Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T09:05:29.187Z Has data issue: false hasContentIssue false

Online parameter identification in time-dependent differential equations as a non-linear inverse problem

Published online by Cambridge University Press:  01 October 2008

PHILIPP KÜGLER*
Affiliation:
Industrial Mathematics Institute, University of Linz, and RICAM, Austrian Academy of Sciences, A–4040 Linz, Austria email: [email protected]

Abstract

Online parameter identification in time-dependent differential equations from time course observations related to the physical state can be understood as a non-linear inverse and ill-posed problem and appears in a variety of applications in science and engineering. The feature as well as the challenge of online identification is that sensor data have to be continuously processed during the operation of the real dynamic process in order to support simulation-based decision making. In this paper we present an online parameter identification method that is based on a non-linear parameter-to-output operator and, as opposed to methods available so far, works both for finite- and infinite-dimensional dynamical systems, e.g., both for ordinary differential equations and time-dependent partial differential equations. A further advantage of the method suggested is that it renders typical restrictive assumptions such as full state observability, linear parametrisation of the underlying model and data differentiation or filtering unnecessary. Assuming existence of a solution for exact data, a convergence analysis based on Lyapunov theory is presented. Numerical illustrations given are by means of online identification both of aerodynamic coefficients in a 3DoF-longitudinal aircraft model and of a (distributed) conductivity coefficient in a heat equation.

Type
Papers
Copyright
Copyright © Cambridge University Press 2008

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

[1]Aström, K. J. & Wittenmark, B. (1995) Adaptive Control, Addison-Wesley, Reaching, Massachusetts.Google Scholar
[2]Banks, H. T. & Kunisch, K. (1989) Estimation Techniques for Distributed Parameter Systems, Birkhäuser, Boston.CrossRefGoogle Scholar
[3]Baumeister, J., Scondo, W., Demetriou, M. A. & Rosen, I. G. (1997) Online parameter estimation for infinite-dimensional dynamical systems. SIAM J. Control Optim. 35, 678713.CrossRefGoogle Scholar
[4]Böhm, M., Demetriou, M. A., Reich, S. & Rosen, I. G. (1998) Model reference adaptive control of distributed parameter systems. SIAM J. Control Optim. 36, 3381.CrossRefGoogle Scholar
[5]Demetriou, M. A. & Rosen, I. G. (1994) On the persistence of excitation in the adaptive identification of distributed parameter systems. IEEE Trans. Automat. Control 39, 11171123.CrossRefGoogle Scholar
[6]Demetriou, M. A. & Rosen, I. G. (2001) On-line robust parameter identification for parabolic systems. Int. J. Adapt. Control Signal Process. 15, 615631.CrossRefGoogle Scholar
[7]Deuflhard, P. & Bornemann, F. (2002) Scientific Computing with Ordinary Differential Equations, Springer, New York.CrossRefGoogle Scholar
[8]Engl, H. W., Hanke, M. & Neubauer, A. (1996) Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht.CrossRefGoogle Scholar
[9]Franklin, G. F., Powell, J. D. & Workman, M. (1997) Digital Control of Dynamic Systems, Addison Wesley Longman, Menlo Park, California.Google Scholar
[10]Grewal, M. S. & Andrews, A. P. (2001) Kalman Filtering–Theory and Practice Using MATLAB, John Wiley & Sons, New York.Google Scholar
[11]Haltmeier, M., Leitao, A. & Scherzer, O. (2007) Kaczmarz methods for regularizing nonlinear ill-posed equations I: Convergence analysis. Inverse Probl. Imaging 1, 289298.CrossRefGoogle Scholar
[12]Hoffmann, K. H. & Sprekels, J. (1984–85) On the identification of coefficients of elliptic problems by asymptotic regularization. Numer. Funct. Anal. Optimiz. 7, 157177.CrossRefGoogle Scholar
[13]Hoffmann, K. H. & Sprekels, J. (1986) Inequalities by asymptotic regularization. SIAM J. Math. Anal. 17, 11981217.CrossRefGoogle Scholar
[14]Ioannou, P. & Sun, J. (1996) Robust Adaptive Control, Prentice Hall, Upper Saddle River, New Jersey.Google Scholar
[15]Jategaonkar, R. & Thielecke, F. (2002) ESTIMA–An integrated software tool for nonlinear parameter estimation. Aerosp. Sci. Technol. 6, 565578.CrossRefGoogle Scholar
[16]Kügler, P. & Engl, H. W. (2002) Identification of a temperature-dependent heat conductivity by Tikhonov regularization. J. Inverse Ill-posed Probl. 10, 6790.CrossRefGoogle Scholar
[17]Landau, I. D., Lozano, R. & M'Saad, M. (1998) Adaptive Control, Springer, London.CrossRefGoogle Scholar
[18]Ljung, L. (1999) System Identification–Theory for the User, Prentice Hall, Upper Saddle River, New Jersey.Google Scholar
[19]Ljung, L. & Söderström, T. (1987) Theory and Practice of Recursive Identification, MIT Press, Boston.Google Scholar
[20]Narendra, K. S. & Annaswamy, A. M. (1989) Stable Adaptive Systems, Dover Publications, New York.Google Scholar
[21]Orlov, Y. & Bentsman, J. (2003) Adaptive distributed parameter systems identification with enforceable identifiability conditions and reduced-order spatial differentiation. IEEE Trans. Automat. Control 45, 203216.CrossRefGoogle Scholar
[22]Sastry, S. & Bodson, M. (1989) Adaptive Control–Stability, Convergence and Robustness, Prentice Hall, Englewood Cliffs, New Jersey.Google Scholar
[23]Showalter, R. (1997) Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs 49, American Mathematical Society, Providence, Rhode Island.Google Scholar
[24]Slotine, J. E. & Li, W. (1991) Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, New Jersey.Google Scholar
[25]Stevens, B. L. & Lewis, F. L. (2003) Aircraft Control and Simulation, John Wiley & Sons, Hoboken, New Jersey.Google Scholar
[26]Tao, G. (2003) Adaptive Control, Design and Analysis, John Wiley & Sons, Hoboken, New Jersey.CrossRefGoogle Scholar
[27]Tautenhahn, U. (1994) On the asymptotical regularization of nonlinear ill-posed problems. Inverse Probl. 10, 14051418.CrossRefGoogle Scholar
[28]Temam, R. (1988) Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68, Springer, New York.CrossRefGoogle Scholar
[29]Xu, G. & Yung, S. P. (2003) Lyapunov stability of abstract nonlinear dynamic in Banach space. IMA J. Math. Control Inf. 20, 105127.CrossRefGoogle Scholar