Update Content - 2020-10-15
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@@ -49,7 +49,7 @@ The noise source has a PSD given by:
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\\[ S\_T(f) = 4 k T \text{Re}(Z(f)) \ [V^2/Hz] \\]
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with \\(k = 1.38 \cdot 10^{-23} \,[J/K]\\) the Boltzmann's constant, \\(T\\) the temperature [K] and \\(Z(f)\\) the frequency dependent impedance of the system.
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<div class="examp">
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<div class="bgreen">
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<div></div>
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A kilo Ohm resistor at 20 degree Celsius will show a thermal noise of \\(0.13 \mu V\\) from zero up to one kHz.
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@@ -62,7 +62,7 @@ It has a white spectral density:
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\\[ S\_S = 2 q\_e i\_{dc} \ [A^2/Hz] \\]
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with \\(q\_e\\) the electronic charge (\\(1.6 \cdot 10^{-19}\, [C]\\)), \\(i\_{dc}\\) the average current [A].
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<div></div>
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An averable current of 1 A will introduce noise with a STD of \\(10 \cdot 10^{-9}\,[A]\\) from zero up to one kHz.
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@@ -97,7 +97,7 @@ The corresponding PSD is white up to the Nyquist frequency:
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\\[ S\_Q = \frac{q^2}{12 f\_N} \\]
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with \\(f\_N\\) the Nyquist frequency [Hz].
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<div></div>
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Let's take the example of a 16 bit ADC which has an electronic noise with a SNR of 80dB.
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@@ -129,7 +129,7 @@ The disturbance force acting on a body, is the **difference of pressure between
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To have a pressure difference, the body must have a certain minimum dimension, depending on the wave length of the sound.
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For a body of typical dimensions of 100mm, only frequencies above 800 Hz have a significant disturbance contribution.
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<div></div>
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Consider a cube with a rib size of 100 mm located in a room with a sound level of 80dB, distributed between one and ten kHz, then the force disturbance PSD equal \\(2.2 \cdot 10^{-2}\,[N^2/Hz]\\)
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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](#orga43f7f1), the generalized plant maps the disturbances to the performance channels.
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In Figure [1](#org322128e), 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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<a id="orga43f7f1"></a>
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<a id="org322128e"></a>
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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](#orga43f7f1) 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](#org322128e) 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](#org906705e) 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](#orgd4f3b10) 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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<a id="org906705e"></a>
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<a id="orgd4f3b10"></a>
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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](#org58a8c87) 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](#orgae97f26) is obtained.
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<a id="org58a8c87"></a>
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<a id="orgae97f26"></a>
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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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@@ -4,7 +4,7 @@ author = ["Thomas Dehaeze"]
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draft = false
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+++
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### Backlinks {#backlinks}
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Backlinks:
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- [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
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@@ -12,7 +12,7 @@ Tags
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: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
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Reference
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: ([Monkhorst 2004](#org7d7f3f5))
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: ([Monkhorst 2004](#orgecd1ca1))
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Author(s)
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: Monkhorst, W.
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@@ -99,9 +99,9 @@ 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](#org7321040)).
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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](#orgf9cf8f6)).
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<a id="org7321040"></a>
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<a id="orgf9cf8f6"></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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@@ -112,23 +112,23 @@ 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](#orgea16f0b)).
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Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#org226c943)).
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<a id="orgea16f0b"></a>
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<a id="org226c943"></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\\)" >}}
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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](#org95f6ca6)) are:
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In this research, the outputs of the closed loop system (Figure [3](#org3e048a8)) 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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<a id="org95f6ca6"></a>
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<a id="org3e048a8"></a>
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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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@@ -154,4 +154,4 @@ 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="org7d7f3f5"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
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<a id="orgecd1ca1"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
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