nass-simscape/org/control_requirements.org

Control Requirements

• Introduction
• Simplify Model for the Nano-Hexapod
  • Model of the nano-hexapod
  • How to include Ground Motion in the model?
  • Motion of the micro-station
• Control with the Stiff Nano-Hexapod
  • Definition of the values
  • Control using $d$
    • Control Architecture
    • Analytical Analysis
  • Control using $F_m$
    • Control Architecture
    • Pure Integrator
    • Low pass filter
  • Comparison
  • Control using $x$
    • Analytical analysis
    • Control implementation
    • Results
• Comparison with the use of a Soft nano-hexapod
• Estimate the level of vibration
• Requirements on the norm of closed-loop transfer functions
  • Approximation of the ASD of perturbations
  • Wanted ASD of outputs
  • Limiting the bandwidth
  • Generalized Weighted plant
  • Synthesis
  • Loop Gain
  • Results
  • Requirements

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{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}

Simplified mechanical system for the nano-hexapod

The signals are described in table tab:general_plant_signals.

	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

Signals definition for the generalized plant

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{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}

Block representation of the nano-hexapod

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{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}

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{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}

Ground Motion $w$ included

$T_w$ is the mechanical transmissibility of the micro-station. We can approximate this transfer function by a second order low pass filter: \[ T_w = \frac{1}{1 + 2 \xi s/\omega_0 + s^2/\omega_0^2} \]

Control with the Stiff Nano-Hexapod

Definition of the values

Let's define the mass and stiffness of the nano-hexapod.

  m = 50; % [kg]
  k = 1e7; % [N/m]

Let's define the Plant as shown in Figure fig:nano_station_inputs_outputs:

  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'};

Now, define the transmissibility transfer function $T_\mu$ corresponding to the micro-station motion.

  wmu = 2*pi*50; % [rad/s]

  Tmu = 1/(1 + s/wmu);
  Tmu.InputName = {'r1'};
  Tmu.OutputName = {'ymu'};

  w0 = 2*pi*40;
  xi = 0.5;
  Tw = 1/(1 + 2*xi*s/w0 + s^2/w0^2);
  Tw.InputName = {'w1'};
  Tw.OutputName = {'dw'};

We add the fact that $x_\mu = d_\mu + T_\mu r + T_w w$:

  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'});

Control using $d$

Control Architecture

Let's consider a feedback loop using $d$ as shown in Figure fig:nano_station_control_d.

  \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}

Feedback diagram using $d$

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: \[ x = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} x_\mu \]

  K_dvf = 2*sqrt(k*m)*s;
  K_dvf.InputName = {'d'};
  K_dvf.OutputName = {'F'};

  Gpz_dvf = feedback(Gpz, K_dvf, 'name');

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 $\epsilon = r - x$: \[ \epsilon = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} d_\mu + \frac{ms^2 + gs + k - T_\mu (gs + k)}{ms^2 + gs + k} r \]

\[ \epsilon = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} d_\mu + \frac{ms^2 - S_\mu(gs + k)}{ms^2 + gs + k} r \]

Control using $F_m$

Control Architecture

Let's consider a feedback loop using $Fm$ as shown in Figure fig:nano_station_control_Fm.

  \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}

Feedback diagram using $F_m$

Pure Integrator

Let's apply integral force feedback by integration $F_m$: \[ F = F^\prime - \frac{g}{s} F_m \]

And we finally obtain: \[ x = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} x_\mu \]

  K_iff = 2*sqrt(k/m)/s;
  K_iff.InputName = {'Fm'};
  K_iff.OutputName = {'F'};

  Gpz_iff = feedback(Gpz, K_iff, 'name');

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 $\epsilon = r - x$: \[ \epsilon = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} d_\mu + \frac{ms^2 + mgs + k - T_\mu k}{ms^2 + mgs + k} r \]

\[ \epsilon = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} d_\mu + \frac{ms^2 + mgs + S_\mu k}{ms^2 + mgs + k} r \]

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 $\omega_mu$

  % 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');

Comparison

<<plt-matlab>>

Obtained transfer functions for DVF and IFF (png, pdf)

	$d_\mu$	$F_d$	$w$
IFF	Better filtering of the vibrations	More sensitive to External forces
DVF	Opposite	Opposite	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{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}

Feedback diagram using $x$

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:

\[ \epsilon = \frac{-G_{\frac{x}{F_d}}}{1 + G_{\frac{x}{F}}K} F_d + \frac{-G_{\frac{x}{x_\mu}}}{1 + G_{\frac{x}{F}}K} d_\mu + \frac{1 - G_{\frac{x}{x_\mu}} T_\mu}{1 + G_{\frac{x}{F}}K} r \]

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.

Exogenous Outputs	Design recommendation
$F_d$	high $k$, high $m$
$d_\mu$	low $k$, high $m$
$r$	no influence if $\omega_\nu > \omega_\mu$

Design recommendation

Control implementation

Controller for the damped plant using DVF.

  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'};

Controller for the damped plant using IFF.

  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'};

Loop gain

<<plt-matlab>>

Loop Gain (png, pdf)

Let's connect all the systems as shown in Figure fig:nano_station_control_x.

  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'});

Results

<<plt-matlab>>

Obtained closed-loop transfer functions (png, pdf)

	Reference Tracking	Vibration Filtering	Compliance
DVF	Similar behavior		Better for $\omega < \omega_\nu$
IFF	Similar behavior	Better for $\omega > \omega_\nu$

Comparison with the use of a Soft nano-hexapod

  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'};

  wmu = 2*pi*50; % [rad/s]

  Tmu = 1/(1 + s/wmu);
  Tmu.InputName = {'r1'};
  Tmu.OutputName = {'ymu'};

  w0 = 2*pi*40;
  xi = 0.5;
  Tw = 1/(1 + 2*xi*s/w0 + s^2/w0^2);
  Tw.InputName = {'w1'};
  Tw.OutputName = {'dw'};

  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'});

  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');

  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'};

  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'});

<<plt-matlab>>

Obtained closed-loop transfer functions (png, pdf)

	Reference Tracking	Vibration Filtering	Compliance
DVF	Similar behavior		Better for $\omega < \omega_\nu$
IFF	Similar behavior	Better for $\omega > \omega_\nu$

<<plt-matlab>>

Comparison of the closed-loop transfer functions for Soft and Stiff nano-hexapod (png, pdf)

	Soft	Stiff
Reference Tracking	=	=
Ground Motion	=	=
Vibration Isolation	+	-
Compliance	-	+

Estimate the level of vibration

  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');

If we note the PSD $\Gamma$: \[ \Gamma_y = |G_{\frac{y}{w}}|^2 \Gamma_w + |G_{\frac{y}{x_\mu}}|^2 \Gamma_{x_\mu} \]

  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);

<<plt-matlab>>

ASD of the position error due to Ground Motion and Vibration (png, pdf)

Actuator usage

  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);

  sqrt(trapz(f, F_pz))
  sqrt(trapz(f, F_vc))

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

  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));

  G_gm = 1e-8*1/s^2*(s + 2*pi*1)^2*(1/((1 + s/2/pi/10)^3));

Wanted ASD of outputs

Wanted ASD of motion error

  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);

  sqrt(trapz(f, abs(squeeze(freqresp(G_y, f, 'Hz'))).^2))

sqrt(trapz(f, abs(squeeze(freqresp(G_y, f, 'Hz'))).^2))
ans =
      9.47118350214793e-08

Limiting the bandwidth

  wF = 2*pi*10;
  G_F = 100000*(wF + s)^2;

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.

  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'})

Normalization.

We multiply the plant input by $G_{rz}$ and the plant output by $G_y^{-1}$:

  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'};

Synthesis

  [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'};

Loop Gain

  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'});

Results

Requirements

reference tracking	$\epsilon/r$	-120dB at 1Hz
vibration isolation	$x/x_\mu$	-60dB above 10Hz
compliance	$x/F_d$