Time-invariant system

File:Time invariance block diagram for a SISO system.png

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 Template:Math for all time Template:Mvar, for all real constant Template:Math and for all input Template:Math.<ref name="Bessai_2005">Template:Cite book</ref><ref name="Sundararajan_2008">Template:Cite book</ref><ref name="Roberts_2018">Template:Cite book</ref> 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:<ref>Template:Cite book</ref>Template:Rp

Given a system with a time-dependent output function Template:Tmath, and a time-dependent input function Template:Tmath, the system will be considered time-invariant if a time-delay on the input Template:Tmath directly equates to a time-delay of the output Template:Tmath function. For example, if time Template:Tmath is "elapsed time", then "time-invariance" implies that the relationship between the input function Template:Tmath and the output function Template:Tmath is constant with respect to time Template:Tmath

<math>y(t) = f( x(t), t ) = f( x(t)).</math>

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: <math>y(t) = t x(t)</math>
• System B: <math>y(t) = 10 x(t)</math>

Since the System Function <math>y(t)</math> for system A explicitly depends on t outside of <math>x(t)</math>, 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 <math>x(t)</math>. 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 <math>x_d(t) = x(t + \delta)</math>

<math>y(t) = t x(t)</math>

<math>y_1(t) = t x_d(t) = t x(t + \delta)</math>

Now delay the output by <math>\delta</math>

<math>y(t) = t x(t)</math>

<math>y_2(t) = y(t + \delta) = (t + \delta) x(t + \delta)</math>

Clearly <math>y_1(t) \ne y_2(t)</math>, therefore the system is not time-invariant.

System B: Start with a delay of the input <math>x_d(t) = x(t + \delta)</math>

<math>y(t) = 10 x(t)</math>

<math>y_1(t) = 10 x_d(t) = 10 x(t + \delta)</math>

Now delay the output by <math>\delta</math>

<math>y(t) = 10 x(t)</math>

<math>y_2(t) = y(t + \delta) = 10 x(t + \delta)</math>

Clearly <math>y_1(t) = y_2(t)</math>, therefore the system is time-invariant.

More generally, the relationship between the input and output is

<math> y(t) = f(x(t), t),</math>

and its variation with time is

<math>\frac{\mathrm{d} y}{\mathrm{d} t} = \frac{\partial f}{\partial t} + \frac{\partial f}{\partial x} \frac{\mathrm{d} x}{\mathrm{d} t}.</math>

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

<math> \frac{\partial f}{\partial t} =0.</math>

Applied to Systems A and B above:

<math> f_A = t x(t) \qquad \implies \qquad \frac{\partial f_A}{\partial t} = x(t) \neq 0 </math> in general, so it is not time-invariant,

<math> f_B = 10 x(t) \qquad \implies \qquad \frac{\partial f_B}{\partial t} = 0 </math> so it is time-invariant.

Abstract example

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

<math>x(t+1) = \delta(t+1) * x(t)</math>

can be represented in this abstract notation by

<math>\tilde{x}_1 = \mathbb{T}_1 \tilde{x}</math>

where <math>\tilde{x}</math> is a function given by

<math>\tilde{x} = x(t) \forall t \in \R</math>

with the system yielding the shifted output

<math>\tilde{x}_1 = x(t + 1) \forall t \in \R</math>

So <math>\mathbb{T}_1</math> is an operator that advances the input vector by 1.

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

<math>\mathbb{T}_r \mathbb{H} = \mathbb{H} \mathbb{T}_r \forall r</math>

If our system equation is given by

<math>\tilde{y} = \mathbb{H} \tilde{x}</math>

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

Applying the system operator first gives

<math>\mathbb{T}_r \mathbb{H} \tilde{x} = \mathbb{T}_r \tilde{y} = \tilde{y}_r</math>

Applying the shift operator first gives

<math>\mathbb{H} \mathbb{T}_r \tilde{x} = \mathbb{H} \tilde{x}_r</math>

If the system is time-invariant, then

<math>\mathbb{H} \tilde{x}_r = \tilde{y}_r</math>

See also

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