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hurak committed Oct 30, 2024
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2 changes: 1 addition & 1 deletion .nojekyll
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<h2 id="toc-title">On this page</h2>

<ul>
<li><a href="#approximation-of-smooth-systems" id="toc-approximation-of-smooth-systems" class="nav-link active" data-scroll-target="#approximation-of-smooth-systems">Approximation of smooth systems</a></li>
<li><a href="#approximation-of-nonlinear-systems" id="toc-approximation-of-nonlinear-systems" class="nav-link active" data-scroll-target="#approximation-of-nonlinear-systems">Approximation of nonlinear systems</a></li>
</ul>
<div class="toc-actions"><ul><li><a href="https://github.com/hurak/hys/issues/new" class="toc-action"><i class="bi bi-github"></i>Report an issue</a></li></ul></div></nav>
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Expand All @@ -654,7 +654,7 @@ <h1 class="title">Piecewise affine (PWA) systems</h1>


<p>This is a subclass of switched systems where the functions on the right-hand side of the differential equations are affine functions of the state. For some (historical) reason these systems are also called <em>piecewise linear</em> (PWL).</p>
<p>We are going to restrict ourselves to state-driven switching.</p>
<p>We are going to reformulate such systems as switched systems with state-driven switching.</p>
<p>First, we consider the autonomous case, that is, systems without inputs: <span class="math display">
\dot{\bm x}
=
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\end{aligned}
</span></p>
</div>
<section id="approximation-of-smooth-systems" class="level2">
<h2 class="anchored" data-anchor-id="approximation-of-smooth-systems">Approximation of smooth systems</h2>
<section id="approximation-of-nonlinear-systems" class="level2">
<h2 class="anchored" data-anchor-id="approximation-of-nonlinear-systems">Approximation of nonlinear systems</h2>
<p>While the example with the saturated linear state feedback can be modelled as a PWA system exactly, there are many practical cases, in which the system is not exactly PWA affine but we want to approximate it as such.</p>
<div id="exm-smooth-approximation" class="theorem example">
<p><span class="theorem-title"><strong>Example 2 (Smooth system approximated by a PWA system)</strong></span> Consider the following nonlinear system <span class="math display">
<div id="exm-nonlinear-approximation" class="theorem example">
<p><span class="theorem-title"><strong>Example 2 (Nonlinear system approximated by a PWA system)</strong></span> Consider the following nonlinear system <span class="math display">
\begin{bmatrix}
\dot x_1\\\dot x_2
\end{bmatrix}
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