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hurak committed Nov 18, 2024
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2 changes: 1 addition & 1 deletion .nojekyll
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"><head>

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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"><head>

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</section>
<section id="piecewise-affine-pwa-systems" class="level2">
<h2 class="anchored" data-anchor-id="piecewise-affine-pwa-systems">Piecewise affine (PWA) systems</h2>
<p>In our course we based our treatment of PWA systems on the monograph <span class="citation" data-cites="johanssonPiecewiseLinearControl2003"><a href="#ref-johanssonPiecewiseLinearControl2003" role="doc-biblioref">[12]</a></span>. It is not freely available online, but it is based on the author’s PhD thesis <span class="citation" data-cites="johanssonPiecewiseLinearControl1999"><a href="#ref-johanssonPiecewiseLinearControl1999" role="doc-biblioref">[13]</a></span>, which is available online. While these resources are a bit outdated (in particular, when it comes to stability analysis, back then they were not aware of the possibility to extend the S-procedure to higher-degree polynomials), they still a good starting point. From about the same time, the paper <span class="citation" data-cites="hassibiQuadraticStabilizationControl1998"><a href="#ref-hassibiQuadraticStabilizationControl1998" role="doc-biblioref">[14]</a></span> reads well (as usual in the case of the second author). A bit more up-to-date book dedicated purely to PWA control <span class="citation" data-cites="rodriguesPiecewiseAffineControl2019"><a href="#ref-rodriguesPiecewiseAffineControl2019" role="doc-biblioref">[15]</a></span>, but again, no free online version. The book refers to the Matlab toolbox documented in <span class="citation" data-cites="fekriPWATOOLSMATLABToolbox2012"><a href="#ref-fekriPWATOOLSMATLABToolbox2012" role="doc-biblioref">[16]</a></span>. While the toolbox is rather dated and will hardly run on the current versions of Matlab (perhaps an opportunity for nice student project), the tutorial paper gives some insight into how the whole concept of a PWA approximation can be used in control design.</p>
<p>In our course we based our treatment of PWA systems on the monograph <span class="citation" data-cites="johanssonPiecewiseLinearControl2003"><a href="#ref-johanssonPiecewiseLinearControl2003" role="doc-biblioref">[12]</a></span>. It is not freely available online, but it is based on the author’s PhD thesis <span class="citation" data-cites="johanssonPiecewiseLinearControl1999"><a href="#ref-johanssonPiecewiseLinearControl1999" role="doc-biblioref">[13]</a></span>, which is available online. While these resources are a bit outdated (in particular, when it comes to stability analysis, back then they were not aware of the possibility to extend the S-procedure to higher-degree polynomials), they still constitute a good starting point. Published at about the same time, the paper <span class="citation" data-cites="hassibiQuadraticStabilizationControl1998"><a href="#ref-hassibiQuadraticStabilizationControl1998" role="doc-biblioref">[14]</a></span> reads well (as usual in the case of the second author). A bit more up-to-date book dedicated purely to PWA control is <span class="citation" data-cites="rodriguesPiecewiseAffineControl2019"><a href="#ref-rodriguesPiecewiseAffineControl2019" role="doc-biblioref">[15]</a></span>, but again, no free online version. The book refers to the Matlab toolbox documented in <span class="citation" data-cites="fekriPWATOOLSMATLABToolbox2012"><a href="#ref-fekriPWATOOLSMATLABToolbox2012" role="doc-biblioref">[16]</a></span>. While the toolbox is rather dated and will hardly run on the current versions of Matlab (perhaps an opportunity for nice student project), the tutorial paper gives some insight into how the whole concept of a PWA approximation can be used in control design.</p>
</section>
<section id="piecewise-affine-linear-approximation" class="level2">
<h2 class="anchored" data-anchor-id="piecewise-affine-linear-approximation">Piecewise affine (linear) approximation</h2>
<p>There is quite a lot of relevant know-how available even outside the domain of (control) systems, in particular, search for piecewise affine (-linear) approximation or fitting (using optimization): <span class="citation" data-cites="magnaniConvexPiecewiselinearFitting2008"><a href="#ref-magnaniConvexPiecewiselinearFitting2008" role="doc-biblioref">[17]</a></span> (although it is only restricted to convex functions), <span class="citation" data-cites="huchetteNonconvexPiecewiseLinear2022"><a href="#ref-huchetteNonconvexPiecewiseLinear2022" role="doc-biblioref">[18]</a></span>, …</p>
<p>There is quite a lot of relevant know-how available even outside the domain of (control) systems, in particular, search for piecewise affine (-linear) approximation or fitting (using optimization): <span class="citation" data-cites="magnaniConvexPiecewiselinearFitting2008"><a href="#ref-magnaniConvexPiecewiselinearFitting2008" role="doc-biblioref">[17]</a></span> (although it is only restricted to convex functions), <span class="citation" data-cites="huchetteNonconvexPiecewiseLinear2022"><a href="#ref-huchetteNonconvexPiecewiseLinear2022" role="doc-biblioref">[18]</a></span>, <span class="citation" data-cites="torielloFittingPiecewiseLinear2012"><a href="#ref-torielloFittingPiecewiseLinear2012" role="doc-biblioref">[19]</a></span>, </p>



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<div id="ref-huchetteNonconvexPiecewiseLinear2022" class="csl-entry" role="listitem">
<div class="csl-left-margin">[18] </div><div class="csl-right-inline">J. Huchette and J. P. Vielma, <span>“Nonconvex <span>Piecewise Linear Functions</span>: <span>Advanced Formulations</span> and <span>Simple Modeling Tools</span>,”</span> <em>Operations Research</em>, May 2022, doi: <a href="https://doi.org/10.1287/opre.2019.1973">10.1287/opre.2019.1973</a>.</div>
</div>
<div id="ref-torielloFittingPiecewiseLinear2012" class="csl-entry" role="listitem">
<div class="csl-left-margin">[19] </div><div class="csl-right-inline">A. Toriello and J. P. Vielma, <span>“Fitting piecewise linear continuous functions,”</span> <em>European Journal of Operational Research</em>, vol. 219, no. 1, pp. 86–95, May 2012, doi: <a href="https://doi.org/10.1016/j.ejor.2011.12.030">10.1016/j.ejor.2011.12.030</a>.</div>
</div>
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text: getTextToCopy,
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