42 lines
1.3 KiB
Markdown
42 lines
1.3 KiB
Markdown
|
+++
|
||
|
|
title = "Relative Gain Array"
|
||
|
|
author = ["Dehaeze Thomas"]
|
||
|
|
draft = false
|
||
|
|
+++
|
||
|
|
|
||
|
|
Tags
|
||
|
|
:
|
||
|
|
|
||
|
|
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.')
|
||
|
|
```
|
||
|
|
|
||
|
|
|
||
|
|
## Bibliography {#bibliography}
|
||
|
|
|
||
|
|
<./biblio/references.bib>
|