Accessibility setting

Section: Scientific Foundations

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 [32]  :

Assuming that ${\theta }^{*}$ is the true parameter value, and that ${𝐄}_{{\theta }^{*}}\left[{𝒦}_N\left(\theta \right)\right]=0$ if and only if $\theta ={\theta }^{*}$ with ${\theta }^{*}$ fixed (identifiability condition), then ${\widehat{\theta }}_N$ converges towards ${\theta }^{*}$. Thanks to the central limit theorem, the vector ${𝒦}_N\left({\theta }^{*}\right)$ is asymptotically Gaussian with zero mean, with covariance matrix $\Sigma$ which can be either computed or estimated. If, additionally, the matrix ${𝒥}_N=-{𝐄}_{{\theta }^{*}}\left[{𝒦}_N^{\text{'}}\left({\theta }^{*}\right)\right]$ is invertible, then using a Taylor expansion and the constraint ${𝒦}_N\left({\widehat{\theta }}_N\right)=0$, the asymptotic normality of the estimate is obtained :

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.