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==Classical vs modern==
A Physical system can be modeled in the "[[time ___domain]]", where the response of a given system is a function of the various inputs, the previous system values, and time. As time progresses, the state of the system and its response change. However, time-___domain models for systems are frequently modeled using high-order differential equations which can become impossibly difficult for humans to solve and some of which can even become impossible for modern computer systems to solve efficiently.
To counteract this problem, classical control theory uses the [[Laplace transform]] to change an Ordinary Differential Equation (ODE) in the time ___domain into a regular algebraic polynomial in the frequency ___domain. Once a given system has been converted into the frequency ___domain it can be manipulated with greater ease.
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and get <math>H(s)=1</math> identically.
For practical PID controllers, a pure differentiator is neither physically realisable nor desirable<ref>Ang, K.H., Chong, G.C.Y., and Li, Y. (2005). [
==Tools==
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