Manuel de La Sen

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Stability criteria for linear time-invariant systems withpoint delays based on one-dimensional RouthHurwitz tests

M. De la Sen

Department of Electricity and Electronics, Faculty of Science and Technology, University of Basque Country,

Campus of Leioa (Bizkaia), Aptdo. 644, Bilbao, Spain

Abstract

This brief deals with the asymptotic stability of a class of linear time-invariant systems subject to point constant uncom-mensurate delays. Results are obtained dependent on and independent of the delays which may be tested be performing afinite set of 1-D RouthHurwitz tests on a corresponding set of auxiliary delay-free linear time-invariant systems plus somesupplementary conditions related to either commutation pairwise of the associate dynamics matrices or their norm close-ness or some simple testable conditions if the matrices have particular forms and there is only one single delay. Theobtained results are applicable to the case when a finite set of matrices describing the dynamics of a set of delay-free aux-iliary dynamical systems are Hurwitz and either commute pair-wise or their norms are sufficiently close to each other.Some extensions are given to robust stability of point time-delay time invariant systems with commensurate delays param-

etrized by an uncertain parameter vector whose components are in real intervals of known limits. 2006 Elsevier Inc. All rights reserved.

Keywords: Time-delay systems; Uncommensurate delays; Asymptotic stability

1. Introduction

The stability and stabilizability of time-delay systems has received important attention in the last years (see,for instance [115]) including derivation of criteria based on Lyapunov theory [37,12,13] or in the frequencydomain [2,13,14]. A bridge in-between those approaches is taken in [1] and [3] where the analysis for linear

time-invariant time-delay systems subject to point delays is performed via sets of Lyapunov matrix identitiesor inequalities each involving an auxiliary matrix built with an algebraic sum of those describing the delayeddynamics. The set of those auxiliary matrices is constructed for all the set of possible alternating sign combi-nations in their algebraic sum. The stability of such systems might be also discussed under strong sufficiency-type conditions from parallel results obtained in the analysis of 2-D systems (see, for instance, [1619]) byintroducing two variables, namely, the Laplace transform s and l = exp(hs) which are considered as inde-pendent in a bivariate characteristic polynomial p(s,l) being isomorphic to the characteristic quasi-polynomial

0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2006.09.033

E-mail address: [email protected]

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p(s,exp(hs)). The stability problem might be then discussed by applying a bilinear transformation to convertthe complex open left-half plane into the open unit circle for the variable s, or viceversa for the variable l. Inthe first case, the stability is performed for a 2-D auxiliary digital system while in the second one, it is per-formed for a 2-D auxiliary continuous-time system. The obtained stability conditions are only sufficient sincepotentially unstable characteristic zeros have to fulfil the a priori constraint l = exp(hs) since, otherwise,

they are not in fact characteristic zeros of the original system and have to be excluded from the analysis. Thestability of such systems is of interest in a wide class of applications like, for instance, transmission problems,population dynamics, neural networks, active displacement control, sliding mode controller synthesis etc.,[8,13,15,16,2024]. In this brief, the set of 2r (r being the number of delays) Lyapunov matrix inequalitiesof [1] is taken as the basis to derive the set of 1-D sufficiency-type stability tests by having in mind that theexistence of such a set of inequalities implies and is implied by an associate set of 2 r stable auxiliary time-invariant delay-free systems possessing a common Lyapunov function. The practical application of the stabil-ity tests may be performed trough a set of standard 1-D RouthHurwitz stability criteria plus some extraconditions related to pair-wise commutation of matrices of dynamics or their closeness in norm or alternativeconditions simply to test if there is only one single delay present. The existence of such a common Lyapunovfunction is investigated in two cases, namely, when the matrices of dynamics of those systems commute pair-wise and when they are all closed in norm to each other, [25,26]. The dynamics of each of those 2r auxiliary

delay-free systems is built by the algebraic sum for one set of possible alternating sign in the algebraic sum ofthe matrices associated with each of the delays. Some direct extensions are then given to investigate the robuststability of systems parametrized by an uncertain parameter vector whose components belong to real intervalsof known limits. The proposed method seems to be promising for obtaining parallel results being applicable tomore general classes of both dynamic linear and nonlinear systems including both point and distributed delaysas well as to linear hybrid systems potentially including also delays (see, for instance, [2737]).

Notation. The square matrix A is a a-stability matrix (or a a-Hurwitz matrix) if and only if all its eigenvalueshave real parts in Re s < a 6 0. Its characteristic polynomial is said to be a-Hurwitz. A dynamical system issaid to be a-globally asymptotically stable (a-GAS) if and only if all its characteristic zeros are in Re s < -a6 0.The properties a-Hurwitz and a-GAS also apply for characteristic quasi-polynomials of a time-delay systemand for its global asymptotic stability with all characteristic zeros in Re s < a 6 0. A 0-stability matrix is sim-

ply called a stability or Hurwitz matrix. Similarly, a 0-Hurwitz polynomial (or quasi-polynomial) is said to beHurwitz and a 0-GAS system is simply said to be GAS. If s is a positive integer number then s f1; 2; . . . ;sg.

The set of real (real positive) numbers is denoted by R (R+) and R0 R [ f0g.

Mi Mj denotes that the square real or complex n-matrices Mi, Mj commute; i.e. MiMj = MjMi. The set ofpolynomials of real coefficients of degree k is denoted by P(k).

2. Preliminaries

Consider the linear and time-invariant system:

_xt

Xr

k0

Akxt hk; 1

where x(t) 2 Rn the state vector subject to initial conditions x(t) u(t) for t2 h; 0 with h : Maxk2rhk andu : h; 0 ! Rn being any piecewise continuous vector function with x(0) = u(0) = x0. The real and constantmatrices Ak 2 R

nn are related to the dynamics of the delays hkk2 r and h0 = 0. A set of sufficiency-typeasymptotic stability results both dependent on and independent of delays were obtained in [1] which are com-pactly summarized as follows.

Lemma 1. The system (1) is a-GAS if there is a constant real square n-matrix P = PT > 0 such that thesubsequent set of Lyapunov matrix inequalities holds:

AT0 aIPPA0 aI

Xr

k1

bkPAk ATkP

" #m

< 0; 2

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with

bk :eahkM; hk 2 0; hkM; h Maxk2rhkM;

1 if a > 0; hk 2 0;1

&

for m 2 2r denoting all 2r possible cases of alternating sign.

Note that if(1) is a-GAS then it is a0-GAS for any a0 2 [0,a]. Note that the above result includes a-stabilitydependent on the delays (i.e. (1) is a-GAS) when bk> 1 for at least one k2 rand stability independent of thedelays (i.e. (1) is GAS) when bk= 1 for all k2 r. In [1], convexity arguments prove the stability of the matricesQk= A0 + gkAk for gk 2 gkM; gkM k2 r provided that the corner matrices defined at the vertices of thehyper-rectangle:

H : fg g1; g2; . . . ; grT

: gk 2 gkM; gkM \ R; k2 rg 3

are stable. Note that the constraints (2) are a set of 2r Lyapunov matrix inequalities which have to be satisfiedby a common matrix P= PT > 0. This idea is addressed via Lyapunov stability theory what first leads to theimmediate subsequent result:

Lemma 2. Lemma 1 holds if and only if the 2r delay-free auxiliary systems:

_zmt A0 aIXrk1

bkAk

" #m

!zmt m 2 2

r 4

are all GAS while sharing a common Lyapunov function Vzmt zTmtPzmt, P = P

T > 0, m 2 2r. Equivalently,Lemma 1 holds if and only if the 2r delay-free auxiliary systems:

_ymt A0 Xrk1

bkAk

" #m

!ymt m 2 2

r 5

are alla-GAS (i.e. GAS with stability margin (a) 6 0 so that all the characteristic zeros of those auxiliary sys-tems lie in Re s < a 6 0) while sharing a common Lyapunov function

Vymt yTmtPymt; P P

T > 0; m 2 2r:

Proof. It follows directly since the sets of dynamic systems (4) (or (5)) are all GAS [a-GAS] if and only if theyshare a common Lyapunov function with associate matrix inequalities (2).

Note that Lemmas 1 and 2 imply that:

The auxiliary delay-free system _za1 A0za1t is GAS what occurs if (1) is GAS independent of delays; i.e.for hk 2 0;1; k2 r.

The auxiliary delay-free system _za2 P

rk0Akza2t is GAS what occurs if (1) is GAS for the first stabilitydelay intervals ; i.e. if there exist hkM> 0 such that (1) is GAS for hk 2 0; hkM; k2 r. h

Note that Lemma 2, and then Lemma 1, holds if and only if the matrices A0 Pr

k1 bkAk

m

m 2 2r

are Hurwitz and satisfy the Lyapunov matrix inequalities (2) with the same matrix P under conditions dis-cussed in the next section which lead to sufficiency-type conditions to guarantee that the delay system (1) isGAS by the only inspection of this set of matrices.

3. Main result

The main result is enounced as follows:

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Theorem 1. The subsequent items hold:

(i) The system (1) is a-GAS for all delays hk 2 0; hkM; k2 rwith all its characteristic zeros in if all the matri-

ces A0

Pr

k1 eahkMAk

m

m 2 2r are a-Hurwitz and commute pair-wise (so that their associate

delay-free systems possess a common Lyapunov function). The system (1) is GAS independent of the delays

with characteristic zeros in Re s < 0 (i.e. a = 0) if A0 Pr

k1 Ak

m

m 2 2r

are all Hurwitz and com-mute pair-wise.

(ii) The system (1) is a-GAS for all hk 2 b0; hkM; k2 r with all its characteristic zeros in if

A0 Pr

k1 eahkMAk

m

m 2 2r are alla-Hurwitz and, furthermore,

(a) they are either in Jordan canonical form or in real canonical form, or(b) they are all transformed by the same (non-singular transformation) into one of those forms, or(c) r = 1 and A0 and A1 are in companion form with A0. A1 having no real negative eigenvalue. The system (1) is

GAS independent of the delays with characteristic zeros in Re s < 0 if A0 Pr

k1 Ak

m

m 2 2r are

Hurwitz and subject to any of the conditions (a), (b) or (c).(iii) Let A 2 Rn Rn be any real Hurwitz matrix whose associate fundamental matrix norm satisfies

keAtk2 6 Keqt for all tP 0 some real constants KP 1 and a > 0. Then the system (1) is a-GAS for all

hk2 [0,hkM) and given hkMk2 r with a Mink2r ln b

khkM

andbk eahkM for k2 r if any of the constraintsbelow hold:

Maxk2r

kDAkk2 ki1 i 2 r and k0 = 0 being described by

_xt Xri0

Aiqxt kih; tP 0: 9

The following assumption is made.Assumption 1

As; q; h Xri0

Aiqekihs

0 In1

Prn1i0

an1;iqekihs; . . . ;

Pr0i0

a0;iqekihs

24

35;

i.e. A(s, q, h) is in companion form with r : Maxi2n1ri and the coefficients aik(q) depend linearly on theuncertain parameter vector q 2 Q.

Assumption 1 is not restrictive since such a companion form may always be obtainedfrom an external descrip-tion. The characteristic quasi-polynomial associate with (9) isps; q; h sn

Pr

0

Pn1i0 aikqe

khssn1i. Defined characteristic quasi-polynomialspi(s, h) at the corner values of the parametric box Q. Consider the whole non-

polytopic family formed by the union of polytopes Ph for each fix delay h 2 0; h

P :[

h20;h

Ph fps; q; h : q 2 Q; h 2 0; hg [

h20;h

Convex hullfp1s; h;p2s; h; . . . ;pds; hg; 10

with d 6 2s being the number of vertices of p(s, q, h) as q equates each vertex Q(i) of Q. Since the coefficientsaik(q) depend linearly on q but P is not a polytope then the stability of any quasi-polynomial in P is ensuredby the stability at any vertex quasi-polynomial pi(s, 0) for h = 0 if all the exposed edges of two-parameter func-tions of s joining each part of vertices pijs; h;l lpis; h 1 lpjs; h;l 2 0; 1; h 2 0; h; i;j 2

d areHurwitz [2]. Then:

Theorem 2. The following items hold:

(i) The system (9) is a-GAS for all delays hi2 [0,kih), i 2~r, h 2 0;h if all the matrices in the convex set

sa;l : Ai0

Pr1 e

akhA

i

h im

n 1 lA

j0

Pr1 e

akhA

j

h im

: m 2 2r; i;j 2 d

are a-Hurwitz

and commute pair-wise for all reall 2 [0,1]. The system (9) is GAS independent of the delays if the matri-

ces in S(0,l) are Hurwitz for l 2 [0,1] and commute pair-wise.

(ii) Assume a : Maxi2dMaxk2rkAik k2 < c=2e

akhr some aP 0 (with h 1 if a = 0) where

c : Mini2dqi=Ki and qi> 0 and KiP 1 are real constants such that Supl20;1keSia;ltk2 6 Kie

qit for

each Sia;l 2 Sa;li 2 d and all tP 0. Then (9) is a-GAS for hi 2 0; kih; i 2 r. A weaker conditionis that (9) is a-GAS for hi 2 0; kih; i 2 r if

Sup

l20;1Xr

1

eakhlkAik k2 ! 1 lkAjk k2 0, the system (1) isa-GAS with all its characteristic zeros located in Res < a lnb=h for all h1;2 2 0; h. Since h ! 1 impliesthat all the characteristic zeros are in Re s < 0 then (1) is GAS independent of the delays. Now, assume a con-stant A0 with identical companion form as above with a

T0 3:5;13:25;15:75;7:0 there is only one sin-

gle delay h1 2 0; h with associate dynamics defined by the family of matrices A1 A1q; a1q T

parameterized by q with AT1 03; 10 I3 being fixed and aT1 q having corner values defined by (0.042,

0.159, 0.189, 0.084). According to Theorem 2(ii), the system is a-GAS for h1 2 0; h if all the one-para-metrical edge matrices A0 e

ahlAi1 1 lA

j1 are Hurwith for all l 2 [0,1] where each of the 32 vertex

matrices Ai

1 is formed for its corresponding vertex aiT

1 aT

1 Qi

of the vector aT

1 q. Thus, the system isa-GAS with characteristic zeros allocated in Res < a ln1:48=h for all h1 2 0; h and any givenh > 0. It can be pointed out that Theorem 2(i) might be tested through graphical 1D-tests using a result provedin [18] for 2-D systems yielding that the system is a-GAS for h1 2 0; h if all elements in S(a,l) are a-Hurwitz

and the 32 hodographs Gijjx DetjxI4A0e

ahAi

1

DetjxI4A0eahA

j

1

do not cross the point (1, 0] what implies thatAij(jx, h,l)5 0 for all real x.

Acknowledgements

The author is very grateful to the Spanish Ministry of Education and UPV/EHU by their partial support ofthis work through Projects DPI2006-00714/ and 9/UPV/EHU 00I06.I06-EB15263/2003, respectively.


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Appendix A. A.1. Auxiliary results on Lyapunov matrix inequalities

Theorem A.1. The following items hold:

(i) If M1 and M2 are real Hurwitz n-matrices in companion form then there exists a real square n-matrix

P = PT

> 0 such that MTi PPMi < 0; i = 1,2 if and only if M1M2 has no real negative eigenvalue.

(ii) Let Mi i 2 N be a set of Hurwitz real n-matrices. Then they satisfy MTi PPMi < 0 i 2 N if any of the

two conditions below holds:

(C1) Mi Mj; i;j 2 N.(C2) It exists a Hurwitz real square n-matrix M with stability abscissa (q) < (q0) < 0 subject to

keMtk2 6 Keq0t for some real constant KP 1 so that DMi : Mi M i 2 N satisfies the constraint

Maxi2 NkDMik2 < q0=K.

Proof. (i) is proved in [26]. To prove (ii), subject to C1, compute Pi PTi > 0 i 2 N satisfying the set of

Lyapunov matrix equalities MTi Pi PiMi Pi1 i 2 N with P0 = In which satisfy uniquely the recursionPi R1

0

eMTisPi1e

Mis ds since Mii 2 N are all Hurwitz. Mi Mj implies Mi eMjt for all i;j 2 N and tP 0.

Then, the proof follows since the unique solution P: PN of the matrix Lyapunov equationMTNPPMN PN1 satisfies also via direct calculations the set of matrix Lyapunov equations:

MTi PPMi Qi :

Z10

eMTNsMTi PN1Mie

MNs ds < 0 i 2 N 1: A:1

Note that P R1

0eM

TseMs ds is the unique solution to the matrix Lyapunov equation MTP+ PM= In, sinceM is Hurwitz, so that MTi PPMi In DM

TkPPDMk < 0i 2 N if C2 holds since

1 > KkDMik2=q0 P 2kPk2kDMik2 i 2 N. Item (ii), subject to C2 has been proved. h

A.2. Auxiliary results for commutation of matrices

Theorem A.2. The following items hold:

(i) IfMi i 2 N are complex n-matrices in Jordan or real canonical forms then Mi Mj; i;j 2 N.(ii) Let M1 and M2 be complex n-matrices for which a common similarity transformation T exists leading to

canonical Jordan forms Ji= T1MiT (i = 1,2). Then, M1 M2.

(iii) Assume that B is a real n-matrix. If M is a non-derogatory n-matrix then M B if and only if B = p(A) forsome polynomial of real coefficients p 2 P(j) of degree j at most (n 1).

(iv) If A and B are real n-matrices with B = p(A) for some polynomial of real coefficients p 2 P(j) of arbitrarydegree j then M B.

(v) Let MN : fMi : i 2 Ng MN 1 [ fMNg with NP 3 be a set of non-derogatory real n-matrices suchthat Mi Mj for alli;j 2 N 1. Then, MN M(N 1) (a compact definition of MN Mj for allj 2 N 1)if and only if MN M for at least one 2 N 1.

(vi) Let M(N) be defined as in (v). If all the matrices in M(N 1) commute pair-wise then all the matrices inM(N) commute pair-wise if and only if MN M for at least one 2 N 1.

Proof. Items (i)(ii) follow from direct calculus. Item (iii) is proved in [25]. Item (iv) follows from the if partproof of (iii). Items (v) and (vi) are trivially identical from MN M(N 1) and MN MN () MN M(N) sothat only Item (v) is now proved.

If Part of (v). From (iii), Mi Mj for all i;j 2 N 1 and MN M for at least one 2 N 1 with Mi i 2 Nbeing non-derogatory, it follows that MN= pN(M) = pN(pj(Mj)) = pNj(Mj) for at least one 2 N 1, all

j 2 N 1 for some polynomials pN 2 PnN, pj 2 P

nj with 0 6Max(nN,nj) 6 n 1 such that


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pNj pNpj 2 PnNj with the degree constraint 0 6 nNj6 nN + nj6 2(n 1) obtained from direct calculus.

It remains to prove that MN Mj for all j 2 N 1 with 0 6 nNj6 n 1. It is now proved by complete

induction that Mkj Pn1

i0 akij M

ij for some set a

kij 2 R. From the CayleyHamilton theorem, [25], such an

identity holds ifk= n. Direct calculus using the CayleyHamilton theorem to expand Mkj yields that it holdsfor k+ 1 if it holds for any integer kP n since:

Mk1j Xn1i0

akij M

i1j

Xn1i0

aki1;jM

ij

Xn1i0

akn1;ja

nij M

ij

Xn1i0

ak1ij M

ij A:2

for all j 2 N 1 with ak1ij a

ki1;j a

kn1;ja

nij with a

k1j 0. Now,

MN pNjMj :Xn1i0

anNjiMij

XnNjin

Xn10

anNjiaij M

j

Xn10

anNj XnNjin

anNjiaij M

j

so that pNj 2 PnNjwith 0 6 nNj6 n 1 with nNj= n 1 if the leading coefficient is not zero.

Only If Part of (v). Since all matrix in M(N) is non-derogatory then MN M(N) is false if there is noj 2 N 1 such that MN Mj. h

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