tdehaeze/dehaeze20_optim_robus_compl_filte
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Remove unused plant inverse

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Thomas Dehaeze
2020-09-22 10:17:50 +02:00
parent 86ad85f17c
commit cdb103b3f5
1 changed files with 8 additions and 11 deletions
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@@ -109,7 +109,7 @@ $\tilde{n}_1$ and $\tilde{n}_2$ are white noise with unitary power spectral dens
\begin{equation}
\begin{split}
\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} N_1 \right) \tilde{n}_1 + \left( H_2 \hat{G}_2^{-1} N_2 \right) \tilde{n}_2
&+ \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
\end{split}
\end{equation}
@@ -123,21 +123,18 @@ Complementary Filters
H_1(s) + H_2(s) = 1
\end{equation}
\begin{equation}
\hat{x} = x + \left( H_1 \hat{G}_1^{-1} N_1 \right) \tilde{n}_1 + \left( H_2 \hat{G}_2^{-1} N_2 \right) \tilde{n}_2
\hat{x} = x + \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2
\end{equation}
Perfect dynamics + filter noise
** Super Sensor Noise
Let's note $n$ the super sensor noise.
Its PSD is determined by:
\begin{equation}
\Phi_n(\omega) = \left| H_1 \hat{G}_1^{-1} N_1 \right|^2 + \left| H_2 \hat{G}_2^{-1} N_2 \right|^2
\Phi_n(\omega) = \left| H_1 N_1 \right|^2 + \left| H_2 N_2 \right|^2
\end{equation}
** $\mathcal{H}_2$ Synthesis of Complementary Filters
@@ -147,18 +144,18 @@ The goal is to design $H_1(s)$ and $H_2(s)$ such that the effect of the noise so
And the goal is the minimize the Root Mean Square (RMS) value of $n$:
#+name: eq:rms_value_estimation
\begin{equation}
\sigma_{n} = \sqrt{\int_0^\infty \Phi_{\hat{n}}(\omega) d\omega} = \left\| \begin{matrix} \hat{G}_1^{-1} N_1 H_1 \\ \hat{G}_2^{-1} N_2 H_2 \end{matrix} \right\|_2
\sigma_{n} = \sqrt{\int_0^\infty \Phi_{\hat{n}}(\omega) d\omega} = \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2
\end{equation}
Thus, the goal is to design $H_1(s)$ and $H_2(s)$ such that $H_1(s) + H_2(s) = 1$ and such that $\left\| \begin{matrix} \hat{G}_1^{-1} N_1 H_1 \\ \hat{G}_2^{-1} N_2 H_2 \end{matrix} \right\|_2$ is minimized.
Thus, the goal is to design $H_1(s)$ and $H_2(s)$ such that $H_1(s) + H_2(s) = 1$ and such that $\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2$ is minimized.
\begin{equation}
\begin{pmatrix}
z_1 \\ z_2 \\ v
\end{pmatrix} = \begin{bmatrix}
\hat{G}_1^{-1} N_1 & -\hat{G}_1^{-1} N_1 \\
0 & \hat{G}_2^{-1} N_2 \\
1 & 0
N_1 & N_1 \\
0 & N_2 \\
1 & 0
\end{bmatrix} \begin{pmatrix}
w \\ u
\end{pmatrix}