Update Content - 2021-08-23
This commit is contained in:
@@ -5,10 +5,10 @@ draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Multivariable Control]({{< relref "multivariable_control" >}})
|
||||
: [Multivariable Control](multivariable_control.md)
|
||||
|
||||
Reference
|
||||
: ([Albertos and Antonio 2004](#orga1617be))
|
||||
: ([Albertos and Antonio 2004](#org22a156f))
|
||||
|
||||
Author(s)
|
||||
: Albertos, P., & Antonio, S.
|
||||
@@ -17,19 +17,14 @@ Year
|
||||
: 2004
|
||||
|
||||
|
||||
## Introduction to Multivariable Control {#introduction-to-multivariable-control}
|
||||
## [Decentralized Control](decentralized_control.md) and Decoupled Control {#decentralized-control--decentralized-control-dot-md--and-decoupled-control}
|
||||
|
||||
Decentralized control is decomposed into two steps:
|
||||
|
||||
## Linear System Representation: Models and Equivalence {#linear-system-representation-models-and-equivalence}
|
||||
1. decoupled the plant into several subsystems
|
||||
2. control the subsystems
|
||||
|
||||
|
||||
## Linear Systems Analysis {#linear-systems-analysis}
|
||||
|
||||
|
||||
## Solutions to the Control Problem {#solutions-to-the-control-problem}
|
||||
|
||||
|
||||
## Decentralised and Decoupled Control {#decentralised-and-decoupled-control}
|
||||
The initial effort of decoupling the system results in subsequent easier design, implementation and tuning.
|
||||
|
||||
|
||||
### Decoupling {#decoupling}
|
||||
@@ -37,7 +32,7 @@ Year
|
||||
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:
|
||||
[Decoupled Control](decoupled_control.md) can be achieved in two ways:
|
||||
|
||||
- feedforward cancellation of the cross-coupling terms
|
||||
- based on state measurements, via a feedback law
|
||||
@@ -66,7 +61,7 @@ Although at first glance, decoupling seems an appealing idea, there are some dra
|
||||
|
||||
#### SVD Decoupling {#svd-decoupling}
|
||||
|
||||
A matrix \\(M\\) can be expressed, using the [Singular Value Decomposition]({{< relref "singular_value_decomposition" >}}) as:
|
||||
A matrix \\(M\\) can be expressed, using the [Singular Value Decomposition](singular_value_decomposition.md) as:
|
||||
|
||||
\begin{equation}
|
||||
M = U \Sigma V^T
|
||||
@@ -77,10 +72,10 @@ 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.
|
||||
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](#orgd447864), 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 [1](#orgbba6502), 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="orgd447864"></a>
|
||||
<a id="orgbba6502"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="Figure 1: SVD decoupling: \\(K\_D\\) is a diagonal controller designed for \\(\Sigma\\)" >}}
|
||||
|
||||
@@ -104,29 +99,47 @@ If some of the vectors in \\(V\\) (input directions) have a significant componen
|
||||
|
||||
## Implementation and Other Issues {#implementation-and-other-issues}
|
||||
|
||||
There are two main categories for the implementation of MIMO control:
|
||||
|
||||
## Appendices {#appendices}
|
||||
- Decentralized, Decoupled, Cascade
|
||||
- Centralized, optimization based
|
||||
|
||||
A fundamental reason to use cascade and decentralized control in most practical applications is because they require less modelling effort.
|
||||
Other advantages of cascade and decentralized control are:
|
||||
|
||||
- its behaviour can be easily understood
|
||||
- standard equipment can be used (PID controllers, etc.)
|
||||
- their decoupled behavior enables easier tuning with model-free strategies
|
||||
- decentralized implementation tends to be more fault-tolerant, as individual loops will try to keep their set-points even in the case some other components have failed.
|
||||
|
||||
|
||||
### Summary of SISO System Analysis {#summary-of-siso-system-analysis}
|
||||
### [Anti-Windup Control](anti_windup_control.md) {#anti-windup-control--anti-windup-control-dot-md}
|
||||
|
||||
In practice, it is possible that an actuator saturate.
|
||||
In such case, the feedback path is broken, and this has several implications:
|
||||
|
||||
- unstable processes: the process output might go out of control
|
||||
- multi-loop and centralized control: even with stable plants, opening a feedback path may cause the overall loop to become unstable
|
||||
|
||||
The wind-up problem can appear with integral action regulators: during significative step changes in the set point, the integral of the error keeps accumulation and when reaching the desired set-point the accumulated integral action produces a significant overshoot increment.
|
||||
In SISO PID regulators, anti-windup schemes are implemented by either stopping integration if the actuator is saturated or by implementing the following control law:
|
||||
|
||||
\begin{equation}
|
||||
u = K(r - y) - K T\_D \frac{dy}{dt} + \int K T\_i^{-1} (r - y) + T\_t^{-1} (u\_m - u) dt \label{eq:antiwindup\_pid}
|
||||
\end{equation}
|
||||
|
||||
where \\(u\\) is the calculated control action and \\(u\_m\\) is the actual control action applied to the plant.
|
||||
In non-saturated behaviour, \\(u=u\_m\\) and the equation is the ordinary PID.
|
||||
In saturation, \\(u\_m\\) is a constant and the resulting equations drive \\(u\\) down towards \\(u\_m\\) dynamically, with time constant \\(T\_T\\).
|
||||
|
||||
|
||||
### Matrices {#matrices}
|
||||
### [Bumpless Transfer](bumpless_transfer.md) {#bumpless-transfer--bumpless-transfer-dot-md}
|
||||
|
||||
|
||||
### Signal and System Norms {#signal-and-system-norms}
|
||||
|
||||
|
||||
### Optimisation {#optimisation}
|
||||
|
||||
|
||||
### Multivariable Statistics {#multivariable-statistics}
|
||||
|
||||
|
||||
### Robust Control Analysis and Synthesis {#robust-control-analysis-and-synthesis}
|
||||
When switching on the regulator, significant transient behavior can be seen and the controller may saturate the actuators.
|
||||
The solution is similar to that of the wind-up phenomenon: the regulator should be always on, carrying out calculations by using \eqref{eq:antiwindup_pid}.
|
||||
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<a id="orga1617be"></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>.
|
||||
<a id="org22a156f"></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>.
|
||||
|
||||
Reference in New Issue
Block a user