Stochastic Nonlinear Model Predictive Control based on Gaussian Mixture Approximations

  • Florian Weissel
  • Marco F. Huber
  • Uwe D. Hanebeck
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 24)

Abstract

In this paper, a framework for stochastic Nonlinear Model Predictive Control (NMPC) that explicitly incorporates the noise influence on systems with continuous state spaces is introduced. By the incorporation of noise, which results from uncertainties during model identification and measurement, the quality of control can be significantly increased. Since stochastic NMPC requires the prediction of system states over a certain horizon, an efficient state prediction technique for nonlinear noise-affected systems is required. This is achieved by using transition densities approximated by axis-aligned Gaussian mixtures together with methods to reduce the computational burden. A versatile cost function representation also employing Gaussianmixtures provides an increased freedom of modeling. Combining the rediction technique with this value function representation allows closed-form calculation of the necessary optimization problems arising from stochastic NMPC. The capabilities of the framework and especially the benefits that can be gained by considering the noise in the controller are illustrated by the example of a mobile robot following a given path.

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References

  1. 1.
    Qin, S.J., Badgewell, T.A.: An Overview of Industrial Model Predictive Control Technology. Chemical Process Control 93(316) (1997) 232–256Google Scholar
  2. 2.
    Findeisen, R., Allgöwer, F.: An Introduction to Nonlinear Model Predictive Control. In: 21st Benelux Meeting on Systems and Control. (March 2002) 119–141Google Scholar
  3. 3.
    Ohtsuka, T.: A Continuation/GMRES Method for Fast Computation of Nonlinear Receding Horizon Control. Automatica 40(4) (April 2004) 563–574Google Scholar
  4. 4.
    Camacho, E.F., Bordons, C.: Model Predictive Control. 2 edn. Springer-Verlag London Ltd. (June 2004)Google Scholar
  5. 5.
    Kappen, H.J.: Path integrals and symmetry breaking for optimal control theory. Journal of Statistical Mechanics: Theory and Experiments 2005(11) (November 2005) P11011Google Scholar
  6. 6.
    Deisenroth, M.P., Weissel, F., Ohtsuka, T., Hanebeck, U.D.: Online-Computation Approach to Optimal Control of Noise-Affected Nonlinear Systems with Continuous State and Control Spaces. In: Proceedings of the European Control Conference (ECC 2007), Kos, Greece (July 2007)Google Scholar
  7. 7.
    Nikovski, D., Brand, M.: Non-Linear Stochastic Control in Continuous State Spaces by Exact Integration in Bellman’s Equations. In: Proceedings of the 2003 International Conference on Automated Planning and Scheduling. (June 2003) 91–95Google Scholar
  8. 8.
    Marecki, J., Koenig, S., Tambe, M.: A Fast Analytical Algorithm for Solving Markov Decision Processes with Real-Valued Resources. In: Proceedings of the Twentieth International Joint Conference on Artificial Intelligence (IJCAI-07). (January 2007)Google Scholar
  9. 9.
    Huber, M., Brunn, D., Hanebeck, U.D.: Closed-Form Prediction of Nonlinear Dynamic Systems by Means of Gaussian Mixture Approximation of the Transition Density. In: Proceedings of the IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2006). (September 2006) 98–103Google Scholar
  10. 10.
    Weissel, F., Huber, M.F., Hanebeck, U.D.: A Closed–Form Model Predictive Control Framework for Nonlinear Noise–Corrupted Systems. In: 4th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2007). Volume SPSMC., Angers, France (May 2007) 62–69Google Scholar
  11. 11.
    Weissel, F., Huber, M.F., Hanebeck, U.D.: Test-Environment based on a Team of Miniature Walking Robots for Evaluation of Collaborative Control Methods. In: IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2007). (November 2007)Google Scholar
  12. 12.
    Schweppe, F.C.: Uncertain Dynamic Systems. Prentice-Hall (1973)Google Scholar
  13. 13.
    Kalman, R.E.: A new Approach to Linear Filtering and Prediction Problems. Transactions of the ASME, Journal of Basic Engineering 82 (March 1960) 35–45Google Scholar
  14. 14.
    Lee, J.H., Ricker, N.L.: Extended Kalman Filter Based Nonlinear Model Predictive Control. In: Industrial & Engineering Chemistry Research. Volume 33., ACS (1994) 1530–1541Google Scholar
  15. 15.
    Maz’ya, V., Schmidt, G.: On Approximate Approximations using Gaussian Kernels. IMA Journal of Numerical Analysis 16(1) (1996) 13–29MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    de Freitas, N.: Rao-Blackwellised Particle Filtering for Fault Diagnosis. In: IEEE Aerospace Conference Proceedings. Volume 4. (2002) 1767–1772Google Scholar
  17. 17.
    Weissel, F., Huber, M.F., Hanebeck, U.D.: A Nonlinear Model Predictive Control Framework Approximating Noise Corrupted Systems with Hybrid Transition Densities. In: IEEE Conference on Decision and Control (CDC 2007), New Orleans, USA (December 2007)Google Scholar
  18. 18.
    Bertsekas, D.P.: Dynamic Programming and Optimal Control. 2nd edn. Athena Scientific, Belmont, Massachusetts, U.S.A. (2000)Google Scholar
  19. 19.
    Weissel, F., Huber, M.F., Hanebeck, U.D.: Efficient Control of Nonlinear Noise–Corrupted Systems Using a Novel Model Predictive Control Framework. In: Proceedings of the 2007 American Control Conference (ACC 2007), New York City, USA (July 2007)Google Scholar
  20. 20.
    He, Y., Chong, E.K.P.: Sensor Scheduling for Target Tracking in Sensor Networks. In: Proceedings of the 43rd IEEE Conference on Decision and Control (CDC 2004). Volume 1. (December 2004) 743–748Google Scholar
  21. 21.
    Savkin, A.V., Evans, R.J., Skafidas, E.: The Problem of Optimal Robust Sensor Scheduling. In: Proceedings of the 39th IEEE Conference on Decision and Control (CDC 2000). Volume 4. (December 2000) 3791–3796Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Florian Weissel
    • 1
  • Marco F. Huber
    • 1
  • Uwe D. Hanebeck
    • 1
  1. 1.Intelligent Sensor-Actuator-Systems Laboratory Institute of Computer Science and EngineeringUniversität Karlsruhe (TH)Germany

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