Time-invariant representation of discrete periodic systems

A,rromar;c~, Vol. 32. No. 2. pp. 267-212, 19% Copyright 0 lYY6 El&&r Science Ltd Printed in Great Britain. All rights reserved 0(xl5-I0Y8/% $15.00 + om

ooos1098(95)00130-1

Brief Paper

Time-invariant

of Discrete Periodic

Representation Systems* PRADEEP

Key Words-Periodic

MISRAt

systems; system equivalence: generalized state-space systems.

tionally expensive, the use of state transition matrix for time invariant representations can (conceivably) introduce significant numerical errors. The second approach is based on a discrete version of the FIoquet transform (Callier and Desoer, 1991: Hench and Laub, 1992: Van Dooren and Sreedhar, 1994). The results in Van Dooren and Sreedhar (1994) are of particular interest. The authors use a periodic version of the well-known Schur decomposition, and are able to address the case of singular periodic systems as well as those described by (1). Of course the Floquet transform deals with homogeneous differential equations, and hence does not take into account the input (B,), output (C,) and input-output (Dk) matrices. The third approach is based on a lifted representation of the periodic systems as developed by Verriest and co-workers (Verriest, 1988, Park and Verriest, 1989; Verriest and Park, 1994: Verriest and Kullstam, 1995). Related work by Flamm (1991) is also of interest. In this paper a new procedure to obtain a time-invariant representation of the periodic system (1) is presented. The proposed method starts with a lifted (w-stacked) representation and, using system equivalence transformations, obtains a realization similar to that given in Meyer and Burrus (1975). However, unlike the algorithms described in Meyer and Burrus (1975) and Grasselli and Longhi (1991a), it does not require the computation of the state transition matrix of the periodic system. The algorithm proposed in this paper uses only unitary transformations and solutions of systems of algebraic equations. Both of these operations are known to be numerically very reliable; hence the computational method based on the main results of this paper has attractive numerical properties.

large number of results from linear timeinvariant system theory can be extended to periodic systems provided an equivalent time-invariant system can be found. This paper presents a simple and numerically reliable procedure to achieve the same. It is shown that, using a stacked representation of periodic systems, under system equivalence, a minimal-order generalized state-space description can always be obtained. Abstract-A

1. Introduction In recent years there has been considerable interest in the study of processes that can be modeled by linear difference equations having periodic coefficients with period o. A state-space description of periodic systems is given by x(k + 1) = A,x(k)

+ B,u(k),

y(k) = C,x(k) + &u(k),

(1)

where k EZ, XEC”, UE@II’, ye@“, Ak, B,, Ck and Dk (k = 0, I,. , w - 1) are (possibly) complex periodic matrices of compatible dimensions with period w (following the notation in Grasselli & Longhi (1991a), we shall refer to systems characterized by (1) as o-periodic systems). Such equations are encountered in the study of periodically time-varying digital filters (Meyer and Burrus, 1975), representation of multirate sampled data systems (Araki and Yamamoto, 1986), robust control of time-invariant systems (Khargonekar et al., 1985) etc. (for other applications, see Richards, 1983: Bittanti, 1986). The system in (1) has been investigated by researchers, most notably by Grasselli and co-workers (see Grasselli and Longhi, 1991a-c; Grasselli and Tornambe, 1992), where a control theory (based on a state-space description) is including eigenvalue assignment, state and developed, output dead-beat control, disturbance localization, model matching, robust tracking and regulation, etc. (see also the work of Verriest and co-workers: Verriest, 1988: Park and Verriest, 1989). In several of these papers the periodic system must first be transformed to a time-invariant representation under system equivalence transformations (Rosenbrock, 1970: Pernebo, 1977). At present, there are three possible approaches to obtain a time-invariant representation of discrete periodic systems. The first makes use of the state transition matrix (Meyer and Burrus, 1975). Determination of the state transition matrix involves computation of products of matrices, which can lead to numerical difficulties. Hence, apart from being computa-

2. Preliminaries In this section we introduce the notation to be followed in the rest of the paper, and review some essential results pertaining to time-invariant state representation of wperiodic systems. 2.1. Associated time-invariant systems. Given the periodic system in (l), one can associate with it the following time-invariant representation: f,(h + 1) = k&(h) y,(h) =&f,(h)

+ &l&z), (2)

+ &u,(h),

where

* Received 19 January 1995; received in final form 3 July 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor A. L. Tits under the direction of Editor Tamer BaSar. Corresponding author Professor Pradeep Misra. Tel. +l 513 873 5062: Fax +I 513 873 5009: E-mail [email protected]. t Department of Electrical Engineering, Wright State University, Dayton, OH 45435, U.S.A.

A, :=

1 “5’ I, lH,,_,

i,:=

L k.0.1, L is.I .o

[, L.:-I.o 267

(3)

1 L k.0. I L k.l.1

L-l.,“’

L !..o.cu

:”

L !..l.<”

I I

” : Lk.,” ‘Lo I I

,

268

Brief Papers dimensions (Grasselli et al., 1992). Then (9) represents Rosenbrock’s strict system equivalence relation. Further, it can be shown that the following holds.

and Vi,jEE,

@(i,j):=A,_,...A,+,A, *(j, j) := I,

i>j,

Vj E Z, j=O ,...,

G,,j:=@(k+o,k+j+l)B,+j, j=O ,...,

H,,j:=C,+j@(k+j,k), L,,,,;:=O,

j=l,...,

(Pkk,&, fiik, Lk)

w-l,

(4)

Lk,;., := C,+jcP(k + i, k + j + l)B*+,, j=O ,..., $(h)

w-2,

and

systems described

by the

(kk, e,, fii,, &)

in (2) and (9) respectively are system-equivalent, i.e. there exists a constant nonsingular transformation matrix T such that

i
w-l,

j=O,...,o-1,

L.,., := Dk,,,

Theorem 2.1. The time-invariant 4-tuples

w-l,

i=j+l,,..,

fik = T-‘&T, Proof.

See the Appendix. Grasselli et al. (1995).

w-l,

8, = i&T,

& = T-‘ek

An alternative

i, = Lk.

proof appears in

:= x(k + ho),

Ilk(h) := [u’(k + hw)

.

In the sequel, all of Rosenbrock’s strict system equivalence transformations will be referred to as strict system equivalence transformations. 2.4. Reachability/observability of periodic systems. The concepts of reachability and observability are well known for linear discrete periodic systems. Let

UT(k + hw + 1)

U-yk + hw + w - l)]‘,

y&l) := [yT(k + ho)

yT(k + hw + 1)

yT(k + ho + w - l)]‘.

P’,(A) = [@- AI,

The system (3), with the matrices and vectors as defined in (4) is known as the associated system at the initial time k of the given o-periodic system (Meyer and Burrus, 1975). 2.2. w-stacked form. The periodic system in (1) can also be expressed as the following time invariant state-space form: ~(A)%(h) =&&(h)

+ %fnk(h)> (5)

Yk(h) = %5,(h) + %nf#), where u&(h) and yk(h) are defined as in (4), but &(h) := [XT@+ ho)

Pg(A) =

i;,],

[’ ;;I”],

Theorem 2.2. The periodic system (2) is reachable (observable) at time k if and only if the matrix Pi(A) or pi(A) (Pg(A) or e(A)) has full row (column) rank for all A E C. Proof

with A denoting the one-step-forward time operator in the variable h or the w-step-forward time operator in the variable k. Further, the matrices _&, Sk, %‘kand Sk are block-diagonal matrices defined as

, Aktw_,},

Ak+,,

&:=blockdiag{B,,Bk+,

,..., Ck+,,

Bk+“_,}, , Cktw_,},

See Grasselli and Longhi (1991a).

2.5. Column compression by unitary transformations. Given an arbitrary row vector x E 62, it is always possible to find a unitary n X n matrix W such that

i;(A) 21, with

det [Sa, - %(A)] # 0. (8)

Then it can be shown that

where U(A) and V(A) are unimodular Pk, ek, fi, and Lk are constant matrices

0

. . . 0 &J.

(10)

The matrix W can be obtained using Householder transformation or a sequence of Givens rotations between the adjacent elements of the vector x (Golub and Van Loan, 1989). 2.6. Transformation of matrix pencils. For a regular matrix pencil (A - AE), i.e. det (A - AE) is not uniformly zero, where A and B E Cnx”, we can always find two unitary transformation matrices U and V E Cnxn, such that

(7)

Equations (5)-(7) define the w-stacked form at the initial time k (=k,,) of the given w-periodic system (Grasselli and Longhi, 1991a). 2.3. Equivalence of associated systems and w-stacked form. Let y’(A) denote the system matrix for the o-stacked realization, where 5&(h) := p

$‘$(A) = [ 9pk ;;(^)].

Then the following holds.

xW=[O

%$:= block diag{&

2&],

x”‘(k + ho + 1)

(6)

~4~:= blockdiag{Ak,

??;(A) = [x.&- %(A)

matrices, and of appropriate

“‘EV=[++],

UTAV=[w],

(11)

where, because of the assumption that det (A - AE) ZO, AZ2 E @Cn~‘)xC”-r~has full rank (= n - r) (Misra and Patek 1989). 3. State-space representation of o-stacked form In this section we present the main results of the paper. In the sequel it is assumed that the periodic system under consideration is reachable and observable. 3.1. o-stacked form to singular descriptor form. Consider the nw X no matrix pencil [s& -%(A)] of the w-stacked form (8). When expanded, the pencil has the following structure:

=Id LO

..:

Ak+‘_?

...

6.

0

Ak+w_ll

ji,@)> (12) I

269

Brief Papers where Q(h) is defined in (6). On premultiplying the block permutation matrix

ro

0 0.’

1”

Cd:n(w-‘)X+-l),

B,

E

@?Ix”W,

B,

E

@w-‘)xmw,

C,

E

cP:p,xn

and C2 E C~)puxn(w-i). Note that the system equivalence transformation to obtain (18) makes E,, a non-zero matrix. The corresponding partitioned system matrix is given by

. . . 0 1,~

I . ‘:

g3=

(12) with

I

(13)

we get rhI,

0 .

01

I

I,,

0

0

..

Ixdh)

.

Lo

I;,1

0 ..:

Since AZ2 is an invertible matrix, the strict sytem equivalence transformations

_

0

f,(h). :1

(i) premultiplication

...

0

(ii) addition of the second block column postmultiplied (-AFiB,) to the third block column, and

Ak+“_, 0

0 .

0 .

.

D

...

the elements

0

0 ..

(iii) interchanging produces

the first two block rows and columns

of the matrix pencil, (14) is

G(h)

OJ

0

0

0

0

A,

-I,,

0

0

6

.:

-;,

0

-0

..

or, equivalently In+,,)?

&+,-I

(by eliminating

Ak+w z

0

0

-I,,

Finally, on applying the block permutation transformation $!7’ to the entire state equation, we get a state-space model in the descriptor form AEx(h) = Ax(h) + Bu(h), \-‘-I

y(h) = Cx(h) + Du(h),

where the matrices E and A are as described in (15) C and D are as described in (7) and the input matrix is given by .

0 0

the block column below

Y(A) :=

0

It+-‘) Ak +*,m3

0 B,

Bkiw_, 0 1

In the sequel, where there is no possibility of ambiguity, we shall not explicitly show the dependence of various to simplify the notation, parameters on k. Further, x(h) := E(h). Note that the system matrix corresponding to (16) is given by Y(A)= [*I

. A,, - AE,, : B, - A,2A;:B2 + AE,,A;:B, ______-____c____________________________ D - C,A;:B, c, : j (21)

Remark 3.2. In view of the proof in the Appendix, the resulting least-order system in (21) has order n. This is possible only if El1 has full rank. Otherwise, as shown in Misra and Pate1 (1989). further reduction in the order of the system can be achieved.

Since E,, is nonsingular, transformed as

the system

matrix

AE,,x,(h) = Ai+,

+ (AIIEIIIEIzA;~~B~

-A,+Gz’B:! + B,)u(h),

(17)

Remnrk 3.1. It should be noted that the descriptor matrix E in (16) is singular. Further. the output equation is left unchanged. 3.1. The w-stacked

form in (5) is strict systemequivalent to the singular descriptor form in (16). Proof: This follows from the transformations obtaining (16) from (5).

employed

in 0

3.2. Minimal state representation of singular descriptor form. Next. consider the system described by (16). By virtue

of (11). we obtain the following conformably system:

A[~w]x(h)

= /+++]x(h)

E,,.

+ [z]u(h), (18)

y(h) = IC, where

partitioned

A,, t C”x”.

1W(h) + Wh), EL*. A,? E Cnxn(“~‘),

AZ2 E

may be

The generalized state-space representation of the resulting system matrix (ignoring the I,,,_ ,r and corresponding block row and column) is given by

y(h) = C+,(h) + (GK;‘E,P%~‘Bz

Lemma

by

(14)

0 By rearranging equivalent to

of the second block row by AT;,

(22)

-C2A~;BZ + D)u(h). Remark

3.3. Since E,, is invertible, the above descriptor form can be easily converted to a state-space form by premultiplying the state equation by Efi’.

4. Algorithm implementation and examples 4.1. Implementation and operations count. The results presented in the previous section may also be used as a possible computational procedure for determining the minimal-order time-invariant realization of the w-stacked representation of periodic systems. However, it is clear from (20)-(22) that a direct implementation would require inverses of E,, and AT2. While, theoretically. the inverses exist, any ill-conditioning of the matrices El, and AZ2 can introduce significant rounding errors in the resulting time-invariant representation. In this section an alternative is developed such that the same realization may be obtained more reliably.

270

Brief Papers

Consider the partitioned system matrix in (19). It may be written as the following (possibly) nonsquare matrix pencil:

Finally, rewriting the pencil as a system matrix and interchanging the first two block rows and columns, we get

~~~,_[~~~_~~_~_~[_~~~_~~_~_~_I.

.3-(h):= (23)

To elimimate & we perform column compression matrix [A= &J with unitary Z such that

We define an elementary transformation 1, 0

v=

0 Z,zz;:

0

kf,

9

where the block denoted by * is irrelevant. Lemma 4.2. The system matrix in system-equivalence to the system in (17)

(28)

is

strict

Proofi Since (A - hE) is regular, the matrix AZ2 is nonsingular. Then the system matrix in (28) can be brought to the form described in (21). Further, (28) and (21) are similar within a constant-state coordinate transformation. 0

matrix as

0 &+I-1)

C0

on the

(28)

,

(25)

I

and postmultiply the system matrix pencil Y{‘(h)by V. It is worth mentioning that the system matrix was written as a matrix pencil for clarity of presentation. The same operation can obviously be performed directly on the system matrix. The transformation pencil has the structure

Remark 43. There are two advantages in forming the (1,2) block of the elementary transformation as Z&&’ instead of -A;zlBp

1. Eficiency. m x [w -

The

former

is equivalent

to

solving

an

l)th order system of equations, while the latter

corresponds to solving an n X (w - 1)th order system of equations. Since m is usually considerably smaller than n, the resulting algorithm is more efficient+ 2. Aecrcrucy. If X is computed as Z12Z2;_1 then

CW Remark 4.1. Note that since (A - AE) is a regular pencil, the submatrix AzZ has full rank. Hence the column compression in (26) always exists. Further, since AZ2 is nonsingular, the elementary transformation in (25) is unique, with the (1,2) block matrix corresponding to the solution of the linear algebraic equations A& = -B2. Clearly, to form X, the matrix 2, must be invertible. The following Iemma ensures its invertibility. Lemma 4.1. Given a full-rank matrix AZ2 and a matrix B2 and a unitary matrix 2 such that

&

is invertible {has full rank).

where cr@ represents the last singular value of 2&Z. Therefore Z can be chosen such that u8(Z2,) is maximized (Golub and Van Loan, 1989). Clearly, this provides us some control over the construction of X, which is not the case if the transformation is formed as -A&‘B2. Operations count. On first sight, it may appear that the column compression characterized by (19) would require 0((0n)~) operations, since the dimension of the block of A being column-compressed is (w - I)n X wn. However, on closer inspection, it is evident that this compression can be performed very efficiently because of the block bidiagonal structure of the state matrix in (16). Specifically, starting at the bottom block row, we only need perform the compression

IO Z

PruoJ Note that since 2 is unitary, l2 has full column [ z22 I rank. We assume that there exists a nonzero vector y such that &$ = 0. Then, since AZ2 has full rank,

212 has full coIumn rank, Zlzy = 0 implies y = 0. I &22I q The proof follows by contradiction. Since

Next, we eliminate the & the column compression [&I

term from (26) by performing

OI:= [E,,

&&,

using a different unitary transformation matrix 2 and forming a corresponding elementary transformation as in (25). On completing the system equivalence transformation, the pencil in (2.6) becomes

(27) Remark 4.2. Since the submatrix E,, has full rank, the coIumn compression in (27) can always be accomplished. Further, in view of Lemma 4.1, the existence and uniqueness of the elementary transformation to get (27) is assured.

&+w-&= [Ak+w--2

-MZ.

On updating the (block) first and (w - 1)th row, the structure of -1, in the (w - l)th row is destroyed. However, Ak+w-3 may still be column-compressed using the matrix replacing -I,. This is easily continued until the entire (o - 1)~ X wn block is column-compressed. Then clearly the operations count for this step is O(on’). The total cost of determining least-order realization may be broken down into the following categories. Column compression in (19) and system update. As discussed above, the column compression may be accomplished in O(wn’) operations. Because of the sparseness of various matrices in the o-stacked representation, update of the system parameters takes O(wn3) operations. 2. Column

compression

in (24) and system update.

result of the column compression of the state submatrix AZ2 is (block) upper-triangular. column compression in (24) cam be performed operations. The system parameters can be another O(u(m + p)n”) operations.

As a matrix, the Hence, the in O(wnn’) updated in

3. Culumn compression in (27) and system update. Here again, the column compression can be performed in 0(omn2) operations, and update of the system parameters requires an additional O(ompn). The overall operations count is then O(wrz”). 4.2. lllustradve example. For the purpose of illustration, we consider the periodic system studied in Grasselli and

271

Brief Papers Longhi (1991a). The system has the following parameters: n = 3, m =p = 1 and o = 2. The matrices characterizing the models are &=[;

6

;],

b#+],

A,=”

0 01, do =0,

q= [l

The state representation

;

c, = [0

;I,

b,=[pl.

1 O], d, = 0.

of the w-stacked form for k = 0 is

given by

systems has been presented. Unlike the existing methods requiring the state transition matrix, which is computationally expensive and can be numerically unreliable, the proposed method uses numerically reliable unitary transformations and solution of linear systems of equations. While conditioning of elementary transformation can cause numerical errors, it has been shown that some control on construction of these transformations can be exercised to make them better conditioned. The results presented in this paper can be extended to singular periodic systems, and this is currently under investigation. Acknowledgements-The

On substituting the matrices and rearranging described in (15), we get

the system as

1

100000 010000 *001000

References

%@I

I 000000

I

%dh)+

-0

1

0

0

0

1

0

0

h(h),

1 0 -0

[o

’ ’ ye(h)= 0

’

0

0

’ O]&(h) + [o” i+,,(h). 1 0

’

0

For clarity of presentation, we use elementary transformations instead of unitary ones. Following the proof in the Appendix, we define a transformation matrix T as 13

T= On performing the Appendix, we get

[ 41

0 13

1

transformations

described

in

the

100000 001000 000000 000000 000000

, B=

A=

Araki, M. and K. Yamamoto (1986). Multivariable multirate sampled-data systems: state space description, transfer characteristics, and Nyquist criterion. IEEE Trans. Aurom. Control, AC-31, 145-154. Bittanti, S. (Ed.) (1986). Time Series and Linenr Systems. Springer-Verlag, Berlin. Calher. F. M. and C. A. Desoer (1991). Linear Svstem Theory. Springer-Verlag, New York: ’ Flamm, D. S. (1991). A new shift-invariant representation of periodic linear systems. Syst. Control Len., 17, 9-14. Golub, G. H. and C. Van Loan (1989). Matrix Computations, 2nd ed. The Johns Hopkins University Press, Baltimore. Grasselli, 0. M. and S. Longhi (1991a). Finite zero structure of linear periodic discrete-time systems. Inf. .I. Syst. Sci., 22,178.5-1806. Grasselli, 0. M. and S. Longhi (1991b). The geometric approach for linear periodic discrete-time systems. Lin. Alg. Applic., 158, 27-60. Grasselli, 0. M. and S. Longhi (1991~). Pole placement for non-reachable periodic discrete-time systems. Mathe. Control, Sig. Syst., 4,439-455.

Grasselli, 0. M. and A. Tornambe (1992) On obtaining a realization of a polynomial matrix description of a system. IEEE

’

010000 E=

author would like to thank Antonio Tornambb for providing him a copy of the report by Grasselli et al. (1992) and the preprints of Grasselli et nl. (1995). Constructive criticism and detailed review of the manuscript by the referees is also gratefully acknowledged.

Trans. Autom. Control, AC-37,852-856.

Grasselli, 0. M., S. Longhi and A. Tornambe (1992). System equivalence for periodic models and systems. Tech. rept. R-92-03. Dipart. di Ing. Elettronica. 11 Universita degli Studi di Roma, ‘Tor Vergata’, Roma, Italy. Grasselli, 0. M., S. Longhi and A. Tornambe (1995). On the computation of the time-invariant associated system of a periodic system. In Proc. American Confrol Conf, Seattle, WA, pp. 574-575. Hench, J. J. and A. J. Laub (1992). An extension of Floquet theory to discrete-time periodic systems. Technical report, CCEC-92-0914, University of California, Santa Barbara. Khargonekar, P. P., K. Poolla and A. Tannenbaum (1985). Robust control of linear time-invariant plants using periodic compensation. IEEE Trans. Autom. Control, AC-30, 1088-1096. Meyer, R. A. and C. S. Burrus (1975). A unified analysis of multirate and periodically time-varying digital filters. IEEE Trans. Circuits Syst., CAS-22, 162-168. Misra, P. and R. V. Pate1 (1989). Computation of minimal order realizations of generalized state space systems. Circuits, Syst. Sig. Process., 8, 49-70.

wherebv the associated time-invariant

A[; H

+(h)=[;

ye(h) = [; 5. Concluding

svstem is obtained as

1 ,0]%(h) +[; ,O]aW 8

j%,(h)-+

0 0 a(h), -1 1 -0

1”

remarks

A computational procedure for determining the timeinvariant representations of linear discrete time-periodic

Park, B. and E. I. Verriest (1989). Canonical forms of discrete linear periodically time-varying systems and a control application. In Proc. 28th IEEE Conj on Decision and Control, Tampa, FL. pp. 1220-1225. Pernebo, L. A. (1977). Note on strict system equivalence. Int. I. Control, 25, 21-38.

Richards, J. A. (1983). Analysis of Periodically Time-varying Systems. Springer-Verlag, Berlin. Rosenbrock, H. H. (1970). State Space and Multivariable Theory. Wiley, New York. Van Dooren, P. and J. Sreedhar (1994). When is a periodic discrete-time system equivalent to a time-invariant one? Lin. Alg. Applic., 2121213, 131-151.

272

Brief Papers

Verriest, E. I. (1988). The operational transfer function and parametrization of N-periodic systems. In Proc. 27th IEEE Conf. on Decision and Control, Austin, TX, pp. 1994-1999. Verriest, E. I. and J. Kullstam (1995). Realization of discrete-time periodic systems from input-output data. In M. Moonen (Ed.), SVD and Signal Processing. NorthHolland, New York. Verriest, E. I. and P. B. Park (1994). Periodic system realization theory with applications. In Proc. 1st IFAC Workshop

Ck+2&+,

1

kfl

Ck+W-l

Ii

A; Cli+w-,

n

Ai

i=k+om2

i=k+o-2

(A.4)

on New Trends in Design of Control Systems,

Smolenice, pp. 356-361. Appendix-Proof

of Theorem 2.1

Using the block permutation matrix P in (13), the system in w-stacked form may be transformed so that its descriptor and state matrices are given by

1

ph

0

...

0

o-

0

0 . .

..

0 .

0 .

.

0

0

..:

0 0

ii 0

0

. ..

0

0.

The resulting matrices are

&+,,-I 0 $(h).

-1,

(A.l)

0 -1,

A k+U-2

0

0

...

0

I,

0

...

0

Ak+,

I,

i

f

. ..o . .

Ak+tAk .

.

.

I,

I:

kt2

kfl

Under strict system equivalence, we postmultiply the above matrices by a block elementary transformation such that the matrices Ak, . , Ak+o-2 are eliminated using the identity matrices of the same block rows as pivots. It is easily verified that the transformation matrix that will accomplish this elimination is Ak

1. Eliminate the block matrices B,, i = 0,. , o - 2, by adding Bi X the (i + 1)th block column of (A - AE) to the (i + 1)th column of B. the block matrices in the first block row of (A - hE). This block elementary operation does not affect the other elements of the system matrix.

=

I”

Next, the following (strict system equivalent) transformations are perfromed on the system matrix defined by the above matrices.

2. Eliminate

4th)

0 0

T=

0 Ck+l

Bk

n i=k+wml

Ai

B=

Bk+,

B k+2

n i=k+o--l k+3

Ai

Ai

l-l

..

(A.51

Bk+,_,

L

0 PwxP(--l)

k+2 fl k=k+w-2

Ai

r=k+w-2

i=k+w-2

64.2) To complete the similarity transformations, the matrices E, A and C are postmultiplied by T. While E remains unaffected, the transformed matrices A and C are given by

0

0

D ktl

0

k+w-l

A=

Ck+Wml n 0, ,=k+om2

0

-I,

kf2

k+1

0

IT 64.3)

C,+,-,

11

0

Ck+$k+,

Ai

n i=k+w-I

n

,=k+w-2

A,&+,

B,+,-,

J

The proof follows on comparing the elements of the 5-tuple (E, A, B, C, D) in (A.5) with the associated system matrix 0 description in (2) and (3).