Optimal Control of Inequality State Constrained Systems

Abstract

To handle inequality state constraints in nonlinear optimal control problems, we propose a method of introducing an auxiliary state variable for each constraint. The derivatives of these state constraint variables are made positive if the constraint is violated, and zero if there is no constraint violation. By incorporating these state variables then as penalty functions in an augmented performance index, we can ensure that the inequality state constraints are satisfied everywhere inside the given time interval. The procedure, as illustrated and tested with three nonlinear optimal control problems, is found to work well even in the presence of many state constraints.

An iterative multi-objective particle swarm optimization-based control vector parameterization for state constrained chemical and biochemical engineering problems

“…Thereafter, he pointed out that the optimum value of the performance index was very sensitive to the time of switching of the coolant flow rate. Based on this, Mekarapiruk and Luus proposed a method of introducing an auxiliary state variable for each constraint and then solved the new system by applying IDP [46], values of 0.67707 to 0.67727…”

“…This case considers a plug-flow reactor as studied by Reddy and Husain [44], Luus [45] and Mekarapiruk and Luus [46]. The objective is to find the optimal control trajectory of the normalized coolant flow rate ( u ) that maximizes the normalized concentration of the desired product.…”

Direct transcription solution of optimal control problems with higher order state constraints: theory vs practice

“…In this type of approach needed gradients can often be found by integrating another set of differential equations called the sensitivity equations. This idea can be pushed even further by using control parameterization in which case the control is viewed as some type of spline (Goh and Teo 1988;Mekarapiruk and Luus 1997) and the problem size is reduced even further. For some types of problems these approaches have proven very successful.…”

Evaluation of Penalty Functions for Optimal Control

“…The same approach could be used in optimal control where flexible stage lengths are used and the sum of the stage lengths must give the specified final time Luus (1996b). In fact, the quadratic penalty function with shifting terms led to an easy way of dealing with equality constraints Luus and Storey (1997) , Mekarapiruk and Luus (1997a). The goal here is to evaluate and compare these two types of penalty functions in the determination of the optimal control policy for three optimal control problems.…”