Linearize Non-linear Equations
The nonlinear system that represents the ball balancer is obtained using the Lagrangian Method. The Lagrangian is equal to the kinetic energy minus the potential energy.
The Lagrangian equations for this 2-body system are:
From the Lagrangian equations above follow the nonlinear equations for the ball balancer system.
Where position is x1 = p and the beam angle is x2 =
q :
By substituting the equations for p’’ and
q ’’ above we can obtain the following system for x’.
Now these assumptions are implemented to obtain the linear state-space equations:
1) nominal point, p = pn
The output equation comes from the measurements of the ball position and beam angle:
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Linearize Non-linear Equations
State Variable Feedback Controller