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+++ title = "Relative Gain Array" author = ["Dehaeze Thomas"] draft = false +++
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Imagine a 2x2 plant: \[ G(s) = \begin{bmatrix} g_{11}(s) & g_{12}(s)\\ g_{21}(s) & g_{22}(s) \end{bmatrix} \]
Now suppose a controller \(k_{1}(s)\) is closed on the first loop. The new transfer function from \(u_2\) to \(y_2\) (while the first loop is closed) is :
\begin{equation} y_2 = g_{22} \left[ 1 - \underbrace{\frac{g_{21}g_{12}}{g_{11}g_{22}}}_{\phi} \underbrace{\frac{g_{11} k_{11}}{1 + g_{11}k_{11}}}_{T_{11}} \right] \end{equation}
\(\phi\) is called the interaction index. \(T_{11}\) is the complementary sensitivity of the first loop (equal to 1 in the bandwidth of the first controller). Therefore, we want \(\phi \approx 0\) to have no interaction.
Similarly, the relative gain is defined as: \[ \Lambda = \frac{1}{1 - \phi} \] And we want \(\Lambda \approx 1\) to have no interaction.
Note that the scaling of inputs or outputs of the MIMO plant has no effect on \(\phi\) or \(\Lambda\).
The relative gain array is defined as: \[ \Lambda(G) = G \star (G^{-1})^{T} \] where \(\star\) means element wise multiplication.
RGA = G.*pinv(G.')
Bibliography
<./biblio/references.bib>