Nihtilä, Markku T. Optimal finite-dimensional recursive identification in a polynomial output mapping class. (English) Zbl 0539.93019 Syst. Control Lett. 3, 341-348 (1983). The search for optimal finite-dimensional recursive filters in nonlinear differential systems was successful in some restricted system classes [e.g. refer to S. Marcus and A. Willsky, SIAM J. Math. Anal. 9, 312-327 (1978; Zbl 0377.93056)]. This paper considers a new system class admitting an optimal finite dimensional recursive estimator. An explicit filter structure is constructed and two examples are discussed to show the application. A time-varying polynomial output equation expressing an implicit dependence of the observed signal on the parameter vector admits a finite-dimensional recursive representation for the optimal on-line estimator. Both continuous and discrete observations result in differential systems which give the estimate. An exponentially weighted least-squares formulation is used. Exponential stability of the identifier and convergence to the true parameter value are proved. A time-continuous algorithm for real time identification of two dissociation constants of a pH-control process is presented. The discrete identification is applied in a dynamic ARMA-type difference model which is quadratic in the parameter to be estimated. Reviewer: R.K.Bose Cited in 1 ReviewCited in 3 Documents MSC: 93B30 System identification 93C10 Nonlinear systems in control theory 93C99 Model systems in control theory 60G35 Signal detection and filtering (aspects of stochastic processes) 93B40 Computational methods in systems theory (MSC2010) 93E10 Estimation and detection in stochastic control theory 93E11 Filtering in stochastic control theory Keywords:deterministic least-squares identification; polynomial systems; exponential weighting; recursive finite-dimensional estimation Citations:Zbl 0377.93056 PDFBibTeX XMLCite \textit{M. T. Nihtilä}, Syst. Control Lett. 3, 341--348 (1983; Zbl 0539.93019) Full Text: DOI References: [1] Beneš, V., Exact finite dimensional filters for certain diffusions with nonlinear drift, (Proc. 18th IEEE Conf. on Decision and Control. Proc. 18th IEEE Conf. on Decision and Control, Fort Lauderdale, FL (Dec. 1979)) · Zbl 0458.60030 [2] Marcus, S. I.; Willsky, A. S., Algebraic structure and finite dimensional nonlinear estimation, SIAM J. Math. Anal., 9, 312-327 (1978) · Zbl 0377.93056 [3] M.T. Nihtilä, Finite dimensional deterministic nonlinear filters via Riccati transformation and Volterra series, SIAM J. Control Optim.; M.T. Nihtilä, Finite dimensional deterministic nonlinear filters via Riccati transformation and Volterra series, SIAM J. Control Optim. · Zbl 0539.93033 [4] Nihtilä, M. T., Optimal finite dimensional solution for a class of nonlinear observation problems, J. Optim. Theory Appl., 38, 231-240 (1982) · Zbl 0471.93014 [5] Whitcombe, D. W., Pseudo state measurements applied to recursive nonlinear filtering, (Proc. 3rd Symposium on Nonlinear Estimation Theory and its Applications (1972), Western Periodicals, Co), 278-281 [6] Greub, W. H., Linear Algebra (1936), Springer: Springer New York · Zbl 0147.27408 [7] Johnstone, P. M.; Johnson, C. R.; Bitmead, R. R.; Anderson, B. D.O., Exponential convergence of recursive least squares with exponential forgetting factor, Systems Control Lett., 2, 77-82 (1982) · Zbl 0537.93027 [8] Kallianpur, G., Stochastic Filtering Theory (1980), Springer: Springer New York · Zbl 0458.60001 [9] Halme, A., Polynomial operators for nonlinear systems analysis, Acta Polytech. Scand. Math. Ser., 24, 1-64 (1972) [10] Jutila, P.; Orava, J., Control and estimation algorithms for physico-chemical models of pH-processes in stirred tank reactors, Internat. J. Systems Sci., 12, 855-875 (1981) 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.