In modeling physical systems, it often happens that a system of nonlinear differential equations is obtained. Since all of the methods studies in this course apply only to linear systems, we must obtain a linear approximation to our nonlinear system. This linearization process starts by finding a certain nominal trajectory and then rewriting the equations in terms of variables that describe the deviation from this nominal trajectory.

Let us first give an example of a nonlinear system.

Example: Pendulum

The figure below shows the pendulum system. A mass m is attached to a cable of length L. The angular position of the mass measured from the vertical axis is . The positive direction for the angular position is counterclockwise. We will assume that there is no external force or air resistance on the system. This means that the mass will oscillate freely forever about the vertical axis.

Click here to see the undamped pendulum system animation

We first obtain the differential equation that governs the dynamics of the pendulum system. The free-body-diagram that shows all the forces on the mass is drawn below.

We will write newton’s second law in the tangential direction.

Note that the above differential equation is nonlinear. We choose the state variables as follows (z is usually used for nonlinear systems and x for linear systems):

The state equation can be written as

This (nonlinear) state equation cannot written in the linear form

In general, the state equation for a nonlinear system with n state variables z₁, z₂, ... z_n and input v (we use v for nonlinear systems and u for linear systems) can be written as

Equivalently,

Assume that we have a nominal solution that satisfies

Define , the deviation of the state from its nominal value, and , the deviation of the input from its nominal value. We can write

In order to obtain an approximation for our system, we express dz/ dt as its Taylor series expansion about the nominal trajectory as follows:

H.O.T. denotes higher order terms which will be ignored. We have

This can be rewritten in the form of a linear differential equation as follows:

We now summarize how to perform the linearization.

Linearization: Given a nonlinear system

Choose a nominal state vector z₀ and a nominal input v_0. We usually choose the nominal state vector and input at the equilibrium or steady state condition.

Linearization of fthe system around the nominal point gives

where, (deviation from the nominal state vector), (deviation from the nominal input),

Note that matrix A is the Jacobian of  f.

Example: Linearize the pendulum system given above.

We will linearize the system around the equilibrium point where the two state variables are equal to zero:

Note that there is no input in this example.