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Below is a system of linear differential equations that describe the motion of a control moment gyroscope:

$$ \begin{aligned} \dot{v}_1 &= - \left( \dfrac{5 \left(200 \tau_{3} \sin{\left(q_{2} \right)} + \sin{\left(2 q_{2} \right)} v_{1} v_{2} + 2 \cos{\left(q_{2} \right)} v_{2} v_{3}\right)}{10 \sin^{2}{\left(q_{2} \right)} - 511} \right) \\[1em] \dot{v}_2 &= \dfrac{10 \left(100 \tau_{2} - \cos{\left(q_{2} \right)} v_{1} v_{3}\right)}{11} \\[1em] \dot{v}_3 &= - \left( \dfrac{51100 \tau_{3} + 5 \sin{\left(2 q_{2} \right)} v_{2} v_{3} + 511 \cos{\left(q_{2} \right)} v_{1} v_{2}}{10 \sin^{2}{\left(q_{2} \right)} - 511} \right) \end{aligned} $$

How can I linearize this system of equations and put them in the state-space model ?

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  • $\begingroup$ Can't anything not involving $v_1,v_2,v_3$ can be replaced by a constant? Then each equation looks like something of the form $\dot{v}_2 = Av_1v_3 + B$, etc. $\endgroup$
    – David Roberts ♦
    Feb 18, 2021 at 5:32
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    $\begingroup$ Hi and welcome to the MathOverflow. Echoing and generalizing a bit the comment of @DavidRoberts, I would say that linearizing an equation usually means to see what is its behavior in the neighborhood of a (supposedly) known solution $\mathbf{v}_o=(v_{o1},v_{o2},v_{o3})$, be it constant or not. Substituting in your equation the variation of the solution $\mathbf{v}_\varepsilon=\mathbf{v}_o + \varepsilon \mathbf{v}$ should give you some insight on what you are searching for. $\endgroup$
    – Daniele Tampieri
    Feb 18, 2021 at 6:45

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