Assume a system with dynamics:
$\dot{\omega}(t) = \alpha \omega^2(t) + \beta i(t)$,
where $\dot{\omega}(t), \omega(t)$ are system's states and $i(t)$ is the system's input. I'd like to approximate the system's transfer function $P(s) = \frac{\omega(s)}{I(s)}$ around some operating point $\xi_0 = \{\omega_0, i_0\}$.
I assumed I could achieve this by linearizing as follows:
$\alpha_0 = \ddot{\omega}(\omega_0) = 2 \alpha \omega_0$
giving:
$\dot{\omega}(t) \approx \alpha_0 \omega(t) + \beta i(t) = 2 \alpha \omega_0 \omega(t) + \beta i(t)$,
hence the transfer function would be:
$\omega(s)(s – 2 \alpha \omega_0) = \beta I(s)$
$P(s) = \frac{\omega(s)}{I(s)} = \frac{\beta}{s – 2 \alpha \omega_0}$.
I tried to simulate step response of the system, but the effects are not as I expected. So there are two solutions: either I made a mistake programming, or the whole thought-process is wrong. Now, which one is it? And why?
Any hints appreciated.
EDIT: The real coefficient and operating point values are given below.
$\alpha = -2182.5$
$\beta = 358.8825$
$\omega_0 = 517.8056$
$i_0 = 6.0814$
EDIT2: I already figured out, what I did is actually a Taylor expansion of the function $\dot{\omega}(t)$. A Taylor expansion also includes the constant term, ie:
$\dot{\omega}(t) = 2 \alpha \omega_0 \omega(t) + \beta i(t) – 2 \alpha \omega^2_0 – \beta i_0$
As @copper.hat commented below, $I(s)$ in the transfer function $P(s)$ reflects perturbations around $i_0$. I already tried testing the transfer function's behaviour in Matlab. I defined a system using:
motor = tf(beta, [1 -2*alpha*omega_0])
To obtain correct amplitude of the step response I had to issue a command
step(motor+i_0)
Is there a way to include the constant term in the transfer function $P(s)$? What does adding $i_0$ really mean in terms of Laplace transform?
What $P(s)$ really is is a model of a BLDC motor with propeller and the non-linear term reflects aerodynamic drag ($\omega$ is shaft angular velocity and $i$ is applied current). I'd like to include the motor in a bigger, linear, model and apply PID control for the whole system. Is it possible?
Best Answer
You did some mistake when linearizing. The equation is already linear in $i(t)$. Thus, you should obtain $$\dot{\omega}(t) \approx 2 \alpha \omega_0 \omega(t) + \beta i(t).$$ The transfer function reads $$ P(s) = \frac{\beta}{s - 2 \alpha \omega_0}.$$