抄録
Discrete control is widely found in various field, including the control by digital computers. Systems whose control inputs are restricted to discrete values are called the quantized control systems.
A state x is called reachable if there is an input sequence which carries the system from zero to x. In quantized control system, the best situation for reachability is that within any neighborhood of x there is a reachable state, in which case x is called to be “almost reachable”.
In this paper, a structure analysis of the state space is developed for the 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.
Concerning with reachability, three A-invariant subspaces (X2+X3+X4, X3+X4 and X4 below) are characterized, resulting the following canonical form of the quantized control system.
X2+X3+X4: A-invariant subspace spanned by the set of reachable states.
X3+X4: A-invariant subspace in which every state is almost reachable.
X4: A-invariant subspace in which every state is almost reachable in finite time.
A2, B2: matrices over the ring of all integers.
A32, A3, B3: matrices over the field of rational numbers, A2 has no algebraic integers as its characteristic values.
A4, B4: matrices over the field of real numbers.
