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:
Given a system with a time-dependent output function y ( t ) {\displaystyle y(t)} , and a time-dependent input function x ( t ) {\displaystyle x(t)} , the system will be considered time-invariant if a time-delay on the input x ( t + δ ) {\displaystyle x(t+\delta )} directly equates to a time-delay of the output y ( t + δ ) {\displaystyle y(t+\delta )} function. For example, if time t {\displaystyle t} is "elapsed time", then "time-invariance" implies that the relationship between the input function x ( t ) {\displaystyle x(t)} and the output function y ( t ) {\displaystyle y(t)} is constant with respect to time t : {\displaystyle t:}
y ( t ) = f ( x ( t ) , t ) = f ( x ( t ) ) . {\displaystyle 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 ) {\displaystyle y(t)=tx(t)}
System B: y ( t ) = 10 x ( t ) {\displaystyle y(t)=10x(t)}
Since the System Function y ( t ) {\displaystyle y(t)} for system A explicitly depends on t outside of x ( t ) {\displaystyle 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 ) {\displaystyle 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 + δ ) {\displaystyle x_{d}(t)=x(t+\delta )}
y ( t ) = t x ( t ) {\displaystyle y(t)=tx(t)}
y 1 ( t ) = t x d ( t ) = t x ( t + δ ) {\displaystyle y_{1}(t)=tx_{d}(t)=tx(t+\delta )}
Now delay the output by δ {\displaystyle \delta }
y ( t ) = t x ( t ) {\displaystyle y(t)=tx(t)}
y 2 ( t ) = y ( t + δ ) = ( t + δ ) x ( t + δ ) {\displaystyle y_{2}(t)=y(t+\delta )=(t+\delta )x(t+\delta )}
Clearly y 1 ( t ) ≠ y 2 ( t ) {\displaystyle 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 + δ ) {\displaystyle x_{d}(t)=x(t+\delta )}
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