By Dr. Walter Murray Wonham (auth.)
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Extra resources for Linear Multivariable Control: A Geometric Approach
Generic, as claimed. On the other hand, if m"' n+1, (37) holds only if all (n+1) x(n+1) minors of [C,y] vanish. ) = 0 for E. E WC , hence n cannot be generic. Finally, if RankC =m at E. then (equivalently) at least one mx m minor of C is nonzero at ß hence nonzero in a neighborhood of ß so n is well-posed at E.. Conversely if Rank C < m at E. then a suitable y, with Iy - y I arbitrarily small, will make Rank[C, y] = Rank C + 1 ; namely, if P ~ (C, y), then n(p) = 0, hence n is not well-posed at E..
P. of A could be used in this calculation, if the auxiliary polynomials are defined accordingly. 17 Transfer Matrices. ) has the Laplace transform A y(s) ~ J'" 0 e -st L'" y(t) dt = 0 e 1'" [Lt =C 0 e defined for -st t-r 0 e( ) -st A l Jt C 0 e( t-r A ) J Bu(r) dr J Bu(r) dr dt dt = C(sI-A) -1 A Bu(s) , Re s sufficiently large. The matrix b. H(s) = C(sI - A) -1 B is the transfer matrix of the tripIe of matrices (C, A, B), and is defined for s E ~ - o(A). For such s, H(s) can be viewed as the matrix of a map t( -'11, taken as linear spaces over ~.
To verify these statements note first that (37) fails only if RankC = d(ImC)