D A Figure 4.1: State space rep res entation of a system. Since C adj(sI-A)B is a poly nomial matrix, obviously all poles of G(s)(i.e. the values of s for which G(s)=oo) have to be roots of the polynomial det(sI-A). The roots of det(sI-A) coincide with the eigenvalues of A. Hence, all poles of G(s) have to be eigenvalues of A. Th opposite needs not be the case al ways, since roots of det(sI - A)might be cancelled in(4.38) and consequently they will not appear as poles of G(s). This is the case, when the realiz ation (A, B, C, D)is uncontrollable, unob servable, or both. On the other hand, if the realiz (A, B, C, D)is both controllable and observable, the roots of det(sI- A)equals the poles of G(s and the pole polynomial p(s) will be given by ps=det(sI-A) 441) This means, that the dimension of A can not be smaller than the McMillan degree of G(s) Hence, a st ate sp ace realization which is both controllable and observable is called a minimal realization. These results can be summarized as the following theorem ThId li maldlaaaw K aaamranlavpacta Let G(s)be a tra fer matric with a minimal realization(A, B, C, D)and let p(s be the Smith-McMillan pole polynomial of G(s. Then dim a= degp(s (4.42) Hence, the McMillan degree of G(s equals the dimension cf a minimal realization. Moreot the eigenvalues f A equal the poles af G(s) Note, that if(A, B, C, D) is a non-minimal realization, then the poles of G(s) constitute a proper sub set of the eigenvalues of A aea Fdh rrtahrvelce dbaspranAauact The transformation of a state space description to a transfer function description is unique given by(4.38 ). In contrast, there are several ways in which a transfer function can be transformed into a st ate space description. A straightforward approach would be to derive separate st ate space descriptions for each column in G(s), i.e. for each input, and then collect these separate state space descriptions into an overall state space model. Let Gp) be the ith ch that G(s)=(Gn(s), Gt(s).,Ghs) O=tt C)ntr)
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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
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