An unbiased equation error identifier and reduced-order approximations

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Abstract
The equation error (EE) identification technique is modified to remove the parameter bias problem induced by uncorrelated measurement errors. The modification replaces a “monic” constraint with a “unit-norm” constraint; the asymptotic solution replaces a normal equation with an eigenequation. The resulting algorithm is simpler than previous schemes, while at the same time preserving the desirable properties of the conventional EE method: simplicity of an on-line algorithm, unimodality of the performance surface, and consistent identification in the sufficient-order case. In the more realistic undermodeled case, a robustness result shows that the mean optimal parameter values of both the monic and unit-norm EE schemes correspond to a stable transfer function for all degrees of undermodeling, and for all stationary output disturbances, provided the input sequence satisfies an autoregressive constraint; otherwise an unstable model may result. Model approximation properties for the undermodeled case are exposed in detail for the case of autoregressive inputs; although both the monic and unit-norm variants provide Pade approximation properties, the unit-norm version is capable of autocorrelation matching properties as as well, and yields the optimal solution to a first- and second-order interpolation problem. Finally, the mismodeling error for the undermodeled case is shown to be a well-behaved function of the Hankel singular values of the unknown system. This modification allows EE methods to be admitted to the class of unbiased identification and approximation techniques