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 2 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 3 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 9 rithms presented in the b o ok [6]. ~ A = diagA ;:::;A = 1 k In this pap er we address the usefulness of the block- ~ 5 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 1 Gs=C sI A B 6 1 ~ A = T AT ; 1 in the form where T is a real transformation matrix, to a BDF k X Gs= G s; 7 i ~ A = diag A ;:::;A 2 1 k i=1 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: 1 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 1 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 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 2 excepting 2n storage lo cations is necessary to p er- Initially, A in 8 is an 1 1or2 2 blo ck dep en- 11 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- 12 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 11 same storage area. Moreover, exp erimental tests indi- A as A . If however, at least one element of X ex- 22 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 12 ginal metho ds. Finally, b ecause the submatrices of the A is formed by incorp orating in it a new 1 1or2 2 11 BDF resulting from the algorithm [2] are in an up- of A whose eigenvalues are the nearest to the mean 22 p er real Schur form and the eigenvalues of each blo ck of eigenvalues of A . Standard blo ck-interchange al- 11 usually form a cluster of nearby eigenvalues, sp ecial al- gorithms 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- 3 n =2, when each eigenvalue is real and can be de a- thogonal reordering techniques [3]. We assume that A 4 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 3 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 22 matrix to the BDF. An attempt to annihilate the o -diagonal blo ck A is 12 made by using a transformation T of the form 3.