On the Approximation of Constrained Linear Quadratic Regulator Problems and their Application to Model Predictive Control

By parametrizing input and state trajectories with basis functions different approximations to the constrained linear quadratic regulator problem are obtained. These notes present and discuss technical results that are intended to supplement a corresponding journal article. The results can be applied in a model predictive control context.

[1]  Jay H. Lee,et al.  Model predictive control: past, present and future , 1999 .

[2]  Christopher J. BISHOPAbstra,et al.  Orthogonal Functions , 2022 .

[3]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[4]  Franco Blanchini,et al.  Suboptimal receding horizon control for continuous-time systems , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[5]  L. Young.  Lectures on the Calculus of Variations and Optimal Control Theory , 1980 .

[6]  H. Halkin.  Necessary conditions for optimal control problems with infinite horizons , 1974 .

[7]  Jürgen Pannek,et al.  Receding Horizon Control: A Suboptimality-based Approach , 2009 .

[8]  Jeffrey C. Kantor,et al.  Improving computational efficiency of model predictive control algorithm using wavelet transformation , 1995 .

[9]  Manfred Morari,et al.  A Multiresolution Approximation Method for Fast Explicit Model Predictive Control , 2011, IEEE Transactions on Automatic Control.

[10]  W. Rudin.  Principles of mathematical analysis , 1964 .

[11]  Sabine Pickenhain,et al.  Infinite Horizon Optimal Control Problems in the Light of Convex Analysis in Hilbert Spaces , 2015 .

[12]  Raffaello D'Andrea,et al.  Implementation of a parametrized infinite-horizon model predictive control scheme with stability guarantees , 2017, 2017 IEEE International Conference on Robotics and Automation (ICRA).

[13]  Daniel Kuhn,et al.  Polynomial Approximations for Continuous Linear Programs , 2012, SIAM J. Optim..

[14]  L. Grüne,et al.  Nonlinear Model Predictive Control : Theory and Algorithms. 2nd Edition , 2011 .

[15]  Bilal Khan,et al.  Alternative parameterisation within predictive control: a systematic selection , 2013, Int. J. Control.

[16]  W. Marsden.  I and J , 2012 .

[17]  Marcus Reble,et al.  Model predictive control for nonlinear continuous-time systems with and without time-delays , 2013 .

[18]  Michael Muehlebach,et al.  Parametrized infinite-horizon model predictive control for linear time-invariant systems with input and state constraints , 2016, 2016 American Control Conference (ACC).

[19]  Yurii Nesterov,et al.  Squared Functional Systems and Optimization Problems , 2000 .

[20]  J. Aubin.  Set-valued analysis , 1990 .

[21]  James B. Rawlings,et al.  Constrained linear quadratic regulation , 1998, IEEE Trans. Autom. Control..

[22]  Knut Sydsæter,et al.  Optimal control theory with economic applications , 1987 .

[23]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[24]  Martin Guay,et al.  A Real-Time Framework for Model-Predictive Control of Continuous-Time Nonlinear Systems , 2007, IEEE Transactions on Automatic Control.

[25]  Lars Grüne,et al.  On the Infinite Horizon Performance of Receding Horizon Controllers , 2008, IEEE Transactions on Automatic Control.

[26]  Michael Muehlebach,et al.  Application of an approximate model predictive control scheme on an unmanned aerial vehicle , 2016, 2016 IEEE International Conference on Robotics and Automation (ICRA).

[27]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[28]  Liuping Wang,et al.  Exploiting Laguerre functions to improve the feasibility/performance compromise in MPC , 2008, 2008 47th IEEE Conference on Decision and Control.

[29]  W. Rudin.  Real and complex analysis , 1968 .

[30]  Pierre-Brice Wieber,et al.  Fast Direct Multiple Shooting Algorithms for Optimal Robot Control , 2005 .

[31]  Karin Rothschild,et al.  A Course In Functional Analysis , 2016 .

[32]  J. Primbs,et al.  Constrained finite receding horizon linear quadratic control , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[33]  Liuping Wang,et al.  Model Predictive Control System Design and Implementation Using MATLAB , 2009 .

[34]  Okko H. Bosgra,et al.  Model predictive control with generalized input parametrization , 1999, 1999 European Control Conference (ECC).

[35]  Sergei Yakovenko,et al.  On Functions and Curves Defined by Ordinary Dierential Equations , 1999 .

[36]  A. Göpfert.  Mathematische Optimierung in allgemeinen Vektorräumen , 1973 .

[37]  Mazen Alamir,et al.  Stabilization of nonlinear systems using receding-horizon control schemes , 2013 .