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Work on Control (HAC-LAC) + Models
2020-03-13 17:40:22 +01:00
#+TITLE: Control Requirements
:DRAWER:
#+STARTUP: overview
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas
#+HTML_LINK_HOME: ./index.html
#+HTML_LINK_UP: ./index.html
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/zenburn.css"/>
#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.stickytableheaders.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="./js/readtheorg.js"></script>
#+HTML_MATHJAX: align: center tagside: right font: TeX
#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:matlab+ :tangle no
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:shell :eval no-export
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/org/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results file raw replace
#+PROPERTY: header-args:latex+ :buffer no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports results
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:
* Introduction :ignore:
The goal here is to write clear specifications for the NASS.
This can then be used for the control synthesis and for the design of the nano-hexapod.
Ideal, specifications on the norm of closed loop transfer function should be written.
* Simplify Model for the Nano-Hexapod
** Model of the nano-hexapod
Let's consider the simple mechanical system in Figure [[fig:nass_simple_model]].
#+begin_src latex :file nass_simple_model.pdf
\begin{tikzpicture}
% Parameters
\def\massw{3}
\def\massh{1}
\def\spaceh{2}
% Ground
\draw[] (-0.5*\massw, 0) -- (0.5*\massw, 0);
% Mass
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5](m){$m$};
% Spring, Damper, and Actuator
\draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k$};
\draw[actuator] ( 0.3*\massw, 0) -- ( 0.3*\massw, \spaceh) node[midway, right=0.1](F){$F$};
% Force Sensor
\node[forcesensor={\massw}{0.2}] (fsens) at (0, \spaceh){};
\node[right] at (fsens.east) {$F_m$};
% Displacements
\draw[dashed] (0.5*\massw, 0) -- ++(0.2*\massw, 0);
\draw[->] (0.6*\massw, 0) -- ++(0, 0.2*\spaceh) node[below right]{$x_\mu$};
% Displacements
\draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(0.2*\massw, 0);
\draw[->] (0.6*\massw, \spaceh+\massh) -- ++(0, 0.2*\spaceh) node[below right]{$x$};
\draw[<->] (-0.5*\massw+-1.5*\dispw, 0) -- node[midway, left]{$d$} ++(0, \spaceh);
% Direct forces
\draw[->] (0, \spaceh+\massh) node[branch]{} -- ++(0, 0.4*\spaceh) node[below right]{$F_d$};
\end{tikzpicture}
#+end_src
#+name: fig:nass_simple_model
#+caption: Simplified mechanical system for the nano-hexapod
#+RESULTS:
[[file:figs/nass_simple_model.png]]
The signals are described in table [[tab:general_plant_signals]].
#+name: tab:general_plant_signals
#+caption: Signals definition for the generalized plant
| | *Symbol* | *Meaning* |
|---------------------+----------+----------------------------------------|
| *Exogenous Inputs* | $x_\mu$ | Motion of the $\nu$-hexapod's base |
| | $F_d$ | External Forces applied to the Payload |
| | $r$ | Reference signal for tracking |
|---------------------+----------+----------------------------------------|
| *Exogenous Outputs* | $x$ | Absolute Motion of the Payload |
|---------------------+----------+----------------------------------------|
| *Sensed Outputs* | $F_m$ | Force Sensors in each leg |
| | $d$ | Measured displacement of each leg |
| | $x$ | Absolute Motion of the Payload |
|---------------------+----------+----------------------------------------|
| *Control Signals* | $F$ | Actuator Inputs |
For the nano-hexapod alone, we have the following equations:
\[ \begin{align*}
x &= \frac{1}{ms^2 + k} F + \frac{1}{ms^2 + k} F_d + \frac{k}{ms^2 + k} x_\mu \\
F_m &= \frac{ms^2}{ms^2 + k} F - \frac{k}{ms^2 + k} F_d + \frac{k m s^2}{ms^2 + k} x_\mu \\
d &= \frac{1}{ms^2 + k} F + \frac{1}{ms^2 + k} F_d - \frac{ms^2}{ms^2 + k} x_\mu
\end{align*} \]
We can write the equations function of $\omega_\nu = \sqrt{\frac{k}{m}}$:
\[ \begin{align*}
x &= \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F + \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F_d + \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu \\
F_m &= \frac{\frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} F - \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} F_d + \frac{k \frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu \\
d &= \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F + \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F_d - \frac{\frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu
\end{align*} \]
*Assumptions*:
- the forces applied by the nano-hexapod have no influence on the micro-station, specifically on the displacement of the top platform of the micro-hexapod.
This means that the nano-hexapod can be considered separately from the micro-station and that the motion $x_\mu$ is imposed and considered as an external input.
The nano-hexapod can thus be represented as in Figure [[fig:nano_station_inputs_outputs]].
#+begin_src latex :file nano_station_inputs_outputs.pdf
\begin{tikzpicture}
\node[block={2.0cm}{2.0cm}] (P) {};
\node[above] at (P.north) {$\nu$-hexapod};
% Input and outputs coordinates
\coordinate[] (inputFd) at ($(P.south west)!0.8!(P.north west)$);
\coordinate[] (inputw) at ($(P.south west)!0.5!(P.north west)$);
\coordinate[] (inputF) at ($(P.south west)!0.2!(P.north west)$);
\coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$);
\coordinate[] (outputd) at ($(P.south east)!0.5!(P.north east)$);
\coordinate[] (outputx) at ($(P.south east)!0.2!(P.north east)$);
% Connections and labels
\draw[<-] (inputFd) -- ++(-0.8, 0) node[above right]{$F_d$};
\draw[<-] (inputw) -- ++(-0.8, 0) node[above right]{$x_\mu$};
\draw[<-] (inputF) -- ++(-0.8, 0) node[above right]{$F$};
\draw[->] (outputF) -- ++(0.8, 0) node[above left]{$F_m$};
\draw[->] (outputd) -- ++(0.8, 0) node[above left]{$d$};
\draw[->] (outputx) -- ++(0.8, 0) node[above left]{$x$};
\end{tikzpicture}
#+end_src
#+name: fig:nano_station_inputs_outputs
#+caption: Block representation of the nano-hexapod
Minor updates
2020-03-17 11:23:47 +01:00
#+RESULTS:
[[file:figs/nano_station_inputs_outputs.png]]
Work on Control (HAC-LAC) + Models
2020-03-13 17:40:22 +01:00
** How to include Ground Motion in the model?
What we measure is not the absolute motion $x$, but the relative motion $x - w$ where $w$ is the motion of the granite.
Also, $w$ induces some motion $x_\mu$ through the transmissibility of the micro-station.
#+begin_src latex :file nano_station_inputs_outputs_ground_motion.pdf
\begin{tikzpicture}
\node[block={3.0cm}{3.0cm}] (P) {};
\node[above] at (P.north) {$\nu$-hexapod};
% Input and outputs coordinates
\coordinate[] (inputFd) at ($(P.south west)!0.95!(P.north west)$);
\coordinate[] (inputw) at ($(P.south west)!0.65!(P.north west)$);
\coordinate[] (inputxm) at ($(P.south west)!0.35!(P.north west)$);
\coordinate[] (inputF) at ($(P.south west)!0.05!(P.north west)$);
\coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$);
\coordinate[] (outputd) at ($(P.south east)!0.5!(P.north east)$);
\coordinate[] (outputx) at ($(P.south east)!0.2!(P.north east)$);
% Connections and labels
\draw[<-] (inputFd) -- ++(-1.4, 0) node[above right]{$F_d$};
\draw[<-] (inputw) -- ++(-1.4, 0) node[above right]{$w$};
\draw[<-] (inputxm) -- ++(-1.4, 0) node[above right]{$x_\mu$};
\draw[<-] (inputF) -- ++(-1.4, 0) node[above right]{$F$};
\draw[->] (outputF) -- ++(1.4, 0) node[above left]{$F_m$};
\draw[->] (outputd) -- ++(1.4, 0) node[above left]{$d$};
\draw[->] (outputx) -- ++(1.4, 0) node[above]{$y = x-w$};
\end{tikzpicture}
#+end_src
** Motion of the micro-station
As explained, we consider $x_\mu$ as an external input ($F$ has no influence on $x_\mu$).
$x_\mu$ is the motion of the micro-station's top platform due to the motion of each stage of the micro-station.
We consider that $x_\mu$ has the following form:
\[ x_\mu = T_\mu r + d_\mu \]
where:
- $T_\mu r$ corresponds to the response of the stages due to the reference $r$
- $d_\mu$ is the motion of the hexapod due to all the vibrations of the stages
$T_\mu$ can be considered to be a low pass filter with a bandwidth corresponding approximatively to the bandwidth of the micro-station's stages.
To simplify, we can consider $T_\mu$ to be a first order low pass filter:
\[ T_\mu = \frac{1}{1 + s/\omega_\mu} \]
where $\omega_\mu$ corresponds to the tracking speed of the micro-station.
What is important to note is that while $x_\mu$ is viewed as a perturbation from the nano-hexapod point of view, $x_\mu$ *does* depend on the reference signal $r$.
Also, here, we suppose that the granite is not moving.
If we now include the motion of the granite $w$, we obtain the block diagram shown in Figure [[fig:nano_station_ground_motion]].
#+begin_src latex :file nano_station_ground_motion.pdf
\begin{tikzpicture}
\node[block={4.0cm}{4.0cm}] (P) {};
\node[above] at (P.north) {$\nu$-hexapod};
% Input and outputs coordinates
\coordinate[] (inputFd) at ($(P.south west)!0.8!(P.north west)$);
\coordinate[] (inputw) at ($(P.south west)!0.5!(P.north west)$);
\coordinate[] (inputF) at ($(P.south west)!0.2!(P.north west)$);
\coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$);
\coordinate[] (outputd) at ($(P.south east)!0.5!(P.north east)$);
\coordinate[] (outputx) at ($(P.south east)!0.2!(P.north east)$);
% Connections and labels
\draw[<-] (inputFd) -- ++(-0.8, 0) node[above right]{$F_d$};
\draw[<-] (inputF) -- ++(-0.8, 0) node[above right]{$F$};
\draw[->] (outputF) -- ++(0.8, 0) node[above left]{$F_m$};
\draw[->] (outputd) -- ++(0.8, 0) node[above left]{$d$};
\node[addb, left= of inputw] (add) {};
\node[block, left= of add] (Tmu) {$T_\mu$};
\node[addb, above= of add] (addw) {};
\node[block, left= of addw] (Tw) {$T_w$};
\node[addb={+}{}{-}{}{}, right= of outputx] (addx) {};
\draw[->] (outputx) -- (addx.west) node[above left]{$x$};
\draw[->] (addx.east) -- ++(0.8, 0) node[above left]{$y$};
\draw[<-] (Tmu.west) -- ++(-0.8, 0)node[above right]{$r$};
\draw[->] (Tmu.east) -- (add.west);
\draw[->] (add.east) -- (inputw) node[above left]{$x_\mu$};
\draw[->] ($(addw.north) + (0, 0.8)$) node[below right]{$d_\mu$} -- (addw.north);
\draw[->] (addw.south) -- (add.north);
\draw[->] ($(Tw.west) + (-1, 0)$) node[above right]{$w$} -- (Tw.west);
\draw[->] ($(Tw.west) + (-0.4, 0)$) node[branch]{} -- ++(0, 1.4) -| (addx.north);
\draw[->] (Tw.east) -- (addw.west);
\end{tikzpicture}
#+end_src
#+name: fig:nano_station_ground_motion
#+caption: Ground Motion $w$ included
#+RESULTS:
[[file:figs/nano_station_ground_motion.png]]
$T_w$ is the mechanical transmissibility of the micro-station.
We can approximate this transfer function by a second order low pass filter:
\[ T_w = \frac{1}{1 + 2 \xi s/\omega_0 + s^2/\omega_0^2} \]
** Notes on the signals :noexport:
Let's consider the reference tracking problem:
The reference $r$ is a sinusoidal signal at 1Hz with an amplitude up to 10mm.
We want to position error $\epsilon = x - r$ to be less than 10nm.
Thus, if we don't take into account the fact that the perturbation $x_\mu$ does depend on the reference $r$, we would conclude that we need the sensitivity function of the nano-hexapod $S$:
\[ |S| < - 60dB\quad\text{at 1 Hz} \]
And thus we would need a bandwidth of approximatively 1kHz which is not realistic.
* Control with the Stiff Nano-Hexapod
** 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
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab
freqs = logspace(-1, 3, 1000);
#+end_src
** Definition of the values
Let's define the mass and stiffness of the nano-hexapod.
#+begin_src matlab
m = 50; % [kg]
k = 1e7; % [N/m]
#+end_src
Let's define the Plant as shown in Figure [[fig:nano_station_inputs_outputs]]:
#+begin_src matlab
Gn = 1/(m*s^2 + k)*[-k, k*m*s^2, m*s^2; 1, -m*s^2, 1; 1, k, 1];
Gn.InputName = {'Fd', 'xmu', 'F'};
Gn.OutputName = {'Fm', 'd', 'x'};
#+end_src
Now, define the transmissibility transfer function $T_\mu$ corresponding to the micro-station motion.
#+begin_src matlab
wmu = 2*pi*50; % [rad/s]
Tmu = 1/(1 + s/wmu);
Tmu.InputName = {'r1'};
Tmu.OutputName = {'ymu'};
#+end_src
#+begin_src matlab
w0 = 2*pi*40;
xi = 0.5;
Tw = 1/(1 + 2*xi*s/w0 + s^2/w0^2);
Tw.InputName = {'w1'};
Tw.OutputName = {'dw'};
#+end_src
We add the fact that $x_\mu = d_\mu + T_\mu r + T_w w$:
#+begin_src matlab
Wsplit = [tf(1); tf(1)];
Wsplit.InputName = {'w'};
Wsplit.OutputName = {'w1', 'w2'};
S = sumblk('xmu = ymu + dmu + dw');
Sw = sumblk('y = x - w2');
Gpz = connect(Gn, S, Wsplit, Tw, Tmu, Sw, {'Fd', 'dmu', 'r1', 'F', 'w'}, {'Fm', 'd', 'y'});
#+end_src
** Control using $d$
*** Control Architecture
Let's consider a feedback loop using $d$ as shown in Figure [[fig:nano_station_control_d]].
#+begin_src latex :file nano_station_control_d.pdf
\begin{tikzpicture}
\node[block={4.0cm}{4.0cm}] (P) {};
\node[above] at (P.north) {$\nu$-hexapod};
% Input and outputs coordinates
\coordinate[] (inputFd) at ($(P.south west)!0.8!(P.north west)$);
\coordinate[] (inputw) at ($(P.south west)!0.5!(P.north west)$);
\coordinate[] (inputF) at ($(P.south west)!0.2!(P.north west)$);
\coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$);
\coordinate[] (outputd) at ($(P.south east)!0.5!(P.north east)$);
\coordinate[] (outputx) at ($(P.south east)!0.2!(P.north east)$);
% Connections and labels
\draw[<-] (inputFd) -- ++(-0.8, 0) node[above right]{$F_d$};
\draw[<-] (inputw) -- ++(-0.8, 0) node[above right]{$x_\mu$};
\draw[->] (outputF) -- ++(0.8, 0) node[above left]{$F_m$};
\draw[->] (outputd) -- ++(1.6, 0) node[above left]{$d$};
\draw[->] (outputx) -- ++(0.8, 0) node[above left]{$x$};
\node[block, below=0.3 of P] (K) {$K_d$};
\node[addb={+}{}{}{}{-}, left= of inputF] (addF) {};
\draw[->] ($(outputd) + (1.1, 0)$) node[branch]{} |- (K.east);
\draw[->] (K.west) -| (addF.south);
\draw[->] (addF.east) -- (inputF) node[above left]{$F$};
\draw[<-] (addF.west) -- ++(-0.8, 0)node[above right]{$F^\prime$};
\end{tikzpicture}
#+end_src
#+name: fig:nano_station_control_d
#+caption: Feedback diagram using $d$
#+RESULTS:
[[file:figs/nano_station_control_d.png]]
*** Analytical Analysis
Let's apply a direct velocity feedback by deriving $d$:
\[ F = F^\prime - g s d \]
Thus:
\[ d = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d - \frac{ms^2}{ms^2 + gs + k} x_\mu \]
\[ F = \frac{ms^2 + k}{ms^2 + gs + k} F^\prime - \frac{gs}{ms^2 + gs + k} F_d + \frac{mgs^3}{ms^2 + gs + k} x_\mu \]
and
\[ x = \frac{1}{ms^2 + k} (\frac{ms^2 + k}{ms^2 + gs + k} F^\prime - \frac{gs}{ms^2 + gs + k} F_d + \frac{mgs^3}{ms^2 + gs + k} x_\mu) + \frac{1}{ms^2 + k} F_d + \frac{k}{ms^2 + k} x_\mu \]
\[ x = \frac{ms^2 + k}{(ms^2 + k) (ms^2 + gs + k)} F^\prime + \frac{ms^2 + k}{(ms^2 + k) (ms^2 + gs + k)} F_d + \frac{mgs^3 + k(ms^2 + gs + k)}{(ms^2 + k) (ms^2 + gs + k)} x_\mu \]
And we finally obtain:
\[ x = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} x_\mu \]
#+begin_src matlab
K_dvf = 2*sqrt(k*m)*s;
K_dvf.InputName = {'d'};
K_dvf.OutputName = {'F'};
Gpz_dvf = feedback(Gpz, K_dvf, 'name');
#+end_src
Now let's consider that $x_\mu = d_\mu + T_\mu r$
\[ x = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} d_\mu + T_\mu \frac{gs + k}{ms^2 + gs + k} r \]
And $\epsilon = r - x$:
\[ \epsilon = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} d_\mu + \frac{ms^2 + gs + k - T_\mu (gs + k)}{ms^2 + gs + k} r \]
#+begin_important
\[ \epsilon = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} d_\mu + \frac{ms^2 - S_\mu(gs + k)}{ms^2 + gs + k} r \]
#+end_important
** Control using $F_m$
*** Control Architecture
Let's consider a feedback loop using $Fm$ as shown in Figure [[fig:nano_station_control_Fm]].
#+begin_src latex :file nano_station_control_Fm.pdf
\begin{tikzpicture}
\node[block={4.0cm}{4.0cm}] (P) {};
\node[above] at (P.north) {$\nu$-hexapod};
% Input and outputs coordinates
\coordinate[] (inputFd) at ($(P.south west)!0.8!(P.north west)$);
\coordinate[] (inputw) at ($(P.south west)!0.5!(P.north west)$);
\coordinate[] (inputF) at ($(P.south west)!0.2!(P.north west)$);
\coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$);
\coordinate[] (outputd) at ($(P.south east)!0.5!(P.north east)$);
\coordinate[] (outputx) at ($(P.south east)!0.2!(P.north east)$);
% Connections and labels
\draw[<-] (inputFd) -- ++(-0.8, 0) node[above right]{$F_d$};
\draw[<-] (inputw) -- ++(-0.8, 0) node[above right]{$x_\mu$};
\draw[->] (outputF) -- ++(1.6, 0) node[above left]{$F_m$};
\draw[->] (outputd) -- ++(0.8, 0) node[above left]{$d$};
\draw[->] (outputx) -- ++(0.8, 0) node[above left]{$x$};
\node[block, below=0.3 of P] (K) {$K_{F_m}$};
\node[addb={+}{}{}{}{-}, left= of inputF] (addF) {};
\draw[->] ($(outputF) + (1.1, 0)$) node[branch]{} |- (K.east);
\draw[->] (K.west) -| (addF.south);
\draw[->] (addF.east) -- (inputF) node[above left]{$F$};
\draw[<-] (addF.west) -- ++(-0.8, 0)node[above right]{$F^\prime$};
\end{tikzpicture}
#+end_src
#+name: fig:nano_station_control_Fm
#+caption: Feedback diagram using $F_m$
#+RESULTS:
[[file:figs/nano_station_control_Fm.png]]
*** Pure Integrator
Let's apply integral force feedback by integration $F_m$:
\[ F = F^\prime - \frac{g}{s} F_m \]
And we finally obtain:
\[ x = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} x_\mu \]
#+begin_src matlab
K_iff = 2*sqrt(k/m)/s;
K_iff.InputName = {'Fm'};
K_iff.OutputName = {'F'};
Gpz_iff = feedback(Gpz, K_iff, 'name');
#+end_src
Now let's consider that $x_\mu = d_\mu + T_\mu r$
\[ x = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} d_\mu + \frac{T_\mu k}{ms^2 + mgs + k} r \]
And $\epsilon = r - x$:
\[ \epsilon = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} d_\mu + \frac{ms^2 + mgs + k - T_\mu k}{ms^2 + mgs + k} r \]
#+begin_important
\[ \epsilon = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} d_\mu + \frac{ms^2 + mgs + S_\mu k}{ms^2 + mgs + k} r \]
#+end_important
*** Low pass filter
Instead of a pure integrator, let's use a low pass filter with a cut-off frequency above the bandwidth of the micro-station $\omega_mu$
#+begin_src matlab
% K_iff = (2*sqrt(k/m)/(2*wmu))*(1/(1 + s/(2*wmu)));
% K_iff.InputName = {'Fm'};
% K_iff.OutputName = {'F'};
% Gpz_iff = feedback(Gpz, K_iff, 'name');
#+end_src
** Comparison
#+begin_src matlab :exports none
figure;
subplot(2, 2, 1);
title('Effect of Ground motion ($\epsilon/w$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gpz('y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_iff('y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_dvf('y', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
subplot(2, 2, 2);
title('Compliance ($\epsilon/F_d$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gpz('y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_iff('y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_dvf('y', 'Fd'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
subplot(2, 2, 3);
title('Vibration Filtering ($\epsilon/d_\mu$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gpz('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_iff('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_dvf('y', 'dmu'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
subplot(2, 2, 4);
title('Reference Tracking ($\epsilon/r$)');
hold on;
plot(freqs, abs(squeeze(freqresp(1 - Gpz('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'OL' );
plot(freqs, abs(squeeze(freqresp(1 - Gpz_iff('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'IFF');
plot(freqs, abs(squeeze(freqresp(1 - Gpz_dvf('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'DVF');
plot(freqs, abs(squeeze(freqresp(1 - Tmu, freqs, 'Hz'))), 'k--', 'DisplayName', '$S_\mu$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
legend('location', 'northwest');
xlim([freqs(1), freqs(end)]);
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/comp_iff_dvf_simplified.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:comp_iff_dvf_simplified
#+caption: Obtained transfer functions for DVF and IFF ([[./figs/comp_iff_dvf_simplified.png][png]], [[./figs/comp_iff_dvf_simplified.pdf][pdf]])
[[file:figs/comp_iff_dvf_simplified.png]]
| | $d_\mu$ | $F_d$ | $w$ |
|-----+------------------------------------+-----------------------------------+--------------------------------------|
| IFF | Better filtering of the vibrations | More sensitive to External forces | |
| DVF | inverse | inverse | Little bit better at low frequencies |
** Control using $x$
*** Analytical analysis
Let's first consider that only the output $x$ is used for feedback (Figure [[fig:nano_station_control_x]])
#+begin_src latex :file nano_station_control_x.pdf
\begin{tikzpicture}
\node[block={4.0cm}{4.0cm}] (P) {};
\node[above] at (P.north) {$\nu$-hexapod};
% Input and outputs coordinates
\coordinate[] (inputFd) at ($(P.south west)!0.8!(P.north west)$);
\coordinate[] (inputw) at ($(P.south west)!0.5!(P.north west)$);
\coordinate[] (inputF) at ($(P.south west)!0.2!(P.north west)$);
\coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$);
\coordinate[] (outputd) at ($(P.south east)!0.5!(P.north east)$);
\coordinate[] (outputx) at ($(P.south east)!0.2!(P.north east)$);
% Connections and labels
\draw[<-] (inputFd) -- ++(-0.8, 0) node[above right]{$F_d$};
\draw[->] (outputF) -- ++(0.8, 0) node[above left]{$F_m$};
\draw[->] (outputd) -- ++(0.8, 0) node[above left]{$d$};
\draw[->] (outputx) -- ++(0.8, 0) node[above left]{$x$};
\node[block, left= of inputF] (K) {$K$};
\node[addb={+}{}{}{}{-}, left= of K] (addfb) {};
\node[addb, left= of inputw] (add) {};
\node[block, left= of add] (Tmu) {$T_\mu$};
\draw[->] ($(outputx) + (0.3, 0)$) node[branch]{} -- ++(0, -1) -| (addfb.south);
\draw[->] (addfb.east) -- (K.west) node[above left]{$\epsilon$};
\draw[->] (K.east) -- (inputF) node[above left]{$F$};
\draw[->] ($(addfb.west) + (-1.6, 0)$)node[above right]{$r$} -- (addfb.west);
\draw[->] ($(addfb.west) + (-0.8, 0)$)node[branch]{} |- (Tmu.west);
\draw[->] (Tmu.east) -- (add.west);
\draw[->] (add.east) -- (inputw) node[above left]{$x_\mu$};
\draw[->] ($(add.north) + (0, 0.8)$) node[below right]{$d_\mu$} -- (add.north);
\end{tikzpicture}
#+end_src
#+name: fig:nano_station_control_x
#+caption: Feedback diagram using $x$
#+RESULTS:
[[file:figs/nano_station_control_x.png]]
We then have:
\[ \epsilon &= r - G_{\frac{x}{F}} K \epsilon - G_{\frac{x}{F_d}} F_d - G_{\frac{x}{x_\mu}} d_\mu - G_{\frac{x}{x_\mu}} T_\mu r \]
And then:
#+begin_important
\[ \epsilon = \frac{-G_{\frac{x}{F_d}}}{1 + G_{\frac{x}{F}}K} F_d + \frac{-G_{\frac{x}{x_\mu}}}{1 + G_{\frac{x}{F}}K} d_\mu + \frac{1 - G_{\frac{x}{x_\mu}} T_\mu}{1 + G_{\frac{x}{F}}K} r \]
#+end_important
With $S = \frac{1}{1 + G_{\frac{x}{F}} K}$, we have:
\[ \epsilon = - S G_{\frac{x}{F_d}} F_d - S G_{\frac{x}{x_\mu}} d_\mu + S (1 - G_{\frac{x}{x_\mu}} T_\mu) r \]
We have 3 terms that we would like to have small by design:
- $G_{\frac{x}{F_d}} = \frac{1}{ms^2 + k}$: thus $k$ and $m$ should be high to lower the effect of direct forces $F_d$
- $G_{\frac{x}{x_\mu}} = \frac{k}{ms^2 + k} = \frac{1}{1 + \frac{s^2}{\omega_\nu^2}}$: $\omega_\nu$ should be small enough such that it filters out the vibrations of the micro-station
- $1 - G_{\frac{x}{x_\mu}} T_\mu$
\[ 1 - G_{\frac{x}{x_\mu}} T_\mu = 1 - \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} T_\mu \]
We can approximate $T_\mu \approx \frac{1}{1 + \frac{s}{\omega_\mu}}$ to have:
\begin{align*}
1 - G_{\frac{x}{x_\mu}} T_\mu &= 1 - \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} \frac{1}{1 + \frac{s}{\omega_\mu}} \\
&\approx \frac{\frac{s}{\omega_\mu}}{1 + \frac{s}{\omega_\mu}} = S_\mu \text{ if } \omega_\nu > \omega_\mu \\
&\approx \frac{\frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} = \text{ if } \omega_\nu < \omega_\mu
\end{align*}
In our case, we have $\omega_\nu > \omega_\mu$ and thus we cannot lower this term.
Some implications on the design are summarized on table [[tab:design_decommentation]].
#+name: tab:design_decommentation
#+caption: Design recommendation
| Exogenous Outputs | Design recommendation |
|-------------------+-------------------------------------------|
| $F_d$ | high $k$, high $m$ |
| $d_\mu$ | low $k$, high $m$ |
| $r$ | no influence if $\omega_\nu > \omega_\mu$ |
*** Control implementation
Controller for the damped plant using DVF.
#+begin_src matlab
wb = 2*pi*50; % control bandwidth [rad/s]
% Lead
h = 2.0;
wz = wb/h; % [rad/s]
wp = wb*h; % [rad/s]
H = 1/h*(1 + s/wz)/(1 + s/wp);
% Integrator until 10Hz
Hi = (1 + s/2/pi/10)/(s/2/pi/10);
K = Hi*H*(1/s);
Kpz_dvf = K/abs(freqresp(K*Gpz_dvf('y', 'F'), wb));
Kpz_dvf.InputName = {'e'};
Kpz_dvf.OutputName = {'Fi'};
#+end_src
Controller for the damped plant using IFF.
#+begin_src matlab
wb = 2*pi*50; % control bandwidth [rad/s]
% Lead
h = 2.0;
wz = wb/h; % [rad/s]
wp = wb*h; % [rad/s]
H = 1/h*(1 + s/wz)/(1 + s/wp);
% Integrator until 10Hz
Hi = (1 + s/2/pi/10)/(s/2/pi/10);
K = Hi*H*(1/s);
Kpz_iff = K/abs(freqresp(K*Gpz_iff('y', 'F'), wb));
Kpz_iff.InputName = {'e'};
Kpz_iff.OutputName = {'Fi'};
#+end_src
Loop gain
#+begin_src matlab :exports none
figure;
title('Transfer function from $F^\prime$ to $\epsilon$');
subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(Kpz_dvf*Gpz_dvf('y', 'F'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Kpz_iff*Gpz_iff('y', 'F'), freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Loop Gain'); xlabel('Frequency [Hz]');
subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Kpz_dvf*Gpz_dvf('y', 'F'), freqs, 'Hz'))), 'DisplayName', '$L_{DVF}$');
plot(freqs, 180/pi*angle(squeeze(freqresp(Kpz_iff*Gpz_iff('y', 'F'), freqs, 'Hz'))), '--', 'DisplayName', '$L_{IFF}$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
legend('location', 'northwest');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/simple_loop_gain_pz.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:simple_loop_gain_pz
#+caption: Loop Gain ([[./figs/simple_loop_gain_pz.png][png]], [[./figs/simple_loop_gain_pz.pdf][pdf]])
[[file:figs/simple_loop_gain_pz.png]]
Let's connect all the systems as shown in Figure [[fig:nano_station_control_x]].
#+begin_src matlab
Sfb = sumblk('e = r2 - y');
R = [tf(1); tf(1)];
R.InputName = {'r'};
R.OutputName = {'r1', 'r2'};
F = [tf(1); tf(1)];
F.InputName = {'Fi'};
F.OutputName = {'F', 'Fu'};
Gpz_fb_dvf = connect(Gpz_dvf, Kpz_dvf, R, Sfb, F, {'r', 'dmu', 'Fd', 'w'}, {'y', 'e', 'Fm', 'd', 'Fu'});
Gpz_fb_iff = connect(Gpz_iff, Kpz_iff, R, Sfb, F, {'r', 'dmu', 'Fd', 'w'}, {'y', 'e', 'Fm', 'd', 'Fu'});
#+end_src
*** Results
#+begin_src matlab :exports none
figure;
subplot(2, 2, 1);
title('Effect of Ground Motion ($\epsilon/w$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gpz('y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_iff('y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_dvf('y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_fb_iff('y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_fb_dvf('y', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
subplot(2, 2, 2);
title('Compliance ($\epsilon/F_d$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gpz('y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_iff('y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_dvf('y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_fb_iff('y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_fb_dvf('y', 'Fd'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
subplot(2, 2, 3);
title('Vibration Filtering ($\epsilon/d_\mu$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gpz('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_iff('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_dvf('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_fb_iff('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_fb_dvf('y', 'dmu'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
subplot(2, 2, 4);
title('Reference Tracking ($\epsilon/r$)');
hold on;
plot(freqs, abs(squeeze(freqresp(1 - Gpz('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'OL' );
plot(freqs, abs(squeeze(freqresp(1 - Gpz_iff('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'IFF');
plot(freqs, abs(squeeze(freqresp(1 - Gpz_dvf('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'DVF');
plot(freqs, abs(squeeze(freqresp(1 - Gpz_fb_iff('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'HAC-IFF');
plot(freqs, abs(squeeze(freqresp(1 - Gpz_fb_dvf('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'HAC-DVF');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
legend('location', 'southeast');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/simple_hac_lac_results.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:simple_hac_lac_results
#+caption: Obtained closed-loop transfer functions ([[./figs/simple_hac_lac_results.png][png]], [[./figs/simple_hac_lac_results.pdf][pdf]])
[[file:figs/simple_hac_lac_results.png]]
| | Reference Tracking | Vibration Filtering | Compliance |
|-----+--------------------+----------------------------------+----------------------------------|
| DVF | Similar behavior | | Better for $\omega < \omega_\nu$ |
| IFF | Similar behavior | Better for $\omega > \omega_\nu$ | |
** Two degree of freedom control :noexport:
Let's try to implement the control architecture shown in Figure [[fig:nano_station_control_2dof_x]].
The pre-filter $K_r$ is added in order to improve the reference tracking performances.
#+begin_src latex :file nano_station_control_2dof_x.pdf
\begin{tikzpicture}
\node[block={4.0cm}{4.0cm}] (P) {};
\node[above] at (P.north) {$\nu$-hexapod};
% Input and outputs coordinates
\coordinate[] (inputFd) at ($(P.south west)!0.8!(P.north west)$);
\coordinate[] (inputw) at ($(P.south west)!0.5!(P.north west)$);
\coordinate[] (inputF) at ($(P.south west)!0.2!(P.north west)$);
\coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$);
\coordinate[] (outputd) at ($(P.south east)!0.5!(P.north east)$);
\coordinate[] (outputx) at ($(P.south east)!0.2!(P.north east)$);
% Connections and labels
\draw[<-] (inputFd) -- ++(-0.8, 0) node[above right]{$F_d$};
\draw[->] (outputF) -- ++(0.8, 0) node[above left]{$F_m$};
\draw[->] (outputd) -- ++(0.8, 0) node[above left]{$d$};
\draw[->] (outputx) -- ++(0.8, 0) node[above left]{$x$};
\node[block, left= of inputF] (K) {$K$};
\node[addb={+}{}{}{}{-}, left= of K] (addfb) {};
\node[block, left= of addfb] (Kr) {$K_r$};
\node[addb, left= of inputw] (add) {};
\node[block, left= of add] (Tmu) {$T_\mu$};
\draw[->] ($(outputx) + (0.3, 0)$) node[branch]{} -- ++(0, -1) -| (addfb.south);
\draw[->] (addfb.east) -- (K.west) node[above left]{$\epsilon$};
\draw[->] (K.east) -- (inputF) node[above left]{$F$};
\draw[->] ($(Kr.west) + (-1.6, 0)$)node[above right]{$r$} -- (Kr.west);
\draw[->] (Kr.east) -- (addfb.west);
\draw[->] ($(Kr.west) + (-0.8, 0)$)node[branch]{} |- (Tmu.west);
\draw[->] (Tmu.east) -- (add.west);
\draw[->] (add.east) -- (inputw) node[above left]{$x_\mu$};
\draw[->] ($(add.north) + (0, 0.8)$) node[below right]{$d_\mu$} -- (add.north);
\end{tikzpicture}
#+end_src
#+name: fig:nano_station_control_2dof_x
#+caption: Two degrees of freedom feedback control
#+RESULTS:
[[file:figs/nano_station_control_2dof_x.png]]
In order to design the pre-filter $K_r$, the dynamics of the system should be known quite precisely (Dynamics of the nano-hexapod + $T_\mu$).
#+begin_src matlab
Krpz = inv(Gpz_fb('y', 'r'));
Krpz.InputName = {'r2'};
Krpz.OutputName = {'r3'};
#+end_src
#+begin_src matlab
Sfb = sumblk('e = r3 - y');
R = [tf(1); tf(1)];
R.InputName = {'r'};
R.OutputName = {'r1', 'r2'};
Gpz_2dof = connect(Gpz_dvf, Krpz, Kpz, R, Sfb, {'r', 'dmu', 'Fd', 'w'}, {'y', 'e', 'Fm', 'd'});
#+end_src
#+begin_src matlab :exports none
figure;
subplot(2, 2, 1);
title('Effect of Ground Motion ($\epsilon/w$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gpz('y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_iff('y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_dvf('y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_fb('y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_2dof('y', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
subplot(2, 2, 2);
title('Compliance ($\epsilon/F_d$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gpz('y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_iff('y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_dvf('y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_fb('y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_2dof('y', 'Fd'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
subplot(2, 2, 3);
title('Vibration Filtering ($\epsilon/d_\mu$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gpz('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_iff('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_dvf('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_fb('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gpz_2dof('y', 'dmu'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
subplot(2, 2, 4);
title('Reference Tracking ($\epsilon/r$)');
hold on;
plot(freqs, abs(squeeze(freqresp(1 - Gpz('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'OL' );
plot(freqs, abs(squeeze(freqresp(1 - Gpz_iff('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'IFF');
plot(freqs, abs(squeeze(freqresp(1 - Gpz_dvf('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'DVF');
plot(freqs, abs(squeeze(freqresp(1 - Gpz_fb('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'FB');
plot(freqs, abs(squeeze(freqresp(1 - Gpz_2dof('y', 'r'), freqs, 'Hz'))), 'DisplayName', '2-DOF');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
legend('location', 'northwest');
#+end_src
* Comparison with the use of a Soft nano-hexapod
#+begin_src matlab
m = 50; % [kg]
k = 1e3; % [N/m]
Gn = 1/(m*s^2 + k)*[-k, k*m*s^2, m*s^2; 1, -m*s^2, 1; 1, k, 1];
Gn.InputName = {'Fd', 'xmu', 'F'};
Gn.OutputName = {'Fm', 'd', 'x'};
#+end_src
#+begin_src matlab
wmu = 2*pi*50; % [rad/s]
Tmu = 1/(1 + s/wmu);
Tmu.InputName = {'r1'};
Tmu.OutputName = {'ymu'};
#+end_src
#+begin_src matlab
w0 = 2*pi*40;
xi = 0.5;
Tw = 1/(1 + 2*xi*s/w0 + s^2/w0^2);
Tw.InputName = {'w1'};
Tw.OutputName = {'dw'};
#+end_src
#+begin_src matlab
Wsplit = [tf(1); tf(1)];
Wsplit.InputName = {'w'};
Wsplit.OutputName = {'w1', 'w2'};
S = sumblk('xmu = ymu + dmu + dw');
Sw = sumblk('y = x - w2');
Gvc = connect(Gn, S, Wsplit, Tw, Tmu, Sw, {'Fd', 'dmu', 'r1', 'F', 'w'}, {'Fm', 'd', 'y'});
#+end_src
#+begin_src matlab
Kvc_dvf = 2*sqrt(k*m)*s;
Kvc_dvf.InputName = {'d'};
Kvc_dvf.OutputName = {'F'};
Gvc_dvf = feedback(Gvc, Kvc_dvf, 'name');
Kvc_iff = 2*sqrt(k/m)/s;
Kvc_iff.InputName = {'Fm'};
Kvc_iff.OutputName = {'F'};
Gvc_iff = feedback(Gvc, Kvc_iff, 'name');
#+end_src
#+begin_src matlab
wb = 2*pi*100; % control bandwidth [rad/s]
% Lead
h = 2.0;
wz = wb/h; % [rad/s]
wp = wb*h; % [rad/s]
H = 1/h*(1 + s/wz)/(1 + s/wp);
Kvc_dvf = H/abs(freqresp(H*Gvc_dvf('y', 'F'), wb));
Kvc_dvf.InputName = {'e'};
Kvc_dvf.OutputName = {'Fi'};
Kvc_iff = H/abs(freqresp(H*Gvc_iff('y', 'F'), wb));
Kvc_iff.InputName = {'e'};
Kvc_iff.OutputName = {'Fi'};
#+end_src
#+begin_src matlab
Sfb = sumblk('e = r2 - y');
R = [tf(1); tf(1)];
R.InputName = {'r'};
R.OutputName = {'r1', 'r2'};
F = [tf(1); tf(1)];
F.InputName = {'Fi'};
F.OutputName = {'F', 'Fu'};
Gvc_fb_dvf = connect(Gvc_dvf, Kvc_dvf, R, Sfb, F, {'r', 'dmu', 'Fd', 'w'}, {'y', 'e', 'Fm', 'd', 'Fu'});
Gvc_fb_iff = connect(Gvc_iff, Kvc_iff, R, Sfb, F, {'r', 'dmu', 'Fd', 'w'}, {'y', 'e', 'Fm', 'd', 'Fu'});
#+end_src
#+begin_src matlab :exports none
figure;
subplot(2, 2, 1);
title('Effect of Ground Motion ($\epsilon/w$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gvc('y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_iff('y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_dvf('y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_fb_iff('y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_fb_dvf('y', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
subplot(2, 2, 2);
title('Compliance ($\epsilon/F_d$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gvc('y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_iff('y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_dvf('y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_fb_iff('y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_fb_dvf('y', 'Fd'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
subplot(2, 2, 3);
title('Vibration Filtering ($\epsilon/d_\mu$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gvc('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_iff('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_dvf('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_fb_iff('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_fb_dvf('y', 'dmu'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
subplot(2, 2, 4);
title('Reference Tracking ($\epsilon/r$)');
hold on;
plot(freqs, abs(squeeze(freqresp(1 - Gvc('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'OL' );
plot(freqs, abs(squeeze(freqresp(1 - Gvc_iff('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'IFF');
plot(freqs, abs(squeeze(freqresp(1 - Gvc_dvf('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'DVF');
plot(freqs, abs(squeeze(freqresp(1 - Gvc_fb_iff('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'HAC-IFF');
plot(freqs, abs(squeeze(freqresp(1 - Gvc_fb_dvf('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'HAC-DVF');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
legend('location', 'southeast');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/simple_hac_lac_results_soft.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:simple_hac_lac_results_soft
#+caption: Obtained closed-loop transfer functions ([[./figs/simple_hac_lac_results_soft.png][png]], [[./figs/simple_hac_lac_results_soft.pdf][pdf]])
[[file:figs/simple_hac_lac_results_soft.png]]
| | Reference Tracking | Vibration Filtering | Compliance |
|-----+--------------------+----------------------------------+----------------------------------|
| DVF | Similar behavior | | Better for $\omega < \omega_\nu$ |
| IFF | Similar behavior | Better for $\omega > \omega_\nu$ | |
#+begin_src matlab :exports none
figure;
subplot(2, 2, 1);
title('Effect of Ground Motion ($\epsilon/w$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gpz_fb_dvf('y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_fb_dvf('y', 'w'), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Gpz_fb_iff('y', 'w'), freqs, 'Hz'))), '--');
plot(freqs, abs(squeeze(freqresp(Gvc_fb_iff('y', 'w'), freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
subplot(2, 2, 2);
title('Compliance ($\epsilon/F_d$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gpz_fb_dvf('y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_fb_dvf('y', 'Fd'), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Gpz_fb_iff('y', 'Fd'), freqs, 'Hz'))), '--');
plot(freqs, abs(squeeze(freqresp(Gvc_fb_iff('y', 'Fd'), freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
subplot(2, 2, 3);
title('Vibration Filtering ($\epsilon/d_\mu$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gpz_fb_dvf('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_fb_dvf('y', 'dmu'), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Gpz_fb_iff('y', 'dmu'), freqs, 'Hz'))), '--');
plot(freqs, abs(squeeze(freqresp(Gvc_fb_iff('y', 'dmu'), freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
subplot(2, 2, 4);
title('Reference Tracking ($\epsilon/r$)');
hold on;
plot(freqs, abs(squeeze(freqresp(1 - Gpz_fb_dvf('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'HAC-DVF PZ');
plot(freqs, abs(squeeze(freqresp(1 - Gvc_fb_dvf('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'HAC-DVF VC');
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(1 - Gpz_fb_iff('y', 'r'), freqs, 'Hz'))), '--', 'DisplayName', 'HAC-IFF PZ');
plot(freqs, abs(squeeze(freqresp(1 - Gvc_fb_iff('y', 'r'), freqs, 'Hz'))), '--', 'DisplayName', 'HAC-IFF VC');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
legend('location', 'southeast');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/simple_comp_vc_pz.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:simple_comp_vc_pz
#+caption: Comparison of the closed-loop transfer functions for Soft and Stiff nano-hexapod ([[./figs/simple_comp_vc_pz.png][png]], [[./figs/simple_comp_vc_pz.pdf][pdf]])
[[file:figs/simple_comp_vc_pz.png]]
| | <c> | <c> |
| | *Soft* | *Stiff* |
|-----------------------+--------+---------|
| *Reference Tracking* | = | = |
| *Ground Motion* | = | = |
| *Vibration Isolation* | + | - |
| *Compliance* | - | + |
* Estimate the level of vibration
#+begin_src matlab
gm = load('./mat/psd_gm.mat', 'f', 'psd_gm');
rz = load('./mat/pxsp_r.mat', 'f', 'pxsp_r');
tyz = load('./mat/pxz_ty_r.mat', 'f', 'pxz_ty_r');
#+end_src
#+begin_src matlab :exports none
f = gm.f(2:end);
psd_gm = gm.psd_gm(2:end); % PSD of ground motion [m^2/Hz]
psd_rz = rz.pxsp_r(2:end)./(2*pi*f.^2); % PSD of hexapod's motion due to Rz [m^2/Hz]
psd_ty = tyz.pxz_ty_r(2:end)./(2*pi*f.^2); % PSD of hexapod's motion due to Ty [m^2/Hz]
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(f, sqrt(psd_rz), 'DisplayName', 'Rz');
plot(f, sqrt(psd_ty), 'DisplayName', 'Ty');
plot(f, sqrt(psd_gm), 'DisplayName', 'Gm');
hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD of the displacement $\left[\frac{m}{\sqrt{Hz}}\right]$')
legend('Location', 'southwest');
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(f, flip(sqrt(-cumtrapz(flip(f), flip(psd_rz)))), 'DisplayName', 'Rz');
plot(f, flip(sqrt(-cumtrapz(flip(f), flip(psd_ty)))), 'DisplayName', 'Ty');
plot(f, flip(sqrt(-cumtrapz(flip(f), flip(psd_ty + psd_rz)))), 'k-', 'DisplayName', 'tot');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('CAS of $x_\mu$ [m]')
xlim([freqs(1), freqs(end)]);
legend('Location', 'southwest');
#+end_src
If we note the PSD $\Gamma$:
\[ \Gamma_y = |G_{\frac{y}{w}}|^2 \Gamma_w + |G_{\frac{y}{x_\mu}}|^2 \Gamma_{x_\mu} \]
#+begin_src matlab
x_pz = abs(squeeze(freqresp(Gpz_fb_iff('y', 'dmu'), f, 'Hz'))).^2.*(psd_rz + psd_ty) + abs(squeeze(freqresp(Gpz_fb_iff('y', 'w'), f, 'Hz'))).^2.*(psd_gm);
x_vc = abs(squeeze(freqresp(Gvc_fb_iff('y', 'dmu'), f, 'Hz'))).^2.*(psd_rz + psd_ty) + abs(squeeze(freqresp(Gvc_fb_iff('y', 'w'), f, 'Hz'))).^2.*(psd_gm);
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(f, sqrt(x_pz));
plot(f, sqrt(x_vc));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD of the displacement $\left[\frac{m}{\sqrt{Hz}}\right]$')
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(f, flip(sqrt(-cumtrapz(flip(f), flip(x_pz)))), 'DisplayName', 'PZ');
plot(f, flip(sqrt(-cumtrapz(flip(f), flip(x_vc)))), 'DisplayName', 'VC');
plot(f, flip(sqrt(-cumtrapz(flip(f), flip(psd_rz + psd_ty + psd_gm.*abs(squeeze(freqresp(Tw, f, 'Hz')).^2))))), 'DisplayName', 'OL');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('CAS of the relative displacement $[m]$')
xlim([freqs(1), freqs(end)]);
legend();
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/simple_asd_motion_error.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:simple_asd_motion_error
#+caption: ASD of the position error due to Ground Motion and Vibration ([[./figs/simple_asd_motion_error.png][png]], [[./figs/simple_asd_motion_error.pdf][pdf]])
[[file:figs/simple_asd_motion_error.png]]
Actuator usage
#+begin_src matlab
F_pz = abs(squeeze(freqresp(Gpz_fb_iff('Fu', 'dmu'), f, 'Hz'))).^2.*(psd_rz + psd_ty) + abs(squeeze(freqresp(Gpz_fb_iff('Fu', 'w'), f, 'Hz'))).^2.*(psd_gm);
F_vc = abs(squeeze(freqresp(Gvc_fb_iff('Fu', 'dmu'), f, 'Hz'))).^2.*(psd_rz + psd_ty) + abs(squeeze(freqresp(Gvc_fb_iff('Fu', 'w'), f, 'Hz'))).^2.*(psd_gm);
#+end_src
#+begin_src matlab :results output replace
sqrt(trapz(f, F_pz))
sqrt(trapz(f, F_vc))
#+end_src
#+RESULTS:
: sqrt(trapz(f, F_pz))
: ans =
: 84.8961762069446
: sqrt(trapz(f, F_vc))
: ans =
: 0.0387785981815527
* Requirements on the norm of closed-loop transfer functions
** Approximation of the ASD of perturbations
#+begin_src matlab
G_rz = 1e-9*1/(1 + s/2/pi/0.5)^2*(s + 2*pi*1)*(s + 2*pi*10)*(1/((1 + s/2/pi/100)^2));
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(f, sqrt(psd_rz), 'DisplayName', 'Rz');
plot(freqs, abs(squeeze(freqresp(G_rz, freqs, 'Hz'))));
hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD of the displacement $\left[\frac{m}{\sqrt{Hz}}\right]$')
legend('Location', 'southwest');
#+end_src
#+begin_src matlab
G_gm = 1e-8*1/s^2*(s + 2*pi*1)^2*(1/((1 + s/2/pi/10)^3));
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(f, sqrt(psd_gm), 'DisplayName', 'Rz');
plot(freqs, abs(squeeze(freqresp(G_gm, freqs, 'Hz'))));
hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD of the displacement $\left[\frac{m}{\sqrt{Hz}}\right]$')
legend('Location', 'southwest');
#+end_src
** Wanted ASD of outputs
Wanted ASD of motion error
#+begin_src matlab
y_wanted = 100e-9; % 10nm rms wanted
y_bw = 2*pi*100; % bandwidth [rad/s]
G_y = 2*y_wanted/sqrt(y_bw) * (1 + s/y_bw/10) / (1 + s/y_bw);
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(G_y, freqs, 'Hz'))));
hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD of the displacement $\left[\frac{m}{\sqrt{Hz}}\right]$')
legend('Location', 'southwest');
#+end_src
#+begin_src matlab :results output replace
sqrt(trapz(f, abs(squeeze(freqresp(G_y, f, 'Hz'))).^2))
#+end_src
#+RESULTS:
: sqrt(trapz(f, abs(squeeze(freqresp(G_y, f, 'Hz'))).^2))
: ans =
: 9.47118350214793e-08
** Limiting the bandwidth
#+begin_src matlab
wF = 2*pi*10;
G_F = 100000*(wF + s)^2;
#+end_src
** Generalized Weighted plant
Let's create a generalized weighted plant for controller synthesis.
Let's start simple:
| | *Symbol* | *Meaning* |
|---------------------+----------+------------------------------------|
| *Exogenous Inputs* | $x_\mu$ | Motion of the $\nu$-hexapod's base |
|---------------------+----------+------------------------------------|
| *Exogenous Outputs* | $y$ | Motion error of the Payload |
|---------------------+----------+------------------------------------|
| *Sensed Outputs* | $y$ | Motion error of the Payload |
|---------------------+----------+------------------------------------|
| *Control Signals* | $F$ | Actuator Inputs |
Add $F$ as output.
#+begin_src matlab
F = [tf(1); tf(1)];
F.InputName = {'Fi'};
F.OutputName = {'F', 'Fu'};
P_pz = connect(F, Gpz_dvf, {'dmu', 'Fi'}, {'y', 'Fu', 'y'})
P_vc = connect(F, Gvc_dvf, {'dmu', 'Fi'}, {'y', 'Fu', 'y'})
#+end_src
Normalization.
We multiply the plant input by $G_{rz}$ and the plant output by $G_y^{-1}$:
#+begin_src matlab
P_pz_norm = blkdiag(inv(G_y), inv(G_F), 1)*P_pz*blkdiag(G_rz, 1);
P_pz_norm.OutputName = {'z', 'F', 'y'};
P_pz_norm.InputName = {'w', 'u'};
P_vc_norm = blkdiag(inv(G_y), inv(G_F), 1)*P_vc*blkdiag(G_rz, 1);
P_vc_norm.OutputName = {'z', 'F', 'y'};
P_vc_norm.InputName = {'w', 'u'};
#+end_src
** Synthesis
#+begin_src matlab
[Kpz_dvf,CL_vc,~] = hinfsyn(minreal(P_pz_norm), 1, 1, 'TOLGAM', 0.001, 'METHOD', 'LMI', 'DISPLAY', 'on');
Kpz_dvf.InputName = {'e'};
Kpz_dvf.OutputName = {'Fi'};
[Kvc_dvf,CL_pz,~] = hinfsyn(minreal(P_vc_norm), 1, 1, 'TOLGAM', 0.001, 'METHOD', 'LMI', 'DISPLAY', 'on');
Kvc_dvf.InputName = {'e'};
Kvc_dvf.OutputName = {'Fi'};
#+end_src
** Loop Gain
#+begin_src matlab :exports none
figure;
title('Transfer function from $F^\prime$ to $\epsilon$');
subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(Kpz_dvf*Gpz_dvf('y', 'F'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Kvc_dvf*Gvc_dvf('y', 'F'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Loop Gain'); xlabel('Frequency [Hz]');
subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(-Kpz_dvf*Gpz_dvf('y', 'F'), freqs, 'Hz'))), 'DisplayName', '$PZ$');
plot(freqs, 180/pi*angle(squeeze(freqresp(-Kvc_dvf*Gvc_dvf('y', 'F'), freqs, 'Hz'))), 'DisplayName', '$VC$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
legend('location', 'northwest');
#+end_src
#+begin_src matlab
Sfb = sumblk('e = r2 - y');
R = [tf(1); tf(1)];
R.InputName = {'r'};
R.OutputName = {'r1', 'r2'};
F = [tf(1); tf(1)];
F.InputName = {'Fi'};
F.OutputName = {'F', 'Fu'};
Gpz_fb_dvf = connect(Gpz_dvf, -Kpz_dvf, R, Sfb, F, {'r', 'dmu', 'Fd', 'w'}, {'y', 'e', 'Fm', 'd', 'Fu'});
Gvc_fb_dvf = connect(Gvc_dvf, -Kvc_dvf, R, Sfb, F, {'r', 'dmu', 'Fd', 'w'}, {'y', 'e', 'Fm', 'd', 'Fu'});
#+end_src
** Results
#+begin_src matlab :exports none
freqs = logspace(-6, 8, 1000);
figure;
subplot(2, 2, 1);
title('Effect of Ground Motion ($\epsilon/w$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gpz_fb_dvf('y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_fb_dvf('y', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
subplot(2, 2, 2);
title('Compliance ($\epsilon/F_d$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gpz_fb_dvf('y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_fb_dvf('y', 'Fd'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
subplot(2, 2, 3);
title('Vibration Filtering ($\epsilon/d_\mu$)');
hold on;
plot(freqs, abs(squeeze(freqresp(Gpz_fb_dvf('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gvc_fb_dvf('y', 'dmu'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_y*inv(G_rz), freqs, 'Hz'))), 'k--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
subplot(2, 2, 4);
title('Reference Tracking ($\epsilon/r$)');
hold on;
plot(freqs, abs(squeeze(freqresp(1 - Gpz_fb_dvf('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'HAC-DVF PZ');
plot(freqs, abs(squeeze(freqresp(1 - Gvc_fb_dvf('y', 'r'), freqs, 'Hz'))), 'DisplayName', 'HAC-DVF VC');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
xlim([freqs(1), freqs(end)]);
legend('location', 'southeast');
#+end_src
** Requirements
| reference tracking | $\epsilon/r$ | -120dB at 1Hz |
| vibration isolation | $x/x_\mu$ | -60dB above 10Hz |
| compliance | $x/F_d$ | |
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