An Introduction to System Modeling and Control

3Differential Equations and Stability

In later chapters we will derive models of motors, carts, inverted pendulums, etc. using first principles of physics. The model of each of these systems will turn out to be a differential equation. To later deal with these models we use this chapter to see how differential equations are solved using Laplace transforms. We will next look at the stability of these differential equation system models. For example, keeping an inverted pendulum upright is the problem of making sure the closed‐loop differential equation describing the inverted pendulum is stable.

3.1 Differential Equations

Consider the differential equation

(3.1)

Recall that if then It then follows that

We next take the Laplace transform of both sides of (3.1) to obtain

or

Collecting terms this becomes

or finally

(3.2)

The zero input response is the inverse Laplace transform ...