Direct and indirect least squares methods in continuous-time parameter estimation. (English) Zbl 0641.93065
This paper extends well-established discrete-time parameter estimation techniques to continuous-time nonlinear models. It is shown that, in most cases, the direct integral least squares (DILS) and symmetric bootstrap (SB) estimators are numerically efficient and are statistically sound alternatives to the conventional indirect least squares (ILS) approach. In most applications a few iterations of the SB algorithm, started at the DILS estimates, yield results as good as the computationally more expensive ILS method. Thus the direct approach often results in a considerable saving of computational effort.
Reviewer: M.Tibaldi
MSC:
| 93E12 | Identification in stochastic control theory |
| 93C10 | Nonlinear systems in control theory |
| 93E25 | Computational methods in stochastic control (MSC2010) |
Keywords:
parameter estimation; continuous-time nonlinear models; direct integral least squares; symmetric bootstrap; continuous-timeReferences:
| [1] | Åström, K. J.; Eykhoff, P., System identification—a survey, Automatica, 7, 123 (1971) · Zbl 0219.93004 |
| [2] | Bard, Y., Comparison of gradient methods for the solution of nonlinear parameter estimation problems, SIAM J. Numer. Anal., 7, 157 (1970) · Zbl 0202.16904 |
| [3] | Bard, Y., (Nonlinear Parameter Estimation (1974), Academic Press: Academic Press New York) · Zbl 0345.62045 |
| [4] | Bates, D. M.; Watts, D. G., Multiresponse estimation with special application to linear systems of differential equations, Technometrics, 27, 329 (1985) · Zbl 0599.93052 |
| [5] | Cobelli, C., Identification of endocrine-metabolic and pharmacokinetic systems, (Barker, H. A.; Young, P. C., 7th IFAC/IFORS Symp.. 7th IFAC/IFORS Symp., Pergamon, Oxford. 7th IFAC/IFORS Symp.. 7th IFAC/IFORS Symp., Pergamon, Oxford, Identification and System Parameter Estimation, Vol. 1 (1985)), 45-54 |
| [6] | (Endrenyi, L., Kinetic Data Analysis (1981), Plenum: Plenum New York) |
| [7] | Gelinas, R. J., Stiff systems of kinetic equations—a practitioner’s view, J. Comput. Phys., 9, 222 (1972) · Zbl 0231.65067 |
| [8] | Gevers, M.; Tsoi, A. C., Structural identification of linear multivariable systems using overlapping forms: a new parametrization, Int. J. Control, 40, 971 (1984) · Zbl 0555.93012 |
| [9] | Godfrey, K. R.; Fitch, W. R., On the identification of Michaelis-Menten elimination parameters from a single dose-response curve, J. Pharmocokin. Biopharm., 12, 193 (1984) |
| [10] | Goodwin, G. C.; Payne, R. L., (Dynamic System Identification: Experiment Design and Data Analysis (1977), Academic Press: Academic Press New York) · Zbl 0578.93060 |
| [11] | Hamming, R. W., (Numerical Methods for Scientists and Engineers (1962), McGraw-Hill: McGraw-Hill New York) · Zbl 0952.65500 |
| [12] | Himmelblau, D. M., (Process Analysis by Statistical Methods (1970), Wiley: Wiley New York) |
| [13] | Himmelblau, D. M.; Jones, C. R.; Bischoff, K. B., Determination of rate constants for complex kinetic models, Ind. Eng. Chem. Fund., 6, 539 (1967) |
| [14] | Hocking, R. R., Developments in linear regression methodology: 1959-1982, Technometrics, 25, 219 (1983) · Zbl 0515.62062 |
| [15] | Hoerl, A. E.; Kennard, R. W., Ridge regression: biased estimation for non-orthogonal problems, Technometrics, 12, 55 (1970) · Zbl 0202.17205 |
| [16] | Holmberg, A.; Ranta, J., Procedures for parameter and state estimation of microbial growth process models, Automatica, 18, 181-193 (1982) · Zbl 0487.93052 |
| [17] | Hosten, L. H., A comparative study of short cut procedures for parameter estimation in differential equations, Comput. Chem. Engng., 3, 117 (1979) |
| [18] | Jennrich, R. I., Asymptotic properties of non-linear least squares estimators, Ann. Math. Statist, 40, 633 (1969) · Zbl 0193.47201 |
| [19] | Jennrich, R. I.; Sampson, P. F., Application of stepwise regression to non-linear estimation, Technometrics, 10, 63 (1968) |
| [20] | Joseph, P.; Lewis, J.; Tou, J., Plant identification in the presence of disturbances and application to digital adaptive systems, AIEE Trans. App. Ind., 80, 18 (1961) |
| [21] | Klonowski, W., Simplifying principles for chemical and enzyme reaction kinetics, Biophys. Chem., 18, 73 (1983) |
| [22] | Loeb, J. H.; Cahen, G. M., More about process identification, IEEE Trans. Aut. Control, AC-10, 359 (1965) |
| [23] | Mori, F.; DiStefano, J. J., Optimal nonuniform sampling interval and test-input design for identification of physiological systems from very limited data, IEEE Trans. Aut. Control, AC-24, 893 (1979) · Zbl 0416.93047 |
| [24] | Nazareth, L., Some recent approaches to solving large residual nonlinear least squares problems, SIAM Rev., 22, 1 (1980) · Zbl 0424.65031 |
| [25] | Nihtilä, M.; Virkkunen, J., Practical identifiability of growth and substrate consumption models, Biotech. Bioengng, 19, 1831 (1977) |
| [26] | Sinha, N. K.; Qijie, Z., Identification of continuous-time systems from sampled data: a comparison of 3 direct methods, (Barker, H. A.; Young, P. C., 7th IFAC/IFORS Symp.. 7th IFAC/IFORS Symp., Pergamon, Oxford. 7th IFAC/IFORS Symp.. 7th IFAC/IFORS Symp., Pergamon, Oxford, Identification and System Parameter Estimation, Vol. 1 (1985)), 1575-1578 |
| [27] | Söderström, T.; Stoica, P., Comparison of some instrumental variable methods—consistency and accuracy aspects, Automatica, 17, 101-115 (1981) · Zbl 0451.93026 |
| [28] | Söderström, T.; Stoica, P. G., (Instrumental Variable Methods for System Identification (1983), Springer: Springer Berlin) · Zbl 0522.93003 |
| [29] | Sorenson, H. W., (Parameter Estimation—Principles and Problems (1980), Marcel Dekker: Marcel Dekker New York) · Zbl 0466.62029 |
| [30] | Stoica, P.; Söderström, T.; Ahlen, A.; Solbrand, G., On the convergence of pseudo-linear regression algorithms, Int. J. Control, 41, 1429 (1985) · Zbl 0581.93057 |
| [31] | Strejc, V., Least squares parameter estimation, Automatica, 16, 535-550 (1980) · Zbl 0441.93043 |
| [32] | Tong, D. D.M.; Metzler, C. M., Mathematical properties of compartment models with Michaelis-Menten type elimination, Math. Biosci., 48, 293 (1980) · Zbl 0431.92015 |
| [33] | Uno, F. K.; Ralston, M. L.; Jennrich, R. I.; Sampson, P. F., Test problems from the pharmacokinetic literature requiring fitting models defined by differential equations, (Tech. Report No. 61, BMDP (1979), Statistical Software, Department of Biomathematics, University of California: Statistical Software, Department of Biomathematics, University of California Los Angeles) |
| [34] | Vajda, S., Identifiability of polynomial and rational systems: structural and numerical aspects, (Barker, H. A.; Young, P. C., 7th IFAC/IFORS Symp.. 7th IFAC/IFORS Symp., Pergamon, Oxford. 7th IFAC/IFORS Symp.. 7th IFAC/IFORS Symp., Pergamon, Oxford, Identification and System Parameter Estimation, Vol. 1 (1985)), 537-542 |
| [35] | Vajda, S.; Valko, P.; Turanyi, T., Principal component analysis of kinetic models, Int. J. Chem. Kinet., 17, 55 (1985) |
| [36] | Vajda, S.; Valko, P.; Yermakova, A., A direct-indirect procedure for estimating kinetic parameters, Comput. Chem. Engng., 10, 49 (1986) |
| [37] | Wong, K. Y.; Polak, E., Identification of linear discrete time systems using the instrumental variable approach, IEEE Trans. Aut. Control, AC-12, 707 (1967) |
| [38] | Yermakova, A.; Umbetov, A. S., Diffusion kinetics of selective hydrogenation of fatty oils in a stationary catalyst bed reactor, Reactor Kinet. Catal. Lett., 14, 187 (1980) |
| [39] | Yermakova, A.; Vajda, S.; Valko, P., Direct integral method via spline approximation for estimating rate constants, Appl. Catal., 2, 139 (1982) |
| [40] | Young, P. C., An instrumental variable method for real time identification of a noisy process, Automatica, 6, 271 (1970) |
| [41] | Young, P. C., Parameter estimation for continuous-time models—a survey, Automatica, 17, 23-39 (1981) · Zbl 0451.93052 |
| [42] | Young, P. C.; Jakeman, A., Refined instrumental variable methods of recursive time-series analysis, Part III. Extensions, Int. J. Control, 31, 741 (1980) · Zbl 0468.93089 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
