Does this article’s appendix contain an error (is the heating system described integral control)

Controlling temperature (it depends upon your medium) isn't terribly hard. That was my first project when I started. Pardon me, if I repeat things you already know.

I assume you already have a way of controlling the system (ie, a heater or cooler unit), and a way of getting feedback from the system (a temperature sensor like a thermistor or something). You'll need both to implement a PID loop, which is a type of closed loop control. All you really need to do after that is write a bit of software to send control commands, read feedback, and make decisions upon that feedback.

I'd start out by reading PID without a PhD. It's the article I used when I first had to regulate temperature in a science experiment. It provides some easy-to-understand pictures, and nice sample code (a basic loop that you can tweak only needs 30 lines) that explains how to control your 'plant' - in this case, the thing you want to control the temperature of.

The gist of PID - Proportional-Integral-Differential - control is to use instantaneous, past, and predicted future performance (respectively) of the system to determine how to control a system at a given point in time to reach a specified set point. In many cases, you'll have to tune the algorithm's gain factors to get the desired performance you need - how quickly the temperature will rise, how much you want to avoid overshoot, etc. You might even find you don't need the differential or even integral control to get where you want to be!

You've entered into the wide and deep field of control theory. Here, Matlab with its comprehensive documentation will be your best guide. Matlab's help is a comprehensive compilation of many many textbooks. Where its help does not suffice, you can take a look at the references used to write the Help section you're interested in.

An impulse response is a unique way of describing a linear system. This means that two LTI systems with an identical impulse response can be judged to be mathematically identical - even if one is an automobile shock absorbeer and the other an electronic filter!

The impulse response is a useful way of testing the parameters of the system because it contains a wide range of frequencies. An ideal impulse contains ALL the frequencies from zero to infinity. In contrast, a sine wave has only a single frequency, so it's a very bad way of identifying a system. If your system is linear and time-invariant, a sinewave input will produce a sinewave on the output.

The problem you're describing is of a dual nature: you first identify a system. The knowledge of its parameters allows you to adjust the parameters of your controller. How do we work out these parameters? I recommend reading about System Identification in the Matlab Documentation. There are many textbooks on how to tackle this, but you're going to end up using a tool such as Matlab to learn this, anyway.

How do you adjust those control parameters? It's simpler than identifying a system, but there is a myriad of methods. There is no single way of doing this. There are dozens of controller topologies and many different ways of calculating the set of parameters that give you your desired response. For more information, read on about pole placement. There are many textbooks describing this. One of the best resources on this is the Matlab Control Toolbox Documentation.

For a simple SISO system you can derive the equations by hand. This tutorial explains more using a relatively simple example.

I hope this explains something.