AN UNBIASED EQUATION ERROR IDENTIFIER AND REDUCED-ORDER APPROXIMATIONS

The equation error 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'' con straint; the asymptotic solution replaces a normal equation with an ei genequation. The resulting algorithm is simpler than previous schemes which have attempted to remove the bias problem, while at the same tim e preserving the desirable properties of the conventional equation err or method: simplicity of an on-line algorithm, unimodality of the perf ormance surface, and consistent identification in the sufficient-order case. In the more realistic undermodeled case, a robustness result sh ows that the mean optimal parameter values of both the monic and unit- norm equation error schemes correspond to a stable transfer function f or all degrees of undermodeling, and for all stationary output disturb ances, provided the input sequence satisfies an autoregressive constra int; otherwise an unstable model may result. Model approximation prope rties 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 capabl e of autocorrelation matching properties as well, and yields the optim al solution to a first and second-order interpolation problem. Finally , the mismodeling error for the undermodeled case is shown to be a wel l-behaved function of the Hankel singular values of the unknown system . This modification finally allows equation error methods to be admitt ed to the class of unbiased identification and approximation technique s.