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