Optimal Control Problems With Constraints On The State And Control And Their Applications

In this thesis, we consider several types of optimal control problems with constraintson the state and control variables. These problems have many engineering applications.Our aim is to develop efficient numerical methods for solving these optimal controlproblems.In the first problem, we consider a class of discrete time nonlinear optimal controlproblems with time delay and subject to constraints on states and controls at each timepoint. These constraints are called all-time-step constraints. A constraint transcriptiontechnique in conjunction with a local smoothing method is used to construct a sequenceof approximate discrete time optimal control problems involving time delay in statesand controls and subject to nonlinear inequality constraints in canonical form. Theseapproximate optimal control problems are special cases of a general discrete time optimalcontrol problems with time delay appearing in the state and control and subject tononlinear inequality constraints in canonical form. Thus, we devise an e?cient gradient-based computational method for solving this general optimal control problem. Thegradient formulas needed for the cost and the canonical constraint functions are derived.With these gradient formulas, the discrete time optimal control problem with timedelay appearing in states and controls and subject to nonlinear inequality constraintsin canonical form is solvable as an optimization problem with inequality constraints bythe Sequential Quadratic Programming (SQP) method. With this computational method,each of the approximate problems constructed from the original optimal control problemcan be solved. A practical problem arising from the study of a tactical logistic decisionanalysis problem is considered and solved by using the computational method that wehave developed.In the second problem, we consider a general class of maximin optimal controlproblems, where the violation avoidance of the continuous state inequality constraintsis to be maximized. An e?cient computational method is developed for solving thisgeneral maximin optimal control problem. In this computational method, the constrainttranscription method is used to construct a smooth approximate function for each ofthe continuous state inequality constraints, where the accuracy of the approximationis controlled by an accuracy parameter. We then obtain a sequence of smooth approximate optimal control problems, where the integral of the summation of thesesmooth approximate functions is taken as its cost function. A necessary condition anda sufficient condition are derived showing the relationship between the original maximinproblem and the sequence of the smooth approximate problems. We then construct aviolation avoidance function from the solution of each of the smooth approximate optimalcontrol problems and the original continuous state inequality constraints in such a waythat the problem of finding an optimal control of the maximin optimal control problem isequivalent to the problem of finding the largest root of the violation avoidance function.The control parameterization technique and a time scaling transform are applied to thesesmooth approximate optimal control problems. Two practical problems are considered asapplications. The first one is an obstacle avoidance problem of an autonomous mobilerobot, while the second one is the abort landing of an aircraft in a windshear downburst.The proposed computational method is then applied to solve these problems.In the third problem, we consider a class of optimal PID control problems subjectto continuous inequality constraints and terminal equality constraint. By applying theconstraint transcription method and a local smoothing technique to these continuousinequality constraint functions, we construct the corresponding smooth approximatefunctions. We use the concept of the penalty function to append these smooth approximatefunctions to the cost function, forming a new cost function. Then, the constrained optimalPID control problem is approximated by a sequence of optimal parameter selectionproblems subject to only terminal equality constraint. Each of these optimal parameterselection problems can be viewed and hence solved as a nonlinear optimization problem.The gradient formulas of the new appended cost function and the terminal equalityconstraint function are derived, and a reliable computation algorithm is given. The methodproposed is used to solve a ship steering control problem.In the fourth problem, we consider a class of optimal control problems subjectto equality terminal state constraints and continuous inequality constraints on the stateand/or control variables. After the control parameterization together with a time scalingtransformation, the problem is approximated by a sequence of optimal parameter selectionproblems with equality terminal state constraints and continuous inequality constraintson the state and/or control. An exact penalty function is constructed for these terminalequality constraints and continuous inequality constraints. It is appended to the costfunction to form a new cost function, giving rise to an unconstrained optimal parameterselection problem. The convergence analysis shows that, for a su?ciently large penalty parameter, a local minimizer of the unconstrained optimization problem is a localminimizer of the optimal parameter selection problem with terminal equality constraintsand continuous inequality constraints. The relationships between the approximate optimalparameter selection problems and the original optimal control problem are also discussed.Finally, the method proposed is applied to solve three nontrivial optimal control problems.l...