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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

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Equilibrium Point enter image description here

Any help is greatly appreciated

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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\$

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