By Hans Blomberg and Raimo Ylinen (Eds.)
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Extra resources for Algebraic Theory for Multivariable Linear Systems
Step 2. The degree of the entries in the positions ( 2 , l ) and ( 3 , l ) are now lowered in the following way. The first row multiplied by - 1 is added to the second row. The third row is multiplied by 3 (this multiplication is performed only to avoid fractions in the next operation). 15 Algebraic Theory for Multiuariable Linear Systems The first row multiplied by -2p is added to the third row. Through these operations [ A @ )i I ] is brought to the form 6p2 + 3p 4p3 + 9p p2 + 3 -2p4 [ 20 p + l 5 8p2 j 2p -4p3+6 j 1 -1 O1 -2p 0 3.
3 Algebraic foundations The introductory discussion presented above showed that in general not all the interesting results concerning a composition of differential inputoutput relations can be obtained just by forming the relevant equations. In addition we need a machinery which allows us to transform the equations obtained in a suitable way. As it happens such a machinery is provided by the abstract algebra. The signal space as a module Let Clp] denote the set of all polynomial operators of the form ( l .
In real parameter systems f(p) can conveniently be formed as a product of factors of the form (P + 4 46 and 47 ((p + a)2 + $) = ( p 2 + 2ap + a2 + /32) with a and /3 real, a > 0. A factor of the form (46) adds a root at -a,and a factor of the form (47) a pair of complex conjugate roots at -a 2 i/3, to the roots of the determinant of the candidate under consideration. 48 It is easily seen that (48) formally qualifies as a candidate for (29). It is, however, no more satisfactory than the original candidate.