Concept

# Time-invariant system

Summary
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), and a time-dependent input function x(t), the system will be considered time-invariant if a time-delay on the input x(t+\delta) directly equates to a time-delay of the output y(t+\delta) function. For example, if time t is "elapsed time", then "time-invariance" implies that the relationship between the input function x(t) and the output function y(t) is constant with respect to time 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.
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