Abstract
Recurrent estimation algorithms (Kalman filter) is widespread in solving a wide range of problems of control, identification and filtering.These algorithms have several advantages over other algorithms for solving these problems. The main of these advantages is an optimal estimate of the state vector (minimization of the mean square error of estimation) and the recurrence property when a "new" estimate of the state vector is obtained from an “old” estimate (evaluation at the previous step) and processing a "new" measurement. This scheme has received the predictor-corrector name. The recurrence property significantly reduces the computational cost to build estimates of the state vector and (in most cases)allows implementing the estimation process in real-time.However, when these algorithms are used in practice, the problem of divergence arises when the "true" error of estimation of the state vector is significantly higher than the "design" values calculated through the estimated correlations of the estimation algorithm, i.e. the estimation algorithm does not worksoptimally. To overcome the divergence, the estimation algorithm must detect the violation of the optimal mode and to correctthe computational schemeof the algorithm.In this paper we construct a simple divergence criterionof the recurrent algorithm estimation well implemented in practice, which accurately detects the time when the algorithm loses the optimality property, i.e. the beginning of a sharp increase in the root mean square error of estimation. This criterion is based on testing statistical hypotheses of properties of the updating process that represents the difference between the current measurement and the predicted value of this measurement. When such a moment occurs, the proposed adaptation algorithm which modifies the variance-covariance matrix of the prediction error intervenes so as to giveback the optimality property to the estimation algorithm. The implementation of the computing experiment showeda high efficiency of the constructed criterion and the proposed adaptation algorithm, even with large errors in the matrix model of the dynamic system.
Keywords: recursive estimation algorithm, the Kalman filter, updating process, the optimality property of the estimation algorithm, the divergence of the estimation algorithm, the criterion of divergence, prediction error of the state vector, statistical properties of the prediction error, adaptation algorithm, efficiency of the adaptation algorithm
References
1. Covan C.F., Grant P.M. Adaptive filter. Englewood Cliffs, New Jersey, Prentice-Hall, 1985. 382 p. (Russ. ed.: Kouen K.F., Grant P.M. Adaptivnyefil'try. Moscow, Mir Publ., 1988. 392 p.).
2. Balakrishnan A.V. Kalman filtering theory. New York, Optimization software, 1982. 182 p. (Russ. ed.: Balakrishnan A.V. Teoriyafil'tratsiiKalmana. Moscow, Nauka Publ., 1984. 186 p.).
3. Brammer K., Siffling G. Kalman-Busi filter. New York, Artech House Publishers, 1985. 202 p. (Russ. ed.: Brammer L., Zifling G. Fil'trKalmana-B'yusi. Мoscow, Nauka Publ., 1982. 199 p.).
4. VoskoboinikovYu.E. Rekurrentnoeotsenivanievektorasostoyaniyadinamicheskikhsistem [Recurrent estimation of state vector of dynamic systems]. Novosibirsk, NSTU Publ., 2014. 136 p.
5. VoskoboinikovYu.E. Ustoichivyemetodyialgoritmyparametricheskoiidentifikatsii [The stable methods and algorithms for parametric identification]. Novosibirsk, NGASU (Sibstrin) Publ., 2006. 168 p.
6. VoskoboinikovYu.E. Ustoichivyealgoritmyresheniyaobratnykhzadach [The stable algorithms for solving inverse problems]. Novosibirsk, NGASU (Sibstrin) Publ., 2007. 188 p.
7. VoskoboinikovYu.E., Khanin A.G. Rekurrentnyialgoritmotsenivaniyagradienta v sistemakhekstremal'nogoregulirovaniya[Recursive algorithm for gradient estimating in the systemsextreme control]. Nauchnyivestnik NGTU – Science Bulletin of the Novosibirsk State Technical University, 2012, no. 1 (46), pp. 3–9.
8. Karelin A.E. Sintez, issledovanieiprimenenierekurrentnykhalgoritmovotsenivaniyaparametrovmatematicheskikhmodeleiob"ektov v avtomatizirovannykhsistemakhupravleniya.Avtoref.diss. kand. tekhn. nauk [Synthesis, study and application of recursive algorithms for estimating the parameters of mathematical models of objects in automated control systems. Author's abstract of PhD eng. sci. diss.]. Tomsk, 2007.18 p.
9. Ljung L. System identification – theory for the user. Upper Saddle River, New Jersey, Prentice Hall, 1988. 446 p. (Russ. ed.: L'yung L. Identifikatsiyasistem. Teoriyadlyapol'zovatelya.Translated from English.Мoscow, Nauka Publ., 1991. 432 p.).
10. Meditch J.S. Stochastic optimal linear estimation and control.New York, Mc. Graw Hill, 1969. 425 p. (Russ. ed.: MedichDzh. Statisticheskieoptimal'nyelineinyeotsenkiiupravlenie. Moscow, Energiya Publ., 1973. 340 p.).
11. Fomin V.N. Rekurrentnoeotsenivanieiadaptivnayafil'tratsiya [Recurrent estimation and adaptive filtration].Мoscow, Nauka Publ., 1984. 288 p.
12. Doblinger G. An adaptive Kalman filter for the enhancement of noisy AR signals. Proceedings of the 1998 IEEE International Symposium on Circuits and Systems, ISCAS’98, Monterey, California, 31 May – 3 June 1998, vol. 5, pp. 305–308. doi: 10.1109/ISCAS.1998.694474
13. Naykin S. Adaptive filter theory.Chap. 9.New York, Prentice-Hall, 1987.323 p.
14. Xiong S., Zhou Z., Zhong L., Xu C., Zhang W. Adaptive filtering of color noise using the Kalman filter algorithm. Proceedings of the 17th Instrumentation and Measurement Technology Conference, IMTC 2000, Baltimore, Maryland, USA, 1–4 May, 2000, vol. 2, pp. 1009–1012. doi: 10.1109/IMTC.2000.848893
15. Spall J.C. Introductionto stochastic search and optimization: estimation, simulation, and control. Hoboken, New Jersey, Wiley, 2003.618 p.
