ABSTRACT
Based upon Eqs. (7.34) through (7.36) e?"~'(s) must be a factor ofn^(s)w. The following development illustrates that a sufficient condition for the ex-
istence of a m.p. det Pe(s) is that at least one det Pt(s) must be m.p. Assume
det I «;,(»; I kl —— ' '^ ' = kl Nl (5) (7.38)
Also, let the summation of the remaining terms, after factoring out dm~'(s) from each term, be expressed as
where k2 is a scalar. Note: kj, k2, and A^s) are now functions of w,-,- which are to be selected in order to try to achieve a m.p. Pe(s). Based upon Eqs. (7.38) and (7.39), Eq. (7.29) can be expressed as follows:
det Pe (s) = —— [kiNi (s) + k2N2 (s)] (7.40)
In order for det Pe(s) to be m.p. then the zeros of the polynomial
must all be in the LHP. Equation (7.41) is manipulated to the mathematical format of
Ni(s) _N2(s)_
which permits a root-locus analysis of Eq. (7.40). Since the zeros of Eq. (7.42) are in the LHP then the weighting factors wff are selected in hopes that all roots of Eq. (7.41) lie in the LHP for all Pb e P. This assumes that throughout the region of plant parameter uncertainty, the initially chosen m.p. submatrix in Eq. (7.34) is m.p. for all Pb e P and be expressed by Eq. (7.38). To enhance the achievability of m.p. det Pe(s) the following guidelines may be used:
1. Determine the number Oy submatrices of P*w of Eq. (7.34) that are rap. 2. Select one of the m.p. submatrices to be identified as Eq. (7.38). 3. The values of w,j of W^ associated with
(a) The remaining Oy -1 m.p. Pbtt, be altered in such a manner as to increase the values of their corresponding det W^.
