Update Content - 2024-12-13

content/zettels/nyquist_stability_criterion.md

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+title = "Nyquist stability criterion"
+author = ["Dehaeze Thomas"]
+draft = false
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+
+Tags
+:
+
+
+## Theory {#theory}
+
+The main reason why the Nyquist plot is used it that it can be used with the experimental FRF data!
+
+The zeros and pole of a MIMO system are the zeros and pole of the determinant of \\(G(s)\\).
+\\[ \det(G(s)) = \frac{z\_G(s)}{p\_G(s)} \\]
+The polynomial \\(p\_G(s)\\) is normally called the **open-loop characteristic** polynomial.
+
+In a MIMO feedback system:
+
+-   The transfer function matrix open-loop is:
+    \\[ L(s) = G(s) K(s) \neq K(s) G(s) \\]
+-   The transfer function matrix closed-loop is:
+    \\[ T(s) = [I + L(s)]^{-1} L(s) \\]
+-   **Return difference matrix**:
+    \\[ F(s) = [I + L(s)] \\]
+
+The closed-loop system is stable if the zeros of the closed-loop characteristic polynomial lie in the complex open left half plane.
+There are the zeros of:
+\\[ \det(I + GK) \\]
+
+<div class="important">
+
+**MIMO Nyquist stability criteria**:
+\\[ \det(I + G(s)K(s)) = 0 \quad \text{for} \quad \text{Re}(s)<0 \\]
+To check the closed-loop stability graphically, plot the Nyquist of \\(\det(I + GK)\\) and evaluate the encirclement with respect to the point \\((0,0)\\).
+The Nyquist plot is the image of the imaginary axis (\\(j\omega\\)) under \\(\det(I + GK)\\), i.e. it is the evolution of \\(\det(I + G(j\omega)K(j\omega))\\) in the complex plane.
+Note that there is a single plot, even in the MIMO case.
+
+</div>
+
+<div class="important">
+
+**Eigenvalue loci**:
+The eigenvalue loci (sometimes called the characteristic loci) are defined as the eigenvalues of the frequency response function of the open-loop transfer function matrix \\(G(s)K(s)\\).
+This time, there are \\(n\\) plots, where \\(n\\) is the size of the system.
+
+</div>
+
+
+## Matlab Example {#matlab-example}
+
+Sure we have identified a system with 6 inputs and 6 outputs.
+The Matlab object has dimension `6 x 6 x n` with `n` is the number of frequency points.
+
+First, compute the open-loop gain:
+
+```matlab
+L  = zeros(6, 6, length(f));
+
+for i_f = 1:length(f)
+    L(:,:,i_f)  = squeeze(G(:,:,i_f))*freqresp(K, f(i_f), 'Hz');
+end
+```
+
+Then, compute the eigenvalues of this open-loop gain:
+Finally, plot the (complex) eigenvalues in the complex plane:
+
+
+## Bibliography {#bibliography}
+
+<./biblio/references.bib>