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**Least-squares identification of a class of multivariable systems with correlated disturbances.**
*(English)*
Zbl 0967.93093

The paper is devoted to the problem of parameter estimation of a multivariable system, given by a certain type of model representation with known structure. The main objective of the paper is to extend the recently proposed new version of the bias-eliminated least-squares method [C.-B. Feng and the author, IEE Proc., Part D 138, No. 5, 484-492 (1991; Zbl 0753.93078); the author, “Unbiased identification of multivariable systems subject to colored noise”, Proc. 33rd IEEE Conf. on Decision and Control (CDC’94), Vol. 3, Lake Buena Vista, FL, USA, 2864-2865 (1994)] to the identification of a class of multivariable stochastic systems, namely, multi-input-single-output systems corrupted by correlated noise. It is shown that the bias in least-square estimators can be eliminated if the cross-covariance vector between the disturbance acting on the multivariable system and the vector of lagged observed input-output data can be estimated exactly. A set of digital prefilters has been designed and connected to the identified multivariable system at the multi-input channels. Certain linear equality constraints, with respect to the system parameters, have been derived, by the use of artificially inserted known zeros in the identified system. The consistent parameter estimates are obtained on the base of the established relations, in conjunction with the bias-correction principle.

The main advantage of the developed method is that there is no need to model the process noise, so the method can work well without a description of the correlated noise dynamic. A batch and recursive approach is applied in the estimation procedure presented here. It has been shown that the developed identification algorithm is straightforward, efficient and promising. The paper could be of interest to all specialists and experts involved in multivariable systems parameter identification.

The main advantage of the developed method is that there is no need to model the process noise, so the method can work well without a description of the correlated noise dynamic. A batch and recursive approach is applied in the estimation procedure presented here. It has been shown that the developed identification algorithm is straightforward, efficient and promising. The paper could be of interest to all specialists and experts involved in multivariable systems parameter identification.

Reviewer: Tzvetan Semerdjiev (Sofia)

### MSC:

93E12 | Identification in stochastic control theory |

93E24 | Least squares and related methods for stochastic control systems |

### Keywords:

least-squares method; multivariable systems; unbiased estimations; correlated noise; system identification### Citations:

Zbl 0753.93078
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\textit{W. X. Zheng}, J. Franklin Inst. 336, No. 8, 1309--1324 (1999; Zbl 0967.93093)

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### References:

[1] | Glover, K.; Willems, J. C., Parametrizations of linear dynamical systems: canonical forms and identifiability, IEEE Trans. Automat. Control, 19, 640-646 (1974) · Zbl 0296.93008 |

[2] | T. Kailath, Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980.; T. Kailath, Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980. · Zbl 0454.93001 |

[3] | N.K. Sinha, B. Kuszta, Modeling and Identification of Dynamic Systems, Van Nostrand, New York, 1983.; N.K. Sinha, B. Kuszta, Modeling and Identification of Dynamic Systems, Van Nostrand, New York, 1983. · Zbl 0365.93011 |

[4] | A.K. Shaw, P. Misra, R. Kumaresan, Optimal identification of discrete multivariable systems from noisy impulse response data, Proceedings of 30th IEEE Conference on Decision and Control (CDC’91), Vol. 2, Brighton, UK, December 1991, pp. 953-958.; A.K. Shaw, P. Misra, R. Kumaresan, Optimal identification of discrete multivariable systems from noisy impulse response data, Proceedings of 30th IEEE Conference on Decision and Control (CDC’91), Vol. 2, Brighton, UK, December 1991, pp. 953-958. |

[5] | M.H.A. Davis, R.B. Vinter, Stochastic Control and Modeling, Chapman & Hall, London, UK, 1985.; M.H.A. Davis, R.B. Vinter, Stochastic Control and Modeling, Chapman & Hall, London, UK, 1985. · Zbl 0654.93001 |

[6] | L. Ljung, System Identification: Theory for the User, Prentice-Hall, Englewood Cliffs, NJ, 1987.; L. Ljung, System Identification: Theory for the User, Prentice-Hall, Englewood Cliffs, NJ, 1987. · Zbl 0615.93004 |

[7] | J. Schoukens, R. Pintelon, Identification of Linear Systems: A Practical Guideline to Accurate Modeling, Pergamon Press, Oxford, UK, 1991.; J. Schoukens, R. Pintelon, Identification of Linear Systems: A Practical Guideline to Accurate Modeling, Pergamon Press, Oxford, UK, 1991. · Zbl 0816.93005 |

[8] | T. Söderström, P. Stoica, System Identification, Prentice-Hall, London, UK, 1989.; T. Söderström, P. Stoica, System Identification, Prentice-Hall, London, UK, 1989. · Zbl 0695.93108 |

[9] | James, P. N.; Souter, P.; Dixon, D. C., Suboptimal estimation of the parameters of discrete systems in the presence of correlated noise, Electron. Lett., 8, 411-412 (1972) |

[10] | Sagara, S.; Wada, K., On-line modified least-squares parameters estimation of linear discrete dynamic systems, Int. J. Control, 25, 329-343 (1977) · Zbl 0349.93053 |

[11] | Stoica, P.; Söderström, T., Bias correction in least-squares identification, Int. J. Control, 35, 449-457 (1982) · Zbl 0479.93070 |

[12] | Feng, C. B.; Zheng, W. X., Robust identification of stochastic linear systems with correlated noise, IEE Proc.-Control Theory Appl., 138, 484-492 (1991) · Zbl 0753.93078 |

[13] | Stoica, P.; Söderström, T.; Simonyte, V., Study of a bias-free least squares parameter estimator, IEE Proc.-Control Theory Appl., 142, 1-6 (1995) · Zbl 0816.93084 |

[14] | Zhang, Y.; Lie, T. T.; Soh, C. B., Consistent parameter estimation of systems disturbed by correlated noise, IEE Proc.-Control Theory Appl., 144, 40-44 (1997) · Zbl 0876.93098 |

[15] | W.X. Zheng, Unbiased identification of multivariable systems subject to coloured noise, Proceedings of 33rd IEEE Conference on Decision and Control (CDC’94), Vol. 3, Lake Buena Vista, FL, USA, December 1994, pp. 2864-2865.; W.X. Zheng, Unbiased identification of multivariable systems subject to coloured noise, Proceedings of 33rd IEEE Conference on Decision and Control (CDC’94), Vol. 3, Lake Buena Vista, FL, USA, December 1994, pp. 2864-2865. |

[16] | Zheng, W. X., On a least-squares based algorithm for identification of stochastic linear systems, IEEE Trans. Signal Process., 46, 1631-1638 (1998) · Zbl 1039.93065 |

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