Time-invariant system

In control theory, a time-invariant (TI) system has a time-dependent system function that is not a direct function of time.

Mathematically speaking, "time-invariance" of a system is the following property:[4]: p. 50 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 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.

In contrast, system B's time-dependence is only a function of the time-varying input

A more formal proof of why systems A and B above differ is now presented.

More generally, the relationship between the input and output is and its variation with time is For time-invariant systems, the system properties remain constant with time, Applied to Systems A and B above: We can denote the shift operator by

is the amount by which a vector's index set should be shifted.

For example, the "advance-by-1" system can be represented in this abstract notation by where

is a function given by with the system yielding the shifted output So

, with the two computations yielding equivalent results.

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)

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