Control Variable Parameterization-based Optimal Control Problems Computational Methods Research

As the main part of modern control theory,optimal control is one of the most important ways for system efficiency improvement,economic benefits growth and energy conservation.However,solving the optimal control problem is the bottleneck of optimal control from theory to application.The research on the core problem of solving efficiency and solving accuracy has always been the frontier and difficulty of domestic and foreign research.Control variable parameterization(CVP)is one of the most popular methods for sovling optimal control problems,the main idea of which is to transform the original optimal control problem into an approximate mathematical programming problem by discreting control horizon,then choose a mathematical programming solver to get an approximate solution.The advantage of CVP method is that the implementation is simple and feasible.However,there are two disadvantages in constraints handling,equality constraints may lead to higher order DAEs,inequality constraints may cause combinatorial problem.Meanwhile,since the discretization time grid of CVP method is decided artificially and usually invariant during the optimization,there is a challenge for high-quality solution.Furthremore,as the state variables in the process model remain in the form of continuous differential equations,the dynamic system should be solved during each iteration,which has significant effect on the efficiency of CVP method.On this basis,this dissertation focuses on the constraints handling,computation efficiency improvement and high-quality solution of CVP method.The main work and contributions of this dissertation include:(1)Considering the nonlinear programming(NLP)problem obtained by CVP method has important influence on the solving of optimal control problem,a novel Karush-Kuhn-Tucker(KKT)correction-based SQP method is proposed to solve these NLP problems.A standard NLP testing set named CathOPT is established.Test results on this set show that,by using the KKT correction,more than 5%solving rate is improved and the performance is extremly improved on iterations and computation efficiency when compared with the FMINCON solver in MATLAB and the original SQP method.Thus,the proposed SQP method will be one of the basic supporting techniques for CVP.(2)To conquer the difficulties that arise from multiple constraints,two penalty methods are presented for handling equality and inequality constraints.The error anslysis and convergence of the proposed methods are proved strictly.By using the proposed methods,the constraints of the optimal control problem are transformed into the objective function,thus the difficulty of solving the problem is reduced.Three optimal control problems with multiple constraints in industrial processes are tested and results show that no violation occurs during the optimization,which is more effective than the well-known DOTcvp software.(3)To obtain high-quality solutions for CVP method and balance the computation cost with precision,two time grid refinement CVP methods are proposed,where the important time grids are subdivided and the unnecessary time grids are eliminated.By using the proposed time grid refiment strategies,the presented CVP method is able to achieve better performance indexes with small number of parameters when compared with traditional CVP method,whereas the computational costs are lower.The above performance is verified in the testing of several calassical optimal control problems.(4)A fast CVP method based on variable time nodes is proposed for solving multivariable optimal control problems.By using the variable time nodes method,each control variable can be discretized by independent time grid.Meanwhile,the fast solution method can effectively reduce the solving time of dynamic system.The instance tests are carried out on continuous stirred tank reactor optimization and container crane optimal control problems.Test results show that,by using the fast calculation method for dynamic system,each control variable can obtain its high-quality individual time grid so as to achieve optimal switching structure.Finally,the computation time of the proposed method can be effectively reduced when compared with traditional CVP method.