#+TITLE: Effect of Uncertainty on the payload's dynamics on the isolation platform dynamics
:DRAWER:
#+STARTUP: overview

#+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:
In this document we will consider an *isolation platform* (e.g. the nano-hexapod) with a *payload* on top (e.g. the the sample to be positioned).

The goal is to study:
- how does the dynamics of the payload influence the dynamics of the isolation platform
- similarly: how does the uncertainty on the payload's dynamics will be transferred to uncertainty on the plant
- what design choice should be made in order to minimize the resulting uncertainty on the plant

Two models are made to study these effects:
- In section [[sec:introductory_example]], simple mass-spring-damper systems are chosen to model both the isolation platform and the payload
- In section [[sec:arbitrary_dynamics]], we consider arbitrary payload dynamics with multiplicative input uncertainty to study the unmodelled dynamics of the payload

* Simple Introductory Example
<<sec:introductory_example>>

** Introduction                                                     :ignore:
Let's consider the system shown in Figure [[fig:2dof_system_stiffness_uncertainty_payload]] consisting of:
- An *isolation platform* represented by a mass $m$, a stiffness $k$ and a dashpot $c$ and an actuator $F$
- A *payload* represented by a mass $m^\prime$, a stiffness $k^\prime$ and a dashpot $c^\prime$

The goal is to stabilize $x$ using $F$ in spite of uncertainty on the payload mechanical properties.

#+begin_src latex :file 2dof_system_stiffness_uncertainty_payload.pdf
  \begin{tikzpicture}
    % ====================
    % Parameters
    % ====================
    \def\massw{2.2}  % Width of the masses
    \def\massh{0.8}  % Height of the masses
    \def\spaceh{1.2} % Height of the springs/dampers
    \def\dispw{0.3}  % Width of the dashed line for the displacement
    \def\disph{0.5}  % Height of the arrow for the displacements
    \def\bracs{0.05} % Brace spacing vertically
    \def\brach{-10pt} % Brace shift horizontaly
    % ====================


    % ====================
    % Ground
    % ====================
    \draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
    \draw[dashed] (0.5*\massw, 0) -- ++(\dispw, 0) coordinate(dlow);
    % \draw[->] (0.5*\massw+0.5*\dispw, 0) -- ++(0, \disph) node[right]{$x_{w}$};

    % ====================
    % Micro Station
    % ====================
    \begin{scope}[shift={(0, 0)}]
      % Mass
      \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m$};

      % Spring, Damper, and Actuator
      \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k$};
      \draw[damper] (0, 0)           -- ( 0, \spaceh)          node[midway, left=0.2]{$c$};
      \draw[actuator] ( 0.4*\massw, 0) -- (0.4*\massw, \spaceh) node[midway, left=0.1](F){$F$};

      % Displacements
      \draw[dashed] (0.5*\massw, \spaceh) -- ++(\dispw, 0);
      \draw[->] (0.5*\massw+0.5*\dispw, \spaceh) -- ++(0, \disph) node[right]{$x$};

      % Legend
      \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
        (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
        node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Isolation\\Platform};
    \end{scope}

    % ====================
    % Nano Station
    % ====================
    \begin{scope}[shift={(0, \spaceh+\massh)}]
      % Mass
      \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m^\prime$};

      % Spring, Damper, and Actuator
      \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k^\prime$};
      \draw[damper] (0, 0)           -- ( 0, \spaceh)          node[midway, left=0.2]{$c^\prime$};

      % Displacements
      \draw[dashed] (0.5*\massw, \spaceh) -- ++(\dispw, 0) coordinate(dhigh);
      \draw[->] (0.5*\massw+0.5*\dispw, \spaceh) -- ++(0, \disph) node[right]{$x^\prime$};

      % Legend
      \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
        (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
        node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Payload};
    \end{scope}
  \end{tikzpicture}
#+end_src

#+name: fig:2dof_system_stiffness_uncertainty_payload
#+caption: Two degrees-of-freedom system
#+RESULTS:
[[file:figs/2dof_system_stiffness_uncertainty_payload.png]]

** 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

** Equations of motion
If we write the equation of motion of the system in Figure [[fig:2dof_system_stiffness_uncertainty_payload]], we obtain:
\begin{align}
  ms^2 x &= F - (cs + k) x + (c^\prime s + k^\prime) (x^\prime - x) \\
  m^\prime s^2 x^\prime &= - (c^\prime s + k^\prime) (x^\prime - x)
\end{align}

After eliminating $x^\prime$, we obtain:
#+name: eq:plant_simple_system
\begin{equation}
  \frac{x}{F} = \frac{m^\prime s^2 + c^\prime s + k^\prime}{(ms^2 + cs + k)(m^\prime s^2 + c^\prime s + k^\prime) + m^\prime s^2(c^\prime s + k^\prime)}
\end{equation}

** Initialization of the payload dynamics
Let the payload have:
- a nominal mass of $m^\prime = 50\ [kg]$
- a nominal stiffness of $k^\prime = 5 \cdot 10^6\ [N/m]$
- a nominal damping of $c^\prime = 3 \cdot 10^3\ [N/(m/s)]$

#+begin_src matlab
  mpi = 50;
  kpi = 5e6;
  cpi = 3e3;

  kpi = (2*pi*50)^2*mpi;
  cpi = 0.2*sqrt(kpi*mpi);
#+end_src

Let's also consider some uncertainty in those parameters:
#+begin_src matlab
  mp = ureal('m', mpi, 'Range', [1, 100]);
  cp = ureal('c', cpi, 'Percentage', 30);
  kp = ureal('k', kpi, 'Percentage', 30);
#+end_src

The compliance of the payload without the isolation platform is $\frac{1}{m^\prime s^2 + c^\prime s + k^\prime}$ and its bode plot is shown in Figure [[fig:nominal_payload_compliance_dynamics]].

One can see that the payload has a resonance frequency of $\omega_0^\prime = 250\ Hz$.

#+begin_src matlab :exports none
  Gps = usample(1/(mp*s^2 + cp*s + kp), 50);

  freqs = logspace(1, 3, 1000);

  figure;

  hold on;
  for i = 1:length(Gps)
    plot(freqs, abs(squeeze(freqresp(Gps(:,:,i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
  end
  plot(freqs, abs(squeeze(freqresp(1/(mpi*s^2 + cpi*s + kpi), freqs, 'Hz'))), 'k-');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]');
  ylabel('Magnitude [dB]');

  xlim([freqs(1), freqs(end)]);
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/nominal_payload_compliance_dynamics.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:nominal_payload_compliance_dynamics
#+caption: Nominal compliance of the payload ([[./figs/nominal_payload_compliance_dynamics.png][png]], [[./figs/nominal_payload_compliance_dynamics.pdf][pdf]])
[[file:figs/nominal_payload_compliance_dynamics.png]]

** Initialization of the isolation platform
Let's first fix the mass of the isolation platform:
#+begin_src matlab
  m = 10;
#+end_src

And we generate three isolation platforms:
- A soft one with $\omega_0 = 0.1 \omega_0^\prime = 5\ Hz$
- A medium stiff one with $\omega_0 = \omega_0^\prime = 50\ Hz$
- A stiff one with $\omega_0 = 10 \omega_0^\prime = 500\ Hz$

#+begin_src matlab :exports none
  k_soft = m*(2*pi*5)^2;
  c_soft = 0.1*sqrt(m*k_soft);
  G_soft = (mp*s^2 + cp*s + kp)/(mp*s^2*(cp*s + kp) + (m*s^2 + c_soft*s + k_soft)*(mp*s^2 + cp*s + kp));

  k_mid = m*(2*pi*50)^2;
  c_mid = 0.1*sqrt(m*k_mid);
  G_mid = (mp*s^2 + cp*s + kp)/(mp*s^2*(cp*s + kp) + (m*s^2 + c_mid*s + k_mid)*(mp*s^2 + cp*s + kp));

  k_stiff = m*(2*pi*500)^2;
  c_stiff = 0.1*sqrt(m*k_stiff);
  G_stiff = (mp*s^2 + cp*s + kp)/(mp*s^2*(cp*s + kp) + (m*s^2 + c_stiff*s + k_stiff)*(mp*s^2 + cp*s + kp));
#+end_src

** Comparison
The obtained dynamics from $F$ to $x$ for the three isolation platform are shown in Figure [[fig:plant_dynamics_uncertainty_payload_variability]].

#+begin_src matlab :exports none
  freqs  = logspace(0, 3, 1000);

  Gs_soft  = usample(G_soft,  10);
  Gs_mid   = usample(G_mid,   10);
  Gs_stiff = usample(G_stiff, 10);

  figure;

  ax1 = subplot(2,3,1);
  hold on;
  for i = 1:length(Gs_soft)
    plot(freqs, abs(squeeze(freqresp(Gs_soft(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
  end
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('Magnitude [m/N]');
  title('$\omega_0 \ll \omega_0^\prime$');
  hold off;

  ax4 = subplot(2,3,4);
  hold on;
  for i = 1:length(Gs_soft)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_soft(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
  end
  set(gca,'xscale','log');
  yticks(-360:90:180);
  ylim([-270 90]);
  ylabel('Phase [deg]');
  hold off;

  ax2 = subplot(2,3,2);
  hold on;
  for i = 1:length(Gs_mid)
    plot(freqs, abs(squeeze(freqresp(Gs_mid(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
  end
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  title('$\omega_0 \approx \omega_0^\prime$');
  hold off;

  ax5 = subplot(2,3,5);
  hold on;
  for i = 1:length(Gs_mid)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_mid(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
  end
  set(gca,'xscale','log');
  yticks(-360:90:180);
  ylim([-270 90]);
  xlabel('Frequency [Hz]');
  hold off;

  ax3 = subplot(2,3,3);
  hold on;
  for i = 1:length(Gs_stiff)
    plot(freqs, abs(squeeze(freqresp(Gs_stiff(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
  end
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  title('$\omega_0 \gg \omega_0^\prime$');
  hold off;

  ax6 = subplot(2,3,6);
  hold on;
  for i = 1:length(Gs_stiff)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_stiff(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
  end
  set(gca,'xscale','log');
  yticks(-360:90:180);
  ylim([-270 90]);
  hold off;

  linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
  xlim([freqs(1), freqs(end)]);

  linkaxes([ax1,ax2,ax3],'y');
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/plant_dynamics_uncertainty_payload_variability.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:plant_dynamics_uncertainty_payload_variability
#+caption: Obtained plant for the three isolation platforms considered ([[./figs/plant_dynamics_uncertainty_payload_variability.png][png]], [[./figs/plant_dynamics_uncertainty_payload_variability.pdf][pdf]])
[[file:figs/plant_dynamics_uncertainty_payload_variability.png]]

** Conclusion
#+begin_important
  The stiff platform dynamics does not seems to depend on the dynamics of the payload.
#+end_important

* Generalization to arbitrary dynamics
<<sec:arbitrary_dynamics>>
** Introduction
Let's now consider a general payload described by its *impedance* $G^\prime(s) = \frac{F^\prime}{x}$ as shown in Figure [[fig:general_payload_impedance]].

#+begin_note
Note here that we use the term /impedance/, however, the mechanical impedance is usually defined as the ratio of the force over the velocity $F^\prime/\dot{x}$. We should refer to /resistance/ instead of /impedance/.
#+end_note

#+begin_src latex :file general_payload_impedance.pdf
  \begin{tikzpicture}
    \def\massw{2.2}  % Width of the masses
    \def\massh{0.8}  % Height of the masses
    \def\spaceh{1.2} % Height of the springs/dampers
    \def\dispw{0.3}  % Width of the dashed line for the displacement
    \def\disph{0.5}  % Height of the arrow for the displacements
    \def\bracs{0.05} % Brace spacing vertically
    \def\brach{-10pt} % Brace shift horizontaly

    \node[piezo={2.2}{3}{10}] (piezo) at (0, 0){};
    \draw[] ($(piezo.north)+(-1.2, 0)$) -- ++(2.4, 0);
    \draw[] ($(piezo.south)+(-1.2, 0)$) -- ++(2.4, 0);
    \draw[dashed] (piezo.south east) -- ++(\dispw, 0) coordinate(dhigh);
    \draw[->] ($(piezo.south east)+(0.5*\dispw, 0)$) -- ++(0, \disph) node[right]{$x$};
    \draw[->] (piezo.south) node[branch]{} -- ++(0, -1) node[above right]{$F^\prime$};

    \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
      ($(piezo.south west) + (-10pt, 0)$) -- ($(piezo.north west) + (-10pt, 0)$) %
      node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{General Payload};
  \end{tikzpicture}
#+end_src

#+name: fig:general_payload_impedance
#+caption: General support
#+RESULTS:
[[file:figs/general_payload_impedance.png]]

Now let's consider the system consisting of a mass-spring-system (the isolation platform) supporting the general payload as shown in Figure [[fig:general_payload_with_isolator]].
#+begin_src latex :file general_payload_with_isolator.pdf
  \begin{tikzpicture}
    \def\massw{2.2}  % Width of the masses
    \def\massh{0.8}  % Height of the masses
    \def\spaceh{1.2} % Height of the springs/dampers
    \def\dispw{0.3}  % Width of the dashed line for the displacement
    \def\disph{0.5}  % Height of the arrow for the displacements
    \def\bracs{0.05} % Brace spacing vertically
    \def\brach{-10pt} % Brace shift horizontaly

    % Mass
    \draw[fill=white] (-0.5*\massw, \spaceh) rectangle node[left=6pt]{$m$} (0.5*\massw, \spaceh+\massh);

    % Spring, Damper, and Actuator
    \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k$};
    \draw[damper] (0, 0)           -- ( 0, \spaceh)          node[midway, left=0.2]{$c$};
    \draw[actuator] ( 0.4*\massw, 0) -- (0.4*\massw, \spaceh) node[midway, left=0.1](F){$F$};

    % Ground
    \draw (-0.5*\massw, 0) -- (0.5*\massw, 0);

    % Displacements
    \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(\dispw, 0) coordinate(dhigh);
    \draw[->] (0.5*\massw+0.5*\dispw, \spaceh+\massh) -- ++(0, \disph) node[right]{$x$};

    % Legend
    \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
      (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
      node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Isolation\\Platform};

    \begin{scope}[shift={(0, \spaceh+\massh)}]
      \node[piezo={2.2}{1.5}{6}, anchor=south] (piezo) at (0, 0){};
      \draw[->] (0,0)node[branch]{} -- ++(0, -0.6)node[above right]{$F^\prime$}

      \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
        ($(piezo.south west) + (-10pt, 0)$) -- ($(piezo.north west) + (-10pt, 0)$) %
        node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Payload};
    \end{scope}
  \end{tikzpicture}
#+end_src

#+name: fig:general_payload_with_isolator
#+caption: Mass-Spring-Damper (isolation platform) supporting a general payload
#+RESULTS:
[[file:figs/general_payload_with_isolator.png]]

** Equations of motion
We have to following equations of motion:
\begin{align}
  ms^2 x &= F - (cs + k) x - F^\prime \\
  F^\prime &= G^\prime(s) x
\end{align}

And by eliminating $F^\prime$, we find the plant dynamics $G(s) = \frac{x}{F}$.

#+begin_important
#+name: eq:plant_dynamics_general_payload
\begin{equation}
  \frac{x}{F} = \frac{1}{ms^2 + cs + k + G^\prime(s)}
\end{equation}
#+end_important

We can learn few things about the obtained transfer function:
- the zeros of $x/F$ will be the poles of $G^\prime(s)$.
- if the impedance of the payload is small $|G^\prime(s)| \ll |ms^2 + cs + k|$, then the payload will have small influence on the obtained dynamics

** 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
** Impedance $G^\prime(s)$ of a mass-spring-damper payload
In order to verify that the formula is correct, let's take the same mass-spring-damper system used in the system shown in Figure [[fig:2dof_system_stiffness_uncertainty_payload]]:
\begin{align*}
  m^\prime s^2 x^\prime &= (x - x^\prime) (c^\prime s + k^\prime) \\
  F^\prime &= (x - x^\prime) (c^\prime s + k^\prime)
\end{align*}

By eliminating $x^\prime$ of the equations, we obtain:
#+begin_important
\begin{equation}
  G^\prime(s) = \frac{F^\prime}{x} = \frac{m^\prime s^2 (c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime} \label{eq:impedance_mass_spring_damper}
\end{equation}

The impedance of a 1dof mass-spring-damper system is described by Eq. eqref:eq:impedance_mass_spring_damper.
#+end_important

And we obtain
\begin{align*}
  \frac{x}{F} &= \frac{1}{ms^2 + cs + k + G^\prime(s)} \\
              &= \frac{1}{ms^2 + cs + k + \frac{m^\prime s^2 (c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime}} \\
              &= \frac{m^\prime s^2 + c^\prime s + k^\prime}{(ms^2 + cs + k) (m^\prime s^2 + c^\prime s + k^\prime) + m^\prime s^2 (c^\prime s + k)}
\end{align*}

Which is the same transfer function that was obtained in section [[sec:introductory_example]] (Eq. [[eq:plant_simple_system]]).

The impedance of the mass-spring-damper system is shown in Figure [[fig:example_impedance_mass_spring_damper]].
- Before the resonance frequency $\omega_0^\prime$, the impedance follows the mass line
- After the resonance, the impedance will follow the stiffness line (depending on the relative values of the stiffness and damping)
- At high frequency, it will follow the damping line

#+begin_src matlab :exports none
  mp = 50;
  kp = 1e7;
  cp = 5e3;

  freqs = logspace(1, 3, 1000);

  figure;
  hold on;
  plot(freqs, abs(squeeze(freqresp(mp*s^2*(cp*s+kp)/(mp*s^2+cp*s+kp), freqs, 'Hz'))), 'k-', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(mp*s^2, freqs, 'Hz'))), '--', 'DisplayName', 'Mass Line');
  plot(freqs, abs(squeeze(freqresp(cp*s, freqs, 'Hz'))), '--', 'DisplayName', 'Damping Line');
  plot(freqs, abs(squeeze(freqresp(tf(kp), freqs, 'Hz'))), '--', 'DisplayName', 'Stiffness Line');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Impedance [N/m]');
  hold off;
  legend('location', 'southeast');
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/example_impedance_mass_spring_damper.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:example_impedance_mass_spring_damper
#+caption: Example of the impedance of a mass-spring-damper system ([[./figs/example_impedance_mass_spring_damper.png][png]], [[./figs/example_impedance_mass_spring_damper.pdf][pdf]])
[[file:figs/example_impedance_mass_spring_damper.png]]

** First Analytical analysis
To summarize, we consider:
- an Isolation platform represented by a mass $m$, a damper $c$ and a stiffness $k$.
  This system resonate at $\omega_0 = \sqrt{\frac{k}{m}}$
- A payload represented by a mass $m^\prime$, a damper $c^\prime$ and a stiffness $k^\prime$.
  The payload resonate at $\omega_0^\prime = \sqrt{\frac{k^\prime}{m^\prime}}$

The "impedance" of the payload is represented by:
\[ G^\prime(s) = \frac{m^\prime s^2 (c^\prime s + k^\prime)}{m^\prime s^2 + c^\prime s + k^\prime} \]

And the plant is:
\[ G(s) = \frac{x}{F} = \frac{1}{ms^2 + cs + k + G^\prime(s)} \]

Let's write the asymptotic behavior of $|G^\prime(j\omega)|$:
- $\lim_{\omega \to 0} |G^\prime(j\omega)| = m^\prime s^2$
- $|G^\prime(j\omega_0)| = \frac{k^\prime \sqrt{1 + (2\xi^\prime)^2}}{2 \xi^\prime}$
- $\lim_{\omega \to \infty} |G^\prime(j\omega)| = c^\prime s + k$

Let's find some conditions in order to have that the dynamics of the payload does not influence to much the dynamics of the plant:
\[ |G^\prime(s)| \ll |ms^2 + cs + k| \]

Let's take the case of a *stiff payload* ($\omega_0^\prime \gg \omega_0$).

Below $\omega_0$, the condition becomes:
\[ |G^\prime(s)| \ll k \Leftrightarrow m^\prime \omega_0^2 \ll k \Leftrightarrow m^\prime \ll m \]
The *payload mass should be small with respect to the isolation platform mass*.

Above $\omega_0$:
\[ |G^\prime(j\omega)| \ll m \omega^2 \]

Until $\omega_0^\prime$, we have $m^\prime \ll m$ which is the same condition as before.
Above $\omega_0^\prime$, we obtain $|jc^\prime \omega + k| \ll m \omega^2$.

#+begin_important
When using a soft isolation platform and a stiff payload such that the payload resonate above the first resonance of the isolation platform, the mass of the payload should be small compared to the isolation platform mass in order to not disturb the dynamics of the isolation platform.
#+end_important

** Impedance of the Payload and Dynamical Uncertainty
We model the payload by a mass-spring-damper model with some uncertainty.

Let the payload have:
- a nominal mass of $m^\prime = 50\ [kg]$
- a nominal stiffness of $k^\prime = 5 \cdot 10^6\ [N/m]$
- a nominal damping of $c^\prime = 3 \cdot 10^3\ [N/(m/s)]$

The main resonance of the payload is then $\omega^\prime = \sqrt{\frac{m^\prime}{k^\prime}} \approx 50\ Hz$.

#+begin_src matlab
  m0 = 10;
  c0 = 3e2;
  k0 = 5e5;

  Gp0 = (m0*s^2 * (c0*s + k0))/(m0*s^2 + c0*s + k0);
#+end_src

Let's represent the uncertainty on the impedance of the payload by a multiplicative uncertainty (Figure [[fig:input_uncertainty_set]]):
\[ G^\prime(s) = G_0^\prime(s)(1 + w_I^\prime(s)\Delta_I(s)) \quad |\Delta_I(j\omega)| < 1\ \forall \omega \]

This could represent *unmodelled dynamics* or unknown parameters of the payload.

#+name: fig:input_uncertainty_set
#+caption: Input Multiplicative Uncertainty
#+RESULTS:
[[file:figs/input_uncertainty_set.png]]

We choose a simple uncertainty weight:
\[ w_I(s) = \frac{\tau s + r_0}{(\tau/r_\infty) s + 1} \]
where $r_0$ is the relative uncertainty at steady-state, $1/\tau$ is the frequency at which the relative uncertainty reaches $100\ \%$, and $r_\infty$ is the magnitude of the weight at high frequency.

The parameters are defined below.
#+begin_src matlab
  r0 = 0.5;
  tau = 1/(50*2*pi);
  rinf = 10;

  wI = (tau*s + r0)/((tau/rinf)*s + 1);
#+end_src

We then generate a complex $\Delta$.
#+begin_src matlab
  DeltaI = ucomplex('A',0);
#+end_src

We generate the uncertain plant $G^\prime(s)$.
#+begin_src matlab
  Gp = Gp0*(1+wI*DeltaI);
#+end_src

A set of uncertainty payload's impedance transfer functions is shown in Figure [[fig:payload_impedance_uncertainty]].

#+begin_src matlab :exports none
  Gps = usample(Gp, 50);

  freqs  = logspace(-1, 4, 1000);

  figure;
  hold on;
  plot(freqs, abs(squeeze(freqresp(Gp0, freqs, 'Hz'))), 'k--');
  for i = 1:length(Gps)
    plot(freqs, abs(squeeze(freqresp(Gps(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
  end
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]');
  ylabel('Magnitude [dB]');
  hold off;
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/payload_impedance_uncertainty.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:payload_impedance_uncertainty
#+caption: Uncertainty of the payload's impedance ([[./figs/payload_impedance_uncertainty.png][png]], [[./figs/payload_impedance_uncertainty.pdf][pdf]])
[[file:figs/payload_impedance_uncertainty.png]]

** Equivalent Inverse Multiplicative Uncertainty
Let's express the uncertainty of the plant $x/F$ as a function of the parameters as well as of the uncertainty on the platform's compliance:
\begin{align*}
  \frac{x}{F} &= \frac{1}{ms^2 + cs + k + G_0^\prime(s)(1 + w_I(s)\Delta(s))}\\
              &= \frac{1}{ms^2 + cs + k + G_0^\prime(s)} \cdot \frac{1}{1 + \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} \Delta(s)}\\
\end{align*}

#+begin_important
We can the plant dynamics that as an inverse multiplicative uncertainty (Figure [[fig:inverse_uncertainty_set]]):
\begin{equation}
  \frac{x}{F} = G_0(s) (1 + w_{iI}(s) \Delta(s))^{-1}
\end{equation}
with:
- $G_0(s) = \frac{1}{ms^2 + cs + k + G_0^\prime(s)}$
- $w_{iI}(s) = \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} = G_0(s) G_0^\prime(s) w_I(s)$
#+end_important

#+name: fig:inverse_uncertainty_set
#+caption: Inverse Multiplicative Uncertainty
#+RESULTS:
[[file:figs/inverse_uncertainty_set.png]]

** Effect of the Isolation platform Stiffness
Let's first fix the mass of the isolation platform:
#+begin_src matlab
  m = 20;
#+end_src

And we generate three isolation platforms:
- A soft one with $\omega_0 = 5\ Hz$
- A medium stiff one with $\omega_0 = 50\ Hz$
- A stiff one with $\omega_0 = 500\ Hz$

Soft Isolation Platform:
#+begin_src matlab
  k_soft = m*(2*pi*5)^2;
  c_soft = 0.1*sqrt(m*k_soft);

  G_soft = 1/(m*s^2 + c_soft*s + k_soft + Gp);
  G0_soft = 1/(m*s^2 + c_soft*s + k_soft + Gp0);
  wiI_soft = Gp0*G0_soft*wI;
#+end_src

Mid Isolation Platform
#+begin_src matlab
  k_mid = m*(2*pi*50)^2;
  c_mid = 0.1*sqrt(m*k_mid);

  G_mid = 1/(m*s^2 + c_mid*s + k_mid + Gp);
  G0_mid = 1/(m*s^2 + c_mid*s + k_mid + Gp0);
  wiI_mid = Gp0*G0_mid*wI;
#+end_src

Stiff Isolation Platform
#+begin_src matlab
  k_stiff = m*(2*pi*500)^2;
  c_stiff = 0.1*sqrt(m*k_stiff);

  G_stiff = 1/(m*s^2 + c_stiff*s + k_stiff + Gp);
  G0_stiff = 1/(m*s^2 + c_stiff*s + k_stiff + Gp0);
  wiI_stiff = Gp0*G0_stiff*wI;
#+end_src

The obtained transfer functions $x/F$ for each of the three platforms are shown in Figure [[fig:plant_uncertainty_impedance_payload]].

#+begin_src matlab :exports none
  freqs  = logspace(0, 3, 1000);

  Gs_soft  = usample(G_soft, 10);
  Gs_mid   = usample(G_mid, 10);
  Gs_stiff = usample(G_stiff, 10);

  figure;

  ax1 = subplot(2,3,1);
  hold on;
  for i = 1:length(Gs_soft)
    plot(freqs, abs(squeeze(freqresp(Gs_soft(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
  end
  plot(freqs, abs(squeeze(freqresp(G0_soft, freqs, 'Hz'))), 'r-');
  plot(freqs, abs(squeeze(freqresp(G0_soft, freqs, 'Hz')))./(1 + abs(squeeze(freqresp(wiI_soft, freqs, 'Hz')))), 'r--');
  plot(freqs, abs(squeeze(freqresp(G0_soft, freqs, 'Hz')))./max(0, (1 - abs(squeeze(freqresp(wiI_soft, freqs, 'Hz'))))), 'r--');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('Magnitude [m/N]');
  hold off;
  title('$\omega_0 \ll \omega_0^\prime$');

  ax4 = subplot(2,3,4);
  hold on;
  for i = 1:length(Gs_soft)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_soft(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
  end
  plot(freqs, 180/pi*angle(squeeze(freqresp(G0_soft, freqs, 'Hz'))), 'r-');
  Dphi = 180/pi*asin(abs(squeeze(freqresp(wiI_soft, freqs, 'Hz'))));
  Dphi(find(abs(squeeze(freqresp(wiI_soft, freqs, 'Hz'))) > 1, 1):end) = 360;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G0_soft, freqs, 'Hz')))+Dphi, 'r--');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G0_soft, freqs, 'Hz')))-Dphi, 'r--');
  set(gca,'xscale','log');
  yticks(-360:90:180);
  ylim([-180 180]);
  ylabel('Phase [deg]');
  hold off;

  ax2 = subplot(2,3,2);
  hold on;
  for i = 1:length(Gs_mid)
    plot(freqs, abs(squeeze(freqresp(Gs_mid(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
  end
  plot(freqs, abs(squeeze(freqresp(G0_mid, freqs, 'Hz'))), 'r-');
  plot(freqs, abs(squeeze(freqresp(G0_mid, freqs, 'Hz')))./(1 + abs(squeeze(freqresp(wiI_mid, freqs, 'Hz')))), 'r--');
  plot(freqs, abs(squeeze(freqresp(G0_mid, freqs, 'Hz')))./max(0, (1 - abs(squeeze(freqresp(wiI_mid, freqs, 'Hz'))))), 'r--');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  hold off;
  title('$\omega_0 \approx \omega_0^\prime$');

  ax5 = subplot(2,3,5);
  hold on;
  for i = 1:length(Gs_mid)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_mid(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
  end
  plot(freqs, 180/pi*angle(squeeze(freqresp(G0_mid, freqs, 'Hz'))), 'r-');
  Dphi = 180/pi*asin(abs(squeeze(freqresp(wiI_mid, freqs, 'Hz'))));
  Dphi(find(abs(squeeze(freqresp(wiI_mid, freqs, 'Hz'))) > 1, 1):end) = 360;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G0_mid, freqs, 'Hz')))+Dphi, 'r--');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G0_mid, freqs, 'Hz')))-Dphi, 'r--');
  set(gca,'xscale','log');
  yticks(-360:90:180);
  ylim([-180 180]);
  xlabel('Frequency [Hz]');
  hold off;

  ax3 = subplot(2,3,3);
  hold on;
  for i = 1:length(Gs_stiff)
    plot(freqs, abs(squeeze(freqresp(Gs_stiff(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
  end
  plot(freqs, abs(squeeze(freqresp(G0_stiff, freqs, 'Hz'))), 'r-');
  plot(freqs, abs(squeeze(freqresp(G0_stiff, freqs, 'Hz')))./(1 + abs(squeeze(freqresp(wiI_stiff, freqs, 'Hz')))), 'r--');
  plot(freqs, abs(squeeze(freqresp(G0_stiff, freqs, 'Hz')))./max(0, (1 - abs(squeeze(freqresp(wiI_stiff, freqs, 'Hz'))))), 'r--');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  hold off;
  title('$\omega_0 \gg \omega_0^\prime$');

  ax6 = subplot(2,3,6);
  hold on;
  for i = 1:length(Gs_stiff)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_stiff(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
  end
  plot(freqs, 180/pi*angle(squeeze(freqresp(G0_stiff, freqs, 'Hz'))), 'r-');
  Dphi = 180/pi*asin(abs(squeeze(freqresp(wiI_stiff, freqs, 'Hz'))));
  Dphi(find(abs(squeeze(freqresp(wiI_stiff, freqs, 'Hz'))) > 1, 1):end) = 360;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G0_stiff, freqs, 'Hz')))+Dphi, 'r--');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G0_stiff, freqs, 'Hz')))-Dphi, 'r--');
  set(gca,'xscale','log');
  yticks(-360:90:180);
  ylim([-180 180]);
  hold off;

  linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
  xlim([freqs(1), freqs(end)]);

  linkaxes([ax1,ax2,ax3],'y');
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/plant_uncertainty_impedance_payload.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:plant_uncertainty_impedance_payload
#+caption: Obtained plant for the three isolators ([[./figs/plant_uncertainty_impedance_payload.png][png]], [[./figs/plant_uncertainty_impedance_payload.pdf][pdf]])
[[file:figs/plant_uncertainty_impedance_payload.png]]

** Reduce the Uncertainty on the plant
*** Introduction                                                    :ignore:
Now that we know the expression of the uncertainty on the plant, we can wonder what parameters of the isolation platform would lower the plant uncertainty, or at least bring the uncertainty to reasonable level.

The uncertainty of the plant is described by an inverse multiplicative uncertainty with the following weight:
\[ w_{iI}(s) = \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} \]

Let's study separately the effect of the platform's mass, damping and stiffness.

*** Effect of the platform's stiffness $k$
Let's fix $\xi = \frac{c}{2\sqrt{km}} = 0.1$, $m = 100\ [kg]$ and see the evolution of $|w_{iI}(j\omega)|$ with $k$.

This is first shown for few values of the stiffness $k$ in figure [[fig:inverse_multiplicative_uncertainty_payload_few_k]]

#+begin_src matlab :exports none
  m = 20;
  freqs = logspace(0, 3, 1000);

  figure;
  hold on;
  for k = logspace(3,9,7)
    c = 0.1*sqrt(m*k);
    G0 = 1/(m*s^2 + c*s + k + Gp0);
    plot(freqs, abs(squeeze(freqresp(G0*Gp0*wI, freqs, 'Hz'))), 'DisplayName', sprintf('$k = %.0e$', k))
  end
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Uncertainty');
  hold off;
  legend('location', 'southeast');
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/inverse_multiplicative_uncertainty_payload_few_k.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:inverse_multiplicative_uncertainty_payload_few_k
#+caption: Norm of the inverse multiplicative uncertainty weight for various values of the the isolation platform's stiffness ([[./figs/inverse_multiplicative_uncertainty_payload_few_k.png][png]], [[./figs/inverse_multiplicative_uncertainty_payload_few_k.pdf][pdf]])
[[file:figs/inverse_multiplicative_uncertainty_payload_few_k.png]]

The norm of the uncertainty weight $|w_iI(j\omega)|$ is displayed as a function of $\omega$ and $k$ in Figure [[fig:inverse_multiplicative_payload_uncertainty_norm_k]].

#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  m = 20;

  Ks = logspace(3, 9, 100);

  wiI_k = zeros(length(freqs), length(Ks));

  for i = 1:length(Ks)
    k = Ks(i);
    c = 0.1*sqrt(m*k);

    G0 = 1/(m*s^2 + c*s + k + Gp0);

    wiI_k(:, i) = abs(squeeze(freqresp(G0*Gp0*wI, freqs, 'Hz')));
  end
#+end_src

#+begin_src matlab :exports none
  figure;
  surf(freqs, Ks, wiI_k', 'FaceColor', 'interp', 'EdgeColor', 'none')
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'ZScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Platform Stiffness [N/m]'); zlabel('$|w_{iI}(j\omega)|$');
  xlim([freqs(1), freqs(end)]); ylim([Ks(1), Ks(end)]);
  view([0 0 1]);
  colorbar('location', 'west');
  set(gca,'ColorScale','log')
  caxis([1e-3, 1]);
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/inverse_multiplicative_payload_uncertainty_norm_k.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:inverse_multiplicative_payload_uncertainty_norm_k
#+caption: Evolution of the norm of the uncertainty weight $|w_{iI}(j\omega)|$ as a function of the platform's stiffness $k$ ([[./figs/inverse_multiplicative_payload_uncertainty_norm_k.png][png]], [[./figs/inverse_multiplicative_payload_uncertainty_norm_k.pdf][pdf]])
[[file:figs/inverse_multiplicative_payload_uncertainty_norm_k.png]]

Instead of plotting as a function of the platform's stiffness, we can plot as a function of $\omega_0/\omega_0^\prime$ where:
- $\omega_0$ is the resonance of the platform alone
- $\omega_0^\prime$ is the resonance of the support alone

The obtain plot is shown in Figure [[fig:inverse_multiplicative_payload_uncertainty_k_normalized_frequency]].
In that case, we can see that with a platform's resonance frequency 10 times higher than the resonance of the payload, we get less than $1\%$ uncertainty until some fairly high frequency.

#+begin_src matlab :exports none
  Ws = sqrt(Ks./m);
  Wn = Ws./sqrt(k0/m0); % Normalized Frequency

  figure;
  contour(freqs, Wn, wiI_k', [0.001, 0.01, 0.1, 1], 'LineWidth', 2, 'ShowText', 'on')
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Normalized Frequency $\frac{\omega_0}{\omega_0^\prime}$');
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/inverse_multiplicative_payload_uncertainty_k_normalized_frequency.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:inverse_multiplicative_payload_uncertainty_k_normalized_frequency
#+caption: Evolution of the norm of the uncertainty weight $|w_{iI}(j\omega)|$ as a function of the frequency ratio $\omega_0/\omega_0^\prime$ ([[./figs/inverse_multiplicative_payload_uncertainty_k_normalized_frequency.png][png]], [[./figs/inverse_multiplicative_payload_uncertainty_k_normalized_frequency.pdf][pdf]])
[[file:figs/inverse_multiplicative_payload_uncertainty_k_normalized_frequency.png]]

*** Effect of the platform's damping $c$
Let's fix $k = 10^7\ [N/m]$, $m = 100\ [kg]$ and see the evolution of $|w_{iI}(j\omega)|$ with the isolation platform damping $c$ (Figure [[fig:inverse_multiplicative_payload_uncertainty_c]]).

#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  m = 20;
  xi = logspace(-2, 1, 100);

  wiI_c_soft  = zeros(length(freqs), length(xi));
  wiI_c_mid   = zeros(length(freqs), length(xi));
  wiI_c_stiff = zeros(length(freqs), length(xi));
  for i = 1:length(xi)
    k = m*(2*pi*5)^2;
    c = 2*xi(i)*sqrt(m*k);
    G0 = 1/(m*s^2 + c*s + k + Gp0);
    wiI_c_soft(:, i) = abs(squeeze(freqresp(G0*Gp0*wI, freqs, 'Hz')));

    k = m*(2*pi*50)^2;
    c = 2*xi(i)*sqrt(m*k);
    G0 = 1/(m*s^2 + c*s + k + Gp0);
    wiI_c_mid(:, i) = abs(squeeze(freqresp(G0*Gp0*wI, freqs, 'Hz')));

    k = m*(2*pi*500)^2;
    c = 2*xi(i)*sqrt(m*k);
    G0 = 1/(m*s^2 + c*s + k + Gp0);
    wiI_c_stiff(:, i) = abs(squeeze(freqresp(G0*Gp0*wI, freqs, 'Hz')));
  end
#+end_src

#+begin_src matlab :exports none
  figure;

  ax1 = subplot(1, 3, 1);
  surf(freqs, xi, wiI_c_soft', 'FaceColor', 'interp', 'EdgeColor', 'none')
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Damping Ratio');
  title('$\omega_0 \ll \omega_0^\prime$');
  view([0 0 1]);
  set(gca,'ColorScale','log')
  caxis([1e-3, 1]);

  ax2 = subplot(1, 3, 2);
  surf(freqs, xi, wiI_c_mid', 'FaceColor', 'interp', 'EdgeColor', 'none')
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]');
  title('$\omega_0 \approx \omega_0^\prime$');
  view([0 0 1]);
  set(gca,'ColorScale','log')
  caxis([1e-3, 1]);

  ax3 = subplot(1, 3, 3);
  surf(freqs, xi, wiI_c_stiff', 'FaceColor', 'interp', 'EdgeColor', 'none')
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  title('$\omega_0 \gg \omega_0^\prime$');
  view([0 0 1]);
  set(gca,'ColorScale','log')
  colorbar('location', 'west');
  caxis([1e-3 1e0]);
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/inverse_multiplicative_payload_uncertainty_c.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:inverse_multiplicative_payload_uncertainty_c
#+caption: Evolution of the norm of the uncertainty weight $|w_{iI}(j\omega)|$ as a function of the platform's damping ratio $\xi$ ([[./figs/inverse_multiplicative_payload_uncertainty_c.png][png]], [[./figs/inverse_multiplicative_payload_uncertainty_c.pdf][pdf]])
[[file:figs/inverse_multiplicative_payload_uncertainty_c.png]]

*** Effect of the platform's mass $m$
Let's fix $k = 10^7\ [N/m]$, $\xi = \frac{c}{2\sqrt{km}} = 0.1$ and see the evolution of $|w_{iI}(j\omega)|$ with the payload mass $m$ (Figure [[fig:inverse_multiplicative_payload_uncertainty_m]]).

#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  Ms = logspace(0, 3, 100);

  wiI_m_soft  = zeros(length(freqs), length(Ms));
  wiI_m_stiff = zeros(length(freqs), length(Ms));

  for i = 1:length(Ms)
    m = Ms(i);

    k = 5e4;
    c = 2*0.1*sqrt(m*k);
    G0 = 1/(m*s^2 + c*s + k + Gp0);
    wiI_m_soft(:, i) = abs(squeeze(freqresp(G0*Gp0*wI, freqs, 'Hz')));

    k = 5e7;
    c = 2*0.1*sqrt(m*k);
    G0 = 1/(m*s^2 + c*s + k + Gp0);
    wiI_m_stiff(:, i) = abs(squeeze(freqresp(G0*Gp0*wI, freqs, 'Hz')));
  end
#+end_src

#+begin_src matlab :exports none
  figure;

  ax1 = subplot(1, 2, 1);
  surf(freqs, Ms, wiI_m_soft', 'FaceColor', 'interp', 'EdgeColor', 'none')
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Platform Mass [kg]');
  xlabel('Frequency [Hz]');
  title('$\omega_0 \ll \omega_0^\prime$');
  view([0 0 1]);
  set(gca,'ColorScale','log')
  caxis([1e-3, 1]);

  ax2 = subplot(1, 2, 2);
  surf(freqs, Ms, wiI_m_stiff', 'FaceColor', 'interp', 'EdgeColor', 'none')
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  title('$\omega_0 \gg \omega_0^\prime$');
  xlabel('Frequency [Hz]');
  view([0 0 1]);
  colorbar('location', 'west');
  set(gca,'ColorScale','log')
  caxis([1e-3, 1]);
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/inverse_multiplicative_payload_uncertainty_m.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:inverse_multiplicative_payload_uncertainty_m
#+caption: Evolution of the norm of the uncertainty weight $|w_{iI}(j\omega)|$ as a function of the payload mass $m$ ([[./figs/inverse_multiplicative_payload_uncertainty_m.png][png]], [[./figs/inverse_multiplicative_payload_uncertainty_m.pdf][pdf]])
[[file:figs/inverse_multiplicative_payload_uncertainty_m.png]]

** Conclusion
#+begin_important
As was expected from Eq. [[eq:plant_dynamics_general_payload]], it is usually a good idea to maximize the mass, damping and stiffness of the isolation platform in order to be less sensible to the payload dynamics.

The best thing to do is to have a stiff isolation platform.

If a soft isolation platform is to be used, it is first a good idea to damp the isolation platform as shown in Figure [[fig:inverse_multiplicative_payload_uncertainty_c]].
This can make the uncertainty quite low until the first resonance of the payload.
In that case, maximizing the stiffness of the payload is a good idea.
#+end_important