# Day 10: Control Systems – Part 1

I have been watching Brian Douglas’ playlist on control systems, and this is what I have learnt today. Control systems is a mechanism to alter the future state of a system, while control theory is a strategy to select the appropriate input for the system.

There are open-loop systems; as an example, let’s take a dishwasher. Now if we input the time for the dishwasher to clean the dishes, say ten minutes, it will stop exactly when ten minutes is over even if the dishes don’t seem clean. When we use closed-loop systems, like the same dishwasher example, we can add a sensor that checks how clean the dish is. Additionally, we can give the reference of a clean dish for the dishwasher to refer to. Now a comparison occurs between the reference dish and the dish that’s being cleaned currently. Depending on the error that’s been evaluated, the input varies, maybe increasing the time in the plant (dishwasher).

Fourier transforms can represent any signal, repeating or not, into an infinite summation of sinusoids. A transform is a mapping between two domains, in this case time and frequency. While predicting a system's response to a jagged, complex input is incredibly hard, predicting its response to a simple sine wave is easy. And if we can break our complex input into a bunch of simple sine waves, we've just turned a hard problem into a series of easy ones.

The LTI (linear time invariant) system which represents most systems we want to control has a property that works well with the broken-up sinusoids. These systems don’t change their fundamental frequency while being able to change the amplitude and its phase. There is a lot of math that I haven’t explained in the blog, but basically f(t) is a summation of scaled amplitudes and phases and a bunch of sinusoids.

In our last section, we saw that the Fourier Transform is a fantastic tool. But it has one major limitation: it struggles with signals that grow infinitely over time, like the response of an unstable system. This is where the Laplace transform comes in.

Laplace improves on Fourier by multiplying the time-domain signal f(t) with a damping factor (e^(-σt)). This allows it to handle the unstable or growing signals that Fourier can't. In controls, its primary purpose is to convert a system's impulse response h(t) into its algebraic transfer function H(s). We then find the poles and zeros of this transfer function. The location of the poles is critical; they directly tell us about the system's fundamental characteristics, like its stability, speed, and whether it will oscillate

Well, I’ve tried my best in explaining what I could understand from the videos in words, but yes, the math itself I could not really talk about much. That’s a wrap for day 10.