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Incontrol theory, atime-invariant (TI)system has a time-dependentsystem function that is not a directfunction of time. Suchsystems are regarded as a class of systems in the field ofsystem analysis. The time-dependent system function is a function of the time-dependentinput function. If this function dependsonly indirectly on thetime-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${\displaystyle y(t)}$⁠, 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 ofsignal processing, this property can be satisfied if thetransfer 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 blockcommutes with an arbitrary delay.

If a time-invariant system is alsolinear, it is the subject oflinear time-invariant theory (linear time-invariant) with direct applications inNMR 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 asshift-invariant systems. Systems which lack the time-invariant property are studied astime-variant systems.

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

• System A:${\displaystyle y(t)=tx(t)}$
• System B:${\displaystyle y(t)=10x(t)}$

Since theSystem Function${\displaystyle y(t)}$ for system A explicitly depends ont outside of${\displaystyle x(t)}$, it is nottime-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${\displaystyle x(t)}$. This makes system Btime-invariant.

TheFormal 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.

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${\displaystyle x_d(t)=x(t+\delta )}$

${\displaystyle y(t)=tx(t)}$

${\displaystyle y_1(t)=t{x}_d(t)=tx(t+\delta )}$

Now delay the output by${\displaystyle \delta }$

${\displaystyle y(t)=tx(t)}$

${\displaystyle y_2(t)=y(t+\delta )=(t+\delta )x(t+\delta )}$

Clearly${\displaystyle y_1(t)\ne y_2(t)}$, therefore the system is not time-invariant.

System B: Start with a delay of the input${\displaystyle x_d(t)=x(t+\delta )}$

${\displaystyle y(t)=10x(t)}$

${\displaystyle y_1(t)=10x_d(t)=10x(t+\delta )}$

Now delay the output by${\displaystyle \delta }$

${\displaystyle y(t)=10x(t)}$

${\displaystyle y_2(t)=y(t+\delta )=10x(t+\delta )}$

Clearly${\displaystyle y_1(t)=y_2(t)}$, therefore the system is time-invariant.

More generally, the relationship between the input and output is

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

and its variation with time is

${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}=\frac{\mathrm{\partial }f}{\mathrm{\partial }t}+\frac{\mathrm{\partial }f}{\mathrm{\partial }x}\frac{\mathrm{d}x}{\mathrm{d}t}.}$

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

${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }t}=0.}$

Applied to Systems A and B above:

${\displaystyle f_A=tx(t)⟹\frac{\mathrm{\partial }{f}_A}{\mathrm{\partial }t}=x(t)\ne 0}$ in general, so it is not time-invariant,

${\displaystyle f_B=10x(t)⟹\frac{\mathrm{\partial }{f}_B}{\mathrm{\partial }t}=0}$ so it is time-invariant.

We can denote theshift operator by${\displaystyle {\mathbb{T}}_r}$ where${\displaystyle r}$ is the amount by which a vector'sindex set should be shifted. For example, the "advance-by-1" system

${\displaystyle x(t+1)=\delta (t+1)\ast x(t)}$

can be represented in this abstract notation by

${\displaystyle {\tilde{x}}_1={\mathbb{T}}_1\tilde{x}}$

where${\displaystyle \tilde{x}}$ is a function given by

${\displaystyle \tilde{x}=x(t)\mathrm{\forall }t\in \mathbb{R}}$

with the system yielding the shifted output

${\displaystyle {\tilde{x}}_1=x(t+1)\mathrm{\forall }t\in \mathbb{R}}$

So${\displaystyle {\mathbb{T}}_1}$ is an operator that advances the input vector by 1.

Suppose we represent a system by anoperator${\displaystyle \mathbb{H}}$. This system istime-invariant if itcommutes with the shift operator, i.e.,

${\displaystyle {\mathbb{T}}_r\mathbb{H}=\mathbb{H}{\mathbb{T}}_r\mathrm{\forall }r}$

If our system equation is given by

${\displaystyle \tilde{y}=\mathbb{H}\tilde{x}}$

then it is time-invariant if we can apply the system operator${\displaystyle \mathbb{H}}$ on${\displaystyle \tilde{x}}$ followed by the shift operator${\displaystyle {\mathbb{T}}_r}$, or we can apply the shift operator${\displaystyle {\mathbb{T}}_r}$ followed by the system operator${\displaystyle \mathbb{H}}$, with the two computations yielding equivalent results.

Applying the system operator first gives

${\displaystyle {\mathbb{T}}_r\mathbb{H}\tilde{x}={\mathbb{T}}_r\tilde{y}={\tilde{y}}_r}$

Applying the shift operator first gives

${\displaystyle \mathbb{H}{\mathbb{T}}_r\tilde{x}=\mathbb{H}{\tilde{x}}_r}$

If the system is time-invariant, then

${\displaystyle \mathbb{H}{\tilde{x}}_r={\tilde{y}}_r}$

• Finite impulse response
• Sheffer sequence
• State space (controls)
• Signal-flow graph
• LTI system theory
• Autonomous system (mathematics)

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• Signal processing

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