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

Institution: | Curtin University of Technology |
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Department: | Mathematics and Statistics |

Year: | 2011 |

Keywords: | Optimal control problems, the state and control variables, engineering applications, numerical methods |

Record ID: | 1056476 |

Full text PDF: | http://espace.library.curtin.edu.au:80/R/?func=dbin-jump-full&object_id=167908&local_base=gen01-era02 |

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…