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In contrast to the frequency-___domain analysis of the classical control theory, modern control theory utilizes the time-___domain [[state space (controls)|state space]] representation,{{citation needed|date=December 2022}} a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. To abstract from the number of inputs, outputs, and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the latter only being possible when the dynamical system is linear). The state space representation (also known as the "time-___domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the information about a system. Unlike the frequency ___domain approach, the use of the state-space representation is not limited to systems with linear components and zero initial conditions. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a point within that space.<ref>{{cite book|title=State space & linear systems|series=Schaum's outline series |publisher=McGraw Hill|author=Donald M Wiberg|year=1971 |isbn=978-0-07-070096-3}}</ref><ref>{{cite journal|author=Terrell, William|title=Some fundamental control theory I: Controllability, observability, and duality —AND— Some fundamental control Theory II: Feedback linearization of single input nonlinear systems|journal=American Mathematical Monthly|volume=106|issue=9|year=1999|pages=705–719 and 812–828|url=http://www.maa.org/programs/maa-awards/writing-awards/some-fundamental-control-theory-i-controllability-observability-and-duality-and-some-fundamental|doi=10.2307/2589614|jstor=2589614}}</ref>
==System interfacing
Control systems can be divided into different categories depending on the number of inputs and outputs.
* [[Single-input single-output system|Single-input single-output]] (SISO) – This is the simplest and most common type, in which one output is controlled by one control signal. Examples are the cruise control example above, or an [[audio system]], in which the control input is the input audio signal and the output is the sound waves from the speaker.
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