In this lecture we give another example on linearization. The example is an advanced control application that demonstrates the idea of linearization both mathematically and physically.


Example: Electromagnetic Levitation

Consider the following magnetic levitation circuit consisting of an electromagnetic with a mass directly under it. In this application the controlled variable is the mass position y, and the input to the system is the voltage v .

Click here to see a picture of the electromagnet and ball
Click here to see a movie of the electromagnet and ball

We first obtain a mathematical model of the above system. The force exerted on the mass by the electromagnetic is

Our system dynamical equations are:

We next define the three state variables

and obtain a state a space representation for the system. Since we have a second order differential equation in y, we choose y and its derivative as state variables. Since we also have a first order differential equation in i, we to choose i as a third state variable.

Now, our state equations are:

We would like to write this in a linear state space form, but we cannot because of the presence of the nonlinearities. Therefore, we proceed by finding a linear approximation to this system. The first step in this process is to define a nominal trajectory for the system. In this case, we choose a nominal trajectory based on a steady-state condition. Consider the Case where v and y are constants, and denote their values by and . Our nominal conditions become:

Our condition for equilibrium: the weight of the ball is equal to the magnetic force. Written in terms of our state variables we have

In the steady state case, we assume that the system input is also a constant. Therefore, we have

To arrive at a linearization model, we calculate the matrices A and B :