133 lines
5.4 KiB
Markdown
133 lines
5.4 KiB
Markdown
+++
|
|
title = "Multivariable control systems: an engineering approach"
|
|
author = ["Thomas Dehaeze"]
|
|
draft = false
|
|
+++
|
|
|
|
Tags
|
|
: [Multivariable Control]({{< relref "multivariable_control" >}})
|
|
|
|
Reference
|
|
: ([Albertos and Antonio 2004](#orgb06343d))
|
|
|
|
Author(s)
|
|
: Albertos, P., & Antonio, S.
|
|
|
|
Year
|
|
: 2004
|
|
|
|
|
|
## Introduction to Multivariable Control {#introduction-to-multivariable-control}
|
|
|
|
|
|
## Linear System Representation: Models and Equivalence {#linear-system-representation-models-and-equivalence}
|
|
|
|
|
|
## Linear Systems Analysis {#linear-systems-analysis}
|
|
|
|
|
|
## Solutions to the Control Problem {#solutions-to-the-control-problem}
|
|
|
|
|
|
## Decentralised and Decoupled Control {#decentralised-and-decoupled-control}
|
|
|
|
|
|
### Decoupling {#decoupling}
|
|
|
|
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.
|
|
This strategy is called **decoupling**.
|
|
|
|
[Decoupled Control]({{< relref "decoupled_control" >}}) can be achieved in two ways:
|
|
|
|
- feedforward cancellation of the cross-coupling terms
|
|
- based on state measurements, via a feedback law
|
|
|
|
|
|
#### Feedforward Decoupling {#feedforward-decoupling}
|
|
|
|
A pre-compensator 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.
|
|
|
|
**Approximate decoupling**:
|
|
To design low-bandwidth loops, insertion of the inverse DC-gain before the loop ensures decoupling at least at steady-state.
|
|
If further bandwidth extension is desired, an approximation of \\(G^{-1}\\) valid in low frequencies can be used.
|
|
|
|
|
|
#### Feedback Decoupling {#feedback-decoupling}
|
|
|
|
Although at first glance, decoupling seems an appealing idea, there are some drawbacks:
|
|
|
|
- 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.
|
|
- a decoupling design (inverse-based controller) may not be desirable for all disturbance-rejection tasks.
|
|
- many MIMO non-minimum phase systems, when feedforward decoupled, increase the RHP-zero multiplicity so performance limitations due to its presence are exacerbated.
|
|
- decoupling may be very sensitive to modeling errors, specially for ill-conditionned plants
|
|
- feedback decoupling needs full state measurements
|
|
|
|
|
|
#### SVD Decoupling {#svd-decoupling}
|
|
|
|
A matrix \\(M\\) can be expressed, using the [Singular Value Decomposition]({{< relref "singular_value_decomposition" >}}) as:
|
|
|
|
\begin{equation}
|
|
M = U \Sigma V^T
|
|
\end{equation}
|
|
|
|
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.
|
|
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 [1](#org3d5b40c), 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.
|
|
|
|
<a id="org3d5b40c"></a>
|
|
|
|
{{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="Figure 1: SVD decoupling: \\(K\_D\\) is a diagonal controller designed for \\(\Sigma\\)" >}}
|
|
|
|
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".
|
|
|
|
In ill-conditioned plants, the ratio between the biggest and lower singular value is large (for reference, greater than 20).
|
|
They are very sensitive to input uncertainty as some "input directions" have much bigger gain than other ones.
|
|
|
|
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).
|
|
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.
|
|
|
|
|
|
## Fundamentals of Centralised Closed-loop Control {#fundamentals-of-centralised-closed-loop-control}
|
|
|
|
|
|
## Optimisation-based Control {#optimisation-based-control}
|
|
|
|
|
|
## Designing for Robustness {#designing-for-robustness}
|
|
|
|
|
|
## Implementation and Other Issues {#implementation-and-other-issues}
|
|
|
|
|
|
## Appendices {#appendices}
|
|
|
|
|
|
### Summary of SISO System Analysis {#summary-of-siso-system-analysis}
|
|
|
|
|
|
### Matrices {#matrices}
|
|
|
|
|
|
### Signal and System Norms {#signal-and-system-norms}
|
|
|
|
|
|
### Optimisation {#optimisation}
|
|
|
|
|
|
### Multivariable Statistics {#multivariable-statistics}
|
|
|
|
|
|
### Robust Control Analysis and Synthesis {#robust-control-analysis-and-synthesis}
|
|
|
|
|
|
|
|
## Bibliography {#bibliography}
|
|
|
|
<a id="orgb06343d"></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>.
|