Exact pole assignment using direct or dynamic output feedback

Abstract

This note addresses the problem of the assignability of the eigenvalues of the matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A + BPC</tex> by choice of the matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> . This mathematical problem corresponds to pole assignment in the direct output feedback control problem, and by proper changes of variables it also represents the pole assignment problem with dynamic feedback controllers. The key to our solution is the introduction of the new concept of local complete assignability which in loose terms is the arbitrary perturbability, of the eigenvalues of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A + BPC</tex> by perturbations of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> . If n <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</inf> is the order of the system, we show that if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A + BP_{0}C</tex> has distinct eigenvalues, a necessary and sufficient condition for local complete assignability at P <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> is that the matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C[A + BP_{0}C]^{i-1}B</tex> be linearly independent, for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1 \leq i \leq n_{x}</tex> . In special cases, this condition reduces to known criteria for controllability and observability. Although these latter properties are necessary conditions for assignability, we also address the question of the assignability of uncontrollable or unobservable systems both by direct output feedback and dynamic compensation. The main result of this note yields an algorithm that assigns the closed-loop poles to arbitrarily chosen values in the direct and in the dynamic output feedback control problems.