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Controllability, Feedback and Pole Assignment

  • Walter Murray Wonham
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 101)

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
where v(.) is a new external input, and F: χu (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.

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Notes and References

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    N. N. KRASOVSKII On the stabilization of unstable motions by additional forces when the feedback loop is incomplete. Priklad. Mat, i Mekh. 27 (4), 1963, pp. 641–663. [Engl. trans. Appl. Math. and Mech. 27 (4), 1963, pp. 971–1004. 1MathSciNetGoogle Scholar
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    N. N. KRASOVSKII On the stabilization of dynamic systems by supplementary forces. Diff. Uray. 1 (1), 1965, pp. 5–16. [Engl. trans. Diff. Eqns. 1 (1), 1966, pp. 1–9.]Google Scholar
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Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • Walter Murray Wonham
    • 1
  1. 1.Department of Electrical EngineeringUniversity of TorontoTorontoCanada

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