Time-invariant system

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Short description: Dynamical system whose system function is not directly dependent on time
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 y2(t) = y1(t – t0) for all time t, for all real constant t0 and for all input x1(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 [math]\displaystyle{ y(t) }[/math], and a time-dependent input function [math]\displaystyle{ x(t) }[/math], the system will be considered time-invariant if a time-delay on the input [math]\displaystyle{ x(t+\delta) }[/math] directly equates to a time-delay of the output [math]\displaystyle{ y(t+\delta) }[/math] function. For example, if time [math]\displaystyle{ t }[/math] is "elapsed time", then "time-invariance" implies that the relationship between the input function [math]\displaystyle{ x(t) }[/math] and the output function [math]\displaystyle{ y(t) }[/math] is constant with respect to time [math]\displaystyle{ t: }[/math]
[math]\displaystyle{ 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]\displaystyle{ y(t) = t x(t) }[/math]
  • System B: [math]\displaystyle{ y(t) = 10 x(t) }[/math]

Since the System Function [math]\displaystyle{ y(t) }[/math] for system A explicitly depends on t outside of [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ x_d(t) = x(t + \delta) }[/math]
[math]\displaystyle{ y(t) = t x(t) }[/math]
[math]\displaystyle{ y_1(t) = t x_d(t) = t x(t + \delta) }[/math]
Now delay the output by [math]\displaystyle{ \delta }[/math]
[math]\displaystyle{ y(t) = t x(t) }[/math]
[math]\displaystyle{ y_2(t) = y(t + \delta) = (t + \delta) x(t + \delta) }[/math]
Clearly [math]\displaystyle{ 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]\displaystyle{ x_d(t) = x(t + \delta) }[/math]
[math]\displaystyle{ y(t) = 10 x(t) }[/math]
[math]\displaystyle{ y_1(t) = 10 x_d(t) = 10 x(t + \delta) }[/math]
Now delay the output by [math]\displaystyle{ \delta }[/math]
[math]\displaystyle{ y(t) = 10 x(t) }[/math]
[math]\displaystyle{ y_2(t) = y(t + \delta) = 10 x(t + \delta) }[/math]
Clearly [math]\displaystyle{ y_1(t) = y_2(t) }[/math], therefore the system is time-invariant.

More generally, the relationship between the input and output is

[math]\displaystyle{ y(t) = f(x(t), t), }[/math]

and its variation with time is

[math]\displaystyle{ \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]\displaystyle{ \frac{\partial f}{\partial t} =0. }[/math]

Applied to Systems A and B above:

[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \mathbb{T}_r }[/math] where [math]\displaystyle{ r }[/math] is the amount by which a vector's index set should be shifted. For example, the "advance-by-1" system

[math]\displaystyle{ x(t+1) = \delta(t+1) * x(t) }[/math]

can be represented in this abstract notation by

[math]\displaystyle{ \tilde{x}_1 = \mathbb{T}_1 \tilde{x} }[/math]

where [math]\displaystyle{ \tilde{x} }[/math] is a function given by

[math]\displaystyle{ \tilde{x} = x(t) \forall t \in \R }[/math]

with the system yielding the shifted output

[math]\displaystyle{ \tilde{x}_1 = x(t + 1) \forall t \in \R }[/math]

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

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

[math]\displaystyle{ \mathbb{T}_r \mathbb{H} = \mathbb{H} \mathbb{T}_r \forall r }[/math]

If our system equation is given by

[math]\displaystyle{ \tilde{y} = \mathbb{H} \tilde{x} }[/math]

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

Applying the system operator first gives

[math]\displaystyle{ \mathbb{T}_r \mathbb{H} \tilde{x} = \mathbb{T}_r \tilde{y} = \tilde{y}_r }[/math]

Applying the shift operator first gives

[math]\displaystyle{ \mathbb{H} \mathbb{T}_r \tilde{x} = \mathbb{H} \tilde{x}_r }[/math]

If the system is time-invariant, then

[math]\displaystyle{ \mathbb{H} \tilde{x}_r = \tilde{y}_r }[/math]

See also

References

  1. ↑ Bessai, Horst J. (2005). MIMO Signals and Systems. Springer. p. 28. ISBN 0-387-23488-8. 
  2. ↑ Sundararajan, D. (2008). A Practical Approach to Signals and Systems. Wiley. p. 81. ISBN 978-0-470-82353-8. 
  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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