Observability and Mathematics

7 Observability and matrix Lie groups

Let us consider now the matrix Lie groups from a ${\mathit{W}}_{\mathit{n}}$-observer point of view.

We remind first the classic definitions of groups – see the book [9], and for Lie groups in particular, see the book [2].

We call a set G a group, if for any two elements $\mathit{a}\mathrm{,}\mathit{b}\in \mathit{G}$ there is a third element $\mathit{d}\in \mathit{G}$, and we call this third element the product of first two and write

$\begin{array}{c}\mathit{a}\ast \mathit{b}=\mathit{d}\mathrm{,}\end{array}$

if the following conditions are satisfied:

1.

(Associativity)

$\begin{array}{c}\mathrm{(}\mathit{a}\ast \mathit{b}\mathrm{)}\ast \mathit{c}=\mathit{a}\ast \mathrm{(}\mathit{b}\ast \mathit{c}\mathrm{)}\end{array}$

for any $\mathit{a}\mathrm{,}\mathit{b}\mathrm{,}\mathit{c}\in \mathit{G}$.

2.

(Existence of a left unit e)

$\begin{array}{c}\mathit{e}\ast \mathit{a}=\mathit{a}\end{array}$

for any $\mathit{a}\in \mathit{G}$.

3.

(Existence of a left inverse element)

$\begin{array}{c}{\mathit{a}}^{-1}\ast \mathit{a}=\mathit{e}\end{array}$

for any $\mathit{a}\in \mathit{G}$. Notation ${\mathit{a}}^{-1}$ means a left inverse element of a.

Let us note that the group associativity property permits us to write ...