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.

Simple example

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

System A: $y(t)=tx(t)$
System B: $y(t)=10x(t)$

Since the System Function $y(t)$ for system A explicitly depends on t outside of $x(t)$ , 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 $x(t)$ . 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

x_{d}(t)=x(t+\delta )

y(t)=tx(t)

y_{1}(t)=tx_{d}(t)=tx(t+\delta )

Now delay the output by

\delta

y(t)=tx(t)

y_{2}(t)=y(t+\delta )=(t+\delta )x(t+\delta )

Clearly

y_{1}(t)\neq y_{2}(t)

, therefore the system is not time-invariant.

System B: Start with a delay of the input

x_{d}(t)=x(t+\delta )

y(t)=10x(t)

y_{1}(t)=10x_{d}(t)=10x(t+\delta )

Now delay the output by

\delta

y(t)=10x(t)

y_{2}(t)=y(t+\delta )=10x(t+\delta )

Clearly

y_{1}(t)=y_{2}(t)

, therefore the system is time-invariant.

More generally, the relationship between the input and output is

y(t)=f(x(t),t),

and its variation with time is

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

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

{\frac {\partial f}{\partial t}}=0.

Applied to Systems A and B above:

f_{A}=tx(t)\qquad \implies \qquad {\frac {\partial f_{A}}{\partial t}}=x(t)\neq 0

in general, so it is not time-invariant,

f_{B}=10x(t)\qquad \implies \qquad {\frac {\partial f_{B}}{\partial t}}=0

so it is time-invariant.

Abstract example

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

x(t+1)=\delta (t+1)*x(t)

can be represented in this abstract notation by

{\tilde {x}}_{1}=\mathbb {T} _{1}{\tilde {x}}

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

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

with the system yielding the shifted output

{\tilde {x}}_{1}=x(t+1)\forall t\in \mathbb {R}

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

Suppose we represent a system by an operator $\mathbb {H}$ . This system is time-invariant if it commutes with the shift operator, i.e.,

\mathbb {T} _{r}\mathbb {H} =\mathbb {H} \mathbb {T} _{r}\forall r

If our system equation is given by

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

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

Applying the system operator first gives

\mathbb {T} _{r}\mathbb {H} {\tilde {x}}=\mathbb {T} _{r}{\tilde {y}}={\tilde {y}}_{r}

Applying the shift operator first gives

\mathbb {H} \mathbb {T} _{r}{\tilde {x}}=\mathbb {H} {\tilde {x}}_{r}

If the system is time-invariant, then

\mathbb {H} {\tilde {x}}_{r}={\tilde {y}}_{r}

References

^ 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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