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Thomas Dehaeze
2020-09-18 11:43:07 +02:00
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content/phdthesis/jabben07_mechat.md
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@@ -49,9 +49,12 @@ 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.
```text
A kilo Ohm resistor at 20 degree Celsius will show a thermal noise of $0.13 \mu V$ from zero up to one kHz.
```
<div class="examp">
<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.
</div>
**Shot Noise**.
Seen with junctions in a transistor.
@@ -59,9 +62,12 @@ 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].
```text
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 class="examp">
<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.
</div>
**Excess Noise** (or \\(1/f\\) noise).
It results from fluctuating conductivity due to imperfect contact between two materials.
@@ -91,24 +97,28 @@ The corresponding PSD is white up to the Nyquist frequency:
\\[ S\_Q = \frac{q^2}{12 f\_N} \\]
with \\(f\_N\\) the Nyquist frequency [Hz].
```text
<div class="examp">
<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 $\frac{0.5}{\sqrt{2}} = 0.354\,[mm]$.
With a SNR of 80dB, the electronic noise from the ADC becomes:
\[ \sigma_{ADn} = 35\,[nm] \]
Let's suppose the ADC is used to measure a sensor with an electronic noise having a standard deviation of $\sigma_{sn} = 17\,[nm]$.
- 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]\\).
With a SNR of 80dB, the electronic noise from the ADC becomes:
\\[ \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]\\).
The PSD of this digitalized sensor noise is:
\[ \sigma_s = \sqrt{\sigma_{sn}^2 + \sigma_{ADq}^2 + \sigma_{ADn}^2} = 39\,[nm]\]
from which the PSD of the total sensor noise $S_s$ is calculated:
\[ S_s = \frac{\sigma_s^2}{f_N} = 1.55\,[nm^2/Hz] \]
with $f_N$ is the Nyquist frequency of 1kHz.
```
\\[ \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] \\]
with \\(f\_N\\) is the Nyquist frequency of 1kHz.
</div>
#### Acoustic Noise {#acoustic-noise}
@@ -119,9 +129,12 @@ The disturbance force acting on a body, is the **difference of pressure between
To have a pressure difference, the body must have a certain minimum dimension, depending on the wave length of the sound.
For a body of typical dimensions of 100mm, only frequencies above 800 Hz have a significant disturbance contribution.
```text
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 class="examp">
<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]\\)
</div>
#### Brownian Noise {#brownian-noise}
@@ -148,21 +161,21 @@ 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](#org30a4301), the generalized plant maps the disturbances to the performance channels.
In Figure [1](#orga43f7f1), 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="org30a4301"></a>
<a id="orga43f7f1"></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\\)" >}}
#### Using Weighting Filters for Disturbance Modelling {#using-weighting-filters-for-disturbance-modelling}
Since disturbances are generally not white, the system of Figure [1](#org30a4301) needs to be augmented with so called **disturbance weighting filters**.
Since disturbances are generally not white, the system of Figure [1](#orga43f7f1) 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](#org3b94947) 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](#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\\) .
The generalized plant framework also allows to include **weighting filters for the performance channels**.
This is useful for three reasons:
@@ -171,7 +184,7 @@ 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="org3b94947"></a>
<a id="org906705e"></a>
{{< figure src="/ox-hugo/jabben07_weighting_functions.png" caption="Figure 2: Control system with the generalized plant \\(G\\) and weighting functions" >}}
@@ -196,9 +209,9 @@ So, to obtain feasible controllers, the performance channel is a combination of
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 \\]
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](#orgb0b1e78) 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](#org58a8c87) is obtained.
<a id="orgb0b1e78"></a>
<a id="org58a8c87"></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\\)." >}}