Problem statement
$i(t) = \frac{1}{b_0}(y(t)^{(4)} + a_3y(t)^{(3)} + a_2y(t)^{(2)} + a_1y(t)^{(1)} -i(t)^{(1)})$
$i(t) = w_1(t)u_1(t) + w_2(t)u_2(t)$
$e(t) = 0.5\times(r(t)-y(t))^2$
$w_n(t+1) = w_n(t)-\eta \frac{\partial{e}}{\partial{w}}, n=1,2$
$\eta(t+1) = \eta(t)-\alpha \frac{\partial{e}}{\partial{\eta}}$
Five equations are given above. I need to solve them to obtain the variations of weight $w(t)$ in terms of $b_i$'s and $a_i$'s,(which can be considered to be constants or system constants) and derivatives of $y(t).$ The numbers in the top of bracket in equation one symbolise different derivatives
What I did
If there were only first 4 equations and $\eta$ is a constant, then the above system is simple to solve by manipulation of partial derivatives as $\frac{\partial{e}}{\partial{w}} = \frac{\partial{e}}{\partial{y}}\frac{\partial{y}}{\partial{i}}\frac{\partial{i}}{\partial{w}}$ and then using first 3 equations to solve it.That part I was able to do easily.
But the problem here is that $\eta$ is also varying in the manner of gradient descent. How should I solve it now? Please give some hints so that I can reach to the solution
Notations $y(t)$ is the output of the system, $i(t)$ is the input to the system, $r(t)$ is the reference input, $e(t)$ is the error, $w_1$ and $w_2$ are the weights, $\alpha$ is a constant, but \eta is varying as can be seen in equation 5. Equation 4 and 5 are gradient descent equations