On global identifiability for arbitrary model parametrizations. (English) Zbl 0795.93026

Ljung, Lennart; Glad, Torkel

Automatica 30, No. 2, 265-276 (1994).

Summary: It is a fundamental problem of identification to be able – even before the data have been analyzed – to decide if all the free parameters of a model structure can be uniquely recovered from data. This is the issue of global identifiability. In this contribution we show how global identifiability for an arbitrary model structure (basically with analytic nonlinearities) can be analyzed using concepts and algorithms from differential algebra. It is shown how the question of global structural identifiability is reduced to the question of whether the given model structure can be rearranged as a linear regression. An explicit algorithm to test this is also given. Furthermore, the question of ‘persistent excitation’ for the input can also be tested explicitly in a similar fashion. The algorithms involved are very well suited for implementation in computer algebra. One such implementation is also described.

MSC:

93B30

System identification

Keywords:

global identifiability

References:

[1]	Bellman, R.; Åström, K. J., On structural identifiability, Math. Biosci., 7, 329-339 (1970)
[2]	Chapell, M.; Godfrey, K.; Vajda, S., Global identifiability of non-linear systems with specified inputs: a comparison of methods, Math. Biosci., 102, 41-73 (1990)
[3]	Dasgupta, S.; Gevers, M.; Bastin, G.; Campion, G.; Chen, L., Identifiability of scalar linearly parametrized polynomial systems, (Proc. 9th IFAC Symp. on System Parameter Estimation. Proc. 9th IFAC Symp. on System Parameter Estimation, Budapest (1991)), 374-378
[4]	Diop, S.; Fliess, M., Nonlinear observability, identifiability and persistent trajectories, (30th CDC. 30th CDC, Brighton, U.K. (1991)), 714-719
[5]	Fliess, M., Quelques définitions de la théorie des systèmes à la lumière des corps différentials, C.R. Acad. Sci. Paris, I-304, 91-93 (1987) · Zbl 0615.93006
[6]	Fliess, M., Quelques remarques sur le observateurs non linearies, (Onzième Colloque sur le Traitement du Signal et des Images (1987), Nice: Nice France), 169-172
[7]	Fliess, M., Automatique et corps differéntials, Forum Mathematicum, 1, 227-238 (1989) · Zbl 0701.93048
[8]	Glad, S., Differential algebraic modeling of nonlinear systems, (Proceedings of MTNS-89. Proceedings of MTNS-89, Amsterdam (1989)), 97-105
[9]	Glad, S.; Ljung, L., Model structure identifiability and persistence of excitation, (Proc. 29th IEEE Conf. on Decision and Control. Proc. 29th IEEE Conf. on Decision and Control, Honolulu, Hawaii (1990)), 5.12-5.7
[10]	Glad, S.; Ljung, L., Parametrization of nonlinear model structures as linear regressions, (Preprints of 11th IFAC World Congress. Preprints of 11th IFAC World Congress, Tallinn, Estonia (1990)), 67-71
[11]	Glad, S.; Ljung, L., An algorithmic differential algebraic approach to identifiability of nonlinear systems, (Technical report (1993), Department of Electrical Engineering, Linköping University: Department of Electrical Engineering, Linköping University Sweden)
[12]	Glad, S. T., Nonlinear state space and input-output descriptions using differential polynomials, (Descusse, J.; Fliess, M.; Isidori, A.; Leborgne, D., Lecture Notes in Control and Information Science, Vol. 122 (1989), Springer: Springer Berlin) · Zbl 0682.93030
[13]	Glad, S. T., Differential algebraic modelling of nonlinear systems, (Kaashoek, M.; van Schuppen, J.; Ranå, A., Realization and Modelling in System Theory (1990), Birkhäuser: Birkhäuser Boston, MA), 97-105 · Zbl 0726.93040
[14]	Glad, S. T., Implementing Ritt’s algorithm of differential algebra, (IFAC Symposium on Control Systems Design, NOLCOS’92. IFAC Symposium on Control Systems Design, NOLCOS’92, Bordeaux, France (1992)), 398-401
[15]	Godfrey, K. R., (Compartmental Models and their Application (1983), Academic Press: Academic Press London)
[16]	Kolchin, E., Extensions of differential fields, J. Am. Math. Soc., 53, 397-401 (1947) · Zbl 0037.18601
[17]	Kolchin, E., (Differential Algebra and Algebraic Groups (1973), Academic Press: Academic Press New York) · Zbl 0264.12102
[18]	Ljung, L., (System Identification—Theory for the User (1987), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ) · Zbl 0615.93004
[19]	Ljung, L.; Glad, S., Testing global identifiability for arbitrary model parametrization, (Proc. IFAC Symp. on Identification and System Parameter Estimation. Proc. IFAC Symp. on Identification and System Parameter Estimation, Budapest, Hungary (1991)), 1077-1082
[20]	Raksanyi, A.; Lecourtier, Y.; Walter, E.; Venot, A., Identifiability and distinguishability testing via computer algebra, Math. Biosci., 77, 245-266 (1985) · Zbl 0574.93019
[21]	Ritt, J., (Differential Algebra (1950), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0037.18501
[22]	Söderström, T.; Stoica, P., (System Identification (1989), Prentice-Hall: Prentice-Hall London) · Zbl 0714.93056
[23]	Walter, E., (Identification of State Space Models (1982), Springer: Springer Berlin)
[24]	(Walter, E., Identifiability of Parametric Models (1987), Pergamon Press: Pergamon Press Oxford)
[25]	Walter, E.; Pronzato, L., Qualitative and quantitative experiment design for phenomenological models—a survey, Automatica, 26, 195-213 (1990) · Zbl 0703.62072