By Hans Blomberg and Raimo Ylinen (Eds.)

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**Sample text**

The number of elementary row operations required to obtain the final result could in some cases be reduced by choosing the operations on the basis of a complete division algorithm as outlined in Polak, 1969, rather than on the basis of a partial division algorithm as applied above. e. 2p 1, is a greatest common divisor of the entries in the first column of the original matrix A @ ) as given by (4). e. p3,is a greatest common divisor of the entries in the first column of (7). Quite generally the triangularization procedure amounts to finding greatest common divisors of certain polynomials, and the final form depends on the appearance of such divisors.

In Fig. 4 S1 is a regular differential input-output relation representing a given system whose dynamic properties we are trying to modify with the aid of a feedback compensator represented by the regular differential input-output relation Sz. If nothing else is explicitly stated, it will be assumed that all the signals indicated in Fig. 4 have one or more com27 Algebraic Theory for Multivariable Linear Systems ponents. We shall in this context consider only certain aspects concerning the internal input-output relation Si determined by the composition.

Determine the integers q ,q ,. . , t, according 37 Algebraic Theory for Multivariable Linear Systems to the rule, for i = 1, 2, . . ,q: { r 39 ti= Denote tm 40 0 if ti = 0, sk - ri otherwise. A min iE{l,Z, . . , q } ti. If tm L 0, go to Step 4. Otherwise, go to Step 5 . Step 4. ' . ' Ol ii' 0 . p" ComDute It is easily found that the above choice of T&) leads indeed to cancellations of leading terms in the kth row of T3(p)A(p)+ X ( p ) . Transform, if necessary, the lower part of (43) to row proper form by means of elementary row operations.