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32 of Write each column as T(8)+ 444) where di(s)is the common denominator polynomial of G:(s) d(s)=s4+a4-1+…+4 Note, that di(s) is monic. ni (s is a vector of polynomials, each having a degree strictly less than ki. The jth entry of n; (s) can be written as the polynom n1()=n,一1+-2+…+m is a vector consisting entirely of constants a state sp ace description in controllable canonical form of the column Gi(s) is then given by the realization(Ai, Bi, Ci,&)where n B 448) Finally, a realiz ation(A, B, C, D)of G(s) can be found as B=diag D=[51,2,…,na This realization is controllable, but not necessarily observable. If a minimal is required, there exist algorithms to remove the unob servable modes, see e.g. [Mac89, Section 8.3.5 The MATLAB function tfm2ss m from MATLABs Robust Control Toolbox pro duces a similar st ate sp ace realization and minreal m from MATLABTM's Control Toolbox can extract a minimal realiz ation from a non-minimal one Remark In computer aided design, and especially in MATLABTM, it is easier in general to work with st ate space descriptions, since it is difficult to represent transfer matrices, as this requires three dimensional structures. Robust control. howe sponse analysis of a number of transfer matrices, such as the sensitivity function S(s)and the complementary sensitivity function T(s). It is no problem, however, to compute the resp onse based on a st ate sp ace description of the system. Hence, a multivariable often be represented in st ate space representation, although the analy sis is performed in th frequency domain. This mixture of time and frequency domain is actually quite typical for < Æ """! ' ""/! ' ""-! Æ 6 Æ ""1! ""2! ""3! Æ Æ Æ "/5! % 8&23 2,/: = > ? % = ? % % % & + 0 > ) %