G Kalogeropulos - On the spectral representation of solutions for generalized autonomous continuous and discrete linear systems - страница 1

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ВІСНИК ЛЬВІВ. УН-ТУ

Сер. прикл. матем. та інформ.

2008. Вип. 14. C. 17-31

VISNYKLVIV UNIV Ser. Appl. Math. Comp. Sci. 2008. No 14. P. 17-31

УДК 512.83:512.64

ON THE SPECTRAL REPRESENTATION OF SOLUTIONS FOR GENERALIZED AUTONOMOUS CONTINUOUS AND DISCRETE LINEAR SYSTEMS

G. Kalogeropulos*, O. Kossak**

Athens National Kapodistrian University, Athens, Greece; **Ivan Franko National University of Lviv Universytetska str. 1, Lviv, 79000, e-mail: olhakossak@yahoo.com

In this paper, the representation of the solution of generalized autonomous continuous and discrete linear systems in spectral form is under consideration. We assume that the system matrix pencil sF G is regular and det F = 0 . For the exact solution of both problems, the Weierstrass canonical form of the matrix pencil described above is used. The canonical form can be constructed by means of finite and infinite elementary devisors.

Key words: Canonical form, linear systems, generalized autonomous continuous and discrete linear systems regular matrix pencil, finite and infinite elementary devisors

1. INTRODUCTION

Let us consider the autonomous continuous control system [2]

Fx (t ) = Gx (t) + Bu (t) (1.1)

where F, G, B, are respectively n x n, n x n , and n xl - matrices with time independent coefficients; state x(t) and input u(t) are nxl and lxl vector of t function, respectively.

Also consider the corresponding of (1.1) system in discrete form, which has a form

Fxk+1 = Gxk + Buk (1.2) where, xk = x(tk) and uk = u(tk) the state and input vector, respectively, at the time tk = k T, k є Z and T is the sampling period [3].

In both systems (1.1) and (1.2), the relative matrices can be complex, although in many applications are appeared to be real. In our particular case [1], both systems have associated regular pencil sF G and det F = 0.

The study and investigation of (1.1) and (1.2) are connected with the investigation of the homogeneous continuous and discrete equations

Fx (t ) = Gx (t) (1.3)

Fxk+1 = Gxk (1.4)

which are called generalized autonomous continuous and discrete systems, respectively.

This paper provides the solution of the generalized autonomous continuous and discrete systems for the regular case, in a spectral form.

2. THE CONTINUES TIME STATE

Let us regard system (1.3) in the regular case when det (sF G)^ 0, and det F = 0, for an arbitrary initial condition

© Kalogeropulos G., Kossak O., 2008

x(t0) = x0 (2.1) and, now, let LTnn be the set of the n x n - regular matrix pencils

LTnn ={sF- G : F, G є Dnxn,det (sF - G) ф 0} .

In the case where sF - G є LTnn and det F = det G = 0 , we have elementary devisors of the following types [5]:

1) zero elementary divisors are those of the type sp;

2) non - zero finite elementary divisors are those of the type (s - a)e, with a ф 0 ;

3) infinite elementary divisors are those of the type sq

The complex Weierstrass canonical form sFW - GW of the regular pencil sF - G where det F = 0 is defined by

P (sF - G )Q = sFw - Gw = blockdiag (sIp- J p(a); sI„- J¥(0); sHq - Iq) (2.2)

where P and Q are non singular n xn -matrices, i.e. |P| ф 0,Q| ф 0 .

The first normal Jordan-type block sIp - Jв (a) is uniquely defined by the set of nonzero finite elementary divisors

V

(s))  (s-av))),   = P

j=1

it is possible aj = aj,  i, j є {1,2,...,v} .

The second Jordan - type block sI¥- J¥(0) is uniquely defined by the set of zero finite elementary divisors

sP1,...,sPf     Pj = \//,   p1 < p2 < ... < Pf , р + у = т

j=1

and it has the following format

sIT - Jr(a)= bl0ck diag {sIT - JT (a1 ),..., sIV - JT sIP1 - JP1 sIPf - JPf (0)}

The last q blocks of the (2.2), i.e. sHq -1 , are correspond to the infinite elementary divisors

Y

s"Y , ^ qj = q, q1 < ... < qY

j=1

of the pencil sF - G associated with the blocks

sHq - Iq = bl0Ck diag {sHqi - Iq1,.-sHq, ~ Iq,}

where Hq is a nilpotent block matrix.

The regular pencil sF - G has nonzero finite elementary divisors (s -a) if and only if there exists a maximal chain of linearly independent vectors (x1, x2,..., xd) such that [4]

Gxt = aFxj + Fxj-1,1 < i < d, x0 = 0

The regular pencil sF -G has a zero finite elementary divisors sP if and only if

there exists a maximal chain of linearly independent vectors (x1, x2,..., xP) such that

Gxi=Fxi-1,1<i <P,x0=0

The regular pencil sF - G has a infinite elementary devisors s'q if and only if there

exists a maximal chain of linearly independent vectors (x1, x2,..., xq) such that

Gxi-1=Fxi,1<i <q,x0=0.

Also it can be proved that the above chains define a set of n linearly independent vectors over.

Let (s-a)T ,i = 1,2,...,V, sPi,i = 1,2..., f and sq,i = 1,2...,, are finite and infinite elementary divisors [7] of the regular pencil sF - G. It is known that for each nonzero finite elementary divisors (s -at )T' , the corresponding block of the Weierstrass canonical form is IPi- JPi(ai); for each zero elementary divisors, the block of the Weierstrass canonical form is IP - JP (0); and for each infinite elementary divisors sq the block is sHq - Iq , respectively. Moreover,

Г1 0 ...  01

0 1 ... 0

0 0 ... 1

T

і ~ai   1   ...   01 t

0 a ... 0

|_ 0 0 ... a

    T

і

[0   1   0 ... 01

0   0   1 ... 0

0   0   0 ... 1

0   0   0 ... 0

q

(2.3)

where Hq is a nilpotent matrix of annihilation index qi and also

hi

-Hq Hq ...Hq (for 1 < i < qi -1)

Additionally, H = blockdiag(H      Hq ) is qxq nilpotent matrix of nilpotency

index q = max {qt, i = 1,2,...,,} . We can also notice that Jt (0) = Ht. From the above is clear that [6]

 

 

0

0 "

 

 

0

0"

PFQ = Fw =

0

 

0

, PGQ = Gw =

0

 

0

 

0

0

 

 

0

0

 

(2.4)

where

Hq = blockdiag (Hql, Hq2,..., HqY ) ,   Jp = blockdiag (J\ (a1 ),..., Jrv (av)) ,

= blockdiag (H

j=1 i=1 i=1

Returning to the equation (1.3), we can write

I

і

Fx = G(x), x(t) = Qy(t).

(2.5)

Multiplying both sides of (2.5) on the left, by the non singular matrix P, we obtain

FwQy (t ) = GwQy (t), y (t0 ) = Q- x (t0

(2.6)

Furthermore, Q is an invertible matrix being divided into n xft,n x p, n x q relative

sub-matrices \_Q„e Qn,¥ Qnq ], where Qn p is the matrix having as columns the vectors

which defined by the nonzero finite elementary devisors, Qn¥ is the matrix having as

columns the vectors which defined by the zero finite elementary divisors and Qn p defined

by the set of infinite elementary divisors.

The existence of elementary divisors guaranties the existence of maximal and linearly independent chains of vectors that satisfy the following equations

Ui = (uil,ui2 un, )R = (r^Гі2 rp, )Z< =(w^l,Wi2,...,w4 ) where Qn,= [Qn,e   Qn,r] = [U   ...  Uv   R   ...  R,] . Since the rankQm=T< n , there exists a left inverse matrix of QnT denoted by (Q„T) ) such that

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