In the recent years various iterative procedures have been developed in the time optimal control computations, because the solution cannot be obtained analytically when the order of the system becomes equal to or higer than the fourth. From a practical point of view, the rate of convergence of iterative methods is very important for control engineers.
This paper presents an iterative procedure for computing time optimal controls. This procedure may be applied in analysing an
n-th order system described by the vector differential equations
x=
Ax+
Bu, where
A is the
n×n constant matrix with real and distinct eigenvalues,
B, the
n th order vector and
u(t), a scalar, is a control function with constraint |
u(t)|≤1.
The computational method prescented here has the advantage such that the computed value generally converges very rapidly starting from an arbitrary initial point.
The principle of this method is as follows. Suppose that a control
u(t) takes only the values of +1 or -1 and that its switching times are
t1,
t2, ……,
tn, where
tn is the final time of
u(t), that is,
u(t)≡0, for
t>
tn. Pulse perturbations Δ
u1, Δ
u2, ……, Δ
un are added to
u(t) at every switching time in such a way that
C(tn)(=∫
tn0Φ
-1(τ)
Bu(τ)
dτ) arrives at -
x0, where Φ
(t) is the transition matrix for the system,
x0 the initial state of
x(t) and the final state is zero. The pulses Δ
uj,
j=1∼
n, are then approximated by rectangular waves so that the perturbed control takes the form of a Bang-Bang control. The optimal control can be obtained if the iteration is continued until
C(tn) becomes equal to -
x0. This method can be easily extented to the case that the switching number is less than
n-1, and the optimal solution can be obtaind starting from an arbitrary initial value.
View full abstract