  Several theorems on the volume computing of the polyhedron spanned by a
n-dimensional vector set with the finite-interval parameters are presented and
proved firstly, and then are used in the analysis of the controllable regions
of the linear discrete time-invariant systems with saturated inputs. A new
concept and continuous measure on the control ability, control efficiency of
the input variables, and the diversity of the control laws, named as the
controllable abundance, is proposed based on the volume computing of the
regions and is applied to the actuator placing and configuring problems, the
optimizing problems of dynamics and kinematics of the controlled plants, etc..
The numerical experiments show the effectiveness of the new concept and methods
for investigating and optimizing the control ability and efficiency.
