Time-invariant system explained

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:^[1]

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

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) = t x(t)

• System B:

y(t) = 10 x(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) = t x(t) y _{1(t)} = t x _{d(t)} = t x(t + \delta)

Now delay the output by

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

Clearly

y _{1(t)} \ne 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) = 10 x(t) y _{1(t)} = 10 x _{d(t)} = 10 x(t + \delta)

Now delay the output by

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

(d y)/(d t) = (\partial f)/(\partial t) + (\partial f)/(\partial x) (d x)/(d t) .

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

(\partial f)/(\partial t) =0 .

Applied to Systems A and B above:

f _{A} = t x(t) \implies (\partial f _{A})/(\partial t) = x(t) ≠ 0 in general, so it is not time-invariant,f _{B} = 10 x(t) \implies (\partial f _{B})/(\partial t) = 0 so it is time-invariant.

Abstract example

We can denote the shift operator by

T _{r} where r is the amount by which a vector's index set should be shifted. For example, the "advance-by-1" systemx(t +1 ) = \delta(t +1 ) * x(t)

can be represented in this abstract notation by

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

where

\tilde{x} is a function given by\tilde{x} = x(t) \forall t \in \R

with the system yielding the shifted output

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

So

T _{1} is an operator that advances the input vector by 1.H . This system is time-invariant if it commutes with the shift operator, i.e.,T _{r} H = H T _{r} \forall r

If our system equation is given by

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

then it is time-invariant if we can apply the system operator

H on \tilde{x} followed by the shift operator T _{r} , or we can apply the shift operator T _{r} followed by the system operator H , with the two computations yielding equivalent results.

Applying the system operator first gives

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

Applying the shift operator first gives

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

If the system is time-invariant, then

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

See also

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

Notes and References

1. Book: Alan . Oppenheim . Alan . Willsky . Signals and Systems. Prentice Hall . 1997. second .