A structure exploiting algorithm for approximate robust optimal control with application to power generating kites

In the present paper we discuss numerical solution strategies for robust optimal control problems. Here, our motivation is to optimize the dynamic open-loop behaviour of a kite, which is affected by unknown wind turbulence. After reviewing some existing strategies for robust optimal control, we specialize on a particular approach which uses a Lyapunov differential equation in order to approximate the influence of the uncertainties on the state of the dynamic system and to guarantee stability of periodic systems. We propose strategies to exploit the structure of the corresponding formulation and provide an open-source implementation of the algorithm. The practical advantage of this implementation is illustrated by applying it to a robustness and stability optimization problem for a power generating kite system. Here, the model equations and system parameters are inspired from a real-world application.

[1]  Moritz Diehl,et al.  Real-Time Optimization for Large Scale Nonlinear Processes , 2001 .

[2]  A. Laub,et al.  Generalized eigenproblem algorithms and software for algebraic Riccati equations , 1984, Proceedings of the IEEE.

[3]  James Demmel,et al.  LAPACK Users' Guide, Third Edition , 1999, Software, Environments and Tools.

[4]  A. M. Li︠a︡punov Problème général de la stabilité du mouvement , 1949 .

[5]  Peter Benner,et al.  Accelerating Newton's Method for Discrete-Time Algebraic Riccati Equations , 1998 .

[6]  Johannes P. Schlöder,et al.  Combined NMPC and MHE for a copolymerization process , 2006 .

[7]  Z. Nagy,et al.  Distributional uncertainty analysis using power series and polynomial chaos expansions , 2007 .

[8]  M. Canale,et al.  Control of tethered airfoils for a new class of wind energy generator , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[9]  Moritz Diehl,et al.  Efficient NMPC of unstable periodic systems using approximate infinite horizon closed loop costing , 2004, Annu. Rev. Control..

[10]  R. Kálmán LYAPUNOV FUNCTIONS FOR THE PROBLEM OF LUR'E IN AUTOMATIC CONTROL. , 1963, Proceedings of the National Academy of Sciences of the United States of America.

[11]  Moritz Diehl,et al.  An approximation technique for robust nonlinear optimization , 2006, Math. Program..

[12]  P. Parrilo Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[13]  Stephen P. Boyd,et al.  Extending Scope of Robust Optimization: Comprehensive Robust Counterparts of Uncertain Problems , 2006, Math. Program..

[14]  B. Lansdorp,et al.  Comparison of concepts for high-altitude wind energy generation with ground based generator , 2005 .

[15]  Johannes P. Schlöder,et al.  Combined nonlinear model predictive control and moving horizon estimation for a copolymerization process , 2006 .

[16]  Sanket Sanjay Diwale Optimal control for power generating kites , 2014 .

[17]  L. Vandenberghe,et al.  Algorithms and software for LMI problems in control , 1997 .

[18]  Arkadi Nemirovski,et al.  Robust Convex Optimization , 1998, Math. Oper. Res..

[19]  Lorenzo Fagiano,et al.  High Altitude Wind Energy Generation Using Controlled Power Kites , 2010, IEEE Transactions on Control Systems Technology.

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

[21]  Lorenzo Fagiano,et al.  KiteGen : A revolution in wind energy generation , 2009 .

[22]  P. Williams,et al.  Optimal Cross-Wind Towing and Power Generation with Tethered Kites , 2007 .

[23]  Moritz Diehl,et al.  Optimal control for power generating kites , 2007, 2007 European Control Conference (ECC).

[24]  Wubbo Ockels,et al.  Ship propulsion by Kites combining energy production by Laddermill principle and direct kite propulsion , 2006 .

[25]  M. L. Loyd,et al.  Crosswind kite power (for large-scale wind power production) , 1980 .

[26]  Lorenzo Fagiano,et al.  Power kites for wind energy generation , 2007 .

[27]  S. Bittanti,et al.  The Periodic Riccati Equation , 1991 .

[28]  Lorenzo Fagiano,et al.  Power Kites for Wind Energy Generation [Applications of Control] , 2007, IEEE Control Systems.

[29]  Moritz Diehl,et al.  Robustness and stability optimization of power generating kite systems in a periodic pumping mode , 2010, 2010 IEEE International Conference on Control Applications.

[30]  Moritz Diehl,et al.  Robust nonlinear optimal control of dynamic systems with affine uncertainties , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[31]  A. Liapounoff,et al.  Problème général de la stabilité du mouvement , 1907 .

[32]  Richard D. Braatz,et al.  Open-loop and closed-loop robust optimal control of batch processes using distributional and worst-case analysis , 2004 .

[33]  Paolo Bolzern,et al.  The periodic Lyapunov equation , 1988 .

[34]  Marco A. López,et al.  Semi-infinite programming , 2007, Eur. J. Oper. Res..

[35]  Lorenz T. Biegler,et al.  Optimization approaches to nonlinear model predictive control , 1991 .

[36]  Kenneth O. Kortanek,et al.  Semi-Infinite Programming: Theory, Methods, and Applications , 1993, SIAM Rev..

[37]  H. Bock,et al.  Recent Advances in Parameteridentification Techniques for O.D.E. , 1983 .