Eigenvalue-eigenvector sensitivity formulae for linear descriptor systems

Systems & Control Letters 15 (1990) 227-235 North-Holland

227

Eigenvalue-eigenvector sensitivity formulae for linear descriptor systems Jacek Kabziflski Institute of Automatic Control, TU Lbd~, Stefanowskiego 18/22, PL 90-924, Poland Received 9 April 1990 Revised 11 July 1990

Abstract: This paper considers the problem of eigenvalue-eigenvector sensitivity analysis of linear multivariable descriptor systems. The formulae for derivatives of eigenvalues and eigenvectors of a parameterized matrix pencil are derived using only elementary eigenvalue-eigenvector calculus. The problem of selecting suitable eigenvectors normalization is considered. Results for regular state-space systems follow as a special case.

Keywords: Matrix algebra; linear systems; descriptor systems; eigenvalue-eigenvector sensitivity.

1. Introduction

Eigensensitivity has been a topic of research interest in linear system theory for a long time. There are several formulae for computations of first- and second-order sensitivities of eigenvalues and eigenvectors. For example Morgan [9] and Crossley and Porter [3] studied sensitivity to a single matrix entry variations. Paraskevopoulos et al. derived the sensitivity formulae expanding the characteristic polynomial in Taylor series [11,12], and in [8] an algebra of Kronecker products is used. Kalaba et al. [5] developed a complete system of differential equations for eigenvalues and eigenvectors of a parameterized matrix. The purpose of this paper is to generalize results from [5] and derive formulae applicable to eigenstructure sensitivity analysis of descriptor systems. The formulae for derivatives of eigenvalues and eigenvectors of a parameterized matrix pencil are derived using only elementary eigenvalue-eigenvector calculus. The problem of selecting a suitable eigenvector normalisation is considered. Results for regular state-space systems follow as a special case. The formulae developed in this paper can be utilized in many control problems and some of possible applications are mentioned in Section 7, although a more elaborate study of these problems is not the subject of the present paper. The presentation starts in Section 2 from some basic facts from descriptor systems theory. The proofs are given in [6] and [7] and the references there.

2. Preliminaries

Consider a time-invariant linear descriptor system described by the equation E(p)~(t)

= A(p)x(t)

(2.1)

where x ( t ) is the n-dimensional state vector and E ( p ) , A ( p ) are n × n complex, matrix-valued, analytical functions of a complex parameter p. Assume that for a certain P0, 0 < rank E ( p o ) = deg [ s E ( p o ) - A ( p o ) [ = m <_ n ,

(2.2)

nullity E ( po ) = 1 = n - m ,

(2.3)

0167-6911/90/$3.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

J. Kabzihski / Eigem;alue-eigenvector sensitivity

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and that the regular matrix pencil [sE(po)-A(po) ] possesses m distinct finite eigenvalues s~(po ), i = 1 . . . . . m, namely the roots of the characteristic equation

IsE(po) - A(po) I = 0

(2.4)

and ! = n - m infinite eigenvalues. Assume that the index of nilpotency of the matrix pencil [sE(po)A(p0) ] equals one - so the eigenvalues at infinity are distinct. Assume also that there exists an open neighborhood J¢" of P0 such that for any p ~ ~/', rank E ( p ) = d e g l s E ( p ) - A ( p )

I = m.

(2.5)

Then for any p in a certain neighborhood ~/'o c JV': (i) the matrix pencil [sE(p) - A ( p ) ] possesses m distinct, finite eigenvalues si(p) , i = 1..... m, and / infinite eigenvalues, where the functions si(.) are differentiable in JIro; (ii) there exist the following sets of vector-valued functions that are differentiable in ~/'o:

( v l ( p ) . . . . . vm(p)}, {t,(p) . . . . . h ( P ) } ,

(wl(p) . . . . . win(p)}, {ul(p) ..... ut(p)},

(2.6) (2.7)

satisfying

A ( p ) v , ( p ) = s~(p)E(p)v~(p),

i=l,...,m,

wT(p)A(p) = s i ( p ) w T ( p ) E ( p ) , E(p)ti(p)=O,

i = 1 .... ,l,

uT(p)E(p)=O,

i = 1 . . . . . l, ){ = 0 S0

wT(p)E(P)v~(P

=0

uT(p)A(p)ti(p)

~0

ifj~i, ifj=i,

if j ~ i , ifj=i,

i=l,...,rn,

(2.8) (2.9) (2.10) (2.11) (2.12) (2.13)

and such that the vectors ( v l ( p ) , . . . , v,~(p), q(p) . . . . . tt(p) ) are linearly independent and so are the vectors {wl(p) . . . . . win(p), ux(p),.., ul(p)}. The vectors vi(p), w~(p) are generally referred to as the right and left eigenvectors (respectively) associated with finite eigenvalues s~(p), and the vector ti(p), u,(p) as the right and left eigenvectors associated with infinite eigenvalues. The eigenstructure of the matrix

E(p) = (c(p)E(p) -A(p))-lE(p) (where c(p) is any real number such that I c ( p ) E ( p ) - A ( p ) l ~ 0) and the eigenstructure of the matrix pencil [sE(p) - A ( p ) ] are related in the well known way [7]. According to above assumptions the structural properties of the matrix pencil [sE(p) - A ( p ) ] such as distinct finite and infinite eigenvalues and the number of finite and infinite eigenvalues are preserved for P ~ ~ 0 . Notice that only distinct finite eigenvalues and associated eigenvectors are differentiable [6]. To consider derivatives of finite eigenvalues and eigenvectors it is not necessary to assume that eigenvalues at infinity are distinct. Under the assumption that infinite eigenvalues are distinct, the infinite eigenvectors are also differentiable. The considered matrix pencil possesses the maximal possible number of finite eigenvalues. The solution of the system (2.1) is unique and an (undesirable) impulsive response at the initial time t = 0 is impossible. The infinite eigenvalues and eigenvectors are associated with the non-dynamical response and do not influence the dynamical response. These are the typical requirements imposed in descriptor system design - see for example [4].

J. Kabzihski / Eigenvalue-eigenvector sensitivity

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The exact form of the formulae for eigenvector derivatives depends on the normalization selected for the eigenvectors. The general normalization will be imposed for the finite right eigenvectors:

vTi(p)vi(p)=f~(p),

i = l ..... m,

(2.14)

where ~ ( . ) are arbitrary functions differentiable at P0. It is natural to assume that the remaining eigenvectors are normalized such that the matrices

V(p):[vl(p)

...

W(p)=[wa(p)

vm(p)

...

tl(p)..,

w,~(p)

t/(p)],

u1(p)..,

u,(p)]

transform the matrix pencil [sE(p) - A ( p ) ] into canonical form:

ri,,,i

o1 [_D_i_o_]

W(p)[sE(p)-A(p)]V(p) [0, 0] tO', I,]

where D = diag(sl(p) .... , Sm(p)} and Im, I/ are m- and /-dimensional identity matrices. This means that (2.12) and (2.13) take the form

{ Io if i = j , if i4:j,

wT(p)E(p)vi(P)= uf(p)A(P)q(P)

(2.15)

i=j, = { Io if if i ~ j ,

(2.16)

and

w(p)=[E(p)ol(p)

""

E(p)vm(p)

A(p)tl(p)"'"

A(p)tt(p)].

(2.17)

In the following sections a prime ' denotes derivatives with respect to p at the point P0. The notation of the p-dependence will be omitted for brevity.

3. Derivatives of finite eigenvalues Differentiating (2.8) with respect to p one obtains /

t

e

e

/

(3.1)

A t)i + A O i = s i E o i + S i E oi + S i E v i.

After left-side multiplication by w,X, because of (2.9), it follows that

s: = ( w?F.v, )- ~wJ( A' - s , e ' ) v ,

(3.2)

and if (2.15) holds then

(3.3)

s: = w?( A' - s,E') v,.

4. Derivatives of eigenvectors Since the set of vectors {vl . . . . . Vm, t l . . . . . j = 1. . . . . l, such that

h} spans C n, there exist coefficients bij, j = 1. . . . . m and c~j,

v'~ = bnv 1 + • • • +b.~v., + CntI + • • • +cab.

(4.1)

J. Kabzihski / Eigenvalue-eigenvector sensitivity

230

It follows from (4.1), (2.10) and (2.12) that for any k 4: i, T

t

Wk Ev i =

T

T

(4.2)

T

b~,w~Evj + E ci,w k Etj = bik w k Ev k j=l j=l

and from (2.11), (2.13) that for any k = 1 ..... l, T

t

T

ukAv~ =

T

T

(4.3)

b i j u k A v j + ~_~ c i j U k A t j = CikUkAtk . j=l

j=l

So

v;=

T

(wjEvj)

'

'

(wjTEv;)vj+b,

vi + Y'~ ( u ~ A t j )

j=l j=~l

1(

(4.4)

ujXAv;)tj.

j=l

Multiplying through (3.1) by w7, j 4= i, and using (2.12), (2.9) one obtains T

t

t

(4.5)

wj Ev i = w T ( A ' - s i E ) v i (s i - s j ) - i .

Multiplying through (3.1) by ujX and using (2.11), UjTA y it = _ u f ( A ' - s i E ' ) v i.

(4.6)

The coefficient b i is obtained after multiplication through (4.4) by v~: -

-

(wjEvi)viv jj=l

(4.7)

Y'~ ( u ~ A t j ) - a ( u ~ A v ; ) v T i t j . j=l

jqi

Finally (2.14) implies that T t

v~vi

1

t

7f, ,

(4.8)

Substituting (4.5), (4.6), (4.7), (4.8) into (4.4) yields the desired formula for the right eigenvector derivative: l

U:

~ [(S i

Sj)wTEuj] -1

T

,

(uTAtj)-I[

T

'

j=l

j=l

j*i

+¢;-.1(½ , _

)

E ( u T A ' j ) - I [ ujMi T tvi] uTlj o,, [(Si_ Sj)wTEuj] -lr [wjMiui]uTiuj"IT t ' j=l

j=l

j4=i

(4.9)

where

M i' = A' -- siE'. Formula (4.9) is considerably simplified if the selected normalization is f~=l

(4.10)

f,'=0

and (2.15), (2.16) hold. In this case,

,,=[I-,A]

1[

,

j=l

jq=i

where I denotes the identity matrix.

T- , ]

g J[u'Moiltj t -- ' j=l

(4.11)

J. Kabzihski / Eigenvalue-eigenvector sensitivity

231

The derivative of the left eigenvector can be expressed in the form (4.12)

w i' = d~awI + . . . + d , , , w , , + e n u I + . . . + e , t u t.

It follows from (4.12), (2.15), (2.16) that for any k = 1 . . . . . m, /

uTETwi ' = ~ dijuTkETwj -b E e i j u T E T u j = dik j=l

(4.13)

j=l

and that for any k = 1 . . . . . l, m

1

tTATwi ' = E dijtTATwj + E eijtTATuj = eik" j=l

(4.14)

j=l

Because of (2.15) for any k = 1 , . . . , m, T ~ t T wi + v T E T w / 0 = ( o T E T w i ) t = ok,TET Wi + vktz

(4.15)

and so oTETwi , = - v

kt TLr T w i - v k ~T r-,tT w i.

(4.16)

Differentiating (2.9) with respect to p and multiplying by t k one obtains for any k = 1,..., l, T t

tT

t T

T t

pT

(4.17)

w i A t k + wi A t k = si w~ Etk + s~w~ E t k + s,w~ E t k

and so AT ,~T

t

t k A Wi

T

--tk(A

=

t

t T

- s i E ) w i,

(4.18)

k = l . . . . l.

Substituting (4.18), (4.16), (4.14), (4.13) into (4.12) yields Wit =

(--W, EUj'

w i e , uj)wj

j=l

E

[wT(At_siet)tj]uj.

(4.19)

j=l

It follows from (4.9) under (2.15), (2.16) that: -1wT

f~-i 1 ~fi 1 t -- E

t

for i ~ j ,

( Si -- Sk ) [WkMiuiluTuk T ' _}_

E

[ukMeo'loTtk T '

for i = j ,

and so

w,

=-

E(s,

--

sj)-lr t w iT., ' , ° + l w1j - -

E'

j=l

[wiMitj]uj

j=l

jq:i T

--fi-i ' ½fi'--

j=l

p

(Si--Sj)-I[wjMivi]vTvj j=l

+ E

T

t

[ujMivilvTtj

Wi"

(4.20)

j=l

jq=i

This formula can be simplified by a suitable choice of the function f , - similarly to (4.11). Changing the selected normalization (2.14), (2.15), (2.16) one obtains the derivatives in other forms. For example, imposing wf(p)wi(p)-=-gi(p),

i = 1 . . . . . m,

instead of (2.14) yields the dual form of (4.9), (4.20).

J. Kabzihski / Eigenvalue-eigenvector sensitivity

232

It is easy to notice that the imposed conditions (2.10), (2.11), (2.16) do not determine the vectors ti, u i exactly - there are 21n elements of t , ui, i = 1 . . . . . l, and 2ln - 12 equations (2.10), (2.11), (2.16). In this case it is impossible to obtain explicit formulae for the derivatives t ' , u~, i = 1 . . . . . I. Proceeding similarly as with the derivation of (4.10) or (4.20) one obtains m

/

, : : - Y'~ (wTE't,)vj - Y" [(u~A' + u;TA)t,ltj, j=l m

u;=-

l

~. (u~E'vi)wj- ~_~ [uVi(A'tj+Atj)]uj. j=l

Computing

(4.22)

j=l

ujt TAt, from (4.22) and substituting into (4.21) one obtains

(,) I-

(4.21)

j=l

.

Z tju~4 t / : - Z (wTE'ti)vj.

j=l

(4.23)

j=l

It is easy to notice that rank

(' I-

~ tjuyA =m.

j=l

T h e flexibility offered in (4.23) for the vectors t{ follows f r o m the flexibility in the choice of the vectors t~. If the vectors t~ are chosen then the vectors u i are determined from (2.17). Very often it is possible to choose the vectors ti, i = 1 . . . . . l, independently from the p a r a m e t e r value p. In this case we can take t~ = 0 and c o m p u t e uf from (4.22).

5. Second derivatives of finite eigenvalues If the matrices A ( p ) , E(p) are functions of a p a r a m e t e r vector p = [Pa . . . . . pq] then the formulae derived in Sections 3 and 4 can be used for c o m p u t a t i o n of the first derivative matrices:

opj

I j=l

.. .. ..... . q. .

Lapj]j=,.

..,q

'

i wl

[0pj]j=,

.....

q

These formulae can also be used to derive the second derivatives of finite eigenvalues. Assume that the p a r a m e t e r p is a complex vector p = [Pl, P:]. Let ' and * denote derivatives at the nominal point with respect to p~ and Pz respectively. Differentiating (3.2) yields

s fw vi) aE [I l,

=

wi*T

_

1

_ wiXEvi (A'-siE')vi+w?(A'-siE')

[I viw T ] , Wi Evi _

_

+ w T ( ( A ' * - s i E ' * ) - E * W T ( A ' - s i]E ' ) Viw' - E ' WTi T ( A *E- s i E w *vTEv7 )vi)

U~,

vi .

(5.1)

If the condition (2.15) holds the above formula can be simplified. U n d e r condition (2.15) the first derivative is given by (3.3). Differentiating (3.3) one obtains

s'* = wi*T(A ' - siE')v i+ wV(A ' - s i E ' ) v * + wiV(A '* - siE'* - (w•(A* - s i E * ) v i ) E ' ) v i. (5.2) T h e derivatives of eigenvectors necessary to c o m p u t e (5.1) or (5.2) are given by (4.9) or (4.11) and (4.20). Of course '* m a y also denote the second derivative with respect to the same variable Pl or P2-

233

J. Kabzihski / Eigenvalue-eigenvector sensitivity

6. Derivatives of eigenvalues and eigenvectors of regular state-space systems If r = n, l = 0 the formulae derived in Sections 3 and 4 give derivatives of eigenvalues and eigenvectors of the matrix E - l ( p ) A ( p ) . Assuming that E = I and E ' = 0 one obtains derivatives of eigenvalues and eigenvectors of the matrix A ( p ) . If follows from (3.2) that St'

:(@,)-, Wi A T,

vi,

i = 1 .....

(6.1)

n,

(the Jacobi formula), from (4.11) that

~,= '

~l

=

(s,-sJ)@J wjA v i ~j+E1 ½f/-

Jjq:i

wjAo, " T j=l (s,-si)wjvj

(6.2)

vivjlvi

]

jq=i

(formula (16.6) from [5]), from (4.21) that

t

[ i

wiAuj

W?A"UiT ]

wi =j~lj~i '= s'-s------fwJ+f71 ½ f i ' - j=lj~i------viv,]Wi,si_sj ~ from (5.1) that s'* = (wi vi)

l[[Vi l

(6.3)

[vi l

wi*T 1 - - - T - I A v i + wTA ' I -- - T - I o * w~ v~ j wi vi J

+

wfA'*v,

] ,

(6.4)

and from (5.2) that

Si'* = wi, TAtvi q_ w~A'v?

(6.5)

+ w~TaA t * v,.

7. Applications The formulae for derivatives of eigenvalues and eigenvectors of regular state-space systems have already been used to solve many control problems concerning robustness and performance optimization. For example these formulae were utilized for eigenvalue tracking [5], developing approximate techniques for pole placement [1], and control of flexible structures [2]. The results of the present paper can be used to solve analogous problems for descriptor systems. The presented formulae can be also used for solving eigenvalue sensitivity minimization problems. For discussion of these problems for regular state-space systems, see for example [10]. To demonstrate the possibilities for descriptor systems let us consider the following example. Example. Consider the descriptor system where [0

e(p)=

0 0 0

0 0 0 0 2

A(p,d)= 1. 1 1

0 0 0 0 -1 2 0 1

1 p 0 0

(7.1)

-2+p -2 -1 -2

2 0 + 1 1

1 1 1 1

-1 0 -1

-d 0

0 0

° '

(7.2)

J. Kabzihski / Eigenvalue-eigenvectorsensitivity

234

and assume that the designer can select the value of the real parameter d from the interval [ - 3, 3] while p is a real disturbing parameter and its nominal values is P0 = 0. This system was investigated in [4] (Example 1) and the parameter d represents design freedom in feedback controller design. It follows from [4] that for any real d and for p = 0 the matrix pencil (7.1), (7.2) possesses the only finite eigenvalue s I = - 1 , associated finite eigenvector V1 = 3-1/210 - 1

-1

-11 T,

(7.3)

three infinite eigenvalues and the infinite eigenvectors can be selected as: t 1 = [ 1 0 0 0] v,

t 2 = [ 0 1 0 0] v,

t 3 = [ 0 0 1 0] v.

(7.4)

Indeed, as for any d o, Odp=O vI = 0,

(7.5)

d= d o

one obtains from (3.3) and (4.11) that for any do,

~Sl -Od ~=~o = 0,

~Vl a=ao 0d p=o = 0,

(7.6)

compatibly with [4]. We want to select the value d o to minimize the sensitivity of the finite eigenvalue: 0p ~S1

p=O. d=do

(7.7)

For d o = 0 we have Op d-O p=0 = 2 0S1

(7.8)

(the matrix W(p, d) was calculated from (2.17)). Computing the second derivative from (5.2) for different d o we find that 02Sl =_ 1, ~p3d ~-oo

(7.9)

so the sensitivity (7.7) is linear function of the design parameter d. For d = 2 we have 0s 1

=o-

8. Conclusions

The formulae for derivatives of eigenvalues and eigenvectors of linear, time invariant, parameterized descriptor systems were derived. The presented approach gives insight into the problem of eigenvalueeigenvector sensitivity, especially highlighting the influence of system eigenstructure. The obtained formulae can be used for solving many control problems and especially sensitivity minimization problems as demonstrated in Section 7.

J. Kabzihski / Eigenvalue-eigenvector sensitivity

235

References

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