Optimal control problems with constraints on the state and control and their applications

Abstract

In this thesis, we consider several types of optimal control problems with constraints on the state and control variables. These problems have many engineering applications. Our aim is to develop efficient numerical methods for solving these optimal control problems.In the first problem, we consider a class of discrete time nonlinear optimal control problems with time delay and subject to constraints on states and controls at each time point. These constraints are called all-time-step constraints. A constraint transcription technique in conjunction with a local smoothing method is used to construct a sequence of approximate discrete time optimal control problems involving time delay in states and controls and subject to nonlinear inequality constraints in canonical form. These approximate optimal control problems are special cases of a general discrete time optimal control problems with time delay appearing in the state and control and subject to nonlinear inequality constraints in canonical form. Thus, we devise an efficient gradient-based computational method for solving this general optimal control problem.The gradient formulas needed for the cost and the canonical constraint functions are derived. With these gradient formulas, the discrete time optimal control problem with time delay appearing in states and controls and subject to nonlinear inequality constraints in canonical form is solvable as an optimization problem with inequality constraints by the Sequential Quadratic Programming (SQP) method. With this computational method, each of the approximate problems constructed from the original optimal control problem can be solved. A practical problem arising from the study of a tactical logistic decision analysis problem is considered and solved by using the computational method that we have developed.In the second problem, we consider a general class of maximin optimal control problems, where the violation avoidance of the continuous state inequality constraints is to be maximized. An efficient computational method is developed for solving this general maximin optimal control problem. In this computational method, the constraint transcription method is used to construct a smooth approximate function for each of the continuous state inequality constraints, where the accuracy of the approximation is controlled by an accuracy parameter. We then obtain a sequence of smooth approximate optimal control problems, where the integral of the summation of these smooth approximate functions is taken as its cost function.A necessary condition and a sufficient condition are derived showing the relationship between the original maximin problem and the sequence of the smooth approximate problems. We then construct a violation avoidance function from the solution of each of the smooth approximate optimal control problems and the original continuous state inequality constraints in such a way that the problem of finding an optimal control of the maximin optimal control problem is equivalent 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 these smooth approximate optimal control problems. Two practical problems are considered as applications. The first one is an obstacle avoidance problem of an autonomous mobile robot, 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 subject to continuous inequality constraints and terminal equality constraint. By applying the constraint transcription method and a local smoothing technique to these continuous inequality constraint functions, we construct the corresponding smooth approximate functions. We use the concept of the penalty function to append these smooth approximate functions to the cost function, forming a new cost function. Then, the constrained optimal PID control problem is approximated by a sequence of optimal parameter selection problems subject to only terminal equality constraint. Each of these optimal parameter selection problems can be viewed and hence solved as a nonlinear optimization problem. The gradient formulas of the new appended cost function and the terminal equality constraint function are derived, and a reliable computation algorithm is given. The method proposed is used to solve a ship steering control problem.In the fourth problem, we consider a class of optimal control problems subject to equality terminal state constraints and continuous inequality constraints on the state and/or control variables. After the control parameterization together with a time scaling transformation, the problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous inequality constraints on the state and/or control. An exact penalty function is constructed for these terminal equality constraints and continuous inequality constraints. It is appended to the cost function to form a new cost function, giving rise to an unconstrained optimal parameter selection problem. The convergence analysis shows that, for a sufficiently large penalty parameter, a local minimizer of the unconstrained optimization problem is a local minimizer of the optimal parameter selection problem with terminal equality constraints and continuous inequality constraints. The relationships between the approximate optimal parameter selection problems and the original optimal control problem are also discussed. Finally, the method proposed is applied to solve three nontrivial optimal control problems.