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{{Being merged|spacetype=|Control theory|Talk:Control theory#Proposed merge of Classical control theory into Control theory|section=|inactive=|date=March 2023|nocat=|dir=to}}
{{Multiple issues|{{more citations needed|date=May 2016}}{{expert needed|1=Engineering|date=May 2016|reason=Need more sources and attention of experts in field for information verification.}}{{one source|date=May 2016}}}}
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The usual objective of control theory is to control a system, often called the ''[[Plant (control theory)|plant]]'', so its output follows a desired control signal, called the ''[[reference]]'', which may be a fixed or changing value. To do this a ''[[Controller (control theory)|controller]]'' is designed, which monitors the output and compares it with the reference. The difference between actual and desired output, called the ''error'' signal, is applied as [[feedback]] to the input of the system, to bring the actual output closer to the reference.
Classical control theory deals with [[linear time-invariant system|linear time-invariant]] (LTI) [[single-input single-output]] (SISO) systems.<ref>{{cite book|last1=Zhong|first1=Wan-Xie|title=Duality System in Applied Mechanics and Optimal Control|url=https://archive.org/details/dualitysystemapp00zhon_389|url-access=limited|date=2004|publisher=Kluwer|isbn=978-1-4020-7880-4|page=[https://archive.org/details/dualitysystemapp00zhon_389/page/n295 283]|quote=The classical controller design methodology is iterative, and is effective for single-input, single-output linear time-invariant system analysis and design.}}</ref> The Laplace transform of the input and output signal of such systems can be calculated. The [[transfer function]] relates the Laplace transform of the input and the output.
==Feedback==
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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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