Cercalo Test Bench

Table of Contents

1 Introduction

1.1 Signals

Signal Name Unit Number
Voltage Sent to Cercalo - Horizontal Uch [V] 1
Voltage Sent to Cercalo - Vertical Ucv [V] 2
Voltage Sent to Newport - Horizontal Unh [V] 3
Voltage Sent to Newport - Vertical Unv [V] 4
4Q Photodiode Measurement - Horizontal Vph [V] 5
4Q Photodiode Measurement - Vertical Vpv [V] 6
Measured Voltage across the Inductance - Horizontal Vch [V] 7
Measured Voltage across the Inductance - Vertical Vcv [V] 8
Newport Metrology - Horizontal Vnh [V] 9
Newport Metrology - Vertical Vnv [V] 10
Attocube Measurement Va [m] 11

1.2 Cercalo

Table 1: Cercalo Parameters
  Maximum Stroke [deg] Resonance Frequency [Hz] DC Gain [mA/deg] Gain at resonance [deg/V] RC Resistance [Ohm]
AX1 5 411.13 28.4 382.9 9.41
AX2 5 252.5 35.2 350.4  

L = 0.1mH on both directions

Resistance= 41 Ohm

Ax2 = 8.3 Ax1 = 9.3

1.3 Optical Setup

1.4 4 quadrant Diode

Front view of the photodiode.

4qd_naming.png

Signals going to the amplifier on the board. The amplifier is located on the bottom right of the board.

4qd_amplifier.png

1.5 ADC/DAC

Let's compute the theoretical noise of the ADC/DAC.

\begin{align*} \Delta V &= 20 V \\ n &= 16bits \\ q &= \Delta V/2^n = 305 \mu V \\ f_N &= 10kHz \\ \Gamma_n &= \frac{q^2}{12 f_N} = 7.76 \cdot 10^{-13} \frac{V^2}{Hz} \end{align*}

with \(\Delta V\) the total range of the ADC, \(n\) its number of bits, \(q\) the quantization, \(f_N\) the sampling frequency and \(\Gamma_n\) its theoretical Power Spectral Density.

1.6 Amplifier for the Cercalo

cercalo_amplifier.png

The values of the parameters are:

  • \(R_h = 41 \Omega\)
  • \(L_{c,h} = 0.1 mH\)
  • \(R_{c,h} = 9.3 \Omega\)
  • \(R_v = 41 \Omega\)
  • \(L_{c,v} = 0.1 mH\)
  • \(R_{c,v} = 8.3 \Omega\)

We want to find the transfer function from \(U_c\) to \(V_L\) and from \(U_c\) to \(i_c\).

We have that:

\begin{align*} V_C &= R_c i + L_c s i \\ U_c &= (R + R_c) i + L_c s i \end{align*}

Thus:

\begin{align} \frac{i_c}{U_c} &= \frac{1}{(R + R_c) + L_c s} \\ &= \frac{G_0}{1 + s/\omega_0} \end{align}

with

  • \(G_{0,i} = \frac{1}{R + R_c}\)
  • \(\omega_0 = \frac{R + R_c}{L_c}\)

And

\begin{align} \frac{V_c}{U_c} &= \frac{R_c + L_c s}{(R + R_c) + L_c s} \\ &= \frac{\frac{R_c}{R + R_c} + \frac{L_c}{R + R_c} s}{1 + \frac{L_c}{R + R_c} s} \\ &= \frac{G_0 + s/\omega_0}{1 + s/\omega_0} \\ \end{align}

with

  • \(G_0 = \frac{R_c}{R + R_c}\)
  • \(\omega_0 = \frac{R + R_c}{L_c}\)
Rh = 41; % [Ohm]
Lch = 0.1e-3; % [H]
Rch = 9.3; % [Ohm]

Rv = 41; % [Ohm]
Lcv = 0.1e-3; % [H]
Rcv = 8.3; % [Ohm]

Gih = 1/(Rh + Rch + Lch * s);
Gvh = (Rch + Lch * s)/(Rh + Rch + Lch * s);

Giv = 1/(Rv + Rcv + Lcv * s);
Gvv = (Rcv + Lcv * s)/(Rv + Rcv + Lcv * s);

The current amplifier has a constant gain over all the frequency band of interest. 

Gih0 = freqresp(Gih, 0);
Gvh0 = freqresp(Gvh, 0);
Giv0 = freqresp(Giv, 0);
Gvv0 = freqresp(Gvv, 0);

1.7 Block Diagram

\begin{tikzpicture}
  \node[DAC] (dac) at (0, 0) {};
  \node[block, right=1.2 of dac] (Gi) {$G_i$};
  \node[block, right=1.5 of Gi] (Gc) {$G_c$};
  \node[addb, right=0.5 of Gc] (add) {};
  \node[block, above= of add] (Gn) {$G_n$};
  \node[ADC, right=1.2 of add] (adc) {};
  \coordinate (GiGc) at ($0.8*(Gi.east) + 0.2*(Gc.west)$);
  \node[block] (Amp) at (GiGc|-Gn) {$G_a$};

  \node[above, align=center] at (Gi.north) {Current\\Amplifier};
  \node[above, align=center] at (Gc.north) {Cercalo};
  \node[left, align=right] at (Gn.west) {Newport};

  \draw[->] ($(dac.west) + (-1, 0)$) --node[midway, sloped]{$/$} (dac.west);
  \draw[->] (dac.east) -- (Gi.west) node[above left]{$\begin{bmatrix}U_{c,h} \\ U_{c,v}\end{bmatrix}$};
  \draw[->] (Gi.east) -- (Gc.west) node[above left]{$\begin{bmatrix}\tilde{U}_{c,h} \\ \tilde{U}_{c,v}\end{bmatrix}$};
  \draw[->] (Gc.east) -- (add.west);
  \draw[->] (Gn.south) -- (add.north);
  \draw[->] (GiGc)node[branch]{} -- (Amp.south);
  \draw[->] (Amp.north) -- ++(0, 1.5) node[below right]{$\begin{bmatrix}V_{c,h} \\ V_{c,v}\end{bmatrix}$};
  \draw[->] ($(Gn.north) + (0, 1.5)$) -- (Gn.north) node[above right]{$\begin{bmatrix}U_{n,h} \\ U_{n,v}\end{bmatrix}$};
  \draw[->] (add.east) -- (adc.west) node[above left]{$\begin{bmatrix}V_{p,h} \\ V_{p,v}\end{bmatrix}$};
  \draw[->] (adc.east) --node[midway, sloped]{$/$} ++(1, 0);
\end{tikzpicture}

cercalo_diagram_simplify.png

  • Transfer function for the Current Amplifier: \[ G_i = \begin{bmatrix} G_{i,h} & 0 \\ 0 & G_{i,v} \end{bmatrix} \]
  • Transfer function for the Voltage Amplifier: \[ G_a = \begin{bmatrix} G_{a,h} & 0 \\ 0 & G_{a,v} \end{bmatrix} \]
  • Transfer function from the Voltage across the Cercalo inductors to the 4 quadrant measurement \[ G_c = \begin{bmatrix} G_{\frac{V_{p,h}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,h}}{\tilde{U}_{c,v}}} \\ G_{\frac{V_{p,v}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,v}}{\tilde{U}_{c,v}}} \end{bmatrix} \]
  • Transfer function from the Newport command signal to the 4 quadrant measurement \[ G_n = \begin{bmatrix} G_{\frac{V_{p,h}}{U_{n,h}}} & G_{\frac{V_{p,h}}{U_{n,v}}} \\ G_{\frac{V_{p,v}}{U_{n,h}}} & G_{\frac{V_{n,v}}{U_{n,v}}} \end{bmatrix} \]
\begin{tikzpicture}
  \node[DAC] (dac) at (0, 0) {};
  \node[block, right=1.5 of dac] (Gi) {$\begin{bmatrix} G_{i,h} & 0 \\ 0 & G_{i,v} \end{bmatrix}$};
  \node[block, right=1.8 of Gi] (Gc) {$\begin{bmatrix} G_{\frac{V_{p,h}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,h}}{\tilde{U}_{c,v}}} \\ G_{\frac{V_{p,v}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,v}}{\tilde{U}_{c,v}}} \end{bmatrix}$};
  \node[addb, right= of Gc] (add) {};
  \node[block, above= of add] (Gn) {$\begin{bmatrix} G_{\frac{V_{p,h}}{U_{n,h}}} & G_{\frac{V_{p,h}}{U_{n,v}}} \\ G_{\frac{V_{p,v}}{U_{n,h}}} & G_{\frac{V_{n,v}}{U_{n,v}}} \end{bmatrix}$};
  \node[ADC, right = 1.5 of add] (adc) {};
  \coordinate (GiGc) at ($0.7*(Gi.east) + 0.3*(Gc.west)$);
  \node[block] (Amp) at (GiGc|-Gn) {$\begin{bmatrix} G_{a,h} & 0 \\ 0 & G_{a,v} \end{bmatrix}$};

  \node[above, align=center] at (Gi.north) {Current\\Amplifier};
  \node[above, align=center] at (Gc.north) {Cercalo};
  \node[left, align=right] at (Gn.west) {Newport};

  \draw[->] ($(dac.west) + (-1, 0)$) --node[midway, sloped]{$/$} (dac.west);
  \draw[->] (dac.east) -- (Gi.west) node[above left]{$\begin{bmatrix}U_{c,h} \\ U_{c,v}\end{bmatrix}$};
  \draw[->] (Gi.east) -- (Gc.west) node[above left]{$\begin{bmatrix}\tilde{U}_{c,h} \\ \tilde{U}_{c,v}\end{bmatrix}$};
  \draw[->] (Gc.east) -- (add.west);
  \draw[->] (Gn.south) -- (add.north);
  \draw[->] (GiGc)node[branch]{} -- (Amp.south);
  \draw[->] (Amp.north) -- ++(0, 1.5) node[below right]{$\begin{bmatrix}V_{c,h} \\ V_{c,v}\end{bmatrix}$};
  \draw[->] ($(Gn.north) + (0, 1.5)$) -- (Gn.north) node[above right]{$\begin{bmatrix}U_{n,h} \\ U_{n,v}\end{bmatrix}$};
  \draw[->] (add.east) -- (adc.west) node[above left]{$\begin{bmatrix}V_{p,h} \\ V_{p,v}\end{bmatrix}$};
  \draw[->] (adc.east) --node[midway, sloped]{$/$} ++(1, 0);
\end{tikzpicture}

cercalo_diagram.png

2 Identification

All the files (data and Matlab scripts) are accessible here.

2.1 Excitation Data

fs = 1e4;
Ts = 1/fs;

We generate white noise with the "random number" simulink block, and we filter that noise. 

Gi = (1)/(1+s/2/pi/100);

c2d(Gi, Ts, 'tustin')

c2d(Gi, Ts, 'tustin')

ans =

  0.030459 (z+1)
  --------------
    (z-0.9391)

Sample time: 0.0001 seconds
Discrete-time zero/pole/gain model.

2.2 Signals

Signal Name Unit
Voltage Sent to Cercalo - Horizontal Uch [V]
Voltage Sent to Cercalo - Vertical Ucv [V]
Voltage Sent to Newport - Horizontal Unh [V]
Voltage Sent to Newport - Vertical Unv [V]
4Q Photodiode Measurement - Horizontal Vph [V]
4Q Photodiode Measurement - Vertical Vpv [V]
Measured Voltage across the Inductance - Horizontal Vch [V]
Measured Voltage across the Inductance - Vertical Vcv [V]
Newport Metrology - Horizontal Vnh [V]
Newport Metrology - Vertical Vnv [V]
Attocube Measurement Va [m]

2.3 Huddle Test

We load the data taken during the Huddle Test. 

load('mat/data_test.mat', ...
     't', 'Uch', 'Ucv', ...
     'Unh', 'Unv', ...
     'Vph', 'Vpv', ...
     'Vch', 'Vcv', ...
     'Vnh', 'Vnv', ...
     'Va');

We remove the first second of data where everything is settling down. 

t0 = 1;

Uch(t<t0) = [];
Ucv(t<t0) = [];
Unh(t<t0) = [];
Unv(t<t0) = [];
Vph(t<t0) = [];
Vpv(t<t0) = [];
Vch(t<t0) = [];
Vcv(t<t0) = [];
Vnh(t<t0) = [];
Vnv(t<t0) = [];
Va(t<t0)  = [];
t(t<t0)   = [];

t = t - t(1); % We start at t=0

We compute the Power Spectral Density of the horizontal and vertical positions of the beam as measured by the 4 quadrant diode. 

[psd_Vph, f] = pwelch(Vph, hanning(ceil(1*fs)), [], [], fs);
[psd_Vpv, ~] = pwelch(Vpv, hanning(ceil(1*fs)), [], [], fs);

figure;
hold on;
plot(f, sqrt(psd_Vph), 'DisplayName', '$\Gamma_{Vp_h}$');
plot(f, sqrt(psd_Vpv), 'DisplayName', '$\Gamma_{Vp_v}$');
hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD $\left[\frac{V}{\sqrt{Hz}}\right]$')
legend('Location', 'southwest');
xlim([1, 1000]);

We compute the Power Spectral Density of the voltage across the inductance used for horizontal and vertical positioning of the Cercalo. 

[psd_Vch, f] = pwelch(Vch, hanning(ceil(1*fs)), [], [], fs);
[psd_Vcv, ~] = pwelch(Vcv, hanning(ceil(1*fs)), [], [], fs);

figure;
hold on;
plot(f, sqrt(psd_Vch), 'DisplayName', '$\Gamma_{Vc_h}$');
plot(f, sqrt(psd_Vcv), 'DisplayName', '$\Gamma_{Vc_v}$');
hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD $\left[\frac{V}{\sqrt{Hz}}\right]$')
legend('Location', 'southwest');
xlim([1, 1000]);

2.4 Input / Output data

The identification data is loaded 

uh = load('mat/data_uh.mat', ...
          't', 'Uch', 'Ucv', ...
          'Unh', 'Unv', ...
          'Vph', 'Vpv', ...
          'Vch', 'Vcv', ...
          'Vnh', 'Vnv', ...
          'Va');

uv = load('mat/data_uh.mat', ...
          't', 'Uch', 'Ucv', ...
          'Unh', 'Unv', ...
          'Vph', 'Vpv', ...
          'Vch', 'Vcv', ...
          'Vnh', 'Vnv', ...
          'Va');

We remove the first seconds where the Cercalo is turned on. 

t0 = 1;

uh.Uch(t<t0) = [];
uh.Ucv(t<t0) = [];
uh.Unh(t<t0) = [];
uh.Unv(t<t0) = [];
uh.Vph(t<t0) = [];
uh.Vpv(t<t0) = [];
uh.Vch(t<t0) = [];
uh.Vcv(t<t0) = [];
uh.Vnh(t<t0) = [];
uh.Vnv(t<t0) = [];
uh.Va(t<t0)  = [];
uh.t(t<t0)   = [];

uh.t = uh.t - uh.t(1); % We start at t=0

t0 = 1;

uv.Uch(t<t0) = [];
uv.Ucv(t<t0) = [];
uv.Unh(t<t0) = [];
uv.Unv(t<t0) = [];
uv.Vph(t<t0) = [];
uv.Vpv(t<t0) = [];
uv.Vch(t<t0) = [];
uv.Vcv(t<t0) = [];
uv.Vnh(t<t0) = [];
uv.Vnv(t<t0) = [];
uv.Va(t<t0)  = [];
uv.t(t<t0)   = [];

uv.t = uv.t - uv.t(1); % We start at t=0

identification_uh.png

Figure 6: Identification signals when exciting the horizontal direction (png, pdf)

identification_uv.png

Figure 7: Identification signals when exciting in the vertical direction (png, pdf)

2.5 Estimation of the Frequency Response Function Matrix

win = hanning(ceil(1*fs));

We compute an estimate of the transfer functions. 

[tf_Uch_Vph, f] = tfestimate(uh.Uch, uh.Vph, win, [], [], fs);
[tf_Uch_Vpv, ~] = tfestimate(uh.Uch, uh.Vpv, win, [], [], fs);
[tf_Ucv_Vph, ~] = tfestimate(uv.Ucv, uv.Vph, win, [], [], fs);
[tf_Ucv_Vpv, ~] = tfestimate(uv.Ucv, uv.Vpv, win, [], [], fs);

frequency_response_matrix.png

Figure 8: Frequency Response Matrix (png, pdf)

2.6 Coherence

[coh_Uch_Vph, f] = mscohere(uh.Uch, uh.Vph, win, [], [], fs);
[coh_Uch_Vpv, ~] = mscohere(uh.Uch, uh.Vpv, win, [], [], fs);
[coh_Ucv_Vph, ~] = mscohere(uv.Ucv, uv.Vph, win, [], [], fs);
[coh_Ucv_Vpv, ~] = mscohere(uv.Ucv, uv.Vpv, win, [], [], fs);

identification_coherence.png

Figure 9: Coherence (png, pdf)

2.7 Extraction of a transfer function matrix

First we define the initial guess for the resonance frequencies and the weights associated. 

freqs_res_uh = [410]; % [Hz]
freqs_res_uv = [250]; % [Hz]

From the number of resonance frequency we want to fit, we define the order N of the system we want to obtain. 

N = 2;

We then make an initial guess on the complex values of the poles. 

xi = 0.001; % Approximate modal damping
poles_uh = [2*pi*freqs_res_uh*(xi + 1i), 2*pi*freqs_res_uh*(xi - 1i)];
poles_uv = [2*pi*freqs_res_uv*(xi + 1i), 2*pi*freqs_res_uv*(xi - 1i)];

We then define the weight that will be used for the fitting. Basically, we want more weight around the resonance and at low frequency (below the first resonance). Also, we want more importance where we have a better coherence. 

weight_Uch_Vph = coh_Uch_Vph';
weight_Uch_Vpv = coh_Uch_Vpv';
weight_Ucv_Vph = coh_Ucv_Vph';
weight_Ucv_Vpv = coh_Ucv_Vpv';

alpha = 0.1;

for freq_i = 1:length(freqs_res)
  weight_Uch_Vph(f>(1-alpha)*freqs_res_uh(freq_i) & f<(1 + alpha)*freqs_res_uh(freq_i)) = 10;
  weight_Uch_Vpv(f>(1-alpha)*freqs_res_uh(freq_i) & f<(1 + alpha)*freqs_res_uh(freq_i)) = 10;
  weight_Ucv_Vph(f>(1-alpha)*freqs_res_uv(freq_i) & f<(1 + alpha)*freqs_res_uv(freq_i)) = 10;
  weight_Ucv_Vpv(f>(1-alpha)*freqs_res_uv(freq_i) & f<(1 + alpha)*freqs_res_uv(freq_i)) = 10;
end

Ignore data above some frequency. 

weight_Uch_Vph(f>1000) = 0;
weight_Uch_Vpv(f>1000) = 0;
weight_Ucv_Vph(f>1000) = 0;
weight_Ucv_Vpv(f>1000) = 0;

weights.png

Figure 10: Weights amplitude (png, pdf)

When we set some options for vfit3. 

opts = struct();

opts.stable = 1;    % Enforce stable poles
opts.asymp = 1;     % Force D matrix to be null
opts.relax = 1;     % Use vector fitting with relaxed non-triviality constraint
opts.skip_pole = 0; % Do NOT skip pole identification
opts.skip_res = 0;  % Do NOT skip identification of residues (C,D,E)
opts.cmplx_ss = 0;  % Create real state space model with block diagonal A

opts.spy1 = 0;      % No plotting for first stage of vector fitting
opts.spy2 = 0;      % Create magnitude plot for fitting of f(s)

We define the number of iteration. 

Niter = 5;

An we run the vectfit3 algorithm. 

for iter = 1:Niter
  [SER_Uch_Vph, poles, ~, fit_Uch_Vph] = vectfit3(tf_Uch_Vph.', 1i*2*pi*f, poles_uh, weight_Uch_Vph, opts);
end
for iter = 1:Niter
  [SER_Uch_Vpv, poles, ~, fit_Uch_Vpv] = vectfit3(tf_Uch_Vpv.', 1i*2*pi*f, poles_uh, weight_Uch_Vpv, opts);
end
for iter = 1:Niter
  [SER_Ucv_Vph, poles, ~, fit_Ucv_Vph] = vectfit3(tf_Ucv_Vph.', 1i*2*pi*f, poles_uv, weight_Ucv_Vph, opts);
end
for iter = 1:Niter
  [SER_Ucv_Vpv, poles, ~, fit_Ucv_Vpv] = vectfit3(tf_Ucv_Vpv.', 1i*2*pi*f, poles_uv, weight_Ucv_Vpv, opts);
end

identification_matrix_fit.png

Figure 11: Transfer Function Extraction of the FRF matrix (png, pdf)

identification_matrix_fit_phase.png

Figure 12: Transfer Function Extraction of the FRF matrix (png, pdf)

And finally, we create the identified state space model: 

G_uh_xh = minreal(ss(full(SER_uh_xh.A),SER_uh_xh.B,SER_uh_xh.C,SER_uh_xh.D));
G_uv_xh = minreal(ss(full(SER_uv_xh.A),SER_uv_xh.B,SER_uv_xh.C,SER_uv_xh.D));
G_uh_xv = minreal(ss(full(SER_uh_xv.A),SER_uh_xv.B,SER_uh_xv.C,SER_uh_xv.D));
G_uv_xv = minreal(ss(full(SER_uv_xv.A),SER_uv_xv.B,SER_uv_xv.C,SER_uv_xv.D));

G = [G_uh_xh, G_uv_xh;
     G_uh_xv, G_uv_xv];

save('mat/plant.mat', 'G');

3 Sensor Noise

4 Plant Analysis

4.1 Load Plant

load('mat/plant.mat', 'G');

4.2 RGA-Number

freqs = logspace(2, 4, 1000);
G_resp = freqresp(G, freqs, 'Hz');
A = zeros(size(G_resp));
RGAnum = zeros(1, length(freqs));

for i = 1:length(freqs)
  A(:, :, i) = G_resp(:, :, i).*inv(G_resp(:, :, i))';
  RGAnum(i) = sum(sum(abs(A(:, :, i)-eye(2))));
end
% RGA = G0.*inv(G0)';

figure;
plot(freqs, RGAnum);
set(gca, 'xscale', 'log');

U = zeros(2, 2, length(freqs));
S = zeros(2, 2, length(freqs))
V = zeros(2, 2, length(freqs));

for i = 1:length(freqs)
  [Ui, Si, Vi] = svd(G_resp(:, :, i));
  U(:, :, i) = Ui;
  S(:, :, i) = Si;
  V(:, :, i) = Vi;
end

4.3 Rotation Matrix

G0 = freqresp(G, 0);

5 Control Objective

The maximum expected stroke is \(y_\text{max} = 3mm \approx 5e^{-2} rad\) at \(1Hz\). The maximum wanted error is \(e_\text{max} = 10 \mu rad\).

Thus, we require the sensitivity function at \(\omega_0 = 1\text{ Hz}\):

\begin{align*} |S(j\omega_0)| &< \left| \frac{e_\text{max}}{y_\text{max}} \right| \\ &< 2 \cdot 10^{-4} \end{align*}

In terms of loop gain, this is equivalent to: \[ |L(j\omega_0)| > 5 \cdot 10^{3} \]

6 Plant Scaling

  • measured noise
  • expected perturbations
  • maximum input usage
  • maximum wanted error

7 Control Design

Author: Dehaeze Thomas

Created: 2019-09-16 lun. 11:59

Validate