bibliography: => #+BIBLIOGRAPHY: here
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@@ -161,21 +161,21 @@ Three factors influence the performance:
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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.
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The modelling of disturbance as stochastic variables, is by excellence suitable for the optimal stochastic control framework.
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In Figure [1](#org7b34df5), the generalized plant maps the disturbances to the performance channels.
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In Figure [1](#orgbf22b5e), the generalized plant maps the disturbances to the performance channels.
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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\\)" >}}
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#### Using Weighting Filters for Disturbance Modelling {#using-weighting-filters-for-disturbance-modelling}
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Since disturbances are generally not white, the system of Figure [1](#org7b34df5) needs to be augmented with so called **disturbance weighting filters**.
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Since disturbances are generally not white, the system of Figure [1](#orgbf22b5e) needs to be augmented with so called **disturbance weighting filters**.
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A disturbance weighting filter gives the disturbance PSD when white noise as input is applied.
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This is illustrated in Figure [2](#org5013433) 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\\) .
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This is illustrated in Figure [2](#org27e9aeb) 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\\) .
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The generalized plant framework also allows to include **weighting filters for the performance channels**.
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This is useful for three reasons:
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@@ -184,7 +184,7 @@ This is useful for three reasons:
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- some performance channels may be of more importance than others
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- 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" >}}
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@@ -209,9 +209,9 @@ So, to obtain feasible controllers, the performance channel is a combination of
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By choosing suitable weighting filters for \\(y\\) and \\(u\\), the performance can be optimized while keeping the controller effort limited:
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\\[ \\|z\\|\_{rms}^2 = \left\\| \begin{bmatrix} y \\ \alpha u \end{bmatrix} \right\\|\_{rms}^2 = \\|y\\|\_{rms}^2 + \alpha^2 \\|u\\|\_{rms}^2 \\]
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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](#org47370f3) is obtained.
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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](#org5ae58f0) 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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@@ -8,7 +8,7 @@ Tags
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: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
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Reference
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: ([Monkhorst 2004](#org5371372))
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: ([Monkhorst 2004](#org7671be3))
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Author(s)
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: Monkhorst, W.
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@@ -95,9 +95,9 @@ Find a controller \\(C\_{\mathcal{H}\_2}\\) which minimizes the \\(\mathcal{H}\_
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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.
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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](#orgbc6c1df)).
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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)).
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<a id="orgbc6c1df"></a>
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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)\\)." >}}
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@@ -108,23 +108,23 @@ The PSD \\(S\_w(f)\\) of the weighted signal is:
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Given \\(S\_w(f)\\), \\(V\_w(f)\\) can be obtained using a technique called _spectral factorization_.
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However, this can be avoided if the modelling of the disturbances is directly done in terms of weighting filters.
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Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#org5bbc30c)).
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Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#orgc49109b)).
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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\\)" >}}
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#### Output scaling and the Pareto curve {#output-scaling-and-the-pareto-curve}
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In this research, the outputs of the closed loop system (Figure [3](#orgff3035c)) are:
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In this research, the outputs of the closed loop system (Figure [3](#org8b8bb94)) are:
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- the performance (error) signal \\(e\\)
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- the controller output \\(u\\)
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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." >}}
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@@ -151,4 +151,4 @@ Drawbacks however are, that no robustness guarantees can be given and that the o
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## Bibliography {#bibliography}
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<a id="org5371372"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
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<a id="org7671be3"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
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