Arbitrary pole placement with the extended Kautsky-Nichols-van Dooren parametric form with minimum gain

Abstract

Abstract-We consider the classic problem of pole placement by state feedback. We revisit the well-known eigenstructure assignment algorithm of Kautsky, Nichols and van Dooren [1] and extend it to obtain a novel parametric form for the pole-placing feedback matrix that can deliver any set of desired closed-loop eigenvalues, with any desired multiplicities. This parametric formula is then employed to introduce an unconstrained nonlinear optimisation algorithm to obtain a feedback matrix that delivers the desired pole placement with minimum gain.

Robust arbitrary pole placement with the extended Kautsky-Nichols-van Dooren parametric form

“…For a given real m × n parameter matrix K, we obtain the eigenvector matrix X(K) and gain matrix F(K) by building the Jordan chains of closedloop generalised eigenvectors; the chains commence with the selection of eigenvectors from the kernel of certain matrix pencils. Thus the results of [1] neatly parallel the achievements of [11]- [12] in providing another novel parametric form to achieve pole placement with arbitrary multiplicities, while employing an m × n-dimensional parameter matrix. The virtue of having a comprehensive parametric formula for the matrices X and F that solve (2) is that they invite the consideration of optimal pole placement problems, in which one seeks a gain matrix that will deliver the desired closed-loop eigenstructure and also provide some other desirable features.…”

“…Byers and Nash used Corollary 2.1 to obtain a parametric form for the matrix of eigenvectors X satisfying (2), and then employed it to consider the REPP problem for the case of a diagonal Λ matrix. In [1] we adapted Corollary 2.1 to obtain a parametric form for X and F that can accommodate any admissible Jordan structure (L , M , P) for (A, B), and we now briefly summarise this method. We begin by noting that for each i ∈ {1, .…”

“…In [1] the present authors revisited the pole-placing feedback method of Kautsky, Nichols and van Dooren [2] and generalised it to obtain a parametric formula that can also accommodate arbitrary pole-placement, in terms an m × n dimensional parameter matrix. For a given real m × n parameter matrix K, we obtain the eigenvector matrix X(K) and gain matrix F(K) by building the Jordan chains of closedloop generalised eigenvectors; the chains commence with the selection of eigenvectors from the kernel of certain matrix pencils.…”

“…Papers [11] and [17] employed a parametric formula based on the Klein-Moore method to consider the MGEPP and REPP problems, respectively. In [1] the present authors employed the parametric formula based on the Kautsky-Nichols-van Dooren method to consider the MGEPP problem. In this paper we complete this quartet of results by using the Kautsky-Nichols-van Dooren method to consider the REPP problem.…”

“…X ν ] ordered such that any complex submatrices occur consecutively in complex conjugate pairs, and so that, for some integer s, the first 2s submatrices are complex while the remaining are real. We define a real matrix Re{X} of the same dimension as X thus: if X i and X i+1 are consecutive complex conjugate submatrices of X, then the corresponding submatrices of Re{X} are 1 2 (X i + X i+1 ) and 1 2 j (X i − X i+1 ).…”