Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-04T21:33:11.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Uncertainty propagation in model extraction by system identification and its implication for control design

Published online by Cambridge University Press:  17 February 2016

Nicolas Dovetta ,
Peter J. Schmid  and
Denis Sipp
Nicolas Dovetta*
Affiliation:
Laboratoire d’Hydrodynamique (LadHyX), Ecole Polytechnique, F-91128 Palaiseau, France
Peter J. Schmid
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Denis Sipp
Affiliation:
ONERA-DAFE, 8 rue des Vertugadins, F-92190 Meudon, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In data-based control design, system-identification techniques are used to extract low-dimensional representations of the input–output map between actuators and sensors from observed data signals. Under realistic conditions, noise in the signals is present and is expected to influence the identified system representation. For the subsequent design of the controller, it is important to gauge the sensitivity of the system representation to noise in the observed data; this information will impact the robustness of the controller and influence the stability margins for a closed-loop configuration. Commonly, full Monte Carlo analysis has been used to quantify the effect of data noise on the system identification and control design, but in fluid systems, this approach is often prohibitively expensive, due to the high dimensionality of the data input space, for both numerical simulations and physical experiments. Instead, we present a framework for the estimation of statistical properties of identified system representations given an uncertainty in the processed data. Our approach consists of a perturbative method, relating noise in the data to identified system parameters, which is followed by a Monte Carlo technique to propagate uncertainties in the system parameters to error bounds in Nyquist and Bode plots. This hybrid approach combines accuracy, by treating the system-identification part perturbatively, and computational efficiency, by applying Monte Carlo techniques to the low-dimensional input space of the control design and performance/stability evaluation part. This combination makes the proposed technique affordable and efficient even for large-scale flow-control problems. The ARMarkov/LS identification procedure has been chosen as a representative system-identification technique to illustrate this framework and to obtain error bounds on the identified system parameters based on the signal-to-noise ratio of the input–output data sequence. The procedure is illustrated on the control design for flow over an idealized aerofoil with a trailing-edge splitter plate.

JFM classification

Mathematical Foundations: Computational methods Flow Control: Control theory Flow Control: Flow Control
Type
Papers
Information
Journal of Fluid Mechanics , Volume 791 , 25 March 2016 , pp. 214 - 236
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Akaike, H. 1974 Fitting autoregressive models for prediction. IEEE Trans. Autom. Control 19, 716–723.Google Scholar
Akers, J. C. & Bernstein, D. S. 1997 ARMarkov least-squares identification. In Proceedings Amer. Contr. Conference, Albuquerque, NM, pp. 191–195. Institute of Electrical and Electronics Engineers.Google Scholar
Brighenti, C., Wahlberg, B. & Rojas, C. R. 2009 Input design using Markov chains for system identification. In Joint 48th IEEE Conference Dec. and Contr. and 28th Chin. Contr. Conference, pp. 1557–1562. Institute of Electrical and Electronics Engineers.Google Scholar
Fledderjohn, M. S., Holzel, M. S., Palanthandalam-Madapusi, H. J., Fuentes, R. J. & Bernstein, D. S. 2010 A comparison of least squares algorithm for estimating Markov parameters. In Amer. Control Conference (ACC), pp. 3735–3740. Institute of Electrical and Electronics Engineers.Google Scholar
Fleming, J.2011 Generalized Tikhonov regularization: basic theory and comprehensive results on convergence rates. PhD thesis, Techn. Univ. Chemnitz.Google Scholar
Gerencser, L., Hjalmarsson, H. & Martensson, J. 2009 Identification of ARX systems with non-stationary inputs – asymptotic analysis with application to adaptive input design. Automatica 45, 623–633.CrossRefGoogle Scholar
Glowinski, R. 2003 Finite element methods for incompressible viscous flow. In Handbook of Numerical Analysis, vol. 9, pp. 3–1176. Elsevier.Google Scholar
Hervé, A., Sipp, D., Schmid, P. J. & Samuelides, M. 2012 A physics-based approach to flow control using system identification. J. Fluid Mech. 702, 26–58.CrossRefGoogle Scholar
Hjalmarsson, H. 2005 From experiment design to closed-loop control. Automatica 41, 393–438.Google Scholar
Huang, S. C. & Kim, J. 2008 Control and system identification of a separated flow. Phys. Fluids 20, 101509.CrossRefGoogle Scholar
Illingworth, S. J., Morgans, A. S. & Rowley, C. W. 2011 Feedback control of flow resonances using balanced reduced-order models. J. Sound Vib. 330 (8), 1567–1581.Google Scholar
Juang, J. N. & Pappa, R. S. 1985 Eigensystem realization algorithm for modal parameter identification and model reduction. J. Guid. Control Dyn. 8 (5), 620–627.Google Scholar
Juillet, F., McKeon, B. J. & Schmid, P. J. 2014 Experimental control of natural perturbations in channel flow. J. Fluid Mech. 752, 296–309.CrossRefGoogle Scholar
Juillet, F., Schmid, P. J. & Huerre, P. 2013 Control of amplifier flows using subspace identification. J. Fluid Mech. 725, 522–565.CrossRefGoogle Scholar
Kamrunnahar, M., Huang, B. & Fisher, D. G. 2000 Estimation of Markov parameters and time-delay/interactor matrix. Chem. Engng Sci. 55 (17), 3353–3363.Google Scholar
Katayama, T. 2005 Subspace Methods for System Identification. Springer.CrossRefGoogle Scholar
Lew, J.-S., Juang, J.-N. & Longman, R. W. 1993 Comparison of several system identification methods for flexible structures. J. Sound Vib. 167 (3), 461–480.CrossRefGoogle Scholar
Ljung, L. 1987 System Identification: Theory for the User. Prentice-Hall.Google Scholar
Longman, R. W., Lew, J.-S., Tseng, D.-H. & Juang, J.-N. 1991 Variance and bias computation for improved modal identification using ERA/DC. In Amer. Control Conference, pp. 3013–3018. Institute of Electrical and Electronics Engineers.Google Scholar
Ma, Z., Ahuja, S. & Rowley, C. W. 2011 Reduced order models for control of fluids using the eigenvalue realization algorithm. Theor. Comput. Fluid Dyn. 25 (1), 233–247.Google Scholar
Mehra, R. K. 1974 Optimal inputs for linear system identification. IEEE Trans. Autom. Control 19 (3), 192–200.Google Scholar
Rathnasingham, R. & Breuer, K. S. 2003 Active control of turbulent boundary layers. J. Fluid Mech. 495, 209–233.CrossRefGoogle Scholar
Rissanen, J. 1983 A universal prior for integers and estimation by minimum description length. Ann. Stat. 11 (2), 416–431.CrossRefGoogle Scholar
Skogestad, S. & Postlethwaite, I. 2001 Multivariable Feedback Control – Analysis and Design. John Wiley.Google Scholar
Stewart, G. W. 1977 On the perturbation of pseudo-inverses, projection and linear least squares problems. SIAM Rev. 19 (4), 634–662.Google Scholar
Stewart, G. W. 1990 Perturbation theory and least squares with errors in the variables. Contemp. Maths 112 (4), 171–181.CrossRefGoogle Scholar
Van Pelt, T. & Bernstein, D. S. 1998 Least squares identification using ${\it\mu}$ -Markov parameterizations. In Proceedings of the 37th IEEE Conference Dec. and Contr. 1998, vol. 1, pp. 618–619. Institute of Electrical and Electronics Engineers.Google Scholar
Wallace, R. D. & McKeon, B. J. 2012 Laminar separation bubble manipulation with dynamic roughness. In 6th AIAA Flow Control Conference, New Orleans, Louisiana, pp. 2012–2680. Institute of Electrical and Electronics Engineers.Google Scholar
Wedin, P. A. 1973 Perturbation theory for pseudo-inverses. BIT Num. Math. 13 (2), 217–232.CrossRefGoogle Scholar