Linear Multivariable Control pp 46-54 | Cite as

# Controllability, Feedback and Pole Assignment

Chapter

## Abstract

Consider as usual the system
Suppose we are free to modify (1) by setting

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

(1)

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.## Preview

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## Notes and References

- [1]M. HEYMANN Pole assignment in multi-input linear systems. IEEE Trans. Aut. Control AC-13 (6), 1968, pp. 748–749.Google Scholar
- [1]A. S. MORSE, W. M. WONHAM Decoupling and pole assignment by dynamic compensation. SIAM J. Control 8 (3), 1970, pp. 317–337.MathSciNetMATHGoogle Scholar
- [1]C.E. LANGENHOP On the stabilization of linear systems. Proc. Amer. Math. Soc. 15 (5), 1964, pp. 735–742.MathSciNetMATHGoogle Scholar
- [2]V. M. POPOV Hyperstability and optimality of automatic systems with several control functions. Rev. Roum. Sci. Electrotech. et Energ. 9 1964, pp. 629–690.Google Scholar
- [1]O.A. SEBAKHY, W.M. WONHAM A design procedure for multivariable regulators. Proc. IFAC Third MultivariableGoogle Scholar
- [1]W. M. WONHAM On pole assignment in multi-input controllable linear systems. IEEE Trans. Aut. Control AC-12 (6), 1967, pp. 660–665.Google Scholar
- [1]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 - [2]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
- [1]E.J. DAVISON, S.G. CHOW An algorithm for the assignment of closed-loop poles using output feedback in large linear multivariable systems. IEEE Trans. Aut. Control AC-18 (1), 1973, pp. 74–75.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1974