Editing Time-invariant system

{{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.

== Simple example ==

To demonstrate how to determine if a system is time-invariant, consider the two systems:

* System A: <math>y(t) = t x(t)</math>
* System B: <math>y(t) = 10 x(t)</math>

Since the '''System Function''' <math>y(t)</math> for system A explicitly depends on ''t'' outside of <math>x(t)</math>, it is not [[time-invariant]] because the time-dependence is not explicitly a function of the input function.

In contrast, system B's time-dependence is only a function of the time-varying input <math>x(t)</math>. This makes system B [[time-invariant]].

The '''Formal Example''' below shows in more detail that while System B is a Shift-Invariant System as a function of time, ''t'', System A is not.

== Formal example ==
A more formal proof of why systems A and B above differ is now presented. To perform this proof, the second definition will be used.

:<u>System A:</u> Start with a delay of the input <math>x_d(t) = x(t + \delta)</math>
::<math>y(t) = t x(t)</math>
::<math>y_1(t) = t x_d(t) = t x(t + \delta)</math>
:Now delay the output by <math>\delta</math>
::<math>y(t) = t x(t)</math>
::<math>y_2(t) = y(t + \delta) = (t + \delta) x(t + \delta)</math>
:Clearly <math>y_1(t) \ne y_2(t)</math>, therefore the system is not time-invariant.

:<u>System B:</u> Start with a delay of the input <math>x_d(t) = x(t + \delta)</math>
::<math>y(t) = 10 x(t)</math>
::<math>y_1(t) = 10 x_d(t) = 10 x(t + \delta)</math>
:Now delay the output by <math>\delta</math>
::<math>y(t) = 10 x(t)</math>
::<math>y_2(t) = y(t + \delta) = 10 x(t + \delta)</math>
:Clearly <math>y_1(t) = y_2(t)</math>, therefore the system is time-invariant.

More generally, the relationship between the input and output is 

:<math> y(t) = f(x(t), t),</math>

and its variation with time is

:<math>\frac{\mathrm{d} y}{\mathrm{d} t} = \frac{\partial f}{\partial t} + \frac{\partial f}{\partial x} \frac{\mathrm{d} x}{\mathrm{d} t}.</math>

For time-invariant systems, the system properties remain constant with time, 

:<math> \frac{\partial f}{\partial t} =0.</math> 

Applied to Systems A and B above:

:<math> f_A = t x(t) \qquad \implies \qquad \frac{\partial f_A}{\partial t} = x(t) \neq 0 </math> in general, so it is not time-invariant,
:<math> f_B = 10 x(t) \qquad \implies \qquad \frac{\partial f_B}{\partial t} = 0 </math> so it is time-invariant.

== Abstract example ==

We can denote the '''[[shift operator]]''' by <math>\mathbb{T}_r</math> where <math>r</math> is the amount by which a vector's [[parameter|index set]] should be shifted. For example, the "advance-by-1" system

:<math>x(t+1) = \delta(t+1) * x(t)</math>

can be represented in this abstract notation by

:<math>\tilde{x}_1 = \mathbb{T}_1 \tilde{x}</math>

where <math>\tilde{x}</math> is a function given by

:<math>\tilde{x} = x(t) \forall t \in \R</math>

with the system yielding the shifted output

:<math>\tilde{x}_1 = x(t + 1) \forall t \in \R</math>

So <math>\mathbb{T}_1</math> is an operator that advances the input vector by 1.

Suppose we represent a system by an [[Operator (mathematics)|operator]] <math>\mathbb{H}</math>. This system is '''time-invariant''' if it [[Commutative operation|commutes]] with the shift operator, i.e.,

:<math>\mathbb{T}_r \mathbb{H} = \mathbb{H} \mathbb{T}_r \forall r</math>

If our system equation is given by

:<math>\tilde{y} = \mathbb{H} \tilde{x}</math>

then it is time-invariant if we can apply the system operator <math>\mathbb{H}</math> on <math>\tilde{x}</math> followed by the shift operator <math>\mathbb{T}_r</math>, or we can apply the shift operator <math>\mathbb{T}_r</math> followed by the system operator <math>\mathbb{H}</math>, with the two computations yielding equivalent results.

Applying the system operator first gives

:<math>\mathbb{T}_r \mathbb{H} \tilde{x} = \mathbb{T}_r \tilde{y} = \tilde{y}_r</math>

Applying the shift operator first gives

:<math>\mathbb{H} \mathbb{T}_r \tilde{x} = \mathbb{H} \tilde{x}_r</math>

If the system is time-invariant, then

:<math>\mathbb{H} \tilde{x}_r = \tilde{y}_r</math>

== See also ==
*[[Finite impulse response]]
*[[Sheffer sequence]]
*[[State space (controls)]]
*[[Signal-flow graph]]
*[[LTI system theory]]
*[[Autonomous system (mathematics)]]

==References==
{{reflist}}

[[Category:Control theory]]
[[Category:Signal processing]]