133 lines
5.4 KiB
Markdown
133 lines
5.4 KiB
Markdown
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title = "Multivariable control systems: an engineering approach"
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author = ["Thomas Dehaeze"]
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draft = false
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: [Multivariable Control]({{< relref "multivariable_control" >}})
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Reference
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: ([Albertos and Antonio 2004](#orgbcb0991))
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Author(s)
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: Albertos, P., & Antonio, S.
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Year
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: 2004
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## Introduction to Multivariable Control {#introduction-to-multivariable-control}
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## Linear System Representation: Models and Equivalence {#linear-system-representation-models-and-equivalence}
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## Linear Systems Analysis {#linear-systems-analysis}
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## Solutions to the Control Problem {#solutions-to-the-control-problem}
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## Decentralised and Decoupled Control {#decentralised-and-decoupled-control}
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### Decoupling {#decoupling}
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In cases when multi-loop control is not effective in reaching the desired specifications, a possible strategy for tackling the MIMO control could be to transform the transfer function matrix into a diagonal dominant one.
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This strategy is called **decoupling**.
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[Decoupled Control]({{< relref "decoupled_control" >}}) can be achieved in two ways:
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- feedforward cancellation of the cross-coupling terms
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- based on state measurements, via a feedback law
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#### Feedforward Decoupling {#feedforward-decoupling}
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A pre-compensator can be added to transform the open-loop characteristics into a new one as chosen by the designer.
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This decoupler can be taken as the inverse of the plant provided it does not include RHP-zeros.
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**Approximate decoupling**:
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To design low-bandwidth loops, insertion of the inverse DC-gain before the loop ensures decoupling at least at steady-state.
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If further bandwidth extension is desired, an approximation of \\(G^{-1}\\) valid in low frequencies can be used.
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#### Feedback Decoupling {#feedback-decoupling}
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Although at first glance, decoupling seems an appealing idea, there are some drawbacks:
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- as decoupling is achieved via the coordination of sensors and actuators to achieve an "apparent" diagonal behavior, the failure of one the actuators may heavily affects all loops.
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- a decoupling design (inverse-based controller) may not be desirable for all disturbance-rejection tasks.
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- many MIMO non-minimum phase systems, when feedforward decoupled, increase the RHP-zero multiplicity so performance limitations due to its presence are exacerbated.
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- decoupling may be very sensitive to modeling errors, specially for ill-conditionned plants
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- feedback decoupling needs full state measurements
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#### SVD Decoupling {#svd-decoupling}
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A matrix \\(M\\) can be expressed, using the [Singular Value Decomposition]({{< relref "singular_value_decomposition" >}}) as:
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\begin{equation}
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M = U \Sigma V^T
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\end{equation}
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where \\(U\\) and \\(V\\) are orthogonal matrices and \\(\Sigma\\) is diagonal.
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The SVD can be used to obtain decoupled equations between linear combinations of sensors and linear combinations of actuators.
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In this way, although losing part of its intuitive sense, a decoupled design can be carried out even for non-square plants.
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If sensors are multiplied by \\(U^T\\) and control actions multiplied by \\(V\\), as in Figure [1](#org335191d), then the loop, in the transformed variables, is decoupled, so a diagonal controller \\(K\_D\\) can be used.
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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.
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<a id="org335191d"></a>
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{{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="Figure 1: SVD decoupling: \\(K\_D\\) is a diagonal controller designed for \\(\Sigma\\)" >}}
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The transformed sensor-actuator pair corresponding to the maximum singular value is the direction with biggest "gain" on the plant, that is, the combination of variables being "easiest to control".
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In ill-conditioned plants, the ratio between the biggest and lower singular value is large (for reference, greater than 20).
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They are very sensitive to input uncertainty as some "input directions" have much bigger gain than other ones.
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SVD decoupling produces the most suitable combinations for independent "multi-loop" control in the transformed variables, so its performance may be better than RGA-based design (at the expense of losing physical interpretability).
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If some of the vectors in \\(V\\) (input directions) have a significant component on a particular input, and the corresponding output direction is also significantly pointing to a particular output, that combination is a good candidate for an independent multi-loop control.
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## Fundamentals of Centralised Closed-loop Control {#fundamentals-of-centralised-closed-loop-control}
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## Optimisation-based Control {#optimisation-based-control}
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## Designing for Robustness {#designing-for-robustness}
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## Implementation and Other Issues {#implementation-and-other-issues}
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## Appendices {#appendices}
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### Summary of SISO System Analysis {#summary-of-siso-system-analysis}
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### Matrices {#matrices}
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### Signal and System Norms {#signal-and-system-norms}
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### Optimisation {#optimisation}
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### Multivariable Statistics {#multivariable-statistics}
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### Robust Control Analysis and Synthesis {#robust-control-analysis-and-synthesis}
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## Bibliography {#bibliography}
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<a id="orgbcb0991"></a>Albertos, P., and S. Antonio. 2004. _Multivariable Control Systems: An Engineering Approach_. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. <https://doi.org/10.1007/b97506>.
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