By Hans Blomberg and Raimo Ylinen (Eds.)
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Extra info for Algebraic Theory for Multivariable Linear Systems
To obtain a generator for So, equation ( 6 ) must first be transformed to a suitable form. 1. The block diagram in Fig. 1 illustrates equation (6) and the various input-output relations mentioned above. The interpretation of the diagram should be quite obvious. Below we shall illustrate the points presented above by means of a few examples. 9 Example. Fig. 2 shows a series composition of two regular differential input-output relations S1 and S2 generated by the regular generators [ A l @ )i -Bl(p)] and [A2@)i -&@)I respectively.
At this stage we therefore augment ( 8 ) with the previously deleted row and column to get 2p +1 0 0 9 16 p2+3 2p P3 0 2P2 6 -1 3p +1 6P2 -3p 1 -6p2 :]. 3 Step 8. The first column in (9) is satisfactory. The second column is also satisfactory, because the entry in the position (1,2) is of lower degree than the corresponding diagonal entry in the position (2,2). The third column is not yet in a satisfactory form, because the entries in the positions (1,3) and (2,3) are of higher degree than the corresponding diagonal entry in the position (3,3).
Such a form can accordingly be called a canonical upper triangular form, or a CUT-form for short (cf. appendix A2). Correspondingly, if we choose a lower left triangular form, then the normalization leads to a canonical lower triangular form,or CLT-form. 12 4. Example. The CUT-form of A @ ) as given by (4)is (cf. ________. , where Al(p) is of CUT-form (Al(p) square, det Al(p) # 0) coincide. Hence such matrices can also be said to be of CUT-form. Let P ( p ) be a unimodular matrix and let B(p) be a suitable polynomial matrix so that P(p)B(p) is of CUT-form.