Lossless convexification of non-convex optimal control problems for state constrained linear systems

This paper analyzes a class of finite horizon optimal control problems with mixed non-convex and convex control constraints and linear state constraints. A convex relaxation of the problem is proposed, and it is proved that a solution of the relaxed problem is also a solution of the original problem. This process is called lossless convexification, and its generalization for problems with state constraints is the primary contribution of the paper. Doing so enables the use of interior point methods of convex optimization to obtain global optimal solutions of the original non-convex problem. The approach is also demonstrated on example problems.

[1]  D. Hull.  Conversion of optimal control problems into parameter optimization problems , 1996 .

[2]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[3]  Kok Lay Teo,et al.  Optimal control problems with a continuous inequality constraint on the state and the control , 2009, Autom..

[4]  Kim-Chuan Toh,et al.  SDPT3 -- A Matlab Software Package for Semidefinite Programming , 1996 .

[5]  Harry L. Trentelman,et al.  Control theory for linear systems , 2002 .

[6]  Andrew E. Johnson,et al.  Flight testing of trajectories computed by G-FOLD: Fuel optimal large divert guidance algorithm for planetary landing , 2013 .

[7]  Suresh P. Sethi,et al.  A Survey of the Maximum Principles for Optimal Control Problems with State Constraints , 1995, SIAM Rev..

[8]  Stephen P. Boyd,et al.  CVXGEN: a code generator for embedded convex optimization , 2011, Optimization and Engineering.

[9]  Matthias Gerdts,et al.  Global Convergence of a Nonsmooth Newton Method for Control-State Constrained Optimal Control Problems , 2008, SIAM J. Optim..

[10]  Jiming Peng,et al.  Self-regularity - a new paradigm for primal-dual interior-point algorithms , 2002, Princeton series in applied mathematics.

[11]  Guang-Ren Duan,et al.  An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem , 2011, J. Optim. Theory Appl..

[12]  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 .

[13]  Behcet Acikmese,et al.  Convex programming approach to powered descent guidance for mars landing , 2007 .

[14]  Max Donath,et al.  American Control Conference , 1993 .

[15]  Behçet Açikmese,et al.  Lossless convexification of a class of optimal control problems with non-convex control constraints , 2011, Autom..

[16]  Behcet Acikmese,et al.  Minimum-Landing-Error Powered-Descent Guidance for Mars Landing Using Convex Optimization , 2010 .

[17]  John M. Carson,et al.  Lossless convexification of control constraints for a class of nonlinear optimal control problems , 2012, 2012 American Control Conference (ACC).

[18]  J. Betts.  Survey of Numerical Methods for Trajectory Optimization , 1998 .

[19]  David Q. Mayne,et al.  Constrained model predictive control: Stability and optimality , 2000, Autom..

[20]  D. Jacobson,et al.  A transformation technique for optimal control problems with a state variable inequality constraint , 1969 .

[21]  W. Wonham.  Linear Multivariable Control: A Geometric Approach , 1974 .

[22]  W. H. Clohessy,et al.  Terminal Guidance System for Satellite Rendezvous , 2012 .

[23]  Behcet Acikmese,et al.  Minimum Time Rendezvous of Multiple Spacecraft Using Differential Drag , 2014 .

[24]  R. V. Dooren,et al.  A Chebyshev technique for solving nonlinear optimal control problems , 1988 .

[25]  M. Mesbahi,et al.  Spacecraft reorientation in presence of attitude constraints via logarithmic barrier potentials , 2011, Proceedings of the 2011 American Control Conference.

[26]  H. Robbins,et al.  Junction phenomena for optimal control with state-variable inequality constraints of third order , 1980 .

[27]  David Bell.  A transformation of partially singular trajectories to state constrained arcs , 1978 .

[28]  A. A. Mili︠u︡tin,et al.  Calculus of variations and optimal control , 1998 .

[29]  Manfred Morari,et al.  Model predictive control: Theory and practice - A survey , 1989, Autom..

[30]  Matthias Gerdts.  A nonsmooth Newton's method for control-state constrained optimal control problems , 2008, Math. Comput. Simul..

[31]  S. Ploen,et al.  A Powered Descent Guidance Algorithm for Mars Pinpoint Landing , 2005 .

[32]  鍋谷 清治,et al.  The Mathematical Theory of Optimal Processes by L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko , 1964 .

[33]  Ping Lu,et al.  Autonomous Trajectory Planning for Rendezvous and Proximity Operations by Conic Optimization , 2012 .

[34]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[35]  K. Teo,et al.  Global impulsive optimal control computation , 2006 .