Orbits of controllable and observable systems
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Let a time-invariant linear system $\left .\aligned \dot x(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)\endaligned \right \}$ corresponding to a realization of a prescribed transfer function matrix can be represented by triples of matrices $(A,B,C)$. The permitted transformations of basis changes in the space state on the systems can be seen in the space of triples of matrices as similarity equivalence. In this paper we give a geometric characteriaztion of controllable and observable systems as orbits under a Lie group action. As a corollary we obtain a lower bound of the distance between a controllable and observable triple and the nearest uncontrollable one.