Control Variable Parameterization Based Computational Method For Constrained Optimal Control Problems

Computational methods for optimal control are important for dealing with bottlenecks in industries. By implementing optimal operation stratigeies that are obtained from the computational methods into dynamic systems, objectives such as saving energy, reducing cost, exploiting protential, improving efficiency can be achieved. Computational methods for optimal control have been used in a broad range of diciplines, including aeronautics and astronautics, oil and chemical engineering, power system, clean energy, biomedical engineering, economicas and management. Due to the great application value of optimal control, its computational methods have drawn wordwide attentions.Direct methods are most effective in computational methods for optimal control. Direct methods transform an optimal control problem into an approximate mathematical programming problem, then choose a mathematical programming solver to get an approximate solution to the original problem. Two representative direct methods are:the control variable parameterization (CVP) method and the orthogonal collocation (OC) method. Both of the two methods have their own advantages and disadvantages. The main disadvantages of the CVP method are:(1) inequality path constraints are difficult to handle;(2) in order to get gradient information, the CVP method needs to solve differential equations repeatly, which will slow down the solution process. This thesis only studies the CVP method, aming to conquer the two main disadvantages.The main work and contribution of this thesis include:1. To conquer the difficulties that arise from inequality path constraints, four smooth approximation functions of max function are studied, and four smoothed penalty function methods are proposed. The validities and convergences of these smoothed penalty function methods are proved rigorously. Results for the challenging problems that contain both of end-point constraints and path constraints (such as constrained car braking problem and constrained brachistochrone problem) show that the proposed methods are reliable and effective.2. To reduce the computational cost and to increase the computational efficiency, a fast approximation algorithm for the CVP method is proposed. Combing with the smoothed penalty function methods, the effect of the fast approximation algorithm is tested and compared. It is surprising to find that, the computational cost for the benchmark problems are reduced below5%, which means the proposed fast approximation algorithm is promising for online optimization.3. The basic time-scaling transformation technique is improved, where the initial time can be nonzero. The improved basic time-scaling transformation techinique is appropriate for solving time-optimal control problems. Furthermore, the enhanced time-scaling transformation technique is also improved, where the initial time can be nonxzero too. The improved enhanced time-scaling transformation technique is appropriate for solving Bang-Bang control problems, especially for catching exact switching time points.4. Based on the above theoretical contribution, a practical optimal control software is developed in MATLAB enviroment, called H-Optimizer-CVP. The smoothed penalty function methods, the improved basic time-scaling transformation technique, and the improved enhanced time-scaling transformation technique are realized in this software. After inputing the mathematical model of an optimal control problem, H-Optimizer-CVP will finish the solving process automatically.5. Mathematical programming theories and algorithms are the basis of optimal control. A robust mathematical programming software called H-Optimizer is also developed in MATLAB environment. H-Optimizer can solve many kinds of mathematical programming problems stably and efficiently.