The Dead Zone in System Identification

Abstract

A prediction error method for parameter estimation in a dynamical system is studied.

$$ \hat \vartheta = \arg {\mkern 1mu} \mathop {\min }\limits_\vartheta \mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{t = 1}^N {{\text{E}}l\left( {\varepsilon \left( {t,\vartheta } \right)} \right)} $$

where ε are the prediction errors of a linear regression. A quadratic norm l is zero within an interval [−c, c]. This kind of a dead zone (DZ) criterion is very common in robust adaptive control. The following problems are treated in this chapter:

• When is the DZ estimate inconsistent, and what is the set of parameters which minimizes the criterion in the case of inconsistency?

• What happens to the variance of the estimate as the DZ is introduced?

• Does the DZ give a better estimate than least squares (LS) when there are unmodeled deterministic disturbances present?

• What are the relations between identification with a dead zone criterion and so called set membership identification?