tdehaeze/digital-brain
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Update Content - 2020-12-11

Browse Source
This commit is contained in:
Thomas Dehaeze
2020-12-11 16:00:37 +01:00
parent 86eb3b2a13
commit 8f368b7515
11 changed files with 325 additions and 211 deletions
Download Patch File Download Diff File Expand all files Collapse all files

22
content/phdthesis/jabben07_mechat.md
View File

@@ -49,7 +49,7 @@ 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.
<div class="bgreen">
<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.
@@ -62,7 +62,7 @@ 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].
<div class="bgreen">
<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.
@@ -97,7 +97,7 @@ 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].
<div class="bgreen">
<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.
@@ -129,7 +129,7 @@ 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.
<div class="bgreen">
<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]\\)
@@ -161,21 +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](#org322128e), the generalized plant maps the disturbances to the performance channels.
In Figure [1](#org7b34df5), 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="org322128e"></a>
<a id="org7b34df5"></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](#org322128e) needs to be augmented with so called **disturbance weighting filters**.
Since disturbances are generally not white, the system of Figure [1](#org7b34df5) 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](#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\\) .
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\\) .
The generalized plant framework also allows to include **weighting filters for the performance channels**.
This is useful for three reasons:
@@ -184,7 +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="orgd4f3b10"></a>
<a id="org5013433"></a>
{{< figure src="/ox-hugo/jabben07_weighting_functions.png" caption="Figure 2: Control system with the generalized plant \\(G\\) and weighting functions" >}}
@@ -209,9 +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](#orgae97f26) 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](#org47370f3) is obtained.
<a id="orgae97f26"></a>
<a id="org47370f3"></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\\)." >}}