Add all bibliographic notes and zettels

content/zettels/multivariable_control.md

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+title = "Multivariable Control"
+author = ["Dehaeze Thomas"]
+draft = false
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+
+Tags
+: [Norms]({{< relref "norms.md" >}})
+
+A very nice book about Multivariable Control is (<a href="#citeproc_bib_item_1">Skogestad and Postlethwaite 2007</a>)
+
+
+## Transfer functions for Multi-Input Multi-Output systems {#transfer-functions-for-multi-input-multi-output-systems}
+
+{{< figure src="/ox-hugo/mimo_tf.png" >}}
+
+\\[ T\_i = -\frac{u}{d\_i} = (I + KG)^{-1} KG \\]
+\\[ T\_o = -\frac{p\_o}{d\_o} = (I + GK)^{-1} GK \\]
+\\[ S\_i = \frac{p\_i}{d\_i} = (I + KG)^{-1} \\]
+\\[ S\_o = \frac{y}{d\_o} = (I + GK)^{-1} \\]
+
+
+## Measures of interaction {#measures-of-interaction}
+
+-   Interaction index (for \\(2 \times 2\\) plant):
+    \\[ \phi = \frac{g\_{12}g\_{21}}{g\_{11}g\_{22}} \\]
+    When \\(\phi\\) is close to zero, this means there is no interaction.
+-   The **relative gain array** of a square matrix:
+    \\[ \text{RGA}(G) \triangleq G \times ( G^{-1})^T \\]
+
+
+## Stability {#stability}
+
+-   **Characteristic Loci**: Eigenvalues of \\(G(j\omega)\\) plotted in the complex plane
+-   **Generalized Nyquist Criterion**: If \\(G(s)\\) has \\(p\_0\\) unstable poles, then the closed-loop system with return ratio \\(kG(s)\\) is stable if and only if the characteristic loci of \\(kG(s)\\), taken together, encircle the point \\(-1\\), \\(p\_0\\) times anti-clockwise, assuming there are no hidden modes
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Skogestad, S., and I. Postlethwaite. 2007. <i>Multivariable Feedback Control: Analysis and Design - Second Edition</i>. John Wiley.</div>
+</div>