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content/inbook/albertos04_decen_decoup_contr.md
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@ -8,7 +8,7 @@ Tags
: [Decentralized Control](decentralized_control.md), [Multivariable Control](multivariable_control.md) : [Decentralized Control](decentralized_control.md), [Multivariable Control](multivariable_control.md)
Reference Reference
: ([Albertos and Antonio 2004](#orgea6b25f)) : ([Albertos and Antonio 2004](#org9f7e5c2))
Author(s) Author(s)
: Albertos, P., & Antonio, S. : Albertos, P., & Antonio, S.
@ -76,10 +76,10 @@ This strategy is called **decoupling**.
### Feedforward Decoupling {#feedforward-decoupling} ### Feedforward Decoupling {#feedforward-decoupling}
A pre-compensator (Figure [1](#orgaa5070b)) can be added to transform the open-loop characteristics into a new one as chosen by the designer. A pre-compensator (Figure [1](#org92f4052)) can be added to transform the open-loop characteristics into a new one as chosen by the designer.
This decoupler can be taken as the inverse of the plant provided it does not include RHP-zeros. This decoupler can be taken as the inverse of the plant provided it does not include RHP-zeros.
<a id="orgaa5070b"></a> <a id="org92f4052"></a>
{{< figure src="/ox-hugo/albertos04_pre_compensator_decoupling.png" caption="Figure 1: Decoupler pre-compensator" >}} {{< figure src="/ox-hugo/albertos04_pre_compensator_decoupling.png" caption="Figure 1: Decoupler pre-compensator" >}}
@ -109,10 +109,10 @@ where \\(U\\) and \\(V\\) are orthogonal matrices and \\(\Sigma\\) is diagonal.
The SVD can be used to obtain decoupled equations between linear combinations of sensors and linear combinations of actuators. The SVD can be used to obtain decoupled equations between linear combinations of sensors and linear combinations of actuators.
In this way, although losing part of its intuitive sense, a decoupled design can be carried out even for non-square plants. In this way, although losing part of its intuitive sense, a decoupled design can be carried out even for non-square plants.
If sensors are multiplied by \\(U^T\\) and control actions multiplied by \\(V\\), as in Figure [2](#orge470e39), then the loop, in the transformed variables, is decoupled, so a diagonal controller \\(K\_D\\) can be used. If sensors are multiplied by \\(U^T\\) and control actions multiplied by \\(V\\), as in Figure [2](#org9e1f260), then the loop, in the transformed variables, is decoupled, so a diagonal controller \\(K\_D\\) can be used.
Usually, the sensor and actuator transformations are obtained using the DC gain, or a real approximation of \\(G(j\omega)\\), where \\(\omega\\) is around the desired closed-loop bandwidth. Usually, the sensor and actuator transformations are obtained using the DC gain, or a real approximation of \\(G(j\omega)\\), where \\(\omega\\) is around the desired closed-loop bandwidth.
<a id="orge470e39"></a> <a id="org9e1f260"></a>
{{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="Figure 2: SVD decoupling: \\(K\_D\\) is a diagonal controller designed for \\(\Sigma\\)" >}} {{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="Figure 2: SVD decoupling: \\(K\_D\\) is a diagonal controller designed for \\(\Sigma\\)" >}}
@ -187,4 +187,4 @@ The solution is similar to that of the wind-up phenomenon: the regulator should
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgea6b25f"></a>Albertos, P., and S. Antonio. 2004. “Decentralized and Decoupled Control.” In _Multivariable Control Systems: An Engineering Approach_, 125–62. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. <https://doi.org/10.1007/b97506>. <a id="org9f7e5c2"></a>Albertos, P., and S. Antonio. 2004. “Decentralized and Decoupled Control.” In _Multivariable Control Systems: An Engineering Approach_, 125–62. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. <https://doi.org/10.1007/b97506>.