Tuning PI controller in non-linear uncertain closed-loop systems with interval analysis
The tuning of a PI controller is usually done through simulation, except for few classes of problems, e.g., linear systems. With a new approach for validated integration allowing us to simulate
dynamical systems with uncertain parameters, we are able to design guaranteed PI controllers. In practical, we propose a new method to identify the parameters of a PI controller for non-linear plants with bounded uncertain parameters using tools from interval analysis and validated simulation. This work relies on interval computation and guaranteed numerical integration of ordinary differential equations based on Runge-Kutta methods. Our method is applied to the
well-known cruise-control problem, under a simplified linear version and with the aerodynamic force taken into account leading to a non-linear formulation.
PID Tuning
Guaranteed numerical integration
non-linear ordinary differential equations
91-102
Regular Paper
Julien
Alexandre dit Sandretto
Julien Alexandre dit Sandretto
Alexandre
Chapoutot
Alexandre Chapoutot
Olivier
Mullier
Olivier Mullier
10.4230/OASIcs.SynCoP.2015.91
K. J. Ãström and T. Hägglund. PID controllers: theory, design, and tuning. Instrument Society of America, Research Triangle Park, NC, 1995.
J. Alexandre dit Sandretto and A. Chapoutot. Validated Solution of Initial Value Problem for Ordinary Differential Equations based on Explicit and Implicit Runge-Kutta Schemes. Research report, ENSTA ParisTech, January 2015.
E. Auer and A. Rauh. Vericomp: a system to compare and assess verified IVP solvers. Computing, 94(2-4):163-172, 2012.
J. Bondia, M. Kieffer, E. Walter, J. Monreal, and J. Picó. Guaranteed tuning of PID controllers for parametric uncertain systems. In Decision and Control, page 2948endash2953. IEEE, 2004.
O. Bouissou, A. Chapoutot, and A. Djoudi. Enclosing temporal evolution of dynamical systems using numerical methods. In NASA Formal Methods, number 7871 in LNCS, pages 108-123. Springer, 2013.
O. Bouissou and M. Martel. GRKLib: a Guaranteed Runge Kutta Library. In Scientific Computing, Computer Arithmetic and Validated Numerics, 2006.
V. Broida. Extrapolation des résponses indicielles apériodiques. Automatisme, XVI, 1969.
J. C. Butcher. Coefficients for the study of Runge-Kutta integration processes. Journal of the Australian Mathematical Society, 3:185-201, 5 1963.
L. H. de Figueiredo and J. Stolfi. Self-Validated Numerical Methods and Applications. Brazilian Mathematics Colloquium monographs. IMPA/CNPq, 1997.
L. Desborough and R. Miller. Increasing customer value of industrial control performance monitoring - Honeywell’s experience. In AIChE Symposium Series, pages 169-189, 2002.
A. Dowling. Modeling and PID controller example - cruise control for an electric vehicle.
E. Hairer, Syvert P. Norsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer-Verlag, 2nd edition, 2009.
L. Jaulin, M. Kieffer, O. Didrit, and E. Walter. Applied Interval Analysis. Springer, 2001.
Y. Lin and M. A. Stadtherr. Validated solutions of initial value problems for parametric odes. Applied Numerical Mathematics, 57(10):1145-1162, 2007.
R. J. Lohner. Enclosing the solutions of ordinary initial and boundary value problems. Computer Arithmetic, pages 255-286, 1987.
R. Moore. Interval Analysis. Prentice Hall, 1966.
N. Nedialkov, K. Jackson, and G. Corliss. Validated solutions of initial value problems for ordinary differential equations. Appl. Math. and Comp., 105(1):21 - 68, 1999.
A. Neumaier. The wrapping effect, ellipsoid arithmetic, stability and confidence regions. Computing Supplementum, 9, 1993.
J. Vehì, I. Ferrer, and M. À. Sainz. A survey of applications of interval analysis to robust control. In IFAC World Congress, 2002.
J.G. Ziegler and N.B. Nichols. Optimum settings for automatic controllers. Journal of dynamic systems, measurement, and control, 115(2B):220-222, 1993.
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