Linearization

This experiment demonstrates the process of linearizing a nonlinear system. This is often done for nonlinear control systems because there are a wide variety of well-known and well-understood algorithms for designing controllers for linear systems. So if we can linearize a nonlinear system, then we can design a control system using a wide variety of methods. The caveat is that the system must remain "near" its linearization point, or else the linearization is not valid.

This web page shows the results of the simulation of the classic inverted pendulum (as described in many books, including page 33 of John Bay's book). In the below simulation you can choose the initial pendulum angle, click on the "Run Simulation" button, and then view the results of the nonlinear simulation and the linearized simulation. Notice that as the initial angle gets smaller, the two simulations approximate each other more closely. As the inital angle gets larger, the two simulations drift farther apart. This is because the linear simulation assumes that the angle is "small" and so the the linear simulation becomes less accurate as the angle increases.

Simulation Details: The spherical mass at the end of the pendulum m = 0.2, its radius r = 0.02, the mass of the cart M = 1, acceleration due to gravity g = 9.81, the coefficient of viscous friction on the cart wheels F = 0.1, the length of the pendulum L = 1. This is a nonlinear system, but it can be linearized if we assume that the angle of the pendulum is "small." Note that the control input used for this simulation is u = -k*x, where k was chosen to place the closed loop poles at -1+j, -1-j, -1.5+0.5j, and -1.5-0.5j. This regulates the pendulum angle and the cart position to zero.