Verification techniques for sensitivity analysis and design of controllers for nonlinear dynamic systems with uncertainties

Rauh, A. ; Minisini, J. ; Hofer, E. P.

International Journal of Applied Mathematics and Computer Science, Tome 19 (2009) no. 3, pp. 425-439.

Résumé

Control strategies for nonlinear dynamical systems often make use of special system properties, which are, for example, differential flatness or exact input-output as well as input-to-state linearizability. However, approaches using these properties are unavoidably limited to specific classes of mathematical models. To generalize design procedures and to account for parameter uncertainties as well as modeling errors, an interval arithmetic approach for verified simulation of continuoustime dynamical system models is extended. These extensions are the synthesis, sensitivity analysis, and optimization of open-loop and closed-loop controllers. In addition to the calculation of guaranteed enclosures of the sets of all reachable states, interval arithmetic routines have been developed which verify the controllability and observability of the states of uncertain dynamic systems. Furthermore, they assure asymptotic stability of controlled systems for all possible operating conditions. Based on these results, techniques for trajectory planning can be developed which determine reference signals for linear and nonlinear controllers. For that purpose, limitations of the control variables are taken into account as further constraints. Due to the use of interval techniques, issues of the functionality, robustness, and safety of dynamic systems can be treated in a unified design approach. The presented algorithms are demonstrated for a nonlinear uncertain model of biological wastewater treatment plants.

Détail

Keywords: interval arithmetic, reachability analysis, observability analysis, robust stability, model-based design of optimal controllers
Mots-clés : arytmetyka przedziałowa, analiza osiągalności, sterowanie odporne

Bibliographie

[1] Ackermann, J., Blue, P., Bünte, T., Güvenc, L., Kaesbauer, D., Kordt, M., Muhler, M. and Odenthal, D. (2002). Robust Control: The Parameter Space Approach, 2nd Edn., Springer-Verlag, London.

[2] Aschemann, H., Rauh, A., Kletting, M. and Hofer, E. P. (2006). Flatness-based control of a simplified wastewater treatment plant, Proceedings of the IEEE International Conference on Control Applications CCA 2006, Munich, Germany, pp. 2243-2248.

[3] Auer, E., Rauh, A., Hofer, E. P. and Luther,W. (2008). Validated modeling of mechanical systems with SMARTMOBILE: Improvement of performance by VALENCIA-IVP, Proceedings of the Dagstuhl Seminar 06021: Reliable Implementation of Real Number Algorithms. Theory and Practice, Dagstuhl, Germany, Lecture Notes in Computer Science, Vol. 5045, Springer-Verlag, Berlin/Heidelberg, pp. 1-27.

[4] Bendsten, C. and Stauning, O. (2007). FADBAD++, Version 2.1, available at: http://www.fadbad.com.

[5] Bünte, T. (2000). Mapping of Nyquist/Popov theta-stability margins into parameter space, Proceedings of the 3rd IFAC Symposium on Robust Control Design, Prague, Czech Republic.

[6] Delanoue, N. (2006). Algoritmes numériques pour l'analyse topologique-Analyse par intervalles et théorie des graphes, Ph.D. thesis, École Doctorale d'Angers, Angers, (in French).

[7] Fliess, M., Lévine, J., Martin, P. and Rouchon, P. (1995). Flatness and defect of nonlinear systems: Introductory theory and examples, International Journal of Control 61(6): 1327-1361.

[8] Hammersley, J. M. and Handscomb, D. C. (1964). Monte-Carlo Methods, John Wiley Sons, New York, NY.

[9] Henze, M., Harremoës, P., Arvin, E. and la Cour Jansen, J. (2002). Wastewater Treatment, 3rd Edn., Springer-Verlag, Berlin.

[10] Hermann, R. and Krener, A. J. (1977). Nonlinear controllability and observability, IEEE Transactions on Automatic Control 22(5): 728-740.

[11] Isidori, A. (1989). Nonlinear Control Systems, 2nd Edn., Springer-Verlag, Berlin.

[12] Keil, C. (2007). PROFIL/BIAS, Version 2.0.4, Available at: www.ti3.tu-harburg.de/keil/profil/.

[13] Khalil, H. K. (2002). Nonlinear Systems, 3rd Edn., Prentice-Hall, Upper Saddle River, NJ.

[14] Marquez, H. J. (2003). Nonlinear Control Systems, John Wiley Sons, Inc., Hoboken, NJ.

[15] Odenthal, D. and Blue, P. (2000). Mapping of frequency response magnitude specifications into parameter space, Proceedings of the 3rd IFAC Symposium on Robust Control Design, Prague, Czech Republic.

[16] Office for Official Publications of the European Communities (2003). Council Directive of 21 May 1991 Concerning Urban Waste Water Treatment (91/271/EEC), Available at: http://ec.europa.eu/environment/water/water-urbanwaste/directiv.html.

[17] Pepy, R., Kieffer, M. and Walter, E. (2008). Reliable robust path planner, Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Nice, France, pp. 1655-1660.

[18] Rauh, A., Auer, E. and Hofer, E. P. (2007a). VALENCIA-IVP:A comparison with other initial value problem solvers, Proceedings of the 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2006, Duisburg, Germany, IEEE Computer Society, Los Alamitos, CA, (on CDROM).

[19] Rauh, A., Auer, E., Minisini, J. and Hofer, E. P. (2007b). Extensions of VALENCIA-IVP for reduction of overestimation, for simulation of differential algebraic systems, and for dynamical optimization, PAMM 7(1): 1023001-1023002.

[20] Rauh, A., Kletting, M., Aschemann, H. and Hofer, E. P. (2007c). Reduction of overestimation in interval arithmetic simulation of biological wastewater treatment processes, Journal of Computational and Applied Mathematics 199(2): 207-212.

[21] Rauh, A., Minisini, J. and Hofer, E. P. (2007d). Interval techniques for design of optimal and robust control strategies, Proceedings of the 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2006, Duisburg, Germany, IEEE Computer Society, Los Alamitos, CA, (on CDROM).

[22] Rauh, A. and Hofer, E. P. (2005). Interval arithmetic optimization techniques for uncertain discrete-time systems, Proceedings of the 13th International Workshop on Dynamics and Control, Modeling and Control of Autonomous Decision Support Based Systems, Wiesensteig, Germany, Shaker Verlag, Aachen, pp. 141-148.

[23] Rauh, A. and Hofer, E. P. (2009). Interval methods for optimal control, Proceedings of the 47th Workshop on Variational Analysis and Aerospace Engineering 2007, Erice, Italy, Springer-Verlag, New York, NY, pp. 397-418.

[24] Rauh, A., Minisini, J. and Hofer, E. P. (2009). Towards the development of an interval arithmetic environment for validated computer-aided design and verification of systems in control engineering, Proceedings of the Dagstuhl Seminar 08021: Numerical Validation in Current Hardware Architectures, Dagstuhl, Germany, Lecture Notes in Computer Science, Vol. 5492, Springer-Verlag, Berlin/Heidelberg, pp. 175-188.

[25] Röbenack, K. (2002). On the efficient computation of higher order maps adkfg(x) using Taylor arithmetic and the Campbell-Baker-Hausdorff formula, in A. Zinober and D. Owens (Eds.), Nonlinear and Adaptive Control, Lecture Notes in Control and Information Science, Vol. 281, Springer, Berlin/Heidelberg, pp. 327-336.

[26] Rohn, J. (1994). Positive definiteness and stability of interval matrices, SIAM Journal on Matrix Analysis and Applications 15(1): 175-184.

[27] Rump, S. M. (2007). INTLAB, Version 5.4, available at: http://www.ti3.tu-harburg.de/~rump/intlab/.

[28] Sienel,W., Bünte, T. and Ackermann, J. (1996). PARADISE-Parametric robust analysis and design interactive software environment: A MATLAB-based robust control toolbox, Proceedings of the 1996 IEEE International Symposium on Computer-Aided Control System Design, Dearborn, MI, USA, pp. 380-385.

[29] Sontag, E. D. (1998). Mathematical Control Theory-Deterministic Finite Dimensional Systems, Springer-Verlag, New York, NY.