Consider the nonlinear system

$x^{}=f(x)+g(x)u,$

where the state $x$ and the input $u$ are available for measurement and $f(x),g(x)$

are smooth but unknown functions of $x.$ In addition, it is known that $g(x)>0$

AAx. We want to estimate the unknown functions $f,g$ online using neural network

approximation techniques. It is known that there exist constant parameters $W_f^{**},W_g^{**},$

referred to as weights, such that

$f(x)$~~${\displaystyle \underset{i=1}{\overset{m}{}}}{W}_{fi}^{**}{}_{fi}(x),$

$g(x)$~~${\displaystyle \underset{i=1}{\overset{n}{}}}{W}_{gi}^{**}{}_{gi}(x),$

where ${}_{fi}(*),{}_{gi}(*)$ are some basis functions that are known and $n,m$ are known

integers representing the number of nodes of the neural network. Obtain a parame$-$

terization of the system in the form of SPM that can be used to identify the weights

$W_{fi}^{**},W_{gi}^{**}$ online.