digital-brain/content/zettels/nyquist_stability_criterion.md

+++ title = "Nyquist stability criterion" author = ["Dehaeze Thomas"] draft = false +++

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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: \[ \det(I + GK) \]

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.

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.

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:

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

<./biblio/references.bib>