Parameter estimation is a discipline that provides tools for the efficient use of data for aiding in mathematically modeling of phenomena and the estimation of constants appearing in these models [2]. It can thus be visualized as a study of inverse problems. Much of parameter estimation can be related to four optimization problems:

**criterion:**- the choice of the best function to optimize (minimize or maximize)
**estimation:**- the optimization of the chosen function
**design:**- optimal design to obtain the best parameter estimates
**modeling:**- the determination of the mathematical model
which best describes the system from which data are measured.

Let be the (state/parameter) vector containing the parameters to be
estimated. The dimension of , say *m*, is the number of parameters to
be estimated. Let be the (measurement) vector which is the
output of the system to be modeled. The system is described by a vector
function which relates to such that

In practice, observed measurements are only available for the system output corrupted with noise , i.e.,

We usually make a number of measurements for the system, say ( ), and we want to estimate using . As the data are noisy, the function is not valid anymore. In this case, we write down a function

which is to be optimized (without loss of generality, we will minimize
the function). This function is usually called *the cost function*
or *the objective function*.

If there are no constraints on and the function has first and second partial derivatives everywhere, necessary conditions for a minimum are

and

By the last, we mean that the -matrix is positive definite.

Thu Feb 8 11:42:20 MET 1996