Nonlinear stabilization via Control Contraction Metrics: A pseudospectral approach for computing geodesics

Real-time nonlinear stabilization techniques are often limited by inefficient or intractable online and/or offline computations, or a lack guarantee for global stability. In this paper, we explore the use of Control Contraction Metrics (CCM) for nonlinear stabilization because it offers tractable offline computations that give formal guarantees for global stability. We provide a method to solve the associated online computation for a CCM controller - a pseudospectral method to find a geodesic. Through a case study of a stiff nonlinear system, we highlight two key benefits: (i) using CCM for nonlinear stabilization and (ii) rapid online computations amenable to real-time implementation. We compare the performance of a CCM controller with other popular feedback control techniques, namely the Linear Quadratic Regulator (LQR) and Nonlinear Model Predictive Control (NMPC). We show that a CCM controller using a pseudospectral approach for online computations is a middle ground between the simplicity of LQR and stability guarantees for NMPC.

[1]  F. G. Shinskey,et al.  2.19 Nonlinear and Adaptive Control , 2008 .

[2]  Vladimir Kolmogorov,et al.  Computing geodesics and minimal surfaces via graph cuts , 2003, Proceedings Ninth IEEE International Conference on Computer Vision.

[3]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[4]  Anil V. Rao,et al.  Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method , 2006 .

[5]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

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

[7]  Ian R. Manchester,et al.  An Amendment to "Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design" , 2017, ArXiv.

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

[9]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Nonlinear Systems Analyzing stability differentially leads to a new perspective on nonlinear dynamic systems , 1999 .

[10]  Frank Allgöwer,et al.  Nonlinear Model Predictive Control , 2007 .

[11]  Gamal N. Elnagar,et al.  Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems , 1998, Comput. Optim. Appl..

[12]  Eduardo Sontag.  A universal construction of Artstein's theorem on nonlinear stabilization , 1989 .

[13]  Qi Gong,et al.  A Chebyshev pseudospectral method for nonlinear constrained optimal control problems , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[14]  Pablo A. Parrilo,et al.  Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming , 2006, at - Automatisierungstechnik.

[15]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[16]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..

[17]  Ian R. Manchester,et al.  Control Contraction Metrics and Universal Stabilizability , 2013, ArXiv.

[18]  John M. Lee.  Riemannian Manifolds: An Introduction to Curvature , 1997 .

[19]  Lloyd N. Trefethen,et al.  Is Gauss Quadrature Better than Clenshaw-Curtis? , 2008, SIAM Rev..

[20]  Christophe Prieur,et al.  Uniting Two Control Lyapunov Functions for Affine Systems , 2008, IEEE Transactions on Automatic Control.

[21]  I. Michael Ross,et al.  Direct trajectory optimization by a Chebyshev pseudospectral method , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[22]  J A Sethian,et al.  Computing geodesic paths on manifolds. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[23]  I. Michael Ross,et al.  Pseudospectral Methods for Infinite-Horizon Nonlinear Optimal Control Problems , 2005 .

[24]  Miroslav Krstic,et al.  Nonlinear and adaptive control de-sign , 1995 .

[25]  Paul Williams,et al.  Application of Pseudospectral Methods for Receding Horizon Control , 2004 .

[26]  William W. Hager,et al.  A unified framework for the numerical solution of optimal control problems using pseudospectral methods , 2010, Autom..

[27]  Pablo A. Parrilo,et al.  Semidefinite programming relaxations for semialgebraic problems , 2003, Math. Program..

[28]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[29]  Lexing Ying,et al.  Fast geodesics computation with the phase flow method , 2006, J. Comput. Phys..

[30]  I. Michael Ross,et al.  A review of pseudospectral optimal control: From theory to flight , 2012, Annu. Rev. Control..