Linearized State Space Controller for Mobile Inverted Pendulum

For this project, my team and I designed a state-space feedback controller for a simulated inverted pendulum mounted to a mobile cart, with the goal of keeping the pendulum in a vertical position by translating the cart from side to side.

To accomplish this, we first determined the nonlinear equations of motion describing the behavior of our chosen system, and converted the equations to state-space form, respecting the position, velocity and acceleration (angular and linear) of the cart. In linearizing the system, we shifted the state variables such that the equilibrium state vector was x = {0} , and took a multivariable Taylor Series Expansion of our equations, evaluated at the point x = {0}.

To escape the "perfection" of a simulation, the parameters used in controlling the cart (cart mass, and pendulum length and mass, etc.) were determined using a least squares approximation drawn from a noise-injected dynamics simulation. In this way, the parameters were estimated, and so more like the not perfectly accurate properties one would likely obtain for a physical system. With some data manipulation, the least squares data, used in the original dynamics simulations, tracked well with the declared "ground truth."

A feedback controller was selected to control the linearized system so that the team could focus on its dynamics rather than sensor noise. The state-space representation of the system was used to determine its controllability, and pole placement functions were used to generate a closed-loop system, with results as plotted in the linked report.