The control parameterization method for nonlinear optimal control: A survey

Abstract

The control parameterization method is a popular numerical technique for solving optimal control problems. The main idea of control parameterization is to discretize the control space by approximating the control function by a linear combination of basis functions. Under this approximation scheme, the optimal control problem is reduced to an approximate nonlinear optimization problem with a finite number of decision variables. This approximate problem can then be solved using nonlinear programming techniques. The aim of this paper is to introduce the fundamentals of the control parameterization method and survey its various applications to non-standard optimal control problems. Topics discussed include gradient computation, numerical convergence, variable switching times, and methods for handling state constraints. We conclude the paper with some suggestions for future research.

Dynamic optimization of open-loop input signals for ramp-up current profiles in tokamak plasmas

“…These constraints are special cases of the well-known canonical form in the optimal control literature (see [9]). Now, under the approximation (28), the cost functional (25) becomes…”

“…However, it is well known that variable switching points cause computational difficulties [12]. To overcome these difficulties, we will employ the so-called time-scaling transformation [9,12] to map the variable switching points to fixed points in a new time horizon. This yields a new optimization problem in which the switching times are fixed.…”

“…For more accurate results, the control switching points should be variable along with the control parameters [9]. However, it is well known that variable switching points cause computational difficulties [12].…”

“…To solve Problem P N , we will apply the control parameterization method [9,23] We impose the following constraints:…”

Local optimization of dynamic programs with guaranteed satisfaction of path constraints

“…Remark 1 Optimal control problems with control trajectories as their decision variables can be approximated (restricted) into (PCDP) via the control vector parameterization technique [6,31,32,50]. Moreover, problems with an integral term as part of their objective function or with explicit time dependence can be transformed into PCDP via the introduction of extra variables and equations in the dynamic system [12,50].…”