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## Homework Statement

Inverted pendulum on a cart on a frictionless surface, write the state-space description of a nonlinear system. outputs are theta(t), theta'(t), y(t), y'(t). Everything is 2-dimensional.

## Homework Equations

theta(t) is the angle between the gravity vector and pendulum

y(t) is the horizontal position of the cart.

These were given:

sum of the forces in the horizontal direction:

[tex]

\ddot{\theta} = \frac{(M+m)gsin(\theta) - cos(\theta)[u + ml \dot{\theta}^2sin(\theta)}{\frac{4}{3}(m+M)l-mlcos(\theta)^2}

[/tex]

sum of the torques about the pivot point:

[tex]

\ddot{y} = \frac{u + ml[\dot{\theta}^2sin(\theta) - \ddot{\theta}cos(\theta)}{m+M}

[/tex]

If it matters,

M = mass of cart

m = mass of pendulum

l = arm length of pendulum

u(t) = input force applied to cart

## The Attempt at a Solution

I have only done linear systems in state-space description with nice simplified answers and the A matrix was full of constants....not sure what to do in the non-linear case. I did think about linearizing it, but the next problem tells me to linearize it and then compare with the result of this problem....so I'm going to have to do it legit.

I got far enough to define the state-variables...but I have no clue what to do about all the sines and cosines in the equations, or how to properly form the state-space matrix.

Anyone care to hold my hand for a bit?