The multiple-input minimal time regulator problem (General theory)

This work considers a two-input linear time-invariant discrete system whose state transition equation is given by
X_{k+1} = AX_{k} + Du_{k+1}
where
A = n \times n
constant nonsingular matrix; x
k
is an
n
-rowed state vector of the system at
t=kT
;
D
is an
n \times 2
constant control matrix with columns d
1
and d
2
; and
u_{k+1}
is a 2-rowed control vector with components
u^{1}_{k+1}
and
u^{2}_{k+1}
. The control vector
u_{k+1}
is restricted to be an admissible control, i.e.,
|u^{i}_{k+1}| \leq 1
for
i=1, 2
and
k=0, 1, ...
. The two-input minimal time regulator problem may be stated as follows 1) Given any arbitrary initial state of the system, find admissible control vectors
u_{1}, u_{2}, ...
which will bring the system to equilibrium (i.e., the state
x=0
) in the minimum number of sampling periods. 2) Determine an optimal strategy, i.e., determine a vector valued function
u^{0}(x)
of the state
x
such that if the system is in state
x
at a sampling instant,
u^{0}(x)
is an admissible optimal control for the next sampling period. First, the general necessary and sufficient conditions for the system to be controllable with admissible controls are established. For a controllable system it is shown that the optimal strategy at each sampling instant requires the following: For each component
u^{i}_{k+1}
,
i=1, 2,
there exists a unique
(n-1)
-dimensional hypersurface
\varepsilon^{i}
,
i=1, 2
. The optimal strategy
u^{0}(x_{k+1})
is then a simple nonlinear function of each of the λ
i
's where λ
i
is the distance of x
0
from
\zeta^{i}
along a direction parallel to
A^{-1}d_{i}
, for
i=1, 2
. This optimal strategy therefore satisfies the operations of the feedback computer in order that the system returns to equilibrium in minimum time after any arbitrary disturbances. The results of this work are applicable to all discrete systems of the above form which are controllable by admissible controls irrespective of whether the eigenvalues of
A
are distinct or multiple, real or occur in complex conjugate pairs. Furthermore, the theory is directly extendable to the case where
D=n \times m
constant matrix and
u_{k+1}
is an
m
-rowed control vector;
m > 2
, subject to the admissibility constraint
|u^{i}_{k+1}| \leq 1, i=1, 2, . . ., m
.