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It is an equations of inverted pendulum. Example of controlling him with pole placement.

\text{eqns}=(M+m)x\text{''}[t]-m l \text{Sin}[\theta [t]] \theta '[t]^2+m l \text{Cos}[\theta [t]] \theta \text{''}[t]\text{==}F[t]+d[t] \text{Cos}[\theta [t]],

m x\text{''}[t] \text{Cos}[\theta [t]]+m l \theta \text{''}[t]==m g \text{Sin}[\theta [t]]+d[t];

\text{invPendulum} = \text{StateSpaceModel}[\text{eqns},\{\theta [t],\theta '[t],x[t],x'[t]\},\{F[t],d[t]\},\{\theta [t],x[t]\},t];

$\text{Eigenvalues}[\text{First}[\text{Normal}[\text{invPendulum}]]]\text{/.}\{M\to 5.6,m\to 0.53,l\to 1.7,g\to 9.8\};$

$\text{feedbackgains} = \text{StateFeedbackGains}[\text{invPendulum},\{-1+ 5I, -1-5I, -3+I, -3-I\}, \text{Method}\to \text{Ackermann}]; $

$ \text{output}=\text{OutputResponse}[\{\text{SystemsModelStateFeedbackConnect}[\text{invPendulum},\text{feedbackgains}]\{0.12,0,0,0\}\},\{0\},\{t,4\}]; $

Where {-1+ 5I, -1-5I, -3+I, -3-I} is poles. How to choose an optimal poles?

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    I changed your `align` environment into a code block, at least it is readable this way.2011-07-21

1 Answers 1

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The inverted pendulum is a nonlinear system. The pole placement method applies to linear systems. So i suggest you linearize your nonlinear differential equations and then apply pole placement. This would yield a controller that is locally acceptable (by locally i mean a neighborhood of the linearization point).