Time-invariant system

In <a href="/facts/Control_theory/aIuERpn6">control theory</a>, a time-invariant (TI) system has a time-dependent system function that is not a direct <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> of time. Such <a href="/facts/Dynamical_system/782gREl5">systems</a> are regarded as a class of systems in the field of <a href="/facts/System_analysis/9jLtmbxT">system analysis</a>. The time-dependent system function is a function of the time-dependent input function. If this function depends only indirectly on the <a href="/facts/Time-domain/1w6mP2MU">time-domain</a> (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:: p. 50 

Given a system with a time-dependent output function ⁠
 
 
 
 y
 (
 t
 )
 
 
 {\displaystyle y(t)}
 
⁠, and a time-dependent input function ⁠
 
 
 
 x
 (
 t
 )
 
 
 {\displaystyle x(t)}
 
⁠, the system will be considered time-invariant if a time-delay on the input ⁠
 
 
 
 x
 (
 t
 +
 δ
 )
 
 
 {\displaystyle x(t+\delta )}
 
⁠ directly equates to a time-delay of the output ⁠
 
 
 
 y
 (
 t
 +
 δ
 )
 
 
 {\displaystyle y(t+\delta )}
 
⁠ function. For example, if time ⁠
 
 
 
 t
 
 
 {\displaystyle t}
 
⁠ is "elapsed time", then "time-invariance" implies that the relationship between the input function ⁠
 
 
 
 x
 (
 t
 )
 
 
 {\displaystyle x(t)}
 
⁠ and the output function ⁠
 
 
 
 y
 (
 t
 )
 
 
 {\displaystyle y(t)}
 
⁠ is constant with respect to time ⁠
 
 
 
 t
 :
 
 
 {\displaystyle t:}
 
⁠

y
        (
        t
        )
        =
        f
        (
        x
        (
        t
        )
        ,
        t
        )
        =
        f
        (
        x
        (
        t
        )
        )
        .
      
    
    {\displaystyle y(t)=f(x(t),t)=f(x(t)).}

In the language of <a href="/facts/Signal_processing/Npaja6zb">signal processing</a>, this property can be satisfied if the <a href="/facts/Transfer_function/slDsh1lH">transfer function</a> 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 <a href="/facts/Commutative/WaYxJLxd">commutes</a> with an arbitrary delay.
If a time-invariant system is also <a href="/facts/Linear_system/uEl20Kc1">linear</a>, it is the subject of <a href="/facts/Linear_time-invariant_theory/kEmj0QDC">linear time-invariant theory</a> (linear time-invariant) with direct applications in <a href="/facts/NMR_spectroscopy/SFKgdthI">NMR spectroscopy</a>, <a href="/facts/Seismology/Lony1LYX">seismology</a>, <a href="/facts/Electrical_network/zt94kTW0">circuits</a>, <a href="/facts/Signal_processing/Npaja6zb">signal processing</a>, <a href="/facts/Control_theory/aIuERpn6">control theory</a>, and other technical areas. <a href="/facts/Nonlinear_system/4aYItALC">Nonlinear</a> time-invariant systems lack a comprehensive, governing theory. <a href="/facts/Discrete-time_signal/RR8wpS9e">Discrete</a> time-invariant systems are known as <a href="/facts/Shift-invariant_system/zaKY72py">shift-invariant systems</a>. Systems which lack the time-invariant property are studied as <a href="/facts/Time-variant_system/sDEWpWGg">time-variant systems</a>.

Time-invariant system open-in-new

Time-invariant system