41 lines
1.4 KiB
Markdown
41 lines
1.4 KiB
Markdown
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title = "Multivariable Control"
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author = ["Dehaeze Thomas"]
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draft = false
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: [Norms]({{< relref "norms.md" >}})
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A very nice book about Multivariable Control is <skogestad07_multiv_feedb_contr>
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## Transfer functions for Multi-Input Multi-Output systems {#transfer-functions-for-multi-input-multi-output-systems}
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{{< figure src="/ox-hugo/mimo_tf.png" >}}
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\\[ T\_i = -\frac{u}{d\_i} = (I + KG)^{-1} KG \\]
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\\[ T\_o = -\frac{p\_o}{d\_o} = (I + GK)^{-1} GK \\]
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\\[ S\_i = \frac{p\_i}{d\_i} = (I + KG)^{-1} \\]
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\\[ S\_o = \frac{y}{d\_o} = (I + GK)^{-1} \\]
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## Measures of interaction {#measures-of-interaction}
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- Interaction index (for \\(2 \times 2\\) plant):
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\\[ \phi = \frac{g\_{12}g\_{21}}{g\_{11}g\_{22}} \\]
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When \\(\phi\\) is close to zero, this means there is no interaction.
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- The **relative gain array** of a square matrix:
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\\[ \text{RGA}(G) \triangleq G \times ( G^{-1})^T \\]
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## Stability {#stability}
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- **Characteristic Loci**: Eigenvalues of \\(G(j\omega)\\) plotted in the complex plane
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- **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
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## Bibliography {#bibliography}
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<./biblio/references.bib>
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