A Splitting Method for Optimal Control

We apply an operator splitting technique to a generic linear-convex optimal control problem, which results in an algorithm that alternates between solving a quadratic control problem, for which there are efficient methods, and solving a set of single-period optimization problems, which can be done in parallel, and often have analytical solutions. In many cases, the resulting algorithm is division-free (after some off-line pre-computations) and can be implemented in fixed-point arithmetic, for example on a field-programmable gate array (FPGA). We demonstrate the method on several examples from different application areas.

[1]  Alberto Bemporad,et al.  The explicit linear quadratic regulator for constrained systems , 2003, Autom..

[2]  Richard G. Baraniuk,et al.  Fast Alternating Direction Optimization Methods , 2014, SIAM J. Imaging Sci..

[3]  Stephen P. Boyd,et al.  AUTOMATIC GENERATION OF HIGH-SPEED SOLVERS , 2011 .

[4]  Hans Joachim Ferreau,et al.  An online active set strategy to overcome the limitations of explicit MPC , 2008 .

[5]  Patrick L. Combettes,et al.  Proximal Splitting Methods in Signal Processing , 2009, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[6]  I. Duff,et al.  Direct Methods for Sparse Matrices , 1987 .

[7]  Stephen J. Wright.  Interior point methods for optimal control of discrete time systems , 1993 .

[8]  Shiqian Ma,et al.  Fast alternating linearization methods for minimizing the sum of two convex functions , 2009, Math. Program..

[9]  Brett Ninness,et al.  Fast Linear Model Predictive Control Via Custom Integrated Circuit Architecture , 2012, IEEE Transactions on Control Systems Technology.

[10]  Eric C. Kerrigan,et al.  Predictive Control of a Boeing 747 Aircraft using an FPGA , 2012 .

[11]  Nicholas I. M. Gould,et al.  A weighted gram-schmidt method for convex quadratic programming , 1984, Math. Program..

[12]  Stephen J. Wright,et al.  Fast, large-scale model predictive control by partial enumeration , 2007, Autom..

[13]  Anders Hansson,et al.  Efficient solution of second order cone program for model predictive control , 2004 .

[14]  Anders Hansson,et al.  A primal-dual interior-point method for robust optimal control of linear discrete-time systems , 2000, IEEE Trans. Autom. Control..

[15]  A. Hansson,et al.  Robust optimal control of linear discrete-time systems using primal-dual interior-point methods , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).

[16]  Bingsheng He,et al.  On non-ergodic convergence rate of Douglas–Rachford alternating direction method of multipliers , 2014, Numerische Mathematik.

[17]  Bingsheng He,et al.  On the O(1/n) Convergence Rate of the Douglas-Rachford Alternating Direction Method , 2012, SIAM J. Numer. Anal..

[18]  Manfred Morari,et al.  Efficient interior point methods for multistage problems arising in receding horizon control , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[19]  Rolf Findeisen,et al.  Fast predictive control of linear systems combining Nesterov's gradient method and the method of multipliers , 2011, IEEE Conference on Decision and Control and European Control Conference.

[20]  Xiaoming Yuan,et al.  Convergence rate and iteration complexity on the alternating direction method of multipliers with a substitution procedure for separable convex programming , 2012 .

[21]  Patrick R. Amestoy,et al.  An Approximate Minimum Degree Ordering Algorithm , 1996, SIAM J. Matrix Anal. Appl..

[22]  P. L. Combettes,et al.  The Convex Feasibility Problem in Image Recovery , 1996 .

[23]  B. Mercier,et al.  A dual algorithm for the solution of nonlinear variational problems via finite element approximation , 1976 .

[24]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[25]  Y. Nesterov.  A method for solving the convex programming problem with convergence rate O(1/k^2) , 1983 .

[26]  Toshiyuki Ohtsuka,et al.  A continuation/GMRES method for fast computation of nonlinear receding horizon control , 2004, Autom..

[27]  L. Biegler,et al.  QPSchur: A dual, active-set, Schur-complement method for large-scale and structured convex quadratic programming , 2006 .

[28]  Stephen P. Boyd,et al.  Receding Horizon Control , 2011, IEEE Control Systems.

[29]  Akira Kodama,et al.  Automatic Code Generation System for Nonlinear Receding Horizon Control , 2002 .

[30]  Eric C. Kerrigan,et al.  Towards a fixed point QP solver for predictive control , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[31]  Alberto Bemporad,et al.  An algorithm for multi-parametric quadratic programming and explicit MPC solutions , 2003, Autom..

[32]  R. Glowinski,et al.  Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires , 1975 .

[33]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[34]  Manfred Morari,et al.  Real-time input-constrained MPC using fast gradient methods , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[35]  J. L. Nazareth,et al.  Cholesky-based Methods for Sparse Least Squares : The Benefits of Regularization ∗ , 1996 .

[36]  Michael A. Saunders,et al.  Procedures for optimization problems with a mixture of bounds and general linear constraints , 1984, ACM Trans. Math. Softw..

[37]  Robert J. Vanderbei,et al.  Symmetric Quasidefinite Matrices , 1995, SIAM J. Optim..

[38]  Rolf Findeisen,et al.  A fast gradient method for embedded linear predictive control , 2011 .

[39]  R. Vanderbei.  Symmetric Quasi-Definite Matrices , 2006 .

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

[41]  Michael C. Ferris,et al.  Operator-Splitting Methods for Monotone Affine Variational Inequalities, with a Parallel Application to Optimal Control , 1998, INFORMS J. Comput..

[42]  Bo Wahlberg,et al.  An ADMM algorithm for solving ℓ1 regularized MPC , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[43]  Timothy A. Davis,et al.  Direct methods for sparse linear systems , 2006, Fundamentals of algorithms.

[44]  Dimitri P. Bertsekas,et al.  On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators , 1992, Math. Program..

[45]  Shiqian Ma,et al.  Fast Multiple-Splitting Algorithms for Convex Optimization , 2009, SIAM J. Optim..

[46]  Donald Goldfarb,et al.  A numerically stable dual method for solving strictly convex quadratic programs , 1983, Math. Program..

[47]  Manfred Morari,et al.  High-speed online MPC based on a fast gradient method applied to power converter control , 2010, Proceedings of the 2010 American Control Conference.

[48]  Jan M. Maciejowski,et al.  Embedded Model Predictive Control (MPC) using a FPGA , 2008 .

[49]  T. Johansen,et al.  An algorithm for multi-parametric quadratic programming and explicit MPC solutions , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[50]  Manfred Morari,et al.  Computational Complexity Certification for Real-Time MPC With Input Constraints Based on the Fast Gradient Method , 2012, IEEE Transactions on Automatic Control.

[51]  Eric C. Kerrigan,et al.  An FPGA implementation of a sparse quadratic programming solver for constrained predictive control , 2011, FPGA '11.

[52]  Stephen P. Boyd,et al.  Graph Implementations for Nonsmooth Convex Programs , 2008, Recent Advances in Learning and Control.

[53]  Stephen J. Wright,et al.  Application of Interior-Point Methods to Model Predictive Control , 1998 .

[54]  Moritz Diehl,et al.  ACADO toolkit—An open‐source framework for automatic control and dynamic optimization , 2011 .

[55]  Eric C. Kerrigan,et al.  Parallel MPC for Real-Time FPGA-based Implementation , 2011 .

[56]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .

[57]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[58]  Fu Lin,et al.  Design of Optimal Sparse Feedback Gains via the Alternating Direction Method of Multipliers , 2011, IEEE Transactions on Automatic Control.

[59]  Timothy A. Davis,et al.  Algorithm 837: AMD, an approximate minimum degree ordering algorithm , 2004, TOMS.

[60]  D. Gabay.  Applications of the method of multipliers to variational inequalities , 1983 .

[61]  Timothy A. Davis,et al.  Algorithm 8 xx : a concise sparse Cholesky factorization package , 2004 .

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

[63]  Eric C. Kerrigan,et al.  Parallel move blocking Model Predictive Control , 2011, IEEE Conference on Decision and Control and European Control Conference.

[64]  Jan M. Maciejowski,et al.  A comparison of interior point and active set methods for FPGA implementation of model predictive control , 2009, 2009 European Control Conference (ECC).

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

[66]  Y. Nesterov.  Gradient methods for minimizing composite objective function , 2007 .

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

[68]  J. Bunch,et al.  Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations , 1971 .

[69]  Leonidas G. Bleris,et al.  A System-on-a-Chip Implementation for Embedded Real-Time Model Predictive Control , 2009, IEEE Transactions on Control Systems Technology.

[70]  M. Best.  An Algorithm for the Solution of the Parametric Quadratic Programming Problem , 1996 .

[71]  Stephen P. Boyd,et al.  An ADMM Algorithm for a Class of Total Variation Regularized Estimation Problems , 2012, 1203.1828.

[72]  Stephen P. Boyd,et al.  Fast Model Predictive Control Using Online Optimization , 2010, IEEE Transactions on Control Systems Technology.

[73]  Jonathan Eckstein.  Parallel alternating direction multiplier decomposition of convex programs , 1994 .

[74]  Rolf Findeisen,et al.  Fast predictive control of linear, time-invariant systems using an algorithm based on the fast gradient method and augmented Lagrange multipliers , 2011, 2011 IEEE International Conference on Control Applications (CCA).

[75]  Philipp Birken,et al.  Numerical Linear Algebra , 2011, Encyclopedia of Parallel Computing.

[76]  Stephen P. Boyd,et al.  Plenary talk: Performance bounds and suboptimal policies for multi-period investment , 2013, 22nd Mediterranean Conference on Control and Automation.