Controllability, Feedback and Pole Assignment

Abstract

Consider as usual the system

$$ \dot x\left( t \right) = Ax\left( t \right) + Bu\left( t \right),t \geqslant 0. $$

(1)

Suppose we are free to modify (1) by setting

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacaqG1bGaaeikaiaabshacaqGPaGaaeyp % aiaabAeacaqG4bGaaeikaiaabshacaqGPaGaae4kaiaabYcacaqG0b % GaeyyzImRaaeimaiaabYcaaaa!4CB4!</EquationSource><EquationSource Format="TEX"><![CDATA[$${\text{u(t) = Fx(t) + ,t}} \geqslant {\text{0,}}$$

where v(.) is a new external input, and F: χ → u (is a constant map. We refer to F as the state feedback. The obvious result of introducing state feedback is to change the pair (A, B) in (1) into the pair (A +BF, B). We shall explore the effect of such a transformation of pairs on controllability and on the spectrum of A+BF. Our main result is that if (A, B) is controllable then σ(A +BF) can be assigned arbitrarily by suitable choice of F, and this property in turn implies controllability.