An Accelerated Dual Gradient-Projection Algorithm for Embedded Linear Model Predictive Control

This paper proposes a dual fast gradient-projection method for solving quadratic programming problems that arise in model predictive control of linear systems subject to general polyhedral constraints on inputs and states. The proposed algorithm is well suited for embedded control applications in that: 1) it is extremely simple and easy to code; 2) the number of iterations to reach a given accuracy in terms of optimality and feasibility of the primal solution can be tightly estimated; and 3) the computational cost per iteration increases only linearly with the prediction horizon.

[1]  Alberto Bemporad,et al.  Model Predictive Control Toolbox™ User’s Guide , 2004 .

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

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

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

[5]  Alberto Bemporad,et al.  Stabilizing embedded MPC with computational complexity guarantees , 2013, 2013 European Control Conference (ECC).

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

[7]  Jan M. Maciejowski,et al.  Predictive control : with constraints , 2002 .

[8]  Alberto Bemporad,et al.  Simple and Certifiable Quadratic Programming Algorithms for Embedded Linear Model Predictive Control , 2012 .

[9]  Bart De Schutter,et al.  Equivalence of hybrid dynamical models , 2001, Autom..

[10]  Asuman E. Ozdaglar,et al.  Approximate Primal Solutions and Rate Analysis for Dual Subgradient Methods , 2008, SIAM J. Optim..

[11]  N. L. Ricker.  Use of quadratic programming for constrained internal model control , 1985 .

[12]  Panagiotis Patrinos,et al.  A new algorithm for solving convex parametric quadratic programs based on graphical derivatives of solution mappings , 2010, Autom..

[13]  Shunsuke Hayashi,et al.  Celis-dennis-tapia based approach to quadratic fractional programming problems with two quadratic constraints , 2011 .

[14]  Manfred Morari,et al.  Towards computational complexity certification for constrained MPC based on Lagrange Relaxation and the fast gradient method , 2011, IEEE Conference on Decision and Control and European Control Conference.

[15]  Dimitri P. Bertsekas,et al.  Convex Analysis and Optimization , 2003 .

[16]  Alberto Bemporad,et al.  An accelerated dual gradient-projection algorithm for linear model predictive control , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

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

[18]  A. Bemporad,et al.  Model Predictive Control Design: New Trends and Tools , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[19]  Pontus Giselsson,et al.  Optimal preconditioning and iteration complexity bounds for gradient-based optimization in model predictive control , 2013, 2013 American Control Conference.

[20]  Yurii Nesterov,et al.  Double Smoothing Technique for Large-Scale Linearly Constrained Convex Optimization , 2012, SIAM J. Optim..

[21]  Alberto Bemporad,et al.  Fixed-point dual gradient projection for embedded model predictive control , 2013, 2013 European Control Conference (ECC).

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

[23]  Panos M. Pardalos,et al.  GLOBAL OPTIMIZATION ALGORITHMS FOR LINEARLY CONSTRAINED INDEFINITE QUADRATIC PROBLEMS , 1991 .

[24]  Yurii Nesterov,et al.  Smooth minimization of non-smooth functions , 2005, Math. Program..

[25]  Alberto Bemporad,et al.  Control of systems integrating logic, dynamics, and constraints , 1999, Autom..

[26]  Samuel Burer,et al.  A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations , 2008, Math. Program..

[27]  Hanif D. Sherali,et al.  A reformulation-convexification approach for solving nonconvex quadratic programming problems , 1995, J. Glob. Optim..

[28]  Pantelis Sopasakis,et al.  A global piecewise smooth Newton method for fast large-scale model predictive control , 2011, Autom..

[29]  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).

[30]  Christodoulos A. Floudas,et al.  αBB: A global optimization method for general constrained nonconvex problems , 1995, J. Glob. Optim..

[31]  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.

[32]  Panos M. Pardalos,et al.  Quadratic programming with one negative eigenvalue is NP-hard , 1991, J. Glob. Optim..

[33]  Yurii Nesterov,et al.  Introductory Lectures on Convex Optimization - A Basic Course , 2014, Applied Optimization.

[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]  Li-Li Zhang,et al.  Two-step modulus-based matrix splitting iteration method for linear complementarity problems , 2011, Numerical Algorithms.

[36]  Johan A. K. Suykens,et al.  Application of a Smoothing Technique to Decomposition in Convex Optimization , 2008, IEEE Transactions on Automatic Control.

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

[38]  Dimitri P. Bertsekas,et al.  Convex Optimization Theory , 2009 .

[39]  Joaquim J. Júdice,et al.  Algorithms for linear programming with linear complementarity constraints , 2011, TOP.

[40]  H. Tuy.  Convex analysis and global optimization , 1998 .

[41]  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.

[42]  Pontus Giselsson,et al.  Execution time certification for gradient-based optimization in model predictive control , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[43]  Jing Hu,et al.  On the Global Solution of Linear Programs with Linear Complementarity Constraints , 2008, SIAM J. Optim..

[44]  L. Biegler,et al.  Quadratic programming methods for reduced Hessian SQP , 1994 .

[45]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..