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+
+  <h1>Source code for spikes.utils</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">spikes</span> 
+<span class="kn">import</span> <span class="nn">sympy</span>
+
+<div class="viewcode-block" id="characteristic_polynomial">
+<a class="viewcode-back" href="../../index.html#spikes.utils.characteristic_polynomial">[docs]</a>
+<span class="k">def</span> <span class="nf">characteristic_polynomial</span><span class="p">(</span><span class="n">matrix</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    Computes the characteristic polynomial of a given matrix.</span>
+
+<span class="sd">    Parameters</span>
+<span class="sd">    ----------</span>
+<span class="sd">    matrix: (list of lists) </span>
+<span class="sd">        A 2D list representing the matrix.</span>
+
+<span class="sd">    Returns</span>
+<span class="sd">    -------</span>
+<span class="sd">    sympy.Poly: </span>
+<span class="sd">        The characteristic polynomial of the matrix.</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="c1"># Define the symbolic variable</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
+
+    <span class="c1"># Convert the input matrix to a sympy.Matrix</span>
+    <span class="n">A</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span><span class="p">(</span><span class="n">matrix</span><span class="p">)</span>
+
+    <span class="c1"># Compute the characteristic polynomial</span>
+    <span class="n">char_poly</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">charpoly</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+
+    <span class="c1"># Get the polynomial coefficients</span>
+    <span class="n">coefficients</span> <span class="o">=</span> <span class="n">char_poly</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span>
+
+    <span class="c1"># Create the polynomial using sympy.Poly</span>
+    <span class="n">p</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Poly</span><span class="p">(</span><span class="n">coefficients</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">p</span></div>
+
+
+
+<div class="viewcode-block" id="routh">
+<a class="viewcode-back" href="../../index.html#spikes.utils.routh">[docs]</a>
+<span class="k">def</span> <span class="nf">routh</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot; Construct the Routh-Hurwitz array given a polynomial in s</span>
+
+<span class="sd">    Parameters</span>
+<span class="sd">    ----------</span>
+<span class="sd">    p: sympy.Poly</span>
+<span class="sd">        The characteristic polynomial of coefficient matrix</span>
+<span class="sd">    </span>
+<span class="sd">    Returns</span>
+<span class="sd">    -------</span>
+<span class="sd">    value: sympy.Matrix</span>
+<span class="sd">        The Routh-Hurwitz array</span>
+<span class="sd">    </span>
+<span class="sd">    References https://github.com/alchemyst/Dynamics-and-Control/blob/master/tbcontrol/symbolic.py</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="n">coefficients</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span>
+    <span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coefficients</span><span class="p">)</span>
+    <span class="n">M</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+
+    <span class="n">r1</span> <span class="o">=</span> <span class="n">coefficients</span><span class="p">[</span><span class="mi">0</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span>
+    <span class="n">r2</span> <span class="o">=</span> <span class="n">coefficients</span><span class="p">[</span><span class="mi">1</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span>
+    <span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:</span><span class="nb">len</span><span class="p">(</span><span class="n">r1</span><span class="p">)]</span> <span class="o">=</span> <span class="p">[</span><span class="n">r1</span><span class="p">]</span>
+    <span class="n">M</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:</span><span class="nb">len</span><span class="p">(</span><span class="n">r2</span><span class="p">)]</span> <span class="o">=</span> <span class="p">[</span><span class="n">r2</span><span class="p">]</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">N</span><span class="p">):</span>
+        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="o">//</span><span class="mi">2</span><span class="p">):</span>
+            <span class="n">S</span> <span class="o">=</span> <span class="n">M</span><span class="p">[[</span><span class="n">i</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]]</span>
+            <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span><span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">det</span><span class="p">()</span><span class="o">/</span><span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
+        <span class="c1"># If a row of the routh table becomes zero,Take the derivative of the previous row and substitute it instead</span>
+        <span class="c1"># Ref: Norman S. Nise, Control Systems Engineering, 8th Edition, Chapter 6, Section 3</span>
+        <span class="k">if</span> <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">==</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">])]):</span>
+            <span class="c1"># Find the coefficients on taking the derivative of the Auxiliary polynomial</span>
+            <span class="n">diff_arr</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="o">-</span><span class="n">i</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">))</span>
+            <span class="n">diff_arr</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">diff_arr</span><span class="p">)))</span>
+            <span class="n">diff_arr</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span><span class="p">([</span><span class="n">diff_arr</span><span class="p">])</span>
+            <span class="c1"># Multiply the coefficients with the value in previous row</span>
+            <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">matrix_multiply_elementwise</span><span class="p">(</span><span class="n">diff_arr</span><span class="p">,</span> <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:])</span>
+    <span class="k">return</span> <span class="n">M</span><span class="p">[:,</span> <span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span></div>
+
+</pre></div>
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