Inexact Newton based Lifted Implicit Integrators for fast Nonlinear MPC

Nonlinear Model Predictive Control (NMPC) requires the online solution of an Optimal Control Problem (OCP) at every sampling instant. In the context of multiple shooting, a numerical integration is needed to discretize the continuous time dynamics. For stiff, implicitly defined or differential-algebraic systems, implicit schemes are preferred to carry out the integration. The Newton-type optimization method and the implicit integrator then form a nested Newton scheme, solving the optimization and integration problem on two different levels. In recent research, an exact lifting technique was proposed to improve the computational efficiency of the latter framework. Inspired by that work, this paper presents a novel class of lifted implicit integrators, using an inexact Newton method. An additional iterative scheme for computing the sensitivities is proposed, which provides similar properties as the exact lifted integrator at considerably reduced computational costs. Using the example of an industrial robot, computational speedups of up to factor 8 are reported. The proposed methods have been implemented in the open-source ACADO code generation software.

[1]  Hans Joachim Ferreau,et al.  Efficient Numerical Methods for Nonlinear MPC and Moving Horizon Estimation , 2009 .

[2]  Moritz Diehl,et al.  Autogenerating microsecond solvers for nonlinear MPC: A tutorial using ACADO integrators , 2015 .

[3]  G. J. Cooper,et al.  Some schemes for the implementation of implicit Runge-Kutta methods , 1993 .

[4]  Moritz Diehl,et al.  An auto-generated real-time iteration algorithm for nonlinear MPC in the microsecond range , 2011, Autom..

[5]  M. Diehl,et al.  Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations , 2000 .

[6]  John C. Butcher,et al.  On the implementation of implicit Runge-Kutta methods , 1976 .

[7]  Hans Bock,et al.  Efficient Numerics for Nonlinear Model Predictive Control , 2010 .

[8]  Juan I. Montijano,et al.  Implementation of high-order implicit runge-kutta methods , 2001 .

[9]  Juan I. Montijano,et al.  Iterative schemes for three-stage implicit Runge-Kutta methods , 1995 .

[10]  Alain Codourey,et al.  Dynamic Modeling of Parallel Robots for Computed-Torque Control Implementation , 1998, Int. J. Robotics Res..

[11]  Moritz Diehl,et al.  The Lifted Newton Method and Its Application in Optimization , 2009, SIAM J. Optim..

[12]  J. E. Dennis,et al.  On Newton-like methods , 1968 .

[13]  Theodore A. Bickart,et al.  An Efficient Solution Process for Implicit Runge–Kutta Methods , 1977 .

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