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content/phdthesis/jabben07_mechat.md
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title = "Mechatronic design of a magnetically suspended rotating platform"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
ref_author = "Jabben, L."
ref_year = 2007
+++
Tags
: [Dynamic Error Budgeting]({{<relref "dynamic_error_budgeting.md#" >}})
: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting.md" >}})
Reference
: ([Jabben 2007](#org6250919))
: (<a href="#citeproc_bib_item_1">Jabben 2007</a>)
Author
: Jabben, L.
@@ -49,10 +49,9 @@ This approach allows frequency dependent error budgeting, which is why it is ref
This noise can be modeled as a voltage source in series with the system impedance.
The noise source has a PSD given by:
\\[ S\_T(f) = 4 k T \text{Re}(Z(f)) \ [V^2/Hz] \\]
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.
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.
<div class="exampl">
<div></div>
A kilo Ohm resistor at 20 degree Celsius will show a thermal noise of \\(0.13 \mu V\\) from zero up to one kHz.
@@ -62,12 +61,11 @@ A kilo Ohm resistor at 20 degree Celsius will show a thermal noise of \\(0.13 \m
Seen with junctions in a transistor.
It has a white spectral density:
\\[ S\_S = 2 q\_e i\_{dc} \ [A^2/Hz] \\]
with \\(q\_e\\) the electronic charge (\\(1.6 \cdot 10^{-19}\, [C]\\)), \\(i\_{dc}\\) the average current [A].
with \\(q\_e\\) the electronic charge (\\(1.6 \cdot 10^{-19}\\, [C]\\)), \\(i\_{dc}\\) the average current [A].
<div class="exampl">
<div></div>
An averable current of 1 A will introduce noise with a STD of \\(10 \cdot 10^{-9}\,[A]\\) from zero up to one kHz.
An averable current of 1 A will introduce noise with a STD of \\(10 \cdot 10^{-9}\\,[A]\\) from zero up to one kHz.
</div>
@@ -100,24 +98,23 @@ The corresponding PSD is white up to the Nyquist frequency:
with \\(f\_N\\) the Nyquist frequency [Hz].
<div class="exampl">
<div></div>
Let's take the example of a 16 bit ADC which has an electronic noise with a SNR of 80dB.
Let's suppose the ADC is used to measure a position over a range of 1 mm.
- ADC quantization noise: it has 16 bots over the 1 mm range.
The standard diviation from the quantization is:
\\[ \sigma\_{ADq} = \frac{1 \cdot 10^6/2^16}{\sqrt{12}} = 4.4\,[nm] \\]
- ADC electronic noise: the RMS value of a sine that covers to full range is \\(\frac{0.5}{\sqrt{2}} = 0.354\,[mm]\\).
- ADC quantization noise: it has 16 bits over the 1 mm range.
The standard deviation from the quantization is:
\\[ \sigma\_{ADq} = \frac{1 \cdot 10^6/2^{16}}{\sqrt{12}} = 4.4\\,[nm] \\]
- ADC electronic noise: the RMS value of a sine that covers to full range is \\(\frac{0.5}{\sqrt{2}} = 0.354\\,[mm]\\).
With a SNR of 80dB, the electronic noise from the ADC becomes:
\\[ \sigma\_{ADn} = 35\,[nm] \\]
\\[ \sigma\_{ADn} = 35\\,[nm] \\]
Let's suppose the ADC is used to measure a sensor with an electronic noise having a standard deviation of \\(\sigma\_{sn} = 17\,[nm]\\).
Let's suppose the ADC is used to measure a sensor with an electronic noise having a standard deviation of \\(\sigma\_{sn} = 17\\,[nm]\\).
The PSD of this digitalized sensor noise is:
\\[ \sigma\_s = \sqrt{\sigma\_{sn}^2 + \sigma\_{ADq}^2 + \sigma\_{ADn}^2} = 39\,[nm]\\]
\\[ \sigma\_s = \sqrt{\sigma\_{sn}^2 + \sigma\_{ADq}^2 + \sigma\_{ADn}^2} = 39\\,[nm]\\]
from which the PSD of the total sensor noise \\(S\_s\\) is calculated:
\\[ S\_s = \frac{\sigma\_s^2}{f\_N} = 1.55\,[nm^2/Hz] \\]
\\[ S\_s = \frac{\sigma\_s^2}{f\_N} = 1.55\\,[nm^2/Hz] \\]
with \\(f\_N\\) is the Nyquist frequency of 1kHz.
</div>
@@ -132,9 +129,8 @@ To have a pressure difference, the body must have a certain minimum dimension, d
For a body of typical dimensions of 100mm, only frequencies above 800 Hz have a significant disturbance contribution.
<div class="exampl">
<div></div>
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]\\)
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]\\)
</div>
@@ -163,21 +159,21 @@ 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](#orgcc56194), the generalized plant maps the disturbances to the performance channels.
In Figure [1](#figure--fig:jabben07-general-plant), 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.
<a id="orgcc56194"></a>
<a id="figure--fig:jabben07-general-plant"></a>
{{< 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\\)" >}}
{{< figure src="/ox-hugo/jabben07_general_plant.png" caption="<span class=\"figure-number\">Figure 1: </span>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](#orgcc56194) needs to be augmented with so called **disturbance weighting filters**.
Since disturbances are generally not white, the system of Figure [1](#figure--fig:jabben07-general-plant) 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](#org772dfb7) 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](#figure--fig:jabben07-weighting-functions) 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:
@@ -186,9 +182,9 @@ 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
<a id="org772dfb7"></a>
<a id="figure--fig:jabben07-weighting-functions"></a>
{{< figure src="/ox-hugo/jabben07_weighting_functions.png" caption="Figure 2: Control system with the generalized plant \\(G\\) and weighting functions" >}}
{{< figure src="/ox-hugo/jabben07_weighting_functions.png" caption="<span class=\"figure-number\">Figure 2: </span>Control system with the generalized plant \\(G\\) and weighting functions" >}}
The weighting filters should be stable transfer functions.
@@ -209,13 +205,13 @@ By making the \\(\mathcal{H}\_2\\) norm of \\(V\_h\\) equal to the RMS-value of
IF only the output \\(y\\) are considered in the performance channel \\(z\\), the resulting optimal controller might result in very large actuator signals.
So, to obtain feasible controllers, the performance channel is a combination of controller output \\(u\\) and system output \\(y\\).
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 \\]
\\[ \\|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](#orgeab38dd) 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](#figure--fig:jabben07-pareto-curve-H2) is obtained.
<a id="orgeab38dd"></a>
<a id="figure--fig:jabben07-pareto-curve-H2"></a>
{{< 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\\)." >}}
{{< figure src="/ox-hugo/jabben07_pareto_curve_H2.png" caption="<span class=\"figure-number\">Figure 3: </span>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\\)." >}}
## Conclusion {#conclusion}
@@ -237,8 +233,3 @@ By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha
> To use the measured PSDs in an optimal control design, such as H2-control, the disturbances must be modelled using linear time invariant models with multiple white noise input.
> To derive such models, spectral factorization is used.
> It is recommended to investigate which methods for spectral factorization are currently available and numerically robust.
## Bibliography {#bibliography}
<a id="org6250919"></a>Jabben, Leon. 2007. “Mechatronic Design of a Magnetically Suspended Rotating Platform.” Delft University.