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hurak · Dec 17, 2024 · 8c0767f · 8c0767f
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diff --git a/.nojekyll b/.nojekyll
@@ -1 +1 @@
-ab5b338b
+62f3a585
diff --git a/classes_reset.html b/classes_reset.html
diff --git a/classes_switched.html b/classes_switched.html
diff --git a/complementarity_references 25.html b/complementarity_references 25.html
diff --git a/complementarity_simulations.html b/complementarity_simulations.html
diff --git a/des_automata.html b/des_automata.html
@@ -2131,16 +2131,16 @@ <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) = [2, 0, 2, 3]
-output(dtr) = [0, 1, 1, 1]
-(update!(dtr), output(dtr)) = ([2, 2, 0, 2], [1, 0, 1, 1])
-(update!(dtr), output(dtr)) = ([3, 2, 2, 0], [0, 1, 0, 1])
-(update!(dtr), output(dtr)) = ([3, 3, 2, 2], [0, 0, 1, 0])
-(update!(dtr), output(dtr)) = ([3, 3, 3, 2], [0, 0, 0, 1])
-(update!(dtr), output(dtr)) = ([3, 3, 3, 3], [1, 0, 0, 0])</code></pre>
+<pre><code>x_initial = rand(0:k, n) = [0, 2, 2, 3]
+output(dtr) = [0, 1, 0, 1]
+(update!(dtr), output(dtr)) = ([0, 0, 2, 2], [0, 0, 1, 0])
+(update!(dtr), output(dtr)) = ([0, 0, 0, 2], [0, 0, 0, 1])
+(update!(dtr), output(dtr)) = ([0, 0, 0, 0], [1, 0, 0, 0])
+(update!(dtr), output(dtr)) = ([1, 0, 0, 0], [0, 1, 0, 0])
+(update!(dtr), output(dtr)) = ([1, 1, 0, 0], [0, 0, 1, 0])</code></pre>
 </div>
 <div class="cell-output cell-output-display" data-execution_count="1">
-<pre><code>([3, 3, 3, 3], [1, 0, 0, 0])</code></pre>
+<pre><code>([1, 1, 0, 0], [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>

diff --git a/hybrid_equations.html b/hybrid_equations.html
diff --git a/hybrid_equations_references 25.html b/hybrid_equations_references 25.html
diff --git a/intro_outline 26.html b/intro_outline 26.html
diff --git a/max_plus_algebra.html b/max_plus_algebra.html
diff --git a/mld_DHA 24.html b/mld_DHA 24.html
diff --git a/mld_intro 25.html b/mld_intro 25.html
diff --git a/mld_intro.html b/mld_intro.html
diff --git a/mpc_mld_explicit.html b/mpc_mld_explicit.html
diff --git a/mpc_mld_online.html b/mpc_mld_online.html
@@ -69,8 +69,22 @@
   }
 }</script>
 
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@@ -648,24 +662,24 @@ <h2 class="anchored" data-anchor-id="optimal-control-on-a-finite-horizon">Optima
 <section id="cost-function" class="level3">
 <h3 class="anchored" data-anchor-id="cost-function">Cost function</h3>
 <p>First, we need to set the cost function for the optimal control problem. As usual in optimal control, we want to impose different weights on invididual state and control variables. The most popular is the quadratic cost function well known from the LQ-optimal control</p>
-<p><span class="math display">\[
+<p><span class="math display">
 J_0(x(0),U_0) = x_N^T S_N x_N + \sum_{k=0}^{N-1} \left( x_k^T Q x_k + u_k^T R u_k \right)
-\]</span></p>
+</span></p>
 <p>But other (weighted) norms can also be used, in particular 1-norm and infinity-norm</p>
-<p><span class="math display">\[
+<p><span class="math display">
 J_0(x(0),U_0) = \|S_N x_N\|_1 + \sum_{k=0}^{N-1} \left( \|Q x_k\|_1 + \|R u_k\|_1 \right),
-\]</span></p>
-<p><span class="math display">\[
+</span></p>
+<p><span class="math display">
 J_0(x(0),U_0) = \|S_N x_N\|_{\infty} + \sum_{k=0}^{N-1} \left( \|Q x_k\|_{\infty} + \|R u_k\|_{\infty} \right).
-\]</span></p>
+</span></p>
 </section>
 </section>
 <section id="optimization-problem" class="level2">
 <h2 class="anchored" data-anchor-id="optimization-problem">Optimization problem</h2>
-<p>Combining the cost function with the MLD model, and perhaps we some extra constraints imposed on the control inputs as well as state variables, we get <span class="math display">\[
+<p>Combining the cost function with the MLD model, and perhaps we some extra constraints imposed on the control inputs as well as state variables, we get <span class="math display">
 \operatorname*{minimize}_{u_0, u_1, \ldots, u_{N-1}} J_0(x(0),(u_0, u_1, \ldots, u_{N-1}))
-\]</span></p>
-<p>subject to <span class="math display">\[
+</span></p>
+<p>subject to <span class="math display">
 \begin{aligned}
 x_{k+1} &amp;= Ax_k + B_u u_k + B_\delta\delta_k + B_z z_k + B_0\\
 y_k &amp;= Cx_k + D_u u_k + D_\delta \delta_k + D_z z_k + D_0\\
@@ -675,7 +689,7 @@ <h2 class="anchored" data-anchor-id="optimization-problem">Optimization problem<
 P x_N &amp;\leq r \\
 x_0 &amp;= x(0)
 \end{aligned}
-\]</span></p>
+</span></p>
 
 
 </section>