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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Time-variant system</span></span>

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<p>A <b>time-variant system</b> is a <a href="System" title="System">system</a> whose output response depends on moment of observation as well as moment of input signal application.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In other words, a time delay or time advance of input not only shifts the output signal in time but also changes other parameters and behavior. Time variant systems respond differently to the same input at different times. The opposite is true for <a href="Time-invariant_system" title="Time-invariant system">time invariant</a> systems (TIV).

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<div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div>

<p>There are many well developed <a href="LTI_system_theory" class="mw-redirect" title="LTI system theory">techniques</a> for dealing with the response of linear time invariant systems, such as <a href="Laplace_transform" title="Laplace transform">Laplace</a> and <a href="Fourier_transform" title="Fourier transform">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:

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<ul><li>It has explicit dependence on time.</li>

<li>It does not have an <a href="Impulse_response" title="Impulse response">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 variation with the passage of time. See <a href="Stationarity_(statistics)" class="mw-redirect" title="Stationarity (statistics)">Stationarity (statistics)</a> for in-depth theoretics regarding this property.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Linear_time-variant_systems">Linear time-variant systems</h2></div>

<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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<annotation encoding="application/x-tex">{\displaystyle y(t)=f(x(t),t)}</annotation>

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</math></span><img src="./360c1cb16219b90903d89170ef95c33d2a3df973.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.843ex; height:2.843ex;" alt="{\displaystyle y(t)=f(x(t),t)}" loading="lazy"></span></dd></dl>

<p>In order to solve time-variant systems, the algebraic methods <a href="https://en.wikibooks.org/wiki/Control_Systems/Time_Variant_System_Solutions" class="extiw external" title="wikibooks:Control Systems/Time Variant System Solutions">consider</a> initial conditions of the system i.e. whether the system is zero-input or non-zero input system.

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<div class="mw-heading mw-heading2"><h2 id="Examples_of_time-variant_systems">Examples of time-variant systems</h2></div>

<p>The following time varying systems cannot be modelled by assuming that they are time invariant:

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<ul><li>The Earth's thermodynamic response to incoming <a href="Solar_irradiance" title="Solar irradiance">Solar irradiance</a> varies with time due to changes in the Earth's <a href="Albedo" title="Albedo">albedo</a> and the presence of <a href="Greenhouse_gas" title="Greenhouse gas">greenhouse gases</a> in the atmosphere.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li>

<li><a href="Discrete_wavelet_transform" title="Discrete wavelet transform">Discrete wavelet transform</a>, often used in modern signal processing, is time variant because it makes use of the <a href="Decimation_(signal_processing)" class="mw-redirect" title="Decimation (signal processing)">decimation</a> operation.</li></ul>

<ul><li><a href="Control_system" title="Control system">Control system</a></li>

<li><a href="Control_theory" title="Control theory">Control theory</a></li>

<li><a href="System_analysis" title="System analysis">System analysis</a></li>

<li><a href="Time-invariant_system" title="Time-invariant system">Time-invariant system</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFCherniakov2003" class="citation book cs1">Cherniakov, Mikhail (2003). <i>An Introduction to Parametric Digital Filters and Oscillators</i>. Wiley. pp. <span class="nowrap">47–</span>49. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0470851043</bdi>.</cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFSungYoonKim2015" class="citation journal cs1">Sung, Taehong; Yoon, Sang; Kim, Kyung (2015-07-13). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fen8077058">"A Mathematical Model of Hourly Solar Radiation in Varying Weather Conditions for a Dynamic Simulation of the Solar Organic Rankine Cycle"</a>. <i>Energies</i>. <b>8</b> (7): <span class="nowrap">7058–</span>7069. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fen8077058">10.3390/en8077058</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1996-1073">1996-1073</a>.</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFAlzahraniShamsiDagliFerdowsi2017" class="citation journal cs1">Alzahrani, Ahmad; Shamsi, Pourya; Dagli, Cihan; Ferdowsi, Mehdi (2017). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.procs.2017.09.045">"Solar Irradiance Forecasting Using Deep Neural Networks"</a>. <i>Procedia Computer Science</i>. <b>114</b>: <span class="nowrap">304–</span>313. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.procs.2017.09.045">10.1016/j.procs.2017.09.045</a></span>.</cite></span>

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