## Controllability and Observability by R.E. Kalman (auth.), Prof. E. Evangelisti (eds.)

By R.E. Kalman (auth.), Prof. E. Evangelisti (eds.)

R.E. Kalman: Lectures on controllability and observability.- E. Kulikowski: Controllability and optimal contro.- A. Straszak: Supervisory controllabilityl.- L. Weiss: Lectures on controllability and observability.

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E. :27) PROPOSITION. 28) A system E generates HI COROLLARY. generated by -, H) is completely observable ~*. The observable COstates of E* are precisely the reachable states of E*, ~* = (F, that is, those of the submodule of H'. We have eliminated the abuse of language incurred by talking about "observable states" through introduction of the new notion of "observable COstates". The full explication of why this is necessary (as well as natural) is postponed until Section 10. 13). F But module theory yields many other, less obvious results as well, which derive mainly from the fact that K[z] is a principa1- ideal domain.

B) n is a free K[z]-module with m generators, that is, n ~ IfI[z] also in the K[z]-module sense. (I) «(I). E K[ z ], j J 1, ••. , m). The product of 7T with the components of the vector (I) is the product in K[z]. 6) . on, j (c) On n 1, ••• , m, the action of the shift operator by multiplication by z. rrn is represented This. of course. 4) in the first place. - 46 - R. E. 1), not defined) for yet) E t < 1. 7) series. 8»), we can say also that (e) r is isomorphic to the K-vector subspace of z-l ~~th coefficients in KP) (formal power series in of all power series with 0 first KP[[z-l]] consisting t er~.

10)), We make it into a X= ~ is already a K[zl-module by defining - 49 - R. E. 14) COMMENT. JCl: (7T, x) H 7T(F)x . 13). 16), the state set X can be treated as a K[zl module irrespective as to whether X is constructed from or given a priori as part of the specification of L the (X f (X = Xf) = ~). Thus K[zl-module structure on X is a nice way of uniting the "external" and the "internal" definitions of a dynamical system . Henceforth we shall talk about a (discrete-time, linear, constant dynamical) system L somewhat imprecisely via properties of its ass ociated K[zl -module~.