Change G1 sign

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Thomas Dehaeze 2020-10-01 13:28:49 +02:00
parent e2a7378339
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5 changed files with 80 additions and 10 deletions

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@ -24,7 +24,7 @@
#+PROPERTY: header-args:matlab+ :output-dir figs
:END:
* Introduction :ignore:
* Introduction :ignore:
In this document, the optimal and robust design of complementary filters is studied.
Two sensors are considered with both different noise characteristics and dynamical uncertainties represented by multiplicative input uncertainty.
@ -118,7 +118,7 @@ Its nominal dynamics $\hat{G}_1(s)$ is defined below.
k_acc = 1e5; % Stiffness [N/m]
g_acc = 1e5; % Gain [V/m]
G1 = -g_acc*m_acc*s/(m_acc*s^2 + c_acc*s + k_acc); % Accelerometer Plant [V/(m/s)]
G1 = g_acc*m_acc*s/(m_acc*s^2 + c_acc*s + k_acc); % Accelerometer Plant [V/(m/s)]
#+end_src
The second sensor is a displacement sensor, its nominal dynamics $\hat{G}_2(s)$ is defined below.
@ -305,7 +305,7 @@ All the dynamical systems representing the sensors are saved for further use.
#+end_src
* First Order Complementary Filters :noexport:
** Matlab Init :noexport:ignore:
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
@ -392,7 +392,77 @@ All the dynamical systems representing the sensors are saved for further use.
#+end_src
* Optimal Super Sensor Noise: $\mathcal{H}_2$ Synthesis with Acc and Pos
* Introduction to Sensor Fusion
<<sec:introduction_sensor_fusion>>
** Sensor Fusion Architecture
<<sec:sensor_fusion_architecture>>
The two sensors presented in Section [[sec:sensor_description]] are now merged together using complementary filters $H_1(s)$ and $H_2(s)$ to form a super sensor (Figure [[fig:sensor_fusion_noise_arch]]).
#+name: fig:sensor_fusion_noise_arch
#+caption: Sensor Fusion Architecture
[[file:figs-tikz/sensor_fusion_noise_arch.png]]
The complementary property of $H_1(s)$ and $H_2(s)$ means that the sum of their transfer function is equal to $1$ eqref:eq:complementary_property.
\begin{equation}
H_1(s) + H_2(s) = 1 \label{eq:complementary_property}
\end{equation}
The super sensor estimate $\hat{x}$ is given by eqref:eq:super_sensor_estimate.
\begin{equation}
\hat{x} = \left( H_1 \hat{G}_1^{-1} G_1 + H_2 \hat{G}_2^{-1} G_2 \right) x + \left( H_1 \hat{G}_1^{-1} G_1 N_1 \right) \tilde{n}_1 + \left( H_2 \hat{G}_2^{-1} G_2 N_2 \right) \tilde{n}_2 \label{eq:super_sensor_estimate}
\end{equation}
** Super Sensor Noise
<<sec:super_sensor_noise>>
If we first suppose that the models of the sensors $\hat{G}_i$ are very close to the true sensor dynamics $G_i$ eqref:eq:good_dynamical_model, we have that the super sensor estimate $\hat{x}$ is equals to the measured quantity $x$ plus the noise of the two sensors filtered out by the complementary filters eqref:eq:estimate_perfect_models.
\begin{equation}
\hat{G}_i^{-1}(s) G_i(s) \approx 1 \label{eq:good_dynamical_model}
\end{equation}
\begin{equation}
\hat{x} = x + \underbrace{\left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2}_{n} \label{eq:estimate_perfect_models}
\end{equation}
As the noise of both sensors are considered to be uncorrelated, the PSD of the super sensor noise is computed as follow:
\begin{equation}
\Phi_n(\omega) = \left| H_1 N_1 \right|^2 + \left| H_2 N_2 \right|^2 \label{eq:super_sensor_psd_noise}
\end{equation}
And the Root Mean Square (RMS) value of the super sensor noise $\sigma_n$ is given by eqref:eq:super_sensor_rms_noise.
\begin{equation}
\sigma_n = \sqrt{\int_0^\infty \Phi_n(\omega) d\omega} \label{eq:super_sensor_rms_noise}
\end{equation}
** Super Sensor Dynamical Uncertainty
<<sec:super_sensor_dynamical_uncertainty>>
If we consider some dynamical uncertainty (the true system dynamics $G_i$ not being perfectly equal to our model $\hat{G}_i$) that we model by the use of multiplicative uncertainty (Figure [[fig:sensor_model_uncertainty]]), the super sensor dynamics is then equals to:
\begin{equation}
\begin{aligned}
\frac{\hat{x}}{x} &= \Big( H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + W_1 \Delta_1) + H_2 \hat{G}_2^{-1} \hat{G}_2 (1 + W_2 \Delta_2) \Big) \\
&= \Big( H_1 (1 + W_1 \Delta_1) + H_2 (1 + W_2 \Delta_2) \Big) \\
&= \left( 1 + H_1 W_1 \Delta_1 + H_2 W_2 \Delta_2 \right), \quad \|\Delta_i\|_\infty<1
\end{aligned}
\end{equation}
#+name: fig:sensor_model_uncertainty
#+caption: Sensor Model including Dynamical Uncertainty
[[file:figs-tikz/sensor_model_uncertainty.png]]
The uncertainty set of the transfer function from $\hat{x}$ to $x$ at frequency $\omega$ is bounded in the complex plane by a circle centered on 1 and with a radius equal to $|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|$ as shown in Figure [[fig:uncertainty_set_super_sensor]].
#+name: fig:uncertainty_set_super_sensor
#+caption: Super Sensor model uncertainty displayed in the complex plane
[[file:figs-tikz/uncertainty_set_super_sensor.png]]
* Optimal Super Sensor Noise: $\mathcal{H}_2$ Synthesis
:PROPERTIES:
:header-args:matlab+: :tangle matlab/optimal_comp_filters.m
:header-args:matlab+: :comments org :mkdirp yes
@ -400,10 +470,10 @@ All the dynamical systems representing the sensors are saved for further use.
<<sec:optimal_comp_filters>>
** Introduction :ignore:
In this section, the two sensors are merged together using complementary
The idea is to combine sensors that works in different frequency range using complementary filters.
Doing so, one "super sensor" is obtained that can have better noise characteristics than the individual sensors over a large frequency range.
The complementary filters have to be designed in order to minimize the effect noise of each sensor on the super sensor noise.
#+name: fig:sensor_fusion_noise_arch
@ -455,7 +525,7 @@ If we define $H_2 = 1 - H_1$, we obtain:
Thus, if we minimize the $\mathcal{H}_2$ norm of this transfer function, we minimize the RMS value of $\hat{x}$.
We define the generalized plant $P$ on matlab as shown on figure [[fig:h_infinity_optimal_comp_filters]].
We define the generalized plant $P$ on matlab as shown on Figure
#+begin_src matlab
P = [N1 -N1;
0 N2;
@ -477,7 +547,7 @@ Finally, we define $H_2(s) = 1 - H_1(s)$.
save('./mat/H2_filters.mat', 'H2', 'H1');
#+end_src
The complementary filters obtained are shown on figure [[fig:htwo_comp_filters]].
The complementary filters obtained are shown on Figure
#+begin_src matlab :exports none
figure;
hold on;
@ -670,7 +740,7 @@ From the above complementary filter design with the $\mathcal{H}_2$ and $\mathca
However, the synthesis does not take into account the robustness of the sensor fusion.
* Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis with Acc and Pos
* Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis
:PROPERTIES:
:header-args:matlab+: :tangle matlab/comp_filter_robustness.m
:header-args:matlab+: :comments org :mkdirp yes
@ -1042,7 +1112,7 @@ Using the $\mathcal{H}_\infty$ synthesis, the dynamical uncertainty of the super
However, the RMS of the super sensor noise is not optimized as it was the case with the $\mathcal{H}_2$ synthesis
* Optimal and Robust Sensor Fusion: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis with Acc and Pos
* Optimal and Robust Sensor Fusion: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
:PROPERTIES:
:header-args:matlab+: :tangle matlab/mixed_synthesis_sensor_fusion.m
:header-args:matlab+: :comments org :mkdirp yes