Meeting and Modeling

A Brief Meeting

I sent an email off to my professor with a link to the blog earlier this week. He said it looked very good and interesting! I talked with him after class on Thursday to discuss the software I was using and the details of the project. We determined I should continue with the following steps:

1. Create a linear time-invariant (LTI) state space model of the platform
2. Design a simple state feedback control over the model
   
3. Perform pole placement on the control
   
4. Test the model against a non-linear simulation
5. Construct a discrete-time control to operate on the unit
6. Install the control on the platform
   

The State Space Model


As mentioned in an earlier post, the platform that I've created can be modelled by a cart and pendulum system. A diagram with the dynamics such that the position does not need to be controlled is

where γ is the coefficient of rotational friction, J_t = J + ml², and F is the force applied at the base. For my system, I'm going to neglect damping by setting γ = 0, break apart F into its acceleration and mass components, and assume J = ml² as in a cylindrical mass system. With these modification, the dynamics become

To convert this non-linear system to a LTI, I define the state of this system to be θ and the derivative of θ, and the actuation to be the acceleration a. I assume that θ = 0 is an equilibrium point and that the change in angle θ is small. This allows the system to be approximated as

It follows that the state space matrices should be

Now I must determine the properties of my platform to populate the variables of the state space matrices and find the poles of the system.