Input-Output Pseudospectral Bounds for Transient Analysis of Networked and High-Order Systems Jonas Hansson and Emma Tegling

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Input-Output Pseudospectral Bounds for Transient Analysis of

Networked and High-Order Systems

Jonas Hansson and Emma Tegling

Abstract— Motivated by a need to characterize transient

behaviors in large network systems in terms of relevant signal

norms and worst-case input scenarios, we propose a novel

approach based on existing theory for matrix pseudospectra.

We extend pseudospectral theorems, pertaining to matrix expo-

nentials, to an input-output setting, where matrix exponentials

are pre- and post-multiplied by input and output matrices.

Analyzing the resulting transfer functions in the complex plane

allows us to state new upper and lower bounds on system

transients. These are useful for higher-order matrix differ-

ential equations, and speciﬁcally control of double-integrator

networks such as vehicle formation problems. Therefore, we

illustrate the theory’s applicability to the problem of vehicle

platooning and the question of string stability, and show how

unfavorable transient behaviors can be discerned and quantiﬁed

directly from the input-output pseudospectra.

I. INTRODUCTION

Characterizing dynamic properties of systems with struc-

ture, in particular, network structure, is a long-standing prob-

lem in the ﬁeld. While questions of stability and convergence

have dominated the literature since the early works [1],

[2], important questions pertaining to the performance and

robustness of network systems are increasingly gaining at-

tention. For example, [3] and later [4], [5] have described

fundamental limitations to the performance of large networks

subject to structural (sparsity) constraints, stated in terms of

system norms.

A particular area where dynamic behaviors have received

more attention is that of vehicle platooning, that is, the

control of strings of vehicles, see [6], [7] for early works.

Here, it is fundamentally important to prevent disturbance

propagation through the string (to avoid collisions!), and

therefore, to have uniform bounds on error ampliﬁcations

during transients. This has motivated the notion of string

stability, see e.g., [8], [9] or [10], [11] for more recent

surveys. Conditions for string stability fall, roughly speaking,

into two categories: 1) bounding the ampliﬁcation of a

disturbance from vehicle ito vehicle j, or 2) requiring

that bounded initial errors lead to bounded output errors,

independently of the string length. The choice of signal

norms, however, is central for the bounds in this literature,

and the interpretations they allow for. Many works have

done analyses based on L2to L2string stability, see [9],

[12], [13] while the, as argued e.g. in [14], possibly more

The authors are with the Department of Automatic Control,

Lund University, Lund, Sweden. Email: {jonas.hansson,

emma.tegling}@control.lth.se

This work was partially funded by Wallenberg AI, Autonomous Systems

and Software Program (WASP) funded by the Knut and Alice Wallenberg

Foundation and the Swedish Research Council through Grant 2019-00691.

important L∞to L∞disturbance ampliﬁcation has received

signiﬁcantly less attention even if considered in [7], [8]. In

this work, we shed light on a new approach to analyzing such

bounds for input-output systems in general, and networks and

vehicle strings in particular.

This approach takes off from the literature on pseudospec-

tra. Pseudospectra, which complement spectral analysis of

linear systems, especially for those with non-normal opera-

tors, have seen usage in describing the transient behavior

of both differential and difference equations. The works

are too numerous to mention, but we refer to [15] for an

excellent textbook on the subject. Through pseudospectra

one can state lower and upper bounds on the transient of

the exponential matrix, i.e., on supt≥0ketAk, and thereby

on the solution to a linear differential equation. In other

words, on the transient response of the internal states of a

linear system. The most famous such bounds are given by the

Kreiss theorem [16]. However, in control, and in particular,

network applications including vehicle platooning, we are not

necessarily interested in the transients of the internal states.

For instance, vehicular formation dynamics tend to have a

double integrator rendering certain internal states unbounded,

while inter-vehicular distances may be well-behaved. To

cope with this one can incorporate measurement and input

matrices C,Band then bound supt≥0kCetABk instead.

The extension of pseudospectral bounds to such an input-

output setting is the main focus of the present work. For

this purpose we will deﬁne a notion of input-output pseu-

dospectra. These will, in the case of higher-order systems

(by which we mean systems with more than one integrator),

become closely related to structured pseudospectra, which

have been studied in [17], [18] and applied to mechanical

systems in [19]. In these works the main focus has been on

the robustness of solutions to matrix polynomial equations

including the quadratic eigenvalue problem. The related

analysis of transient behavior of kCetABk has, to the best

of our knowledge, barely received attention, though some

structured Kreiss-like theorems were proven in [20], [21].

This paper aims to highlight the potential usefulness of

the pseudospectral framework for networked systems and

systems with higher-order dynamics. Platooning, where ve-

hicles are modeled as double integrators (the acceleration

is actuated), and which have a string network topology,

is a prototypical example. We ﬁrst generalize certain key

results from [15] to an input-output setting. Furthermore,

we use complex analysis to derive new upper bounds on

the transients of state space realizations, which are espe-

cially useful for systems that have high-order dynamics.

arXiv:2210.08903v1 [math.OC] 17 Oct 2022

The generalizations lead to lower and upper bounds on the

transient supt≥0kCetABk, which under given input scenarios

imply bounds on the output supt≥0ky(t)k(in any p-norm).

Through examples we show how the new bounds can be

applied. For a large-scale platooning problem, we compute

bounds on the deviations from equilibrium for a worst-case

bounded initial condition.

The remainder of this paper is organized as follows. In

Sec. II we introduce the preliminaries of this work. Lower

and upper bounds on the transient of suptky(t)kand simple

examples illustrating how to apply the bounds are presented

in Sec. III. Then we illustrate an application of our results in

the form of vehicle strings in Sec. IV. Lastly our conlusions

are presented in Sec. V.

II. PRELIMINARIES

Consider the linear time-invariant system

ξ(t) = Aξ(t) + Bu(t)

y(t) = Cξ(t),(1)

where the state ξ∈RN,A ∈ RN×N,B ∈ RN×P,C ∈

RQ×N, and output y∈RQ. The initial condition is ξ(0) =

ξ0. We will interpret C(sI −A)−1Bas a transfer matrix and

call the system (1) input-output stable if all poles of this

transfer matrix lie in the open left half plane. Denote by

σ(A)the spectrum, i.e., the set of eigenvalues of A.

We will often let the system in (1) model matrix differen-

tial equations of the form

x(l)(t) + Al−1x(l−1)(t) + ··· +A0x(t) = Bu(t)

y(t) = Cξ(t),(2)

where x(t)∈Rnand x(k)denotes the kth time deriva-

tive of x:x(k)(t) = dkx(t)

dtk. In this case, ξ(t) =

[x, ˙x, . . . , x(l−1)]>∈Rnl, with nl =N. This system can

be equivalently stated on block-companion form as

ξ(t) = 





0In0. . .

.......0

0. . . 0In

−A0−A1. . . −Al−1







| {z }

ξ(t) + 











|{z}

u(t)

y(t) = Cξ(t).

(3)

A. Signal and system norms

Norms are central to this work. Here we will consider the

standard vector p-norms:

kxkp=



PN

k=1 |xk|p1

pif 1≤p < ∞

maxk|xk|if p=∞,

where x∈CN. For matrices we consider the corresponding

induced norms, i.e.

kAk = sup

kxk=1 kAxk,

where A ∈ CM×N.

In general, our results can be interpreted in any of these

norms and we will often omit the subscript to indicate that

the results are valid for all of them. What we need for our

theorems is, more speciﬁcally, that the matrix norms are

submultiplicative, which means that the following inequality

is valid for any two compatible matrices A1, A2

kA1A2k≤kA1kkA2k.

It is well known that this is true for all the p-norms.

B. Input-output scenarios

We will present bounds in terms of the scaled exponential

matrix CeAtB. Its norm can be seen as bounds on the tran-

sient response of the system (1) in the following scenarios:

1) Impulse response: Consider the input signal {u(t) =

δ(t)u0}with u0∈RPand let ||u0|| = 1 in some norm. The

solution of (1) is given by

y(t) = CetABu0(4)

and the worst possible transient of y(t)is given by

sup

tky(t)k= sup

sup

ku0k=1 kCetABu0k= sup

tkCetABk.

2) Response to an initial condition: An initial condition

response is given by

y(t) = CetAξ(0).

To study the worst possible initial condition with respect to

resulting deviations in the output y(t)we may consider

sup

tky(t)k= sup

sup

kξ0k=1 kCetAξ0k= sup

tkCetAk.

The corresponding analysis for the worst-case structured

initial condition is done by multiplying ξ0by B. In this case,

sup

tky(t)k= sup

sup

kξ0k=1 kCetABξ0k= sup

tkCetABk.

For example, B= (I, 0,...,0)Tin (3) corresponds to all

initial derivatives being zero.

C. Complex analysis

The basis for our upcoming theorems is three Laplace

transform results, which were also used to derive key results

in [15]. For completeness they are also presented.

Lemma 1 ( [15, Theorem 15.1] ): Let Abe a matrix.

There exist ω∈Rand M≥1such that

ketAk ≤ Meωt ∀t≥0.(5)

Any s∈Cwith Res > ω is in the resolvent set of A, with

(sI − A)−1=Z∞

e−stetAdt. (6)

If Ais a matrix or bounded operator, then

etA=1

2πi ZΓ

est(sI − A)−1ds, (7)

where Γis any closed and positively oriented contour that

encloses σ(A)once in its interior.

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Input-OutputPseudospectralBoundsforTransientAnalysisofNetworkedandHigh-OrderSystemsJonasHanssonandEmmaTeglingAbstractMotivatedbyaneedtocharacterizetransientbehaviorsinlargenetworksystemsintermsofrelevantsignalnormsandworst-caseinputscenarios,weproposeanovelapproachbasedonexistingtheoryformatrixpseu...

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Input-Output Pseudospectral Bounds for Transient Analysis of Networked and High-Order Systems Jonas Hansson and Emma Tegling

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