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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. – t.b. Jul 21 '11 at 14:15
up vote 2 down vote accepted

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).

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