Systems whose control inputs are restricted to discrete values are called quantized control systems. Discrete control is usually found in various fields, for example, in socio-economical systems, in ecological systems and in the control problems by digital computers.
Even if a given system is controllable by continuous level controls (an arbitrary small input value is available) it can not be expected that the system is controllable by quantized control inputs. “To what extent can we control the system?” is a fundamental quetion in quantized control systems.
In this paper, the system structure and reachability of quantized control systems are studied. 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)∈Z^m and A and B are real matrices.
Since the cardinality of the set of reachable states is at most countably infinite, the best situation for reachability of the system is that the set of reachable states is dense in the state space Rⁿ, in which case the system is called to be “almost reachable.”
The main results are as follows.
(1) The state space is a direct sum of two subspaces: the “almost reachable subspace” in which the reachable states are dense and the “discretely reachable subspace” in which the reachable states are isolated points. The almost reachable subspace is A-invariant.
(2) The “almost reachability”of a system is closely related to the integral properties of the eigenvalues of A.
(3) When A and B are matrices over the field of rational numbers, the “almost reachability” is dominated only by the eigenvalues.
(4) In the general case when A and B are real matrices, the “almost reachability” is influenced also by the input matrix B. However, the almost reachability can be examined through finite procedure.