Overview

The parameter estimation process consists of finding values of unknown model parameters ? in an assumed model structure, based on noisy measurements z. An estimator is a function of the random variable z that produces an estimate of the unknown parameters ?. Since the estimator computes based on noisy measurements z, is a random variable.

Parameter estimation requires specification of the following:

1. A model structure with unknown parameters ? to be estimated;

2. Observations, or measurements, z;

3. A mathematical model for the measurement process;

4. Assumptions about the uncertainty in the model parameters ? and the measurement noise v.

In Chapter 2, a distinction was made between linear and nonlinear dynamic systems, based on the relation between state time derivatives and the state and control variables. For parameter estimation, however, the relation between the measured outputs and model parameters is of much greater importance.

A model is called linear in the parameters if the output y is given by

where the matrix H is assumed to be known. Then the measurement equation can be expressed as

A model that is nonlinear in the parameters has a measurement equation of the form

where the form of the function h( ?) is assumed to be known.

In general, there are n ₀ measured outputs, and a vector of measurements is taken at each sample i, where i=1, 2, ..., N, and