|
20 | 20 | S = ss(matrices_S...) |
21 | 21 | T = ss(matrices_T...) |
22 | 22 | bodeplot([S, T], lab = ["S" "" "T" ""], plot_title = "Bode plot of sensitivity functions", |
23 | | - margin = 5Plots.mm)</code></pre><img src="020c822d.svg" alt="Example block output"/><p>The sensitivity functions obtained this way should be equivalent to the ones obtained with the code below</p><pre><code class="language-julia hljs">using ControlSystemsBase |
| 23 | + margin = 5Plots.mm)</code></pre><img src="b9d8df13.svg" alt="Example block output"/><p>The sensitivity functions obtained this way should be equivalent to the ones obtained with the code below</p><pre><code class="language-julia hljs">using ControlSystemsBase |
24 | 24 | P = tf(1.0, [1, 1]) |> ss |
25 | 25 | C = 1 # Negative feedback assumed in ControlSystems |
26 | 26 | S = sensitivity(P, C) # or feedback(1, P*C) |
|
70 | 70 | ---------- |
71 | 71 | 1.0s + 2.0 |
72 | 72 |
|
73 | | -Continuous-time transfer function model</code></pre><h2 id="Gain-and-phase-margins"><a class="docs-heading-anchor" href="#Gain-and-phase-margins">Gain and phase margins</a><a id="Gain-and-phase-margins-1"></a><a class="docs-heading-anchor-permalink" href="#Gain-and-phase-margins" title="Permalink"></a></h2><p>Further linear analysis can be performed using the <a href="https://juliacontrol.github.io/ControlSystems.jl/stable/lib/analysis/">analysis methods from ControlSystemsBase</a>. For example, calculating the gain and phase margins of a system can be done using</p><pre><code class="language-julia hljs">margin(P)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(wgm = [NaN;;], gm = [Inf;;], wpm = [NaN;;], pm = [Inf;;])</code></pre><p>(they are infinite for this system). A Nyquist plot can be produced using</p><pre><code class="language-julia hljs">nyquistplot(P)</code></pre><img src="ef822bd8.svg" alt="Example block output"/><h2 id="Index"><a class="docs-heading-anchor" href="#Index">Index</a><a id="Index-1"></a><a class="docs-heading-anchor-permalink" href="#Index" title="Permalink"></a></h2><ul><li><a href="#ModelingToolkit.linearize"><code>ModelingToolkit.linearize</code></a></li></ul><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="ModelingToolkit.linearize" href="#ModelingToolkit.linearize"><code>ModelingToolkit.linearize</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">(; A, B, C, D), simplified_sys, extras = linearize(sys, inputs, outputs; t=0.0, op = Dict(), allow_input_derivatives = false, zero_dummy_der=false, kwargs...) |
| 73 | +Continuous-time transfer function model</code></pre><h2 id="Gain-and-phase-margins"><a class="docs-heading-anchor" href="#Gain-and-phase-margins">Gain and phase margins</a><a id="Gain-and-phase-margins-1"></a><a class="docs-heading-anchor-permalink" href="#Gain-and-phase-margins" title="Permalink"></a></h2><p>Further linear analysis can be performed using the <a href="https://juliacontrol.github.io/ControlSystems.jl/stable/lib/analysis/">analysis methods from ControlSystemsBase</a>. For example, calculating the gain and phase margins of a system can be done using</p><pre><code class="language-julia hljs">margin(P)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(wgm = [NaN;;], gm = [Inf;;], wpm = [NaN;;], pm = [Inf;;])</code></pre><p>(they are infinite for this system). A Nyquist plot can be produced using</p><pre><code class="language-julia hljs">nyquistplot(P)</code></pre><img src="328fd252.svg" alt="Example block output"/><h2 id="Index"><a class="docs-heading-anchor" href="#Index">Index</a><a id="Index-1"></a><a class="docs-heading-anchor-permalink" href="#Index" title="Permalink"></a></h2><ul><li><a href="#ModelingToolkit.linearize"><code>ModelingToolkit.linearize</code></a></li></ul><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="ModelingToolkit.linearize" href="#ModelingToolkit.linearize"><code>ModelingToolkit.linearize</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">(; A, B, C, D), simplified_sys, extras = linearize(sys, inputs, outputs; t=0.0, op = Dict(), allow_input_derivatives = false, zero_dummy_der=false, kwargs...) |
74 | 74 | (; A, B, C, D), extras = linearize(simplified_sys, lin_fun; t=0.0, op = Dict(), allow_input_derivatives = false, zero_dummy_der=false)</code></pre><p>Linearize <code>sys</code> between <code>inputs</code> and <code>outputs</code>, both vectors of variables. Return a NamedTuple with the matrices of a linear statespace representation on the form</p><p class="math-container">\[\begin{aligned} |
75 | 75 | ẋ &= Ax + Bu\\ |
76 | 76 | y &= Cx + Du |
77 | | -\end{aligned}\]</p><p>The first signature automatically calls <a href="@ref"><code>linearization_function</code></a> internally, while the second signature expects the outputs of <a href="@ref"><code>linearization_function</code></a> as input.</p><p><code>op</code> denotes the operating point around which to linearize. If none is provided, the default values of <code>sys</code> are used.</p><p>If <code>allow_input_derivatives = false</code>, an error will be thrown if input derivatives (<span>$u̇$</span>) appear as inputs in the linearized equations. If input derivatives are allowed, the returned <code>B</code> matrix will be of double width, corresponding to the input <code>[u; u̇]</code>.</p><p><code>zero_dummy_der</code> can be set to automatically set the operating point to zero for all dummy derivatives.</p><p>The return value <code>extras</code> is a NamedTuple <code>(; x, p, t)</code> containing the result of the initialization problem that was solved to determine the operating point.</p><p>See also <a href="@ref"><code>linearization_function</code></a> which provides a lower-level interface, <a href="@ref"><code>linearize_symbolic</code></a> and <a href="@ref"><code>ModelingToolkit.reorder_unknowns</code></a>.</p><p>See extended help for an example.</p><p>The implementation and notation follows that of <a href="https://ep.liu.se/ecp/043/075/ecp09430097.pdf">"Linear Analysis Approach for Modelica Models", Allain et al. 2009</a></p><p><strong>Extended help</strong></p><p>This example builds the following feedback interconnection and linearizes it from the input of <code>F</code> to the output of <code>P</code>.</p><pre><code class="nohighlight hljs"> |
| 77 | +\end{aligned}\]</p><p>The first signature automatically calls <a href="@ref"><code>linearization_function</code></a> internally, while the second signature expects the outputs of <a href="@ref"><code>linearization_function</code></a> as input.</p><p><code>op</code> denotes the operating point around which to linearize. If none is provided, the default values of <code>sys</code> are used.</p><p>If <code>allow_input_derivatives = false</code>, an error will be thrown if input derivatives (<span>$u̇$</span>) appear as inputs in the linearized equations. If input derivatives are allowed, the returned <code>B</code> matrix will be of double width, corresponding to the input <code>[u; u̇]</code>.</p><p><code>zero_dummy_der</code> can be set to automatically set the operating point to zero for all dummy derivatives.</p><p>The return value <code>extras</code> is a NamedTuple <code>(; x, p, t)</code> containing the result of the initialization problem that was solved to determine the operating point.</p><p>See also <a href="@ref"><code>linearization_function</code></a> which provides a lower-level interface, <a href="@ref"><code>linearize_symbolic</code></a> and <a href="@ref"><code>ModelingToolkit.reorder_unknowns</code></a>.</p><p>See extended help for an example.</p><p>The implementation and notation follows that of <a href="https://ep.liu.se/ecp/043/075/ecp09430097.pdf">"Linear Analysis Approach for Modelica Models", Allain et al. 2009</a></p><p><strong>Extended help</strong></p><p>This example builds the following feedback interconnection and linearizes it from the input of <code>F</code> to the output of <code>P</code>.</p><pre><code class="language-julia hljs"> |
78 | 78 | r ┌─────┐ ┌─────┐ ┌─────┐ |
79 | 79 | ───►│ ├──────►│ │ u │ │ |
80 | 80 | │ F │ │ C ├────►│ P │ y |
|
130 | 130 | ## Symbolic linearization |
131 | 131 | lsys_sym, _ = ModelingToolkit.linearize_symbolic(cl, [f.u], [p.x]) |
132 | 132 |
|
133 | | -@assert substitute(lsys_sym.A, ModelingToolkit.defaults(cl)) == lsys.A</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/ModelingToolkit.jl/blob/v10.24.0/src/linearization.jl#L665-L765">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../hydraulic/">« Hydraulic Components</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.14.1 on <span class="colophon-date" title="Sunday 5 October 2025 13:38">Sunday 5 October 2025</span>. Using Julia version 1.11.7.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> |
| 133 | +@assert substitute(lsys_sym.A, ModelingToolkit.defaults(cl)) == lsys.A</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/ModelingToolkit.jl/blob/v10.26.1/src/linearization.jl#L670-L770">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../hydraulic/">« Hydraulic Components</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.15.0 on <span class="colophon-date" title="Thursday 23 October 2025 14:53">Thursday 23 October 2025</span>. Using Julia version 1.12.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> |
0 commit comments