Abstract
Procedures of active parametric identification of stochastic linear discrete systems based on an optimal design of input signals or initial states have already been developed. In this paper, the authors try to generalize the results obtained earlier and construct new algorithms based on the simultaneous design of input signals and initial states. The procedure of active parametric identification of systems with a preliminary chosen model structure assumes performing the following stages: the calculation of unknown parameter estimates based on the measured data corresponding to some experiment plan; the synthesis of an optimal experiment plan based on the received estimates and the recalculation of estimates of unknown parameter estimates from the measured data corresponding to the optimal plan. A systematic interpretation of the most significant practical issues of the theory and techniques of active identification of multidimensional Gaussian stochastic linear continuous-discrete systems described by a state space model is given in the paper. For the first time the authors consider and solve an urgent problem of the active identification for a general case when unknown parameters appear in state and control equations as well as in the covariance matrices of process noises and measurement errors. A designed calculation algorithm of information matrix derivatives with respect to components of both an input signal vector and an initial state vector is proposed. This algorithm allows us to synthesize input signals and initial states by means of the sequential quadratic programming method and thus to considerably reduce the optimal experiment design search time. The original gradient algorithms of the optimal parameter estimation are designed. They enable us to solve optimal parameter estimation problems for mathematical models using the maximum likelihood method involving direct and dual procedures for synthesizing the А- and D –optimal experiment design. Some theoretical and applied aspects of the active identification of stochastic linear discrete- continuous systems based on designing input signals and initial conditions are considered for the first time. An example of the active parametric identification for one stochastic discrete- continuous model structure is shown.
Keywords: discrete-continuous system, active identification, parameter estimation, maximum likelihood method, experiment design, information matrix, optimality criterion, Kalman filter
References
1. Kashyap R.L., Rao А.R. Dynamic stochastic models from empirical data. New York, Academic Press, 1976. 333 p. (Russ. ed.: Kash'yap R.L., Rao A.R. Postroenie dinamicheskikh stokhasticheskikh modelei po eksperimental'nym dannym. Moscow, Nauka Publ., 1983. 384 p.).
2. Ljung L. Systems identification. Theory for the user. New Jersey, Prentice Hall, 1987. (Russ. ed.: L'yung L. Identifikatsiya system. Teoriya pol'zovatelya. Moscow, Nauka Publ., 1991. 432 p.).
3. Tsypkin Ya.Z. Informatsionnaya teoriya identifikatsii [Information theory of identification]. Moscow, Nauka Publ., 1995. 336 p.
4. Walter E., Pronzato L. Identification of parametric models from experimental data. Berlin, Springer-Verlag, 1997. 413 p.
5. Denisov V.I., Chubich V.M., Chernikova O.S., Bobyleva D.I. Aktivnaya parametricheskaya identifikatsiya stokhasticheskikh lineinykh sistem [Active parametric identification of stochastic linear systems]. Novosibirsk, NSTU Publ., 2009. 192 p.
6. Chubich V.M. Aktivnaya parametricheskaya identifikatsiya stokhasticheskikh nelineinykh nepreryvno-diskretnykh sistem na osnove linearizatsii vo vremennoi oblasti [Active parametric identification of stochastic nonlinear continuous-discrete systems based on time domain linearization]. Informatsionno-upravlyayushchie sistemy – Information and Control Systems, 2010, no. 6 (49), pp. 54–61.
7. Denisov V.I., Voevoda А.А., Chubich V.M., Filippova E.V. [Active parametric identification of stochastic nonlinear continuous-discrete systems based on experimental design]. Trudy 12 Vserossiiskogo soveshchaniya po problemam upravleniya [Proceedings of the 12 All-Russian meeting on problems of management]. Moscow, IPU RАN Publ., 2014, pp. 2795–2806. Available at: http://vspu2014.ipu.ru/node/8581 (accessed 21.11.2014)
8. Astrom K.J. Maximum likelihood and prediction errors methods. Automatica, 1980, vol. 16, iss. 5, pp. 551–574. doi:10.1016/0005-1098(80)90078-3
9. Ogarkov M.A. Metody statisticheskogo otsenivaniya parametrov sluchainykh protsessov [Methods of statistical estimation of parameters of random processes]. Moscow, Energoatomizdat Publ., 1980. 208 p.
10. Chubich V.M. Algoritm vychisleniya informatsionnoi matritsy Fishera v zadache aktivnoi parametricheskoi identifikatsii stokhasticheskikh nelineinykh nepreryvno-diskretnykh sistem [The procedure of the computation of the Fisher information matrix in the problem of active parametric identification for stochastic nonlinear continues-discrete systems]. Nauchnyi vestnik NGTU – Science Bulletin of Novosibirsk State Technical University, 2009, no. 3 (36), pp. 15–22.
11. Gorskii V.G., Adler Yu.P., Talalai A.M. Planirovanie promyshlennykh eksperimentov (modeli dinamiki) [The planning of industrial experiments (dynamic model)]. Moscow, Metallurgiya Publ., 1978. 112 p.
12. Mehra R.K. Optimal input signals for parameter estimation in dynamic systems:survey and new results. IEEE Transactions on Automatic Control, 1974, vol. 19, iss. 6, pp.753–768. doi: 10.1109/TAC.1974.1100701
13. Chubich V.M., Filippova E.V. Vychislenie proizvodnykh informatsionnoi matritsy Fishera po komponentam vkhodnogo signala v zadache aktivnoi parametricheskoi identifikatsii stokhasticheskikh nelineinykh nepreryv-no-diskretnykh sistem [The computation of the derivatives of the Fisherinformation matrix with respect to the components of input signal in the problem of active parametric identification of stochastic nonlinear continuous-discrete systems]. Nauchnyi vestnik NGTU – Science Bulletin of Novosibirsk State Technical University, 2010, no. 2 (39), pp. 53–63.
14. Ermakov S.M., Zhiglyavskii A.A. Matematicheskaya teoriya optimal'nogo eksperimenta [The mathematical theory of optimal experiment]. Moscow, Nauka Publ., 1987. 320 p.
