# Second Order Systems

concept

Second order systems will be the main focus of our analyses in Control theory.
They're complex enough to exhibit several interesting traits of real control systems (which are often higher order) but are simple enough that we can create and use quite a few math tools to make using them so much simpler.
Even when dealing with higher order systems we'll often pretend that they're a second order system and keep in mind that we'll be just a little bit wrong.
In this topic we'll cover what a second order system is, what it looks like, and a few of its properties.
In the next topic we'll cover analysing second order systems like we did with first order systems before.

fact

A second order system has the transfer function:
$$ H(s) = \frac{\omega_n}{s^2 + 2 \zeta \omega_n + \omega_n^2} $$
Where \(\omega_n\) is called the natural frequency and \(\zeta\) is the damping ratio (both defined below)

So a second order system can be completely described with just two numbers, the natural frequency \(\omega_n\) and the damping ratio \(\zeta\).
The information in this topic is probably something you've seen before (or at least most of it) when learning differential equations.
Here the natural frequency and damping ratio mean the same as they did in first year physics classes on oscillations.
Simply put the natural frequency is the frequency the system would operate at without any friction and the like, while the damping ratio tells you something about how much "friction" is present.
Having a large damping ratio tends to create a sluggish system but prevents a lot of oscillating back and forth (called "ringing").

fact

The damping ratio \(\zeta\) is divided into three categories:

- \(\zeta = 1\): critically damped
- \(\zeta \gt 1\): overdamped
- \(0 \lt \zeta \lt 1\): underdamped

Whether the system is critically damped, overdamped or underdamped determines the general shape of the system's output when given a step input.
As such classifying systems by their damping ratio proves very useful and makes communicating about a system much quicker and simpler.

When underdamped the system's closed loop poles are complex conjugates and the transient response is oscillatory, but will die down over time.
If \(\zeta = 0\) the oscillations will not die down but in fact continue forever like a sine wave.

When overdamped the system exhibits no oscillations whatsoever, the output climbs to its steady state value but never reaches it. Overdamped systems are slower than critically damped systems but have the benefit of never overshooting their mark (this might be useful for something like a cruise control system where the user doesn't want to unintentionally speed).

A critically damped system also has no oscillations but it reaches its final value faster than any overdamped system. However if you attempt to design a critically damped system you may find that component tolerances leave you with an underdamped system which does oscillate, this may or may not be satisfactory for your system.

fact

If a second order system is given in the general form:
$$ H(s) = \frac{K}{Js^2 + Bs + K} $$
Then \(\frac{K}{J} = \omega_n^2\) and \(\zeta = \frac{B}{B_c}\) where \(B_c = 2 \sqrt{JK}\) is called the "critical damping coefficient"

practice problems