2.2 KiB
+++ title = "Interaction Analysis" author = ["Dehaeze Thomas"] draft = false category = "Control Theory" subcategory = "Multivariable Analysis" +++
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Two main ways to evaluate the interaction:
- Relative Gain Array
- Structured Singular Value
Relative Gain Array
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.')
RGA-number
The RGA-number is a scalar measure of how far the RGA is from the identity matrix (i.e. from a plant without interaction):
\begin{equation} \text{RGA-number} = \| \Lambda(G) - I \|_{\text{sum}} = \sum_{i,j} \left| \Lambda_{ij} - \delta_{ij} \right| \end{equation}
with \(\delta_{ij}\) the Kronecker delta, and \(\|\cdot\|_{\text{sum}}\) the sum of the absolute values of all the elements.
The RGA-number should be small around the crossover frequency of the decentralized controller, as this is where the interaction matters most.
RGA = G.*pinv(G.');
RGA_number = sum(sum(abs(RGA - eye(size(RGA)))));
For a plant evaluated over frequency, this is computed at each frequency point of the frequency response.