I'm doing an digital control exercise where I need to
Consider a self-balancing motorcycle, which can compensate deviations from equilibrium by rotating an inertial wheel. What is asked is:
"Linearize the nonlinear equation (1) around the equilibrium point θ = 0, obtain the continuous time transfer function, and study the stability of the plant."
But I really don't know how to do. I've tried to search papers but mostly don't give an step by step of how to do it.
The dynamics of the roll angle is defined by the following differential equation:
Iθ¨ = mgh sin θ + τ (1)
where mgh and I are constants (mass, gravity, height of center mass and moment of inertia) and τ it's an input torque that can be arbitrily and directly set.
I think I need to use a Taylor expansions and the derivatives but I am not certain.
If someone could show me how to solve this I would be very grateul. Thanks.
KBS' comment, really, answers your question. To be more precise, since since $\Theta=0$ is an equilibrium, I imagine that $\tau=0$. Given that, I replace $\Theta$ by $\Theta=0+v$, where $v$ is the "small" perturbation to the zero equilibrium. I get $$ v''=mgh \sin(v). $$ I expand $$ v''=mgh\,(v-v^3/(3!)+v^5/(5!)-...) $$ The linearization is obtained by keeping the terms of degree one only. In other words, by eliminating the term of degree two or higher $$ v''=mgh v. $$