Time-variant system

<h2 id="overview">Overview</h2>
<p>There are many well developed <a href="/facts/LTI_system_theory/kEmj0QDC">techniques</a> for dealing with the response of linear time invariant systems, such as <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace</a> and <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transforms</a>. However, these techniques are not strictly valid for time-varying systems.  A system undergoing slow time variation in comparison to its time constants can usually be considered to be time invariant: they are close to time invariant on a small scale.  An example of this is the aging and wear of electronic components, which happens on a scale of years, and thus does not result in any behaviour qualitatively different from that observed in a time invariant system: day-to-day, they are effectively time invariant, though year to year, the parameters may change.  Other linear time variant systems may behave more like nonlinear systems, if the system changes quickly – significantly differing between measurements.
</p><p>The following things can be said about a time-variant system:
</p>
<ul><li>It has explicit dependence on time.</li>
<li>It does not have an <a href="/facts/Impulse_response/l4vaE5er">impulse response</a> in the normal sense. The system can be characterized by an impulse response except the impulse response must be known at each and every time instant.</li>
<li>It is not stationary in the sense of constancy of the signal's distributional frequency. This means that the parameters which govern the signal's process exhibit varaition with the passage of time. See <a href="/facts/Stationarity_(statistics)/vozdgfEX">Stationarity (statistics)</a> for in-depth theoretics regarding this property.</li></ul>
<h2 id="linear-time-variant-systems">Linear time-variant systems</h2>
<p>Linear-time variant (LTV) systems are the ones whose parameters vary with time according to previously specified laws. Mathematically, there is a well defined dependence of the system over time and over the input parameters that change over time.
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        f
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        x
        (
        t
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        ,
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    {\displaystyle y(t)=f(x(t),t)}

<p>In order to solve time-variant systems, the algebraic methods consider initial conditions of the system i.e. whether the system is zero-input or non-zero input system.
</p>
<h2 id="examples-of-time-variant-systems">Examples of time-variant systems</h2>
<p>The following time varying systems cannot be modelled by assuming that they are time invariant:
</p>
<ul><li>The Earth's thermodynamic response to incoming <a href="/facts/Solar_irradiance/SI26I7aD">Solar irradiance</a> varies with time due to changes in the Earth's <a href="/facts/Albedo/TJDRdKEB">albedo</a> and the presence of <a href="/facts/Greenhouse_gas/HXaHqrT8">greenhouse gases</a> in the atmosphere.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li>
<li><a href="/facts/Discrete_wavelet_transform/19DVnk4n">Discrete wavelet transform</a>, often used in modern signal processing, is time variant because it makes use of the <a href="/facts/Decimation_(signal_processing)/gN5Wwrhv">decimation</a> operation[<i>dubious – discuss</i>].</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Control_system/8CEjIDlS">Control system</a></li>
<li><a href="/facts/Control_theory/aIuERpn6">Control theory</a></li>
<li><a href="/facts/System_analysis/9jLtmbxT">System analysis</a></li>
<li><a href="/facts/Time-invariant_system/NpWUk0YG">Time-invariant system</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Cherniakov, Mikhail (2003). An Introduction to Parametric Digital Filters and Oscillators. Wiley. pp. 47–49. ISBN 978-0470851043. <a href="978-0470851043" target="_blank">978-0470851043</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Sung, Taehong; Yoon, Sang; Kim, Kyung (2015-07-13). "A Mathematical Model of Hourly Solar Radiation in Varying Weather Conditions for a Dynamic Simulation of the Solar Organic Rankine Cycle". Energies. 8 (7): 7058–7069. doi:10.3390/en8077058. ISSN 1996-1073. <a href="https://doi.org/10.3390%2Fen8077058" target="_blank">https://doi.org/10.3390%2Fen8077058</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Alzahrani, Ahmad; Shamsi, Pourya; Dagli, Cihan; Ferdowsi, Mehdi (2017). "Solar Irradiance Forecasting Using Deep Neural Networks". Procedia Computer Science. 114: 304–313. doi:10.1016/j.procs.2017.09.045. <a href="https://doi.org/10.1016%2Fj.procs.2017.09.045" target="_blank">https://doi.org/10.1016%2Fj.procs.2017.09.045</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Time-variant system open-in-new

Time-variant system