Im designing a PI controller for a Maglev experiement but im a little stuck on how to linearize the non linear state space model. I have developed the state equations (seen below) and i know the equilibrium point of interest below. But im stuck on the maths behind linearizing it.
Nonlinear State Space Model
Equilibrium Point
Any help is greatly appreciated
The typical steps followed to linearise the system \$\dot{x} = f(x, u)\$ is to split the state variable into two parts; a steady part (operating point) and a small-signal part. This can be done with the help of Taylor series. Only the first derivative contributes to the linearisation.
$$ x \triangleq x_0 + \delta x$$
$$ \dot{x} = f(x,u) = f(x_0, u_0) + \frac{\partial f(x,u)}{\partial x}|_{x=x_0} \delta x + \frac{\partial f(x,u)}{\partial u}|_{u=u_0} \delta u + \dots $$
\$f(x_0, u_0)\$ can be taken as zero if it is a steady operating point. (You need to check it separately)
\$\frac{\partial f}{\partial x}\$ is a \$3\times 3\$ matrix since \$f(x,u)\$ is a \$3 \times 1\$ matrix and \$x\$ is a \$3 \times 1\$ matrix.
\$\frac{\partial f}{\partial u}\$ is a \$3\times 1\$ matrix since \$f(x,u)\$ is a \$3 \times 1\$ matrix and \$u\$ is a \$1 \times 1\$ matrix.
For your system, $$ f(x,u) = \begin{bmatrix} x_2\\ \frac{-K x_3^2}{2Mx_1^2}+g\\ (-Rx_3 + u)/L \end{bmatrix} $$
The linearised equation is $$ \frac{d \delta x}{dt} = \begin{bmatrix} 0 & 1 & 0\\ \frac{K x_3^2 \times -2x_{b0}^{-3}}{2M} & 0 & \frac{-2K x_3}{2M x_{b0}^2}\\ 0 & 0 & -R/L \end{bmatrix} \begin{bmatrix} \delta x_1\\ \delta x_2\\ \delta x_3 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ -1/L \end{bmatrix} \delta u $$
Note that the linearised differential equation is in terms of new variables; small-signal riding on top of the operating point \$(x_{b0}, 0, x_3)\$.
Also note that, for variables which were already having linear relations, the equations effectively remain the same. e.g. \$\dot{\delta x_1} = \delta x_2\$
