Update all files with new citeproc-org package

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@@ -148,21 +148,21 @@ Three factors influence the performance:
The DEB helps identifying which disturbance is the limiting factor, and it should be investigated if the controller can deal with this disturbance before re-designing the plant.
The modelling of disturbance as stochastic variables, is by excellence suitable for the optimal stochastic control framework.
In Figure [1](#orgf051865), the generalized plant maps the disturbances to the performance channels.
In Figure [1](#org30a4301), the generalized plant maps the disturbances to the performance channels.
By minimizing the \\(\mathcal{H}\_2\\) system norm of the generalized plant, the variance of the performance channels is minimized.
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{{< figure src="/ox-hugo/jabben07_general_plant.png" caption="Figure 1: Control system with the generalized plant \\(G\\). The performance channels are stacked in \\(z\\), while the controller input is denoted with \\(y\\)" >}}
#### Using Weighting Filters for Disturbance Modelling {#using-weighting-filters-for-disturbance-modelling}
Since disturbances are generally not white, the system of Figure [1](#orgf051865) needs to be augmented with so called **disturbance weighting filters**.
Since disturbances are generally not white, the system of Figure [1](#org30a4301) needs to be augmented with so called **disturbance weighting filters**.
A disturbance weighting filter gives the disturbance PSD when white noise as input is applied.
This is illustrated in Figure [2](#org35c7d66) where a vector of white noise time signals \\(\underbar{w}(t)\\) is filtered through a weighting filter to obtain the colored physical disturbances \\(w(t)\\) with the desired PSD \\(S\_w\\) .
This is illustrated in Figure [2](#org3b94947) where a vector of white noise time signals \\(\underbar{w}(t)\\) is filtered through a weighting filter to obtain the colored physical disturbances \\(w(t)\\) with the desired PSD \\(S\_w\\) .
The generalized plant framework also allows to include **weighting filters for the performance channels**.
This is useful for three reasons:
@@ -171,7 +171,7 @@ This is useful for three reasons:
- some performance channels may be of more importance than others
- by using dynamic weighting filters, one can emphasize the performance in a certain frequency range
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{{< figure src="/ox-hugo/jabben07_weighting_functions.png" caption="Figure 2: Control system with the generalized plant \\(G\\) and weighting functions" >}}
@@ -196,9 +196,9 @@ So, to obtain feasible controllers, the performance channel is a combination of
By choosing suitable weighting filters for \\(y\\) and \\(u\\), the performance can be optimized while keeping the controller effort limited:
\\[ \\|z\\|\_{rms}^2 = \left\\| \begin{bmatrix} y \\ \alpha u \end{bmatrix} \right\\|\_{rms}^2 = \\|y\\|\_{rms}^2 + \alpha^2 \\|u\\|\_{rms}^2 \\]
By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in Figure [3](#org742f80a) is obtained.
By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in Figure [3](#orgb0b1e78) is obtained.
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{{< figure src="/ox-hugo/jabben07_pareto_curve_H2.png" caption="Figure 3: An illustration of a Pareto curve. Each point of the curve represents the performance obtained with an optimal controller. The curve is obtained by varying \\(\alpha\\) and calculating an \\(\mathcal{H}\_2\\) optimal controller for each \\(\alpha\\)." >}}

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@@ -4,11 +4,15 @@ author = ["Thomas Dehaeze"]
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### Backlinks {#backlinks}
- [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
Tags
: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
Reference
: <sup id="651e626e040250ee71a0847aec41b60c"><a class="reference-link" href="#monkhorst04_dynam_error_budget" title="Wouter Monkhorst, Dynamic Error Budgeting, a design approach (2004).">(Wouter Monkhorst, 2004)</a></sup>
: ([Monkhorst 2004](#org7d7f3f5))
Author(s)
: Monkhorst, W.
@@ -95,9 +99,9 @@ Find a controller \\(C\_{\mathcal{H}\_2}\\) which minimizes the \\(\mathcal{H}\_
In order to synthesize an \\(\mathcal{H}\_2\\) controller that will minimize the output error, the total system including disturbances needs to be modeled as a system with zero mean white noise inputs.
This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise (Figure [1](#org76ddb2c)).
This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise (Figure [1](#org7321040)).
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{{< figure src="/ox-hugo/monkhorst04_weighting_filter.png" caption="Figure 1: The use of a weighting filter \\(V\_w(f)\,[SI]\\) to give the weighted signal \\(\bar{w}(t)\\) a certain PSD \\(S\_w(f)\\)." >}}
@@ -108,23 +112,23 @@ The PSD \\(S\_w(f)\\) of the weighted signal is:
Given \\(S\_w(f)\\), \\(V\_w(f)\\) can be obtained using a technique called _spectral factorization_.
However, this can be avoided if the modelling of the disturbances is directly done in terms of weighting filters.
Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#org425ff37)).
Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#orgea16f0b)).
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{{< figure src="/ox-hugo/monkhorst04_general_weighted_plant.png" caption="Figure 2: The open loop system \\(\bar{G}\\) in series with the diagonal input weightin filter \\(V\_w\\) and diagonal output scaling iflter \\(W\_z\\) defining the generalized plant \\(G\\)" >}}
#### Output scaling and the Pareto curve {#output-scaling-and-the-pareto-curve}
In this research, the outputs of the closed loop system (Figure [3](#orgba842f3)) are:
In this research, the outputs of the closed loop system (Figure [3](#org95f6ca6)) are:
- the performance (error) signal \\(e\\)
- the controller output \\(u\\)
In this way, the designer can analyze how much control effort is used to achieve the performance level at the performance output.
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{{< figure src="/ox-hugo/monkhorst04_closed_loop_H2.png" caption="Figure 3: The closed loop system with weighting filters included. The system has \\(n\\) disturbance inputs and two outputs: the error \\(e\\) and the control signal \\(u\\). The \\(\mathcal{H}\_2\\) minimized the \\(\mathcal{H}\_2\\) norm of this system." >}}
@@ -147,10 +151,7 @@ To achieve the highest degree of prediction accuracy, it is recommended to use t
When an \\(\mathcal{H}\_2\\) controller is synthesized for a particular system, it can give the control designer useful hints about how to control the system best for optimal performance.
Drawbacks however are, that no robustness guarantees can be given and that the order of the \\(\mathcal{H}\_2\\) controller will generally be too high for implementation.
# Bibliography
<a class="bibtex-entry" id="monkhorst04_dynam_error_budget">Monkhorst, W., *Dynamic error budgeting, a design approach* (2004). Delft University.</a> [](#651e626e040250ee71a0847aec41b60c)
## Bibliography {#bibliography}
## Backlinks {#backlinks}
- [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
<a id="org7d7f3f5"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.