Characterizing matrices with eigenvalues in an LMI region A dissipative-Hamiltonian approach Neelam ChoudharyNicolas GillisPunit Sharma

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Characterizing matrices with eigenvalues in an LMI region:

A dissipative-Hamiltonian approach

Neelam Choudhary∗Nicolas Gillis†Punit Sharma‡

Abstract

In this paper, we provide a dissipative Hamiltonian (DH) characterization for the set of matrices

whose eigenvalues belong to a given LMI region. This characterization is a generalization of that of

Choudhary et al. (Numer. Linear Algebra Appl, 2020) to any LMI region. It can be used in various

contexts, which we illustrate on the nearest Ω-stable matrix problem: given an LMI region Ω ⊆C

and a matrix A∈Cn,n, ﬁnd the nearest matrix to Awhose eigenvalues belong to Ω. Finally, we

generalize our characterization to more general regions that can be expressed using LMIs involving

complex matrices.

Keywords: Ω-stability, linear matrix inequalities, dissipative-Hamiltonian systems, nearest stable

matrix

1 Introduction

In this paper, we study matrices A∈Rn,n whose eigenvalues belong to a subset of the complex plane,

Ω⊆C, such matrices are called Ω-stable.

Deﬁnition 1. (Ω-stability) For Ω⊆C, the matrix A∈Rn,n is said to be Ω-stable if every eigenvalue

of Alie inside the region Ω.

The two most famous examples of Ω-stable matrices are Hurwitz stable matrices for which Ω =

{z∈C: Re z < 0}, and Schur stable matrices for which Ω = {z∈C:|z|<1}. Hurwitz stable

matrices play a signiﬁcant role in the study of linear time-invariant (LTI) systems of the form

˙x(t) = Ax(t) + Bu(t), y(t) = Cx(t),

where, for all t∈R,x(t)∈Rn,u(t)∈Rm,y(t)∈Rp,A∈Rn,n,B∈Rn,m, and C∈Rp,n. In fact, such

a system is stable if Ais Hurwitz stable. Moreover, the transient response of such a system is directly

related to the location of its poles [3] in the complex plane. The poles in a speciﬁc region in the

∗School of Engineering and Applied Sciences, Department of Mathematics, Bennett University, Greater Noida-201310,

Uttar Pradesh, India; neelam.choudhary@bennett.edu.in.

†Department of Mathematics and Operational Research, Facult´e Polytechnique, Universit´e de Mons, Rue de

Houdain 9, 7000 Mons, Belgium; nicolas.gillis@umons.ac.be. N. Gillis acknowledges the support by the Fonds

de la Recherche Scientiﬁque - FNRS and the Fonds Wetenschappelijk Onderzoek - Vlanderen (FWO) under EOS Project

no O005318F-RG47, and by the Francqui Foundation.

‡Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi-110016, India;

punit.sharma@maths.iitd.ac.in. P. Sharma acknowledges the support of the DST-Inspire Faculty Award (MI01807-G)

by the Government of India and Institute SEED Grant (NPN5R) by IIT Delhi.

arXiv:2210.07326v1 [math.OC] 13 Oct 2022

complex plane can bound the maximum overshoot, the frequency of oscillatory modes, the delay time,

the rise time, and the settling time. The problem of locating all the closed-loop poles of a controlled

system inside a speciﬁc region Ω ⊆Cis known as the Ω-pole placement problem and has appeared in

several applications [3, 13, 1, 23, 4, 24].

For these reasons, characterizing Ω-stable matrices is an important topic in numerical linear algebra

and control. In this paper, we focus on regions of the complex plane that can be expressed by linear

matrix inequalities (LMIs). Such sets are referred to as LMI regions [3] and deﬁned as follows.

Deﬁnition 2 (LMI Region, [3]).A subset Ω⊆Cis called an LMI region if there exists a symmetric

matrix B∈Rs,s and a matrix C∈Rs,s such that

Ω = {z∈C:fΩ(z)≺0},(1.1)

where

fΩ:C7→ Hs,s is given by z7→ fΩ(z) := B+zC +zCT,(1.2)

where Hs,s is the set of Hermitian matrices with real eigenvalues, that is, Hs,s ={X∈Cs,s :X=X∗}

with X∗the conjugate transpose of X, and for X∈Hs,s,X≺0means that Xis negative deﬁnite,

that is, its eigenvalues are negative.

The function fΩ(z) is called the characteristic function of the LMI region Ω, and sis called the order

of fΩ(z). The characteristic function of an LMI region is not unique [15]. Since fΩ(z)=(fΩ(z))T,

any LMI region is symmetric along the real axis. An LMI region is convex, and so is the intersection

of two or more LMI regions. Due to the strict inequality “≺” in (1.1), the LMI regions are open.

Furthermore, the LMI regions are invariant under congruence transformations. We refer to [15] for

other topological and geometrical properties of the LMI regions. A large number of subsets in the

complex plane can be expressed as LMI regions; for example, conic sectors, vertical half-planes, vertical

strips, discs, horizontal strips, ellipses, parabolas, and hyperbolic sectors; see [2] and Section 3. The

set of LMI regions is dense in the set of convex regions symmetric to the real axis, which are relevant

for control systems [3, 2].

In [9], authors characterized Ω-stable matrices using the so-called dissipative Hamiltonian (DH)

matrices.

Deﬁnition 3 (DH matrix).A matrix A∈Rn,n is said to be a DH matrix if A= (J−R)Qfor some

J, R, Q ∈Rn,n such that JT=−J,R0, and Q0.

A DH matrix is always Hurwitz stable, that is, all its eigenvalues are in the left half of the complex

plane [9]. The term DH is inspired by the DH systems in which the state matrix has the form

A= (J−R)Q, where JT=−Jis the structure matrix describing the ﬂux among energy storage

elements, Ris a positive semideﬁnite matrix describing the energy dissipation in the system, and Qis

a positive deﬁnite matrix that describes the energy of the system [21, 22]. By replacing the constraint

on R, namely R0, by other LMI constraints on (J, R, Q), DH matrices can represent diﬀerent types

of Ω-stable matrices. This was studied in [5] where Ω-stable matrices were written as DH matrices

where Ω could be vertical strips, disks, conic sectors, and their intersection; see the next section for

more details. An application of these characterizations is to solve the nearest Ω-stable matrix problem.

For example, in system identiﬁcation, one needs to identify a Ω-stable system from observations [19].

In fact, sometimes numerical or modelling errors or approximation processes may produce an unstable

system in place of a stable one. The unstable system then has to be approximated by a nearby stable

system without perturbing its entries too much. More precisely, for a region Ω ∈Cand a matrix

A∈Rn,n, this requires to solve the following optimization problem

inf

X∈SΩ

kA−Xk2

F,(1.3)

where k·kFstands for the Frobenius norm and SΩis the set of all Ω-stable matrices. The DH

characterization of stable matrices has been proven very eﬀective in solving several nearness problems

for LTI control systems. For example, distance to Ω-stability [9, 5, 17], nearest admissible descriptor

system problem [8], distance to passivity [10], minimal-norm-static-state feedback problem [11], and

learning data-driven stable Koopman operators [16]. This DH characterization was also used recently

to design an optimization-based algorithm for parametric model order reduction of LTI dynamical

systems [20].

Contribution and outline of the paper This paper is organized as follows. In Section 2, for

a given LMI region Ω ⊆C, we characterize the set of all Ω-stable matrices as DH matrices of the

form A= (J−R)Qwith LMI constraints on the triplet (J, R, Q)∈(Rn,n)3. This characterization

generalizes the work on [5] that only considered three types of LMI regions, namely conic sectors,

vertical strips and disks. In Section 3, we provide several examples of such LMI regions, including

parabolas, ellipsoids, hyperbolas and horizontal strips; this is the ﬁrst time DH characterizations of

such regions are given via LMIs on the triplet (J, R, Q). In Section 4, we illustrate the use of these

characterizations to solve the nearest Ω-stable matrix problem (1.3). Finally, in Section 5, we extend

our characterizations of LMI regions for complex matrices, where the set Ω is not necessarily symmetric

with respect to the real line.

Notation Throughout the paper, XTand kXkstand for the transpose and the spectral norm of a

real matrix X, respectively. We write X0 (X≺0) and X0 (X0) if Xis symmetric and

positive deﬁnite (negative deﬁnite) or positive semideﬁnite (negative semideﬁnite), respectively. By

Imwe denote the identity matrix of size m×m. The Kronecker product is represented by ⊗and we

refer to [14] for the standard properties of the Kronecker product. The set of n×nHermitian matrices

is denoted by Hn,n.

2 DH characterization of matrices with eigenvalues in generic LMI

regions

In this section, we consider matrices with eigenvalues in some generic LMI regions, and for them, we

provide a parametrization using the DH matrices. This will allow us for example in Section 4 to use

standard optimization tools to ﬁnd a nearby matrix to a given matrix with eigenvalues all in the given

LMI region.

The following result from [3] is crucial for our DH formulation of the set of Ω-stable matrices,

where Ω is an LMI region.

Theorem 1. [3, Theorem 2.2] Let Ωbe an LMI region given by (1.1) and let A∈Rn,n. Then Ais

Ω-stable if and only if there exists a symmetric matrix X∈Rn,n such that X0and

MΩ(A, X) := B⊗X+C⊗(AX) + CT⊗(AX)T≺0.

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CharacterizingmatriceswitheigenvaluesinanLMIregion:Adissipative-HamiltonianapproachNeelamChoudhary*NicolasGillisPunitSharmaAbstractInthispaper,weprovideadissipativeHamiltonian(DH)characterizationforthesetofmatriceswhoseeigenvaluesbelongtoagivenLMIregion.Thischaracterizationisageneralizationofthato...

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Characterizing matrices with eigenvalues in an LMI region A dissipative-Hamiltonian approach Neelam ChoudharyNicolas GillisPunit Sharma

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