A minimal time discrete system

Consider a sampled-data control system with the following sequence of components in the forward path: a sampler with period
T
, a zero-order hold circuit, a linear amplifier with saturation limits ±1, and a plant with transfer function
G(s)=\frac{1}{\Pi\min{i=1}\max{n} (s-\lambda_{i})}.
It is assumed that the poles
\lambda_{1}, \lambda_{2}, ... , \lambda_{n}
of
G(s)
are real, distinct, and non-positive (a single integral is permissible). The sampler, zero-order hold, and saturating amplifier constrain
f(t)
, the forcing function of
G(s)
, to be piecewise constant with values between -1 and +1. The forcing function
f(t)
is completely defined, for
t>0
, by the sequence of numbers
f_{1}, f_{2}
, ... , where f
i
is the value of
f(t)
during the i'th sampling period. The minimal time regulator problem for the above system can then be stated as follows: Given
G(s)
with an arbitrary set of initial conditions [i.e., the state vector
\overrightarrow{c(0)}
defined by its components
c(0), \dot{c}(0), ... , c^{n-1}(0)
]; find the forcing function
f(t)
[specified by
f_{1}, f_{2}, ...
and satisfying
|f_{i}| \leq 1]
, and the corresponding computer in the feedback loop which will bring the system to equilibrium in the minimum number of sampling periods. Any such forcing function will be called an optimal control. The first step is to consider
R_{N}'
the set of all initial states
\overrightarrow{c(0)}
from which the origin can be reached in
N
sampling periods or less. From this definition all such states are characterized algebraically and geometrically:
R_{N}'
is shown to be a convex polyhedron with
2 \sum\min{k=1}\max{n} ({N-1}\over{k-1})
vertices. Let R
N
be the set of all initial states
\overrightarrow{c(0)}
from which the origin can be reached in
N
sampling periods and no less. Each point of R
N
is shown to have a unique canonical representation. The coefficients appearing in the canonical representation suggest an optimal control. To obtain this particular optimal control we define a surface in state space called the critical surface. It is shown that this optimal control will be generated by the following procedure: at the beginning of each sampling period the distance φ from the state of the system to the critical surface is measured along a fixed specified direction; if
\phi \geq 1
(or ≤ -1) then the forcing function for that sampling period is +1 (or -1); if
|\phi| < 1
, then the forcing function is φ. For a third-order plant it is shown that the critical surface has certain properties which lead to a simple analog computer simulation.