional chan-nels. Specifically, the infinite-dimensional component ofstate-space (as defined in [6,18]) of the UAV networkin the DDF framework is

∏i L2[−τi, 0]ni as opposed

to∏i L2[−τi, 0]

∑ni using a DDE. In addition, DDFs

allow us to represent difference equations which arise insome network models - See Subsection 7.

From the DDF model we turn to coupled ODE-PDEmodels in Sec. 4. ODE-PDEs can be used to model avariety of systems. However, for the particular class ofODE-PDEs we use in Sec. 4, the solutions to the ODE-PDE are equivalent to those of the DDF (as defined inSec. 3). Backstepping methods have been developed forODE-PDE models of delay (e.g. [12,28]) and the formu-lae we present for conversion of DDFs to ODE-PDEsmay prove useful if the reader is interested in applicationor further development of these backstepping methods.

Next, in Sec. 5, we consider Partial Integral Equa-tions (PIEs) [1]. PIEs are a generalization of integro-differential equations of Barbashin type which havebeen used since the 1950s to model systems in biol-ogy, physics, and continuum mechanics (See chapters19-20 of [1] for a survey). PIEs and ODE-PDEs definean equivalent set of solutions and in this section, weprovide formulae for conversion of DDEs and DDFs toPIEs. PIE models have the advantage that they aredefined by Partial Integral (PI) operators. Unlike Dirac

and differential operators, PI operators are boundedand form an algebra. Furthermore, PIE models do notrequire boundary conditions or continuity constraints- simplifying analysis and optimal control problems.Indeed, it has been recently shown in [22,4] that manyproblems in analysis, optimal estimation and control ofODE-PDE models can be formulated as optimizationover the cone of positive PI operators. In Sec. 8, we showthat the PIE formulation allows for H∞-optimal controlof a 40 user, 80-state, 40-delay, 40-input, 40-disturbancenetwork model of temperature control.

Finally, we emphasize that this paper does not advo-cate for any particular time-domain representation (wedo not consider the literature on analysis and controlin the frequency domain), be it the DDE, DDF, ODE-PDE, or PIE formulation, and does not propose any newalgorithms for analysis and control of delay systems perse. Rather, the purpose of this document is to serve as aguide to representation of delay systems in each frame-work. Specifically, for each representation, we: state themost general form of each representation - allowing fordelays in input, output, process and state; define a no-tion of solution in each case; provide formulae for con-version between representations under which solutionsare equivalent; and briefly list advantages and limita-tions of the representation as applied to network modelsof the form of Eqn. (1). As discussed in the conclusions,these results can be used to establish notions of stabilitywhich are equivalent in all representations and to allowfor conversion of optimal controllers and estimators be-tween representations.

While subsets of the DDF and ODE-PDE represen-tations of delay systems can be found in the litera-ture [3,6,18,13,19,7,14], and some of these equivalencesare known [10,20], previous works do not: consider allinput-output signals and sources of delay; include PIEs;compare the relative advantages of the models as ap-plied to networks; or provide formulae for conversionbetween representations. This guide, then, may be usedas a convenient source of information for researchersinterested in either selection of a representation or con-version of a representation to an alternative format.For convenience and comparison, all representations arelisted in Figure 1. All conversion formulae are listed inFigures 2 and 3. Finally, note that all proofs have beenomitted, but are included in the extended version ofthis paper on Arxiv [16].

Notation In is the identity matrix in Rn×n, ei is theith canonical unit vector, 1n is the dimension n vector ofall ones. 0n,m is the zero matrix of dimension Rn×m andWn,2[X] is the nth-order Sobolev subspace of L2[X].

2 The DDE Representation

We begin by defining the signals in the Delay-DifferentialEquation (DDE) representation:

2

The Class of Delay-Differential Equations (DDEs):x(t)

z(t)

y(t)

=

A0 B1 B2

C10 D11 D12

C20 D21 D22

x(t)

w(t)

u(t)

+

K∑i=1

Ai B1i B2i

C1i D11i D12i

C2i D21i D22i

x(t− τi)w(t− τi)u(t− τi)

+

K∑i=1

0∫−τi

Adi(s) B1di(s) B2di(s)

C1di(s) D11di(s) D12di(s)


x(t+ s)

w(t+ s)

u(t+ s)

ds (2)

The Class of Differential-Difference Equations (DDFs):x(t)

z(t)

y(t)

ri(t)

=

A0 B1 B2

C1 D11 D12

C2 D21 D22

Cri Br1i Br2i

x(t)

w(t)

u(t)

+

Bv

D1v

D2v

Drvi

v(t)

v(t) =

K∑i=1

Cviri(t− τi) +

K∑i=1

∫ 0

−τiCvdi(s)ri(t+ s)ds. (3)

The Class of Neutral-Type Systems (NDS):x(t)

z(t)

y(t)

=

A0 B1 B2

C10 D11 D12

C20 D21 D22

x(t)

w(t)

u(t)

+

K∑i=1

Ai B1i B2i Ei

C1i D11i D12i E1i

C2i D21i D22i E2i

x(t− τi)w(t− τi)u(t− τi)x(t− τi)

+

K∑i=1

∫ 0

−τi

Adi(s) B1di(s) B2di(s) Edi(s)

C1di(s)D11di(s)D12di(s)E1di(s)


x(t+ s)

w(t+ s)

u(t+ s)

x(t+ s)

ds (4)

The Class of ODE-PDE Systems:x(t)

z(t)

y(t)

φi(t, 0)

=

A0 B1 B2

C1 D11 D12

C2 D21 D22

Cri Br1i Br2i

x(t)

w(t)

u(t)

+

Bv

D1v

D2v

Drvi

v(t)

φi(t, s) =1

τiφi,s(t, s), v(t) =

K∑i=1

Cviφi(t,−1) +

K∑i=1

∫ 0

−1τiCvdi(τis)φi(t, s)ds (5)

The Class of Partial Integral Equation (PIE) Systems:

T x(t) + BT1w(t) + BT2

u(t) = Ax(t) + B1w(t) + B2u(t)

z(t) = C1x(t) +D11w(t) +D12u(t),

y(t) = C2x(t) +D21w(t) +D22u(t) (6)

Fig. 1. Formulation of the DDE, DDF, NDS, ODE-PDE, and PIE Representations of Systems with Delay.

3

• The present state x(t) ∈ Rn• The disturbance or exogenous input, w(t) ∈ Rm• The controlled input, u(t) ∈ Rp• The regulated or external output, z(t) ∈ Rq• The observed or sensed output, y(t) ∈ Rr

For convenience, we combine all sources of delay (state,input, output, process) into a single set of delays {τi}Ki=1with 0 < τ1 < · · · < τK . For given u ∈ Lp2, w ∈ Lm2 ,and initial condition x0 ∈ W 1,2[−τK , 0]n, we say thatx : [−τK ,∞] → Rn, z : [0,∞] → Rq, and y : [0,∞] →Rr satisfy the DDE defined by {Ai, Bi, Ci, Dij , · · · } ifx is differentiable on [0,∞] (from the right at t = 0),x(s) = x0(s) for s ∈ [−τK , 0], and Eqns. (2) are sat-isfied for all t ≥ 0. If any B1i, D11i, D21i 6= 0, we re-quire w ∈ W 1,2[0,∞]m and w(s) = 0 for s ≤ 0. If anyB2i, D12i, D22i 6= 0, we require u ∈ W 1,2[0,∞]p andu(s) = 0 for s ≤ 0.

Under the conditions stated above, existence of a classi-cal continuously differentiable solution x is guaranteedas in, e.g. Thm. 3.3 of Chapter 3 in [11] (See also Thm. 1.1of Chapter 6 in [8]). Note that the dimensions of all ma-trices in this representation can be inferred from the di-mension of the respective state and signals.

2.1 Advantages of the DDE Formulation

The DDE formulation is the prima facie modeling toolfor systems with delay and as such is used in almost allnetwork models. The DDE representation has a clear andintuitive meaning. Furthermore, most algorithms andanalysis tools are built for this representation. Specifi-cally, Lyapunov-Krasovskii and Lyapunov-Razumikhinstability tests are naturally formulated in this frame-work. However, the DDE does not allow for the repre-sentation of difference equations and does not allow usto identify which of the states and inputs are delayedby which amount. For this reason, we consider next theDDF representation.

3 The DDF Representation

A generalization of the DDE is the Differential-Difference (DDF) formulation. In addition to the signalsincluded in the DDE, the DDF adds the following.

• The items stored in the signal ri(t) ∈ Rpi are the partsof x, w, u, v which are delayed by amount τi. The riare the infinite-dimensional part of the system.

• The “output” signal v(t) ∈ Rnv extracts informationfrom the infinite-dimensional signals ri and distributesthis information to the state, sensed output, and regu-lated output. This information can also be re-delayedby feeding back directly into the ri.

The governing equations may now be represented in themore compact form of Eqns. (3).

For given u ∈ Lp2, w ∈ Lm2 , and initial conditions x0 ∈Rn, ri0 ∈ W 1,2[−τi, 0]pi satisfying the “sewing condi-tion”

ri0(0) = Crix0

+Drvi

(K∑i=1

Cviri0(−τi) +

K∑i=1

∫ 0

−τiCvdi(s)ri0(s)ds

)for i = 1, · · · ,K, we say that x : [0,∞] → Rn, z :[0,∞]→ Rq, y : [0,∞]→ Rr, ri : [−τi,∞]→ Rpi for i =1, · · · ,K, and v : [0,∞]→ Rnv satisfy the DDF definedby {Ai, Bi, Ci, Dij , · · · } if x is differentiable on [0,∞],ri(s) = ri0(s) for s ∈ [−τi, 0], ri(t+ ·) ∈W 1,2[−τi, 0] fori = 1, · · · ,K, and Eqns. (3) are satisfied for all t ≥ 0. Inthis manuscript, we assume the Cvdi are bounded andin the case where Br1i 6= 0 or Br2i 6= 0, we require w ∈W 1,2[0,∞]m with w(s) = 0 for s ≤ 0 or u ∈W 1,2[0,∞]p

with u(s) = 0 for s ≤ 0, respectively.

Under the conditions stated above, existence of a clas-sical solution x, ri, v is guaranteed as in [8], Chapter 9,Thm. 1.1. Furthermore, the “sewing condition” and con-straints on w and u ensure the solution ri is continuouslydifferentiable as in [5] p. 226; or [11], Thms. 3.1 and 5.4.Note also that the condition ri(t + ·) ∈ W 1,2 may berelaxed to continuity as treated in [9].3.1 DDEs are a special case of DDFs

Although Eqns. (3) are more compact, they are moregeneral than the DDEs in (2). Specifically, if we use theconversion formula defined in Eqn. (8), then the solutionto the DDF is also a solution to the DDE and vice-versa.

Lemma 1 Suppose that Cvi, Cvdi, Cri, Br1i , Br1i,Drvi, Bv, D1v, and D2v are as defined in Eqns. (8).Given u, w, x0, the functions x, y, and z satisfy the DDEdefined by {Ai, Bi, Ci, Dij , · · · } if and only if x, y, z, andri satisfy the DDF defined by {Ai, Bi, Ci, Dij , · · · } where

ri(t) =

x(t)

w(t)

u(t)

, ri0 =

x0

0

0

i = 1, · · · ,K.

3.2 Neutral-Delay Systems (NDSs) are a special caseof DDFs

DDFs are a natural extension of NDSs, which have thegeneral form of Eqn. (4) where for simplicity, we assumex(t), w(t), u(t) = 0 for all t ≤ 0. The conversion fromNDS to DDF is given in Eqn. (9).

Lemma 2 Suppose that Cvi, Cvdi, Cri, Br1i , Br2i,Drvi, Bv, D1v, and D2v are as defined in Eqns. (9).Given u, w, the functions x, y, and z satisfy the NDSdefined by {Ai, Bi, Ci, Dij , · · · } if and only if x, y, z, vand ri satisfy the DDF defined by {Ai, Bi, Ci, Dij , · · · }where ri0 = 0 and

ri(t) =

x(t)

w(t)

u(t)

x(t)

, i = 1, · · · ,K.

4

and

v(t) =

K∑i=1

Ai B1i B2i Ei

C1i D11i D12i C1ei

C2i D21i D22i C2ei


+

K∑i=1

0∫−τi


C1di(s) D11di(s) D12di(s) C1dei(s)


x(t+ s)

w(t+ s)

u(t+ s)

x(t+ s)

ds.

3.3 Advantages of the DDF Representation

The first advantage of the DDF is that it may includedifference equations. To illustrate, suppose we set allmatrices to zero except Drvi and Cvi. Then we have thefollowing set of Difference Equations (DEs)

ri(t) =∑K

j=1DrviCvjrj(t− τj) i = 1, · · · ,K.

Another example of DEs can be found in Subsec. 7,where we provide a model of network control which canbe represented as a DDF, but not a DDE. A related ad-vantage of the DDF is the ability of DDFs to generatediscontinuous solutions if the “sewing condition” on ini-tial conditions is relaxed. This ability is not inheritedusing our formulation of ODE-PDE or PIE.

The second advantage of the DDF occurs when the de-layed channels only include subsets of the state. For ex-ample, if the matrices Ai have low rank (ignoring input

and disturbance delay), then Ai = AiAi for some Ai,

Ai where Ai ∈ Rli×n with li < n and we may chooseCvi = Ai and Cri = Ai. The dimension of ri(t) now be-comes Rli . This decomposition may be used to reducecomplexity in the DDF formulation if li < n. This re-duction is illustrated in detail using the UAV networkmodel in Subsec. 6.2 and the temperature control net-work in Sec. 8.

A disadvantage of the DDF is that fewer tools are avail-able for analysis and control of DDFs. This is partiallybecause the class of DDFs is larger than the DDEs andthus the tools must be more general. However, we do notethat versions of both the Lyapunov-Krasovskii ([6]) andLyapunov-Razumikhin ([27]) stability tests have beenformulated in the DDF framework.

4 The Coupled ODE-PDE Representation

We next consider the coupled ODE-PDE representation.Widely recognized as a physical interpretation of delaysystems [20,8], ODE-PDE representations allow us to

use backstepping methods originally developed for con-trol of PDE models and which have recently been ex-tended to systems with delay - See [12,28,10]. The par-ticular class of ODE-PDE systems, as given in Eqn. (5),is equivalent to the class of DDFs. Since we have shownthat DDEs are a special case of DDFs, we present onlythe conversion between DDF and ODE-PDE. Such con-version is trivial, however, as all matrices in the follow-ing ODE-PDE model are the same ones used to definethe DDF.

For given u ∈ Lp2, w ∈ Lm2 , and initial conditions x0 ∈Rn, φi0 ∈ W 1,2[−1, 0]pi satisfying the “sewing condi-tion”

φi0(0) = Crix0 (7)

+Drvi

(K∑i=1

Cviφi0(−1) +

K∑i=1

∫ 0

−1τiCvdi(τis)φi0(s)ds

)for i = 1, · · · ,K, we say that x : [0,∞] → Rn, z :[0,∞] → Rq, y : [0,∞] → Rr, φi(t) ∈ W 1,2[−1, 0]pi fori = 1, · · · ,K, and v : [0,∞] → Rnv satisfy the ODE-PDE defined by {Ai, Bi, Ci, Dij , · · · } if x is differentiableand φi is Frechet differentiable on [0,∞], x(0) = x0,φi(0, s) = φi0(s) for s ∈ [−1, 0] for i = 1, · · · ,K, andEqns. (5) are satisfied for all t ≥ 0. As for the DDF, ifBr1i 6= 0 or Br2i 6= 0, we require w ∈W 1,2 or u ∈W 1,2,respectively.

In Eqns. (5), the infinite-dimensional part of the state isφi - which represents a pipe through which informationis flowing. Our formulation is somewhat atypical in thatwe have scaled all the pipes to have unit length and accel-erated or decelerated flow through the pipes accordingto the desired delay. Solutions to Eqns. (5) and Eqns. (3)are equivalent, as in the following lemma.

Lemma 3 Suppose for given u, w, ri0, that x, ri, v, y,and z satisfy the DDF defined by {Ai, Bi, Ci, Dij , · · · }.Then for u, w, φi0(s) = ri0(τis), we have that x,v, y, and z also satisfy the ODE-PDE defined by{Ai, Bi, Ci, Dij , · · · } with φi(t, s) = ri(t + τis). Simi-larly, for given u, w, φi0, if x, v, y, φi and z satisfythe ODE-PDE defined by {Ai, Bi, Ci, Dij , · · · }, then x,v, y, and z satisfy the DDF with ri(t) = φi(t, 0) andri0(s) = φi0(s/τi).

4.1 Advantages of the ODE-PDE Representation

In the ODE-PDE representation, the infinite-dimensional

part of the state is φ(t) ∈W 1,2[−1, 0]∑

ipi . Significantly,

by scaling the pipes (and ignoring the distributed delay),the ODE-PDE representation isolates the effect of thedelay parameters to a single term - φi(t, s) = 1

τiφi,s(t, s).

This feature makes it easier to understand the effects ofuncertainty and time-variation in the delay parameter.Additionally, the ODE-PDE is the native representa-tion used for recently developed backstepping methodsfor systems with delay, such as proposed in [12,28,10]and use of the conversion formulae provided may allow

5

these methods to be applied to solve a larger class ofsystems - including difference equations.

5 The PIE Representation

A Partial Integral Equation (PIE) has the form ofEqn. (6), where the operators T ,A,Bi, Ci,Dij are Par-tial Integral (PI) operators and have the form(P[P, Q1

Q2,{Ri}

][x

Φ

])(s) :=

[Px+

∫ 0

−1Q1(s)Φ(s)ds

Q2(s)x+(P{Ri}Φ

)(s)

]where(P{Ri}Φ

)(s) :=

R0(s)Φ(s) +

∫ s

−1R1(s, θ)Φ(θ)dθ +

∫ 0

s

R2(s, θ)Φ(θ)dθ.

For given u ∈ Lp2, w ∈ Lm2 , and initial conditions x0 ∈Rn × L2[−1, 0]p, we say that x(t) ∈ Rn × L2[−1, 0]p,z : [0,∞]→ Rq, y : [0,∞]→ Rr satisfy the PIE definedby {T ,A,Bi, Ci,Dij ,BTi} if x is Frechet differentiable on[0,∞], x(0) = x0 and Eqns. (6) are satisfied for all t ≥ 0.As for the ODE-PDE, if BT1 6= 0 or BT2 6= 0 we requirew ∈ W 1,2 or u ∈ W 1,2, with w(0) = 0 or u(0) = 0,respectively.

Heretofore, we have shown that the DDE is a special caseof the DDF, which is equivalent to a coupled ODE-PDE,where coupling occurs at the boundary. Given a DDFor ODE-PDE representation, it is relatively straight-forward to convert to a PIE by defining the operatorsT ,A,Bi, Ci,Dij ,BTi for which solutions to Eqns. (6) alsodefine solutions to Eqns. (3) (DDF) and Eqns. (5) (ODE-PDE). Specifically, let us define {T ,A,Bi, Ci,Dij ,BTi}as in Eqn. (10) where the required matrices are as de-fined in Eqns. (11). Then we have the following.

Lemma 4 Given u, w, and x0, φi0 satisfying the“Sewing Condition (7)”, Suppose x, φi, v, y, and z satisfythe ODE-PDE defined by {Ai, Bi, Ci, Dij , · · · }. Then yand z also satisfy the PIE defined by {T ,A,Bi, Ci,Dij ,BTi}with T ,A,Bi, Ci,Dij ,BTi as defined in Eqn. (10) and

x(t) :=

x(t)

∂sφ1(t, ·)...

∂sφK(t, ·)

x0 :=

x0

∂sφ10...

∂sφK0

.Furthermore, for given u, w, x0 ∈ Rn×L2[−1, 0]p, if y, zand x satisfy the PIE defined by {T ,A,Bi, Ci,Dij ,BTi},then x, φi, v, y, and z satisfy the ODE-PDE defined by{Ai, Bi, Ci, Dij , · · · } where

x(t)

φ1(t, ·)...

φK(t, ·)

= T x(t)+BT1w(t)+BT2u(t),

x0

φ10...

φK0

= T x0.

Note that while solutions of the ODE-PDE are equiva-lent to those of the PIE, some notions of stability of suchsolutions may not be.

5.1 Advantages of the PIE Representation

Like the DDF and ODE-PDE, PIEs can be used to repre-sent low-dimensional delay channels. An additional ad-vantage is the lack of boundary conditions or the ‘sewing’constraint on the initial condition in, e.g. Eqn. (7). Thisis significant in that the implicit dynamics in an ODE-PDE imposed by boundary conditions on φi complicatestability and optimal control problems. By contrast, inPIEs, the infinite-dimension part of the state is ∂sφiwhich is in L2 but is otherwise unconstrained. Further-more, PIEs are defined using the algebra of Partial In-tegral (PI) operators. The algebraic nature of PI opera-tors implies that most tools developed for matrices canbe extended to PIEs - including the LMI framework.Specifically, the LMIs forH∞-optimal observer and con-troller synthesis have been extended to PIEs, as can befound in [25] and [23], respectively. We refer to Linear PIInequalities (LPIs) as this extension of the LMI frame-work and a Matlab toolbox for solving LPIs can be foundin [21]. An example of these synthesis results can befound in Sec. 8.

5.2 Conversion from DDE to PIE

In this subsection, we bypass the DDF and give a for-mula for direct conversion between the DDE and PIErepresentations. This formula is given in Eqns. (12).

6 Modeling of a Network of UAVs

To compare the DDE, DDF, ODE-PDE and PIE repre-sentations, we return to control of a network of UAVs.In this section, we focus on the DDE and DDF represen-tations, as conversion from DDF to ODE-PDE or PIEis straightforward using the formulae provided. For sim-plicity, we eliminate the state delays τij governing inter-actions between UAVs (we will consider state delays inSec. 8) and map the process, input, and output delaysto a common set of delays, {τj}3Nj=1 where the index forthe process delay for UAV i is as τi = τi, the index forinput delay for UAV i is as τN+i = hi, and the index ofthe output delay from UAV i is as τ2N+i = τi. The pro-cess noise is dimension w(t) ∈ Rm, the common input isdimension u(t) ∈ Rp, all states are dimension xi(t) ∈ Rnand the outputs are all dimension yi(t) ∈ Rr. In thiscase, we re-write the network model in Eqns. (1) as

xi(t) = aixi(t) +∑N

j=1aijxj(t)

+ b1iw(t− τi) + b2iu(t− τN+i)

z(t) = C1x(t) +D12u(t)

yi(t) = c2ixi(t− τ2N+i) + d21iw(t− τ2N+i).

6.1 The DDE Representation

To model this network as a DDE, we consider Eqn. (2)where K = 3N for a given C10 and D12. First, we defineA0 blockwise as

6

Conversion Formula from DDE to DDF:Bv

D1v

D2v

= I, Cvi =

Ai B1i B2i

C1i D11i D12i

C2i D21i D22i

, Cvdi(s) =

Adi(s) B1di(s) B2di(s)



, Drvi = 0,[Cri Br1i Br2i

]= I.

(8)Conversion Formula from NDS to DDF:

Drvi =

0 0 0

0 0 0

0 0 0

I 0 0

,[Cri Br1i Br2i

]=

In 0 0

0 Im 0

0 0 Ip

A0 B1 B2

,Bv

D1v

D2v

= In+q+r,

Cvi =

Ai B1i B2i Ei

C1i D11i D12i E1i

C2i D21i D22i E2i

, Cvdi(s) =


C1di(s) D11di(s) D12di(s) E1di(s)

C2di(s) D21di(s) D22di(s) E2di(s)

(9)

Conversion Formula from ODE-PDE or DDF to PIE:

A = P[A0, A

0, {Iτ , 0, 0}

], T = P

[I, 0

T0,{0,Ta,Tb}

], BT1

= P[

0, ∅

T1,{∅}

], BT2

= P[

0, ∅

T2,{∅}

],

B1 = P[B1, ∅

0, {∅}

], B2 = P

[B2, ∅

0, {∅}

], C1 = P

[C10,C11

∅, {∅}

], C2 = P

[C20,C21

∅, {∅}

], Dij = P

[Dij , ∅

∅, {∅}

](10)

where

Cvi = Cvi +

∫ 0

−1τiCvdi(τis)ds, DI =

(Inv −

(K∑i=1

CviDrvi

))−1, CIi(s) = −DI

(Cvi + τi

∫ s

−1Cvdi(τiη)dη

)

[T0 T1 T2

]=

Cr1 Br11 Br21

......

...

CrK Br1K Br2K

+

Drv1

...

DrvK

[Cvx Dvw Dvu

],[Cvx Dvw Dvu

]= DI

K∑i=1

Cvi

[Cri Br1i Br2i

]

Ta(s, θ) =

Drv1

...

DrvK

[CI1(θ) · · · CIK(θ)], Tb(s, θ) = −I∑

ipi

+ Ta(s, θ), Iτ =

1τ1Ip1

. . .

1τKIpK

,

A(s)

C11(s)

C21(s)

=

Bv

D1v

D2v

[CI1(s) · · · CIK(s)],

A0 B1 B2

C10 D11 D12

C20 D21 D22

=

A0 B1 B2

C10 D11 D12

C20 D21 D22

+

Bv

D1v

D2v

[Cvx Dvw Dvu

]. (11)

Fig. 2. Conversion formulae from DDE to DDF, NDS to DDF, and DDF/ODE-PDE to PIE

[A0]ij =

{ai, i = j

aij otherwise

and define the following matrices blockwise for i =1, · · · , N as

B1,i = ei ⊗ b1i, B2,N+i = ei ⊗ b2i,C2,2N+i = ei ⊗ c2i, D21,2N+i = ei ⊗ d2i.

All other undefined matrices in Eqn. (2) are 0. The DDErepresentation of the network has the obvious disadvan-tage that there are 3N delays and each delayed channel

contains all states and inputs - yielding an aggregate de-layed channel of size R3N(nN+m+p).

6.2 The DDF Representation

To efficiently model the network model as a DDF, we re-tain the matrix A0 from the DDE model in Subsec. 6.1,set C1 = C10 and leave D12 unchanged. Our first step isto define the vectors ri(t) and v(t) using Br1i, Br2i,Cri,Cvi, Bv, and B2v (all other matrices are 0). The first 3sets of matrices are defined for i = 1, · · · , N as Br1,i =b1i, Br1,2N+i = d21i, Br2,N+i = b2i, and Cr,2N+i = c2i.

7

Conversion Formula from DDE to PIE: T ,A,Bi, Ci,Dij ,BTi are as defined in Eqn. (10) where now

Iτ =

1τ1In+m+p

. . .

1τKIn+m+p

, T0 =

[In 0n,m 0n,p

]T...[

In 0n,m 0n,p

]T

, T1 =

[0m,n Im 0m,p

]T...[

0m,n Im 0m,p

]T

, T2 =

[0p,n 0p,m Ip

]T...[

0p,n 0p,m Ip

]T

,Ta = 0(n+m+p)K , Tb = −I(n+m+p)K ,

A(s)

C11(s)

C21(s)

= −[X1(s) · · · XK(s)

], Xi(s) =

Ai B1i B2i

C1i D11i D12i

C2i D21i D22i

+ τi

∫ s

−1

Adi(τiη) B1di(τiη) B2di(τiη)

C1di(τiη) D11di(τiη) D12di(τiη)

C2di(τiη) D21di(τiη) D22di(τiη)

dη,

A0 B1 B2

C10 D11 D12

C20 D21 D22

=

A0 B1 B2

C10 D11 D12

C20 D21 D22

+

K∑i=1

Ai B1i B2i

C1i D11i D12i

C2i D21i D22i

+

∫ 0

−1

K∑i=1

τi

Adi(τis) B1di(τis) B2di(τis)

C1di(τis) D11di(τis) D12di(τis)

C2di(τis) D21di(τis) D22di(τis)

ds(12)

Fig. 3. Direct conversion formula from DDE to PIE, bypassing the DDF.

We presume the UAV state dimensions (n) are less thanthe size of the aggregate input (m) and disturbance vec-tors (p) (i.e. n < m and n < p). In this case it is prefer-able to delay only the part of the input and disturbancesignals which affects each UAV. We now have the follow-ing definition for ri for i = 1, · · · , 3N .

ri(t) =b1iw(t) i ∈ [1, N ]

b2,i−Nu(t) i ∈ [N + 1, 2N ]

c2,i−2Nxi−2N (t) + d21,i−2Nw(t) i ∈ [2N + 1, 3N ].

Next, we construct output v(t) by defining Cvi for i =1, · · · , 3N as Cvi = ei ⊗ Ipi which yields

v(t) =[r1(t− τ1)T · · · r3N (t− τ3N )T

]T.

Finally, we feed v(t) back into the dynamics using

Bv =[I · · · I I · · · I 0

], D2v =

[0 · · · 0 0 · · · 0 I

],

which recovers the network model.

6.3 Complexity of DDEs vs. DDFs

In the DDF model, the infinite-dimensional state is ri.In our DDF formulation of the UAV model: each processdelay adds n states; each input delay adds n states; andeach output delay adds r states to this vector. The ag-gregated infinite-dimensional state is then L2ˆ(

∑pi

=

(2n+ r)N). Assuming that optimal control and estima-tion problems are tractable when the number of infinite-dimensional states is less than 50 [15], and if we supposen = r = 1, then it is possible to control 17 UAVs. Bycontrast, in the DDE model of our UAVs, the infinite-

dimensional state is L3N(m+p+r)2 ( meaning we can con-

trol at most 5 or 6 UAVs).

7 A Network which is a DDF, but not a DDE

In this subsection, we present a network model whichcan be represented using DDFs, ODE-PDEs, and PIEs,but not using DDEs. These models arise from the useof static feedback - i.e. u(t) = Fy(t) where y(t) is theconcatenated vector of outputs from the UAVs. Notethat y may include measurement of all states (the staticstate feedback problem). In this example, let us ignoreoutput, process and state delay, but retain input delayand add a term which models the impact of actuatorinput u(t) on the sensors as

yi(t) = c2ixi(t) + d21iw(t) + d22iu(t− τi).

Let A0, C1, D12, B2i, Cvi be as defined in Subsec. 6.2and define

B1 =

b11...

b1N

, D21 =

d21,1

...

d21,N

C2 = diag(c2,1, · · · , c2,N ), D22i = ei ⊗ d22i.

Aggregating the measurements, we have

y(t) = C2x(t) +D21w(t) +∑N

i=1D22iu(t− τi).

Now, substituting u(t) = Fy(t) into the sensed outputterm, we obtain solutions of the form

x(t) = A0x(t) +B1w(t) +∑N

i=1B2iFy(t− τi)

z(t) = C1x(t) +D12Fy(t) (13)

y(t) = C2x(t) +D21w(t) +∑N

i=1D22iFy(t− τi).

Clearly, there is no DDE model with solutions which sat-isfy Eqns. (13) due to the recursion in the output [9].

8

However (assuming appropriate initial conditions), thesesolutions can be constructed using the DDF (and con-sequently the ODE-PDE and PIE frameworks). To con-struct such a model, we define the following terms.

D12 = D12FD21, D22 = 0, C1 = C1 +D12FC2

Cri = FC2, Br1i = FD21, [Drvi]i = FD22i

Bv =[B21 · · · B2N

], Cvi = ei ⊗ I

D1v = D12FD2v, D2v =[D22,1 · · · D22,N

](14)

Lemma 5 For given ri0, x0, suppose ri, v, y, x, and zsatisfy the DDF defined by

{A0, B1, Bv, C1, D12, D1v, D2v, Cri, Br1i, Drvi, Cvi}given by Eqns. (14). Then x, z and y also satisfyEqns. (13).

8 Optimal Control of a Large Network

To illustrate the computational advantages of the DDFs,ODE-PDEs, and PIEs for controller synthesis problems,we consider the scalable network model with state-delayfor centralized control of water temperature for multipleshowering customers as defined in [15]. If T1i is the tapposition and T2i is the temperature for user i, then thedynamics of this model are given by

T1i(t) = T2i(t)− wi(t) (15)

T2i(t) = −αi (T2i(t− τi)− wi(t))

+∑N

j 6=iγijαj (T2j(t− τj)− wj(t)) + ui(t)

z(t) =[∑N

i=1 T1i(t) .1∑Ni=1 ui(t)

]T.

For N users, we choose αi = 1, γij = 1/N , τi = i, andwi(t) = N .

8.0.1 DDE Formulation of the Network

In [15], we formed the aggregate state vector as

x(t) =[T11(t) · · · T1N (t) T21(t) · · · T2N (t)

]Tand defined the DDE model using

A0 =

[0N×N IN

0N×N 0N×N

], Ai =

[0N×N 0N×N

0N×N Ai

]Ai = Γ ∗ diag(ei) = Γ ∗ diag

([01×i−1 1 01×N−i

])B1 =

[−IN−Γ

], B2 =

[0N×N

IN

]

[Γ]ij =

{γijαj i 6= j

−αi i = ji, j = 1, · · · , N

C1 =

[1TN 01×N

01×N 01×N

], D11 =

[02×N

], D12 =

[01×N

.11TN

].

In this formulation, we have n = 2N states, m = Ndisturbances, p = N inputs, q = 2 regulated outputs

and K = N delays (τij = τj). Using the SOS-basedH∞-optimal controller synthesis algorithm for DDEs aspresented in [15], we were able to design controllers forN = 4 users. This corresponds to an infinite-dimensionalchannel of size LnK=32

2 .

8.0.2 DDE Formulation of the problem

To construct the DDF formulation of the problem, x(t) isunchanged. However, we now define the delayed channelsas

ri(t) =[01×N+i−1 1 01×N−i

]x(t) = T2i(t).

This is done by defining Cri, Br1i, Br2i and Drvi as

Cri =[01×N+i−1 1 01×N−i

]Br1i = 01×N Br2i = 01×N Drvi = 01×N .

We would like the output of the delayed channels to bethe delayed states as

v(t) =[T21(t− τ1) · · · T2N (t− τN )

]T.

This is accomplished by defining

Cvi = ei =[01×i−1 1 01×N−i

]T, Cvdi = 02×N .

Finally, we retain A0, B1, B2, C1, C2, D11, D12 from theDDE formulation, and use Bv and D1v to model howthe delayed terms affect the state dynamics and outputsignal.

Bv =

[0N×N

Γ

], D1v = 0

In the DDF formulation, we have n = 2N states,m = Ndisturbances, p = N inputs, q = 2 regulated outputs,K = N delays (τij = τj), and K delay channels, each ofdimension L1

2.

8.1 H∞-optimal Control Using PIETOOLS 2020a

For H∞-optimal controller synthesis, we used the DDFto PIE converter convert PIETOOLS DDF and H∞-optimal synthesis option in the PIETOOLS 2020a Mat-lab toolbox, as described in [21] and available onlineat [17]. The DDF system input format for this tool-box is described in detail in the user manual [17], as isthe converter and controller synthesis feature. In thistoolbox, the extreme performance option was selectedto decrease computation times and reduce memory us-age. The H∞-optimal controller synthesis feature inPIETOOLS solves the optimal control problem for aPIE and is based on the result in [23]. The numericaltest was performed on a desktop computer with 128GBRAM and a 3 GHz intel processor. CPU seconds is aslisted for the interior-point calculations determined bySedumi. The computation times, indexed by number ofusers, are listed in Table 1. In all cases, the achievedclosed-loop H∞-norm was in the interval [.3, 3]. Practi-cally, we observe that the controller synthesis problemis tractable up to 40 users - a significant improvement

9

Table 1IPM CPU sec vs. # of states (N) forH∞ control of Eqn. (15).

N→ 1 3 5 10 20 30 40

CPU sec .48 .638 2.42 94.7 5455 35k 157k

from the 4 users in [15]. Note that 40 users correspondsto an aggregated infinite-dimensional channel of size

L

∑ipi=N=40

2 . Also recall that for 40 users, we have 80states, 40 inputs, 40 disturbances and 40 delays.

Note that the PIETOOLS 2020a toolbox does notrequire use of the PIE formulation and will con-vert a DDE to a DDF, if desired. There is also afeature for constructing minimal DDF representa-tions of DDEs - which can be very useful for solvinglarge network problems. The conversion from a NDSto DDF is also included in the PIETOOLS libraryexamples DDF library PIETOOLS.m.

9 Conclusion

This paper summarizes four possible representationsfor systems with delay: the Delay-Differential Equation(DDE) form; The Differential Difference (DDF) form;the ODE-PDE form; and the Partial Integral Equation(PIE) form. Formulae are given for conversion betweenthese representations, although direct conversion fromDDE to DDF is not advised if the delayed channels arelow-dimensional (although PIETOOLS 2020a includesa feature for constructing minimal DDF representationsof DDEs). Using the given formulae and definitions ofsolution, we show that the set of solutions for the DDF,ODE-PDE, and PIE are equivalent. These results im-ply that if there is a valid conversion formula, manysolutions to the H∞-optimal control and estimationproblems can be converted between representations byapplying this formula to the closed-loop system. How-ever, this only works if optimality is defined in termsof the finite-dimensional vectors, x0, u, w, x, y, z. Thisis because any input-output pair (u,w, x0) 7→ (y, z, x)which defines a solution to one representation also de-fines a solution for every other representation for whichthere is a valid conversion formula. Likewise, stability ofthe representations is equivalent as long as the stabilitydefinition only involves the finite-dimensional vectors,x0, x, u, w, y, z.

The results and formulae in this paper are meant to pro-vide a convenient reference for researchers interested inexploring alternative representations of delay systems. Asummary of the representations and conversion formu-lae is given in Table 2, along with examples of simulationtools and controller synthesis results. We have shownusing an example of a network of UAVs that some net-works cannot be modeled in the DDE formulation andthat careful choice of representation can significantly re-duce the complexity of the underlying analysis and con-trol problems. Finally, we have shown that H∞-optimalcontrol in the DDF/ODE-PDE/PIE framework allows

up to 40 agents, while formulation in the DDE frame-work only allows for control of 4 agents.

Table 2Conversion formulae (DDF,PDE,PIE), simulation tools(Sim), controller design tools (H∞), and model definitions(Model) for each class of systems (PDE→ODE-PDE).

Need→ DDF PDE PIE Sim H∞ Model

DDE (8) (8)+(5) (12) [2] [15] (2)

Neut. (9) (9)+(5) (9)+(11) [2] [26] (4)

DDF X (5) (11) - - (3)

PDE X X [22] [24] [12] (5)

PIE X X X - [23] (6)

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[4] A. Das, S. Shivakumar, S. Weiland, and M. Peet. H∞ optimalestimation for linear coupled PDE systems. In Proceedingsof the IEEE Conference on Decision and Control, 2019.

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[16] M. Peet. Representation of networks and systems withdelay: DDEs, DDFs, ODE-PDEs and PIEs. Technical report,arXiv.org, 2020. https://arxiv.org/abs/1910.03881.

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Proof of Lemma 1

Lemma 6 (Lemma 1) Suppose that Cvi, Cvdi, Cri,Br1i , Br1i, Drvi, Bv, D1v, and D2v are as definedin Eqns. (8). Given u, w, x0, the functions x, y, andz satisfy the DDE defined by {Ai, Bi, Ci, Dij , · · · } ifand only if x, y, z, and ri satisfy the DDF defined by{Ai, Bi, Ci, Dij , · · · } where

ri(t) =

x(t)

w(t)

u(t)

, ri0 =

x0

0

0

i = 1, · · · ,K.

PROOF. Most of the proof follows immediately fromthe proof of Lemma 2. However, in this case, we a non-zero initial condition. Specifically, suppose x, y, and zsatisfy the DDE for given u, w, x0. If

ri(t) =

x(t)

w(t)

u(t)

, ri0(s) =

x0(s)

0

0

,then as in the proof of Lemma 2, Eqn. (3) is satisfied.Furthermore, since w(t), u(t) = 0 for t ≤ 0, we have fort ≤ 0,

ri(s) =

x(s)

0

0

=

x0(s)

0

0

= ri0(s).

Finally, x0 ∈ W 1,2 implies ri0 ∈ W 1,2 and since Drvi =0, the sewing condition is satisfied since

ri0(0) = Crix0 =

x0

0

0

.Conversely, suppose x, y, z, and ri satisfy the DDF.Then from the proof of Lemma 2, Eqn. (2) is satisfied.Likewise since ri0 ∈ W 1,2[−τi, 0]pi for all i = 1, · · · ,K,we have x0 ∈W 1,2[−τK , 0]n which completes the proof.

11

Proof of Lemma 2

Lemma 7 (Lemma 2) Suppose that Cvi, Cvdi, Cri,Br1i , Br2i, Drvi, Bv, D1v, and D2v are as definedin Eqns. (9). Given u, w, the functions x, y, and zsatisfy the NDS defined by {Ai, Bi, Ci, Dij , · · · } if andonly if x, y, z, v and ri satisfy the DDF defined by{Ai, Bi, Ci, Dij , · · · } where ri0 = 0 and

ri(t) =

x(t)

w(t)

u(t)

x(t)

, i = 1, · · · ,K.

and

v(t) =

K∑i=1

Ai B1i B2i Ei

C1i D11i D12i C1ei

C2i D21i D22i C2ei


+

K∑i=1

0∫−τi




x(t+ s)

w(t+ s)

u(t+ s)

x(t+ s)

ds.

PROOF. Given u, w, suppose x, y, and z satisfy theNDS. Defining ri and v, we have

v(t) =

K∑i=1

Cviri(t− τi) +

K∑i=1

∫ 0

−τiCvdi(s)ri(t+ s)ds

as desired. Furthermore,

x(t)

z(t)

y(t)

=

A0 B1 B2

C1 D11 D12

C2 D21 D22

x(t)

w(t)

u(t)

+

Bv

D1v

D2v

v(t)

Now all that remains is to show

ri(t) =[Cri Br1i Br2i

]x(t)

w(t)

u(t)

+Drviv(t)

ri(t) =

x(t)

w(t)

u(t)

x(t)

=

I 0 0

0 I 0

0 0 I

0 0 0

x(t)

w(t)

u(t)

+

0

0

0

x(t)

=

I 0 0

0 I 0

0 0 I

A0 B1 B2

x(t)

w(t)

u(t)

+

0 0 0

0 0 0

0 0 0

I 0 0

v(t)

=[Cri Br1i Br2i

]x(t)

w(t)

u(t)

+Drviv(t)

as desired. Finally, ri0 = 0 since x(t), w(t), u(t) = 0 fort ≤ 0.

For the converse, given u, w suppose x, y, z, v and risatisfy the DDF where ri0 = 0. Then

x(t)

z(t)

y(t)

=

A0 B1 B2

C1 D11 D12

C2 D21 D22

x(t)

w(t)

u(t)

+ v(t)

and hence

ri(t) =[Cri Br1i Br2i

]x(t)

w(t)

u(t)

+Drviv(t)

=

I 0 0

0 I 0

0 0 I

A0 B1 B2

x(t)

w(t)

u(t)

+

0 0 0

0 0 0

0 0 0

I 0 0

v(t) =

x(t)

w(t)

u(t)

x(t)

.

12

We now have that

v(t) =

K∑i=1

Cviri(t− τi) +

K∑i=1

∫ 0

−τiCvdi(s)ri(t+ s)ds

=

K∑i=1

Ai B1i B2i Ei

C1i D11i D12i C1ei

C2i D21i D22i C2ei

ri(t− τi)+K∑i=1

0∫−τi




ri(t+ s)ds

=

K∑i=1

Ai B1i B2i Ei

C1i D11i D12i C1ei

C2i D21i D22i C2ei


+

K∑i=1

0∫−τi




x(t+ s)

w(t+ s)

u(t+ s)

x(t+ s)

ds.

from which we conclude thatx(t)

z(t)

y(t)

=

A0 B1 B2

C1 D11 D12

C2 D21 D22

x(t)

w(t)

u(t)

+ v(t)

=

A0 B1 B2

C10 D11 D12

C20 D21 D22

x(t)

w(t)

u(t)

+

K∑i=1

Ai B1i B2i Ei

C1i D11i D12i E1i

C2i D21i D22i E2i


+

K∑i=1

∫ 0

−τi




x(t+ s)

w(t+ s)

u(t+ s)

x(t+ s)

dsas desired. Finally, by definition x0 = 0, which concludesthe proof.

Proof of Lemma 3

Lemma 8 Suppose for given u, w, ri0, that x, ri, v, y,and z satisfy the DDF defined by {Ai, Bi, Ci, Dij , · · · },with Cvdi bounded. Then for u, w, φi0(s) = ri0(τis),we have that x, v, y, and z also satisfy the ODE-PDEdefined by {Ai, Bi, Ci, Dij , · · · }with φi(t, s) = ri(t+τis).Similarly, for given u, w, φi0, if x, v, y, φi and z satisfythe ODE-PDE defined by {Ai, Bi, Ci, Dij , · · · }, then x,v, y, and z satisfy the DDF with ri(t) = φi(t, 0) andri0(s) = φi0(s/τi).

PROOF. Suppose x, v, y, and z satisfy the DDF,φi(t, s) = ri(t + τis) and φi0(s) = ri0(τis). By assump-tion, ri0 ∈ W 1,2[−τi, 0] and hence is absolutely con-tinuous with classical derivative ri0 ∈ L2[−τi, 0]. Fur-thermore, φi0 ∈ W 1,2[−1, 0] and is likewise absolutelycontinuous with classical derivative. Since ri(t + ·) ∈W 1,2[−τi, 0], we have φi(t, ·) = ri(t+ τi·) ∈W 1,2[−1, 0]for all t ≥ 0.

Note that the continuity property may be proven directlyand not as an assumption. Specifically, let us examinedifferentiability of φi(t, s) = ri(t + τis). For any i ∈{1,K}, on the interval t ∈ [0, τ1], we have

ri(t) = Crix(t) +Br1iw(t) +Br2iu(t) +Drviv(t)

= Cri (A0x(t) +B1w(t) +B2u(t) +Bvv(t))

+Br1iw(t) +Br2iu(t) +Drviv(t)

= Cri (A0x(t) +B1w(t) +B2u(t) +Bvv(t))

+Br1iw(t) +Br2iu(t)

+Drvi

( K∑j=1

Cvj rj0(t− τj) +

K∑j=1

∫ 0

t−τjCvdj(s− t)rj0(s)ds

+

K∑j=1

∫ t

0

Cvdj(s− t)rj(s)ds).

Now recall Br1iw, Br2iu ∈ L2[0,∞] and ri0 ∈L2[−τi, 0]. Hence, by Gronwall-Bellman, since Cvdi arebounded, we conclude that Pτ1ri ∈ W 1,2[−τi, τ1] wherePT is the truncation operator at time T . Proceedingby steps, we conclude that PT ri ∈ W 1,2[−τi, T ] for anyT ≥ 0. Thus we conclude that φi(t) ∈ W 1,2[−1, 0] forall t ≥ 0.

Now, examining the initial conditions, we find φi(0, s) =ri(τis) = ri0(τis) = φi0(s) and

φi0(0) = ri0(0) = Crix0

+Drvi

(K∑i=1

Cviri0(−τi) +

K∑i=1

∫ 0


)= Crix0

+Drvi

(K∑i=1

Cviφi0(−1) +

K∑i=1

∫ 0


).

13

Furthermore,

v(t) =

K∑i=1

Cviri(t− τi) +

K∑i=1

∫ 0

−τiCvdi(s)ri(t− τis)ds

=

K∑i=1

Cviφi(t,−1) +

K∑i=1

∫ 0

−1τiCvdi(τis)φi(t, s)ds

and

φi(t, s) = ri(t+ τis) =1

τi∂sri(t+ τis) =

1

τiφi,s(t, s).

Finally, x, z, y satisfy the ODE-PDE by inspection and

φi(t, 0) = ri(t) = Crix(t)+Br1iw(t)+Br2iu(t)+Drviv(t).

Conversely, suppose x, v, y, φi and z satisfy the ODE-PDE and ri0(s) = φi0(s/τi). Since φi(t) ∈ W 1,2[−1, 0],ri(t+ s) = φi(t, s/τi) and hence ri(t+ ·) ∈W 1,2[−τi, 0].Likewise, ri0 ∈W 1,2[−τi, 0], and

ri0(0) = φi0(0) = Crix0

+Drvi

(K∑i=1

Cviφi0(−1) +

K∑i=1

∫ 0


)= Crix0

+Drvi

(K∑i=1

Cviri0(−τi) +

K∑i=1

∫ 0


).

Now if ri(t) = φi(t, 0), then

ri(t) = φi(t, 0) = Crix(t)+Br1iw(t)+Br2iu(t)+Drviv(t).

Next, if φi(t, s) = 1τi∂sφi(t, s), then φi(t, s) = φi(t +

sτi, 0) = ri(t+ sτi) and hence

v(t)

=

K∑i=1

Cviφi(t,−1) +

K∑i=1

∫ 0


=

K∑i=1

Cviri(t− τi) +

K∑i=1

∫ 0

−τiCvdi(s)ri(t− τis)ds.

Finally, x, z, y satisfy the DDF by inspection and fors ∈ [−τi, 0], ri(s) = φi(0, s/τi) = φi0(s/τi) = ri0(s).

Proof of Lemma 4

Lemma 9 (Lemma 4) Given u, w, and x0, φi0 satis-fying Condition (7), Suppose x, φi, v, y, and z satisfythe ODE-PDE defined by {Ai, Bi, Ci, Dij , · · · }. Then yand z also satisfy the PIE defined by {A,Bi, Ci,Dij ,BTi}with T ,A,Bi, Ci,Dij as defined in Eqn. (10) and

x(t) :=

x(t)

∂sφ1(t, ·)...

∂sφK(t, ·)

x0 :=

x0

∂sφ10

...

∂sφK0

.

Furthermore, for given u, w, x0 ∈ Rn ×L2[−1, 0]p, if y,z and x satisfy the PIE defined by {A,Bi, Ci,Dij ,BTi},then x, φi, v, y, and z satisfy the ODE-PDE defined by{Ai, Bi, Ci, Dij , · · · } where

x(t)

φ1(t, ·)...

φK(t, ·)


x0

φ10

...

φK0

= T x0.

PROOF. Suppose x, φi, v, y, and z satisfy the ODE-PDE and

x(t) :=

x(t)

∂sφ1(t, ·)...

∂sφK(t, ·)

x0 :=

x0

∂sφ10...

∂sφK0

.

Then, x0 ∈ Rn × L2[−1, 0]p and

φi(t, s) = φi(t, 0)−∫ 0

s

φi,s(t, η)dη. (.1)

Recall from Eqns. (5) that

v(t) =

K∑i=1

Cviφi(t,−1) +

K∑i=1

∫ 0


and

φi(t, 0) = Crix(t) +Br1iw(t) +Br2iu(t) +Drviv(t).

From Eqn. (??),

φi(t,−1) = φi(t, 0)−∫ 0

−1φi,s(t, η)dη

and hence

14

v(t) =

K∑i=1

Cvi

(φi(t, 0)−

∫ 0

−1φi,s(t, η)dη

)

+

K∑i=1

∫ 0

−1τiCvdi(τis)

(φi(t, 0)−

∫ 0

s

φi,s(t, η)dη

)ds

=

(K∑i=1

CviCri

)x(t) +

(K∑i=1

CviBr1i

)w(t)

+

(K∑i=1

CviBr2i

)u(t) +

(K∑i=1

CviDrvi

)v(t)

−

(∫ 0

−1

K∑i=1

Cviφi,s(t, η)dη

)

−

(K∑i=1

τi

∫ 0

−1

∫ 0

s

Cvdi(τis)φi,s(t, η)dηds

).

Eliminating v from the RHS, we obtain

v(t) =Cvxx(t) +Dvww(t) +Dvuu(t)

−DI

(∫ 0

−1

K∑i=1

Cviφi,s(t, η)dη

)

−DI

(K∑i=1

τi

∫ 0

−1

∫ 0

s


).

Using the identity∫ 0

−1

∫ 0

s

f(s, η)dηds =

∫ 0

−1

∫ s

−1f(η, s)dηds,

we obtain

v(t) = Cvxx(t) +Dvww(t) +Dvuu(t)

−DI

(∫ 0

−1

K∑i=1

Cviφi,s(t, s)ds

)

−DI

(K∑i=1

∫ 0

−1

(∫ s

−1τiCvdi(τiη)dη

)φi,s(t, s)ds

)= Cvxx(t) +Dvww(t) +Dvuu(t)

−DI

(∫ 0

−1

K∑i=1

(Cvi + τi

∫ s

−1Cvdi(τiη)dη

)φi,s(t, s)ds

)= Cvxx(t) +Dvww(t) +Dvuu(t)

+

∫ 0

−1

K∑i=1

CIi(s)φi,s(t, s)ds.

The rest of the sufficiency proof is straightforward. Plug-ging this expression for v(t) into Eqns. (5), we obtain

z(t) = C10x(t) +

0∫−1

C11(s)Φ(t, η)dη + D11w(t) + D12u(t)

= C1x(t) +D11w(t) +D12u(t)

y(t) = C20x(t) +

0∫−1

C21(s)Φ(t, η)dη + D21w(t) + D22u(t)

= C2x(t) +D21w(t) +D22u(t)

where

x(t) :=

[x(t)

Φ(t, ·)

], Φ(t) :=

φ1,s(t, ·)

...

φK,s(t, ·)

.Likewise,

x(t)

φ1(t, s)...

φK(t, s)

=

A0x(t) +0∫−1

A(η)Φ(t, η)dη

IτΦ(t, s)

+

[B1

0

]w(t) +

[B2

0

]u(t)

= Ax(t) + B1w(t) + B2u(t).

Finally, we observe that

φi(t, s) = Crix(t) +Br1iw(t) +Br2iu(t) +Drviv(t)

−∫ 0

s

φi,s(t, η)dη

= (Cri +DrviCvx)x(t) + (Br1i +DrviDvw)w(t)

+ (Br2i +DrviDvu)u(t)

−∫ 0

s

φi,s(t, η)dη +

∫ 0

−1

K∑j=1

DrviCIj(s)φj,s(t, s)ds

.

Henceφ1(t, ·)

...

φK(t, ·)

= T0x(t) + T1w(t) + T2u(t)

+

∫ s

−1Ta(η)Φ(t, η)dη +

∫ 0

s

Tb(η)Φ(t, η)dη

and x(t)

φ1(t, ·)...

φK(t, ·)

= T x(t) + BT1w(t) + BT2u(t).

15

Finally, we differentiate and combine these expressionsto obtain

T x(t) + BT1w(t) + BT2 u(t) = Ax(t) + B1w(t) + B2u(t)

z(t) = C1x(t) +D11w(t) +D12u(t),

y(t) = C2x(t) +D21w(t) +D22u(t).

We conclude that x, y and z satisfy the PIE.

Conversely, suppose x, y and z satisfy the PIE. Partitionx as

x(t, s) =

[x1(t)

Φ(t, s)

]and define

x(t)

φ1(t, ·)...

φK(t, ·)


x0

φ10

...

φK0

= T x0,

and

v(t) = Cvxx(t)+Dvww(t)+Dvuu(t)+

0∫−1

K∑j=1

CIjΦj(t, θ)dθ.

Examine

T x(t) =

[x(t)

T0x(t) +∫ 0

−1 Ta(θ)Φ(t, θ)dθ −∫ 0

sΦ(t, θ)dθ

].

This implies thatφ1,s(t, ·)

...

φK,s(t, ·)

= Φ(t, s)

andx(t)

φ1(t, 0)...

φK(t, 0)

=

x1(t)

T0x(t) +0∫−1Ta(θ)Φ(t, θ)dθ

+ BT1w(t) + BT2u(t)

and hence x(t) = x1(t) and

φi(t, 0) = Crix(t) +Br1iw(t) +Br2iu(t)

+Drvi

Cvxx(t) +Dvww(t) +Dvuu(t) +

0∫−1

K∑j=1

CIjφj,s(t, θ)dθ

= Crix(t) +Br1iw(t) +Br2iu(t) +Drviv(t)

as desired. Next, we have

x(t)

φ1(t, ·)...

φK(t, ·)

= T x(t) + BT1w(t) + BT2u(t)

= Ax(t) + B1w(t) + B2u(t)

=

A0x(t) +B1w(t) +B2u(t)

1τ1φ1,s(t, s)

...

1τKφK,s(t, s)

+

[Bv(Cvxx(t) +Dvww(t) +Dvuu(t) +

∑Kj=1 CIjφj,s(t, θ)dθ

)0

]

=

A0x(t) +B1w(t) +B2u(t) +Bvv(t)

1τ1φ1,s(t, s)

...

1τKφK,s(t, s)

as desired. Returning to v, we have

v(t)

= Cvxx(t) +Dvww(t) +Dvuu(t) +

0∫−1

K∑j=1

CIjΦj(t, θ)dθ

= Cvxx(t) +Dvww(t) +Dvuu(t)

−DI

(∫ 0

−1

K∑i=1

(Cvi + τi

∫ s

−1Cvdi(τiη)dη

)φi,s(t, s)ds

)

= DI

((K∑i=1

CviCri

)x(t) +

(K∑i=1

CviBr1i

)w(t)

+

(K∑i=1

CviBr2i

)u(t)−

(∫ 0

−1

K∑i=1

Cviφi,s(t, η)dη

)

−

(K∑i=1

τi

∫ 0

−1

∫ 0

s


))

= DI

(K∑i=1

Cvi (Crix(t) +Br1iw(t) +Br2iu(t))

−K∑i=1

Cvi

(∫ 0

−1φi,s(t, η)dη

)

−K∑i=1

∫ 0

−1τi

(∫ 0

s

Cvdi(τis)φi,s(t, η)dη

)ds

).

Hence

16

(I −

(K∑i=1

CviDrvi

))v(t) =

=

K∑i=1

Cvi (Crix(t) +Br1iw(t) +Br2iu(t))

−K∑i=1

Cvi

(∫ 0

−1φi,s(t, η)dη

)

−K∑i=1

∫ 0

−1τi

(∫ 0

s


)ds.

Hence

v(t) =

=

K∑i=1

Cvi (Crix(t) +Br1iw(t) +Br2iu(t) +Drviv(t))

−K∑i=1

Cvi

(∫ 0

−1φi,s(t, η)dη

)

−K∑i=1

∫ 0

−1τi

(∫ 0

s


)ds

=

K∑i=1

Cviφi(t, 0)−K∑i=1

Cvi

(∫ 0

−1φi,s(t, η)dη

)

−K∑i=1

∫ 0

−1τi

(∫ 0

s


)ds

=

K∑i=1

(Cvi +

∫ 0

−1τiCvdi(τis)ds

)φi(t, 0)

−K∑i=1

Cvi

(∫ 0

−1φi,s(t, η)dη

)

−K∑i=1

∫ 0

−1τi

(∫ 0

s


)ds

=

K∑i=1

Cvi

(φi(t, 0)−

∫ 0

−1φi,s(t, η)dη

)

+

K∑i=1

(τi

∫ 0

−1Cvdi(τis)

(φi(t, 0)−

∫ 0

s

φi,s(t, η)dη

)ds

)

=

K∑i=1

Cviφi(t,−1) +

K∑i=1

(τi

∫ 0

−1Cvdi(τis)φi(t, s)ds

)

as desired. Next, we have

z(t) = C1x(t) +D11w(t) +D12u(t)

= C10x(t) +

0∫−1

C11(s)φ(t, η)dη + D11w(t) + D12u(t)

= C1x(t) +D11w(t) +D12u(t)

+D1v

Cvxx(t) +Dvww(t) +Dvuu(t) +

0∫−1

K∑i=1

C1i(s)φi,s(t, η)dη

= C1x(t) +D11w(t) +D12u(t) +D1vv(t)

as desired. Furthermore,

y(t) = C2x(t) +D21w(t) +D22u(t)

= C20x(t) +

0∫−1

C21(s)Φ(t, η)dη + D21w(t) + D22u(t)

= C2x(t) +D21w(t) +D22u(t)

+D2v

(Cvxx(t) +Dvww(t) +Dvuu(t)

+

0∫−1

K∑i=1

C1i(s)φi,s(t, η)dη

)= C2x(t) +D21w(t) +D22u(t) +D2vv(t)

as desired. Finally, sincex0

φ10...

φK0

= T x0.

as before, we havex0

∂sφ10

...

∂sφK0

= x0 =

[x0

Φ0

].

andx0

φ10(t, 0)...

φK0(t, 0)

=

x(t)

T0x(t) +0∫−1Ta(θ)Φ(t, θ)dθ

+ BT1w(t) + BT2u(t)

and hence

17

φi0(0) = Crix0

+Drvi

Cvxx0 +

0∫−1

K∑j=1

CIjφj0,s(θ)dθ

= Crix0 +Drviv(0)

= Crix0

+Drvi

(K∑i=1

Cviφi0(−1) +

K∑i=1

∫ 0


)

as desired.

Proof of Lemma 5

Lemma 10 (Lemma 5) For given ri0, x0, suppose ri,v, y, x, and z satisfy the DDF defined by

{A0, B1, Bv, C1, D12, D1v, D2v, Cri, Br1i, Drvi, Cvi}

given by Eqns. (14). Then x, z and y also satisfyEqns. (13).

PROOF. Suppose ri, v, y, x, and z satisfy the DDF inEqns. (3). Then

y(t) = C2x(t) +D21w(t) +D2vv(t)

= C2x(t) +D21w(t) +

N∑i=1

D22ivi(t)

and hence

ri(t) = Crix(t) +Br1iw(t) +Drviv(t)

= FC2x(t) + FD21w(t) +

N∑i=1

FD22ivi(t)

= F

(C2x(t) +D21w(t) +

N∑i=1

D22ivi(t)

)= Fy(t).

Next,vi(t) = ri(t− τi) = Fy(t− τi)

and we conclude

y(t) = C2x(t) +D21w(t) +

N∑i=1

D22iFy(t− τi).

Similarly

x(t) = A0x(t) +B1w(t) +Bvv(t)

= A0x(t) +B1w(t) +

N∑i=1

B2ivi(t)

= A0x(t) +B1w(t) +

N∑i=1

B2iFy(t− τi)

and finally,

z(t) = C1x(t) + D12w(t) +D1vv(t)

= C1x(t) +D12F

(C2x(t) +D21w(t) +

N∑i=1

D22ivi(t)

)= C1x(t) +D12Fy(t)

as desired.

18

representationofnetworksandsystemswithdelay: ddes,ddfs,ode

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