Design of quasi-optimal minimum-time controllers

Abstract

Absfroct-This paper presents and evaluates a heuristic method for designing easily implemented quasi-optimal, minimum-time controllers for high-order dynamic systems. The high-performance, or quasi-optimal, controller is obtained by least-squares fitting points on the optimal switching surface with an easily implemented, linear-segment switching surface. This method is attractive because least-squares approximation is an analytic procedure which is readily applicable to high-order dynamic systems since it requires no visualization of the surface for its application. The method is applicable to linear dynamic systems with real characteristic f r e quencies as well as many other dynamic systems.For evaluation, the method has been used to design the switching surface for a triple-integrator dynamic system (Le., l/s3 plant).A third-order system has been used, instead of a more easily portrayed second-order system, since it more typically illustrates the problems encountered in the approximation of higher-order switching surfaces. Typical response times for the q-Jasi-optimal switching surface in this example are 1.5 to 2 times those of the minimum-time optimal switching surface and roughly one-half those of the best linear switching surface.

Minimum time path planning for robot motion in obstacle strewn environment

“…Then we benefit of considering the piecewise linear motion of the ~,/stem. Let us consider the analytical equations for a piecewise time optimal motion• Fcr a straight line motion the time optimal speed traiectory can be obtained (see [ 3g ,40] where the results from [ 47,48] have been developed), and given as illustrated in Figure tO, a. As shown in Figure I0, b the time loss due a turn depends on the stages of decceleration and acceleration from the maximum value of allowed speed at the turning point.…”

Punish/Reward: Learning with a Critic in Adaptive Threshold Systems

“…Thus, through the encoding procedure, a nonlinear function is made perfectly realizable by a single linear threshold logic element. The strategy needed to play the game of blackjack is related to that required for the "bang-bang" (contactor) control of a variety of dynamic systems [38]. In both cases, binary decisions must be made based on the values of several analog or multilevel state variables.…”

“…tI vPmin [I (2 t f)Pmjnj 37 2-7TD (l 2Pmin)V(l i/'4l -(l 1 tf)Pmin] For practical cases having small Pn,in and i/J, (37) simplifies to (38) (P p~)4t--IPmin…”

“…APPLICATIONS OF BOOTSTRAP LEARNING MODEL In the previous sections of this paper, a mathematical model of the bootstrap punish-reward learning process has been proposed and analyzed. The key result, the derivation of (p+ -p ), is given by (37) and (38). It is expected that this derivation will be very useful in understanding the behavior of bootstrap learning, although the set of hypotheses on which the atnalytical imiodel is based muay not always precisely agree with the physical situation in a given application.…”

“…Since z is a function of (p, --p ) and thereby is a fiunction of i/i. this differential equationi becomes, using (38) and 39,…”