Orthogonal Collocation Based Numerical Methods Research For Optimal Control Problems

Optimal control is widely applied in various fields such as petrochemical industry,biomedicine engineering,rail transportation and aerospace engineering,it is an important tool to overcome the bottleneck in engineering fields and crucial means to improve the economic benefit,reduce the production cost and allocate the resources rationally.Optimal control has received considerable attention and research all over the world due to its great theoretical significance and high value on practical applications.Orthogonal collocation(OC)is one of the most important numerical methods for optimal control problems(OCPs),it converts original infinite-dimensional OCPs to finite-dimensional mathematical programming problems(generally nonlinear programming prob?lems)by parameterizing the state vector and the control vector.However,traditional method is hard to satisfy the requirements on efficiency and accuracy simultaneously for complex problems.If the number of collocation points is small,the solution quality may be poor;the scale of NLP,the computational time will increase and the oscillation phenomenon,the misconvergence may occur if large numbers of collocation points are applied,which is one of the frontiers and difficulties for OC to solve the OCPs.To overcome the disadvantages and the international bottleneck problem and promote the utility of OC method,this dissertation mainly focuses on the studies of time grid discretization and sensitivity computation.The main contributions of this dissertation are gives as follows:(1)An adaptive orthogonal collocation method is proposed for OCPs with complex path con?straints.Some local extreme points of the control profiles are applied in designing the number and the location of segments,the number and the distribution of collocation points are determined based on the evaluated approximation errors in the method.To overcome the weakness of tradition?al method where the path constraints are guaranteed to be satisfied only at the collocation points,a detection procedure is proposed to ensure that the constraints are satisfied all the way during the whole optimization process.Moreover,nonlinear state transformation is applied to make the variables more uniform in quantities.The numerical results of a benchmark problem demonstrate the effectiveness of proposed approach.(2)For the OCPs with discontinuous control profiles,considering the disadvantages of uni-form mesh discretization strategy,an adaptive non-uniform mesh refinement approach based on the sensitivity information is proposed in the frarmework of orthogonal collocation on finite el-ement.The proposed method starts with coarse uniform mesh and can converge to reasonable non-uniform mesh gradually.It can guarantee the accuracy of solution and improve the efficiency simultaneously.(3)Considering the OCPs with switching points in the control trajectories,sparse variable time nodes are proposed,it can be regarded as a generalization and extension of many existing optimization methods in terms of mesh discretization.Several variable time points are adopted first,then the interval between the neighbouring variable time points is further equidistantly divided into more subintervals,thus non-uniform mesh is formed in the whole time horizon.The first-order sensitivities of discretized state parameters with respect to control parameters and variable time nodes parameters are derived,respectively.As the initial state of some finite elements should be computed during the optimization process,a linear calculation formula is derived,corresponding error and convergence are analyzed.(4)On the basis of sparse variable time nodes and orthogonal collocation on finite elemen-t,the second-order sensitivities of dependent variables with respect to independent variables are derived.As it is necessary to calculate the dependent variables,first-order and second-order sen-sitivities through the discretized dynamic systems in each iteration of the optimization process,the computational amount is analyzed.The numerical benchmark test results demonstrate that pro-posed approach can capture the switching points of control profiles precisely and has high accuracy.Compared with the approach applying BFGS,proposed method has an advantage on efficiency.