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.

In using penalty functions to handle final state equality constraints, we propose a systematic scheme for adjusting the penalty function factors, so that all the constraints are satisfied within a specified tolerance, and so that the Performance index is minimized. Two typical engineering problems, which are used to test the procedure, show that the proposed method of adjusting the penalty function factors can be used for reasonably complex systems to yield reliable results.A I'aide des fonctions de pbnalite pour traiter les contraintes d'egalite d'etat final, nous proposons une methode systematique dans le but d'ajuster les facteurs de fonction de penalite, de fapon a ce que toutes les contraintes soient satisfaites a I'interieur d'une tolerance specifiee, et de telle sorte que I'indice de performance soit minimise. Deux problemes d'ingenierie typiques, qui sont utilises pour tester la procedure, montrent que la methode proposee pour I'ajustement des facteurs de fonction de penalite peut servir a des systemes raisonablement complexes et donner des resultats fiables.

In addition to randomly chosen candidates for control, we examine the effect of also including deterministic control candidates in iterative dynamic programming (IDP) to improve the chance of achieving the global optimal solution. Two types of deterministic control candidates (shifting and smoothing candidates) are chosen on the basis of the control policy obtained in the previous iteration. The search for the optimal value for control in the subsequent iteration is then made on the combined set of control candidates chosen randomly and deterministically. Three highly nonlinear and multimodal chemical engineering optimal control problems are used to illustrate the advantages of this procedure in obtaining the global optimum.

ptimal control problems encountered in chemical engineering frequently involve situations where the initial values of some of 0 the state variables are not specified or the best initial condition is not known. For example, in optimal control of a fed-batch reactor process, the composition of two reactants at the beginning of the operation may be free to be determined, or the optimal initial volume of solution inside the reactor may not be known. In these situations, apart from establishing the optimal policy for the feed rate, we also want to find the initial condition for the state vector that yields the maximum yield of the desired product.One way of finding the best starting condition is to solve the optimal control problem for different values of initial states. However, a more efficient way is to carry out the optimization simultaneously. Teo et al.(1 991) showed that sequential quadratic programming (SQP) could be used with control parametrization technique to solve for both the optimal control policy and the optimal initial condition. Rehbock et al.(1 999) provided a recent survey of the computational methods based on the control parametrization technique for solving a general class of optimal control problems. Another approach is based on converting differential equations to algebraic equations through discretization of all variables and using SQP to solve the transformed equations (Tanartkit and Kegler, 1995). In this approach, the initial values of state variables are incorporated directly into the main SQP problem. The methods based on solving an approximated problem using SQP are generally very fast, but the global solution may be difficult to achieve when the system under consideration is highly nonlinear or multi-modal, as for example, the fedbatch reactor system studied by Hartig et al. (1995). Also, in the optimization of a bifunctional catalyst reactor problem, Luus et al. (1992) showed that the use of SQP led to over 20 local optima. Therefore, research in the development of alternative methods for solving optimal control problems continues. Recently, Claes et al. (1 999) applied a procedure based on Pontryagin's maximum principle to optimal control of a fed-batch reactor with free initial amount of substrate; Chiou and Wang (1999) used a differential evolution algorithm to establish the optimal solution for a fed-batch polymerization system with some free initial conditions. 'Author to whom correspondence may be addressed: E-mail address: luus@ ecf.utoronto.ca For solving optimal control problems where the initial conditions of some of the state variables are not specified, a procedure based on iterative dynamic programming (IDP) is presented. In this procedure, the free initial conditions are taken to be additional control variables for the first time stage only; then the search for the optimal initial conditions and also the optimal control policy is carried out simultaneously using IDP. The procedure is straightforward, and as illustrated with two nonlinear optimal control problems, for e...

Abstract:To handle final st ate equality constraints, two ty pe s of pen al ty functions are evaluated a nd compared in solving t hree op timal cont rol pr oblems. Both t he a bsol ute valu e pen al ty func t ion and t he quadra t ic pen al ty fun ction with shift ing terms yielded t he opt imal control policy in each cas e. T he qu adrat ic penal ty fun cti on with shift ing te r ms gave t he optimum more accurately, and t he a pp roach to t he optimum is less oscillatory t han with t he use of abso lute value pen alty fun cti on . In solving for t he optimal control in t he third examp le, t he use of the absolute value penalt y function required a series of runs to get t he performanc e ind ex to wit hin 0.035% of t he opti mum . A furt her advantage in usin g the q uadratic penalty function is obtaining sens it ivi ty informat ion wit h resp ect to t he final state const raints from t he shifti ng te r ms.