Open problems in sequential parametric estimation

Abstract

In this paper, we consider a single input-single output discrete-time system

$${{y}_{i}} = \phi _{i}^{T}\theta + {{e}_{i}},\;{\text{ }}{\mkern 1mu} i = {\mkern 1mu} 1,2,...,n$$

where y _i ∈ R is the system output, ϕ _i ∈ R ^m the measurable regressor, θ ∈ R ^m the unknown parameter vector to be identified and e _i ∈ R the output noise. This system can be re-written more compactly as

$$ y = \Phi \theta + e $$

((5.1))

were

$$ y = \left( {\begin{array}{*{20}{c}} {{y_1}} \\ {{y_2}} \\ \vdots \\ {{y_n}} \end{array}} \right),\;\Phi = \left( {\begin{array}{*{20}{c}} {\phi _1^T} \\ {\phi _2^T} \\ \vdots \\ {\phi _n^T} \end{array}} \right)\;and\;e = \left( {\begin{array}{*{20}{c}} {{e_1}} \\ {{e_2}} \\ \vdots \\ {{e_n}} \end{array}} \right). $$