By Hans Blomberg and Raimo Ylinen (Eds.)
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Extra resources for Algebraic Theory for Multivariable Linear Systems
Then A ( p ) is invertible as a polynomial matrix if and only if s = q and det A ( p ) = c = constant # 0. Such a matrix is said to be unimodular. Further, A @ ) and B(p) are said to be row equivalent if there is a unimodular matrix P(p) E C[plSxS such that A ( p ) = P(p)B@),or equivalently, B ( p ) = P ( p ) - ' A ( p ) ,where P(p)-' is the inverse of P ( p ) as a polynomial matrix. Correspondingly, A @ ) and B ( p ) are column equivalent if A @ ) = B(p)Q(p) with Q(p)E C[p]qxq unimodular. Every square polynomial matrix is row equivalent to an upper right (or lower left, as desired) triangular matrix where every entry above a nonzero diagonal entry is of lower degree than the diagonal entry itself.
For every s E C, let %(s) denote the complex matrix obtainable from (28) by substituting s for p in (28). We shall say that S, [ A @ )i -B(p)], as well as %(p)above are proper if 21 Algebraic Theory for Multivariable Linear Systems lim %(s) = K 1sl-m = a constant matrix. If K = 0, then these quantities are said to be strictly proper. 29 Example. Let a regular generator [A@) i -B(p)] be given according to (20) above. The corresponding transfer matrix %(p)= A(p)-' B(p) can be computed as Al(p)-' Bl(p) with [Al@) i - B l ( p ) ] given by (26).
We shall generally not explicitly indicate Si, and its generator. Other obvious simplifications are also possible-thus Fig. 6 shows in a simplified and more 0 customary form the same composition as Fig. 2. 15 Example. As a second example consider a feedback composition, as shown in Fig. 3, around a regular differential input-output relation S generated by the regular generator Y 16 u2 u1 Fig. 3. Feedback composition around a regular differential input-output relation S. The interconnection matrix C is found to be uo ;'1 __ Y Yo ......