From a865af66ce3f3f39ac9fd27469b5f66dd2b9914a Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Mon, 26 Oct 2020 14:04:35 +0100 Subject: [PATCH] Change plot output to matlab figures --- figs/filter_high_pass_first_order.png | Bin 0 -> 104842 bytes figs/filter_high_pass_first_order_add.png | Bin 0 -> 133926 bytes figs/filter_high_pass_second_order.png | Bin 0 -> 159716 bytes figs/filter_lag.png | Bin 0 -> 124551 bytes figs/filter_lag_effect_a_phase.png | Bin 0 -> 20618 bytes figs/filter_lead.png | Bin 0 -> 125919 bytes figs/filter_lead_effect_a_phase.png | Bin 0 -> 18510 bytes figs/filter_low_pass_first_order.png | Bin 0 -> 104712 bytes figs/filter_low_pass_first_order_add.png | Bin 0 -> 129690 bytes figs/filter_low_pass_second_order.png | Bin 0 -> 160512 bytes figs/filter_notch_gc.png | Bin 0 -> 94580 bytes figs/filter_notch_xi.png | Bin 0 -> 90576 bytes figs/weight_first_order.png | Bin 0 -> 18250 bytes index.html | 1095 +++++++------- index.org | 1656 +++++++++------------ 15 files changed, 1189 insertions(+), 1562 deletions(-) create mode 100644 figs/filter_high_pass_first_order.png create mode 100644 figs/filter_high_pass_first_order_add.png create mode 100644 figs/filter_high_pass_second_order.png create mode 100644 figs/filter_lag.png create mode 100644 figs/filter_lag_effect_a_phase.png create mode 100644 figs/filter_lead.png create mode 100644 figs/filter_lead_effect_a_phase.png create mode 100644 figs/filter_low_pass_first_order.png create mode 100644 figs/filter_low_pass_first_order_add.png create mode 100644 figs/filter_low_pass_second_order.png create mode 100644 figs/filter_notch_gc.png create mode 100644 figs/filter_notch_xi.png create mode 100644 figs/weight_first_order.png diff --git a/figs/filter_high_pass_first_order.png b/figs/filter_high_pass_first_order.png new file mode 100644 index 0000000000000000000000000000000000000000..f611b00ad493e8d8be3cf710dc653574e4df6cf1 GIT binary patch literal 104842 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Proportional - Integral - Derivative</a> +<li><a href="#orgb83f79c">1. Low Pass</a> <ul> -<li><a href="#org8aafed7">1.1. Proportional</a></li> -<li><a href="#org37b43e8">1.2. Integral</a></li> -<li><a href="#orgfb65936">1.3. Derivative</a></li> +<li><a href="#orgd88840c">1.1. First Order Low Pass Filter</a></li> +<li><a href="#org0938139">1.2. Second Order</a></li> +<li><a href="#orge580369">1.3. Combine multiple first order filters</a></li> </ul> </li> -<li><a href="#orgfed2e4e">2. Low Pass</a> +<li><a href="#org8c4f98e">2. High Pass</a> <ul> -<li><a href="#orga8faa42">2.1. First Order</a></li> -<li><a href="#org170a9b0">2.2. Second Order</a></li> -<li><a href="#orgf8f61f9">2.3. Combine multiple filters</a></li> -<li><a href="#orge265051">2.4. Nice combination</a></li> +<li><a href="#org946fda0">2.1. First Order</a></li> +<li><a href="#orgdb3a46a">2.2. Second Order</a></li> +<li><a href="#org81b8ec1">2.3. Combine multiple filters</a></li> </ul> </li> -<li><a href="#org7e3ece6">3. High Pass</a> +<li><a href="#org9a4d9f1">3. Band Pass</a> <ul> -<li><a href="#org291d74d">3.1. First Order</a></li> -<li><a href="#org5d4ea15">3.2. Second Order</a></li> -<li><a href="#orge71c2dd">3.3. Combine multiple filters</a></li> +<li><a href="#orgb93eef3">3.1. Second Order</a></li> </ul> </li> -<li><a href="#org791f35b">4. Band Pass</a></li> -<li><a href="#orge054b4d">5. Notch</a></li> -<li><a href="#orgf94d78a">6. Bump</a></li> -<li><a href="#org6255d82">7. Chebyshev</a> +<li><a href="#org06d380e">4. Notch</a> <ul> -<li><a href="#org2bfe9f5">7.1. Chebyshev Type I</a></li> +<li><a href="#org064544f">4.1. Second Order</a></li> </ul> </li> -<li><a href="#org127884a">8. Lead - Lag</a> +<li><a href="#org745749b">5. Chebyshev</a> <ul> -<li><a href="#orgb58ebc8">8.1. Lead</a></li> -<li><a href="#org700da69">8.2. Lag</a></li> -<li><a href="#orgf2125da">8.3. Lead Lag</a></li> +<li><a href="#orgdb4d414">5.1. Chebyshev Type I</a></li> </ul> </li> -<li><a href="#orga776beb">9. Complementary</a></li> -<li><a href="#org18d2496">10. Performance Weight</a></li> -<li><a href="#org2ee1e4e">11. Combine Filters</a> +<li><a href="#org42f9bf3">6. Lead - Lag</a> <ul> -<li><a href="#org890b908">11.1. Additive</a></li> -<li><a href="#org6f10b80">11.2. Multiplicative</a></li> +<li><a href="#org0cb85e5">6.1. Lead</a></li> +<li><a href="#org19e9264">6.2. Lag</a></li> +</ul> +</li> +<li><a href="#org15058b6">7. Complementary</a></li> +<li><a href="#org2d03ba9">8. Performance Weight</a> +<ul> +<li><a href="#orge550845">8.1. Nice combination</a></li> +<li><a href="#org6202dd8">8.2. Alternative</a></li> +</ul> +</li> +<li><a href="#org5e21f67">9. Combine Filters</a> +<ul> +<li><a href="#orga943d8f">9.1. Additive</a></li> +<li><a href="#org6cc036a">9.2. Multiplicative</a></li> +</ul> +</li> +<li><a href="#org80b3ca3">10. Filters representing noise</a> +<ul> +<li><a href="#org579591f">10.1. First Order Low Pass Filter</a></li> </ul> </li> </ul> </div> </div> -<div id="outline-container-orge8f7972" class="outline-2"> -<h2 id="orge8f7972"><span class="section-number-2">1</span> Proportional - Integral - Derivative</h2> +<div id="outline-container-orgb83f79c" class="outline-2"> +<h2 id="orgb83f79c"><span class="section-number-2">1</span> Low Pass</h2> <div class="outline-text-2" id="text-1"> </div> -<div id="outline-container-org8aafed7" class="outline-3"> -<h3 id="org8aafed7"><span class="section-number-3">1.1</span> Proportional</h3> -</div> - -<div id="outline-container-org37b43e8" class="outline-3"> -<h3 id="org37b43e8"><span class="section-number-3">1.2</span> Integral</h3> -</div> - -<div id="outline-container-orgfb65936" class="outline-3"> -<h3 id="orgfb65936"><span class="section-number-3">1.3</span> Derivative</h3> -</div> -</div> - -<div id="outline-container-orgfed2e4e" class="outline-2"> -<h2 id="orgfed2e4e"><span class="section-number-2">2</span> Low Pass</h2> -<div class="outline-text-2" id="text-2"> -</div> -<div id="outline-container-orga8faa42" class="outline-3"> -<h3 id="orga8faa42"><span class="section-number-3">2.1</span> First Order</h3> -<div class="outline-text-3" id="text-2-1"> +<div id="outline-container-orgd88840c" class="outline-3"> +<h3 id="orgd88840c"><span class="section-number-3">1.1</span> First Order Low Pass Filter</h3> +<div class="outline-text-3" id="text-1-1"> <p> \[ H(s) = \frac{1}{1 + s/\omega_0} \] </p> -<div class="org-src-container"> -<pre class="src src-matlab">w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span> +<p> +Parameters: +</p> +<ul class="org-ul"> +<li>\(\omega_0\): cut-off frequency in [rad/s]</li> +</ul> -H = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-1">)</span>; +<p> +Characteristics: +</p> +<ul class="org-ul"> +<li>Low frequency gain of \(1\)</li> +<li>Roll-off equals to -20 dB/dec</li> +</ul> + +<p> +Matlab code: +</p> +<div class="org-src-container"> +<pre class="src src-matlab">w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% Cut-off Frequency [rad/s]</span> + +H = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0); </pre> </div> -<div id="org92da067" class="figure"> -<p><img src="figs/lpf_first_order.png" alt="lpf_first_order.png" /> +<div id="org65bce28" class="figure"> +<p><img src="figs/filter_low_pass_first_order.png" alt="filter_low_pass_first_order.png" /> </p> -<p><span class="figure-number">Figure 1: </span>First Order Low Pass Filter (<a href="./figs/lpf_first_order.png">png</a>, <a href="./figs/lpf_first_order.pdf">pdf</a>.</p> </div> </div> </div> -<div id="outline-container-org170a9b0" class="outline-3"> -<h3 id="org170a9b0"><span class="section-number-3">2.2</span> Second Order</h3> -<div class="outline-text-3" id="text-2-2"> +<div id="outline-container-org0938139" class="outline-3"> +<h3 id="org0938139"><span class="section-number-3">1.2</span> Second Order</h3> +<div class="outline-text-3" id="text-1-2"> <p> \[ H(s) = \frac{1}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \] </p> -<div class="org-src-container"> -<pre class="src src-matlab">w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span> -xi = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">3</span>; +<p> +Parameters: +</p> +<ul class="org-ul"> +<li>\(\omega_0\):</li> +<li>\(\xi\): Damping ratio</li> +</ul> -H = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span>xi<span class="org-type">/</span>w0<span class="org-type">*</span>s <span class="org-type">+</span> s<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>w0<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span>; +<p> +Characteristics: +</p> +<ul class="org-ul"> +<li>Low frequency gain: 1</li> +<li>High frequency roll off: - 40 dB/dec</li> +</ul> + +<p> +Matlab code: +</p> +<div class="org-src-container"> +<pre class="src src-matlab">w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% Cut-off frequency [rad/s]</span> +xi = 0.3; <span class="org-comment">% Damping Ratio</span> + +H = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">/</span>w0<span class="org-type">*</span>s <span class="org-type">+</span> s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2); </pre> </div> -<div id="orgd09d719" class="figure"> -<p><img src="figs/lpf_second_order.png" alt="lpf_second_order.png" /> +<div id="org3d51d53" class="figure"> +<p><img src="figs/filter_low_pass_second_order.png" alt="filter_low_pass_second_order.png" /> </p> -<p><span class="figure-number">Figure 2: </span>Second Order Low Pass Filter (<a href="./figs/lpf_second_order.png">png</a>, <a href="./figs/lpf_second_order.pdf">pdf</a>.</p> </div> </div> </div> - -<div id="outline-container-orgf8f61f9" class="outline-3"> -<h3 id="orgf8f61f9"><span class="section-number-3">2.3</span> Combine multiple filters</h3> -<div class="outline-text-3" id="text-2-3"> +<div id="outline-container-orge580369" class="outline-3"> +<h3 id="orge580369"><span class="section-number-3">1.3</span> Combine multiple first order filters</h3> +<div class="outline-text-3" id="text-1-3"> <p> \[ H(s) = \left( \frac{1}{1 + s/\omega_0} \right)^n \] </p> +<p> +Matlab code: +</p> <div class="org-src-container"> -<pre class="src src-matlab">w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span> -n = <span class="org-highlight-numbers-number">3</span>; +<pre class="src src-matlab">w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% Cut-off frequency [rad/s]</span> +n = 3; <span class="org-comment">% Filter order</span> -H = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span>n; +H = (1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0))<span class="org-type">^</span>n; </pre> </div> -<div id="org05c14e7" class="figure"> -<p><img src="figs/lpf_multiple_first_order.png" alt="lpf_multiple_first_order.png" /> +<div id="org03c708d" class="figure"> +<p><img src="figs/filter_low_pass_first_order_add.png" alt="filter_low_pass_first_order_add.png" /> </p> -<p><span class="figure-number">Figure 3: </span>Combine Multiple First Order Low Pass Filter (<a href="./figs/lpf_multiple_first_order.png">png</a>, <a href="./figs/lpf_multiple_first_order.pdf">pdf</a>.</p> +</div> </div> </div> </div> +<div id="outline-container-org8c4f98e" class="outline-2"> +<h2 id="org8c4f98e"><span class="section-number-2">2</span> High Pass</h2> +<div class="outline-text-2" id="text-2"> +</div> +<div id="outline-container-org946fda0" class="outline-3"> +<h3 id="org946fda0"><span class="section-number-3">2.1</span> First Order</h3> +<div class="outline-text-3" id="text-2-1"> +<p> +\[ H(s) = \frac{s/\omega_0}{1 + s/\omega_0} \] +</p> -<div id="outline-container-orge265051" class="outline-3"> -<h3 id="orge265051"><span class="section-number-3">2.4</span> Nice combination</h3> -<div class="outline-text-3" id="text-2-4"> +<p> +Parameters: +</p> +<ul class="org-ul"> +<li>\(\omega_0\): cut-off frequency in [rad/s]</li> +</ul> + +<p> +Characteristics: +</p> +<ul class="org-ul"> +<li>High frequency gain of \(1\)</li> +<li>Low frequency slow of +20 dB/dec</li> +</ul> + +<p> +Matlab code: +</p> +<div class="org-src-container"> +<pre class="src src-matlab">w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% Cut-off frequency [rad/s]</span> + +H = (s<span class="org-type">/</span>w0)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0); +</pre> +</div> + + +<div id="org5e8496d" class="figure"> +<p><img src="figs/filter_high_pass_first_order.png" alt="filter_high_pass_first_order.png" /> +</p> +</div> +</div> +</div> + +<div id="outline-container-orgdb3a46a" class="outline-3"> +<h3 id="orgdb3a46a"><span class="section-number-3">2.2</span> Second Order</h3> +<div class="outline-text-3" id="text-2-2"> +<p> +\[ H(s) = \frac{s^2/\omega_0^2}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \] +</p> + +<p> +Parameters: +</p> +<ul class="org-ul"> +<li>\(\omega_0\):</li> +<li>\(\xi\): Damping ratio</li> +</ul> + +<p> +Matlab code: +</p> +<div class="org-src-container"> +<pre class="src src-matlab">w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span> +xi = 0.3; + +H = (s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2)<span class="org-type">/</span>(1 <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">/</span>w0<span class="org-type">*</span>s <span class="org-type">+</span> s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2); +</pre> +</div> + + +<div id="orga48ef87" class="figure"> +<p><img src="figs/filter_high_pass_second_order.png" alt="filter_high_pass_second_order.png" /> +</p> +</div> +</div> +</div> + +<div id="outline-container-org81b8ec1" class="outline-3"> +<h3 id="org81b8ec1"><span class="section-number-3">2.3</span> Combine multiple filters</h3> +<div class="outline-text-3" id="text-2-3"> +<p> +\[ H(s) = \left( \frac{s/\omega_0}{1 + s/\omega_0} \right)^n \] +</p> + +<p> +Matlab code: +</p> +<div class="org-src-container"> +<pre class="src src-matlab">w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span> +n = 3; + +H = ((s<span class="org-type">/</span>w0)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0))<span class="org-type">^</span>n; +</pre> +</div> + + +<div id="org99826f5" class="figure"> +<p><img src="figs/filter_high_pass_first_order_add.png" alt="filter_high_pass_first_order_add.png" /> +</p> +</div> +</div> +</div> +</div> + +<div id="outline-container-org9a4d9f1" class="outline-2"> +<h2 id="org9a4d9f1"><span class="section-number-2">3</span> Band Pass</h2> +<div class="outline-text-2" id="text-3"> +</div> +<div id="outline-container-orgb93eef3" class="outline-3"> +<h3 id="orgb93eef3"><span class="section-number-3">3.1</span> Second Order</h3> +</div> +</div> + +<div id="outline-container-org06d380e" class="outline-2"> +<h2 id="org06d380e"><span class="section-number-2">4</span> Notch</h2> +<div class="outline-text-2" id="text-4"> +</div> +<div id="outline-container-org064544f" class="outline-3"> +<h3 id="org064544f"><span class="section-number-3">4.1</span> Second Order</h3> +<div class="outline-text-3" id="text-4-1"> +\begin{equation} + \frac{s^2 + 2 g_c \xi \omega_n s + \omega_n^2}{s^2 + 2 \xi \omega_n s + \omega_n^2} +\end{equation} + +<p> +Parameters: +</p> +<ul class="org-ul"> +<li>\(\omega_n\): frequency of the notch</li> +<li>\(g_c\): gain at the notch frequency</li> +<li>\(\xi\): damping ratio (notch width)</li> +</ul> + +<p> +Matlab code: +</p> +<div class="org-src-container"> +<pre class="src src-matlab">gc = 0.02; +xi = 0.1; +wn = 2<span class="org-type">*</span><span class="org-constant">pi</span>; + +H = (s<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>gm<span class="org-type">*</span>xi<span class="org-type">*</span>wn<span class="org-type">*</span>s <span class="org-type">+</span> wn<span class="org-type">^</span>2)<span class="org-type">/</span>(s<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>wn<span class="org-type">*</span>s <span class="org-type">+</span> wn<span class="org-type">^</span>2); +</pre> +</div> + + +<div id="orgba70cf5" class="figure"> +<p><img src="figs/filter_notch_xi.png" alt="filter_notch_xi.png" /> +</p> +</div> + + +<div id="orga3207e8" class="figure"> +<p><img src="figs/filter_notch_gc.png" alt="filter_notch_gc.png" /> +</p> +</div> +</div> +</div> +</div> + +<div id="outline-container-org745749b" class="outline-2"> +<h2 id="org745749b"><span class="section-number-2">5</span> Chebyshev</h2> +<div class="outline-text-2" id="text-5"> +</div> +<div id="outline-container-orgdb4d414" class="outline-3"> +<h3 id="orgdb4d414"><span class="section-number-3">5.1</span> Chebyshev Type I</h3> +<div class="outline-text-3" id="text-5-1"> +<div class="org-src-container"> +<pre class="src src-matlab">n = 4; <span class="org-comment">% Order of the filter</span> +Rp = 3; <span class="org-comment">% Maximum peak-to-peak ripple [dB]</span> +Wp = 2<span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% passband-edge frequency [rad/s]</span> + +[A,B,C,D] = cheby1(n, Rp, Wp, <span class="org-string">'high'</span>, <span class="org-string">'s'</span>); +H = ss(A, B, C, D); +</pre> +</div> +</div> +</div> +</div> + +<div id="outline-container-org42f9bf3" class="outline-2"> +<h2 id="org42f9bf3"><span class="section-number-2">6</span> Lead - Lag</h2> +<div class="outline-text-2" id="text-6"> +</div> +<div id="outline-container-org0cb85e5" class="outline-3"> +<h3 id="org0cb85e5"><span class="section-number-3">6.1</span> Lead</h3> +<div class="outline-text-3" id="text-6-1"> +\begin{equation} + H(s) = \frac{1 + \frac{s}{w_c/\sqrt{a}}}{1 + \frac{s}{w_c \sqrt{a}}}, \quad a > 1 +\end{equation} + +<p> +Parameters: +</p> +<ul class="org-ul"> +<li>\(\omega_c\): frequency at which the phase lead is maximum</li> +<li>\(a\): parameter to adjust the phase lead, also impacts the high frequency gain</li> +</ul> + +<p> +Characteristics: +</p> +<ul class="org-ul"> +<li>the low frequency gain is \(1\)</li> +<li>the high frequency gain is \(a\)</li> +<li>the phase lead at \(\omega_c\) is equal to (Figure <a href="#org6e073c7">10</a>): +\[ \angle H(j\omega_c) = \tan^{-1}(\sqrt{a}) - \tan^{-1}(1/\sqrt{a}) \]</li> +</ul> + +<p> +Matlab code: +</p> +<div class="org-src-container"> +<pre class="src src-matlab">a = 0.6; <span class="org-comment">% Amount of phase lead / width of the phase lead / high frequency gain</span> +wc = 2<span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% Frequency with the maximum phase lead [rad/s]</span> + +H = (1 <span class="org-type">+</span> s<span class="org-type">/</span>(wc<span class="org-type">/</span>sqrt(a)))<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>(wc<span class="org-type">*</span>sqrt(a))); +</pre> +</div> + + +<div id="orgf77036f" class="figure"> +<p><img src="figs/filter_lead.png" alt="filter_lead.png" /> +</p> +</div> + + +<div id="org6e073c7" class="figure"> +<p><img src="figs/filter_lead_effect_a_phase.png" alt="filter_lead_effect_a_phase.png" /> +</p> +</div> +</div> +</div> + +<div id="outline-container-org19e9264" class="outline-3"> +<h3 id="org19e9264"><span class="section-number-3">6.2</span> Lag</h3> +<div class="outline-text-3" id="text-6-2"> +\begin{equation} + H(s) = \frac{w_c \sqrt{a} + s}{\frac{w_c}{\sqrt{a}} + s}, \quad a > 1 +\end{equation} + +<p> +Parameters: +</p> +<ul class="org-ul"> +<li>\(\omega_c\): frequency at which the phase lag is maximum</li> +<li>\(a\): parameter to adjust the phase lag, also impacts the low frequency gain</li> +</ul> + +<p> +Characteristics: +</p> +<ul class="org-ul"> +<li>the low frequency gain is increased by a factor \(a\)</li> +<li>the high frequency gain is \(1\) (unchanged)</li> +<li>the phase lag at \(\omega_c\) is equal to (Figure <a href="#orge3aeee4">12</a>): +\[ \angle H(j\omega_c) = \tan^{-1}(1/\sqrt{a}) - \tan^{-1}(\sqrt{a}) \]</li> +</ul> + +<p> +Matlab code: +</p> +<div class="org-src-container"> +<pre class="src src-matlab">a = 0.6; <span class="org-comment">% Amount of phase lag / width of the phase lag / high frequency gain</span> +wc = 2<span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% Frequency with the maximum phase lag [rad/s]</span> + +H = (wc<span class="org-type">*</span>sqrt(a) <span class="org-type">+</span> s)<span class="org-type">/</span>(wc<span class="org-type">/</span>sqrt(a) <span class="org-type">+</span> s); +</pre> +</div> + + +<div id="orgcb08a98" class="figure"> +<p><img src="figs/filter_lag.png" alt="filter_lag.png" /> +</p> +</div> + + +<div id="orge3aeee4" class="figure"> +<p><img src="figs/filter_lag_effect_a_phase.png" alt="filter_lag_effect_a_phase.png" /> +</p> +</div> +</div> +</div> +</div> + +<div id="outline-container-org15058b6" class="outline-2"> +<h2 id="org15058b6"><span class="section-number-2">7</span> Complementary</h2> +</div> + +<div id="outline-container-org2d03ba9" class="outline-2"> +<h2 id="org2d03ba9"><span class="section-number-2">8</span> Performance Weight</h2> +<div class="outline-text-2" id="text-8"> +</div> +<div id="outline-container-orge550845" class="outline-3"> +<h3 id="orge550845"><span class="section-number-3">8.1</span> Nice combination</h3> +<div class="outline-text-3" id="text-8-1"> \begin{equation} W(s) = G_c * \left(\frac{\frac{1}{\omega_0}\sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}}{\frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{\left(\frac{G_\infty}{G_c}\right)^{\frac{2}{n}} - 1}} s + 1}\right)^n \end{equation} <div class="org-src-container"> -<pre class="src src-matlab">n = <span class="org-highlight-numbers-number">2</span>; w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span><span class="org-highlight-numbers-number">11</span>; G0 = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-highlight-numbers-number">10</span>; G1 = <span class="org-highlight-numbers-number">1000</span>; Gc = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-highlight-numbers-number">2</span>; -wL = Gc<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-rainbow-delimiters-depth-3">(</span>G1<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">/</span>w0<span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-5">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-rainbow-delimiters-depth-5">(</span>G1<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>s <span class="org-type">+</span> <span class="org-rainbow-delimiters-depth-3">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>w0<span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-5">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-rainbow-delimiters-depth-5">(</span>G1<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>s <span class="org-type">+</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span>n; +<pre class="src src-matlab">n = 2; w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>11; G0 = 1<span class="org-type">/</span>10; G1 = 1000; Gc = 1<span class="org-type">/</span>2; +wL = Gc<span class="org-type">*</span>(((G1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n)<span class="org-type">/</span>w0<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>((G1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n)<span class="org-type">-</span>1))<span class="org-type">*</span>s <span class="org-type">+</span> (G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">/</span>w0<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>((G1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n)<span class="org-type">-</span>1))<span class="org-type">*</span>s <span class="org-type">+</span> 1))<span class="org-type">^</span>n; -n = <span class="org-highlight-numbers-number">3</span>; w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span><span class="org-highlight-numbers-number">9</span>; G0 = <span class="org-highlight-numbers-number">10000</span>; G1 = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>; Gc = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-highlight-numbers-number">2</span>; -wH = Gc<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-rainbow-delimiters-depth-3">(</span>G1<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">/</span>w0<span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-5">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-rainbow-delimiters-depth-5">(</span>G1<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>s <span class="org-type">+</span> <span class="org-rainbow-delimiters-depth-3">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>w0<span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-5">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-rainbow-delimiters-depth-5">(</span>G1<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>s <span class="org-type">+</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span>n; -</pre> -</div> -</div> -</div> -</div> - -<div id="outline-container-org7e3ece6" class="outline-2"> -<h2 id="org7e3ece6"><span class="section-number-2">3</span> High Pass</h2> -<div class="outline-text-2" id="text-3"> -</div> -<div id="outline-container-org291d74d" class="outline-3"> -<h3 id="org291d74d"><span class="section-number-3">3.1</span> First Order</h3> -<div class="outline-text-3" id="text-3-1"> -<p> -\[ H(s) = \frac{s/\omega_0}{1 + s/\omega_0} \] -</p> - -<div class="org-src-container"> -<pre class="src src-matlab">w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span> - -H = <span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-1">)</span>; -</pre> -</div> - - -<div id="orgc492a24" class="figure"> -<p><img src="figs/hpf_first_order.png" alt="hpf_first_order.png" /> -</p> -<p><span class="figure-number">Figure 4: </span>First Order High Pass Filter (<a href="./figs/hpf_first_order.png">png</a>, <a href="./figs/hpf_first_order.pdf">pdf</a>.</p> -</div> -</div> -</div> - - -<div id="outline-container-org5d4ea15" class="outline-3"> -<h3 id="org5d4ea15"><span class="section-number-3">3.2</span> Second Order</h3> -<div class="outline-text-3" id="text-3-2"> -<p> -\[ H(s) = \frac{s^2/\omega_0^2}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \] -</p> - -<div class="org-src-container"> -<pre class="src src-matlab">w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span> -xi = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">3</span>; - -H = <span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>w0<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span>xi<span class="org-type">/</span>w0<span class="org-type">*</span>s <span class="org-type">+</span> s<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>w0<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span>; -</pre> -</div> - - -<div id="orga8fa21b" class="figure"> -<p><img src="figs/hpf_second_order.png" alt="hpf_second_order.png" /> -</p> -<p><span class="figure-number">Figure 5: </span>Second Order High Pass Filter (<a href="./figs/hpf_second_order.png">png</a>, <a href="./figs/hpf_second_order.pdf">pdf</a>.</p> -</div> -</div> -</div> - - -<div id="outline-container-orge71c2dd" class="outline-3"> -<h3 id="orge71c2dd"><span class="section-number-3">3.3</span> Combine multiple filters</h3> -<div class="outline-text-3" id="text-3-3"> -<p> -\[ H(s) = \left( \frac{s/\omega_0}{1 + s/\omega_0} \right)^n \] -</p> - -<div class="org-src-container"> -<pre class="src src-matlab">w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span> -n = <span class="org-highlight-numbers-number">3</span>; - -H = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span>s<span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span>n; -</pre> -</div> - - -<div id="orge98e2ed" class="figure"> -<p><img src="figs/hpf_multiple_first_order.png" alt="hpf_multiple_first_order.png" /> -</p> -<p><span class="figure-number">Figure 6: </span>Combine Multiple First Order High Pass Filter (<a href="./figs/hpf_multiple_first_order.png">png</a>, <a href="./figs/hpf_multiple_first_order.pdf">pdf</a>.</p> -</div> -</div> -</div> -</div> - - - -<div id="outline-container-org791f35b" class="outline-2"> -<h2 id="org791f35b"><span class="section-number-2">4</span> Band Pass</h2> -</div> - -<div id="outline-container-orge054b4d" class="outline-2"> -<h2 id="orge054b4d"><span class="section-number-2">5</span> Notch</h2> -</div> - -<div id="outline-container-orgf94d78a" class="outline-2"> -<h2 id="orgf94d78a"><span class="section-number-2">6</span> Bump</h2> -<div class="outline-text-2" id="text-6"> -<div class="org-src-container"> -<pre class="src src-matlab">n = <span class="org-highlight-numbers-number">4</span>; -w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; -A = <span class="org-highlight-numbers-number">10</span>; - -a = sqrt<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span>A<span class="org-type">^</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type">-</span> <span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span>A<span class="org-type">^</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-2">(</span>A<span class="org-type">^</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-3">)</span> <span class="org-type">-</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>; -G = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span><span class="org-rainbow-delimiters-depth-3">(</span>w0<span class="org-type">/</span>a<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span><span class="org-rainbow-delimiters-depth-3">(</span>w0<span class="org-type">*</span>a<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span>n; -bodeFig<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">{</span>G<span class="org-rainbow-delimiters-depth-2">}</span><span class="org-rainbow-delimiters-depth-1">)</span> +n = 3; w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>9; G0 = 10000; G1 = 0.1; Gc = 1<span class="org-type">/</span>2; +wH = Gc<span class="org-type">*</span>(((G1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n)<span class="org-type">/</span>w0<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>((G1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n)<span class="org-type">-</span>1))<span class="org-type">*</span>s <span class="org-type">+</span> (G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">/</span>w0<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>((G1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n)<span class="org-type">-</span>1))<span class="org-type">*</span>s <span class="org-type">+</span> 1))<span class="org-type">^</span>n; </pre> </div> </div> </div> -<div id="outline-container-org6255d82" class="outline-2"> -<h2 id="org6255d82"><span class="section-number-2">7</span> Chebyshev</h2> -<div class="outline-text-2" id="text-7"> -</div> -<div id="outline-container-org2bfe9f5" class="outline-3"> -<h3 id="org2bfe9f5"><span class="section-number-3">7.1</span> Chebyshev Type I</h3> -<div class="outline-text-3" id="text-7-1"> -<div class="org-src-container"> -<pre class="src src-matlab">n = <span class="org-highlight-numbers-number">4</span>; <span class="org-comment">% Order of the filter</span> -Rp = <span class="org-highlight-numbers-number">3</span>; <span class="org-comment">% Maximum peak-to-peak ripple [dB]</span> -Wp = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% passband-edge frequency [rad/s]</span> - -<span class="org-rainbow-delimiters-depth-1">[</span>A,B,C,D<span class="org-rainbow-delimiters-depth-1">]</span> = cheby1<span class="org-rainbow-delimiters-depth-1">(</span>n, Rp, Wp, <span class="org-string">'high', 's'</span><span class="org-rainbow-delimiters-depth-1">)</span>; -H = ss<span class="org-rainbow-delimiters-depth-1">(</span>A, B, C, D<span class="org-rainbow-delimiters-depth-1">)</span>; -</pre> -</div> - - -<div id="org175fae7" class="figure"> -<p><img src="figs/cheby1_hpf.png" alt="cheby1_hpf.png" /> -</p> -<p><span class="figure-number">Figure 7: </span>First Order Low Pass Filter (<a href="./figs/cheby1_hpf.png">png</a>, <a href="./figs/cheby1_hpf.pdf">pdf</a>.</p> -</div> -</div> -</div> -</div> - -<div id="outline-container-org127884a" class="outline-2"> -<h2 id="org127884a"><span class="section-number-2">8</span> Lead - Lag</h2> -<div class="outline-text-2" id="text-8"> -</div> -<div id="outline-container-orgb58ebc8" class="outline-3"> -<h3 id="orgb58ebc8"><span class="section-number-3">8.1</span> Lead</h3> -<div class="outline-text-3" id="text-8-1"> -<p> -\[ H(s) = \frac{1 + s/\omega_z}{1 + s/\omega_p}, \quad \omega_z < \omega_p \] -</p> - -<ul class="org-ul"> -<li class="off"><code>[&#xa0;]</code> Find a nice parametrisation to be able to specify the center frequency and the phase added</li> -<li class="off"><code>[&#xa0;]</code> Compute also the change in magnitude</li> -</ul> - -<div class="org-src-container"> -<pre class="src src-matlab">h = <span class="org-highlight-numbers-number">2</span>.<span class="org-highlight-numbers-number">0</span>; -wz = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>h; <span class="org-comment">% [rad/s]</span> -wp = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>h; <span class="org-comment">% [rad/s]</span> - -H = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>wz<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>wp<span class="org-rainbow-delimiters-depth-1">)</span>; -</pre> -</div> - - -<div id="orgbcca770" class="figure"> -<p><img src="figs/lead_filter.png" alt="lead_filter.png" /> -</p> -<p><span class="figure-number">Figure 8: </span>Lead Filter (<a href="./figs/lead_filter.png">png</a>, <a href="./figs/lead_filter.pdf">pdf</a>.</p> -</div> -</div> -</div> - -<div id="outline-container-org700da69" class="outline-3"> -<h3 id="org700da69"><span class="section-number-3">8.2</span> Lag</h3> +<div id="outline-container-org6202dd8" class="outline-3"> +<h3 id="org6202dd8"><span class="section-number-3">8.2</span> Alternative</h3> <div class="outline-text-3" id="text-8-2"> -<p> -\[ H(s) = \frac{1 + s/\omega_z}{1 + s/\omega_p}, \quad \omega_z > \omega_p \] -</p> - -<ul class="org-ul"> -<li class="off"><code>[&#xa0;]</code> Find a nice parametrisation to be able to specify the center frequency and the phase added</li> -<li class="off"><code>[&#xa0;]</code> Compute also the change in magnitude</li> -</ul> - <div class="org-src-container"> -<pre class="src src-matlab">h = <span class="org-highlight-numbers-number">2</span>.<span class="org-highlight-numbers-number">0</span>; -wz = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>h; <span class="org-comment">% [rad/s]</span> -wp = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>h; <span class="org-comment">% [rad/s]</span> +<pre class="src src-matlab">w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span> +A = 1e<span class="org-type">-</span>2; +M = 5; -H = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>wz<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>wp<span class="org-rainbow-delimiters-depth-1">)</span>; +H = (s<span class="org-type">/</span>sqrt(M) <span class="org-type">+</span> w0)<span class="org-type">^</span>2<span class="org-type">/</span>(s <span class="org-type">+</span> w0<span class="org-type">*</span>sqrt(A))<span class="org-type">^</span>2; </pre> </div> -<div id="org4ddf70a" class="figure"> -<p><img src="figs/lag_filter.png" alt="lag_filter.png" /> -</p> -<p><span class="figure-number">Figure 9: </span>Lag Filter (<a href="./figs/lag_filter.png">png</a>, <a href="./figs/lag_filter.pdf">pdf</a>.</p> -</div> -</div> -</div> - -<div id="outline-container-orgf2125da" class="outline-3"> -<h3 id="orgf2125da"><span class="section-number-3">8.3</span> Lead Lag</h3> -<div class="outline-text-3" id="text-8-3"> -<p> -\[ H(s) = \frac{1 + s/\omega_z}{1 + s/\omega_p} \frac{1 + s/\omega_z}{1 + s/\omega_p}, \quad \omega_z > \omega_p \] -</p> - -<div class="org-src-container"> -<pre class="src src-matlab">wz1 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span><span class="org-highlight-numbers-number">1</span>; <span class="org-comment">% [rad/s]</span> -wp1 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span><span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>; <span class="org-comment">% [rad/s]</span> -wz2 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span><span class="org-highlight-numbers-number">10</span>; <span class="org-comment">% [rad/s]</span> -wp2 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span><span class="org-highlight-numbers-number">20</span>; <span class="org-comment">% [rad/s]</span> - -H = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>wz1<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>wp1<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>wz2<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>wp2<span class="org-rainbow-delimiters-depth-1">)</span>; -</pre> -</div> - - -<div id="orge8b5da5" class="figure"> -<p><img src="figs/lead_lag_filter.png" alt="lead_lag_filter.png" /> -</p> -<p><span class="figure-number">Figure 10: </span>Lead Lag Filter (<a href="./figs/lead_lag_filter.png">png</a>, <a href="./figs/lead_lag_filter.pdf">pdf</a>.</p> -</div> -</div> -</div> -</div> - -<div id="outline-container-orga776beb" class="outline-2"> -<h2 id="orga776beb"><span class="section-number-2">9</span> Complementary</h2> -</div> -<div id="outline-container-org18d2496" class="outline-2"> -<h2 id="org18d2496"><span class="section-number-2">10</span> Performance Weight</h2> -<div class="outline-text-2" id="text-10"> -<div class="org-src-container"> -<pre class="src src-matlab">w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span> -A = <span class="org-highlight-numbers-number">1e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">2</span>; -M = <span class="org-highlight-numbers-number">5</span>; - -H = <span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">/</span>sqrt<span class="org-rainbow-delimiters-depth-2">(</span>M<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type">+</span> w0<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span>s <span class="org-type">+</span> w0<span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-2">(</span>A<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span>; -</pre> -</div> - - -<div class="figure"> +<div id="org503aff1" class="figure"> <p><img src="figs/weight_first_order.png" alt="weight_first_order.png" /> </p> </div> </div> </div> - -<div id="outline-container-org2ee1e4e" class="outline-2"> -<h2 id="org2ee1e4e"><span class="section-number-2">11</span> Combine Filters</h2> -<div class="outline-text-2" id="text-11"> </div> -<div id="outline-container-org890b908" class="outline-3"> -<h3 id="org890b908"><span class="section-number-3">11.1</span> Additive</h3> -<div class="outline-text-3" id="text-11-1"> + +<div id="outline-container-org5e21f67" class="outline-2"> +<h2 id="org5e21f67"><span class="section-number-2">9</span> Combine Filters</h2> +<div class="outline-text-2" id="text-9"> +</div> +<div id="outline-container-orga943d8f" class="outline-3"> +<h3 id="orga943d8f"><span class="section-number-3">9.1</span> Additive</h3> +<div class="outline-text-3" id="text-9-1"> <ul class="org-ul"> <li class="off"><code>[&#xa0;]</code> Explain how phase and magnitude combine</li> </ul> </div> </div> -<div id="outline-container-org6f10b80" class="outline-3"> -<h3 id="org6f10b80"><span class="section-number-3">11.2</span> Multiplicative</h3> +<div id="outline-container-org6cc036a" class="outline-3"> +<h3 id="org6cc036a"><span class="section-number-3">9.2</span> Multiplicative</h3> +</div> +</div> +<div id="outline-container-org80b3ca3" class="outline-2"> +<h2 id="org80b3ca3"><span class="section-number-2">10</span> Filters representing noise</h2> +<div class="outline-text-2" id="text-10"> +<p> +Let&rsquo;s consider a noise \(n\) that is shaped from a white-noise \(\tilde{n}\) with unitary PSD (\(\Phi_\tilde{n}(\omega) = 1\)) using a transfer function \(G(s)\). +The PSD of \(n\) is then: +\[ \Phi_n(\omega) = |G(j\omega)|^2 \Phi_{\tilde{n}}(\omega) = |G(j\omega)|^2 \] +</p> + +<p> +The PSD \(\Phi_n(\omega)\) is expressed in \(\text{unit}^2/\text{Hz}\). +</p> + +<p> +And the root mean square (RMS) of \(n(t)\) is: +\[ \sigma_n = \sqrt{\int_{0}^{\infty} \Phi_n(\omega) d\omega} \] +</p> +</div> + +<div id="outline-container-org579591f" class="outline-3"> +<h3 id="org579591f"><span class="section-number-3">10.1</span> First Order Low Pass Filter</h3> +<div class="outline-text-3" id="text-10-1"> +<p> +\[ G(s) = \frac{g_0}{1 + \frac{s}{\omega_c}} \] +</p> + +<div class="org-src-container"> +<pre class="src src-matlab">g0 = 1; <span class="org-comment">% Noise Density in unit/sqrt(Hz)</span> +wc = 1; <span class="org-comment">% Cut-Off frequency [rad/s]</span> + +G = g0<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc); + +<span class="org-comment">% Frequency vector [Hz]</span> +freqs = logspace(<span class="org-type">-</span>3, 3, 1000); + +<span class="org-comment">% </span><span class="org-comment"><span class="org-constant">PSD </span></span><span class="org-comment">of n in [unit^2/Hz]</span> +Phi_n = abs(squeeze(freqresp(G, freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2; + +<span class="org-comment">% </span><span class="org-comment"><span class="org-constant">RMS </span></span><span class="org-comment">value of n in [unit, rms]</span> +sigma_n = sqrt(trapz(freqs, Phi_n)) +</pre> +</div> + +<p> +\[ \sigma = \frac{1}{2} g_0 \sqrt{\omega_c} \] +with: +</p> +<ul class="org-ul"> +<li>\(g_0\) the Noise Density of \(n\) in \(\text{unit}/\sqrt{Hz}\)</li> +<li>\(\omega_c\) the bandwidth over which the noise is located, in rad/s</li> +<li>\(\sigma\) the rms noise</li> +</ul> + +<p> +If the cut-off frequency is to be expressed in Hz: +\[ \sigma = \frac{1}{2} g_0 \sqrt{2\pi f_c} = \sqrt{\frac{\pi}{2}} g_0 \sqrt{f_c} \] +</p> + + +<p> +Thus, if a sensor is said to have a RMS noise of \(\sigma = 10 nm\ rms\) over a bandwidth of \(\omega_c = 100 rad/s\), we can estimated the noise density of the sensor to be (supposing a first order low pass filter noise shape): +\[ g_0 = \frac{2 \sigma}{\sqrt{\omega_c}} \quad \left[ m/\sqrt{Hz} \right] \] +</p> + +<div class="org-src-container"> +<pre class="src src-matlab">2<span class="org-type">*</span>10e<span class="org-type">-</span>9<span class="org-type">/</span>sqrt(100) +</pre> +</div> + +<pre class="example"> +2e-09 +</pre> + + +<div class="org-src-container"> +<pre class="src src-matlab">6<span class="org-type">*</span>0.5<span class="org-type">*</span>20e<span class="org-type">-</span>12<span class="org-type">*</span>sqrt(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>100) +</pre> +</div> + +<pre class="example"> +1.504e-09 +</pre> +</div> </div> </div> </div> <div id="postamble" class="status"> <p class="author">Author: Dehaeze Thomas</p> -<p class="date">Created: 2019-10-08 mar. 13:27</p> -<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p> +<p class="date">Created: 2020-10-26 lun. 14:03</p> </div> </body> </html> diff --git a/index.org b/index.org index 881e656..60393ca 100644 --- a/index.org +++ b/index.org @@ -46,310 +46,729 @@ <<matlab-init>> #+end_src -* Proportional - Integral - Derivative -** Proportional - -** Integral - -** Derivative - * Low Pass -** First Order +** First Order Low Pass Filter \[ H(s) = \frac{1}{1 + s/\omega_0} \] +Parameters: +- $\omega_0$: cut-off frequency in [rad/s] + +Characteristics: +- Low frequency gain of $1$ +- Roll-off equals to -20 dB/dec + +Matlab code: #+begin_src matlab - w0 = 2*pi; % [rad/s] + w0 = 2*pi; % Cut-off Frequency [rad/s] H = 1/(1 + s/w0); #+end_src #+begin_src matlab :exports none - freqs = logspace(-1, 1, 1000); - resp = squeeze(freqresp(H, freqs, 'Hz')); - Ha = abs(resp); - Hp = 180/pi*phase(resp); + freqs = logspace(-2, 2, 1000); - T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'}); - writetable(T,'mat/lpf_first_order.csv'); + figure; + tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + % Magnitude + ax1 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(H, freqs, 'Hz'))), 'k-'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); set(gca, 'XTickLabel',[]); + axis padded + hold off; + xline(1, '--', 'LineWidth', 2); + + % Phase + ax2 = nexttile; + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(H, freqs, 'Hz'))), 'k-'); + set(gca,'xscale','log'); + yticks(-90:15:0); ylim([-90 0]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + xline(1, '--', {'\omega_0'}, 'LineWidth', 2, 'LabelVerticalAlignment', 'bottom', 'LabelOrientation', 'horizontal'); + + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); #+end_src -#+begin_src latex :file lpf_first_order.pdf :tangle figs/lpf_first_order.tex :exports results - \setlength\fwidth{8cm} - \setlength\fheight{6cm} - - \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}% - \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}% - - \begin{tikzpicture} - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0.5\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xticklabels={{}}, - xminorticks=true, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymode=log, - ymin=1e-2, - ymax=1e1, - % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6}, - % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}}, - yminorticks=true, - ylabel={Amplitude}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids, - yminorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlpf_first_order.csv}; - \end{axis} - - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xminorticks=true, - xlabel={Frequency [Hz]}, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymin=-270, - ymax=90, - ytick={-360, -270, -180, -90, 0, 90}, - ylabel={Phase [deg]}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlpf_first_order.csv}; - \end{axis} - \end{tikzpicture} +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/filter_low_pass_first_order.pdf', 'width', 'wide', 'height', 'tall'); #+end_src -#+name: fig:lpf_first_order -#+caption: First Order Low Pass Filter ([[./figs/lpf_first_order.png][png]], [[./figs/lpf_first_order.pdf][pdf]]. +#+name: fig:filter_low_pass_first_order +#+caption: #+RESULTS: -[[file:figs/lpf_first_order.png]] +[[file:figs/filter_low_pass_first_order.png]] ** Second Order \[ H(s) = \frac{1}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \] +Parameters: +- $\omega_0$: +- $\xi$: Damping ratio + +Characteristics: +- Low frequency gain: 1 +- High frequency roll off: - 40 dB/dec + +Matlab code: #+begin_src matlab - w0 = 2*pi; % [rad/s] - xi = 0.3; + w0 = 2*pi; % Cut-off frequency [rad/s] + xi = 0.3; % Damping Ratio H = 1/(1 + 2*xi/w0*s + s^2/w0^2); #+end_src #+begin_src matlab :exports none - freqs = logspace(-1, 1, 1000); - resp = squeeze(freqresp(H, freqs, 'Hz')); - Ha = abs(resp); - Hp = 180/pi*phase(resp); + xis = [0.1, 0.3, 0.5, 0.7, 0.9]; + Hs = {zeros(length(xis), 1)}; - T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'}); - writetable(T,'mat/lpf_second_order.csv'); + for xi_i = 1:length(xis) + Hs(xi_i) = {1/(1 + 2*xis(xi_i)/w0*s + s^2/w0^2)}; + end #+end_src -#+begin_src latex :file lpf_second_order.pdf :tangle figs/lpf_second_order.tex :exports results - \setlength\fwidth{8cm} - \setlength\fheight{6cm} +#+begin_src matlab :exports none + freqs = logspace(-2, 2, 1000); - \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}% - \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}% + figure; + tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); - \begin{tikzpicture} - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0.5\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xticklabels={{}}, - xminorticks=true, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymode=log, - ymin=1e-2, - ymax=1e1, - % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6}, - % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}}, - yminorticks=true, - ylabel={Amplitude}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids, - yminorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlpf_second_order.csv}; - \end{axis} + % Magnitude + ax1 = nexttile; + hold on; + for xi_i = 1:length(xis) + plot(freqs, abs(squeeze(freqresp(Hs{xi_i}, freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$\\xi = %.1f$', xis(xi_i))); + end + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); set(gca, 'XTickLabel',[]); + axis padded + legend('location', 'northeast'); + hold off; + xline(1, '--', 'LineWidth', 2, 'HandleVisibility', 'off'); - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xminorticks=true, - xlabel={Frequency [Hz]}, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymin=-270, - ymax=90, - ytick={-360, -270, -180, -90, 0, 90}, - ylabel={Phase [deg]}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlpf_second_order.csv}; - \end{axis} - \end{tikzpicture} + % Phase + ax2 = nexttile; + hold on; + for xi_i = 1:length(xis) + plot(freqs, 180/pi*angle(squeeze(freqresp(Hs{xi_i}, freqs, 'Hz'))), '-'); + end + set(gca,'xscale','log'); + yticks(-180:30:0); ylim([-180 0]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + xline(1, '--', {'\omega_0'}, 'LineWidth', 2, 'LabelVerticalAlignment', 'bottom', 'LabelOrientation', 'horizontal'); + + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); #+end_src -#+name: fig:lpf_second_order -#+caption: Second Order Low Pass Filter ([[./figs/lpf_second_order.png][png]], [[./figs/lpf_second_order.pdf][pdf]]. +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/filter_low_pass_second_order.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:filter_low_pass_second_order +#+caption: #+RESULTS: -[[file:figs/lpf_second_order.png]] +[[file:figs/filter_low_pass_second_order.png]] -** Combine multiple filters +** Combine multiple first order filters \[ H(s) = \left( \frac{1}{1 + s/\omega_0} \right)^n \] +Matlab code: #+begin_src matlab - w0 = 2*pi; % [rad/s] - n = 3; + w0 = 2*pi; % Cut-off frequency [rad/s] + n = 3; % Filter order H = (1/(1 + s/w0))^n; #+end_src #+begin_src matlab :exports none + ns = [1, 2, 3, 4]; + Hs = {zeros(length(ns), 1)}; + + for n_i = 1:length(ns) + Hs(n_i) = {1/(1 + s/w0)^ns(n_i)}; + end +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(-2, 2, 1000); + + figure; + tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + % Magnitude + ax1 = nexttile; + hold on; + for n_i = 1:length(ns) + plot(freqs, abs(squeeze(freqresp(Hs{n_i}, freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$n = %i$', ns(n_i))); + end + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); set(gca, 'XTickLabel',[]); + axis padded + legend('location', 'northeast'); + hold off; + xline(1, '--', 'LineWidth', 2, 'HandleVisibility', 'off'); + + % Phase + ax2 = nexttile; + hold on; + for n_i = 1:length(ns) + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hs{n_i}, freqs, 'Hz')))), '-'); + end + set(gca,'xscale','log'); + yticks(-360:45:0); ylim([-360 0]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + xline(1, '--', {'\omega_0'}, 'LineWidth', 2, 'LabelVerticalAlignment', 'bottom', 'LabelOrientation', 'horizontal'); + + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/filter_low_pass_first_order_add.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:filter_low_pass_first_order_add +#+caption: +#+RESULTS: +[[file:figs/filter_low_pass_first_order_add.png]] + +* High Pass +** First Order +\[ H(s) = \frac{s/\omega_0}{1 + s/\omega_0} \] + +Parameters: +- $\omega_0$: cut-off frequency in [rad/s] + +Characteristics: +- High frequency gain of $1$ +- Low frequency slow of +20 dB/dec + +Matlab code: +#+begin_src matlab + w0 = 2*pi; % Cut-off frequency [rad/s] + + H = (s/w0)/(1 + s/w0); +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(-2, 2, 1000); + + figure; + tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + % Magnitude + ax1 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(H, freqs, 'Hz'))), 'k-'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); set(gca, 'XTickLabel',[]); + axis padded + hold off; + xline(1, '--', 'LineWidth', 2); + + % Phase + ax2 = nexttile; + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(H, freqs, 'Hz'))), 'k-'); + set(gca,'xscale','log'); + yticks(0:15:90); ylim([0, 90]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + xline(1, '--', {'\omega_0'}, 'LineWidth', 2, 'LabelVerticalAlignment', 'bottom', 'LabelOrientation', 'horizontal'); + + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/filter_high_pass_first_order.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:filter_high_pass_first_order +#+caption: +#+RESULTS: +[[file:figs/filter_high_pass_first_order.png]] + +** Second Order +\[ H(s) = \frac{s^2/\omega_0^2}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \] + +Parameters: +- $\omega_0$: +- $\xi$: Damping ratio + +Matlab code: +#+begin_src matlab + w0 = 2*pi; % [rad/s] + xi = 0.3; + + H = (s^2/w0^2)/(1 + 2*xi/w0*s + s^2/w0^2); +#+end_src + +#+begin_src matlab :exports none + xis = [0.1, 0.3, 0.5, 0.7, 0.9]; + Hs = {zeros(length(xis), 1)}; + + for xi_i = 1:length(xis) + Hs(xi_i) = {(s^2/w0^2)/(1 + 2*xis(xi_i)/w0*s + s^2/w0^2)}; + end + + freqs = logspace(-2, 2, 1000); + + figure; + tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + % Magnitude + ax1 = nexttile; + hold on; + for xi_i = 1:length(xis) + plot(freqs, abs(squeeze(freqresp(Hs{xi_i}, freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$\\xi = %.1f$', xis(xi_i))); + end + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); set(gca, 'XTickLabel',[]); + axis padded + legend('location', 'northeast'); + hold off; + xline(1, '--', 'LineWidth', 2, 'HandleVisibility', 'off'); + + % Phase + ax2 = nexttile; + hold on; + for xi_i = 1:length(xis) + plot(freqs, 180/pi*angle(squeeze(freqresp(Hs{xi_i}, freqs, 'Hz'))), '-'); + end + set(gca,'xscale','log'); + yticks(0:30:180); ylim([0 180]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + xline(1, '--', {'\omega_0'}, 'LineWidth', 2, 'LabelVerticalAlignment', 'bottom', 'LabelOrientation', 'horizontal'); + + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/filter_high_pass_second_order.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:filter_high_pass_second_order +#+caption: +#+RESULTS: +[[file:figs/filter_high_pass_second_order.png]] + +** Combine multiple filters +\[ H(s) = \left( \frac{s/\omega_0}{1 + s/\omega_0} \right)^n \] + +Matlab code: +#+begin_src matlab + w0 = 2*pi; % [rad/s] + n = 3; + + H = ((s/w0)/(1 + s/w0))^n; +#+end_src + +#+begin_src matlab :exports none + ns = [1, 2, 3, 4]; + Hs = {zeros(length(ns), 1)}; + + for n_i = 1:length(ns) + Hs(n_i) = {((s/w0)/(1 + s/w0))^ns(n_i)}; + end +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(-2, 2, 1000); + + figure; + tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + % Magnitude + ax1 = nexttile; + hold on; + for n_i = 1:length(ns) + plot(freqs, abs(squeeze(freqresp(Hs{n_i}, freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$n = %i$', ns(n_i))); + end + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); set(gca, 'XTickLabel',[]); + axis padded + legend('location', 'northeast'); + hold off; + xline(1, '--', 'LineWidth', 2, 'HandleVisibility', 'off'); + + % Phase + ax2 = nexttile; + hold on; + for n_i = 1:length(ns) + plot(freqs, 180/pi*angle(squeeze(freqresp(Hs{n_i}, freqs, 'Hz'))), '-'); + end + set(gca,'xscale','log'); + yticks(-180:45:180); ylim([-180 180]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + xline(1, '--', {'\omega_0'}, 'LineWidth', 2, 'LabelVerticalAlignment', 'bottom', 'LabelOrientation', 'horizontal'); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/filter_high_pass_first_order_add.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:filter_high_pass_first_order_add +#+caption: +#+RESULTS: +[[file:figs/filter_high_pass_first_order_add.png]] + +* TODO Band Pass +** Second Order + +* Notch +** Second Order +\begin{equation} + \frac{s^2 + 2 g_c \xi \omega_n s + \omega_n^2}{s^2 + 2 \xi \omega_n s + \omega_n^2} +\end{equation} + +Parameters: +- $\omega_n$: frequency of the notch +- $g_c$: gain at the notch frequency +- $\xi$: damping ratio (notch width) + +Matlab code: +#+begin_src matlab + gc = 0.02; + xi = 0.1; + wn = 2*pi; + + H = (s^2 + 2*gm*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2); +#+end_src + +#+begin_src matlab :exports none + xi = 0.2; + gcs = [0.02, 0.1, 0.5]; + Hs = {zeros(length(gcs), 1)}; + + for g_i = 1:length(gcs) + Hs(g_i) = {(s^2 + 2*gcs(g_i)*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2)}; + end + freqs = logspace(-1, 1, 1000); - resp = squeeze(freqresp(H, freqs, 'Hz')); - Ha = abs(resp); - Hp = 180/pi*phase(resp); - T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'}); - writetable(T,'mat/lpf_multiple_first_order.csv'); + figure; + tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + % Magnitude + ax1 = nexttile; + hold on; + for g_i = 1:length(gcs) + plot(freqs, abs(squeeze(freqresp(Hs{g_i}, freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$g_c = %.2f$', gm(g_i))); + end + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); set(gca, 'XTickLabel',[]); + axis padded + legend('location', 'southeast'); + hold off; + + % Phase + ax2 = nexttile; + hold on; + for g_i = 1:length(gcs) + plot(freqs, 180/pi*angle(squeeze(freqresp(Hs{g_i}, freqs, 'Hz'))), '-'); + end + set(gca,'xscale','log'); + yticks(-90:30:90); ylim([-90 90]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); #+end_src -#+begin_src latex :file lpf_multiple_first_order.pdf :tangle figs/lpf_multiple_first_order.tex :exports results - \setlength\fwidth{8cm} - \setlength\fheight{6cm} - - \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}% - \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}% - - \begin{tikzpicture} - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0.5\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xticklabels={{}}, - xminorticks=true, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymode=log, - ymin=1e-2, - ymax=1e1, - % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6}, - % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}}, - yminorticks=true, - ylabel={Amplitude}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids, - yminorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlpf_multiple_first_order.csv}; - \end{axis} - - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xminorticks=true, - xlabel={Frequency [Hz]}, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymin=-270, - ymax=90, - ytick={-360, -270, -180, -90, 0, 90}, - ylabel={Phase [deg]}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlpf_multiple_first_order.csv}; - \end{axis} - \end{tikzpicture} +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/filter_notch_xi.pdf', 'width', 'wide', 'height', 'tall'); #+end_src -#+name: fig:lpf_multiple_first_order -#+caption: Combine Multiple First Order Low Pass Filter ([[./figs/lpf_multiple_first_order.png][png]], [[./figs/lpf_multiple_first_order.pdf][pdf]]. +#+name: fig:filter_notch_xi +#+caption: +#+RESULTS: +[[file:figs/filter_notch_xi.png]] + +#+begin_src matlab :exports none + xis = [0.05, 0.1, 0.5]; + gc = 0.1; + Hs = {zeros(length(xis), 1)}; + + for xi_i = 1:length(xis) + Hs(xi_i) = {(s^2 + 2*gc*xis(xi_i)*wn*s + wn^2)/(s^2 + 2*xis(xi_i)*wn*s + wn^2)}; + end + + freqs = logspace(-1, 1, 1000); + + figure; + tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + % Magnitude + ax1 = nexttile; + hold on; + for xi_i = 1:length(xis) + plot(freqs, abs(squeeze(freqresp(Hs{xi_i}, freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$\\xi = %.2f$', xis(xi_i))); + end + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); set(gca, 'XTickLabel',[]); + axis padded + legend('location', 'southeast'); + hold off; + + % Phase + ax2 = nexttile; + hold on; + for xi_i = 1:length(xis) + plot(freqs, 180/pi*angle(squeeze(freqresp(Hs{xi_i}, freqs, 'Hz'))), '-'); + end + set(gca,'xscale','log'); + yticks(-90:30:90); ylim([-90 90]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/filter_notch_gc.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:filter_notch_gc +#+caption: +#+RESULTS: +[[file:figs/filter_notch_gc.png]] + +* TODO Chebyshev +** Chebyshev Type I + +#+begin_src matlab + n = 4; % Order of the filter + Rp = 3; % Maximum peak-to-peak ripple [dB] + Wp = 2*pi; % passband-edge frequency [rad/s] + + [A,B,C,D] = cheby1(n, Rp, Wp, 'high', 's'); + H = ss(A, B, C, D); +#+end_src + +* Lead - Lag +** Lead +\begin{equation} + H(s) = \frac{1 + \frac{s}{w_c/\sqrt{a}}}{1 + \frac{s}{w_c \sqrt{a}}}, \quad a > 1 +\end{equation} + +Parameters: +- $\omega_c$: frequency at which the phase lead is maximum +- $a$: parameter to adjust the phase lead, also impacts the high frequency gain + +Characteristics: +- the low frequency gain is $1$ +- the high frequency gain is $a$ +- the phase lead at $\omega_c$ is equal to (Figure [[fig:filter_lead_effect_a_phase]]): + \[ \angle H(j\omega_c) = \tan^{-1}(\sqrt{a}) - \tan^{-1}(1/\sqrt{a}) \] + +Matlab code: +#+begin_src matlab + a = 0.6; % Amount of phase lead / width of the phase lead / high frequency gain + wc = 2*pi; % Frequency with the maximum phase lead [rad/s] + + H = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a))); +#+end_src + +#+begin_src matlab :exports none + wc = 2*pi; + as = [1.2, 2, 5]; + Hs = {zeros(length(as), 1)}; + + for a_i = 1:length(as) + Hs(a_i) = {(1 + s/(wc/sqrt(as(a_i))))/(1 + s/(wc*sqrt(as(a_i))))}; + end + + freqs = logspace(-2, 2, 1000); + + figure; + tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + % Magnitude + ax1 = nexttile; + hold on; + for a_i = 1:length(as) + plot(freqs, abs(squeeze(freqresp(Hs{a_i}, freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$a = %.1f$', as(a_i))); + end + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); set(gca, 'XTickLabel',[]); + axis padded + legend('location', 'northwest'); + hold off; + xline(1, '--', 'LineWidth', 2, 'HandleVisibility', 'off'); + + % Phase + ax2 = nexttile; + hold on; + for a_i = 1:length(as) + plot(freqs, 180/pi*angle(squeeze(freqresp(Hs{a_i}, freqs, 'Hz'))), '-'); + end + set(gca,'xscale','log'); + yticks(0:15:60); ylim([0 60]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + xline(1, '--', {'\omega_c'}, 'LineWidth', 2, 'LabelVerticalAlignment', 'top', 'LabelOrientation', 'horizontal'); + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/filter_lead.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:filter_lead +#+caption: +#+RESULTS: +[[file:figs/filter_lead.png]] + +#+begin_src matlab :exports none + as = logspace(0, 1, 100); + + figure; + plot(as, 180/pi*(atan(sqrt(as)) - atan(1./sqrt(as)))) + xlabel('$a$'); ylabel('Phase added at $\omega_c$ [deg]'); + yticks(0:15:60); ylim([0, 60]); + xticks(1:1:10); xlim([as(1), as(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/filter_lead_effect_a_phase.pdf', 'width', 'normal', 'height', 'normal'); +#+end_src + +#+name: fig:filter_lead_effect_a_phase +#+caption: +#+RESULTS: +[[file:figs/filter_lead_effect_a_phase.png]] + +** Lag +\begin{equation} + H(s) = \frac{w_c \sqrt{a} + s}{\frac{w_c}{\sqrt{a}} + s}, \quad a > 1 +\end{equation} + +Parameters: +- $\omega_c$: frequency at which the phase lag is maximum +- $a$: parameter to adjust the phase lag, also impacts the low frequency gain + +Characteristics: +- the low frequency gain is increased by a factor $a$ +- the high frequency gain is $1$ (unchanged) +- the phase lag at $\omega_c$ is equal to (Figure [[fig:filter_lag_effect_a_phase]]): + \[ \angle H(j\omega_c) = \tan^{-1}(1/\sqrt{a}) - \tan^{-1}(\sqrt{a}) \] + +Matlab code: +#+begin_src matlab + a = 0.6; % Amount of phase lag / width of the phase lag / high frequency gain + wc = 2*pi; % Frequency with the maximum phase lag [rad/s] + + H = (wc*sqrt(a) + s)/(wc/sqrt(a) + s); +#+end_src + +#+begin_src matlab :exports none + wc = 2*pi; + as = [1.2, 2, 5]; + Hs = {zeros(length(as), 1)}; + + for a_i = 1:length(as) + Hs(a_i) = {(wc*sqrt(as(a_i)) + s)/(wc/sqrt(as(a_i)) + s)}; + end + + freqs = logspace(-2, 2, 1000); + + figure; + tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + % Magnitude + ax1 = nexttile; + hold on; + for a_i = 1:length(as) + plot(freqs, abs(squeeze(freqresp(Hs{a_i}, freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$a = %.1f$', as(a_i))); + end + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); set(gca, 'XTickLabel',[]); + axis padded + legend('location', 'southwest'); + hold off; + xline(1, '--', 'LineWidth', 2, 'HandleVisibility', 'off'); + + % Phase + ax2 = nexttile; + hold on; + for a_i = 1:length(as) + plot(freqs, 180/pi*angle(squeeze(freqresp(Hs{a_i}, freqs, 'Hz'))), '-'); + end + set(gca,'xscale','log'); + yticks(-60:15:0); ylim([-60 0]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + xline(1, '--', {'\omega_c'}, 'LineWidth', 2, 'LabelVerticalAlignment', 'bottom', 'LabelOrientation', 'horizontal'); + + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/filter_lag.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:filter_lag +#+caption: +#+RESULTS: +[[file:figs/filter_lag.png]] + +#+begin_src matlab :exports none + as = logspace(0, 1, 100); + + figure; + plot(as, 180/pi*(atan(1./sqrt(as)) - atan(sqrt(as)))) + xlabel('$a$'); ylabel('Phase added at $\omega_c$ [deg]'); + yticks(-60:15:0); ylim([-60, 0]); + xticks(1:1:10); xlim([as(1), as(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/filter_lag_effect_a_phase.pdf', 'width', 'normal', 'height', 'normal'); +#+end_src + +#+name: fig:filter_lag_effect_a_phase +#+caption: +#+RESULTS: +[[file:figs/filter_lag_effect_a_phase.png]] + +* TODO Complementary + +* TODO Performance Weight ** Nice combination \begin{equation} W(s) = G_c * \left(\frac{\frac{1}{\omega_0}\sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}}{\frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{\left(\frac{G_\infty}{G_c}\right)^{\frac{2}{n}} - 1}} s + 1}\right)^n @@ -364,735 +783,8 @@ wH = Gc*(((G1/Gc)^(1/n)/w0*sqrt((1-(G0/Gc)^(2/n))/((G1/Gc)^(2/n)-1))*s + (G0/Gc)^(1/n))/(1/w0*sqrt((1-(G0/Gc)^(2/n))/((G1/Gc)^(2/n)-1))*s + 1))^n; #+end_src -* High Pass -** First Order -\[ H(s) = \frac{s/\omega_0}{1 + s/\omega_0} \] - -#+begin_src matlab - w0 = 2*pi; % [rad/s] - - H = (s/w0)/(1 + s/w0); -#+end_src - -#+begin_src matlab :exports none - freqs = logspace(-1, 1, 1000); - resp = squeeze(freqresp(H, freqs, 'Hz')); - Ha = abs(resp); - Hp = 180/pi*phase(resp); - - T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'}); - writetable(T,'mat/hpf_first_order.csv'); -#+end_src - -#+begin_src latex :file hpf_first_order.pdf :tangle figs/hpf_first_order.tex :exports results - \setlength\fwidth{8cm} - \setlength\fheight{6cm} - - \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}% - \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}% - - \begin{tikzpicture} - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0.5\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xticklabels={{}}, - xminorticks=true, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymode=log, - ymin=1e-2, - ymax=1e1, - % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6}, - % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}}, - yminorticks=true, - ylabel={Amplitude}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids, - yminorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/mathpf_first_order.csv}; - \end{axis} - - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xminorticks=true, - xlabel={Frequency [Hz]}, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymin=-270, - ymax=90, - ytick={-360, -270, -180, -90, 0, 90}, - ylabel={Phase [deg]}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/mathpf_first_order.csv}; - \end{axis} - \end{tikzpicture} -#+end_src - -#+name: fig:hpf_first_order -#+caption: First Order High Pass Filter ([[./figs/hpf_first_order.png][png]], [[./figs/hpf_first_order.pdf][pdf]]. -#+RESULTS: -[[file:figs/hpf_first_order.png]] - - -** Second Order -\[ H(s) = \frac{s^2/\omega_0^2}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \] - -#+begin_src matlab - w0 = 2*pi; % [rad/s] - xi = 0.3; - - H = (s^2/w0^2)/(1 + 2*xi/w0*s + s^2/w0^2); -#+end_src - -#+begin_src matlab :exports none - freqs = logspace(-1, 1, 1000); - resp = squeeze(freqresp(H, freqs, 'Hz')); - Ha = abs(resp); - Hp = 180/pi*phase(resp); - - T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'}); - writetable(T,'mat/hpf_second_order.csv'); -#+end_src - -#+begin_src latex :file hpf_second_order.pdf :tangle figs/hpf_second_order.tex :exports results - \setlength\fwidth{8cm} - \setlength\fheight{6cm} - - \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}% - \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}% - - \begin{tikzpicture} - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0.5\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xticklabels={{}}, - xminorticks=true, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymode=log, - ymin=1e-2, - ymax=1e1, - % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6}, - % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}}, - yminorticks=true, - ylabel={Amplitude}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids, - yminorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/mathpf_second_order.csv}; - \end{axis} - - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xminorticks=true, - xlabel={Frequency [Hz]}, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymin=-270, - ymax=90, - ytick={-360, -270, -180, -90, 0, 90}, - ylabel={Phase [deg]}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/mathpf_second_order.csv}; - \end{axis} - \end{tikzpicture} -#+end_src - -#+name: fig:hpf_second_order -#+caption: Second Order High Pass Filter ([[./figs/hpf_second_order.png][png]], [[./figs/hpf_second_order.pdf][pdf]]. -#+RESULTS: -[[file:figs/hpf_second_order.png]] - - -** Combine multiple filters -\[ H(s) = \left( \frac{s/\omega_0}{1 + s/\omega_0} \right)^n \] - -#+begin_src matlab - w0 = 2*pi; % [rad/s] - n = 3; - - H = ((s/w0)/(1 + s/w0))^n; -#+end_src - -#+begin_src matlab :exports none - freqs = logspace(-1, 1, 1000); - resp = squeeze(freqresp(H, freqs, 'Hz')); - Ha = abs(resp); - Hp = 180/pi*phase(resp); - - T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'}); - writetable(T,'mat/hpf_multiple_first_order.csv'); -#+end_src - -#+begin_src latex :file hpf_multiple_first_order.pdf :tangle figs/hpf_multiple_first_order.tex :exports results - \setlength\fwidth{8cm} - \setlength\fheight{6cm} - - \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}% - \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}% - - \begin{tikzpicture} - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0.5\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xticklabels={{}}, - xminorticks=true, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymode=log, - ymin=1e-2, - ymax=1e1, - % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6}, - % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}}, - yminorticks=true, - ylabel={Amplitude}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids, - yminorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/mathpf_multiple_first_order.csv}; - \end{axis} - - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xminorticks=true, - xlabel={Frequency [Hz]}, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymin=-270, - ymax=90, - ytick={-360, -270, -180, -90, 0, 90}, - ylabel={Phase [deg]}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/mathpf_multiple_first_order.csv}; - \end{axis} - \end{tikzpicture} -#+end_src - -#+name: fig:hpf_multiple_first_order -#+caption: Combine Multiple First Order High Pass Filter ([[./figs/hpf_multiple_first_order.png][png]], [[./figs/hpf_multiple_first_order.pdf][pdf]]. -#+RESULTS: -[[file:figs/hpf_multiple_first_order.png]] - - - -* Band Pass - -* Notch - -* Bump -#+begin_src matlab - n = 4; - w0 = 2*pi; - A = 10; - - a = sqrt(2*A^(2/n) - 1 + 2*A^(1/n)*sqrt(A^(2/n) - 1)); - G = ((1 + s/(w0/a))*(1 + s/(w0*a))/(1 + s/w0)^2)^n; - bodeFig({G}) -#+end_src - - -* Chebyshev -** Chebyshev Type I - -#+begin_src matlab - n = 4; % Order of the filter - Rp = 3; % Maximum peak-to-peak ripple [dB] - Wp = 2*pi; % passband-edge frequency [rad/s] - - [A,B,C,D] = cheby1(n, Rp, Wp, 'high', 's'); - H = ss(A, B, C, D); -#+end_src - -#+begin_src matlab :exports none - freqs = logspace(-1, 1, 1000); - resp = squeeze(freqresp(H, freqs, 'Hz')); - Ha = abs(resp); - Hp = 180/pi*phase(resp); - - T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'}); - writetable(T,'mat/cheby1_hpf.csv'); -#+end_src - -#+begin_src latex :file cheby1_hpf.pdf :tangle figs/cheby1_hpf.tex :exports results - \setlength\fwidth{8cm} - \setlength\fheight{6cm} - - \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}% - \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}% - - \begin{tikzpicture} - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0.5\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xticklabels={{}}, - xminorticks=true, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymode=log, - % ymin=1e-2, - % ymax=1e1, - % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6}, - % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}}, - yminorticks=true, - ylabel={Amplitude}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids, - yminorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matcheby1_hpf.csv}; - \end{axis} - - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xminorticks=true, - xlabel={Frequency [Hz]}, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymin=-270, - ymax=90, - ytick={-360, -270, -180, -90, 0, 90}, - ylabel={Phase [deg]}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matcheby1_hpf.csv}; - \end{axis} - \end{tikzpicture} -#+end_src - -#+name: fig:cheby1_hpf -#+caption: First Order Low Pass Filter ([[./figs/cheby1_hpf.png][png]], [[./figs/cheby1_hpf.pdf][pdf]]. -#+RESULTS: -[[file:figs/cheby1_hpf.png]] - -* Lead - Lag -** Lead -\[ H(s) = \frac{1 + s/\omega_z}{1 + s/\omega_p}, \quad \omega_z < \omega_p \] - -- [ ] Find a nice parametrisation to be able to specify the center frequency and the phase added -- [ ] Compute also the change in magnitude - -#+begin_src matlab - h = 2.0; - wz = 2*pi/h; % [rad/s] - wp = 2*pi*h; % [rad/s] - - H = (1 + s/wz)/(1 + s/wp); -#+end_src - -#+begin_src matlab :exports none - freqs = logspace(-1, 1, 1000); - resp = squeeze(freqresp(H, freqs, 'Hz')); - Ha = abs(resp); - Hp = 180/pi*phase(resp); - - T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'}); - writetable(T,'mat/lead_filter.csv'); -#+end_src - -#+begin_src latex :file lead_filter.pdf :tangle figs/lead_filter.tex :exports results - \setlength\fwidth{8cm} - \setlength\fheight{6cm} - - \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}% - \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}% - - \begin{tikzpicture} - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0.5\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xticklabels={{}}, - xminorticks=true, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymode=log, - % ymin=1e-2, - % ymax=1e1, - % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6}, - % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}}, - yminorticks=true, - ylabel={Amplitude}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids, - yminorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlead_filter.csv}; - \end{axis} - - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xminorticks=true, - xlabel={Frequency [Hz]}, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymin=-90, - ymax=90, - ytick={-360, -270, -180, -90, 0, 90}, - ylabel={Phase [deg]}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlead_filter.csv}; - \end{axis} - \end{tikzpicture} -#+end_src - -#+name: fig:lead_filter -#+caption: Lead Filter ([[./figs/lead_filter.png][png]], [[./figs/lead_filter.pdf][pdf]]. -#+RESULTS: -[[file:figs/lead_filter.png]] - -** Lag -\[ H(s) = \frac{1 + s/\omega_z}{1 + s/\omega_p}, \quad \omega_z > \omega_p \] - -- [ ] Find a nice parametrisation to be able to specify the center frequency and the phase added -- [ ] Compute also the change in magnitude - -#+begin_src matlab - h = 2.0; - wz = 2*pi*h; % [rad/s] - wp = 2*pi/h; % [rad/s] - - H = (1 + s/wz)/(1 + s/wp); -#+end_src - -#+begin_src matlab :exports none - freqs = logspace(-1, 1, 1000); - resp = squeeze(freqresp(H, freqs, 'Hz')); - Ha = abs(resp); - Hp = 180/pi*phase(resp); - - T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'}); - writetable(T,'mat/lag_filter.csv'); -#+end_src - -#+begin_src latex :file lag_filter.pdf :tangle figs/lag_filter.tex :exports results - \setlength\fwidth{8cm} - \setlength\fheight{6cm} - - \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}% - \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}% - - \begin{tikzpicture} - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0.5\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xticklabels={{}}, - xminorticks=true, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymode=log, - % ymin=1e-2, - % ymax=1e1, - % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6}, - % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}}, - yminorticks=true, - ylabel={Amplitude}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids, - yminorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlag_filter.csv}; - \end{axis} - - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.1, - xmax=10, - xminorticks=true, - xlabel={Frequency [Hz]}, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymin=-90, - ymax=90, - ytick={-360, -270, -180, -90, 0, 90}, - ylabel={Phase [deg]}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlag_filter.csv}; - \end{axis} - \end{tikzpicture} -#+end_src - -#+name: fig:lag_filter -#+caption: Lag Filter ([[./figs/lag_filter.png][png]], [[./figs/lag_filter.pdf][pdf]]. -#+RESULTS: -[[file:figs/lag_filter.png]] - -** Lead Lag -\[ H(s) = \frac{1 + s/\omega_z}{1 + s/\omega_p} \frac{1 + s/\omega_z}{1 + s/\omega_p}, \quad \omega_z > \omega_p \] - -#+begin_src matlab - wz1 = 2*pi*1; % [rad/s] - wp1 = 2*pi*0.1; % [rad/s] - wz2 = 2*pi*10; % [rad/s] - wp2 = 2*pi*20; % [rad/s] - - H = (1 + s/wz1)/(1 + s/wp1)*(1 + s/wz2)/(1 + s/wp2); -#+end_src - -#+begin_src matlab :exports none - freqs = logspace(-2, 2, 1000); - resp = squeeze(freqresp(H, freqs, 'Hz')); - Ha = abs(resp); - Hp = 180/pi*phase(resp); - - T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'}); - writetable(T,'mat/lead_lag_filter.csv'); -#+end_src - -#+begin_src latex :file lead_lag_filter.pdf :tangle figs/lead_lag_filter.tex :exports results - \setlength\fwidth{8cm} - \setlength\fheight{6cm} - - \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}% - \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}% - - \begin{tikzpicture} - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0.5\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.01, - xmax=100, - xticklabels={{}}, - xminorticks=true, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymode=log, - % ymin=1e-2, - % ymax=1e1, - % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6}, - % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}}, - yminorticks=true, - ylabel={Amplitude}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids, - yminorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlead_lag_filter.csv}; - \end{axis} - - \begin{axis}[% - width=\fwidth, - height=0.45\fheight, - at={(0\fwidth,0\fheight)}, - scale only axis, - separate axis lines, - every outer x axis line/.append style={black}, - every x tick label/.append style={font=\color{black}}, - every x tick/.append style={black}, - xmode=log, - xmin=0.01, - xmax=100, - xminorticks=true, - xlabel={Frequency [Hz]}, - every outer y axis line/.append style={black}, - every y tick label/.append style={font=\color{black}}, - every y tick/.append style={black}, - ymin=-90, - ymax=90, - ytick={-360, -270, -180, -90, 0, 90}, - ylabel={Phase [deg]}, - axis background/.style={fill=white}, - xmajorgrids, - xminorgrids, - ymajorgrids - ] - \addplot[color=black, mark=none] - table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlead_lag_filter.csv}; - \end{axis} - \end{tikzpicture} -#+end_src - -#+name: fig:lead_lag_filter -#+caption: Lead Lag Filter ([[./figs/lead_lag_filter.png][png]], [[./figs/lead_lag_filter.pdf][pdf]]. -#+RESULTS: -[[file:figs/lead_lag_filter.png]] - -* Complementary -* Performance Weight +** Alternative #+begin_src matlab w0 = 2*pi; % [rad/s] A = 1e-2; @@ -1160,12 +852,12 @@ #+RESULTS: [[file:figs/weight_first_order.png]] -* Combine Filters +* TODO Combine Filters ** Additive - [ ] Explain how phase and magnitude combine ** Multiplicative -* Filters representing noise +* TODO Filters representing noise Let's consider a noise $n$ that is shaped from a white-noise $\tilde{n}$ with unitary PSD ($\Phi_\tilde{n}(\omega) = 1$) using a transfer function $G(s)$. The PSD of $n$ is then: \[ \Phi_n(\omega) = |G(j\omega)|^2 \Phi_{\tilde{n}}(\omega) = |G(j\omega)|^2 \]