I would like to understand if it is possible to demonstrate the stability of this closed-loop nonlinear system
$$a b -K_1 u = a K_2 \dot{a}$$
where $a$ is the variable I am trying to control, $u$ is the control law, and the nonlinearity is represented by the fact that $b=f(a)$, and by the multiplication of $a$ with $\dot{a}$ on the right hand side of the equation.
The control law I am using is
$$ u= \frac{[ k_p(a^*-a) + k_i\psi+b] a}{K_1} \\ \dot{\psi}=a^*-a$$
where $k_p$ and $k_i$ are the coefficients of a proportional-integral controller.
I am not a control expert, so I apologise in advance for any imprecision or mistake I might have made in formulating the question.
The proposed control law makes the resulting dynamics linear. Namely, when substituting the control law in the dynamics one can factor out $a$ which gives
$$ \dot{a} = -\frac{k_p}{K_2}(a^*-a) - \frac{k_i}{K_2} \psi. $$
By using that $\dot{\psi}=a^*-a$ and that $a^*$ is constant, it follows that the second derivative of $\psi$ with respect to time would be $\ddot{\psi}=-\dot{a}$. Substituting the expression for $\dot{a}$ in yields
$$ \ddot{\psi} = \frac{k_p}{K_2} \dot{\psi} + \frac{k_i}{K_2} \psi, $$
which is linear and can be shown to be stable if $\frac{k_p}{K_2},\frac{k_i}{K_2}<0$.