Editing Time-invariant system (section)

{{Short description|Dynamical system whose system function is not directly dependent on time}}
{{More citations needed|date=May 2018}}
[[File:Time invariance block diagram for a SISO system.png|thumb|Block diagram illustrating the time invariance for a deterministic continuous-time single-input single-output system. The system is time-invariant if and only if {{math|1=''y''{{sub|2}}(''t'') = ''y''{{sub|1}}(''t'' – ''t''{{sub|0}})}} for all time {{mvar|t}}, for all real constant {{math|''t''{{sub|0}}}} and for all input {{math|''x''{{sub|1}}(''t'')}}.<ref name="Bessai_2005">{{cite book | title = MIMO Signals and Systems | first = Horst J. | last = Bessai | publisher = Springer | year = 2005 | page = 28 | isbn = 0-387-23488-8}}</ref><ref name="Sundararajan_2008">{{cite book | title = A Practical Approach to Signals and Systems | first = D. | last = Sundararajan | publisher = Wiley | year = 2008 | page = 81 | isbn = 978-0-470-82353-8}}</ref><ref name="Roberts_2018">{{cite book | title = Signals and Systems: Analysis Using Transform Methods and MATLAB® | edition = 3 | first = Michael J. | last = Roberts | publisher = McGraw-Hill | year = 2018 | page = 132 | isbn = 978-0-07-802812-0}}</ref> Click image to expand it.]]

In [[control theory]], a '''time-invariant''' ('''TI''') '''system''' has a time-dependent '''system function''' that is not a direct [[Function (mathematics)|function]] of time. Such [[Dynamical system|system]]s are regarded as a class of systems in the field of [[system analysis]]. The time-dependent system function is a function of the time-dependent '''input function'''. If this function depends ''only'' indirectly on the [[time-domain]] (via the input function, for example), then that is a system that would be considered time-invariant. Conversely, any direct dependence on the time-domain of the system function could be considered as a "time-varying system".

Mathematically speaking, "time-invariance" of a system is the following property:<ref>{{cite book | first1=Alan | last1=Oppenheim | first2=Alan | last2=Willsky | title=Signals and Systems| publisher=Prentice Hall | year=1997| edition=second }}</ref>{{rp|p. 50}}

:''Given a system with a time-dependent output function {{tmath|y(t)}}, and a time-dependent input function {{tmath|x(t)}}, the system will be considered time-invariant if a time-delay on the input {{tmath|x(t+\delta)}} directly equates to a time-delay of the output {{tmath|y(t+\delta)}} function. For example, if time {{tmath|t}} is "elapsed time", then "time-invariance" implies that the relationship between the input function {{tmath|x(t)}} and the output function {{tmath|y(t)}} is constant with respect to time {{tmath|t:}}''
::<math>y(t) = f( x(t), t ) = f( x(t)).</math> 

In the language of [[signal processing]], this property can be satisfied if the [[transfer function]] of the system is not a direct function of time except as expressed by the input and output.

In the context of a system schematic, this property can also be stated as follows, as shown in the figure to the right:

:''If a system is time-invariant then the system block [[commutative|commutes]] with an arbitrary delay.''
<!-- Insert picture showing this 2nd definition pictorially as a block diagram -->

If a time-invariant system is also [[Linear system|linear]], it is the subject of [[linear time-invariant theory]] (linear time-invariant) with direct applications in [[NMR spectroscopy]], [[seismology]], [[electrical network|circuit]]s, [[signal processing]], [[control theory]], and other technical areas. [[Nonlinear system|Nonlinear]] time-invariant systems lack a comprehensive, governing theory. [[Discrete-time signal|Discrete]] time-invariant systems are known as [[shift-invariant system]]s. Systems which lack the time-invariant property are studied as [[time-variant system]]s.