By Hans Blomberg and Raimo Ylinen (Eds.)
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Additional resources for Algebraic Theory for Multivariable Linear Systems
By the equation A(p)y = B(p)u. It may happen that this equation contains purely algebraic relationships between the various components of u and y. The situation is closely related to the observability concept considered above, and it is always possible to extract, from A(p)y = B(p)u, relevant equations representing such algebraic relationships. 1. Controllability Now suppose that the generator [ A @ )i - B ( p ) ] for S above is such that 7 i -B(p)l= L(p)[Al(p)i -B1(p)I, where L(p) is a GCLD of A(p) and B(p).
46p + 4 2 -24 with det Z @ ) = 47. ]=[; ............................ : ..... Xl 12 46 i Hence we have det Yl = Yl = O(Xl # 0), and (52) is, as expected, not a satisfactory candidate for (29). Therefore we shall, starting from (52), construct new candidates according to the general scheme outlined above. 5 Step 1, . . ,step 5 below refer to the corresponding steps as described in the scheme. Step 1. In this case the lower part of (52) has one single row. Therefore k = 1. Go to Step 2. Step 2. From (52), (53) we obtain :] AT1=[: 54 and Xi1 = Xi = 55 [12 461.
Consider, for instance, the bottom block row in (7) and (12). The structure of C2 means simply a renaming of some of the components of y , for instance in (12) it means that y2 is renamed yo in forming the overall input-output relation So. This renaming can, however, equally well be performed already in forming Si and its generator (8). If this is done, then Si, can be replaced by Si without loss of information. In example (9) we can thus replace the label y 2 in (13) by the label yo, which makes (12) superfluous.
Algebraic Theory for Multivariable Linear Systems by Hans Blomberg and Raimo Ylinen (Eds.)