The dynamics of a magnetically suspended steel ball can be described by
$$ m \ddot{h} = mg – c \frac{u^2}{h^2}$$
where $m$ is the mass of the ball, $g$ is gravitational acceleration, $c$ is a positive constant, and $h$ represents the vertical position of the ball. The input $u$ is the current supplied to the electromagnet.
(a) Write down a nonlinear state space model using $x_1 = h $ and $x_2 = \dot{h}$.
(b) Determine the equilibrium control input $u_e$ that has to be applied to suspend the ball at some position $h=h_0 > 0$.
Answer:
For part (a), the system is
$$
\begin{align}
\dot{x}_1 &= x_2 \tag{1} \\
\dot{x}_2 &= g – \frac{c}{m} \frac{u^2}{x^2_1} \tag{2}
\end{align}
$$
For part (b), the equilibrium point $x_e=(h_0,0)$ which is not at the origin, so we need first to make sure the equilibrium point $x_e$ is transformed to the origin by introducing new variables: let $y=x-x_e$, we get:
$$
\begin{align}
y_1 &= x_1 – h_0 &\implies \dot{x}_1 = \dot{y}_1 \tag{3}\\
y_2 &= x_2 – 0 &\implies \dot{x}_2 = \dot{y}_2 \tag{4}
\end{align}
$$
From (1),(2),(3), and (4), the new system is
$$
\begin{align}
\dot{y}_1 &= y_2 \\
\dot{y}_2 &= g – \frac{c}{m} \frac{u^2}{(y_1 + h_0)^2}
\end{align}
$$
The equilibrium point $y_e = (\sqrt{\frac{c}{gm}} u – h_0, 0)$. The control input $u_e$ that makes $y_e$ is zero is $u_e = \frac{ h_0}{\sqrt{\frac{c}{gm}}} $, therefore, the equilibrium control input $u_e$ that has to be applied to suspend the ball at some position $h=h_0 > 0$ is
$$
u_e = \frac{ h_0}{\sqrt{\frac{c}{gm}}}
$$
Is this correct? if not, any suggestions how to tackle this problem.
Best Answer
Looks correct, but you can always check this yourself by plugging your $u_e$ into the expression for either $\dot{x}_2$ or $\dot{y}_2$. Also note that the input only appears as $u^2$, so using $-u$ should have the same effect as $u$.
Lastly it is also possible to solve for $u_e$ by setting $\dot{x}_2=0$ while using $x_1=h_0$. However, you might also have to linearize the system around this equilibrium point, so doing this coordinate translation would probably have to be done anyway in a later stage.