Update content - 2026-10-08

content/zettels/interaction_analysis.md

@@ -11,8 +11,59 @@ Tags
 
 Two main ways to evaluate the interaction:
 
--   [Relative Gain Array]({{< relref "relative_gain_array.md" >}})
--   [Structured Singular Value]({{< relref "structured_singular_value.md" >}})
+-   Relative Gain Array
+-   Structured Singular Value
+
+
+## Relative Gain Array {#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.
+
+```matlab
+RGA = G.*pinv(G.')
+```
+
+
+## RGA-number {#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.
+
+```matlab
+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.
 
 
 ## Bibliography {#bibliography}