# 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:^{[4]}^{: p. 50 }

*Given a system with a time-dependent output function , and a time-dependent input function , the system will be considered time-invariant if a time-delay on the input directly equates to a time-delay of the output function. For example, if time is "elapsed time", then "time-invariance" 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 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.

## Simple example

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

- System A:
- System B:

Since the **System Function** for system A explicitly depends on *t* outside of , 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 . 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.

__System A:__Start with a delay of the input- Now delay the output by
- Clearly , therefore the system is not time-invariant.

__System B:__Start with a delay of the input- Now delay the output by
- Clearly , therefore the system is time-invariant.

More generally, the relationship between the input and output is

and its variation with time is

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

Applied to Systems A and B above:

- in general, so it is not time-invariant,
- so it is time-invariant.

## 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 "advance-by-1" 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 **time-invariant** if it commutes with the shift operator, i.e.,

If our system equation is given by

then it is time-invariant 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 time-invariant, then

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