### PID Compensator: Compensating Second-order Systems

In the previous article, I showed how to compensate simple first-order systems. As a quick recap, a Proportional-Integral (PI) controller is all that is needed for compensating first-order systems perfectly. By perfectly, I mean that you can theoretically achieve any combination of crossover frequencies and phase margins. There is, of course, an entertaining mathematical proof for this, but it is not needed for the purposes of implementation.

### The P, I, and PI in PID Controllers

In my previous column, I showed bode plots for different types of transfer functions. I also touched on the basic principles of closed loop systems. As a refresher, the gain crossover frequency dictates the speed of the controller, and ensuring a phase margin of around 70 degrees achieves a nice trade-off between the overshoot and speed. Now I’d like to look at how different compensators affect the characteristics of the plant. In the course of this discussion though, we will see that not all systems need a full-blown proportional-integral-derivative compensator (PID compensator). Most of the time, a PI compensator or even a simple integral controller can do the job.

### Mixing Signals, Part 2: What Do All the Squiggly Lines Mean?

Before diving into proportional-integral-derivative (PID) design, an important factor to understand is how to interpret frequency plots as a time response. Having a rough idea of some key characteristics like ringing and amount of overshoot will…

### Mixing Signals, Part 1: An Introduction to PID Controllers

Since the early 1900s, PID compensators have been one of the most widely used closed loop controllers in industrial applications. However, the tuning of such controllers is widely considered a difficult art. Talking with anyone who is familiar with tuning often quickly leads to a very distressing discussion about poles, zeros, and margins which makes the casually inclined engineer either stop or, if he’s feeling brave because a deadline is closing in, tune the loop intuitively.