+++ title = "Nyquist stability criterion" author = ["Dehaeze Thomas"] draft = false +++ 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) \\]
**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 {#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>