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Robust Stability of Differential-Algebraic

Equations

Nguyen Huu Du, Vu Hoang Linh, and Volker Mehrmann

Abstract This paper presents a survey of recent results on the robust stability

analysis and the distance to instability for linear time-invariant and time-varying

differential-algebraic equations (DAEs). Different stability concepts such as expo-

nential and asymptotic stability are studied and their robustness is analyzed under

general as well as restricted sets of real or complex perturbations. Formulas for the

distances are presented whenever these are available and the continuity of the dis-

tances in terms of the data is discussed. Some open problems and challenges are

indicated.

Keywords Differential-algebraic equation · Restricted perturbation · Robust

stability · Stability radius · Spectrum · Index

Mathematics Subject Classiﬁcation 93B35 · 93D09 · 34A09 · 34D10

1 Introduction

In many areas of science and engineering one uses mathematical models to simulate,

control or optimize a system or process. These mathematical models, however, are

typically inexact or contain uncertainties and thus, the following question is of major

importance.

How robust is a speciﬁc property of a given system described by differential

or difference equations under perturbations to the data?

N.H. Du · V. H . L i n h (

)

Faculty of Mathematics, Mechanics and Informatics, Vietnam National University,

334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam

e-mail: linhvh@vnu.edu.vn

N.H. Du

e-mail: dunh@vnu.edu.vn

V. Mehrmann

Institut für Mathematik, MA 4-5, Technische Universität Berlin, 10623 Berlin,

Fed. Rep. Germany

e-mail: mehrmann@math.tu-berlin.de

A. Ilchmann, T. Reis (eds.), Surveys in Differential-Algebraic Equations I,

Differential-Algebraic Equations Forum,

DOI 10.1007/978-3-642-34928-7_2, © Springer-Verlag Berlin Heidelberg 2013

64 N.H. Du et al.

Here, we say that a certain property of a system is robust if it is preserved when

an arbitrary (but sufﬁciently small) perturbation affects the system. An important

quantity in this respect is then the distance (measured by an appropriate metric)

between the nominal system and the closest perturbed system that does not possess

the mentioned property, this is typically called the radius of the system property.

In this paper, we deal with robustness and distance problems for differential-

algebraic equations (DAEs), with a focus on robust stability and stability radii. Sys-

tems of DAEs, which are also called descriptor systems in the control literature, are

a very convenient modeling concept in various real-life applications such as me-

chanical multibody systems, electrical circuit simulation, chemical reactions, semi-

discretized partial differential equations, and in general for automatically generated

coupled systems, see [12, 39, 47, 63, 68, 84, 85] and the references therein.

DAEs are generalizations of ordinary differential equations (ODEs) in that cer-

tain algebraic equations constrain the dynamical behavior. Since the dynamics of

DAEs is constrained to a set which often is only given implicitly, many theoreti-

cal and numerical difﬁculties arise, which may lead to a sensitive behavior of the

solution of DAEs to perturbation in the data. The difﬁculties are characterized by

fundamental notions for DAEs such as regularity, index, solution subspace, or hid-

den constraints, which do not arise for ODEs. These properties may be easily lost

when the data are subject to arbitrarily small perturbations. As a consequence, usu-

ally restrictions to the allowed perturbations have to be made, leading to robustness

questions for DAEs that are very different from those for ODEs.

This paper surveys robustness results for linear DAEs with time-invariant or

time-varying coefﬁcients of the form

E(t)˙x(t) =A(t)x(t) +f(t), (1.1)

on the half-line I =[0, ∞), together with an initial condition

x(t

) =x

∈I. (1.2)

Here we assume that E,A ∈C(I, K

n×n

), and f ∈C(I, K

) are sufﬁciently smooth.

We use the notation C(I, K

n×n

) to denote the space of continuous functions from I

to K

n×n

, where K =R or K =C.

Linear systems of the form (1.1) arise directly in many applications and via lin-

earization around solution trajectories [22]. They describe the local behavior in the

neighborhood of a solution for a general implicit nonlinear system of DAEs



t,x(t), ˙x(t)



=0, (1.3)

the constant coefﬁcient case arising in the case of linearization around stationary

solutions.

Deﬁnition 1.1 A function x : I →R

is called a solution of (1.1)ifx ∈ C

(I, R

)

and x satisﬁes (1.1) pointwise. It is called a solution of the initial value problem

(1.1)–(1.2)ifx is a solution of (1.1) and satisﬁes (1.2). An initial condition (1.2)is

called consistent if the corresponding initial value problem has at least one solution.

Robust Stability of Differential-Algebraic Equations 65

We note that, by the tractability index approach [43, 68], the condition on the

smoothness of solutions may be relaxed, namely, only a part of x is required to be

continuously differentiable.

Recall the following classical stability concepts for ordinary differential equa-

tions:

˙x(t) =f



t,x(t)



,t∈I, (1.4)

with initial condition (1.2), see e.g. [55].

Deﬁnition 1.2 A solution x :t →x(t;t

) of (1.4) with initial condition (1.2)is

called

1. stable if for every ε>0 there exists δ>0 such that

(a) the initial value problem (1.4) with initial condition x(t

) =ˆx

is solvable

on I for all ˆx

∈K

with ˆx

−x

<δ;

(b) the solution x(t;t

, ˆx

) satisﬁes x(t;t

, ˆx

) −x(t;t

)<εon I.

2. asymptotically stable if it is stable and there exists ρ>0 such that

(a) the initial value problem (1.4) with initial condition x(t

) =ˆx

is solvable

on I for all ˆx

∈K

with ˆx

−x

<ρ;

(b) the solution x(t;t

, ˆx

) satisﬁes lim

t→∞

x(t;t

, ˆx

) −x(t;t

)=0.

3. exponentially stable if it is stable and exponentially attractive, i.e., if there exist

δ>0, L>0, and γ>0 such that

(a) the initial value problem (1.4) with initial condition x(t

) =ˆx

is solvable

on I for all ˆx

∈K

with ˆx

−x

<δ;

(b) the solution satisﬁes the estimate x(t;t

, ˆx

) − x(t;t

) <Le

−γ(t−t

)

on I.

If δ does not depend on t

, then we say the solution is uniformly (exponentially)

stable.

Note that one can transform the ODE (1.4) in such a way that a given solution

x(t;t

) is mapped to the trivial solution by simply shifting the arguments. When

studying the stability of a selected solution, one may therefore assume without loss

of generality that the selected solution is the trivial solution, and also that t

=0.

In this paper, we restrict the discussion to regular DAEs, i.e., we require that

(1.1)(or(1.3) locally) have a unique solution for sufﬁciently smooth E,A,f (F )

and appropriately chosen (consistent) initial conditions. One can immediately ex-

tend Deﬁnition 1.2 verbatim to regular DAEs. However, one has to be careful with

the initial conditions and the inhomogeneities, since they are restricted due to the

algebraic constraints in the system. This is, in particular, true if one considers the

robustness of the stability concepts under perturbations to the system.

The following examples give an illustration for the possible difﬁculties in the

robustness of the stability concepts for DAEs under small perturbations.

66 N.H. Du et al.

Example 1.1 Consider the homogeneous linear time-invariant DAE





˙x









, (1.5)

which can be written as ˙x

, 0 =x

and has only the trivial solution x

=0.

If we perturb (1.5)byasmallε as





˙x





1 ε





, (1.6)

then solving the second equation of (1.6)forx

and substituting into the ﬁrst equa-

tion, we obtain

˙x

=−(1/ε)x

. (1.7)

Clearly, if ε<0, then the perturbed DAE (1.6) is unstable. If ε>0, then the system

is asymptotically stable, but it qualitatively differs from the solution of the orig-

inal system (1.5). For an arbitrarily prescribed initial value x

(0) = 0, the initial

value problem for (1.7) has a unique solution. Furthermore, the value of x

(0) is not

required and is uniquely determined by x

(0). In fact, this small perturbation has

changed the index of the DAE (1.5), see Deﬁnition 2.2 below.

If we add an inhomogeneity to these DAEs, then more essential differences ap-

pear.

In Example 1.1 the perturbation is affecting only the coefﬁcient A. The situation

is even more complicated if we allow perturbations in the coefﬁcient of ˙x.

Example 1.2 Consider the well-known singularly perturbed system



0 εI



˙x









, (1.8)

where A

, i, j ∈{1, 2}, are constant matrices of appropriate sizes, and ε>0isa

small parameter. Let us assume that A

is invertible. If ε is set to 0, then the leading

matrix becomes singular, i.e., we have a DAE, and we can solve the second equation

for x

, and obtain the so-called underlying ODE

˙x



−A

−1



It is well known that for sufﬁciently small ε, the (asymptotic) stability of (1.8) de-

pends not only on the stability of the so-called slow subsystem associated with the

underlying ODE, but also on that of the so called fast subsystem ˙x

= A

,see

[32, 34, 61], associated with the algebraic equation.

In this example, the rank of the leading matrix is changed, when ε moves from

zero to a nonzero value. In the case ε = 0, the initial condition must be consistent

to ensure the existence of a solution, but obviously this is not required in the case

of a nonzero ε. The difﬁculties increase if A

is singular and/or the leading matrix

involves a singular perturbation of a more general structure.

Robust Stability of Differential-Algebraic Equations 67

The presented examples already indicate some of the possible difﬁculties which

will be discussed in this paper. We present the analysis of robust exponential and

asymptotic stability for linear DAEs with time-invariant or time-varying coefﬁ-

cients. This is a relatively young topic, starting with the work in [20, 82], which

generalized results on the distance to instability and the concept of stability radius

for ODEs in [50] and [92].

The outline of the paper is as follows. In Sect. 2 we summarize recent results

on the robust stability and stability radii for linear time-invariant DAEs. In Sect. 3

we study the robust stability of linear time-varying DAEs. Stability radii, their de-

pendence on the data, and the robustness of stability spectra are analyzed. Some

discussions and topics for future research close the paper.

2 Robust Stability of Linear Time-Invariant DAEs

In this section we study homogeneous linear time-invariant DAEs of the form

E ˙x(t) =Ax(t), t ∈I, (2.1)

where E and A are given constant matrices in K

n×n

, K =C or R.

Deﬁnition 2.1 A matrix pair (E, A), E,A ∈ K

n×n

is called regular if there exists

λ ∈C such that the determinant of (λE −A), denoted by det(λE −A), is different

from zero. Otherwise, if det(λE −A) =0 for all λ ∈C, then we say that (E, A) is

singular.

If (E, A) is regular, then a complex number λ is called a (generalized ﬁnite)

eigenvalue of (E, A) if det(λE −A) =0. The set of all (ﬁnite) eigenvalues of (E, A)

is called the (ﬁnite) spectrum of the pencil (E, A) and denoted by σ(E,A).IfE is

singular and the pair is regular, then we say that (E, A) has the eigenvalue ∞.

In the following we only consider regular pairs (E, A). Such pairs can be trans-

formed to Weierstraß–Kronecker canonical form,see[12, 41, 43], i.e., there exist

nonsingular matrices W,T ∈C

n×n

such that

E =W



0 N



−1

,A=W



J 0

0 I

n−r



−1

, (2.2)

where I

n−r

are identity matrices of indicated size, J ∈ C

r×r

, and N ∈

(n−r)×(n−r)

are matrices in Jordan canonical form and N is nilpotent. If E is

invertible, then r =n, i.e., the second diagonal block does not occur.

Deﬁnition 2.2 Consider a regular pair (E, A) with E, A ∈ K

n×n

in Weierstraß–

Kronecker form (2.2). If r<nand N has nilpotency index ν ∈{1, 2,...}, i.e.,

=0,N

=0fori =1, 2,...,ν −1, then ν is called the index of the pair (E,A)

and the associated DAE (2.1) and we write ind(E, A) = ν.Ifr =n then the DAE

has index ν =0.

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