In control theory, a timeinvariant (TI) system has a timedependent 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 timedependent system function is a function of the timedependent input function. If this function depends only indirectly on the timedomain (via the input function, for example), then that is a system that would be considered timeinvariant. Conversely, any direct dependence on the timedomain of the system function could be considered as a "timevarying system".
Mathematically speaking, "timeinvariance" of a system is the following property:^{[4]}^{: p. 50 }
 Given a system with a timedependent output function , and a timedependent input function , the system will be considered timeinvariant if a timedelay on the input directly equates to a timedelay of the output function. For example, if time is "elapsed time", then "timeinvariance" implies that the relationship between the input function and the output function is constant with respect to time
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 timeinvariant then the system block commutes with an arbitrary delay.
If a timeinvariant system is also linear, it is the subject of linear timeinvariant theory (linear timeinvariant) with direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. Nonlinear timeinvariant systems lack a comprehensive, governing theory. Discrete timeinvariant systems are known as shiftinvariant systems. Systems which lack the timeinvariant property are studied as timevariant systems.
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TimeInvariant and TimeVariant Systems

Signals & Systems  Time Variant & Time Invariant

TimeInvariant and TimeVariant Systems (Solved Problems)  Part 1

Time Invariance: Conceptual Introduction

What is a Linear Time Invariant (LTI) System?
Transcription
Simple example
To demonstrate how to determine if a system is timeinvariant, consider the two systems:
 System A:
 System B:
Since the System Function for system A explicitly depends on t outside of , it is not timeinvariant because the timedependence is not explicitly a function of the input function.
In contrast, system B's timedependence is only a function of the timevarying input . This makes system B timeinvariant.
The Formal Example below shows in more detail that while System B is a ShiftInvariant 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.
 System A: Start with a delay of the input
 Now delay the output by
 Clearly , therefore the system is not timeinvariant.
 System B: Start with a delay of the input
 Now delay the output by
 Clearly , therefore the system is timeinvariant.
More generally, the relationship between the input and output is
and its variation with time is
For timeinvariant systems, the system properties remain constant with time,
Applied to Systems A and B above:
 in general, so it is not timeinvariant,
 so it is timeinvariant.
Abstract example
We can denote the shift operator by where is the amount by which a vector's index set should be shifted. For example, the "advanceby1" system
can be represented in this abstract notation by
where is a function given by
with the system yielding the shifted output
So is an operator that advances the input vector by 1.
Suppose we represent a system by an operator . This system is timeinvariant if it commutes with the shift operator, i.e.,
If our system equation is given by
then it is timeinvariant if we can apply the system operator on followed by the shift operator , or we can apply the shift operator followed by the system operator , with the two computations yielding equivalent results.
Applying the system operator first gives
Applying the shift operator first gives
If the system is timeinvariant, then
See also
 Finite impulse response
 Sheffer sequence
 State space (controls)
 Signalflow graph
 LTI system theory
 Autonomous system (mathematics)
References
 ^ Bessai, Horst J. (2005). MIMO Signals and Systems. Springer. p. 28. ISBN 0387234888.
 ^ Sundararajan, D. (2008). A Practical Approach to Signals and Systems. Wiley. p. 81. ISBN 9780470823538.
 ^ Roberts, Michael J. (2018). Signals and Systems: Analysis Using Transform Methods and MATLAB® (3 ed.). McGrawHill. p. 132. ISBN 9780078028120.
 ^ Oppenheim, Alan; Willsky, Alan (1997). Signals and Systems (second ed.). Prentice Hall.