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Thomas Dehaeze
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content/phdthesis/monkhorst04_dynam_error_budget.md
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@@ -7,10 +7,10 @@ ref_year = 2004
+++
Tags
: [Dynamic Error Budgeting]({{<relref "dynamic_error_budgeting.md#" >}})
: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting.md" >}})
Reference
: <monkhorst04_dynam_error_budget>
: (<a href="#citeproc_bib_item_1">Monkhorst 2004</a>)
Author(s)
: Monkhorst, W.
@@ -106,11 +106,11 @@ 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](#orgfce1d5b)).
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](#figure--fig:monkhorst04-weighting-filter)).
<a id="orgfce1d5b"></a>
<a id="figure--fig:monkhorst04-weighting-filter"></a>
{{< 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)\\)." >}}
{{< figure src="/ox-hugo/monkhorst04_weighting_filter.png" caption="<span class=\"figure-number\">Figure 1: </span>The use of a weighting filter \\(V\_w(f)\\,[SI]\\) to give the weighted signal \\(\bar{w}(t)\\) a certain PSD \\(S\_w(f)\\)." >}}
The white noise input \\(w(t)\\) is dimensionless, and when the weighting filter has units [SI], the resulting weighted signal \\(\bar{w}(t)\\) has units [SI].
The PSD \\(S\_w(f)\\) of the weighted signal is:
@@ -119,25 +119,25 @@ 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 modeling 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](#orgd937879)).
Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#figure--fig:monkhorst04-general-weighted-plant)).
<a id="orgd937879"></a>
<a id="figure--fig:monkhorst04-general-weighted-plant"></a>
{{< 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\\)" >}}
{{< figure src="/ox-hugo/monkhorst04_general_weighted_plant.png" caption="<span class=\"figure-number\">Figure 2: </span>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](#orgf4dc585)) are:
In this research, the outputs of the closed loop system (Figure [3](#figure--fig:monkhorst04-closed-loop-H2)) 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.
<a id="orgf4dc585"></a>
<a id="figure--fig:monkhorst04-closed-loop-H2"></a>
{{< 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." >}}
{{< figure src="/ox-hugo/monkhorst04_closed_loop_H2.png" caption="<span class=\"figure-number\">Figure 3: </span>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." >}}
The resulting problem is a multi-objective control problem: while constraining the variance of the controller output \\(u\\), the variance of the performance channel should be minimized.
This problem can be solved by scaling the controller output \\(u\\) with a factor \\(\alpha\\) during the \\(\mathcal{H}\_2\\) synthesis.
@@ -157,3 +157,10 @@ 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 {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.</div>
</div>