I have following dynamic system
$$ \frac{\mathrm{d}v_C}{\mathrm{d}t} = -\frac{1}{R_b\cdot C}\cdot v_C\cdot\alpha, $$
where $v_C$ is the system state and output, $\alpha$ is the system input and $R_b, C$ are the system parameters. Due to the fact that the differential equation contains product of the state and input variable the system is nonlinear. I have linearized this model in a steady state operating point $(v_{C_{0_e}}, \alpha_{0_e})$.
$$ \frac{\mathrm{d}\widetilde{v_C}}{\mathrm{d}t} = -\frac{1}{R_b\cdot C}\left(v_{C_{0_e}}\cdot\widetilde{\alpha} + \widetilde{v_C}\cdot\alpha_{0_e}\right), $$
where $\widetilde{\alpha}$ and $\widetilde{v_{C}}$ are the perturbations from the steady state operating point. I was curious about difference between both of the models. So I have prepared a simulation in Scilab (free of charge variant of Matlab) with following values $R_b = 1.9, C = 2700\cdot 10^{-6}, v_{C_{0_e}} = 800, \alpha_{0_e} = 0.5$ and $\mathrm{d\_alpha} \sim \widetilde{\alpha}, \mathrm{d\_vC} \sim \widetilde{v_{C}}$.
I have done two runs of the simulation (black trace is the output/state of the nonlinear system, green trace is the output/state of the linearized system)
- $\widetilde{\alpha} = 0.1$
- $\widetilde{\alpha} = 0.2$
The results I have received were surprising for me. Because based on them it seems that increase in the deviation from the steady state operating point used for linearization results in increase of the interval where the linearized model follows the nonlinear one. I would say that I have done a mistake somewhere. May I ask you for help?
I will introduce an additional substitution with
$$\alpha(t) = -R_bC\dfrac{\dot{u}(t)}{u(t)}$$
As mentioned in the comments the system is a linear ODE.
$$\dfrac{dv_C}{v_C}=\dfrac{\dot{u}(t)}{u(t)}dt \implies \dfrac{dv_C}{v_C}=\dfrac{du(t)}{u(t)}$$ $$\ln v_C(t)-\ln v_C(t_0)=\ln u(t) - \ln u(t_0)$$ $$v_C(t) = \dfrac{v_C(t_0)}{u(t_0)}u(t)$$
As soon as you fix the desired trajectory for $v_C(t)$ you can determine $u(t)$. Using $\alpha(t) = -R_bC\dfrac{\dot{u}(t)}{u(t)}$ will give you ideal solution to the problem. But as we all know this will not work in real life due to imperfect dynamics or inaccurate parameter values. But I enjoyed the process of solving the problem :D.