Abstract
The discrete control is widely found in various field, including those related to the control problem by digital computers. Systems whose control inputs are restricted to discrete values are called the quantized control systems.
A state x is reachable if there is an input sequence which carries the system from zero to x. A state x is controllable if there is an input sequence which carries the system from x to zero. An adjective “almost” will be used if the above is true not exactly but within any degree of accuracy.
In this paper, some of the properties concerning the controllability of the system which are true for the systems controlled by continuous level inputs are examined for quantized control systems. The systems under discussion here are linear, constant and discrete-time dynamical systems over the field of real numbers whose control inputs are restricted to the integer vectors:
x(k+1)=Ax(k)+Bu(k)where x(k)∈Rn, u(k)∈Zm and A and B are real matrices.
The main results are as follows:
(1) The set of all controllable states H∞ has a structure of Z-module which spans the subspace of all controllable states by continuous level inputs.
(2) Under the finiteness condition that the set of all reachable states M∞ is a finitely generated Z-module, H∞ is A-invariant and every reachable state is controllable.
(3) The set of all “almost controllable” states is A-invariant but without some finiteness condition it is not a Z-module.
(4) (Without any conditions) Every “almost reachable” state is “almost controllable” and if the system is “almost reachable” we can send the system from any initial state to any final state within any degree of accuracy.
