Section: Research Program

Identification

The behavior of the monitored continuous system is assumed to be described by a parametric model $\{{𝐏}_{\theta },\theta \in \Theta \}$, where the distribution of the observations ($Z_0,...,Z_N$) is characterized by the parameter vector $\theta \in \Theta$. An estimating function, for example of the form :

is such that ${𝐄}_{\theta }\left[{𝒦}_N\left(\theta \right)\right]=0$ for all $\theta \in \Theta$. In many situations, $𝒦$ is the gradient of a function to be minimized : squared prediction error, log-likelihood (up to a sign), .... For performing model identification on the basis of observations $(Z_0,...,Z_N)$, an estimate of the unknown parameter is then [63]  :

In many applications, such an approach must be improved in the following directions :

• Recursive estimation: the ability to compute ${\widehat{\theta }}_{N+1}$ simply from ${\widehat{\theta }}_N$;

• Adaptive estimation: the ability to track the true parameter ${\theta }^{*}$ when it is time-varying.