Built site for gh-pages

.nojekyll

@@ -1 +1 @@
-f63be0ef
+1b4ccf87

des_automata.html

@@ -2143,16 +2143,16 @@ <h3 class="anchored" data-anchor-id="mealy-machine">Mealy machine</h3>
 <span id="cb3-11"><a href="#cb3-11" aria-hidden="true" tabindex="-1"></a><span class="pp">@show</span> <span class="fu">update!</span>(dtr), <span class="fu">output</span>(dtr)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </details>
 <div class="cell-output cell-output-stdout">
-<pre><code>x_initial = rand(0:k, n) = [0, 4, 0, 4]
-output(dtr) = [0, 1, 1, 1]
-(update!(dtr), output(dtr)) = ([0, 0, 4, 0], [1, 0, 1, 1])
-(update!(dtr), output(dtr)) = ([1, 0, 0, 4], [0, 1, 0, 1])
-(update!(dtr), output(dtr)) = ([1, 1, 0, 0], [0, 0, 1, 0])
-(update!(dtr), output(dtr)) = ([1, 1, 1, 0], [0, 0, 0, 1])
-(update!(dtr), output(dtr)) = ([1, 1, 1, 1], [1, 0, 0, 0])</code></pre>
+<pre><code>x_initial = rand(0:k, n) = [4, 2, 3, 3]
+output(dtr) = [0, 1, 1, 0]
+(update!(dtr), output(dtr)) = ([4, 4, 2, 3], [0, 0, 1, 1])
+(update!(dtr), output(dtr)) = ([4, 4, 4, 2], [0, 0, 0, 1])
+(update!(dtr), output(dtr)) = ([4, 4, 4, 4], [1, 0, 0, 0])
+(update!(dtr), output(dtr)) = ([1, 4, 4, 4], [0, 1, 0, 0])
+(update!(dtr), output(dtr)) = ([1, 1, 4, 4], [0, 0, 1, 0])</code></pre>
 </div>
 <div class="cell-output cell-output-display" data-execution_count="1">
-<pre><code>([1, 1, 1, 1], [1, 0, 0, 0])</code></pre>
+<pre><code>([1, 1, 4, 4], [0, 0, 1, 0])</code></pre>
 </div>
 </div>
 <p>We can see that although initially the there can be more tokens, after a few iterations the algorithm achieves the goal of having just one token in the ring.</p>

hybrid_automata.html

@@ -765,7 +765,7 @@ <h1 class="title">Hybrid automata</h1>
 <div id="exm-thermostat" class="theorem example">
 <p><span class="theorem-title"><strong>Example 1 (Thermostat – the hello world example of a hybrid automaton)</strong></span> The thermostat is a device that turns some heater <code>on</code> or <code>off</code> (or sets some valve open or closed) based on the sensed temperature. The goal is to keep the temperature around, say, <span class="math inline">18^\circ</span> C.</p>
 <p>Naturally, the discrete states (modes, locations) are <code>on</code> and <code>off</code>. Initially, the heater is <code>off</code>. We can identify the first two components of the hybrid automaton: <span class="math display">\mathcal Q = \{\text{on}, \text{off}\}, \quad \mathcal Q_0 = \{\text{off}\}</span></p>
-<p>The only continuous state variable is the temperature. The initial temperature not not quite certain, say it is known to be in the interval <span class="math inline">[5,10]</span>. Two more components of the hybrid automaton follow: <span class="math display">\mathcal X = \mathbb R, \quad \mathcal X_0 = \{x:x\in \mathcal X, 5\leq x\leq 10\}</span></p>
+<p>The only continuous state variable is the temperature. The initial temperature is not quite certain, say, it is known to be in the interval <span class="math inline">[5,10]</span>. Two more components of the hybrid automaton follow: <span class="math display">\mathcal X = \mathbb R, \quad \mathcal X_0 = \{x:x\in \mathcal X, 5\leq x\leq 10\}</span></p>
 <p>In the two modes <code>on</code> and <code>off</code>, the evolution of the temperature can be modelled by two different ODEs. Either from first-principles modelling or from system identification (or preferrably from the combination of the two) we get the two differential equations, say: <span class="math display">
 f_\text{off}(x) = -0.1x,\quad f_\text{on}(x) = -0.1x + 5,
 </span> which gives another component for the hybrid automaton.</p>

sitemap.xml

@@ -22,7 +22,7 @@
   </url>
   <url>
     <loc>https://hurak.github.io/hys/stability_recap.html</loc>
-    <lastmod>2025-01-21T19:23:17.008Z</lastmod>
+    <lastmod>2025-01-27T16:25:56.654Z</lastmod>
   </url>
   <url>
     <loc>https://hurak.github.io/hys/petri_nets_timed.html</loc>
@@ -46,7 +46,7 @@
   </url>
   <url>
     <loc>https://hurak.github.io/hys/stability_concepts.html</loc>
-    <lastmod>2024-11-12T17:32:24.731Z</lastmod>
+    <lastmod>2025-01-27T16:31:29.770Z</lastmod>
   </url>
   <url>
     <loc>https://hurak.github.io/hys/max_plus_systems.html</loc>
@@ -74,7 +74,7 @@
   </url>
   <url>
     <loc>https://hurak.github.io/hys/classes_switched.html</loc>
-    <lastmod>2025-01-23T12:07:55.774Z</lastmod>
+    <lastmod>2025-01-27T16:25:14.831Z</lastmod>
   </url>
   <url>
     <loc>https://hurak.github.io/hys/hybrid_system_course_mindmap.html</loc>
@@ -134,15 +134,15 @@
   </url>
   <url>
     <loc>https://hurak.github.io/hys/stability_via_common_lyapunov_function.html</loc>
-    <lastmod>2024-12-23T21:24:32.414Z</lastmod>
+    <lastmod>2025-01-27T16:26:27.094Z</lastmod>
   </url>
   <url>
     <loc>https://hurak.github.io/hys/mld_logic_vs_inequalities.html</loc>
     <lastmod>2025-01-14T15:19:55.746Z</lastmod>
   </url>
   <url>
     <loc>https://hurak.github.io/hys/hybrid_equations.html</loc>
-    <lastmod>2025-01-10T21:39:02.096Z</lastmod>
+    <lastmod>2025-01-27T16:22:03.552Z</lastmod>
   </url>
   <url>
     <loc>https://hurak.github.io/hys/des_references.html</loc>
@@ -214,7 +214,7 @@
   </url>
   <url>
     <loc>https://hurak.github.io/hys/stability_via_multiple_lyapunov_function.html</loc>
-    <lastmod>2025-01-11T11:59:09.118Z</lastmod>
+    <lastmod>2025-01-27T16:29:55.442Z</lastmod>
   </url>
   <url>
     <loc>https://hurak.github.io/hys/complementarity_references.html</loc>
@@ -230,6 +230,6 @@
   </url>
   <url>
     <loc>https://hurak.github.io/hys/hybrid_automata.html</loc>
-    <lastmod>2025-01-10T21:43:46.174Z</lastmod>
+    <lastmod>2025-01-27T16:21:30.700Z</lastmod>
   </url>
 </urlset>

stability_concepts.html

@@ -708,7 +708,7 @@ <h2 class="anchored" data-anchor-id="equilibrium-of-a-hybrid-system-modelled-by-
 </div>
 </div>
 <p>We now consider a hybrid automaton for which the dynamics of each individual <em>mode</em> <span class="math inline">q</span> is given by <span class="math inline">\dot{\bm x} = \mathbf f_q(\bm x)</span>. The <em>invariants</em> (or <em>domains</em>) of each mode are <span class="math inline">\mathcal X_q, \, q=1, \ldots, m</span>.</p>
-<p>The definition of the equilibrium <span class="math inline">\bm x_\mathrm{eq}</span> that is ofter found in the literature imposes these two conditions:</p>
+<p>The definition of the equilibrium <span class="math inline">\bm x_\mathrm{eq}</span> that is often found in the literature imposes these two conditions:</p>
 <ul>
 <li><span class="math inline">\mathbf 0 = \mathbf f_q(\bm x_\mathrm{eq})</span> for all <span class="math inline">q\in \mathcal Q</span>,</li>
 <li>the reset map <span class="math inline">r(q,q',\bm x_\mathrm{eq}) = \bm x_\mathrm{eq}</span>.</li>
@@ -720,7 +720,7 @@ <h2 class="anchored" data-anchor-id="equilibrium-of-a-hybrid-system-modelled-by-
 <h2 class="anchored" data-anchor-id="equilibrium-of-a-hybrid-system-modelled-by-hybrid-equations">Equilibrium of a hybrid system modelled by hybrid equations</h2>
 <p>The state vector within this modelling framework is composed by both the discrete and continuous state variables. The two conditions for the equilibrium of a hybrid automata can be translated into the hybrid equation framework, which means that the equilibrium is not just a single point but rather a set of points.</p>
 <div id="exm-equilibrium-hybrid-equations" class="theorem example">
-<p><span class="theorem-title"><strong>Example 1 (Equilibrium of a hybrid system modelled by hybrid equations)</strong></span> Consider a hybrid system modelled by hybrid equations, for which the state space is given by <span class="math inline">\mathcal X = \{0,1\} \times \mathbb R</span>. The dynamics of the system is given by</p>
+<p><span class="theorem-title"><strong>Example 1 (Equilibrium of a hybrid system modelled by hybrid equations)</strong></span> Consider a hybrid system modelled by hybrid equations, for which the state space is given by <span class="math inline">\mathcal X = \{0,1\} \times \mathbb R</span>. The dynamics of the system is given by [TBD]</p>
 </div>
 <p>This makes the analysis significantly more challenging. Therefore, in our lecture we will only consider stability of hybrid automata.</p>
 </section>

stability_recap.html

@@ -716,7 +716,7 @@ <h1 class="title">Recap of stability analysis for continuous dynamical systems</
 <h2 class="anchored" data-anchor-id="equilibrium">Equilibrium</h2>
 <p>Loosely speaking, equilibrium is a state at which the system can rest indefinitely when undisturbed by external disturbances. More technically speaking, equilibrium is a point in the state space, that is, a vector <span class="math inline">\bm x_\mathrm{eq}\in \mathbb R^n</span>, at which the vector field <span class="math inline">\mathbf f</span> vanishes, that is,<br>
 <span class="math display">\mathbf f(\bm x_\mathrm{eq}) = \mathbf 0.</span></p>
-<p>Without loss of generality we often assume that <span class="math inline">\bm x_\mathrm{eq} = \mathbf 0</span>, because if the equilibrium is considered anywhere else than at the origin, we can alway introduce a new <em>shifted</em> state vector <span class="math inline">\bm x_\mathrm{new}(t) = \bm x(t) - \bm x_\mathrm{eq}</span>.</p>
+<p>Without loss of generality we often assume that <span class="math inline">\bm x_\mathrm{eq} = \mathbf 0</span>, because if the equilibrium is considered anywhere else than at the origin, we can always introduce a new <em>shifted</em> state vector <span class="math inline">\bm x_\mathrm{new}(t) = \bm x(t) - \bm x_\mathrm{eq}</span>.</p>
 <div class="callout callout-style-default callout-caution callout-titled">
 <div class="callout-header d-flex align-content-center">
 <div class="callout-icon-container">
@@ -765,8 +765,8 @@ <h2 class="anchored" data-anchor-id="attractivity">Attractivity</h2>
 </section>
 <section id="asymptotic-stability" class="level2">
 <h2 class="anchored" data-anchor-id="asymptotic-stability">Asymptotic stability</h2>
-<p>Combination of Lyapunov stability and attractivity is called assymptotic stability.</p>
-<p>If the attractivity is global, the assymptotic stability is called global too.</p>
+<p>Combination of Lyapunov stability and attractivity is called asymptotic stability.</p>
+<p>If the attractivity is global, the asymptotic stability is called global too.</p>
 </section>
 <section id="exponential-stability" class="level2">
 <h2 class="anchored" data-anchor-id="exponential-stability">Exponential stability</h2>

stability_via_common_lyapunov_function.html

@@ -857,7 +857,7 @@ <h2 class="anchored" data-anchor-id="solution-set-of-an-lmi-is-convex">Solution
 </section>
 <section id="what-if-quadratic-lf-is-not-enough" class="level2">
 <h2 class="anchored" data-anchor-id="what-if-quadratic-lf-is-not-enough">What if quadratic LF is not enough?</h2>
-<p>So far we considered quadratic Lyapunov functions – and tt may be useful to display their prescription explicitly in the scalar form <span class="math display">
+<p>So far we considered quadratic Lyapunov functions – and it may be useful to display their prescription explicitly in the scalar form <span class="math display">
 \begin{aligned}
 V(\bm x) &amp;= \bm x^\top \mathbf P \bm x\\
 &amp;= \begin{bmatrix}x_1 &amp; x_2\end{bmatrix} \begin{bmatrix} p_{11} &amp; p_{12}\\ p_{12} &amp; p_{22}\end{bmatrix} \begin{bmatrix}x_1\\ x_2\end{bmatrix}\\

stability_via_multiple_lyapunov_function.html

@@ -796,7 +796,7 @@ <h1 class="title">Stability via multiple Lyapunov functions</h1>
    │     └─ Convex.NegateAtom (affine; real)
    │        └─ …
    ├─ PSD constraint (convex)
-   │  └─ 2×2 real variable (id: 232…843)
+   │  └─ 2×2 real variable (id: 726…247)
    ⋮</code></pre>
 </div>
 </div>
@@ -973,7 +973,7 @@ <h2 class="anchored" data-anchor-id="using-comparison-functions-and-nonstrict-in
 <p><span class="math display">
 \bm x^\top \left( \mathbf A_i^\top \mathbf P_i + \mathbf P_i \mathbf A_i \right) \bm x \leq  -\alpha_3 \bm x^\top \mathbf I \bm x\quad \forall \;\bm x\in \Omega_i.
 </span></p>
-<p>The difference now is that these conditions are only reuired to hold on some state regions, some subsets of the state space. It is now time to discuss how to characterize those regions.</p>
+<p>The difference now is that these conditions are only required to hold on some state regions, some subsets of the state space. It is now time to discuss how to characterize those regions.</p>
 </section>
 <section id="characterization-of-subsets-of-state-space-using-lmi" class="level2">
 <h2 class="anchored" data-anchor-id="characterization-of-subsets-of-state-space-using-lmi">Characterization of subsets of state space using LMI</h2>