Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://mailybaev.imec.msu.ru/papers/GarciaMailybaev2003.pdf Äàòà èçìåíåíèÿ: Tue Jun 14 12:09:59 2005 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:40:06 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: space station

SIAM J. MATRIX ANAL. APPL. Vol. 24, No. 4, pp. 943­962

c 2003 Society for Industrial and Applied Mathematics

REDUCTION TO VERSAL DEFORMATIONS OF MATRIX PENCILS AND MATRIX PAIRS WITH APPLICATION TO CONTROL THEORY
M. I. GARC´ IA-PLANAS AND A. A. MAILYBAEV Abstract. Matrix pencils under the strict equivalence and matrix pairs under the state feedback equivalence are considered. It is known that a matrix pencil (or a matrix pair) smoothly dependent on parameters can be reduced locally to a special typically more simple form, called the versal deformation, by a smooth change of parameters and a strict equivalence (or feedback equivalence) transformation. We suggest an explicit recurrent procedure for finding the change of parameters and equivalence transformation in the reduction of a given family of matrix pencils (or matrix pairs) to the versal deformation. As an application, this procedure is applied to the analysis of the uncontrollability set in the space of parameters for a one-input linear dynamical system. Explicit formulae for a tangent plane to the uncontrollability set at its regular point and the perturbation of the uncontrollable mode are derived. A physical example is given and studied in detail. Key words. versal deformation, matrix pencil, matrix pair, feedback equivalence, controllability AMS sub ject classifications. 15A21, 93B05, 93B52 PI I. S0895479801392016

1. Intro duction. The Arnold technique of constructing a local canonical form, called versal deformation, of a differentiable family of square matrices under conjugation [1, 2] has been generalized by several authors to matrix pencils under the strict equivalence [4, 10], pairs or triples of matrices under the action of the general linear group [18], pairs of matrices under the feedback similarity [6], and triples or quadruples of matrices representing linear dynamical systems under the equivalence derived from standard transformations (the change of basis in state, input, and output spaces, state feedback, and output injection) [8, 9]. Versal deformations provide a special parametrization of matrix spaces, which can be effectively applied to perturbation analysis and investigation of complicated ob jects like singularities and bifurcations in multiparameter dynamical systems [1, 2, 3, 4, 5, 12, 14, 15]. The general notion of versality is the following. Let M be a differential manifold with the equivalence relation defined by the action of a Lie group G . The G -action is described by the mapping x - g x, where x, g x M and g G . The classical example is the space of square complex matrices M = Mmâm (C) with the Lie group G = GL(m, C) determining the similarity transformation (the change of basis) A - C -1 AC , where A Mmâm (C) and C GL(m, C). Let us consider a smooth mapping x : U0 - M, where U0 is a neighborhood of the origin of the space F ; F stands for the space of real or complex numbers. The mapping x( ) is called a deformation of x0 = x(0) with the parameter vector F . Introducing a change of parameters : U0 - U0 , where U0 is a neighborhood of the origin in Fk , such that (0) = 0, we obtain the deformation x(( )) of x0 with the parameter vector
Received by the editors July 3, 2001; accepted for publication (in revised form) by P. Van Dooren August 5, 2002; published electronically February 12, 2003. This work was supported by INTAS Young Scientists Fellowship 00-58. http://www.siam.org/journals/simax/24-4/39201.html Dept. de Matem` atica Aplicada I, Universitat Polit` ecnica de Catalunya, Miner´ 1, Esc. C, 1-3, ia 08038 Barcelona, Spain (maria.isabel.garcia@upc.es). Institute of Mechanics, Moscow State University, Michurinsky pr. 1, 117192 Moscow, Russia (mailybaev@imec.msu.ru).

943


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M. I. GARC´ IA-PLANAS AND A. A. MAILYBAEV

U0 Fk . Applying the equivalence transformation g ( ), where g : U0 - G is a smooth mapping such that g (0) = e is the unit element of G , we get the deformation (1.1) z ( ) = g ( ) x(( ))

of z (0) = e x0 = x0 . Then x( ) is called a versal deformation of x0 if any deformation z ( ) of x0 can be represented in the form (1.1) in some neighborhood of the origin U0 Fk . This definition implies that a versal deformation generates all deformations of x0 and, hence, possesses properties (invariant under the equivalence transformation) of all deformations of the given element x0 M. The theorem given by Arnold [1, 2] says that the deformation x( ) of x0 is versal if and only if it is transversal to the orbit of x0 under the action of G . This theorem reduces the problem of finding a versal deformation to solving a specific linear equation determined by x0 . This method allows finding versal deformations x( ) having simple form, which can be treated as local canonical forms. For the reduction of a given deformation z ( ) to this form, one needs to find the change of parameters = ( ) and the equivalence transformation g ( ) smoothly depending on , which satisfy locally equality (1.1). In this paper versal deformations of matrix pencils under the strict equivalence and pairs of matrices under the feedback equivalence are considered. The method of finding the change of basis = ( ) and the equivalence transformation g ( ), which reduce a given deformation z ( ) to the versal deformation, is developed. The mappings ( ) and g ( ) are represented in the form of Taylor series, whose coefficients are found from the explicit recurrent procedure. This approach is the generalization to these particular cases of the one presented by Mailybaev [12, 13] for spaces of square matrices under conjugation; see also [5, 17] for related problems. A pair of matrices (F, G) Mmâm (R) â Mmân (R) determines the linear dynami cal system = F + G with the state vector Rm and input vector Rn . The controllability of this system (the possibility of reaching any state by choosing an appropriate input vector (t)) is an invariant property under the feedback equivalence transformation. Using this fact, we apply the method presented in this paper to study the uncontrollability set of a multiparameter one-input linear dynamical system. As a result, explicit formulae for the tangent plane to the uncontrollability set at its regular point and the perturbation of the uncontrollable mode (the generalized eigenvalue) are derived. Note that this approach provides a simple and systematic way for the perturbation analysis of the uncontrollability set, while the classical controllability condition related to the rank of a certain matrix (called the controllability matrix) is difficult to use for multiparameter perturbation analysis. The organization of the paper is as follows. In section 2 the case of matrix pencils under the strict equivalence is considered. The local structure of the orbit and stabilizer of a matrix pencil is described by a specific linear function (differential of the equivalence transformation mapping) and its adjoint. Using this information, a versal deformation x( ) is determined. Then the change of basis = ( ) and the equivalence transformation g ( ) for the reduction of a given deformation z ( ) to this versal deformation are found in the form of Taylor series. Section 3 studies the case of pairs of matrices under the feedback equivalence. In section 4 the obtained results are applied to the perturbation analysis of the uncontrollability set for a one-input linear dynamical system dependent on parameters. A physical example is given and studied in detail. The conclusion discusses applicability issues of the presented method and its importance for the versal deformation theory.


REDUCTION TO VERSAL DEFORMATIONS

945

2. Matrix p encils and their deformations. Let us consider a space of matrix pencils M = {A - B | A, B Mmân (F)}, where Mmân (F) is a set of m â n matrices with real or complex elements, F {R, C}. In this space we consider the following equivalence relation [7]: two pencils A1 - B1 and A2 - B2 are (strict) equivalent if and only if (2.1) A2 - B2 = P
-1

(A1 - B1 )Q

for some nonsingular square matrices P Gl(m; F), Q Gl(n; F). 2.1. Equivalence as a Lie group action. Equivalence relation (2.1) may be seen as induced by the action of a Lie group G = {(P, Q) | P Gl(m; F),Q Gl(n; F)}. Using the short notation g = (P, Q) G and x = A - B M, we define multiplication in G , action of the group G , and equivalence condition (2.1) as follows: g1 g2 = (P1 P2 ,Q1 Q2 ) G , (2.2) gx=P
-1

(A - B )Q M,

x2 = g x1 . Multiplication in the group corresponds to successive equivalence transformations: g2 (g1 x) = (g1 g2 ) x. The unit element of G has the form e = (Im ,In ), where Im and In are the identity matrices. Let us fix a pencil x0 = A0 - B0 M and define the mapping (2.3) fx0 (g ) = g x0 .

The equivalence class of the pencil x0 with respect to the action of G is the range of the function fx0 . It is called the orbit of x0 and denoted by (2.4) O(x0 ) = Im fx0 = {g x0 | g G }.

The stabilizer of x0 under the G -action is a null-space of the function fx0 - x0 . We denote it by (2.5) S (x0 ) = Ker (fx0 - x0 ) = {g G | g x0 = x0 }.

The mapping fx0 is differentiable, and O(x0 ) and S (x0 ) are smooth submanifolds of M and G , respectively. Let us use the notation Te G for a tangent space to the manifold G at the unit element e. Since G is an open subset of Mmâm (F) â Mnân (F), we have Te G = {(U, V ) | U M and, since M is a linear space, Tx0 M = M. The Euclidean scalar products in the spaces M and Te G considered in this paper are defined as follows: (2.6) x1 ,x2 y1 ,y
1 = trace(A1 A ) + trace(B1 B2 ), 2 mâm

(F),V M

nân

(F)}

xi = Ai - Bi M, yi = (Ui ,Vi ) Te G ,

22

= trace(

U1 U2

) + trace(V

1 V2

),


946

M. I. GARC´ IA-PLANAS AND A. A. MAILYBAEV

Fig. 1. Local structure of the orbit O(x0 ) and stabilizer S (x0 ).

where A denotes the conjugate transpose of a matrix A. Let dfx0 : Te G - M be the differential of fx0 at the unit element e. Using expressions (2.2) and (2.3), we find [4] (2.7) dfx0 (y ) = (A0 V - UA0 ) - (B0 V - UB0 ) M,
= y, dfx0 (z ) 2 ,

y = (U, V ) Te G .

The adjoint linear mapping dfx0 : M - Te G is defined by the relation

(2.8)

dfx0 (y ),z

1

y Te G , z M.

Using expressions (2.6) and (2.7) in (2.8), it is straightforward to find (2.9)
dfx0 (z ) = (-XA - YB0 ,A X + B0 Y ) Te G , 0 0

z = X - Y M.

The mappings dfx0 and dfx0 provide a simple description of the tangent spaces Tx0 O(x0 ), Te S (x0 ) and their normal complements (Tx0 O(x0 )) , (Te S (x0 )) ; see Figure 1. Theorem 2.1. The tangent spaces to the orbit and stabilizer of the matrix pencil x0 and the corresponding normal complementary subspaces with respect to M and Te G can be found in the fol lowing form: 1. Tx0 O(x0 ) = Im dfx0 M. 2. (Tx0 O(x0 )) = Ker dfx0 M. 3. Te S (x0 ) = Ker dfx0 Te G . 4. (Te S (x0 )) = Im dfx0 Te G . Proof. Assertions 1 and 3 follow from (2.4), (2.5), and the definition of dfx0 as the differential of the function fx0 at e. Then assertions 2 and 4 follow from properties of the adjoint function dfx0 [7]. Corollary 2.2. The mappings dfx0 and dfx0 define one-to-one correspondences between the subspaces Tx0 O(x0 ) and (Te S (x0 )) : dfx0 Tx0 O(x0 ) - (Te S (x0 )) . - dfx0

Example 2.1. Let us consider a 0 x0 = A0 - B0 = 0 (2.10) 0

matrix pencil 100 1 0 0 0 - 0 001 0

0 0 0

0 1 0

0 0 . 1


REDUCTION TO VERSAL DEFORMATIONS

947

According to Theorem 2.1, the elements z (Tx0 O(x0 )) can be found by solving the linear system dfx0 (z ) = 0 with dfx0 given by expression (2.9). As a result, we obtain a general element of (Tx0 O(x0 )) in the form 0 0 0 0 000 0 1 0 3 0 - 0 0 0 0 , (2.11) -2 -2 0 -4 2 2 0 4 where 1 ,...,4 F are arbitrary; dim(Tx0 O(x0 )) = 4. Using (2.11), it is straightforward to find a general element of the space Tx0 O(x0 ) as follows:
(2.12) µ1 0 µ7 + µ µ2 µ5 µ8 - µ µ3 0 µ10 µ4 µ6 µ11 - µ12 µ16 µ7 - µ µ13 µ17 µ8 + µ µ µ µ
14 18 20

9

9

9

9

µ µ µ

15 19 11

,

where µ1 ,...,µ20 F are arbitrary; dim Tx0 O(x0 ) = 20. Using (2.12) in Corollary 2.2, we find a general element of the space (Te S (x0 )) = dfx0 (Tx0 O(x0 )) in the form
(2.13)



-µ2 - µ12 -µ5 - µ16 2µ9 - µ8 - µ

7

-µ -µ -µ

14 18 20

- µ4 - µ - µ6 - µ -2µ11

15 19

µ12 µ1 , µ16 2µ7



µ13 µ2 µ17 2µ8

µ14 µ3 µ18 µ10 + µ

20

µ15 µ4 . µ19 2µ11



Finally, we obtain elements follows: 1 0 (2.14) 0

of the space Te S (x0 ) from the equation dfx0 (y ) = 0 as 2 4 0 1 3 0 0 , 0 5 0 0 1 0 0 2 0 4 0 3 3 , 0 5

where 1 ,...,5 F are arbitrary; dim Te S (x0 ) = 5. 2.2. Versal deformation. Let U0 be a neighborhood of the origin of F . A deformation x( ) of x0 is a smooth mapping x : U0 - M such that x(0) = x0 . The vector = (1 ,..., ) U0 is called the parameter vector. The deformation x( ) is also called the family of matrixpencils. The deformation x( ) of x0 is called versal if any deformation z ( ) of x0 , where = (1 ,...,k ) U0 Fk is the parameter vector, can be represented in some neighborhood of the origin in the following form: (2.15) z ( ) = g ( ) x(( )), U0 U0 ,

where : U0 - F and g : U0 - G are differentiable mappings such that (0) = 0 and g (0) = e. Expression (2.15) means that any deformation z ( ) of x0 can be obtained from the versal deformation x( ) of x0 by an appropriate smooth change of parameters = ( ) and equivalence transformation g ( ) smoothly dependent on parameters. The versal deformation with minimal possible number of parameters is called miniversal.


948

M. I. GARC´ IA-PLANAS AND A. A. MAILYBAEV

The following result, proved by Arnold [1, 2] for Gl(n; C) acting on Mnân (C), and generalized by Tannenbaum [18] for a Lie group acting on a complex manifold, provides the relation between the versal deformation of x0 and the local structure of the orbit and stabilizer of x0 . Theorem 2.3. 1. A deformation x( ) of x0 is versal if and only if it is transversal to the orbit O(x0 ) at x0 . 2. The minimal number of parameters of a versal deformation is equal to the codimension of the orbit of x0 in M, = codim O(x0 ). 3. If x( ) is a miniversal deformation and values of the mapping g ( ) are restricted to belong to a smooth submanifold R G , which is transversal to S (x0 ) at e and has the minimal dimension dim R = codim S (x0 ), then the mappings ( ) and g ( ) in representation (2.15) are uniquely determined by z ( ). Note that the third assertion of Theorem 2.3 was not explicitly stated in [1, 2, 18] but proved in the proof of the corresponding theorem. Let us denote by {t1 ,...,td }, d = dim Tx0 O(x0 ), a basis of the tangent space Tx0 O(x0 ); by {n1 ,...,n }, = codim Tx0 O(x0 ), a basis the normal complement (Tx0 O(x0 )) ; by {c1 ,...,c } a basis of an arbitrary complementary subspace (Tx0 O(x0 ))c to Tx0 O(x0 ); and by {r1 ,...,rd } a basis of (Te S (x0 )) . By Corollary 2.2, if we have the basis {t1 ,...,td }, then the basis {r1 ,...,rd } can be chosen in the form {dfx0 (t1 ),...,dfx0 (td )}, and, vice versa, if the basis {r1 ,...,rd } is known, then we can choose the basis {t1 ,...,td } in the form {dfx0 (r1 ),...,dfx0 (rd )}. Corollary 2.4. The deformation (2.16) x( ) = x0 +
i=1

ci i

is a miniversal deformation. The functions ( ) and g ( ) in the versal deformation reduction (2.15) are uniquely determined if the mapping g ( ) is taken in the form
d

(2.17)

g ( ) = e +
j =1

rj µj ( ),

where µj ( ) are smooth functions in F such that µj (0) = 0, j = 1,...,d. If we take ci = ni , i = 1,..., , in (2.16), then the corresponding miniversal deformation is called orthogonal. If the pencil x0 = A0 - B0 is reduced to the Kronecker canonical form (this is not a restriction because of the homogeneity of the orbit), it is possible to write down explicitly the bases {c1 ,...,c }, {n1 ,...,n }, {t1 ,...,td }, and {r1 ,...,rd }. Explicit forms of the bases {c1 ,...,c } and {n1 ,...,n } were given in [4, 10]. Example 2.2. Let us consider a matrix pencil (2.10). The matrix pencils ni , tj and matrix pairs rj can be obtained from (2.11), (2.12), and (2.13), respectively, by taking i = µj = 1 and zeros for other variables. Using the explicit form of the tangent space Tx0 O(x0 ) given in (2.12), we can choose a basis {c1 ,...,c }, = 4, of a complementary subspace (Tx0 O(x0 ))c such that every ci has exactly one nonzero element. This will give us a simplest miniversal deformation, for example, 010 0 1000 - 0 0 1 0 . 0 x( ) = 1 0 3 (2.18) 0001 2 0 0 1 + 4


REDUCTION TO VERSAL DEFORMATIONS

949

2.3. Reduction to miniversal deformation. Let us assume that the pencil x0 and its miniversal deformation x( ) in the form (2.16) are given. To reduce an arbitrary deformation z ( )of x0 to the miniversal deformation, we need to find smooth mappings ( ) and g ( ) satisfying relation (2.15). Recall that these mappings are unique if g ( ) is taken in the form (2.17). Since these mappings are determined in the neighborhood of the origin = 0, they can be represented in Taylor series form. Let h = (h1 ,...,hk ) be a vector with nonnegative integer components hi Z+ . We will use the conventional notation |h| = h1 + ··· + hk , h! = h1 ! ··· hk !,
h Ch =

h! , h !(h - h )! ,

h h h = 1 1 ··· k k ,

(

h)

=

|h| h1 ···

hk

where derivatives are evaluated at = 0; the derivative of zero order denotes the function value at zero, i.e., (0) = (0). Using expression (2.17), we can write the Taylor series for the mappings ( ) and g ( ) as ( ) = (2.19)
|h|s

(h) h + o( s ), h!
d

g ( ) = e +
j =1 (0)

r

j |h|s

µj h + o( s ), h!

(h)

where (0) = 0 and µj = 0; is the norm in the parameter space Fk . Therefore, to find the transformation functions ( ) and g ( ), we need to determine the derivatives (h) (h) (h) (h) (h) = (1 ,..., ) and µ1 ,...,µd . The following theorem provides explicit recurrent formulae for calculation of these derivatives up to an arbitrary order |h|. (h) (h) (h) (h) Theorem 2.5. The derivatives 1 ,..., and µ1 ,...,µd determining transformation functions (2.19), which reduce the deformation z ( ) of x0 to the miniversal deformation (2.16), satisfy the recurrent formulae (h) sh ,n1 1 1 . . . = Z -1 . (2.20) , . . (h) sh ,n 1 (2.21) (h) µ1 . . =W . (h) µd
-1



sh - sh -

(h) i=1 ci i (h) i=1 ci i

. . .

,t1 ,td


1

,

1

where Z and W are nonsingular â and d â d matrices with the elements zij = cj ,ni 1 and wij = dfx0 (rj ),ti 1 = rj ,dfx0 (ti ) 2 , respectively. The pencil sh M has the form (2.22) sh = z
(h)

-
h +h =h |h |>0, |h |>0

h Ch i=1

ci i

(h )

d

,
j =1

rj µj

(h )

,z

(h )

.


950

M. I. GARC´ IA-PLANAS AND A. A. MAILYBAEV

The mapping : Mâ Te G â M - M is defined by the expression (2.23) (x, y , z ) = (AV - UX ) - (BV - UY ), where x = A - B , y = (U, V ), and z = X - Y . Proof. Using the notation x = A - B , g = (P, Q), and z = X - Y , we can write expression (2.15) in the form (2.24) X ( ) - Y ( ) = P
-1

( ) A(( )) - B (( )) Q( ).

Multiplying (2.24) by P ( ) from left and collecting all terms at the left-hand side, we obtain (2.25) P ( ) X ( ) - Y ( ) - A(( )) - B (( )) Q( ) = 0. Taking the derivative of order h of (2.25) and using the Leibniz formula for differentiation of a function product, we get
h Ch P (h )

X

(h )

- Y
h)

(h )

(2.26)

h +h =h

- (A(( )))(

h)

- (B (( )))(

Q(

h)

= 0.

Using expressions (2.16), (2.17), (2.22), (2.23) in (2.26) and taking into account that P (0) = Im , Q(0) = In , A(0) = X (0) = A0 , B (0) = Y (0) = B0 , after permutation of terms we find
d

(2.27)

df

x0 j =1

rj µj

(h)

= sh -
i=1

ci i ,

(h)

where the linear mapping dfx0 is defined in (2.7). Equality (2.27) represents a system of linear equations with respect to + d = 2mn (h) (h) (h) (h) unknowns 1 ,..., and µ1 ,...,µd . The solution of (2.27) exists if and only if its right-hand side belongs to Im dfx0 = Tx0 O(x0 ). Hence, the right-hand side has to be orthogonal to every pencil from the basis {n1 ,...,n } of (Tx0 O(x0 )) . This condition, written in the matrix form, yields (h) (h) sh ,n1 1 1 sh - i=1 ci i ,n1 1 . . . = . . (2.28) - Z . = 0. . . . (h) (h) sh ,n 1 sh - i=1 ci i ,n 1 The solution of this system gives expression (2.20) of the theorem. (h) (h) To determine values of the derivatives µ1 ,...,µd , we take the scalar product of (2.27) and ti . For the left-hand side this yields
d

(2.29)

df

x0 j =1

rj µj

(h)

d

,ti

1

=
j =1

dfx0 (rj ),ti

1

µj

(h)

d

=
j =1

wij µj .

(h)

Recall that dfx0 (rj ),ti 1 = r obtain the following system of (h) µ 1 (2.30) W . . . µd
(h)

f j ,d x0

(ti ) 2 by definition (2.8). linear equations: (h) sh - i=1 ci i ,t1 1 . = . . sh -
(h) i=1 ci i

Taking i = 1,...,d, we .

,td

1


REDUCTION TO VERSAL DEFORMATIONS

951

The solution of (2.30) gives expression (2.21) of the theorem. (h) (h) Note that for evaluation of derivatives i , µj , expressions of Theorem 2.5 require only derivatives i , µj of lower orders |h | < |h| and derivatives z (h ) of orders |h | < |h| and h = h. This makes it possible to use Theorem 2.5 for successive (h) (h) calculation of the derivatives i , µj in order to find the transformation functions ( ) and g ( ) in the form of Taylor series (2.19) up to small terms of arbitrary order. (0) (0) Recall that at the initial step of the recurrent procedure we take i = 0 and µj = 0. The matrices Z -1 and W -1 have to be computed only once in the beginning of the recurrent procedure. The size d of the matrix W is typically close to 2mn and can be big. Nevertheless, this matrix is usually very sparse. Moreover, we can avoid difficulties with the inversion by making the matrices Z and W diagonal. For this purpose, we need to choose the bases {c1 ,...,c }, {n1 ,...,n }, {t1 ,...,td }, and {r1 ,...,rd } such that cj ,ni 1 = 0 and dfx0 (rj ),ti 1 = 0 for i = j . Note that the orthogonal miniversal deformation, represented by the orthonormal basis {n1 ,...,n } of (Tx0 O(x0 )) , keeps the metric information in the normal direction to the orbit O(x0 ). This deformation is useful for the numerical problem of computation of a Kronecker canonical form [4]. In many applications, a metric based on properties of the underlying system is defined in the parameter space rather than in the whole space of matrix pencils. Computation on the mapping = ( ) connecting the parameter spaces allows us to keep the metric information of the original parameter space and transfer this metric into the parameter space of the miniversal deformation. Theorem 2.5 can be used with an arbitrary versal deformation satisfying the requirements of each particular problem. As noted by Arnold [1, 2], a miniversal deformation can be chosen in a simple form, which makes it convenient for applications. To avoid numerical instability in transformation to the miniversal deformation, the angle between the image of the miniversal deformation x( ) and the tangent space to the orbit Tx0 O(x0 ) should not be small, i.e., the transversality condition of Theorem 2.3 should not be affected by numerical uncertainties and round-off. Example 2.3. Let us consider the following two-parameter deformation z ( ), = (1 ,2 ), of matrix pencil (2.10):
(2.31) z ( ) = sin 1 0 1 2 1 sin 2 sin 1 0 2 0
2 2 1 cos 2

(h )

(h )

-

1 0 2 2

0 sin 2 0

0 1+ 2 2

1 2 1 cos 1

.

Using the pencils c1 ,...,c4 , n1 ,...,n4 , t1 ,...,t20 and pairs r1 ,...,r20 , constructed in Examples 2.1, 2.2, and applying Theorem 2.5, we find 32 2 3 2 1 ( ) = -1 2 + 2 + 1 - 21 2 + 1 2 - 2 7 3 12 2 2 2 ( ) = 1 - 1 + 1 2 - 2 + 1 + 1 2 12 2 2 2 3 3 ( ) = 2 - 2 + 1 2 + 2 + o( 3 ), 12 12 12 42 4 ( ) = 1 - 2 + 1 2 - 1 2 + o( 2 2 3 6 13 µ1 ( ) = 1 - 1 + o( 3 ), 6 22 22 µ2 ( ) = - 1 2 - 1 2 + o( 3 ), 9 9 53 + o( 3 ), 22 12 - 1 2 + o( 3 ), 2
3

),


952

M. I. GARC´ IA-PLANAS AND A. A. MAILYBAEV

µ3 ( ) = o( 3 ), 1 22 µ4 ( ) = - 1 2 + 2 + o( 3 ), 3 3
2 2 2 2 µ5 ( ) = 22 - 21 - 32 + 1 2 +21 2 +

1 13 12 92 3 µ6 ( ) = 1 - 1 2 - 1 + 1 2 + 1 2 - 22 + o( 3 ), 2 2 2 4 1 12 1 3 12 µ7 ( ) = - 1 + 2 - 1 - 1 2 + o( 3 ), 2 2 24 4 12 3 µ8 ( ) = - 2 + o( ), 2 3 1 3 32 µ9 ( ) = - 1 - 1 - 1 2 + o( 3 ), 4 16 8 12 12 2 µ10 ( ) = -2 - 2 - 1 2 - 1 2 + o( 3 ), 4 2 12 1 12 µ11 ( ) = - 1 - 1 2 - 1 2 + o( 3 ), 8 4 4 12 12 µ12 ( ) = 1 2 + 1 2 + o( 3 ), 9 9 µ13 ( ) = o( 3 ), 1 2 13 µ14 ( ) = - 1 2 - 2 + o( 3 ), 6 6 2 12 µ15 ( ) = 1 2 - 2 + o( 3 ), 3 3 32 12 3 2 19 3 2 µ16 ( ) = -2 + 1 + 2 - 1 2 - 1 2 - 2 + o( 3 ), 2 2 2 12 12 1 2 13 3 µ17 ( ) = 2 - 2 + 1 2 + 2 + o( ), 2 2 12 1 1 12 1 2 13 µ18 ( ) = 2 - 1 2 - 2 + 1 2 + 2 + o( 3 ), 2 2 4 2 8 13 3 2 3 3 µ19 ( ) = 1 - 1 2 + 2 + o( ), 2 2 12 12 µ20 ( ) = 2 + 1 2 + 1 2 + o( 3 ). 4 2 These expressions determine the change of parameters = ( ) and equivalence transformation g ( ) in the reduction of z ( ) to the miniversal deformation (2.18). 3. Pairs of matrices under the feedback equivalence. In this section we consider the space of pairs of matrices (3.1) M = {(F, G) | F Mmâm (F), G M
mân

19 3 + o( 3 ), 62

(F)}.

Each pair x = (F, G) M represents the time-invariant linear dynamical system = F + G , Fm , with the input vector Fn . The change of basis in the state and input spaces and feedback operation in this system induce an equivalence relation in the space M as follows: two pairs of matrices x1 = (F1 ,G1 ) and x2 = (F2 ,G2 ) are called feedback equivalent if and only if there exist matrices P Gl(m; F), R Gl(n; F), and S Mnâm (F) such that [16] (3.2) F2 = P
-1

(F1 P + G1 S ),

G2 = P

-1

G1 R.


REDUCTION TO VERSAL DEFORMATIONS

953

The feedback equivalence transformation may be seen as the action of the Lie group (3.3) G = {g = (P, R, S ) | P Gl(m; F), R Gl(n; F), S M
nâm

(F)}

with the multiplication of elements g1 ,g2 G determined by the expression (3.4) g1 g2 = (P1 P2 ,R1 R2 ,S1 P2 + R1 S2 ), gi = (Pi ,Ri ,Si ).

The unit element of the group G is e = (Im ,In , 0). We will use the short notation x2 = g x1 for the equivalence relation (3.2). Note that g1 g2 x = g2 (g1 x). Given a pair of matrices x = (F, G) M and a triple g = (P, R, S ) G , we can associate a matrix pencil x M of dimension m â (m + n) and a pair g from the corresponding Lie group G in the following manner: (3.5) x = (F G) - (Im 0), g= P, P S 0 R .

It is easy to see that x2 = g x1 (the pairs x1 and x2 are feedback equivalent) if and only if x2 = g x1 (the associated matrix pencils x1 and x2 are strict equivalent) [11]. Hence, M and G can be seen as the subspace of M and subgroup of G , respectively. Note that the subspace M M is not invariant under the action of the Lie group G defined over the space of matrix pencils. 3.1. Orbit and stabilizer. Let us fix some pair of matrices x0 = (F0 ,G0 ) and define the mapping fx0 (g ) = g x0 , g G . Then the orbit O(x0 ) and stabilizer S (x0 ) of the pair x0 are defined as follows: (3.6) (3.7) O(x0 ) = Im fx0 = {g x0 | g G}, S (x0 ) = Ker (fx0 - x0 ) = {g G | g x0 = x0 }.

The sets O(x0 ) and S (x0 ) are differentiable submanifolds of M and G , respectively. Note that under relations (3.5) we have O(x0 ) O(x0 ) and S (x0 ) S (x0 ). Since G is an open subset of Mmâm (F) â Mnân (F) â Mnâm (F), the tangent space Te G to the manifold G at the unit element e is (3.8) Te G = {(U, V , W ) | U M
mâm

(F), V M

nân

(F), W M

nâm

(F)}.

Since M is a linear space, Tx0 M = M. We consider Euclidean scalar products in M and Te G having the form (3.9) x1 ,x2 y1 ,y
1 = trace(F1 F2 ) + trace(G1 G ), 2 = trace(U1 U2 ) + trace(V1 V2 ) + trace(W1 W2 ),

22

where xi = (Fi ,Gi ) M, yi = (Ui ,Vi ,Wi ) Te G , i = 1, 2. Let dfx0 : Te G - M be the differential of fx0 at the unit element e. Using (3.2), it can be shown [6] that (3.10) dfx0 (y ) = (F0 U - UF0 + G0 W, G0 V - UG0 ) M,


954

M. I. GARC´ IA-PLANAS AND A. A. MAILYBAEV

where y = (U, V , W ) Te G . The adjoint linear mapping dfx0 : M - Te G is determined by the relation

(3.11)

dfx0 (z ) = (F0 X - XF0 - YG , G Y, G X ) Te G , 0 0 0

where z = (X, Y ) M. Analogously to Theorem 2.1, the mappings dfx0 and dfx0 provide the following description for the tangent spaces Tx0 O(x0 ), Te S (x0 ) and their normal complements. Theorem 3.1. The tangent spaces to the orbit and stabilizer of the pair of matrices x0 and corresponding normal complementary subspaces can be found in the fol lowing form: 1. Tx0 O(x0 ) = Im dfx0 M. 2. (Tx0 O(x0 )) = Ker dfx0 M. 3. Te S (x0 ) = Ker dfx0 Te G . 4. (Te S (x0 )) = Im dfx0 Te G . Example 3.1. Let x0 = (F0 ,G0 ) be a pair of matrices with 000 1 (3.12) F0 = 0 0 0 , G0 = 0 . 001 0 Then elements z = (X, Y ) of the space (Tx0 O(x0 )) can be found from the equation dfx0 (z ) = 0 in the form 0 000 1 2 0 , 0 , (3.13) 3 3 0 4 where 1 ,...,4 F are arbitrary, and dim(Tx0 O(x0 )) = 4. The elements of Tx0 O(x0 ) have the form µ1 µ2 µ3 µ7 0 0 µ4 , µ8 , (3.14) µ5 µ6 0 -µ5 where µ1 ,...,µ8 F are arbitrary and dim Tx0 O(x0 ) = 8. Then, by Theorem 3.1, dim(Te S (x0 )) = 8 and elements y = (U, V , W ) of (Te S (x0 )) = dfx0 (Tx0 O(x0 )) take the form -µ7 0 -µ3 -µ8 0 -µ4 , (µ7 ), (µ1 , µ2 , µ3 ) . (3.15) 0 2µ5 µ6 Finally, the space Te S (x0 ) = Ker dfx0 is formed by the triples 1 2 3 0 4 0 , (1 ), (0, 0, 3 ) , (3.16) 0 0 5 where 1 ,...,5 F are arbitrary and dim Te S (x0 ) = 5.


REDUCTION TO VERSAL DEFORMATIONS

955

Note that under relation (3.5), the matrix pencil corresponding to pair (3.12) is equivalent to matrix pencil (2.10) considered in Example 2.1. Dimensions of the tangent space to the stabilizer and normal complement of the tangent space to the orbit are the same for the cases of matrix pairs and matrix pencils. But dimensions of the tangent space to the orbit and the normal complement of the tangent space to the stabilizer are smaller in the case of matrix pairs. 3.2. Versal deformation. Let us consider a deformation x( ) of x0 M in the form (3.17) x( ) = x0 +
i=1

ci i ,
c

where {c1 ,...,c } is a basis of an arbitrary complementary subspace Tx0 O(x0 ) to Tx0 O(x0 ); = codim Tx0 O(x0 ). Analogously to Corollary 2.4, we have the following. Corollary 3.2. The deformation (3.17) is a miniversal deformation; i.e., any deformation z ( ), Fk , of x0 can be represented in the neighborhood of the origin U0 Fk in the form (3.18) z ( ) = g ( ) x(( )),

where : U0 - F and g : U0 - G are smooth mappings such that (0) = 0 and g (0) = e. The functions ( ) and g ( ) are uniquely determined by the deformation z ( ) if g ( ) is taken in the form
d

(3.19)

g ( ) = e +
j =1

rj µj ( ),

where µj ( ) are smooth functions in F such that µj (0) = 0, j = 1,...,d, and {r1 ,...,rd } is a basis of (Te S (x0 )) . Recall that if {t1 ,...,td } is a basis of Tx0 O(x0 ), then {dfx0 (t1 ),...,dfx0 (td )} is a basis of (Te S (x0 )) , and, vice versa, if {r1 ,...,rd } is a basis of (Te S (x0 )) , then {dfx0 (r1 ),...,dfx0 (rd )} is a basis of Tx0 O(x0 ). For pairs of matrices, reduced to the Brunovsky canonical form, explicit expressions for the bases {c1 ,...,c } and {n1 ,...,n } may be found in [6]. Example 3.2. Let x0 = (F0 ,G0 ) be the pair of matrices considered in Example 3.1. Using explicit form of the tangent space Tx0 O(x0 ) given in (3.14), we can choose a basis {c1 ,...,c4 } of the complementary space (Tx0 O(x0 ))c such that every ci has exactly one nonzero element. For example, we can choose the miniversal deformation in the form 1 00 0 , 0 , = (1 ,...,4 ). 0 x( ) = 1 2 (3.20) 0 3 0 1 + 4 3.3. Reduction to miniversal deformation. Let x0 and x( ) be a pair of matrices and its miniversal deformation. In order to reduce a given deformation z ( )


956

M. I. GARC´ IA-PLANAS AND A. A. MAILYBAEV

of x0 to the miniversal deformation, we need to find the smooth mappings ( ) and g ( ) satisfying (3.18). These mappings can be found in Taylor series form: ( ) =
|h|s

(h) h + o( s ), h!
d

(3.21) g ( ) = e +
j =1 (0)

r

j |h|s

µj h + o( s ), h!

(h)

where (0) = 0 and µj = 0. Analogously to Theorem 2.5, we can find explicit recurrent formulae for calcula(h) tion of the derivatives (h) and µj up to an arbitrary order. Theorem 3.3. The derivatives 1 ,..., and µ1 ,...,µd determining transformation functions (3.21), which reduce the deformation z ( ) of x0 to the miniversal deformation (3.17), satisfy the recurrent formulae (h) sh ,n1 1 1 . . . = Z -1 . (3.22) , . . (3.23)
(h) (h) (h) (h) (h)

sh ,n

1

(h) µ1 . . =W . (h) µd


-1



sh - sh -

(h) i=1 ci i (h) i=1 ci i

. . .

,t1 ,td


1

,

1

where Z and W are nonsingular â and d â d matrices with the elements zij = cj ,ni 1 , wij = dfx0 (rj ),ti 1 , respectively. The pair of matrices sh M has the form (3.24) sh = z
(h)

-
h +h =h |h |>0, |h |>0

h Ch i=1

ci i

(h )

d

,
j =1

rj µj

(h )

,z

(h )

.

The mapping : Mâ Te G â M - M is defined as fol lows: (3.25) (x, y , z ) = (FU - UX + GW, GV - UY ), where x = (F, G), y = (U, V , W ), and z = (X, Y ). Analogously to the case of matrix pencils, in order to simplify the calculations we can choose the bases {c1 ,...,c }, {n1 ,...,n }, {t1 ,...,td }, and {r1 ,...,rd } in such a way that cj ,ni 1 = 0 and dfx0 (rj ),ti 1 = 0 for i = j , which implies that Z and W are diagonal matrices. Example 3.3. Let us consider the following two-parameter deformation z ( ), = (1 ,2 ), of the pair of matrices x0 = (F0 ,G0 ) considered in Example 3.1: 3 1 2 2 /6 1+ 1 2 1 2 . 1 1 + 2 , 1 (3.26) z ( ) = 2 2 3 1 2 1 2 1 2


REDUCTION TO VERSAL DEFORMATIONS

957

Using the bases {c1 ,...,c4 }, {n1 ,...,n4 }, {t1 ,...,t8 }, and {r1 ,...,r8 } constructed in Examples 3.1, 3.2 and applying Theorem 3.3, we find
2 2 2 1 ( ) = 2 + 1 2 /2+ o( 3 ), 2 ( ) = 1 - 1 2 - 1 2 + o( 3 ), 2 2 3 2 2 3 ( ) = 1 2 + 1 2 + 2 + o( 3 ), 4 ( ) = 1 2 + 1 2 + o( 3 ), 2 µ1 ( ) = 1 - 1 2 /2+ o( 3 ), µ2 ( ) = 1 2 + o( 3 ),

(3.27)

3 µ3 ( ) = 2 /12 + o( 3 ), 2 3 2 µ4 ( ) = 1 + 2 + 1 + 1 2 + 1 + 1 2 + o( 3 ), 3 2 µ5 ( ) = -2 /2+ o( 3 ), µ6 ( ) = 1 2 + 1 2 + o( 3 ), 2 µ7 ( ) = 1 2 /2+ o( 3 ), µ8 ( ) = 1 + o( 3 ).

Expressions (3.27) determine the change of parameters = ( ) and the equivalence transformation g ( ) given by (3.19) in the reduction of z ( ) to the miniversal deformation (3.20). 4. Lo cal analysis of the uncontrollability set for one-input systems. Let us consider a pair of real matrices z = (F, G) M with n = 1 and arbitrary m. This pair corresponds to the system of differential equations (4.1) (t) = F (t)+ G (t)

with m-dimensional state vector Rm and one input variable R. System (4.1) is called control lable if it is possible to construct a control signal (t) that will transfer an initial state to any final state in finite time [16]. The pair z = (F, G) corresponding to such a system is called controllable. The well-known criterion for controllability says that the pair z is controllable if and only if the control lability matrix C = [G, F G,...,F m-1 G] has full rank [16] (4.2) rank [G, F G,...,F
m-1

G] = m.

For one-input systems, i.e., when the matrix G has dimension m â 1, this criterion takes the form (4.3) det[G, F G,...,F
m-1

G] = 0.

Let us consider a family of matrix pairs z ( ) = (F ( ),G( )) with the parameter vector Rk . The set of values of the parameter vector such that the pair z ( ) is uncontrollable is called the uncontrol lability set and will be denoted by N = { Rk | rank C ( ) < m}. Let us assume that the pair z ( ) is uncontrollable at some point 0 N . We are going to analyze the structure of the uncontrollability set in the neighborhood of this point. Due to the complicated entry of elements of the matrices F and G into the controllability matrix, it is difficult to use the controllability condition (4.3) for analytical analysis of the set N . Using reduction of the family z ( ) to the miniversal deformation, this analysis can be carried out in a more simple and systematic way, as shown below. The matrix pair z0 = z (0 ) can be reduced to the Brunovsky canonical form z0 = g0 z0 by the state feedback transformation g0 G [11, 16]. Let us consider the


958

M. I. GARC´ IA-PLANAS AND A. A. MAILYBAEV

case when the Brunovsky form z0 is as follows: 01 . 0 .. .. (4.4) z0 = .1 0

,
0

0 . . . 0 , 1 0

where 0 R is an arbitrary number called the uncontrol lable mode or the generalized eigenvalue. In the generic (typical) case, the parameter vectors , corresponding to the matrix pairs z ( ) having Brunovsky form (4.4), represent typical elements of the uncontrollability set N and form a codimension 1 smooth submanifold of Rk . Uncontrollable matrix pairs having different Brunovsky structures form submanifolds of higher codimensions [6]. The following proposition gives explicit formulae for the tangent plane to the uncontrollability set N at 0 and the first approximation of the uncontrollable mode. Proposition 4.1. Let z0 = z (0 ), 0 N , be a matrix pair having Brunovsky canonical form (4.4) with the triple g0 = (P0 ,R0 ,S0 ) G providing the feedback equivalence transformation z0 = g0 z0 . Let us define real vectors = (1 ,...,k ) and = (1 ,...,k ) with the components (4.5)
- i = P0 1 (m, :) - i = P0 1 (m, :)

F i

m-1


j =1

j -1 P0 0

(:,j )+

G i

m-1


j =1

j -1 S0 0

(:,j )+

m-1 0

R

0

,

F G P0 (:,m)+ S0 (:,m) , i i

i = 1,...,k ,
- P0 1 , the j th column of the uncontrol lability set is the normal vector to by the equation

- where P0 1 (m, :), P0 (:,j ), and S0 (:,j ) denote the mth row of P0 , and the j th column of S0 , respectively. Then, if = 0, N is a smooth hypersurface in the vicinity of 0 ; the vector this hypersurface at 0 ; the tangent plane to N at 0 is given

(4.6)
k

(, - 0 ) = 0,

where (, ) = i=1 i i is a scalar product in Rk ; and the first order approximation of the uncontrol lable mode on the hypersurface N is given by the relation (4.7) ( ) = 0 +( , - 0 )+ o( - 0 ).

Proof. Without loss of generality, we can take 0 = 0. Let us consider the family z ( ) = g0 z ( ), which is a deformation of the matrix z0 = g0 z0 given by (4.4). The deformation z ( ) can be reduced to the orthogonal miniversal deformation of z0 having the form [6] 0 1 0 .. . . . 0 . , .. (4.8) x( ) = , 0 . 1 1 0 m-1 m-2 0 1 1 0 1 ··· 0 1 0 + 2


REDUCTION TO VERSAL DEFORMATIONS

959

where = (1 ,2 ). Since the controllability property is invariant under the feedback group action [16], the controllability of the pair z ( ) is equivalent to the controllability of the pair x(( )), where = ( ) represents the change of parameters in the reduction of z ( ) to the miniversal deformation x( ). Applying the criterion of controllability (4.2) to matrix pair (4.8), we conclude that for small the pair x( ) is controllable if and only if 1 = 0. Hence, the uncontrollability set in the vicinity of 0 is determined by the equation 1 = 1 ( ) = 0. If 1 = 0, then we find the uncontrollable mode = 0 + 2 = 0 + 2 ( ). Using formula (3.22) of Theorem 3.3 and taking into account that the matrix Z is diagonal, we find 1 z - = z111 ,n1 i i
- = z111 m-1

(4.9)

1


j =1

j -1 0

Fmj + i

m-1 0

Gm1 , i

z11 = n1 ,n1

1

2 = 1 + 0 + ··· +

2m-2 0

,

where derivatives are taken at 0 , the pair n1 was found from the orthogonal miniversal deformation (4.8) as a coefficient corresponding to 1 , and Fmj , Gm1 denote the (m, j )th and (m, 1)th elements of the matrices (F, G) = z . Using expression z ( ) = g0 z ( ), we obtain (4.10) F ( ) = P
-1 0

(F ( )P0 + G( )S0 ),

G( ) = P

-1 0

G( )R0 .

Substitution of (4.10) into (4.9) yields 1 F - - = z111 P0 1 (m, :) i i (4.11) + G i
m-1 m-1


j =1

j -1 P0 0

(:,j )


j =1

j -1 S0 0

(:,j )+

m-1 0

R

0

.

Hence, using the notation of (4.5), we find the gradient vector of the function 1 ( ) at 0 in the form (4.12) 1 = 1 1 ,..., 1 k
- = z111 .

If = 0, then 1 = 0 and, by the implicit function theorem applied to the equation 1 ( ) = 0, we conclude that the uncontrollability set is a smooth hypersurface in the vicinity of 0 with the tangent plane (4.6). The vector is normal to this surface at 0 . Analogously, we find z 2 - = z221 ,n2 i i
- = P0 1 (m, :) 1

=

(4.13)

Fmm i

F G P0 (:,m)+ S0 (:,m) . i i

Hence, using the notation of (4.5), we find the gradient 2 = at 0 , which gives approximation (4.7) for the uncontrollable mode ( ) = 0 + 2 ( ).


960

M. I. GARC´ IA-PLANAS AND A. A. MAILYBAEV

Fig. 2. Elastic system control led by a force F .

Note that Proposition 4.1 provides quantitative local information on the uncontrollability set using only information on the matrix pair z0 = z (0 ) and derivatives of the system matrices F ( ) and G( ) evaluated at the point 0 . Using this information we can choose an optimal change of parameters in order to obtain a good-controllable system. Formula for the tangent plane is useful for numerical computation of the uncontrollability set. A multi-input system is characterized by a vector of real input variables (t) in (4.1). In this case uncontrollable pairs have different Brunovsky forms, and corresponding miniversal deformations are more complicated. The suggested approach can be extended to analysis of the uncontrollability set for a multi-input dynamical system depending on parameters. For this purpose, we need to find the uncontrollability set for that particular versal deformation, and then transfer the result to the original parameter space by means of the mapping = ( ) found by Theorem 2.5. Example 4.1. Let us consider the mechanical system shown in Figure 2. The system consists of a light platform of length L carrying a point mass m in the middle; both ends of the platform are supported on the ground by means of springs with elastic coefficients k1 , k2 and damping coefficients c1 , c2 . The system is controlled by a force F applied to the platform at the distance 1 L from the left end. We assume that the equilibrium of this system for F = 0 corresponds to the horizontal position of the platform. Equations of motion of the system have the form (4.14) m(¨1 + x2 )/4+ c1 x1 + k1 x1 = (1 - 1 )F, x ¨ m(¨1 + x2 )/4+ c2 x2 + k2 x2 = 1 F, x ¨

where x1 and x2 are vertical displacements of the left and right ends of the platform, respectively. Taking m = 1, c1 = c2 = 1, k1 = 2 , k2 = 3 , F = and introducing new state variables 1 = x1 + x2 , 2 = 1 , 3 = x2 , after simple manipulations we obtain system (4.1), depending on the vector of parameters R3 with one control variable , the state vector R3 , and the matrices (4.15) 0 F ( ) = -22 2 /2 1 0 -2 2(2 - 3 ) , 1/2 -(2 + 3 )/2 0 . 2 G( ) = 1 - 1/2

Let us consider a point 0 = (1/4, 3/2, 5/6) in the parameter space. At this point


REDUCTION TO VERSAL DEFORMATIONS

961

Fig. 3. Uncontrol lability set and its tangent plane.

the pair of matrices (4.15) takes the form (4.16) 0 F0 = -3 3/4 1 0 -2 4/3 , 1/2 -7/6 0 G0 = 2 . -1/4

It is straightforward to check that transformed to the Brunovsky form of the following form: 1 0 0 1 0 (4.17) P0 = 0 3/8 -1/8 1

the pair (F0 ,G0 ) is uncontrollable and can be (4.4) with 0 = -1 by the triple (P0 ,R0 ,S0 ) G , R0 = 1/2, S0 = (5/4, 13/12, -2/3).

Using (4.15) and (4.17) in (4.5), we find (4.18) = (2/3, 1/8, -3/8), = -(2/3, 1/4, 3/4).

Hence, by Proposition 4.1, the uncontrollability set is a smooth hypersurface in the vicinity of 0 . The tangent plane to this surface at 0 is given by the equation (4.19) (, - 0 ) = 21 2 33 1 + - - = 0, 3 8 8 24

and the perturbation of the uncontrollable mode on this surface has the form (4.20) ( ) = -1 - 2(1 - 1/4) 2 - 3/2 3(3 - 5/6) - - + o( - 0 ). 3 4 4

The plane (4.19) is plotted in Figure 3 (bold rectangular). For comparison, the uncontrollability set found numerically using (4.3) (determinant of the controllability matrix changes the sign when we cross the uncontrollability set) is shown in Figure 3. Numerical computations confirm the analytical results.


962

M. I. GARC´ IA-PLANAS AND A. A. MAILYBAEV

5. Conclusion. The general idea of any normal form theory is to transform an ob ject under consideration to a form whose properties are easy to analyze. In this process both the normal form and transformation to it are important. For example, the Jordan normal form of a square matrix determines its spectrum, while knowledge of the transformation to the Jordan form (change of basis) allows us to find explicitly a general solution to the corresponding dynamical system. In this paper we have solved the second part of the normal form problem (finding the transformation) in the reduction of families of matrix pencils and matrix pairs to the local normal form (miniversal deformation). Information on the transformation (the change of parameters and equivalence transformation) allows the development of the multi-parameter perturbation theory for multi-input linear dynamical systems. In a similar problem for square matrices, advantages of this approach for the perturbation analysis of the spectrum and stability of linear dynamical systems depending on parameters have been illustrated in [3, 12, 14, 15]. In section 4 of this paper it has been shown that the suggested method is useful for the controllability analysis of single-input dynamical systems dependent on parameters. Acknowledgment. The second author thanks M. I. Garc´ ia-Planas for the hospitality during his staying at the Department of Applied Mathematics I, UPC, Barcelona.
REFERENCES [1] V. I. Arnold, On matrices depending on parameters, Russian Math. Surveys, 26 (1971), pp. 29­43. [2] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1988. [3] J. V. Burke and M. L. Overton, Stable perturbations of nonsymmetric matrices, Linear Algebra Appl., 171 (1992), pp. 249­273. ¨ [4] A. Edelman, E. Elmroth, and B. K° agstrom, A geometric approach to perturbation theory of matrices and matrix pencils. Part I: Versal deformations, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 653­692. [5] A. Edelman and Y. Ma, Staircase failures explained by orthogonal versal forms, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1004­1025. [6] J. Ferrer, M. I. Garcia-Planas, and F. Puerta, Brunowsky local form of a holomorphic family of pairs of matrices, Linear Algebra Appl., 253 (1997), pp. 175­198. [7] F. R. Gantmacher, The Theory of Matrices, Vols. 1, 2, Chelsea, New York, 1959. [8] M. I. Garcia-Planas and M. D. Magret, Deformation and stability of triples of matrices, Linear Algebra Appl., 254 (1997), pp. 159­192. [9] M. I. Garcia-Planas and M. D. Magret, Miniversal deformations of linear systems under the ful l group action, System Control Lett., 35 (1998), pp. 279­286. [10] M. I. Garcia-Planas and V. V. Sergeichuk, Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils, Linear Algebra Appl., 302­303 (1999), pp. 45­61. [11] I. Gohberg, P. Lancaster, and L. Rodman, Invariant Subspaces of Matrices with Applications, Chelsea, New York, 1977. [12] A. A. Mailybaev, Transformation of families of matrices to normal forms and its application to stability theory, SIAM J. Matrix Anal. Appl., 21 (1999), pp. 396­417. [13] A. A. Mailybaev, Transformation to versal deformations of matrices, Linear Algebra Appl., 337 (2001), pp. 87­108. [14] A. A. Mailybaev, On stability domains of nonconservative systems under smal l parametric excitation, Acta Mechanica, 154 (2002), pp. 11­30. [15] A. A. Mailybaev and A. P. Seyranian, On singularities of a boundary of the stability domain, SIAM J. Matrix Anal. Appl., 21 (1999), pp. 106­128. [16] E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, Texts Appl. Math. 6, Springer-Verlag, New York, 1990. [17] L. Stolovitch, On the computation of a versal family of matrices, Numer. Algorithms, 4 (1993), pp. 25­46. [18] A. Tannenbaum, Invariance and System Theory: Algebraic and Geometric Aspects, Lecture Notes in Math. 845, Springer-Verlag, New York, 1981.