Computational Techniques Based on the Block-Diagonal Form for Solving Large Systems Modeling Problems

Home , Transformation matrix

View metadata, citation and similar papers at core.ac.uk brought to you by CORE

provided by Institute of Transport Research:Publications

Computational Techniques Based on the Blo ck-Diagonal

Form for Solving Large Systems Mo deling Problems

A. Varga

German Aerospace Research Establishment DLR

Institute for Rob otics and System Dynamics

Ob erpfa enhofen, 82230 Wessling, Germany.

Abstract in which each matrix A is square of order n . In ap-

i i

plications it is generally desirable that the diagonal

The reduction of the state-matrix of a linear time-

blo cks have orders as small as p ossible. Theoretically

invariant state-space mo del to a blo ck-diagonal form

the blo cks cannot b e smaller than the blo cks in the Jor-

by using a state co ordinate transformation is equiva-

dan canonical form of A. However, the computation of

lent with an additive decomp osition of the corresp on-

the Jordan form is generally a dicult and often, very

ding transfer-function matrix. Computationally invol-

ill-conditioned problem. From numerical p oint of view

ved and large storage demanding algorithms for solving

a sound approachisto determine blo cks with the lo-

several systems mo deling problems can b e conveniently

west achievable dimensions by using a well-conditioned

reformulated such that they p erform exclusively on the

transformation matrix. In this case, the application of

low order subsystems corresp onding to the individual

transformation to other matrices will not cause signi -

terms of suitable additive decomp ositions. Imp ortant

cant accuracy losses. Avery ecient and numerically

reductions of b oth the computational e ort and requi-

reliable algorithm for computing the BDF was prop o-

red memory usually by using the reformulated algo-

sed byBavely and Stewart [2]. Besides simplicity and

rithms and thus, their applicability can be extended

economy, the main advantage of this algorithm is its

to handle higher order systems. The pap er presents

abilitytokeep under control the condition number of

several algorithms suitable to p erform eciently on

the transformation matrix. Therefore, with this algo-

additively decomp osed systems. The e ectiveness of

rithm, the BDF can be computed with a prescrib ed

these algorithms for solving large order systems mo de-

accuracy loss. The computational e ort to compute

3 3

ling problems relies on a reliable numerical algorithm

the BDF is approximately 11n 15n op erations and

to compute the blo ck-diagonal form of a matrix.

all computations can b e p erformed by using only 2n

storage lo cations.

Consider now a state-space mo del of the form

1. Intro duction

x_ t = Axt+But

The use of condensed forms of systems matrices in

y t = Cxt

algorithms for analysis and design of control systems

was also considered by other authors. An overview of

denoted also fA; B ; C g, where x is the n-dimensional

computational techniqes based on the use of conden-

state vector, u is the m-dimensional input vector and

sed forms obtainable by employing orthogonal system

y is the p-dimensional output vector. The reduction

similarity transformations is presented in [10]. The use

of the state matrix A to a BDF by a system similarity

of the Hessenb erg form to enhance the eciency of se-

transformation of the form

veral algorithms for systems analysis and mo deling is

1 1

~ ~ ~

A = T AT ; B = T B; C = CT 4

discussed in [12]. The wide usage of the Schur form

in control computations is illustrated byvarious algo-

where

rithms presented in the b o ok [6].

A = diagA ;:::;A

1 k

In this pap er we address the usefulness of the block-

B = rowB ;:::;B

1 k

;

diagonal form BDF of a square matrix in solving e-

C = colC ;:::;C ;

1 k

ciently computational problems arising in systems mo-

is equivalent with an additive decomp osition of the cor-

deling applications. Any square real matrix A of order

resp onding transfer-function matrix TFM

n can b e reduced by a similarity transformation of the

form

Gs=C sI A B 6

A = T AT ; 1

in the form

where T is a real transformation matrix, to a BDF

Gs= G s; 7

A = diag A ;:::;A 2

1 k

i=1 1

where G s=C sI A B , for i =1;:::;k. equation

i i i i

The pap er presents several computational algo-

A X XA = A ; 10

11 22 12

rithms for manipulating state-space mo dels which can

then, by using T from 9, we obtain

b e reformulated in terms of the additively decomp osed

TFM. The algorithms considered in the pap er are for:

T AT = diagA ;A : 11

11 22

1 discretization of continuous systems [11]; 2 compu-

tation of minimal realizations and mo del reduction by

A standard technique to solve 10 is the numerically

using balancing and balancing related techniques [13];

stable algorithm prop osed in [1].

3 evaluation of the TFM corresp onding to a state-

Notice that

space description [15]; and 4 computation of frequency

resp onses [4].

I X

One of the advantages of using such an approachis

T = 12

0 I

that usually the main computational e ort in the re-

formulated algorithms is the computation of the BDF,

and therefore

all subsequent e orts b eing negligible due to the low

1 2

orders of the subsystems fA ;B ;C g. Another advan-

condT = kT k kT k k = n + kX k : 13

i i i

F F F

tage is that, b ecause of p erforming the computations

Clearly T is ill-conditioned when X has a large norm.

on low order subsystems, usually no additional storage

excepting 2n storage lo cations is necessary to p er- Initially, A in 8 is an 1 1or2 2 blo ck dep en-

form the computations. In contrast, when the mentio- ding on the dimension of the leading diagonal blo ck.

ned algorithms excepting the last one p erform on full An attempt is made rst to annihilate A with a well-

2 2

order mo dels, the necessary storage is ab out 4n 5n conditioned T . If all elements of X are suciently

storage lo cations. Therefore, by using the new ap- small, then T is accepted and the algorithm pro ceeds

proach, problems of higher orders can b e solved in the further by de ating A from A and rede ning a new

same storage area. Moreover, exp erimental tests indi- A as A . If however, at least one element of X ex-

cate that the accuracy of the new metho ds is b etter

ceeds in magnitude a given b ound, then the attempt

sometimes much b etter than the accuracy of the ori- to annihilate A is abandoned and a new, larger blo ck

ginal metho ds. Finally, b ecause the submatrices of the A is formed by incorp orating in it a new 1 1or2 2

BDF resulting from the algorithm [2] are in an up- of A whose eigenvalues are the nearest to the mean

p er real Schur form and the eigenvalues of each blo ck of eigenvalues of A . Standard blo ck-interchange al-

usually form a cluster of nearby eigenvalues, sp ecial algorithms are available to reorder the RSF of A for this

gorithmic enhancements can b e further devised which purp ose [3].

contribute additionally to an increased e ectiveness of

The nal result of the Bavely-Stewart algorithm is a

these metho ds.

BDF in which each diagonal blo ck is in RSF and has

nearly equal eigenvalues. The orders of the blo cks are

usually related to the b ound sp eci ed by the user on

2. Computation of blo ck-diagonal form

the condition numb er of the reducing matrices of the

form 9. This b ound restricts only the elements of X

We review very shortly the main features of the blo ck-

and therefore the algorithm cannot guarantee that the

diagonalization algorithm of [2]. A preliminary step in

nal transformation is well-conditioned. Nonetheless,

this algorithm is to reduce the matrix A to a quasi-

in most of cases there is a very little tendency toward

triangular, the so-called real Schur form RSF, in

excessive growth in the condition of the transformation

which the diagonal blo cks are of orders at most two.

matrix.

The 2 2 blo cks have exclusively complex conjugate

The number of oating-point operations ops for

pairs of eigenvalues. The ordering of blo cks is generally

reducing a matrix in RSF to the BDF ranges from

arbitrary and can b e changed by using appropriate or-

n =2, when each eigenvalue is real and can be de a-

thogonal reordering techniques [3]. We assume that A

ted, to n =12, in the highly improbable extreme case,

is already in a RSF.

when none of them can b e de ated. Several implemen-

The subsequent blo ck-diagonalization is based on

tation tricks can improve the eciency of the overall

partitions of the matrix A in the form

algorithms such that nally we can consider 15n ops

A A

11 12

as a go o d maximal gure for the reduction of a dense

A = : 8

0 A

matrix to the BDF.

An attempt to annihilate the o -diagonal blo ck A is

made by using a transformation T of the form

3. Discretization of continuous mo dels

I X

T = ; 9

Let us consider the sampled-data system

0 I

where the identity matrices are of same orders as A

xt +1 = A xt+B ut

d d

and A . If there exists X satisfying the Sylvester

y t = Cxt 22

resulting from the discretization of the continuous sy- most of other metho ds. The storage needs are 2n lo-

stem 3 with a sampling p erio d . The standard me- cations, which is ab out the half of the storage required

tho d [11] to compute the matrices A and C consists by other approaches. The metho d is suitable for the di-

d d

in evaluating them from the matrix identity scretization of systems with multiple sampling p erio ds,

b ecause the evaluation of the matrices of the sampled-

A B A B

d d

data systems corresp onding to di erentvalues of is

exp = : 15

0 0 0 I

very cheap if the system is already in form 5.

One of the most reliable metho ds to evaluate the

exp onential of a matrix is by using diagonal Pad e ap-

4. Minimal realization and

proximations combined with the scaling and squaring

mo del reduction

technique. A detailed algorithm was prop osed byWard

in [16]. The metho d p erforms ab out q + j + n + m

3 Consider the following minimal realization problem:

ops, where q is the order of the Pad e approximation

for a given non-minimal system fA; B ; C g, determine

and j is the numb er of p erformed squaring op erations.

^ ^ ^

a minimal realization fA; B; C g such that the two sy-

The required memory is ab out 4n + m storage lo-

stems have the same TFMs. There exist two main

cations. In order to employ as small values of q and j

classes of computational metho ds for solving this pro-

as p ossible, Ward prop osed two additional norm redu-

blem.

cing techniques: a diagonal translation of eigenvalues

A rst class of minimal realization algorithms

combined with a row-columns balancing with diagonal

MRAs, based on the use of orthogonal staircase ca-

similarity transformations matrices.

nonical forms, determines the matrices of a minimal

The usage of the BDF for evaluating matrix exp o-

realization by successively removing the uncontrolla-

nentials is discussed in [5]. The corresp onding com-

ble and unobservable parts of the given non-minimal

putational e ort to compute 15 is ab out 17n + m

system. The complete MRA, using for example the or-

ops. Here we discuss a somewhat cheap er approach

thogonal staircase algorithm prop osed in [9], is nume-

which relies on the additive decomp osition form of the

rically stable, p erforms no more than n ops and

system matrices in 5. Let the matrices of the system

needs no additional storage. Although very ecient,

be already as in 5. Then, for each pair A ;B we

i i

this algorithm is very sensitive to the choice of zero-

can compute the corresp onding pair A ;B from

id id

tolerances used in the rank determinations during the

reduction to staircase forms. A single erroneous rank

A B A B

i i id id

decision leads to the failure of the whole MRA.

exp = 16

0 0 0 I

The metho ds in the second class are pro jection me-

tho ds based on balancing [8] or related to balancing

and then A , B and C can b e assembled as

d d d

techniques [7]. Both algorithms determine directly two

pro jection matrices T and L satisfying LT = I such

A = diag A ;:::;A

d 1d kd

that the matrices of the minimal realization can be

B = rowB ;:::;B

d 1d kd

;

computed as

C = colC ;:::;C :

d 1d kd

^ ^ ^

A = LAT ; B = LB ; C = CT 19

If A is non-singular and well-conditioned a very

cheap test b ecause A is already in RSF, then a more

The algorithm of [8] determines the pro jection ma-

ecient way to compute A and B is by using the

id id

trices by using the so-called square-ro ot accuracy en-

formulas

hancing approach, by working exclusively with the

Cholesky factors of gramians. The resulting minimal

A = expA ; B = A A I B : 18

id i id id i

realization is balanced. Accuracy losses can b e howe-

The diagonal translation and balancing techniques ver induced in the resulting minimal realization when

the pro jection matrices are ill-conditioned. The second

prop osed byWard to b e used in conjunction with the

metho d [7] uses a balancing-free approach to determine

Pad e approximation metho d are usually very e ective

T and L and pro duces always well-conditioned pro jec-

when applied to the low order RSF matrices A . This is

tion matrices. However the metho d relies on forming

b ecause often A has nearly equal eigenvalues and after

a diagonal translation with , the mean of eigenvalues the pro duct of the gramians and usually is less accurate

than the square-ro ot metho d. An improved version of

of A , the matrix A I , is nearly nilp otent and its

i i i

b oth metho ds, obtained by combining in a single al-

powers vanish rapidly. Additional diagonal balancing

can b e used to reduce the norm almost arbitrarily,avoi- gorithm b oth the square-ro ot and the balancing-free

approaches, was prop osed recently in [13] and has very

ding thus completely the need for scaling and squaring

satisfactory accuracy prop erties.

op erations. The costs to evaluate A and B by using

id id

either 16 or 18 is therefore negligible.

The MRAs related to balancing have the very at-

The describ ed approach requires ab out 15n ops, tractive prop erty to remove simultaneously the un-

which represent roughly the cost of computing the controllable and unobservable parts by using a single

BDF. Therefore, the metho d compares favorably with rank decision based on a singular value decomp osition.

They are thus reliable in determining correctly the or- p erformed very inaccurate when used on the full or-

der of the minimal realization. The direct application der system, the results obtained in conjunction with

of these metho ds to large systems raises however pro- the blo ck-diagonalization technique were always very

blems. First, the numb er of op erations is considerable. accurate. Due to this very satisfactory numerical b e-

A rough estimation is ab out 40n ops with an order havior, the usage of this metho d is to b e recommended

of magnitude more than for the staircase metho ds. Se- even for low order systems.

cond, the memory requirement is also notable, ab out

5n storage lo cations are necessary. Therefore, the ap-

5. Evaluation of transfer-function matrices

plicability of balancing related metho ds to high order

systems is problematic.

For the computation of the TFM corresp onding to a

The additive decomp osition of the system in the

state-space mo del fA; B ; C g, there exist several algo-

form 5 can be used to compute minimal realizati-

rithms. Anumerically reliable metho d based exclusi-

ons provided any two diagonal blo cks have no com-

vely on orthogonal transformations is the poles-zeros

mon eigenvalues. This condition can be easily ful l-

method prop osed in [15]. This metho d determines the

led by incorp orating in a single, larger blo ck, all in-

elements of the TFM by evaluating for each of them

dividual blo cks pro duced by the blo ck-diagonalization

the corresp onding p oles, zeros and gain. The metho d

algorithm of [2] which have nearly equal eigenvalues

is generally applicable regardless the original system is

complex eigenvalues enter always in pairs in the ne-

minimal or not. The resulting elements are in canceled,

wly formed blo cks. This approach seems to b e more

minimal forms.

economical than a preliminary segregation of the ei-

The applicability of this algorithm to high order sy-

genvalues in clusters by reordering of the RSF of A.

stems encounters the same diculties as discussed pre-

The following straightforward algorithm can be used

viously. The need to compute for each element eigen-

to compute minimal realizations:

values representing p oles or zeros can lead in extreme

1. Reduce the system fA; B ; C g to the additively de-

although highly improbable cases to more than 18n

comp osed form 5, where A \ A 6= for

i j

ops p er element to b e p erformed compare with 15n

i 6= j .

ops necessary to compute the BDF. The required me-

mory is at most 4n +4nmp storage lo cations, which

2. For each subsystem fA ;B ;C g determine a mini-

i i i

is also substantial.

^ ^ ^

mal realization fA ; B ; C g by using the algorithm

i i i

The usage of the blo ck-diagonal decomp osition of

from [13].

system matrices in the form 5 could be advantage-

^ ^ ^

3. Construct the minimal system fA; B; C g as

ous also in this case. Each term G s of the decom-

p osition 7 can be evaluated separately by applying

^ ^ ^

A = diagA ;:::;A ;

1 k the p oles-zeros metho d to the corresp onding subsystem

^ ^ ^

B = rowB ;:::;B ;

fA ;B ;C g. Then, the TFM Gs can be computed

1 k

i i i

^ ^ ^

from its partial fraction decomp osition by summing the

C = colC ;:::;C :

1 k

individual terms. This summation reduces to simple

p olynomial op erations without any need to compute

If the orders of diagonal blo cks A are small, then the

greatest common divisors if we ensure that the dia-

op erations p erformed at step 2 are negligible in com-

gonal blo cks have no common eigenvalues. This can

parison with the cost to reduce A to the BDF. The

b e done similarly as for the algorithm presented in the

additional storage is also substantially reduced.

previous section.

This algorithm is particularly well suited for compu-

The blo ck-diagonalization approachtoevaluate the

ting minimal state-space realizations of TFMs. In this

TFM of a state-space mo del is thusavery e ective al-

case, the initial non-minimal realization can b e direc-

ternative to existing metho ds. The algorithm p erforms

tly constructed in a BDF. If necessary, the grouping of

roughly the usual 15n ops to compute the BDF of A.

eigenvalues of A can b e p erformed by simple row and

The storage requirements are also substantially redu-

column p ermutations p erformed on matrices A, B and

2 2

ced to less than n + maxn ; 2nmp storage lo cations.

C . The numb er of p erformed op eration is comparable

with that p erformed by a staircase algorithm and in

many cases can b e even smaller.

6. Computation of frequency resp onses

The ab ove algorithm can also be used for mo del

reduction by approximating the diagonal subsystems

An ecient metho d to evaluate the frequency resp onse

with lower order approximations or by removing com-

pletely those subsystems whichhave negligible contri-

Gj!=C j!I A B 20

bution to the TFM of the system.

It is worth mentioning an interesting asp ect revealed for many values of the frequency ! was prop osed by

by tests p erformed with the blo ck-diagonal approach Laub [4]. The metho d is based on a preliminary re-

used in conjunction with seven other MRAs related to duction of the state-matrix A to a Hessenb erg form by

balancing [14]. Although some of particular metho ds means of an orthogonal similarity transformation and

to mo dify B and C accordingly. After that, the evalua- [3] J. J. Dongarra, S. Hammarling, and J. Wilkinson.

tion of Gj! for a given frequency value can b e p erfor- Numerical considerations in computing invariant

med relatively cheaply with ab out m +1n + pmn subspaces. SIAM J. Matrix Anal. Appl., 13:145{

complex ops. If the number of frequency values is 161, 1992.

large, a more ecient approach suggested in [10] is

[4] A. J. Laub. Ecientmultivariable frequency re-

to determine rst the TFM Gs given by 6 see the

sp onse computations. IEEE Trans. Autom. Con-

metho d in the previous section and then to evaluate

trol,AC-26:407{408, 1981.

Gj! from Gs directly only simple p olynomial eva-

luations are necessary.

[5] C. B. Moler and C. F. Van Loan. Nineteen du-

Alternatively, the decomp osition 5 with A in a

bious ways to compute the exp onential of a ma-

BDF can be used to evaluate 21 instead the repre-

trix. SIAM Review, 20:801{836, 1978.

sentation with A in the Hessenb erg form. For a given

[6] P.H.Petkov, N. D. Christov, and M. M. Konstan-

frequency value ! , Gj! can b e computed as

tinov. Computational Methods for Linear Control

Systems. Prentice Hall Int., 1991.

Gj!= G j!; 21

[7] M. G. Safonov and R. Y. Chiang. A Schur metho d

i=1

for balanced-truncation mo del reduction. IEEE

Trans. Autom. Control,AC-34:729{733, 1989.

where G j!=C j!I A B . Recall that A is

i i i i i

already in a RSF and thus the evaluation of G j!

[8] M. S. Tombs and I. Postlethwaite. Truncated ba-

is computationally very cheap b ecause of usually very

lanced realization of a stable non-minimal state-

low order of A . The p ossibility to use more condensed

space system. Int. J. Control, 46:1319{1330, 1987.

forms the diagonal form was mentioned in [4], but re-

jected b ecause of p otential accuracy problems due to [9] P. Van Do oren. The generalized eigenstructure

ill-conditioned eigenstructures. Our exp erience indica-

problem in linear systems theory. IEEE Trans.

tes that even when the computed eigenvalues app ear to Autom. Control,AC-26:111{129, 1981.

be inaccurate due to ill-conditioning, the accuracy of

[10] P. Van Do oren and M. Verhaegen. On the use

resulting G j! is still comparable with the accuracy

of unitary state-spacetransformations,volume 47

of the Hessenb erg metho d.

of Special Issue of Contemporary Mathematics in

Linear Algebra and Its Role in Systems Theory.

Amer. Math. So c., Providence, R.I., 1985.

7. Conclusions

[11] C. F. Van Loan. Computing integrals involving

The pap er presented several useful applications of the

matrix exp onentials. IEEE Trans. Autom. Con-

BDF of a square matrix in solving eciently some sy-

trol,AC-24:395{404, 1978.

stems mo deling problems. The blo ck-diagonalization

approach allows to extend the ranges of applicability

[12] C. F. Van Loan. Using the Hessenb erg decomp osi-

of several existing metho ds to higher order systems by

tion in control theory. In D. C. Sorenson and R. J.

reducing simultaneously the asso ciated computational

Wets, editors, Mathematical Programming Study,

burden and storage requirements. The e ectiveness of

volume 18, pages 102{111. North Holland, 1982.

the new approach is guaranteed by the availabilityof

[13] A. Varga. Ecient minimal realization pro cedure

anumerically reliable algorithm to compute the BDF

based on balancing. In A. El Moudni, P. Borne,

with a prescrib ed accuracy loss. For most practical

and S. G. Tzafestas, editors, Prepr. of IMACS

problems, the reformulated algorithms have the addi-

Symp. on Model ling and Control of Technological

tional b ene ts of providing more accurate results than

Systems,volume 2, pages 42{47, 1991.

those resulted by direct application of the original me-

tho ds. The usage of the BDF for ecient on-line im-

[14] A. Varga. Minimal realization pro cedures based

plementation of comp ensators and lters is also an area

on balancing and related techniques. In F. Pich-

of p otential applications of this condensed form.

ler and R. Moreno Diaz, editors, Computer Aided

Systems Theory - EUROCAST'91,volume 585 of

Lect. Notes Comp. Scie., pages 733{761. Springer

References

Verlag, Berlin, 1992.

[1] R. H. Bartels and G. W. Stewart. Algorithm

[15] A. Varga and V. Sima. A numerically stable algo-

432: Solution of the matrix equation AX+XB=C.

rithm for transfer-function matrix evaluation. Int.

Comm. ACM, 15:820{826, 1972.

J. Control, 33:1123{1133, 1981.

[2] C. Bavely and G. W. Stewart. An algorithm

[16] R. C. Ward. Numerical computation of the ma-

for computing reducing subspaces by blo ck dia-

trix exp onential with accuracy estimate. SIAM J.

gonalization. SIAM J. Numer. Anal., 16:359{367,

Numer. Anal., 14:600{610, 1977. 1979.