Controls has become one of the biggest (and most important) areas in electrical engineering. However, many times when implementing a controls solution, we need to work with linear systems. The majority of real life systems are not linear! So how do we fix this? With linearization. In this article I’ll walk you through an example of how to linearize a system around a certain state vector and control input!
Example control system
Okay so let’s say your system (ie, the thing you want to control) is modeled by the following differential equation (y is the output, u is the input):
Say we want to linearize around the state vector x = 0, and the input u = 0. We’re going to need to represent this system in state space form, so let’s do it.
That wasn’t too bad, state space is great. Now we can see it clearly isn’t linear, there is a squared term~~! So the first step to linearization requires us to find the Jacobian matrices. Let’s do it.
Similarly, we want to take partial derivatives for the input, in control systems, we call this the B matrix:
Standard state space representation tells us now, that we basically have linearized our system! You can see this clearly by representing the system in state space form.
Realizing the vectors are just representations for x1 and x2, we now have the following system of equations:
Clearly a linear system. So in summation, to linearize a non linear system, it is quite simple. Represent the system in state space form. Then find the jacobian matrices, and evaluate around the point(s) you want to linearize around. Then you are done! Later (read: when I don’t have interviews), I will show how we can use this to model a maglev system.