Parameter consistency and quadratically constrained errors-in-variables least-squares identification

Abstract

In this article, we investigate the consistency of parameter estimates obtained from least-squares identification with a quadratic parameter constraint. For generality, we consider infinite impulse-response systems with coloured input and output noise. In the case of finite data, we show that there always exists a possibly indefinite quadratic constraint depending on the noise realisation that results in a constrained optimisation problem that yields the true parameters of the system when a persistency condition is satisfied. When the noise covariance matrix is known to within a scalar multiple, we prove that solutions of the quadratically constrained least-squares (QCLs) estimator with a semidefinite constraint matrix are both unbiased and consistent in the sense that the averaged problem and limiting problem produce, respectively, unbiased and true (with probability 1) estimators. In addition, we provide numerical results that illustrate these properties of the QCLS estimator.

Keywords:

• errors-in-variables identification
• least-squares identification
• time-series model identification
• consistency
• unbiasedness
• Koopmans-Levin method

Acknowledgements

We wish to thank Spilios Fassois and Dave Bayard for lengthy discussions and numerous helpful suggestions. This research was supported in part by the National Science Foundation Information Technology Research initiative, through Grant ATM-0325332 and by the Air Force Office of Scientific Research under grants F49620-98-1-0037 and F49620-97-1-0406.