Time-invariant system

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

y 2 (t) = y 1 (t - t 0)

for all time

t

, for all real constant

t 0

and for all input

x 1 (t)

.^[1]^[2]^[3] 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 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:^[4]^{: p. 50}

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

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.

^ Bessai, Horst J. (2005). MIMO Signals and Systems. Springer. p. 28. ISBN 0-387-23488-8.
^ Sundararajan, D. (2008). A Practical Approach to Signals and Systems. Wiley. p. 81. ISBN 978-0-470-82353-8.
^ Roberts, Michael J. (2018). Signals and Systems: Analysis Using Transform Methods and MATLAB® (3 ed.). McGraw-Hill. p. 132. ISBN 978-0-07-802812-0.
^ Oppenheim, Alan; Willsky, Alan (1997). Signals and Systems (second ed.). Prentice Hall.

[Bessai_2005-1] Bessai, Horst J. (2005). MIMO Signals and Systems. Springer. p. 28. ISBN 0-387-23488-8.

[Sundararajan_2008-2] Sundararajan, D. (2008). A Practical Approach to Signals and Systems. Wiley. p. 81. ISBN 978-0-470-82353-8.

[Roberts_2018-3] Roberts, Michael J. (2018). Signals and Systems: Analysis Using Transform Methods and MATLAB® (3 ed.). McGraw-Hill. p. 132. ISBN 978-0-07-802812-0.

[4] Oppenheim, Alan; Willsky, Alan (1997). Signals and Systems (second ed.). Prentice Hall.

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