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2.4 Historical notes and references
The general structure theorem 2.1.1 is due to R.E. Kalman, Canonical structure of linear dynamical systems. Proc. Nat. Acad. Sci. (U.S.A.), 48, pp. 596–600, 1962, and L. Weiss and R.E. Kalman, Contributions to linear system theory, Int. J. Engr. Sci., 3, pp 141–171. 1965. The canonical form given in Theorem 2.1.2 is one of several for completely controllable systems given by D.G. Luenberger, Canonical Forms for linear multi-variable systems. IEEE Trans. Aut. Control AC-12, pp 290–293 (1967). See also the recent survey articles
Maroulas, J. and S. Barnett. Canonical forms for time-invariant linear control systems: a survey with extensions, Part I. Single-input case. Int. J. Systems Sci., 9, pp. 497–514, 1978; Part II. Multivariable Case, Int. J. Systems Sci., 10, pp 33–50, 1979.
For single-input systems J.E. Betram in 1959 first proved that complete controllability implies that closed loop-poles can be freely positioned. For r > 1 the result is due to W.M. Wonham. On pole assignment in multi-input controllable linear systems. IEEE Trans. Aut. Control. AC-12, pp 660–665, (1967). The proof given here, based upon Luenberger's canonical form, is due to M. Heymann, Comments on ‘On pole assignment in multi-input controllable linear systems.’ IEEE Trans Aut. Control, AC-13, p 748, (1968).
Observers having state dimension less than n were first introduced by D.G. Luenberger in two papers: Observing the state of a linear system. IEEE Trans. Military Electronics, MIL-8, pp 74–80, 1964, and Observers for multivariable systems. IEEE Trans. Aut. Control, AC-11, pp 190–197, 1966. See also his paper: An introduction to observers, IEEE Trans. Aut. Control, AC-16, pp 596–602, 1971.
Theorem 2.3.3 and related matters are treated in the following papers: H. Kimura, Pole assignment by gain output feedback. IEEE Trans. Aut. Control. AC-20, pp 509–516, 1975.
E.J. Davison, S.H. Wang, On pole assignment in linear multivariable systems using output feedback. IEEE Trans. Aut. Control, AC-20, pp 516–518, 1975.
S. Barnett, Introduction to Mathematical Control Theory, Clarendon Press, Oxford, 1975.
R.W. Brockett, Finite Dimensional Linear Systems. John Wiley and Sons, Inc., New York, 1970.
R.E. Kalman, P.L. Falb and M.A. Arbib, Topics in Mathematical System Theory. McGraw-Hill Book Company, New York, 1969.
W.M. Wonham, Linear Multivariable Control. Springer-Verlag, New York, 1974.
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(1980). Canonical forms, pole assignment and state observers. In: Jacobson, D.H., Martin, D.H., Pachter, M., Geveci, T. (eds) Extensions of Linear-Quadratic Control Theory. Lecture Notes in Control and Information Sciences, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0004372
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