Grassmannian, central projection, and output feedback pole assignment of linear systems

Degrees of Real Wronski Maps

“…, x mp−k ) in the range, where −x j are the negative roots of W q as in (19). So the i-th row of this matrix ∆ q corresponds to x i , and the j -th column to the j -th term of the sequence (18). When k = 0, so that k = (0, 1, .…”

“…We denote by ∆ q the Jacobi matrix of the map b(k) → Poly mp−k R , q → W q , using coordinates (18) in the domain and x = (x 1 , x 2 , . .…”

The Wronski map and Grassmannians of Real Codimension 2 Subspaces

“…Applying Laplace's expansion along the first p rows to the determinant in (55), we conclude that the map φ S , when expressed in Plücker coordinates, is nothing but a projection of the Grassmann variety G R (m, m + p) into RP mp from some center S . This interpretation of the pole placement map as a projection comes from [14]. We notice that projections associated with linear systems share the property of the Wronski map stated in Remark 1 in the Introduction: φ −1 S (Y ) = X, where X is the big cell of the Grassmann variety and Y is the big cell of RP mp .…”

“…If complex gain matrices are permitted, this condition is also sufficient, see, for example [2]. It was proved by X. Wang [13] that for n < mp the real pole placement map is generically surjective (see also [14,9]. We consider the real pole placement map with n = mp.…”

“…The set of all labelings of γ is identified with a closed convex polytope L γ in the affine subspace A of R 4d−4 described by equations (14), (15), (16). The dimension of A and L γ is 2d − 5.…”