5 Pages

W. P

Fig. 7.4 An mxm effective plant P(s). Case B From the block diagram of Fig. 7.4, where v(s) is a l x1 vector and W is

P (7.34)

where en′ is a polynomial. Since Eq. (7.23) and (7.34) represent the same open-loop system of Fig. 7.4 then


)( =






sn m e

e ′ (7.35)

which results in de(s) = d(s) and


)( = )(


sn sn 1-m

e (7.36)

Based upon Eqs. (7.34) through (7.36) dm-1(s) must be a factor of nij(s)w. The following development illustrates that a sufficient condition for the

existence of a m.p. det Pe(s) is that at least one det Pb(s) must be m.p. Assume

11ijij1ij kss 1 |)(|det||det|)(|det nwn = (7.37)

in Eq. (7.34) is m.p. where k1 = det |wij1| is a scalar. Factor out d m -1(s) from every

term in Eq. (7.36) and let

)( )(

|)(|det sNk


s k 11

n (7.38)

Also, let the summation of the remaining terms, after factoring out d m -1(s) from each term, be expressed as





)( det )(det


sn s

s s


n P (7.43)



)( =


)( det det

w sh

sw s


w W = (7.44)

where the dij(s) and hij(s) are not all the same and where nw(s), dw(s), ww(s) and hw(s) are scalar polynomials. Hence

∑= )()(

)()( )(det


swsns ww

α P (7.45)

For each value of w, nw(s)/dw(s) has a range of uncertainty which may be correlated with that of the remaining αz − 1 sub-matrices of P(s). Thus, for the general case some or all of the nw(s)/dw(s) may have RHP zeros and/or poles. The problem now becomes one of trying to choose fixed ww(s) and hw(s) polynomials so that the det Pe(s) has no RHP zeros over the entire range, or failing that has them as relatively "far-off"; as possible. This problem has been studied by several researchers and the resulting techniques may be used for this purpose.16,23

Thus, the Binet-Cauchy formula permits the determination if an m.p. effective plant det Pe(s) is achievable over the region of plant uncertainty.