Noise – Relation Between Bandwidth, Response Time, and Disturbance Rejection in Control Systems

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.

Some comments to your question:

(1) A frequency response analysis can describe the systems behaviour only if the system is LINEAR (which is not the case for the thermostat system).

(2) For linear systems, it is best to open the feedback loop (however, for finding the conditions and the correct place for the opening requires some deeper knowledge). Then, you can analyze the loop gain response and apply one of the stability crireria (Nyquist, Bode). This method allows to determine the degree (quality) of stability (phase and/or gain margin)

(3) There is a rough method to evaluate the closed-system response (stable yes/no): The magnitude will exhibit a peaking in the "critical" frequency region where the system could oscillate. In this case, the SLOPE of the phase function at the peaking frequency must be NEGATIVE for a stable system (and will be positive in case of instability).

(4) A circuit involving thermostat devices should be analyzed in the time domain only.