A Method for Computing Suboptimal Controls

Abstract

A time optimal control function cannot be obtained analytically when the order of the system becomes equal to or higer than four. In the recent years various iterative procedures have been developed for computing time optimal controls.
This paper presents an iterative procedure for computing suboptimal controls. This procedure is an extension of a pulse perturbation method developed by the author and may be applied in analysing an nth order system described by the vector defferential equations x=Ax+Bu, where A is the n×n constant matrix with complex eigenvalues, B the nth order vector and u(t), a scalar, is a control function with constraint |u(t)|≤1.
The principle of this method is as follows. Suppose that a control u(t) takes only the values of +1 or -1 and that the control time is t_s, that is u(t)≡0, for t>t_s. Pulse perturbations are added to u(t) at every switching time in such a way that C(t_s)(=-∫t_s0Φ^-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. These pulses are then approximated by rectangular waves sb that the perturbed control takes the form of a Bang-Bang control. The suboptimal control can be obtained if the iteration is continued until C(t_s) becomes equal to x₀.