Dynamic Pole Assignment Using Global, Blow Up Linearization: Low Complexity Solutions

Abstract

The problem of pole assignment by low-order dynamics output feedback controllers with McMillan degree n ₁ is studied for minimal systems described by a proper transfer function matrix G(s)ɛR ^m×p(s) with McMillan degree n. The new method is based on an asymptotic linearization of the pole placement map related to the problem. In this context, the problem is reduced to solving a set of linear equations, and this is done without losing any of the degrees of freedom of the controller. In terms of composite left matrix fraction description (MFD) representations, the asymptotic solution of the problem as ɛ→0 is given in closed form in terms of a one-parameter family of feedback compensators A(s)+ɛB(s), where A(s) is the so-called degenerate compensator and B(s) is the solution of the linear equations. The arbitrary pole placement property holds true when the linearization matrix has full rank, and it is shown that this occurs genetically when $n_1 \geqslant n'_1$, where $n'_1$ is the smallest multiple of min(p,m) satisfying $n'_1 \left( {m + p} \right) + mp \geqslant n + n'$