Decomposition strategies for large-scale dynamic optimization problems

Abstract

Abstract Recently, efficient strategies have been developed to solve dynamic simulation and optimization problems in a simultaneous manner. These rely on the ability to obtain an accurate algebraic discretization of the differential equations as well as the ability to solve large optimization problems in an efficient manner. These concerns have been addressed by applying orthogonal collocation on finite elements to these systems and solving the nonlinear program (NLP) with a reduced space successive quadratic programming (SQP) approach. In a recent study we discussed theoretical properties of these differential algebraic equation (DAE) systems and cautioned that application of orthogonal collocation may not yield a stable discretization nor an accurate solution to the control problem. As a result of this, preanalysis of the DAE system is required and appropriate approximation error criteria must be embedded within the nonlinear program. In this paper we tailor this approach to the accurate solution of optimal control problems. The optimal control problem has a natural partitioning of control variables and state variables for the NLP. Note here that partitioned spaces are not orthogonal. We develop a decomposition strategy to: (1) exploit the block matrix form of the discretized differential equations which results from using collocation on finite elements, and (2) allow us to perform the optimization in the control space. Here the state variables for each finite element are determined by linearized differential equations, and a coordination step is used to update the control variables and integration length. Information is passed from element to element by chainruling the state information. While the approach has much in common with earlier quasilinearization approaches, the nonlinear programming strategy has a great deal of flexibility in determining control variable discontinuities, enforcing a wide variety of state and control variable constraints and ensuring the accurate determination of both state and control variable profiles. Two classes of problems are investigated; first we consider problems where the differential equations are linear in the state variables and then we consider the general nonlinear (states and controls) problem. Example problems are illustrated for both classes of problems.