APPLICATION OF A CONTROL VECTOR PARAMETERIZATION METHOD USING AN INTERIOR POINT ALGORITHM T. Hirmajer, M. Fikar Institute of Information Engineering, Automation, and Mathematics Department of Information Engineering and Process Control Faculty of Chemical and Food Technology STU Radlinsk´ eho 9, 812 37 Bratislava, Slovakia E-mails: {tomas.hirmajer, miroslav.fikar}@stuba.sk E. Balsa-Canto and J. R. Banga Process Engineering Group Instituto de Investigaciones Marinas (C.S.I.C.) Spanish Council for Scientific Research C Eduardo Cabello 6, 36208 Vigo, Spain E-mails: {ebalsa, julio}@iim.csic.es ABSTRACT This paper explores the possibilities of using the combina- tion of the state of the art nonlinear programming prob- lem solver IPOPT and the initial value problem solver CVODES (incorporated in SUNDIALS) for the solution of dynamic optimization problems, also regarded as open loop optimal control problems (OCP). Recent works con- firm the potential of IPOPT in solving large scale optimiza- tion problems and particularly in solving optimal control problems in the framework of the complete parameteriza- tion approach [12]. Here the OCP problems are solved using the control vector parameterization scheme and the advantages and disadvantages of the combination CVP + IPOPT + CVODES are evaluated through several exam- ples. KEY WORDS dynamic optimization, control vector parameterization, sensitivity calculations, ordinary differential equations 1. Introduction There are many numerical methods able to solve nonlinear optimal control problems. Analytical methods represented by Bellman’s optimal principle or Pontryagin’s maximum principle [5] that solve two-point boundary value problems (TPBVP) are inappropriate in the case a complicated non- linear system is assumed. The same is true for the iterative methods represented by control vector iteration (CVI) or boundary condition iteration (BCI). Another approach is to transform the optimal control problems into finite dimensional optimization problems us- ing parametrization schemes. One possibility is a complete discretization of state and control variables – orthogonal collocation (OC). Such formulation can be found in [11]. However, the size of the resulting nonlinear programming problem (NLP) is very large. Other possibility is to leave the states intact and approximate only the control variables as piecewise constants, or with some higher order approxi- mations. This approach is known as control vector parame- terization (CVP). The main advantage of the CVP approach rests in parametrization only of control trajectory. This has an impact on the total number of the decision variables. The resultant NLP problems are usually solved by using a gradient based approach. In this regard there are several possibilities for gradient calculation [16]. One of them is a total difference approach which is the easiest to imple- ment. Main drawback of this method is the lowest accuracy in contrast to other gradient methods. Another possibility is to use adjoint equations [7, 9] approach or to solve sen- sitivity equations [6, 19]. Among advantages of sensitiv- ity approach are implementation issues, because the whole system of differential equations is integrated in the same direction of time, whereas adjoint approach needs integra- tions both forward and backward in time. Regarding the selection of the gradient based method, there are several possibilities. However and in the context of optimal control the SQP methods seem to be the most successful. The recent works by Biegler and coworkers [12, 13] show how the recently developed interior point method IPOPT can successfully handle large scale opti- mization problems. In fact [13] presents a dynamic op- timization toolbox based on the use of IPOPT within the complete parameterization scheme. This work addresses the solution of several dynamic optimization problems within the control vector parame- terization method. Particularly, the combination of IPOPT with CVODES and the computation of exact gradients through the use of first order parametric sensitivities is sug- gested as an alternative to obtain refined control profiles with reasonable computational costs. 2. Optimal Control Problem 2.1 System and Cost Description Consider a dynamical system described by the vector of ordinary differential equations (ODEs) ˙ x = f (t, x, u, p) (1) with given initial and terminal conditions x(0) = x 0 , x(t F )= x F (2) where x ∈ R nx is the vector of state variables, u ∈ R nu is the vector of control variables, p ∈ R np is the vector of parameters and t F is the final time of the process. 596-039 122