Efficient NMPC of unstable periodic systems using approximate infinite horizon closed loop costing

Abstract We develop a state-of-the-art nonlinear model predictive controller (NMPC) for periodic unstable systems, and apply the method to a dual line kite that shall fly loops. The kite is described by a nonlinear unstable ODE system (which we freely distribute), and the aim is to let the kite fly a periodic figure. Our NMPC approach is based on the “infinite horizon closed loop costing” scheme to ensure nominal stability. To be able to apply this scheme, we first determine a periodic LQR controller to stabilize the kite locally in the periodic orbit. Then, we formulate a two-stage NMPC optimal control problem penalizing deviations of the system state from the periodic orbit, which also contains a state constraint that avoids that the kite collides with the ground. To solve the optimal control problems reliably and in real-time, we apply the newly developed “real-time iteration scheme” for fast online optimization in NMPC. The optimization based NMPC leads to significantly improved performance compared to the LQR controller, in particular as it respects state constraints. The NMPC closed loop also performs well in the presence of large random disturbances and shows considerable robustness against changes in the wind direction.

[1]  W. C. Li,et al.  Newton-type control strategies for constrained nonlinear processes , 1989 .

[2]  Hans Bock,et al.  An Efficient Algorithm for Nonlinear Model Predictive Control of Large-Scale Systems Part I: Description of the Method (Ein effizienter Algorithmus für die nichtlineare prädiktive Regelung großer Systeme Teil I: Methodenbeschreibung) , 2002 .

[3]  Johannes P. Schlöder,et al.  An Efficient Algorithm for Nonlinear Model Predictive Control of Large-Scale Systems Part II: Experimental Evaluation for a Distillation Column Ein effizienter Algorithmus für die nichtlineare prädiktive Regelung großer Systeme Teil II: Experimentelle Erprobung an einer Destillationskolonne , 2003 .

[4]  H. Bock,et al.  A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems , 1984 .

[5]  Johannes P. Schlöder,et al.  An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part 1: theoretical aspects , 2003, Comput. Chem. Eng..

[6]  S. Joe Qin,et al.  An Overview of Nonlinear Model Predictive Control Applications , 2000 .

[7]  Baake,et al.  Fitting ordinary differential equations to chaotic data. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[8]  M. Diehl,et al.  Nominal stability of real-time iteration scheme for nonlinear model predictive control , 2005 .

[9]  David Q. Mayne,et al.  Nonlinear Model Predictive Control:Challenges and Opportunities , 2000 .

[10]  Irene Bauer.  Numerische Verfahren zur Lösung von Anfangswertaufgaben und zur Generierung von ersten und zweiten Ableitungen mit Anwendungen bei Optimierungsaufgaben in Chemie und Verfahrenstechnik , 1999 .

[11]  Riccardo Scattolini,et al.  A stabilizing model-based predictive control algorithm for nonlinear systems , 2001, Autom..

[12]  G. Nicolao,et al.  Stability and Robustness of Nonlinear Receding Horizon Control , 2000 .

[13]  Hans Bock,et al.  Least squares parameter estimation in chaotic differential equations , 1993 .

[14]  Johannes P. Schlöder,et al.  Newton-Type Methods for the Approximate Solution of Nonlinear Programming Problems in Real-Time , 2003 .

[15]  H. Michalska,et al.  Receding horizon control of nonlinear systems , 1988, Proceedings of the 28th IEEE Conference on Decision and Control,.

[16]  Lorenz T. Biegler,et al.  An extension of Newton-type algorithms for nonlinear process control , 1995, Autom..

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

[18]  H. ChenT,et al.  A Quasi-Infinite Horizon Nonlinear Model Predictive Control Scheme with Guaranteed Stability * , 1998 .

[19]  G. Nicolao,et al.  Stabilizing receding-horizon control of nonlinear time-varying systems , 1998, IEEE Trans. Autom. Control..

[20]  Benjamin Friedlander,et al.  An efficient algorithm , 1983 .

[21]  J. P. Schlöder,et al.  Ein effizienter Algorithmus für die nichtlineare prädiktive Regelung großer Systeme Teil II: Experimentelle Erprobung an einer Destillationskolonne (An Efficient Algorithm for Nonlinear Model Predictive Control of Large-Scale Systems Part II: Experimental Evaluation for a Distillation Column) , 2003 .

[22]  W. E. Stewart,et al.  Sensitivity analysis of initial value problems with mixed odes and algebraic equations , 1985 .

[23]  Johannes P. Schlöder,et al.  An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization: Part II: Software aspects and applications , 2003, Comput. Chem. Eng..

[24]  Karl Johan Åström,et al.  GLOBAL STABILIZATION OF THE INVERTED PENDULUM USING MODEL PREDICTIVE CONTROL , 2002 .

[25]  Stephen J. Wright,et al.  Nonlinear Predictive Control and Moving Horizon Estimation — An Introductory Overview , 1999 .

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