Hierarchical gradient-based identification of multivariable discrete-time systems. (English) Zbl 1073.93012

Ding, Feng; Chen, Tongwen

Automatica 41, No. 2, 315-325 (2005).

Summary: We use a hierarchical identification principle to study identification problems for multivariable discrete-time systems. We propose a hierarchical gradient iterative algorithm and a hierarchical stochastic gradient algorithm and prove that the parameter estimation errors given by the algorithms converge to zero for any initial values under persistent excitation. The proposed algorithms can be applied to identification of systems involving non-stationary signals and have significant computational advantage over existing identification algorithms. Finally, we test the proposed algorithms by simulation and show their effectiveness.

MSC:

93B30	System identification
93C55	Discrete-time control/observation systems
93A13	Hierarchical systems

Keywords:

Recursive identification; Estimation; Least squares; Hierarchical identification principle; Multivariable systems; Convergence properties

References:

[1]	Chen, T.; Qiu, L., \(\mathcal{H}_\infty\) design of general multirate sampled-data control systems, Automatica, 30, 7, 1139-1152, (1994) · Zbl 0806.93038
[2]	Cho, Y.M.; Xu, G.; Kailath, T., Fast recursive identification of state space models via exploitation of displacement structure, Automatica, 30, 1, 45-49, (1994) · Zbl 0787.93020
[3]	Chou, C.T.; Verhaegen, M., Subspace algorithms for the identification of multivariable dynamic error-in-variable models, Automatica, 33, 10, 1857-1869, (1997) · Zbl 0889.93011
[4]	Comon, P.; Golub, G., Tracking a few extreme singular values and vectors in signal processing, Proceedings of the IEEE, 78, 8, 1327-1343, (1990)
[5]	Ding, F., Yang, J. B., & Xu, Y. M. (2000). Hierarchical identification and its convergence for the transfer function matrix. Proceedings of IFAC Symposium on System Identification (SYSID 2000), 21-23 June 2000, Santa Barbara, CA, USA.
[6]	Drouin, M.; Abou-Kandil, H.; Mariton, M., Control of complex systemsmethods and technology, (1991), Plenum Press New York, London, (Applied Information Technology. Series Editor: M. G. Singh)
[7]	El-Sherief, H., Parametric identification of a state-space model of multivariable systems using the extended least-squares method, IEEE transactions on systems man, and cybernetics, 11, 3, 223-227, (1981)
[8]	El-Sherief, H.; Sinha, N.K., Choices of models for the identification of linear multivariable discrete-time systems, IEEE Proceedings, part D, control theory and applications, 126, 12, 1326-1330, (1979)
[9]	Gauthier, A.; Landau, I.D., On the recursive identification of multi-input multi-output systems, Automatica, 14, 6, 609-614, (1978) · Zbl 0396.93022
[10]	Goodwin, G.C.; Sin, K.S., Adaptive filtering prediction and control, (1984), Prentice-Hall Englewood Cliffs, NJ · Zbl 0653.93001
[11]	Gustafsson, T., Instrumental variable subspace tracking using projection approximation, IEEE transactions on signal processing, 46, 3, 669-681, (1998)
[12]	Lai, T.L.; Ying, Z.L., Recursive identification and adaptive prediction in linear stochastic systems, SIAM journal on control and optimization, 29, 5, 1061-1090, (1991) · Zbl 0756.93081
[13]	Li, D.; Shah, S.L.; Chen, T., Identification of fast-rate models from multirate data, International journal of control, 74, 7, 680-689, (2001) · Zbl 1038.93017
[14]	Li, D.; Shah, S.L.; Chen, T., Analysis of dual-rate inferential control systems, Automatica, 38, 6, 1053-1059, (2002) · Zbl 1038.93033
[15]	Li, D.; Shah, S.L.; Chen, T.; Qi, K.Z., Application of dual-rate modeling to CCR octane quality inferential control, IEEE transactions on control systems technology, 11, 1, 43-51, (2003)
[16]	Ljung, L., System identificationtheory for the user, (1999), Prentice-Hall Englewood Cliffs, NJ
[17]	Lovera, M.; Gustafsson, T.; Verhaegen, M., Recursive subspace identification of linear and non-linear Wiener state-space models, Automatica, 36, 11, 1651-1658, (2000)
[18]	McKelvey, T.; Akcay, H.; Ljung, L., Subspace-based multivariable system identification from frequency response data, IEEE transactions on automatic control, 41, 7, 960-979, (1996) · Zbl 0855.93020
[19]	Oku, H.; Kimura, H., Recursive 4SID algorithms using gradient type subspace tracking, Automatica, 38, 6, 1035-1043, (2002) · Zbl 1007.93018
[20]	Overshee, P.V.; De Moor, B., N4sidsubspace algorithm for the identification of combined determine-stochastic systems, Automatica, 30, 1, 75-93, (1994)
[21]	Overshee, P.V.; De Moor, B., Subspace identification for linear systemstheory, implementation and applications, (1996), Kluwer Academic Publisher Boston
[22]	Pintelon, R., Frequency-domain subspace system identification using non-parametric noise models, Automatica, 38, 8, 1295-1311, (2002) · Zbl 1008.93066
[23]	Qiu, L.; Chen, T., \(\mathcal{H}_\infty\)-optimal design of multirate sampled-data systems, IEEE transactions on automatic control, 39, 12, 2506-2511, (1994) · Zbl 0825.93436
[24]	Qiu, L.; Chen, T., Multirate sampled-data systemsall H∞ suboptimal controllers and the minimum entropy controllers, IEEE transactions on automatic control, 44, 3, 537-550, (1999) · Zbl 0958.93031
[25]	Sen, A.; Sinha, N.K., On-line estimation of the parameters of a multivariable system using matrix pseudoinverse, International journal of systems science, 7, 4, 461-471, (1976) · Zbl 0323.93033
[26]	Sheng, J.; Chen, T.; Shah, S.L., Generalized predictive control for non-uniformly sampled systems, Journal of process control, 12, 875-888, (2002)
[27]	Singh, M.G., Dynamical hierarchical control, (1980), North-Holland Publishing Company New York
[28]	Sinha, N.K.; Kwong, Y.H., Recursive estimation of the parameters of linear multivariable systems, Automatica, 15, 471-475, (1979) · Zbl 0415.93034
[29]	Söderström, T.; Stoica, P., System identification, (1988), Prentice-Hall Englewood-Cliffs, NJ · Zbl 0714.93056
[30]	Tamura, H.; Yoshikawa, T., Large-scale systems control and decision making, (1990), Marcel Dekker, Inc New York and Basel · Zbl 0748.93006
[31]	Tangirala, A.K.; Li, D.; Patwardhan, R.S.; Shah, S.L.; Chen, T., Ripple-free conditions for lifted multirate control systems, Automatica, 37, 10, 1637-1645, (2001) · Zbl 0995.93049
[32]	Van Overschee, P.; De Moor, B., Continuous time-frequency domain subspace system identification, Signal processing, 52, 2, 179-194, (1996) · Zbl 0875.93560
[33]	Verhaegen, M., Subspace model identification part 3analysis of the ordinary output-error state space model identification algorithm, International journal of control, 58, 3, 555-586, (1993) · Zbl 0782.93030
[34]	Verhaegen, M., Identification of the deterministic part of MIMO state space models given in innovations form from input-output data, Automatica, 30, 1, 61-71, (1994) · Zbl 0791.93054
[35]	Verhaegen, M., & Deprettere, E. (1991). A fast recursive MIMO state space model identification algorithm. Proceedings of the 1991 conference on decision and control (30th CDC), Brighton, USA (pp. 1349-1354).
[36]	Verhaegen, M.; Dewilde, P., Subspace model identification part 1the output-error state space model identification class of algorithms, International journal of control, 56, 5, 1187-1210, (1992) · Zbl 0772.93020
[37]	Verhaegen, M.; Dewilde, P., Subspace model identification part 2analysis of the elementary output-error state space model identification algorithm, International journal of control, 56, 5, 1211-1241, (1992) · Zbl 0772.93021
[38]	Yang, B., Projection approximation subspace tracking, IEEE transactions on signal processing, 43, 1, 95-107, (1995)
[39]	Yang, B., Asymptotic convergence analysis of the projection approximation subspace tracking algorithm, Signal processing (EURASIP), 50, 1, 123-136, (1996) · Zbl 0875.94068