Time-invariant system

In control theory, a time-invariant (TI) system has a time-dependent system function that is not a direct function of time. Such systems 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:^{: p. 50}

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

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

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 commutes with an arbitrary delay.

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