Solving higher index DAE optimal control problems

A number of methods have been proposed for solving optimal control problems where the process being optimized is described by a differential algebraic equation (DAE). However, many of these methods require special circumstances to hold or the user to have special software. In this paper we go over many of these options and discuss what is usually necessary for them to be successful. We use a nonlinear index three control problem to illustrate many of our observations..

[1]  Volker Mehrmann,et al.  Differential-Algebraic Equations: Analysis and Numerical Solution , 2006 .

[2]  Li Tiancheng,et al.  アルゴリズム906: elrint3d―組み込み格子ルールのシーケンスを用いる三次元非適応自動立体求積法ルーチン , 2011 .

[3]  A. Engelsone,et al.  Order of Convergence in the Direct Transcription Solution of Optimal Control Problems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[4]  W. Hager,et al.  An hp‐adaptive pseudospectral method for solving optimal control problems , 2011 .

[5]  W. Hager,et al.  A new approach to Lipschitz continuity in state constrained optimal control 1 1 This research was su , 1998 .

[6]  W. Hager,et al.  Lipschitzian Stability for State Constrained Nonlinear Optimal Control , 1998 .

[7]  Volker Mehrmann,et al.  Symmetric collocation for unstructured nonlinear differential-algebraic equations of arbitrary index , 2004, Numerische Mathematik.

[8]  Stephen L. Campbell,et al.  Direct transcription solution of high index optimal control problems and regular Euler-Lagrange equations , 2007 .

[9]  F. Ghanbari,et al.  CONVERGENCE ANALYSIS OF THE PSEUDOSPECTRAL METHOD FOR LINEAR DAEs OF INDEX-2 , 2013 .

[10]  Volker Mehrmann,et al.  Optimal control for unstructured nonlinear differential-algebraic equations of arbitrary index , 2008, Math. Control. Signals Syst..

[11]  William W. Hager,et al.  Runge-Kutta methods in optimal control and the transformed adjoint system , 2000, Numerische Mathematik.

[12]  Stephen L. Campbell,et al.  Comments on direct transcription solution of DAE constrained optimal control problems with two discretization approaches , 2016, Numerical Algorithms.

[13]  R. Pytlak,et al.  Runge–Kutta Based Procedure for the Optimal Control of Differential-Algebraic Equations , 1998 .

[14]  D. Jacobson,et al.  New necessary conditions of optimality for control problems with state-variable inequality constraints , 1971 .

[15]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[16]  Volker Mehrmann,et al.  Formal adjoints of linear DAE operators and their role in optimal control , 2011 .

[17]  Peter Kunkel,et al.  Symmetric collocation methods for linear differential-algebraic boundary value problems , 2002, Numerische Mathematik.

[18]  John T. Betts,et al.  Practical Methods for Optimal Control and Estimation Using Nonlinear Programming , 2009 .

[19]  Anil V. Rao,et al.  Algorithm 902: GPOPS, A MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method , 2010, TOMS.

[20]  John T. Betts,et al.  Optimal Low Thrust Trajectories to the Moon , 2003, SIAM J. Appl. Dyn. Syst..

[21]  Paul I. Barton,et al.  Dynamic simulation and optimization with inequality path constraints , 1996 .

[22]  Hans Bock,et al.  A Direct Multiple Shooting Method for Real-Time Optimization of Nonlinear DAE Processes , 2000 .

[23]  J. Betts,et al.  Direct transcription solution of optimal control problems with higher order state constraints: theory vs practice , 2007 .

[24]  J. Baumgarte Stabilization of constraints and integrals of motion in dynamical systems , 1972 .

[25]  Volker Mehrmann,et al.  Chapter 2: Regularization of Linear and Nonlinear Descriptor Systems , 2012 .

[26]  Robert Altmann,et al.  Simulation of multibody systems with servo constraints through optimal control , 2016, Multibody System Dynamics.

[27]  Lorenz T. Biegler,et al.  On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , 2006, Math. Program..

[28]  Stephen L. Campbell,et al.  Direct transcription solution of higher-index optimal control problems and the virtual index , 2007 .

[29]  H. Maurer,et al.  SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control , 2000 .

[30]  E. Weinmüller,et al.  Boundary-Value Problems for Differential-Algebraic Equations: A Survey , 2015 .