Perturbation of Matrix Pairs: Hermite Indices*

Copyright © IFAC System, Structure and Control Oaxaca, Mexico, USA, 8-10 December 2004

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PERTURBATION OF MATRIX PAIRS: HERMITE INDICES * I. Baragaiia

1

V. Fermindez 2

Departamento de Ciencia de la Computacion e lA. Universidad del Pais Vasco. Apdo. 649. 20080. Donostia-San Sebastian. Spain I. Zaballa 3

Departamento de Matematica Aplicada y £10. Universidad del Pats Vasco. Apdo. 644. 48080. Bilbao . Spain

Abstract: \Ve study the relationship hetweeen the Hermite Indices and the Jordan st.ructure of a pair of matrices (A, B) E IF7Ixn X IF"xm (for IF = IR or C) and those of all pairs (A', B') obtained by small aditive perturbatiolls 011 (A, B). Copyright © 2004 IFAC Keywords: Controllability. Perturbation. Invariants. Eigenvalues. Norms .

INTRODUCTION, NOTATION AXD

use IFnxm and IF[s]" xm to denote the set of n x In matrices over IF and IF[s], respectively. Gin (IF) denotes the linear group of order n over IF .

PRELIMINARY RESULTS Consider the following system of differential equations with control:

i:(t)

We will use Greek letters to denote polynomials. If a E IF[s] we will write d(a) for the degree of a and A(a) for the set of roots in C of the polynomial a. If a E IR[s] we will denote by r(a) the sum of the multiplicities of the real roots of the polynomial

= Ax(t) + Bu(t)

where A E IFn xfI , B E IF"xm, IF being the field of real or complex numbers. We will identify the system with the matrix pair (A, B) .

a.

We will use the Hermite -indices as introduced in (Zaballa, 1997). Given a matrix pair (A, B) E IFl1 xn X IF"xm, the matrix

IF[s] will denote the ring of polynomials in the indeterminate s with coefficients in IF . We will

* Partially supported by MCYT , Proyecto de Invest igad6n BFM2001-0081-C03-01 and UPY /EHU, Proyecto de Investigaci6n 9/UPYOOlOO.310-14578/2002 1 email: ccpbagaiCsi . ehu.es 2 email: ccpfegovCsi.ehu.es 3 email: mepzatejClg . ehu.es

where bi E IF nx1 is the ith column of B, will be called the Hermite controllability matrix of

49

(A, B). We will say that (.4, B) is controllable if rank H(A, B) = n .

( ii)

~ [T ~:: •.. ::::, 1 B1111

If rank H(A, B) = r and we select from left to right the first r linearly independent columns in

B,

H(A, B) and we write them as

then hI"", h m are the Hermite indices of the pair (we agree that hi = 0 if the column bi has not been selected).

RH

~ [r1EF''''

B2 = PBI·

If we write for i

IJ'O ] .r), 1 [

In (Popov, 1978) Popov gives a complete system of invariants for the similarity of controllable matrix pairs , but it is not the only complete system of invariants that one can obtain (see (Kailath , 1980, Ch. 6) or (ZabaIla, 1997)).

hi>

0,

if h j > 0, i < j ,

Two matrix pairs (AI , Bd, (A 2 , B 2 ) E JFlIxn X JF1Ixm are similar, and we write (AI, Bd~(A2' B 2 ) , if there exists P E Cl" (IF) such that A2 = PAIP- I ,

if

.fJ'~'-1

E

' JF h i X1 IIj' f=l O ,1' < _ J,

and we agree that if a block has 0 rows or 0 columns. then it vanishes.

If the pair (.4, B) is not controllable, then the Hermite indices are also invariant under similarity. Furthermore, if we call invariant factors of (11, B) those of the pol.vnomial mat rix [sI - A BJ , then these polynomials are also invariant under similarity. Notice that the invariant factors of (A, B) are equal to 1 if and only if the pair (A, B) is controllable (Rosenbrock, H)70).

= 1, ... , m

then, it was proved in (Zaballa, 1997) , following Popov 's idea.~, that the Hermite indices and the scalars .fijl E IF form a complete system of invariants for the similarity of controllable matrix pairs. tdoreover, a canonical form for this relation was provided (see also (Hinrichsen and PriitzelvVolters, 1983; l\Iayne, 1972)).

In Lemma 1, block Aii is a companion matrix of the poly nomial ()i = s h; - X;,hi_ISh;-1 IiitS - X;iO. i = 1, ... , rn. These polynomials will be called t he diagonal Herrnit e polynomials of (AB) .

Lemma 1. Let (A, B) E JF" X1I X JFn xllI be a controllable pair and let hi , . ... h", be its Hermite indices. Then there exists PE CL,,(JF) such that (PAP-I,PB) = (Ac,Bc). where

We will present now the so-called Kalman decomposition of a matrix pair (Basile and I\Iarro, 1992. p.143)

(i) All Al2

I1c = [

o An .

.

Aim] A 2111

o

0

A n l'1n

o0

Aii =

... 0 1 0 .. . 0 0 1 ... 0

o0 Aij =

.. . 1

00 .. . 0 o 0 ... 0 0 0 ... 0

o0

.. . 0

Lemma 2. Let (A. B) E JF" x"xJF" xm and assume that rank H(A, B) = r. Then there exists P E Cl" (JF) such that

(PAP-I.PB) = ([

~l ~~], [~l]) ,

.[iiO

where (AI, Bd E lF rxr x JFr .
Xiii Xii2

Xiihi-l

In general a matrix pair (A. B) may admit many different Kalman decompositions but the sizes of the diagonal blocks Al and A3 in all of them are fixed by rank H(A . B) . j\Ioreover, given any Kalman decomposition of a pair (A , Bl , t.he controllable part (A I, B I) a nd the square block

XjiO

Ijd

:Lji 2

E

JFhi < hj,

i < j,

:rjihi -I

50

zero. The length of a partition a is the number of its components ai different from zero. vVe denote by l(a) the length of a. If a and b are partitions, a + b is the partition whose ith component is ai +b i ·

A3 are determined up to similarity by (A, B) (Zaballa, 1982) . Therefore, the diagonal Hermite polynomials of (A[, Bd are invariant under similarity and, in this paper, will be called diagonal Hermite polynomials of (A , B). On the other hand, the invariant factors different from 1 of (A, B) and those of A3 coincide (Zaballa, 1988) .

Let a and b be partitions and n:= max{l(a),l(b)} . Following (Marshall and Olkin, 1979), we will say that a is majorized by b and write a -< b if

Let (A. B) E IFTl xn x IFTl x m. a complex number >. is said to be an eigenvalue of (A, B) if >. is an eigenvalue of A3 in a Kalman decomposition of (A , B). We will denote by A(A , B) the set of the eigenvalues of (A. B). Similarly the characteristic polynomial of (A. B) will be the characteristic polynomial of A 3 . r.,[oreover, the Segre characteristic corresponding to >. E A(A. B) will be the Segre characteristic corresponding to >. as an eigenvalue of A3 and the algebraic multiplicity of >. E A(A, B) will be the algebraic multiplicity of >. as an eigenvalue of A3 and we will denote it by 1n(A.B)(>') . That is to say. if 0'" 1 . . . 1 0'[ are the invariant factors of (A, B) and

k

L j=l

j ,

=

1, . ..• n. and equality holds

= n.

Lemma 3. (Helmke. 1986, proof of Th. "1.2) Let (A. B) E IF" Xll X IF" x m be a pair with hi ..... h", as Hermite indices. There exists E > 0 such that if 11 [A B ]- [A' B' ]II< E ancl h~, .... h;" are the Hermite indices of (A'. B') , then

1=[

then the Segre characteristic of >'i is (8 i l, · ··.8;T/) and m(A.B)(>';) = L~'= l Sij , i = L .. . ,p o We agree that if >. rt A(A, B), then m (.-I.B)(>') = 0 and (0. 0 • ... ) is the partition corresponding to the Segre characteristic of A.

i

Lhj:S: Lhj, i = )= 1

Given A = (aij) E IF" xm we are going to consider the following matrix norm:

l , .. . , m.

(1)

)=1

I\Ioreover, if (A, B) is controllable, then so is (A'. B') and

laijl

IH

1H

j=l

)=1

Lhj = Lhj.

i.j

The set IF" x m is a metric space with the distance associated to this norm .

(2)

When IF = C , the conditions of the previous lemma are sufficient for the existence of matrix pairs with prescribed Hermite indices in any neighborhood of a given pair.

If :; E C and r is a positive real number, the open ball of C centered in 0' and radius r is denoted by B(z , r).

Let (A . B) E IF7IXl1 X IFl1 X"" A(A , B) = {AI, .. .• >'u} and Jet 77 be a positive real number. We define the l/-neighborhood of the spectrum of (A. B) as the set VI)(A, B) := Ui'=l B(>'i, 1/) whenever the balls B(>';,7]). i = pairwise disjoint.

k

This paper has been motivated by (Gracia et al. , 1989) and (Helmke, 1986). In the last one the author studies the topological properties of the controllable systems under similarity. Among other more general results, he gives conditions that have necessarily to satisfy the Hermite indices of all pairs that are close enough to a given pair.

OJ = IT(s - A;)S ij , j = 1• .. . , n

L

Lb )=1

for k

p

11 A 11=

k

:s:

Uj

L ... u are

Lemma 4. (Helmke, 1986, Th. 4.2) Let (A . B) E C" X 11 X X m be a controllable pair wi th hi, ... . hill as Hermite indices and let h~, ... . h'rr, he nOIlnegative integers. For all E > 0 there exists a controllable matrix pair (A' , B') E C" Xll X C', x m such that

cn

(i) (ii)

From now on T) will always mean a positive real number small enough for the expresion 7] - neighborhood of the spectrum of (A, B) to make sense.

11

[A B]- [A' B']II<

h~,

E,

. . . , h;1I are the Hermite indices of (A', B').

if and only if conditions (1) and (2) hold.

In general. a partition is an infinite sequence = (a1,02 •... ) of non negative integers almost all

In the present work we generalize the above results for IF = R or C and for noncontrollable systems.

a

51

1. NECESSARY CONDITIONS

This theorem generalizes the result in Lemma 3 to noncontrollable systems. Following step by step the idea." of (Gracia et al.• 1989) one can prove that if IF = C. conditions Oi) and (iii) of the above theorem are enough to guarantee the existence of a pair (A', B') as close to (A. B) as one may want with h~ .... , h;n as Hermite indices and s:j as the Segre characteristic of their eigenvalues, all these being in a '7-neighborhood of (A, B). This is, however, no longer true if IF = lR as shown in (Helmke, 1986). Our goal is to find a more general set of necessary conditions, good enough to guarantee the existence of such matrix pair (A', B') when IF = lR or C. The following theorem provides sllch a set of conditions.

Lemma 5. (Gracia et aI., 1989, Th 4.3) Let (A, B) E lFT,xn X IF,,xm and '1'1 > O. Let A E A(A. B) and let s be the partition corresponding to A in the Segre characteristic of (A, B). There exists E > 0 such that if 11 [A B] - [A' B'] 11 < E, then

(i) A(A', B') c V'1(A. B) and (ii) if 111,"" I1t are the eigenvalues of (A', B') in B(A, TJ) and is the partition corresponding to l1i in the Segre characteristic of (A',B'), i = 1, .... t, then there exists a non negative integer q 2: 0 such that

s;

Theorem 7. Let (A,B) E IF" XlI X IF"xm , 11> 0 and let el , ... , em be the diagonal Her'mite polynomials of (A,B). Let A(A,B) = {Al, .... Ap } and let Si be the partition corresponding to Ai, i = L . . . , p in the Segre characteristic of (A, B). There exists E > 0 such that if 11 [A B] [A' B'] 11< E, then conditions (i) and (ii) of Theorem 6 and t.he following condition hold

t

s

-<

2:.5; + (q,O, .. .). j=l

Remarks. 1. Condition (ii) implies

2:

(iii) if h~ , . .. , h"" are the Hermite indices of (At, B'), then there exist monic polynomials /31, ... ,Hm E IF[s] such that d(/3i) = I 1, ... ,m and

m(A' .B')(I1) :::: m(A.B)(A).

/1.EB(.\.7/)ni\(A' .B')

,<.

2. If (.4. B) E lRn~" x lR'I"m and A E C \

lR, then 711 •... , ill are the eigenvalues of (A',B') in BC);. 17) and s; is the partition corresponding to 71i in the Segre characteristic of (A'. B'), i = 1•... , t.

/3j ... /3/11

1

().1 . ..

emu.!, j

=

2, ... , m,

(5)

/31' .. /311' = el ... emu.!,

The following theorem is a consequence of Lemma 3 and Lemma 5.

Sketch of the proof.

Theorem 6. Let (A, B) E IF" xn X lF nxm be a pair with hi. ...• h/1l as Hermite indices and '7 > O. Let A(A,B) = {AI, ...• Ap } and let Si be the partition corresponding to Ai in the Segre characteristic of (A, B), i = 1, ... ,po There exists E > 0 such that if 11 [A B] - [A' B'] 11< E, then

Let B = [b l Bj:= [b l

(i) A(A', B') c V,,(A, B), (ii) if l1il, ' .. ,l1it, are the eigenvalues of (A', B') in B(Ai,II), i = 1, ... ,p and S;j is the partition corresponding to l1ij, i = 1, ... , p , j = 1, ... , ti in the Segre characteristic of (A', B'), then there exist nonnegative integers qi 2: 0, i = 1, . .. ,p such that

...

Let m{ :=

...

bm]. For j

1, . ..• 111.,

put

O, ... ,m, i

=

bj ].

m(A.B,)(A;.),

j

=

l, ... ,p+q.

By Lemma 5, for j = 0•... , m there exists Ej > 0 such that if (A,B j ) satisfies 11 [A,Bj] - [A,B j ] 11< Ej, then

t;

Si

-<

2:

S;j

+ (qi, 0, ... ),

i

=

2:

1, ... , p, (3)

j=1

m(A .B))(I1)::;

m{,i = 1, ... ,p+q.

,LEB(.\i ,'1)nA(A.B) )

(iii) if h~, . .. , h~, are the Hermite indices of (A' , B'), then (1) and m

-m

;=1

,=1

Let

]J

2: h'; - 2: hi = 2:qi.

E

= min (Ej), and let (A', B') 0<;)<;".

E

!F/xm be such that 11 [A' B'] - [A B] 11< (A', B') satisfies conditions (i) and Oi).

(4)

52

lF nxlI x E.

Then

Let B' = [b'l ... b~,] and for j = 1, ... , m , put

Bj

:= [b~

(iii) the Hermite index of (A', B') is h' dew) .

.. . bj].

For i = 1, ... , p

+ q,

j = 0, ... , m ., let

L

ni :=

h+

In the next theorem we will generalize this result when IF'=C.

m(A' ,Bj)(Il),

IIEB(A, ,1/)nA(A' .Bj)

Theorem 10. Let (A, B) E c nxn x c"xm he a pair with hi,· .. , h m as Hermite indices and an I .. . I 01 as invariant factors. Let 0;, . .. , w E q8] be monic polynomials such that a~ 1... 1a~ and let h;, ...,h:" he nonnegative integers. If (7), (1) and

a:"

Then /31 " ..

•

,/3m satisfy (iii).

For the controllable case we have the following corollary. Corollary 8. Let (A, B) E !Rn x" X !Rn x III be a controllable pair with 0 1 , ... ,0", as diagonal Hermite polynomials. There exists E > 0 such that if 11 [A B] ~ [A' B']II< E and h~, . .. , h~, are the Hermite indices of (A', B'), then there exist monic polynomials PI, .. . , .3 ", such t.hat d(;J;) = h~ , I = 1, ... , m. and

3 j ···3", 10j ... Om, j = 2, .. .. In , 3 1 " ./3", = (h· · 'Om '

In.

111

j=1

j=1

Lhj ~ Lh

j

=

d(",,),

then for all E > 0 there exists (A', B') E C" X " X c" X III such that

(i) 11 [A' B'] ~ [A B]II< f. (ii) h;, ... , h;" are the Hennite indices of (A', B'), (iii) a;, ... ,n:, are the invariant factors of (A',B')

(6)

Our next result is the near-converse of Theorem 6 when IF = C. Theorem 11. Let (A,B) E C" >( " X C">(1I1 be a pair with hI, ... , h m as Hermite indices. Let A(A, B) = PI, ... , Ap} and let 8i he the partition corresponding to Ai in the Segre characteristic of (A,B) , i = L .... p . Let S';j, i = l, .. . , p, j = 1, .... ti he given partitions. Let h~, . ... h;" he nonnegative integers.

2. SUFFICIENT CO;\lDITIONS

This section is devoted to proving near-converses of Theorem 6 and Theorem 7. The proof is made by induction on m. Given a matrix pair (A, B), in (Gracia et al., 1989, Theorem 5.6, 5.7 and remarks that follow them) sufficient conditions are provided for the existence of a matrix pair (A', B') as close to (A, Bl as desired and such that (A', B') has prescribed controllability indices and invariant factors. Bearing in mind that when m = 1 the controllability index and the Hermite index coincide, the following lemma is just this result for the case m. = 1.

For all f > 0 there exists a matrix pair (A', B') E C" >( " X cn xm such that

(i) 11 [A' B'] ~ [A B]II< f., (ii) A(A', B') C V,/(A. B), (iii) (A', B') has ti eigenvalues J-lil, . .. , Illt, in B(Ai,7]) and S';j is the partition corresponding to J-lij in the Segre characteristic of (A' , B'), j = 1, .. .. t;, i = l. .... p, (iv) h;, ... , h;" are the Hermite indices of (A', B') .

Lemma 9. Let (A, B) E IF" x" x IFnxI he a pair with h as Hermite index and an I .. . I 0'1 as invariant factors. Let a~, ... , a:" W E IF[8] be I ... I If monic polynomials such that

a:,

01'" ak

I a~ ... a~w , , k

al"

' a"

= L .. .,n ~ L

,

= n 1 ·· · anw,

if and only if (1) and there exist llonnegative integers qi , 'i = 1, ... ,p, such that (3) and (4) hold.

a;.

(7)

The conditions in the previous theorem are not sufficient in the real case. In fact, if IF' = !R Theorem 7 gives us stronger conditions. In the next lemma, we present a similar result to that of Lemma 9. Here we precribe, instead of the Hermite index of (A', B'), the number of real roots of its diagonal Herrnite polynomial.

then for all E > 0 there exists (A', B') E IF'n xn x IF'"xl such that

(i) 11 [A' B']- [A B] II< E, (ii) the invariant factors of (A', B') are

0;, ... ,a:" 53

Lemma 12. Let (A, B) E 1R" xn X IR nx1 be a matrix pair with 8 as diagonal Hermite polynomial and a 1l I .. . I al as invariant factors. Let a~, ... , a:, ' w E lR [s] monic polynomials such that a~ I ... I a~ . If (7) holds, then for all t > 0 there exists (A',B') E lR"xn x IR nx1 such that

(ii) A(A', B') c V,,(A, B) , (iii) (A', B') has ti eigenvalues /-til , "" /-titi in B(Ai , l]) and .s:j is the partition corresponding to /-tij in the Segre characteristic of

(i) /I [A' B'] - [A B] /1< E, (ii) the diagonal Hermite polynomial of (A', B') is (lw, where rev) = r(8) and d(iJ) = d(e), (iii) the invariant factors of (A', B') are a;, ... , a~.

if and only if the following conditions hold

(A',B'), j=l, ... ,ti' i=l , . . . ,p, (iv) h;, ... , h:n are the Hermite indices of (A', B'),

With the help of this lemma, we can solve by induction on m the controllable case.

(a) there exist nonnegative integers qi 1, ... , p such that (3) holds, (b) there exist monic polynomials /31 , ... , Pm E R[8] such that d({3j) = hj, j = 1, . . . ,In and p

(5) holds, where w :=

IT (.5 -

Ai)q,.

i=l

Theorem 13. Let (A, B) E IR n x ll x IR"X1ll be a controllable pair and let el, ... , em be the diagonal Hermite polynomials of (A, B). Let h~ . ... ,h:" be nonnegative integers. If t.here exist monic polynomials i3l , ... ,,3'" E lR[s] such that d(.3i ) = h;, i = L ... , m and (6) holds, then for all t > 0 thE're exists a controllable matrix pair (A', E') E 1R" Xl X lR"xm such that (i)

/I [A B]- [A' B']II<

REFERENCES

t,

(ii) (A' , B') has h;, ... , h:/! as Hermite indices. Theorem 14. Let (A, B) E lR" xn X lR" x 'III be a pair with el , . .. , em as diagonal Hermite polynomials and a" I .. . 1 etl as invariant factors. Let a; .... , a~, w E lR[.s] be monic polynomials such that 1 ... 1 a; and let h'I" ' " h~, be nOllnegative integers. If there exist manic polynomials ;3r, ... ,Pm E lR[s] such that d(pj) = hj, j = 1, . .. ,m and (5) and (7) hold, then for all t > 0 there exists (A' , B') E IR n x lI x lR" x rn such that

a:,

(i) 11 [A' B']- [A B]/I< c (ii) h~, ... , h:1I are the Hermite indices of (A', B'), (iii) a'I, .. . . are the invariant factors of (A', B').

a;,

Our next result is near-converse of Theorem 7. Theorem 15. Let (A, B) E lR" x n x lR"xm with as diagonal Hermite polynomials and let I] > O. Let A(A, B) = P'l,"" Ap} and let .si be the partition corresponding to Ai in the Segre characteristic of (A, B), i = L ... , p. Let 8:), i = 1, ... , p, j = 1, ... , ti be given partitions such that if :\, = Al then ti = tl and 8~j = 8;J' j = L .... t;. Let h~ .. .. , h;n be nonnegative integers.

el , ... , em

For all E > 0 there exists (A',B') E 1R1I x 1I such that

(i) /I [A' B'] - [A B]II<

X

1R1Ixm

t,

54

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