In this paper some definitions of degree of controllability (observability), which is based on the scalar measure of the controllability (observability) Grammian matrix, are presented for discrete-time systems. The emphasis here is in showing the physical and geometrical meanings of the degree of controllability (observability) and the general properties of degree of controllability (observability). The concepts of degree of controllability are applied to the actuator placement problem for the shallow spherical sheel control system. The LQG transient responses for several typical systems with different degrees of controllability are also shown.

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