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Thomas Dehaeze
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content/phdthesis/monkhorst04_dynam_error_budget.md
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Tags
: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
: [Dynamic Error Budgeting]({{<relref "dynamic_error_budgeting.md#" >}})
Reference
: ([Monkhorst 2004](#org7671be3))
: ([Monkhorst 2004](#orgb303aca))
Author(s)
: Monkhorst, W.
@@ -74,10 +74,10 @@ The assumptions when applying DEB are:
In practice, many disturbances will have a normal like distribution.
### \\(\mathcal{H}\_2\\) control, maximizing performance {#mathcal-h-2--control-maximizing-performance}
### \\(\mathcal{H}\_2\\) control, maximizing performance {#mathcal-h-2-control-maximizing-performance}
#### The \\(\mathcal{H}\_2\\) norm and variance of the output {#the--mathcal-h-2--norm-and-variance-of-the-output}
#### The \\(\mathcal{H}\_2\\) norm and variance of the output {#the-mathcal-h-2-norm-and-variance-of-the-output}
The \\(\mathcal{H}\_2\\) norm is a norm defined on a system:
\\[ \\|H\\|\_2^2 = \int\_{-\infty}^\infty |H(j2\pi f)|^2 df \\]
@@ -85,7 +85,7 @@ The \\(\mathcal{H}\_2\\) norm is a norm defined on a system:
Stochastic interpretation of the \\(\mathcal{H}\_2\\) norm: the squared \\(\mathcal{H}\_2\\) norm can be interpreted as the output variance of a system with zero mean white noise input.
#### The \\(\mathcal{H}\_2\\) control problem {#the--mathcal-h-2--control-problem}
#### The \\(\mathcal{H}\_2\\) control problem {#the-mathcal-h-2-control-problem}
Find a controller \\(C\_{\mathcal{H}\_2}\\) which minimizes the \\(\mathcal{H}\_2\\) norm of the closed loop system \\(H\\):
\\[ C\_{\mathcal{H}\_2} \in \arg \min\_C \\|H\\|\_2 \\]
@@ -95,9 +95,9 @@ 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](#org16f42a7)).
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](#orgdc82b09)).
<a id="org16f42a7"></a>
<a id="orgdc82b09"></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)\\)." >}}
@@ -108,23 +108,23 @@ 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](#orgc49109b)).
Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#org624c0f1)).
<a id="orgc49109b"></a>
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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](#org8b8bb94)) are:
In this research, the outputs of the closed loop system (Figure [3](#org1993951)) 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="org8b8bb94"></a>
<a id="org1993951"></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." >}}
@@ -151,4 +151,4 @@ Drawbacks however are, that no robustness guarantees can be given and that the o
## Bibliography {#bibliography}
<a id="org7671be3"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
<a id="orgb303aca"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.