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By Hans Blomberg and Raimo Ylinen (Eds.)
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Extra resources for Algebraic Theory for Multivariable Linear Systems
E. if the whole output yl of S1 is used as input uz to the compensator Sz,then the upper block row along with the first block column on the left are deleted from (l),and detAl(p) in (2) is interpreted as being equal to 1. The problem is to choose Sz generated by a regular generator of the form 3 Y2 u2 -m)l [C(P>i so that the resulting composition behaves in a satisfactory way. Now the first step is to form a generator for the internal input-output relation Si given as the set of all (UO, (yll, y12, yz)) generated by the composition.
0 Hence in trying to solve a problem of the present kind the first thing to do would be to determine a generator (1) for different orderings and different groupings of the components of y, = (yll,y12),and then to choose an ordering and a grouping-if it exists-yielding a satisfactory location of the roots of det Al(p) det L(p). Of course, a corresponding analysis could also be performed with respect to the choice of u 1 7 i t may happen that a satisfactory result can be obtained using only some of the control inputs available as components of 2412.
E. divisors, which remain common divisors for every possible set of parameter values. This problem imposes special requirements on the computational procedures applied for the present purpose. 0 Canonical forms An important question concerns the uniqueness of the triangular equivalent form of a given polynomial matrix. Suppose that A @ ) , B(p) E C[P]qxq, det A @ ) # 0, det B (p) # 0, are row equivalent and of the particular upper right triangular form mentioned above. Then the transformation matrix P ( p ) E C[pIqxqsatisfying A @ ) = P(p)B(p) proves to be a diagonal matrix whose diagonal entries are nonzero constants.