State-space description, nonlinear system, inverted pendulum

[Number2Pencil]

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:
$$\ddot{\theta} = \frac{(M+m)gsin(\theta) - cos(\theta)[u + ml \dot{\theta}^2sin(\theta)}{\frac{4}{3}(m+M)l-mlcos(\theta)^2}$$sum of the torques about the pivot point:
$$\ddot{y} = \frac{u + ml[\dot{\theta}^2sin(\theta) - \ddot{\theta}cos(\theta)}{m+M}$$

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?

 

[Number2Pencil]

Hopefully someone here knows how to "write a state description of the nonlinear system." I did keep moving forward, and simulated the system in MATLAB SIMULINK by just creating a diagram of the given equations.

My new question is, do these plots make sense in a frictionless environment? I'll attach the plots, but basically the position keeps teetering back and forth without actually making any total displacement, and the pendulum keeps swinging perpetually, always reaching a the starting vertical direction (IE, if I set the initial condition at pi/4, it will swing to 7pi/4, then swing in the opposite direction back to pi/4.

I stink at thinking about things without friction.

Does this make sense?