# Incorporating a class of constraints into the dynamics of optimal control problems

04 09 Category : Optimal Control | All

**Authors**: K. Graichen and N. Petit

Optimal Control, Applications and Methods, Volume 30 Issue 6, Pages 537 - 561, 2009. DOI: 10.1002/oca.880

A method is proposed to systematically transform a constrained optimal control problem (OCP) into an unconstrained OCP, which can be treated in the standard calculus of variations. The considered class of constraints comprises up to m input constraints and m state constraints with well-deﬁned relative degree, where m denotes the number of inputs of the given nonlinear system. Starting from an equivalent normal form representation, the constraints are incorporated into a new system dynamics by means of saturation functions and differentiation along the normal form cascade. This procedure leads to a new unconstrained OCP, where an additional penalty term is introduced to avoid the unboundedness of the saturation function arguments if the original constraints are touched. The penalty parameter has to be successively reduced to converge to the original optimal solution. The approach is independent of the method used to solve the new unconstrained OCP. In particular, the constraints cannot be violated during the numerical solution and a successive reduction of the constraints is possible, e.g. to start from an unconstrained solution. Two examples in the single and multiple input case illustrate the potential of the approach. For these examples, a collocation method is used to solve the boundary value problems stemming from the optimality conditions.

Download PDF

**BibTeX:**

@Article{,

author = {Knut Graichen, Nicolas Petit},

title = {Incorporating a class of constraints into the dynamics of optimal control problems},

journal = {Optimal Control, Applications and Methods},

volume = {30},

number = {6},

pages = {537-561},

year = {2009},

abstract = {A method is proposed to systematically transform a constrained optimal control problem (OCP) into an unconstrained OCP, which can be treated in the standard calculus of variations. The considered class of constraints comprises up to m input constraints and m state constraints with well-deﬁned relative degree, where m denotes the number of inputs of the given nonlinear system. Starting from an equivalent normal form representation, the constraints are incorporated into a new system dynamics by means of saturation functions and differentiation along the normal form cascade. This procedure leads to a new unconstrained OCP, where an additional penalty term is introduced to avoid the unboundedness of the saturation function arguments if the original constraints are touched. The penalty parameter has to be successively reduced to converge to the original optimal solution. The approach is independent of the method used to solve the new unconstrained OCP. In particular, the constraints cannot be violated during the numerical solution and a successive reduction of the constraints is possible, e.g. to start from an unconstrained solution. Two examples in the single and multiple input case illustrate the potential of the approach. For these examples, a collocation method is used to solve the boundary value problems stemming from the optimality conditions.},

location = {},

keywords = {optimal control problem, state and input constraints, normal form, saturation functions, calculus of variations, boundary value problem}}