Linearization.pdf

Dynamics:

where

is the input, and

is the output

Define states:

Dynamics using states:

Suppose our system is at the operating point satisfies the dynamics at that point, that is

. Naturally, it

Let us perturb our system from its operating point. Then:

From Eq. (1):

Substitute for

and

from (4) and (5), respectively:

From Eq. (2):

Substitute for

,

and

from (5), (4), and 6, respectively

Small angle approximation:

Substitute dynamics from (3):

The linearized dynamics around the operating point as follows:

Define the state vector:

Then:

is given

Now, let’s discuss a short-cut to linearization… Dynamics in state space form:

Dynamics of the unforced system:

Equilibrium point: The state for which the unforced system is at rest That is, the state for which when That is, the state for which the unforced system has zero dynamics