From 9a276fe64dd5268641b61c3ec893b715f3514cd1 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 17 Sep 2019 15:55:59 +0200 Subject: [PATCH] Identification of all the elements in the system. Decentralized Controller. --- .../NASS-TestProcedure-13Sept2019-V1.pdf | Bin figs/4qd_linear_reg.png | Bin 0 -> 78928 bytes figs/calib_4qd_h.png | Bin 0 -> 100881 bytes figs/calib_4qd_v.png | Bin 0 -> 95482 bytes figs/cercalo_amplifier.png | Bin 7676 -> 9862 bytes figs/cercalo_diagram.png | Bin 35455 -> 46396 bytes figs/cercalo_diagram_simplify.png | Bin 20979 -> 30741 bytes figs/coh_cercalo.png | Bin 0 -> 111171 bytes figs/current_amplifier_comp_theory_id.png | Bin 0 -> 131867 bytes figs/current_amplifier_comp_theory_id_bis.png | Bin 0 -> 129884 bytes figs/current_amplifier_tf.png | Bin 175700 -> 157041 bytes figs/diag_contr_effect_newport.png | Bin 0 -> 100799 bytes figs/diag_contr_loop_gain.png | Bin 0 -> 105197 bytes figs/frf_cercalo_gain.png | Bin 0 -> 185554 bytes figs/frf_cercalo_phase.png | Bin 0 -> 127153 bytes figs/frf_newport_gain.png | Bin 0 -> 182851 bytes figs/frf_newport_phase.png | Bin 0 -> 121773 bytes figs/id_newport_coherence.png | Bin 108221 -> 111144 bytes figs/identification_matrix_fit.png | Bin 188639 -> 189954 bytes figs/identification_matrix_fit_phase.png | Bin 125338 -> 129076 bytes figs/identification_uh.png | Bin 57203 -> 58085 bytes figs/identification_unh.png | Bin 97099 -> 52213 bytes figs/identification_unv.png | Bin 48328 -> 49821 bytes figs/identification_uv.png | Bin 54851 -> 55789 bytes figs/time_delay_newport.png | Bin 125224 -> 131365 bytes figs/weights_cercalo.png | Bin 0 -> 64817 bytes index.html | 1780 +++++++--- index.org | 3013 +++++++++-------- mat/plant.mat | Bin 1203 -> 3671 bytes matlab/cercalo_identification.m | 1039 ++++++ matlab/decentralized_control.m | 77 + matlab/frf_processing.m | 141 + matlab/plant_identification.m | 341 -- run_test.m => matlab/run_test.m | 0 setup.m => matlab/setup.m | 0 matlab/test_bench.slx | Bin 0 -> 21892 bytes test_cercalo.slx => matlab/test_cercalo.slx | Bin 37 files changed, 4177 insertions(+), 2214 deletions(-) rename NASS-TestProcedure-13Sept2019-V1.pdf => 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      Table of Contents

      -
      -

      1 Introduction

      +
      +

      1 Introduction

      -
      -

      1.1 Block Diagram

      +
      +

      1.1 Block Diagram

      -The block diagram of the setup to be controlled is shown in Fig. 1. +The block diagram of the setup to be controlled is shown in Fig. 1.

      -
      +

      cercalo_diagram_simplify.png

      Figure 1: Block Diagram of the Experimental Setup

      @@ -336,22 +371,30 @@ The block diagram of the setup to be controlled is shown in Fig. -
    1. Current Amplifier: from the voltage set by the DAC to the voltage across the Cercalo inductors -\[ G_i = \begin{bmatrix} G_{i,h} & 0 \\ 0 & G_{i,v} \end{bmatrix} \]
    2. +
    3. Current Amplifier: from the voltage set by the DAC to the current going to the Cercalo's inductors +\[ G_i = \begin{bmatrix} G_{i,h} & 0 \\ 0 & G_{i,v} \end{bmatrix} \text{ in } \left[ \frac{A}{V} \right] \] +\[ \begin{bmatrix} I_{c,h} \\ I_{c,v} \end{bmatrix} = G_i \begin{bmatrix} U_{c,h} \\ U_{c,v} \end{bmatrix} \]
    4. +
    5. Impedance of the Cercalo that converts the current going to the cercalo to the voltage across the cercalo: +\[ Z_c = \begin{bmatrix} Z_{c,h} & 0 \\ 0 & Z_{c,v} \end{bmatrix} \text{ in } \left[ \frac{V}{A} \right] \] +\[ \begin{bmatrix} \tilde{V}_{c,h} \\ \tilde{V}_{c,v} \end{bmatrix} = Z_c \begin{bmatrix} I_{c,h} \\ I_{c,v} \end{bmatrix} \]
    6. Voltage Amplifier: from the voltage across the Cercalo inductors to the measured voltage -\[ G_a = \begin{bmatrix} G_{a,h} & 0 \\ 0 & G_{a,v} \end{bmatrix} \]
    7. -
    8. Cercalo: Transfer function from the Voltage across the cercalo inductors to the 4 quadrant measurement -\[ G_c = \begin{bmatrix} G_{\frac{V_{p,h}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,h}}{\tilde{U}_{c,v}}} \\ G_{\frac{V_{p,v}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,v}}{\tilde{U}_{c,v}}} \end{bmatrix} \]
    9. +\[ G_a = \begin{bmatrix} G_{a,h} & 0 \\ 0 & G_{a,v} \end{bmatrix} \text{ in } \left[ \frac{V}{V} \right] \] +\[ \begin{bmatrix} V_{c,h} \\ V_{c,v} \end{bmatrix} = G_a \begin{bmatrix} \tilde{V}_{c,h} \\ \tilde{V}_{c,v} \end{bmatrix} \] +
    10. Cercalo: Transfer function from the current going through the cercalo inductors to the 4 quadrant measurement +\[ G_c = \begin{bmatrix} G_{\frac{V_{p,h}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,h}}{\tilde{U}_{c,v}}} \\ G_{\frac{V_{p,v}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,v}}{\tilde{U}_{c,v}}} \end{bmatrix} \text{ in } \left[ \frac{V}{A} \right] \] +\[ \begin{bmatrix} V_{p,h} \\ V_{p,v} \end{bmatrix} = G_c \begin{bmatrix} I_{c,h} \\ I_{c,v} \end{bmatrix} \]
    11. Newport Transfer function from the command signal of the Newport to the 4 quadrant measurement -\[ G_n = \begin{bmatrix} G_{\frac{V_{p,h}}{U_{n,h}}} & G_{\frac{V_{p,h}}{U_{n,v}}} \\ G_{\frac{V_{p,v}}{U_{n,h}}} & G_{\frac{V_{n,v}}{U_{n,v}}} \end{bmatrix} \]
    12. +\[ G_n = \begin{bmatrix} G_{\frac{V_{p,h}}{U_{n,h}}} & G_{\frac{V_{p,h}}{U_{n,v}}} \\ G_{\frac{V_{p,v}}{U_{n,h}}} & G_{\frac{V_{n,v}}{U_{n,v}}} \end{bmatrix} \text{ in } \left[ \frac{V}{V} \right] \] +\[ \begin{bmatrix} V_{p,h} \\ V_{p,v} \end{bmatrix} = G_c \begin{bmatrix} V_{n,h} \\ V_{n,v} \end{bmatrix} \] +
    13. 4 Quadrant Diode: the gain of the 4 quadrant diode in [V/rad] is inverse in order to obtain the physical angle of the beam +\[ G_d = \begin{bmatrix} G_{d,h} & 0 \\ 0 & G_{d,v} \end{bmatrix} \text{ in } \left[\frac{V}{rad}\right] \]
    14. -The block diagram with each transfer function is shown in Fig. 2. +The block diagram with each transfer function is shown in Fig. 2.

      - -
      +

      cercalo_diagram.png

      Figure 2: Block Diagram of the Experimental Setup with detailed dynamics

      @@ -359,129 +402,14 @@ The block diagram with each transfer function is shown in Fig. -

      1.2 Amplifier for the Cercalo

      +
      +

      1.2 Cercalo

      - -
      -

      cercalo_amplifier.png -

      -
      -

      -The value of the resistor in series with the buffer have been measured for both axis. -

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

      -We want to find the transfer function from \(U_c\) to \(V_L\) and from \(U_c\) to \(i_c\). +From the Cercalo documentation, we have the parameters shown on table 1.

      -

      -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 -

      -
        -
      • \(G_0 = \frac{R_c}{R + R_c}\)
      • -
      • \(\omega_0 = \frac{R + R_c}{L_c}\)
      • -
      - -

      -Let's verify that the electrical circuit behaves as a constant current amplifier in the frequency band of interest. -

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

      current_amplifier_tf.png -

      -

      Figure 4: Transfer function for the current amplifier (png, pdf)

      -
      - - -
      -

      -The current amplifier has a constant gain over all the frequency band of interest. -\[ G_i(s) = \begin{bmatrix} 0.02 & 0 \\ 0 & 0.02 \end{bmatrix}\quad \left[\frac{A}{V}\right] \] -\[ G_a(s) = \begin{bmatrix} 0.185 & 0 \\ 0 & 0.168 \end{bmatrix} \left[\frac{V}{V}\right] \] -

      - -
      -
      -
      - -
      -

      1.3 Cercalo

      -
      -

      -From the Cercalo documentation, we have the parameters shown on table 1. -

      - - +
      @@ -539,14 +467,14 @@ The Inductance and DC resistance of the two axis of the Cercalo have been measur

      -Let's first consider the horizontal direction and we try to model the Cercalo by a spring/mass/damper system (Fig. 5). +Let's first consider the horizontal direction and we try to model the Cercalo by a spring/mass/damper system (Fig. 3).

      -
      +

      mech_cercalo.png

      -

      Figure 5: 1 degree-of-freedom model of the Cercalo

      +

      Figure 3: 1 degree-of-freedom model of the Cercalo

      @@ -582,7 +510,7 @@ The current \(I\) is also proportional to the voltage at the output of the buffe

      Let's try to determine the equivalent mass and spring values. -From table 1, for the horizontal direction: +From table 1, for the horizontal direction: \[ \left| \frac{x}{I} \right|(0) = \left| \alpha \frac{x}{F} \right|(0) = 28.4\ \frac{mA}{deg} = 1.63\ \frac{A}{rad} \]

      @@ -643,18 +571,18 @@ This will be done using the Newport.
      -
      -

      1.4 Optical Setup

      +
      +

      1.3 Optical Setup

      -
      -

      1.5 Newport

      -
      +
      +

      1.4 Newport

      +

      -Parameters of the Newport are shown in Fig. 6. +Parameters of the Newport are shown in Fig. 4.

      -It's dynamics for small angle excitation is shown in Fig. 7. +It's dynamics for small angle excitation is shown in Fig. 5.

      @@ -666,50 +594,50 @@ And we have: \end{align*} -

      +

      newport_doc.png

      -

      Figure 6: Documentation of the Newport

      +

      Figure 4: Documentation of the Newport

      -
      +

      newport_gain.png

      -

      Figure 7: Transfer function of the Newport

      +

      Figure 5: Transfer function of the Newport

      -
      -

      1.6 4 quadrant Diode

      -
      +
      +

      1.5 4 quadrant Diode

      +

      -The front view of the 4 quadrant photo-diode is shown in Fig. 8. +The front view of the 4 quadrant photo-diode is shown in Fig. 6.

      -
      +

      4qd_naming.png

      -

      Figure 8: Front view of the 4QD

      +

      Figure 6: Front view of the 4QD

      -Each of the photo-diode is amplified using a 4-channel amplifier as shown in Fig. 9. +Each of the photo-diode is amplified using a 4-channel amplifier as shown in Fig. 7.

      -
      +

      4qd_amplifier.png

      -

      Figure 9: Wiring of the amplifier. The amplifier is located on the bottom right of the board

      +

      Figure 7: Wiring of the amplifier. The amplifier is located on the bottom right of the board

      -
      -

      1.7 ADC/DAC

      -
      +
      +

      1.6 ADC/DAC

      +

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

      @@ -728,60 +656,16 @@ with \(\Delta V\) the total range of the ADC, \(n\) its number of bits, \(q\) th
      -
      -

      2 Identification

      +
      +

      2 Identification of the system dynamics

      - +

      -

      -All the files (data and Matlab scripts) are accessible here. +In this section, we seek to identify all the blocks as shown in Fig. 1.

      -
      -
      -
      -

      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

      -
      Table 1: Cercalo Parameters
      @@ -869,17 +753,1036 @@ Discrete-time zero/pole/gain model.
      +
      +

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

      + +
      +
      +
      +

      2.1 Calibration of the 4 Quadrant Diode

      +
      +

      +Prior to any dynamic identification, we would like to be able to determine the meaning of the 4 quadrant diode measurement. +For instance, instead of obtaining transfer function in [V/V] from the input of the cercalo to the measurement voltage of the 4QD, we would like to obtain the transfer function in [rad/V]. +This will give insight to physical interpretation. +

      + +

      +To calibrate the 4 quadrant photo-diode, we can use the metrology included in the Newport. +We can choose precisely the angle of the Newport mirror and see what is the value measured by the 4 Quadrant Diode. +We then should be able to obtain the "gain" of the 4QD in [V/rad]. +

      +
      + +
      +

      2.1.1 Input / Output data

      +
      +

      +The identification data is loaded +

      +
      +
      uh = load('mat/data_cal_pd_h.mat', 't', 'Vph', 'Vpv', 'Vnh');
      +uv = load('mat/data_cal_pd_v.mat', 't', 'Vph', 'Vpv', 'Vnv');
      +
      +
      + +

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

      +
      +
      t0 = 1;
      +
      +uh.Vph(uh.t<t0) = [];
      +uh.Vpv(uh.t<t0) = [];
      +uh.Vnh(uh.t<t0) = [];
      +uh.t(uh.t<t0)   = [];
      +uh.t = uh.t - uh.t(1); % We start at t=0
      +
      +t0 = 1;
      +
      +uv.Vph(uv.t<t0) = [];
      +uv.Vpv(uv.t<t0) = [];
      +uv.Vnv(uv.t<t0) = [];
      +uv.t(uv.t<t0) = [];
      +uv.t = uv.t - uv.t(1); % We start at t=0
      +
      +
      + + +
      +

      calib_4qd_h.png +

      +

      Figure 8: Identification signals when exciting the horizontal direction (png, pdf)

      +
      + + + +
      +

      calib_4qd_v.png +

      +

      Figure 9: Identification signals when exciting in the vertical direction (png, pdf)

      +
      -
      -

      2.3 Huddle Test

      +
      +

      2.1.2 Linear Regression to obtain the gain of the 4QD

      +
      +

      +We plot the angle of mirror +

      + +

      +Gain of the Newport metrology in [rad/V]. +

      +
      +
      gn0 = 2.62e-3;
      +
      +
      + +

      +The angular displacement of the beam is twice the angular displacement of the Newport mirror. +

      + +

      +We do a linear regression +\[ y = a x + b \] +where: +

      +
        +
      • \(y\) is the measured voltage of the 4QD in [V]
      • +
      • \(x\) is the beam angle (twice the mirror angle) in [rad]
      • +
      • \(a\) is the identified gain of the 4QD in [rad/V]
      • +
      + +

      +The linear regression is shown in Fig. 10. +

      + +
      +
      bh = [ones(size(uh.Vnh)) 2*gn0*uh.Vnh]\uh.Vph;
      +bv = [ones(size(uv.Vnv)) 2*gn0*uv.Vnv]\uv.Vpv;
      +
      +
      + + +
      +

      4qd_linear_reg.png +

      +

      Figure 10: Linear Regression (png, pdf)

      +
      + +

      +Thus, we obtain the "gain of the 4 quadrant photo-diode as shown on table 2. +

      + + + + +++ ++ + + + + + + + + + + + + +
      Table 2: Identified Gain of the 4 quadrant diode
      Horizontal [V/rad]Vertical [V/rad]
      -31.036.3
      + +
      +
      Gd = tf([bh(2) 0 ;
      +         0     bv(2)]);
      +
      +
      + +

      +We obtain: +

      +\begin{align*} + \frac{V_{qd,h}}{\alpha_{0,h}} &\approx 0.032\ \left[ \frac{rad}{V} \right] \\ + &\approx 32.3\ \left[ \frac{\mu rad}{mV} \right] +\end{align*} + +\begin{align*} + \frac{V_{qd,v}}{\alpha_{0,v}} &\approx 0.028\ \left[ \frac{rad}{V} \right] \\ + &\approx 27.6\ \left[ \frac{\mu rad}{mV} \right] +\end{align*} +
      +
      +
      + +
      +

      2.2 Identification of the Cercalo Impedance, Current Amplifier and Voltage Amplifier dynamics

      +
      +

      +We wish here to determine \(G_i\) and \(G_a\) shown in Fig. 1. +

      + +

      +We ignore the electro-mechanical coupling. +

      +
      + +
      +

      2.2.1 Electrical Schematic

      +
      +

      +The schematic of the electrical circuit used to drive the Cercalo is shown in Fig. 11. +

      + + +
      +

      cercalo_amplifier.png +

      +

      Figure 11: Current Amplifier Schematic

      +
      + +

      +The elements are: +

      +
        +
      • \(U_c\): the voltage generated by the DAC
      • +
      • BUF: is a unity-gain open-loop buffer that allows to increase the output current
      • +
      • \(R\): a chosen resistor that will determine the gain of the current amplifier
      • +
      • \(L_c\): inductor present in the Cercalo
      • +
      • \(R_c\): resistance of the inductor
      • +
      • \(\tilde{V}_c\): voltage measured across the Cercalo's inductor
      • +
      • \(V_c\): amplified voltage measured across the Cercalo's inductor
      • +
      • \(I_c\) is the current going through the Cercalo's inductor
      • +
      + + +

      +The values of the components have been measured for the horizontal and vertical directions: +

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

      +Let's first determine the transfer function from \(U_c\) to \(I_c\). +

      + +

      +We have that: +\[ U_c = (R + R_c) I_c + L_c s I_c \] +

      + +

      +Thus: +

      +\begin{align} + G_i(s) &= \frac{I_c}{U_c} \\ + &= \frac{1}{(R + R_c) + L_c s} \\ + &= \frac{G_{i,0}}{1 + s/\omega_0} +\end{align} +

      +with +

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

      +Now, determine the transfer function from \(I_c\) to \(\tilde{V}_c\): +\[ \tilde{V}_C = R_c I_c + L_c s I_c \] +Thus: +

      +\begin{align} + Z_c(s) &= \frac{\tilde{V}_c}{I_c} \\ + &= R_c + L_c s +\end{align} + + +

      +Finally, the transfer function of the voltage amplifier \(G_a\) is simply a low pass filter: +

      +\begin{align} + G_a(s) &= \frac{V_c}{\tilde{V}_c} \\ + &= \frac{G_{a,0}}{1 + s/\omega_c} +\end{align} +

      +with +

      +
        +
      • \(G_{a,0}\) is the gain 1000 (60dB)
      • +
      • \(\omega_c\) is the cut-off frequency of the voltage amplifier set to 1000Hz
      • +
      +
      +
      + +
      +

      2.2.2 Theoretical Transfer Functions

      +
      +

      +The values of the components in the current amplifier have been measured. +

      +
      +
      Rh = 41; % [Ohm]
      +Lch = 0.1e-3; % [H]
      +Rch = 9.3; % [Ohm]
      +
      +Rv = 41; % [Ohm]
      +Lcv = 0.1e-3; % [H]
      +Rcv = 8.3; % [Ohm]
      +
      +
      + +\begin{align*} + G_i(s) &= \frac{1}{(R + R_c) + L_c s} \\ + Z_c(s) &= R_c + L_c s \\ + G_a(s) &= \frac{1000}{1 + s/\omega_c} +\end{align*} + +
      +
      Gi = blkdiag(1/(Rh + Rch + Lch * s), 1/(Rv + Rcv + Lcv * s));
      +Zc = blkdiag(Rch+Lch*s, Rcv+Lcv*s);
      +Ga = blkdiag(1000/(1 + s/2/pi/1000), 1000/(1 + s/2/pi/1000));
      +
      +
      + + +
      +

      current_amplifier_tf.png +

      +

      Figure 12: Transfer function for the current amplifier (png, pdf)

      +
      + +
      +

      +Over the frequency band of interest, the current amplifier transfer function \(G_i\) can be considered as constant. +This is the same for the impedance \(Z_c\). +

      + +
      + +
      +
      Gi = tf(blkdiag(1/(Rh + Rch), 1/(Rv + Rcv)));
      +Zc = tf(blkdiag(Rch, Rcv));
      +
      +
      +
      +
      + +
      +

      2.2.3 Identified Transfer Functions

      +
      +

      +Noise is generated using the DAC (\([U_{c,h}\ U_{c,v}]\)) and we measure the output of the voltage amplifier \([V_{c,h}, V_{c,v}]\). +From that, we should be able to identify \(G_a Z_c G_i\). +

      + +

      +The identification data is loaded. +

      +
      +
      uh = load('mat/data_uch.mat', 't', 'Uch', 'Vch');
      +uv = load('mat/data_ucv.mat', 't', 'Ucv', 'Vcv');
      +
      +
      + +

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

      +
      +
      win = hanning(ceil(1*fs));
      +[GaZcGi_h, f] = tfestimate(uh.Uch, uh.Vch, win, [], [], fs);
      +[GaZcGi_v, ~] = tfestimate(uv.Ucv, uv.Vcv, win, [], [], fs);
      +
      +
      + + +
      +

      current_amplifier_comp_theory_id.png +

      +

      Figure 13: Identified and Theoretical Transfer Function \(G_a G_i\) (png, pdf)

      +
      + +

      +There is a gain mismatch, that is probably due to bad identification of the inductance and resistance measurement of the cercalo inductors. +Thus, we suppose \(G_a\) is perfectly known (the gain and cut-off frequency of the voltage amplifier is very accurate) and that \(G_i\) is also well determined as it mainly depends on the resistor used in the amplifier that is well measured. +

      +
      +
      Gi_resp_h = abs(GaZcGi_h)./squeeze(abs(freqresp(Ga(1,1)*Zc(1,1), f, 'Hz')));
      +Gi_resp_v = abs(GaZcGi_v)./squeeze(abs(freqresp(Ga(2,2)*Zc(2,2), f, 'Hz')));
      +Gi = tf(blkdiag(mean(Gi_resp_h(f>20 & f<200)), mean(Gi_resp_v(f>20 & f<200))));
      +
      +
      + + +
      +

      current_amplifier_comp_theory_id_bis.png +

      +

      Figure 14: Identified and Theoretical Transfer Function \(G_a G_i\) (png, pdf)

      +
      + +

      +Finally, we have the following transfer functions: +

      +
      +ans = filepath;
      +if ischar(ans), fid = fopen('/tmp/babel-ZKMGJu/matlab-FA7h5L', 'w'); fprintf(fid, '%s\n', ans); fclose(fid);
      +else, dlmwrite('/tmp/babel-ZKMGJu/matlab-FA7h5L', ans, '\t')
      +end
      +'org_babel_eoe'
      +Gi,Zc,Ga
      +'org_babel_eoe'
      +ans = filepath;
      +if ischar(ans), fid = fopen('/tmp/babel-ZKMGJu/matlab-FA7h5L', 'w'); fprintf(fid, '%s\n', ans); fclose(fid);
      +else, dlmwrite('/tmp/babel-ZKMGJu/matlab-FA7h5L', ans, '\t')
      +end
      +'org_babel_eoe'
      +ans =
      +    'org_babel_eoe'
      +Gi,Zc,Ga
      +
      +Gi =
      +
      +  From input 1 to output...
      +   1:  0.01275
      +
      +   2:  0
      +
      +  From input 2 to output...
      +   1:  0
      +
      +   2:  0.01382
      +
      +Static gain.
      +
      +
      +Zc =
      +
      +  From input 1 to output...
      +   1:  9.3
      +
      +   2:  0
      +
      +  From input 2 to output...
      +   1:  0
      +
      +   2:  8.3
      +
      +Static gain.
      +
      +
      +Ga =
      +
      +  From input 1 to output...
      +       6.2832e+06
      +   1:  ----------
      +        (s+6283)
      +
      +   2:  0
      +
      +  From input 2 to output...
      +   1:  0
      +
      +       6.2832e+06
      +   2:  ----------
      +        (s+6283)
      +
      +Continuous-time zero/pole/gain model.
      +
      +
      +
      +
      + +
      +

      2.3 Identification of the Cercalo Dynamics

      +We now wish to identify the dynamics of the Cercalo identified by \(G_c\) on the block diagram in Fig. 1. +

      + +

      +To do so, we inject some noise at the input of the current amplifier \([U_{c,h},\ U_{c,v}]\) (one input after the other) and we measure simultaneously the output of the 4QD \([V_{p,h},\ V_{p,v}]\). +

      + +

      +The transfer function obtained will be \(G_c G_i\), and because we have already identified \(G_i\), we can obtain \(G_c\) by multiplying the obtained transfer function matrix by \({G_i}^{-1}\). +

      +
      + +
      +

      2.3.1 Input / Output data

      +
      +

      +The identification data is loaded +

      +
      +
      uh = load('mat/data_uch.mat', 't', 'Uch', 'Vph', 'Vpv');
      +uv = load('mat/data_ucv.mat', 't', 'Ucv', 'Vph', 'Vpv');
      +
      +
      + +

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

      +
      +
      t0 = 1;
      +
      +uh.Uch(uh.t<t0) = [];
      +uh.Vph(uh.t<t0) = [];
      +uh.Vpv(uh.t<t0) = [];
      +uh.t(uh.t<t0)   = [];
      +uh.t = uh.t - uh.t(1); % We start at t=0
      +
      +t0 = 1;
      +
      +uv.Ucv(uv.t<t0) = [];
      +uv.Vph(uv.t<t0) = [];
      +uv.Vpv(uv.t<t0) = [];
      +uv.t(uv.t<t0)   = [];
      +
      +uv.t = uv.t - uv.t(1); % We start at t=0
      +
      +
      + + +
      +

      identification_uh.png +

      +

      Figure 15: Identification signals when exciting the horizontal direction (png, pdf)

      +
      + + + +
      +

      identification_uv.png +

      +

      Figure 16: Identification signals when exciting in the vertical direction (png, pdf)

      +
      +
      +
      + +
      +

      2.3.2 Coherence

      +
      +

      +The window used for the spectral analysis is an hanning windows with temporal size equal to 1 second. +

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

      coh_cercalo.png +

      +

      Figure 17: Coherence (png, pdf)

      +
      +
      +
      + +
      +

      2.3.3 Estimation of the Frequency Response Function Matrix

      +
      +

      +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);
      +
      +
      + + +
      +

      frf_cercalo_gain.png +

      +

      Figure 18: Frequency Response Matrix (png, pdf)

      +
      + + +
      +

      frf_cercalo_phase.png +

      +

      Figure 19: Frequency Response MatrixPhase (png, pdf)

      +
      +
      +
      + +
      +

      2.3.4 Time Delay

      +
      +

      +Now, we would like to remove the time delay included in the FRF prior to the model extraction. +

      + +

      +Estimation of the time delay: +

      +
      +
      Ts_delay = Ts; % [s]
      +
      +G_delay = tf(1, 1, 'InputDelay', Ts_delay);
      +
      +G_delay_resp = squeeze(freqresp(G_delay, f, 'Hz'));
      +
      +
      + +

      +We then remove the time delay from the frequency response function. +

      +
      +
      tf_Uch_Vph = tf_Uch_Vph./G_delay_resp;
      +tf_Uch_Vpv = tf_Uch_Vpv./G_delay_resp;
      +tf_Ucv_Vph = tf_Ucv_Vph./G_delay_resp;
      +tf_Ucv_Vpv = tf_Ucv_Vpv./G_delay_resp;
      +
      +
      +
      +
      + +
      +

      2.3.5 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]
      +
      +
      + +

      +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. +Finally, we ignore data above some frequency. +

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

      +The weights are shown in Fig. 20. +

      + + +
      +

      weights_cercalo.png +

      +

      Figure 20: 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 21: Transfer Function Extraction of the FRF matrix (png, pdf)

      +
      + + + +
      +

      identification_matrix_fit_phase.png +

      +

      Figure 22: Transfer Function Extraction of the FRF matrix (png, pdf)

      +
      + +

      +And finally, we create the identified \(G_c\) matrix by multiplying by \({G_i}^{-1}\). +

      +
      +
      G_Uch_Vph = tf(minreal(ss(full(SER_Uch_Vph.A),SER_Uch_Vph.B,SER_Uch_Vph.C,SER_Uch_Vph.D)));
      +G_Ucv_Vph = tf(minreal(ss(full(SER_Ucv_Vph.A),SER_Ucv_Vph.B,SER_Ucv_Vph.C,SER_Ucv_Vph.D)));
      +G_Uch_Vpv = tf(minreal(ss(full(SER_Uch_Vpv.A),SER_Uch_Vpv.B,SER_Uch_Vpv.C,SER_Uch_Vpv.D)));
      +G_Ucv_Vpv = tf(minreal(ss(full(SER_Ucv_Vpv.A),SER_Ucv_Vpv.B,SER_Ucv_Vpv.C,SER_Ucv_Vpv.D)));
      +
      +Gc = [G_Uch_Vph, G_Ucv_Vph;
      +      G_Uch_Vpv, G_Ucv_Vpv]*inv(Gi);
      +
      +
      +
      +
      +
      + +
      +

      2.4 Identification of the Newport Dynamics

      +
      +

      +We here identify the transfer function from a reference sent to the Newport \([U_{n,h},\ U_{n,v}]\) to the measurement made by the 4QD \([V_{p,h},\ V_{p,v}]\). +

      + +

      +To do so, we inject noise to the Newport \([U_{n,h},\ U_{n,v}]\) and we record the 4QD measurement \([V_{p,h},\ V_{p,v}]\). +

      +
      + +
      +

      2.4.1 Input / Output data

      +
      +

      +The identification data is loaded +

      +
      +
      uh = load('mat/data_unh.mat', 't', 'Unh', 'Vph', 'Vpv');
      +uv = load('mat/data_unv.mat', 't', 'Unv', 'Vph', 'Vpv');
      +
      +
      + +

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

      +
      +
      t0 = 3;
      +
      +uh.Unh(uh.t<t0) = [];
      +uh.Vph(uh.t<t0) = [];
      +uh.Vpv(uh.t<t0) = [];
      +uh.t(uh.t<t0)   = [];
      +uh.t = uh.t - uh.t(1); % We start at t=0
      +
      +t0 = 1.5;
      +
      +uv.Unv(uv.t<t0) = [];
      +uv.Vph(uv.t<t0) = [];
      +uv.Vpv(uv.t<t0) = [];
      +uv.t(uv.t<t0)   = [];
      +
      +uv.t = uv.t - uv.t(1); % We start at t=0
      +
      +
      + + +
      +

      identification_unh.png +

      +

      Figure 23: Identification signals when exciting the horizontal direction (png, pdf)

      +
      + + +
      +

      identification_unv.png +

      +

      Figure 24: Identification signals when exciting in the vertical direction (png, pdf)

      +
      +
      +
      + +
      +

      2.4.2 Coherence

      +
      +

      +The window used for the spectral analysis is an hanning windows with temporal size equal to 1 second. +

      +
      +
      win = hanning(ceil(1*fs));
      +
      +
      + +
      +
      [coh_Unh_Vph, f] = mscohere(uh.Unh, uh.Vph, win, [], [], fs);
      +[coh_Unh_Vpv, ~] = mscohere(uh.Unh, uh.Vpv, win, [], [], fs);
      +[coh_Unv_Vph, ~] = mscohere(uv.Unv, uv.Vph, win, [], [], fs);
      +[coh_Unv_Vpv, ~] = mscohere(uv.Unv, uv.Vpv, win, [], [], fs);
      +
      +
      + + +
      +

      id_newport_coherence.png +

      +

      Figure 25: Coherence (png, pdf)

      +
      +
      +
      + +
      +

      2.4.3 Estimation of the Frequency Response Function Matrix

      +
      +

      +We compute an estimate of the transfer functions. +

      +
      +
      [tf_Unh_Vph, f] = tfestimate(uh.Unh, uh.Vph, win, [], [], fs);
      +[tf_Unh_Vpv, ~] = tfestimate(uh.Unh, uh.Vpv, win, [], [], fs);
      +[tf_Unv_Vph, ~] = tfestimate(uv.Unv, uv.Vph, win, [], [], fs);
      +[tf_Unv_Vpv, ~] = tfestimate(uv.Unv, uv.Vpv, win, [], [], fs);
      +
      +
      + + +
      +

      frf_newport_gain.png +

      +

      Figure 26: Frequency Response Matrix (png, pdf)

      +
      + + +
      +

      frf_newport_phase.png +

      +

      Figure 27: Frequency Response Matrix Phase (png, pdf)

      +
      +
      +
      + +
      +

      2.4.4 Time Delay

      +
      +

      +Now, we would like to remove the time delay included in the FRF prior to the model extraction. +

      + +

      +Estimation of the time delay: +

      +
      +
      Ts_delay = 0.0005; % [s]
      +
      +G_delay = tf(1, 1, 'InputDelay', Ts_delay);
      +
      +G_delay_resp = squeeze(freqresp(G_delay, f, 'Hz'));
      +
      +
      + +

      +We then remove the time delay from the frequency response function. +

      + +
      +

      time_delay_newport.png +

      +

      Figure 28: Phase change due to time-delay in the Newport dynamics (png, pdf)

      +
      +
      +
      + +
      +

      2.4.5 Extraction of a transfer function matrix

      +
      +

      +From Fig. 26, it seems reasonable to model the Newport dynamics as diagonal and constant. +

      + +
      +
      Gn = blkdiag(tf(mean(abs(tf_Unh_Vph(f>10 & f<100)))), tf(mean(abs(tf_Unv_Vpv(f>10 & f<100)))));
      +
      +
      +
      +
      +
      + +
      +

      2.5 Full System

      +
      +

      +We now have identified: +

      +
        +
      • \(G_i\)
      • +
      • \(G_a\)
      • +
      • \(G_c\)
      • +
      • \(G_n\)
      • +
      • \(G_d\)
      • +
      + +

      +We name the input and output of each transfer function: +

      +
      +
      Gi.InputName = {'Uch', 'Ucv'};
      +Gi.OutputName = {'Ich', 'Icv'};
      +
      +Zc.InputName = {'Ich', 'Icv'};
      +Zc.OutputName = {'Vtch', 'Vtcv'};
      +
      +Ga.InputName = {'Vtch', 'Vtcv'};
      +Ga.OutputName = {'Vch', 'Vcv'};
      +
      +Gc.InputName = {'Ich', 'Icv'};
      +Gc.OutputName = {'Vpch', 'Vpcv'};
      +
      +Gn.InputName = {'Unh', 'Unv'};
      +Gn.OutputName = {'Vpnh', 'Vpnv'};
      +
      +Gd.InputName = {'Rh', 'Rv'};
      +Gd.OutputName = {'Vph', 'Vpv'};
      +
      +
      + +
      +
      Sh = sumblk('Vph = Vpch + Vpnh');
      +Sv = sumblk('Vpv = Vpcv + Vpnv');
      +
      +
      + +
      +
      inputs  = {'Uch', 'Ucv', 'Unh', 'Unv'};
      +outputs = {'Vch', 'Vcv', 'Ich', 'Icv', 'Rh', 'Rv', 'Vph', 'Vpv'};
      +
      +sys = connect(Gi, Zc, Ga, Gc, Gn, inv(Gd), Sh, Sv, inputs, outputs);
      +
      +
      + +

      +The file mat/plant.mat is accessible here. +

      +
      +
      save('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd');
      +
      +
      +
      +
      +
      + +
      +

      3 Active Damping

      +
      +
      +
      +

      3.1 Load Plant

      +
      +
      +
      load('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd');
      +
      +
      +
      +
      + +
      +

      3.2 Test

      +
      +
      +
      bode(sys({'Vch', 'Vcv'}, {'Uch', 'Ucv'}));
      +
      +
      + +
      +
      Kppf = blkdiag(-10000/s, tf(0));
      +
      +Kppf.InputName  = {'Vch', 'Vcv'};
      +Kppf.OutputName = {'Uch', 'Ucv'};
      +
      +
      + +
      +
      inputs  = {'Uch', 'Ucv', 'Unh', 'Unv'};
      +outputs = {'Ich', 'Icv', 'Rh', 'Rv', 'Vph', 'Vpv'};
      +
      +sys_cl = connect(sys, Kppf, inputs, outputs);
      +
      +figure; bode(sys_cl({'Vph', 'Vpv'}, {'Uch', 'Ucv'}), sys({'Vph', 'Vpv'}, {'Uch', 'Ucv'}))
      +
      +
      +
      +
      +
      + +
      +

      4 TODO Huddle Test

      +
      +

      We load the data taken during the Huddle Test.

      -
      load('mat/data_test.mat', ...
      +
      load('mat/data_huddle_test.mat', ...
            't', 'Uch', 'Ucv', ...
            'Unh', 'Unv', ...
            'Vph', 'Vpv', ...
      @@ -958,295 +1861,9 @@ xlim(
      -

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

      -
      - - - -
      -

      identification_uv.png -

      -

      Figure 11: 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 12: 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 13: 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 14: 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 15: Transfer Function Extraction of the FRF matrix (png, pdf)

      -
      - - - -
      -

      identification_matrix_fit_phase.png -

      -

      Figure 16: 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 Calibration of the 4 Quadrant Diode

      -
      -
      -

      4 Plant Scaling

      -
      +
      +

      5 Plant Scaling

      +
      • measured noise
      • expected perturbations
      • @@ -1256,13 +1873,13 @@ G = [G_uh_xh, G_uv_xh;
      -
      -

      5 Plant Analysis

      -
      +
      +

      6 Plant Analysis

      +
      -
      -

      5.1 Load Plant

      -
      +
      +

      6.1 Load Plant

      +
      load('mat/plant.mat', 'G');
       
      @@ -1270,9 +1887,9 @@ G = [G_uh_xh, G_uv_xh;
      -
      -

      5.2 RGA-Number

      -
      +
      +

      6.2 RGA-Number

      +
      freqs = logspace(2, 4, 1000);
       G_resp = freqresp(G, freqs, 'Hz');
      @@ -1310,9 +1927,9 @@ V = zeros(
      -

      5.3 Rotation Matrix

      -
      +
      +

      6.3 Rotation Matrix

      +
      G0 = freqresp(G, 0);
       
      @@ -1321,9 +1938,9 @@ V = zeros( -

      6 Control Objective

      -
      +
      +

      7 Control Objective

      +

      The maximum expected stroke is \(y_\text{max} = 3mm \approx 5e^{-2} rad\) at \(1Hz\). The maximum wanted error is \(e_\text{max} = 10 \mu rad\). @@ -1344,16 +1961,85 @@ In terms of loop gain, this is equivalent to:

      -
      -

      7 Control Design

      +
      +

      8 Decentralized Control

      +
      +

      + +

      +

      +In this section, we try to implement a simple decentralized controller. +

      +
      +

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

      +
      -
      -

      8 Measurement of the non-repeatability

      +
      + +
      +

      8.1 Load Plant

      +
      +
      +
      load('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd');
      +
      +
      +
      +
      + +
      +

      8.2 Diagonal Controller

      +
      +

      +Using SISOTOOL, a diagonal controller is designed. +The two SISO loop gains are shown in Fig. 29. +

      +
      +
      Kh = -0.25598*(s+112)*(s^2 + 15.93*s + 6.686e06)/((s^2*(s+352.5)*(1+s/2/pi/2000)));
      +Kv = 10207*(s+55.15)*(s^2 + 17.45*s + 2.491e06)/(s^2*(s+491.2)*(s+7695));
      +
      +K = blkdiag(Kh, Kv);
      +K.InputName  = {'Rh', 'Rv'};
      +K.OutputName = {'Uch', 'Ucv'};
      +
      +
      + + +
      +

      diag_contr_loop_gain.png +

      +

      Figure 29: Loop Gain using the Decentralized Diagonal Controller (png, pdf)

      +
      + +

      +We then close the loop and we look at the transfer function from the Newport rotation signal to the beam angle (Fig. 30). +

      +
      +
      inputs  = {'Uch', 'Ucv', 'Unh', 'Unv'};
      +outputs = {'Vch', 'Vcv', 'Ich', 'Icv', 'Rh', 'Rv', 'Vph', 'Vpv'};
      +
      +sys_cl = connect(sys, -K, inputs, outputs);
      +
      +
      + + +
      +

      diag_contr_effect_newport.png +

      +

      Figure 30: Effect of the Newport rotation on the beam position when the loop is closed using the Decentralized Diagonal Controller (png, pdf)

      +
      +
      +
      +
      + +
      +

      9 Measurement of the non-repeatability

      Author: Dehaeze Thomas

      -

      Created: 2019-09-16 lun. 15:21

      +

      Created: 2019-09-17 mar. 15:53

      Validate

      diff --git a/index.org b/index.org index 09e482e..d70be4b 100644 --- a/index.org +++ b/index.org @@ -49,22 +49,28 @@ The block diagram of the setup to be controlled is shown in Fig. [[fig:block_dia \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[block] (Zc) at (GiGc|-Gn) {$Z_c$}; + \node[block, above=1.2 of Zc] (Ga) {$G_a$}; + \node[block, right= of adc] (Gd) {${G_{d}}^{-1}$}; \node[above, align=center] at (Gi.north) {Current\\Amplifier}; + \node[left, align=right] at (Zc.west) {Cercalo's\\Impedance}; + \node[left, align=right] at (Ga.west) {Voltage\\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[->] (dac.east) -- (Gi.west) node[above left]{$\begin{bmatrix}U_{c,h} \\ U_{c,v}\end{bmatrix}$} node[below left]{$[V]$}; + \draw[->] (Gi.east) -- (Gc.west) node[above left]{$\begin{bmatrix}I_{c,h} \\ I_{c,v}\end{bmatrix}$} node[below left]{$[A]$}; \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); + \draw[->] (GiGc)node[branch]{} -- (Zc.south); + \draw[->] (Zc.north) -- (Ga.south) node[below right]{$\begin{bmatrix}\tilde{V}_{c,h} \\ \tilde{V}_{c,v}\end{bmatrix}$} node[below left]{$[V]$}; + \draw[->] (Ga.north) -- ++(0, 1.2) node[below right]{$\begin{bmatrix}V_{c,h} \\ V_{c,v}\end{bmatrix}$} node[below left]{$[V]$}; + \draw[->] ($(Gn.north) + (0, 1.2)$) -- (Gn.north) node[above right]{$\begin{bmatrix}U_{n,h} \\ U_{n,v}\end{bmatrix}$} node[above left]{$[V]$}; + \draw[->] (add.east) -- (adc.west) node[above left]{$\begin{bmatrix}V_{p,h} \\ V_{p,v}\end{bmatrix}$} node[below left]{$[V]$}; + \draw[->] (adc.east) --node[midway, sloped]{$/$} (Gd.west); + \draw[->] (Gd.east) --node[midway, sloped]{$/$} ++(1, 0) node[above left]{$\begin{bmatrix} \theta_h \\ \theta_v \end{bmatrix}$} node[below left]{$[rad]$}; \end{tikzpicture} #+end_src @@ -74,17 +80,25 @@ The block diagram of the setup to be controlled is shown in Fig. [[fig:block_dia [[file:figs/cercalo_diagram_simplify.png]] The transfer functions in the system are: -- *Current Amplifier*: from the voltage set by the DAC to the voltage across the Cercalo inductors - \[ G_i = \begin{bmatrix} G_{i,h} & 0 \\ 0 & G_{i,v} \end{bmatrix} \] +- *Current Amplifier*: from the voltage set by the DAC to the current going to the Cercalo's inductors + \[ G_i = \begin{bmatrix} G_{i,h} & 0 \\ 0 & G_{i,v} \end{bmatrix} \text{ in } \left[ \frac{A}{V} \right] \] + \[ \begin{bmatrix} I_{c,h} \\ I_{c,v} \end{bmatrix} = G_i \begin{bmatrix} U_{c,h} \\ U_{c,v} \end{bmatrix} \] +- *Impedance of the Cercalo* that converts the current going to the cercalo to the voltage across the cercalo: + \[ Z_c = \begin{bmatrix} Z_{c,h} & 0 \\ 0 & Z_{c,v} \end{bmatrix} \text{ in } \left[ \frac{V}{A} \right] \] + \[ \begin{bmatrix} \tilde{V}_{c,h} \\ \tilde{V}_{c,v} \end{bmatrix} = Z_c \begin{bmatrix} I_{c,h} \\ I_{c,v} \end{bmatrix} \] - *Voltage Amplifier*: from the voltage across the Cercalo inductors to the measured voltage - \[ G_a = \begin{bmatrix} G_{a,h} & 0 \\ 0 & G_{a,v} \end{bmatrix} \] -- *Cercalo*: Transfer function from the Voltage across the cercalo inductors to the 4 quadrant measurement - \[ G_c = \begin{bmatrix} G_{\frac{V_{p,h}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,h}}{\tilde{U}_{c,v}}} \\ G_{\frac{V_{p,v}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,v}}{\tilde{U}_{c,v}}} \end{bmatrix} \] + \[ G_a = \begin{bmatrix} G_{a,h} & 0 \\ 0 & G_{a,v} \end{bmatrix} \text{ in } \left[ \frac{V}{V} \right] \] + \[ \begin{bmatrix} V_{c,h} \\ V_{c,v} \end{bmatrix} = G_a \begin{bmatrix} \tilde{V}_{c,h} \\ \tilde{V}_{c,v} \end{bmatrix} \] +- *Cercalo*: Transfer function from the current going through the cercalo inductors to the 4 quadrant measurement + \[ G_c = \begin{bmatrix} G_{\frac{V_{p,h}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,h}}{\tilde{U}_{c,v}}} \\ G_{\frac{V_{p,v}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,v}}{\tilde{U}_{c,v}}} \end{bmatrix} \text{ in } \left[ \frac{V}{A} \right] \] + \[ \begin{bmatrix} V_{p,h} \\ V_{p,v} \end{bmatrix} = G_c \begin{bmatrix} I_{c,h} \\ I_{c,v} \end{bmatrix} \] - *Newport* Transfer function from the command signal of the Newport to the 4 quadrant measurement - \[ G_n = \begin{bmatrix} G_{\frac{V_{p,h}}{U_{n,h}}} & G_{\frac{V_{p,h}}{U_{n,v}}} \\ G_{\frac{V_{p,v}}{U_{n,h}}} & G_{\frac{V_{n,v}}{U_{n,v}}} \end{bmatrix} \] + \[ G_n = \begin{bmatrix} G_{\frac{V_{p,h}}{U_{n,h}}} & G_{\frac{V_{p,h}}{U_{n,v}}} \\ G_{\frac{V_{p,v}}{U_{n,h}}} & G_{\frac{V_{n,v}}{U_{n,v}}} \end{bmatrix} \text{ in } \left[ \frac{V}{V} \right] \] + \[ \begin{bmatrix} V_{p,h} \\ V_{p,v} \end{bmatrix} = G_c \begin{bmatrix} V_{n,h} \\ V_{n,v} \end{bmatrix} \] +- *4 Quadrant Diode*: the gain of the 4 quadrant diode in [V/rad] is inverse in order to obtain the physical angle of the beam + \[ G_d = \begin{bmatrix} G_{d,h} & 0 \\ 0 & G_{d,v} \end{bmatrix} \text{ in } \left[\frac{V}{rad}\right] \] The block diagram with each transfer function is shown in Fig. [[fig:block_diagram]]. - #+begin_src latex :file cercalo_diagram.pdf :exports results \begin{tikzpicture} \node[DAC] (dac) at (0, 0) {}; @@ -94,7 +108,9 @@ The block diagram with each transfer function is shown in Fig. [[fig:block_diagr \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[block] (Zc) at (GiGc|-Gn) {$\begin{bmatrix} Z_{c,h} & 0 \\ 0 & Z_{c,v} \end{bmatrix}$}; + \node[block, above=1.2 of Zc] (Ga) {$\begin{bmatrix} G_{a,h} & 0 \\ 0 & G_{a,v} \end{bmatrix}$}; + \node[block, right= of adc] (Gd) {$\begin{bmatrix} G_{d,h}^{-1} & 0 \\ 0 & G_{d,v}^{-1} \end{bmatrix}$}; \node[above, align=center] at (Gi.north) {Current\\Amplifier}; \node[above, align=center] at (Gc.north) {Cercalo}; @@ -102,14 +118,16 @@ The block diagram with each transfer function is shown in Fig. [[fig:block_diagr \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[->] (Gi.east) -- (Gc.west) node[above left]{$\begin{bmatrix}I_{c,h} \\ I_{c,v}\end{bmatrix}$}; \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[->] (GiGc)node[branch]{} -- (Zc.south); + \draw[->] (Zc.north) -- (Ga.south) node[below right]{$\begin{bmatrix}\tilde{V}_{c,h} \\ \tilde{V}_{c,v}\end{bmatrix}$}; + \draw[->] (Ga.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); + \draw[->] (adc.east) --node[midway, sloped]{$/$} (Gd.west); + \draw[->] (Gd.east) --node[midway, sloped]{$/$} ++(1, 0) node[above left]{$\begin{bmatrix} \theta_h \\ \theta_v \end{bmatrix}$} node[below left]{$[rad]$}; \end{tikzpicture} #+end_src @@ -118,129 +136,6 @@ The block diagram with each transfer function is shown in Fig. [[fig:block_diagr #+RESULTS: [[file:figs/cercalo_diagram.png]] -** Amplifier for the Cercalo -#+begin_src latex :file cercalo_amplifier.pdf :exports results - \begin{circuitikz}[] - \ctikzset{bipoles/length=1.0cm} - \draw - (0, -2) node[ground]{} to[vco, V=$U_c$] (0, 0) - to [amp, t={1},i^>=$i_c$, l=BUF] ++(2, 0) - to [R=$R$] ++(2, 0) coordinate(A) - to [L=$L_c$] ++(0, -1) - to [R=$R_c$] ++(0, -1) coordinate(B) node[ground]{} - ; - \draw[->] ($(B)+(0.8, 0)$) -- node[midway, right]{$V_c$} ($(A)+(0.8, 0)$); - \end{circuitikz} -#+end_src - -#+RESULTS: -[[file:figs/cercalo_amplifier.png]] - -The value of the resistor in series with the buffer have been measured for both axis. -- $R_h = 41 \Omega$ -- $L_{c,h} = 0.1 mH$ -- $R_{c,h} = 9.3 \Omega$ -- $R_v = 41 \Omega$ -- $L_{c,v} = 0.1 mH$ -- $R_{c,v} = 8.3 \Omega$ - -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 -- $G_0 = \frac{R_c}{R + R_c}$ -- $\omega_0 = \frac{R + R_c}{L_c}$ - -Let's verify that the electrical circuit behaves as a constant current amplifier in the frequency band of interest. - -#+begin_src matlab - Rh = 41; % [Ohm] - Lch = 0.1e-3; % [H] - Rch = 9.3; % [Ohm] - - Rv = 41; % [Ohm] - Lcv = 0.1e-3; % [H] - Rcv = 8.3; % [Ohm] -#+end_src - -#+begin_src matlab - 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); -#+end_src - -#+begin_src matlab - Gih0 = freqresp(Gih, 0); - Gvh0 = freqresp(Gvh, 0); - Giv0 = freqresp(Giv, 0); - Gvv0 = freqresp(Gvv, 0); -#+end_src - -#+begin_src matlab :exports none - freqs = logspace(2, 5, 1000); - - figure; - - ax1 = subplot(1, 2, 1); - hold on; - plot(freqs, abs(squeeze(freqresp(Gvh, freqs, 'Hz'))), 'DisplayName', ['$\frac{\tilde{U}_{c,h}}{U_{c,h}}$, ', sprintf('$G_{v,h}(0)=%.3f V/V$', Gvh0)]) - plot(freqs, abs(squeeze(freqresp(Gvv, freqs, 'Hz'))), 'DisplayName', ['$\frac{\tilde{U}_{c,v}}{U_{c,v}}$, ', sprintf('$G_{v,v}(0)=%.3f V/V$', Gvv0)]) - hold off; - set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); - xlabel('Frequency [Hz]'); ylabel('Amplitude [V/V]'); - legend('location', 'northwest'); - - ax2 = subplot(1, 2, 2); - hold on; - plot(freqs, abs(squeeze(freqresp(Gih, freqs, 'Hz'))), 'DisplayName', ['$\frac{i_{c,h}}{U_{c,h}}$, ', sprintf('$G_{i,h}(0) = %.3f A/V$', Gih0)]) - plot(freqs, abs(squeeze(freqresp(Giv, freqs, 'Hz'))), 'DisplayName', ['$\frac{i_{c,v}}{U_{c,v}}$, ', sprintf('$G_{i,v}(0) = %.3f A/V$', Giv0)]) - hold off; - set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); - xlabel('Frequency [Hz]'); ylabel('Amplitude [A/V]'); - legend('location', 'southwest'); - - linkaxes([ax1, ax2], 'x'); - xlim([freqs(1), freqs(end)]); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/current_amplifier_tf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:current_amplifier_tf -#+CAPTION: Transfer function for the current amplifier ([[./figs/current_amplifier_tf.png][png]], [[./figs/current_amplifier_tf.pdf][pdf]]) -[[file:figs/current_amplifier_tf.png]] - - -#+begin_important -The current amplifier has a constant gain over all the frequency band of interest. -\[ G_i(s) = \begin{bmatrix} 0.02 & 0 \\ 0 & 0.02 \end{bmatrix}\quad \left[\frac{A}{V}\right] \] -\[ G_a(s) = \begin{bmatrix} 0.185 & 0 \\ 0 & 0.168 \end{bmatrix} \left[\frac{V}{V}\right] \] -#+end_important - ** Cercalo From the Cercalo documentation, we have the parameters shown on table [[tab:cercalo_parameters]]. @@ -359,9 +254,9 @@ This will be done using the Newport. ** Optical Setup ** Newport -Parameters of the Newport are shown in Fig. [[fig:Newport_doc]]. +Parameters of the Newport are shown in Fig. [[fig:newport_doc]]. -It's dynamics for small angle excitation is shown in Fig. [[fig:Newport_gain]]. +It's dynamics for small angle excitation is shown in Fig. [[fig:newport_gain]]. And we have: \begin{align*} @@ -369,13 +264,13 @@ And we have: G_{n, v}(0) &= 2.62 \cdot 10^{-3}\ \frac{rad}{V} \end{align*} -#+name: fig:Newport_doc +#+name: fig:newport_doc #+caption: Documentation of the Newport -[[file:figs/Newport_doc.png]] +[[file:figs/newport_doc.png]] -#+name: fig:Newport_gain +#+name: fig:newport_gain #+caption: Transfer function of the Newport -[[file:figs/Newport_gain.png]] +[[file:figs/newport_gain.png]] ** 4 quadrant Diode The front view of the 4 quadrant photo-diode is shown in Fig. [[fig:4qd_naming]]. @@ -425,71 +320,15 @@ Let's compute the theoretical noise of the ADC/DAC. \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. -* Identification of the Cercalo Dynamics +* Identification of the system dynamics :PROPERTIES: -:header-args:matlab+: :tangle matlab/plant_identification.m +:header-args:matlab+: :tangle matlab/cercalo_identification.m :header-args:matlab+: :comments org :mkdirp yes :END: -<> +<> ** Introduction :ignore: +In this section, we seek to identify all the blocks as shown in Fig. [[fig:block_diagram_simplify]]. -** ZIP file containing the data and matlab files :ignore: -#+begin_src bash :exports none :results none - if [ matlab/plant_identification.m -nt data/plant_identification.zip ]; then - cp matlab/plant_identification.m plant_identification.m; - zip data/plant_identification \ - mat/data_ux.mat \ - mat/data_uy.mat \ - plant_identification.m - rm plant_identification.m; - fi -#+end_src - -#+begin_note - All the files (data and Matlab scripts) are accessible [[file:data/plant_identification.zip][here]]. -#+end_note - -** Matlab Init :noexport:ignore: -#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) - <> -#+end_src - -#+begin_src matlab :exports none :results silent :noweb yes - <> -#+end_src - -#+begin_src matlab - fs = 1e4; - Ts = 1/fs; - freqs = logspace(1, 3, 1000); -#+end_src - -** Excitation Data -We generate white noise with the "random number" simulink block, and we filter that noise. - -#+begin_src matlab - Gi = (1)/(1+s/2/pi/100); -#+end_src - -#+begin_src matlab :results output replace - c2d(Gi, Ts, 'tustin') -#+end_src - -#+RESULTS: -#+begin_example -c2d(Gi, Ts, 'tustin') - -ans = - - 0.030459 (z+1) - -------------- - (z-0.9391) - -Sample time: 0.0001 seconds -Discrete-time zero/pole/gain model. -#+end_example - -** Signals | Signal | Name | Unit | |-----------------------------------------------------+-------+------| | Voltage Sent to Cercalo - Horizontal | =Uch= | [V] | @@ -506,7 +345,1518 @@ Discrete-time zero/pole/gain model. |-----------------------------------------------------+-------+------| | Attocube Measurement | =Va= | [m] | -** Huddle Test +** ZIP file containing the data and matlab files :ignore: +#+begin_src bash :exports none :results none + if [ matlab/cercalo_identification.m -nt data/cercalo_identification.zip ]; then + cp matlab/cercalo_identification.m cercalo_identification.m; + zip data/cercalo_identification \ + mat/data_cal_pd_h.mat \ + mat/data_cal_pd_v.mat \ + mat/data_uch.mat \ + mat/data_ucv.mat \ + mat/data_unh.mat \ + mat/data_unv.mat \ + cercalo_identification.m + rm cercalo_identification.m; + fi +#+end_src + +#+begin_note + All the files (data and Matlab scripts) are accessible [[file:data/cercalo_identification.zip][here]]. +#+end_note + +** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab + fs = 1e4; + Ts = 1/fs; + freqs = logspace(1, 3, 1000); +#+end_src + +** Calibration of the 4 Quadrant Diode +*** Introduction :ignore: +Prior to any dynamic identification, we would like to be able to determine the meaning of the 4 quadrant diode measurement. +For instance, instead of obtaining transfer function in [V/V] from the input of the cercalo to the measurement voltage of the 4QD, we would like to obtain the transfer function in [rad/V]. +This will give insight to physical interpretation. + +To calibrate the 4 quadrant photo-diode, we can use the metrology included in the Newport. +We can choose precisely the angle of the Newport mirror and see what is the value measured by the 4 Quadrant Diode. +We then should be able to obtain the "gain" of the 4QD in [V/rad]. + +*** Input / Output data +The identification data is loaded +#+begin_src matlab + uh = load('mat/data_cal_pd_h.mat', 't', 'Vph', 'Vpv', 'Vnh'); + uv = load('mat/data_cal_pd_v.mat', 't', 'Vph', 'Vpv', 'Vnv'); +#+end_src + +We remove the first seconds where the Cercalo is turned on. +#+begin_src matlab + t0 = 1; + + uh.Vph(uh.t> +#+end_src + +#+NAME: fig:calib_4qd_h +#+CAPTION: Identification signals when exciting the horizontal direction ([[./figs/calib_4qd_h.png][png]], [[./figs/calib_4qd_h.pdf][pdf]]) +[[file:figs/calib_4qd_h.png]] + + +#+begin_src matlab :exports none + figure; + ax1 = subplot(1, 2, 1); + plot(uv.t, uv.Vnv, 'DisplayName', '$Vn_v$'); + xlabel('Time [s]'); + ylabel('Amplitude [V]'); + + ax2 = subplot(1, 2, 2); + hold on; + plot(uv.t, uv.Vpv, 'DisplayName', '$Vp_v$'); + plot(uv.t, uv.Vph, 'DisplayName', '$Vp_h$'); + hold off; + xlabel('Time [s]'); + ylabel('Amplitude [V]'); + legend(); + + linkaxes([ax1,ax2],'x'); + xlim([uv.t(1), uv.t(end)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/calib_4qd_v.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:calib_4qd_v +#+CAPTION: Identification signals when exciting in the vertical direction ([[./figs/calib_4qd_v.png][png]], [[./figs/calib_4qd_v.pdf][pdf]]) +[[file:figs/calib_4qd_v.png]] + +*** Linear Regression to obtain the gain of the 4QD +We plot the angle of mirror + +Gain of the Newport metrology in [rad/V]. +#+begin_src matlab + gn0 = 2.62e-3; +#+end_src + +The angular displacement of the beam is twice the angular displacement of the Newport mirror. + +We do a linear regression +\[ y = a x + b \] +where: +- $y$ is the measured voltage of the 4QD in [V] +- $x$ is the beam angle (twice the mirror angle) in [rad] +- $a$ is the identified gain of the 4QD in [rad/V] + +The linear regression is shown in Fig. [[fig:4qd_linear_reg]]. + +#+begin_src matlab + bh = [ones(size(uh.Vnh)) 2*gn0*uh.Vnh]\uh.Vph; + bv = [ones(size(uv.Vnv)) 2*gn0*uv.Vnv]\uv.Vpv; +#+end_src + +#+begin_src matlab :exports none + figure; + ax1 = subplot(1, 2, 1); + hold on; + plot(2*gn0*uh.Vnh, uh.Vph, 'o', 'DisplayName', 'Exp. data'); + plot(2*gn0*[min(uh.Vnh) max(uh.Vnh)], 2*gn0*[min(uh.Vnh) max(uh.Vnh)].*bh(2) + bh(1), 'k--', 'DisplayName', sprintf('%.1e x + %.1e', bh(2), bh(1))) + hold off; + xlabel('$\alpha_{0,h}$ [rad]'); ylabel('$Vp_h$ [V]'); + legend(); + + ax2 = subplot(1, 2, 2); + hold on; + plot(2*gn0*uv.Vnv, uv.Vpv, 'o', 'DisplayName', 'Exp. data'); + plot(2*gn0*[min(uv.Vnv) max(uv.Vnv)], 2*gn0*[min(uv.Vnv) max(uv.Vnv)].*bv(2) + bv(1), 'k--', 'DisplayName', sprintf('%.1e x + %.1e', bv(2), bv(1))) + hold off; + xlabel('$\alpha_{0,v}$ [rad]'); ylabel('$Vp_v$ [V]'); + legend(); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/4qd_linear_reg.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:4qd_linear_reg +#+CAPTION: Linear Regression ([[./figs/4qd_linear_reg.png][png]], [[./figs/4qd_linear_reg.pdf][pdf]]) +[[file:figs/4qd_linear_reg.png]] + +Thus, we obtain the "gain of the 4 quadrant photo-diode as shown on table [[tab:gain_4qd]]. + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) + data2orgtable([bh(2), bv(2)], {}, {'Horizontal [V/rad]', 'Vertical [V/rad]'}, ' %.1f '); +#+end_src + +#+name: tab:gain_4qd +#+caption: Identified Gain of the 4 quadrant diode +#+RESULTS: +| Horizontal [V/rad] | Vertical [V/rad] | +|--------------------+------------------| +| -31.0 | 36.3 | + +#+begin_src matlab + Gd = tf([bh(2) 0 ; + 0 bv(2)]); +#+end_src + +We obtain: +\begin{align*} + \frac{V_{qd,h}}{\alpha_{0,h}} &\approx 0.032\ \left[ \frac{rad}{V} \right] \\ + &\approx 32.3\ \left[ \frac{\mu rad}{mV} \right] +\end{align*} + +\begin{align*} + \frac{V_{qd,v}}{\alpha_{0,v}} &\approx 0.028\ \left[ \frac{rad}{V} \right] \\ + &\approx 27.6\ \left[ \frac{\mu rad}{mV} \right] +\end{align*} + +** Identification of the Cercalo Impedance, Current Amplifier and Voltage Amplifier dynamics +*** Introduction :ignore: +We wish here to determine $G_i$ and $G_a$ shown in Fig. [[fig:block_diagram_simplify]]. + +We ignore the electro-mechanical coupling. + +*** Electrical Schematic +The schematic of the electrical circuit used to drive the Cercalo is shown in Fig. [[fig:current_amplifier]]. + +#+begin_src latex :file cercalo_amplifier.pdf :exports results + \begin{circuitikz}[] + \ctikzset{bipoles/length=1.0cm} + \draw + (0, -2) node[ground]{} to[vco, V=$U_c$] (0, 0) + to [amp, t={1},i^>=$I_c$, l=BUF] ++(2, 0) + to [R=$R$] ++(2, 0) coordinate(A) + to [L=$L_c$] ++(0, -1) + to [R=$R_c$] ++(0, -1) coordinate(B) node[ground]{} + ; + \draw (A) to [amp, i>^=$0$, l={60dB}] ++ (2, 0); + + \draw[->] ($(B)+(-0.4, 0)$) -- node[midway, left]{$\tilde{V}_c$} ($(A)+(-0.4, 0)$); + \draw[->] ($(B)+(2, 0)$) -- node[midway, left]{$V_c$} ($(A)+(2, 0)$); + \end{circuitikz} +#+end_src + +#+name: fig:current_amplifier +#+caption: Current Amplifier Schematic +#+RESULTS: +[[file:figs/cercalo_amplifier.png]] + +The elements are: +- $U_c$: the voltage generated by the DAC +- BUF: is a unity-gain open-loop buffer that allows to increase the output current +- $R$: a chosen resistor that will determine the gain of the current amplifier +- $L_c$: inductor present in the Cercalo +- $R_c$: resistance of the inductor +- $\tilde{V}_c$: voltage measured across the Cercalo's inductor +- $V_c$: amplified voltage measured across the Cercalo's inductor +- $I_c$ is the current going through the Cercalo's inductor + + +The values of the components have been measured for the horizontal and vertical directions: +- $R_h = 41 \Omega$ +- $L_{c,h} = 0.1 mH$ +- $R_{c,h} = 9.3 \Omega$ +- $R_v = 41 \Omega$ +- $L_{c,v} = 0.1 mH$ +- $R_{c,v} = 8.3 \Omega$ + +Let's first determine the transfer function from $U_c$ to $I_c$. + +We have that: +\[ U_c = (R + R_c) I_c + L_c s I_c \] + +Thus: +\begin{align} + G_i(s) &= \frac{I_c}{U_c} \\ + &= \frac{1}{(R + R_c) + L_c s} \\ + &= \frac{G_{i,0}}{1 + s/\omega_0} +\end{align} +with +- $G_{i,0} = \frac{1}{R + R_c}$ +- $\omega_0 = \frac{R + R_c}{L_c}$ + + +Now, determine the transfer function from $I_c$ to $\tilde{V}_c$: +\[ \tilde{V}_C = R_c I_c + L_c s I_c \] +Thus: +\begin{align} + Z_c(s) &= \frac{\tilde{V}_c}{I_c} \\ + &= R_c + L_c s +\end{align} + + +Finally, the transfer function of the voltage amplifier $G_a$ is simply a low pass filter: +\begin{align} + G_a(s) &= \frac{V_c}{\tilde{V}_c} \\ + &= \frac{G_{a,0}}{1 + s/\omega_c} +\end{align} +with +- $G_{a,0}$ is the gain 1000 (60dB) +- $\omega_c$ is the cut-off frequency of the voltage amplifier set to 1000Hz + +*** Theoretical Transfer Functions +The values of the components in the current amplifier have been measured. +#+begin_src matlab + Rh = 41; % [Ohm] + Lch = 0.1e-3; % [H] + Rch = 9.3; % [Ohm] + + Rv = 41; % [Ohm] + Lcv = 0.1e-3; % [H] + Rcv = 8.3; % [Ohm] +#+end_src + +\begin{align*} + G_i(s) &= \frac{1}{(R + R_c) + L_c s} \\ + Z_c(s) &= R_c + L_c s \\ + G_a(s) &= \frac{1000}{1 + s/\omega_c} +\end{align*} + +#+begin_src matlab + Gi = blkdiag(1/(Rh + Rch + Lch * s), 1/(Rv + Rcv + Lcv * s)); + Zc = blkdiag(Rch+Lch*s, Rcv+Lcv*s); + Ga = blkdiag(1000/(1 + s/2/pi/1000), 1000/(1 + s/2/pi/1000)); +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(1, 4, 1000); + + figure; + + ax1 = subplot(1, 3, 1); + hold on; + plot(freqs, abs(squeeze(freqresp(Gi(1,1), freqs, 'Hz'))), 'DisplayName', '$G_{i, h} = \frac{I_{c,h}}{U_{c,h}}$') + plot(freqs, abs(squeeze(freqresp(Gi(2,2), freqs, 'Hz'))), 'DisplayName', '$G_{i, v} = \frac{I_{c,v}}{U_{c,v}}$') + hold off; + set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Amplitude [A/V]'); + legend('location', 'northwest'); + ylim([0.01, 1]); + + ax2 = subplot(1, 3, 2); + hold on; + plot(freqs, abs(squeeze(freqresp(Zc(1,1), freqs, 'Hz'))), 'DisplayName', '$Z_{c, h} = \frac{\tilde{V}_{c,h}}{I_{c,h}}$') + plot(freqs, abs(squeeze(freqresp(Zc(2,2), freqs, 'Hz'))), 'DisplayName', '$Z_{c, v} = \frac{\tilde{V}_{c,v}}{I_{c,v}}$') + hold off; + set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Amplitude [V/A]'); + legend('location', 'southwest'); + ylim([1, 100]); + + ax3 = subplot(1, 3, 3); + hold on; + plot(freqs, abs(squeeze(freqresp(Ga(1,1), freqs, 'Hz'))), 'DisplayName', '$G_{a, h} = \frac{V_{c,h}}{\tilde{V}_{c,h}}$') + plot(freqs, abs(squeeze(freqresp(Ga(2,2), freqs, 'Hz'))), 'DisplayName', '$G_{a, v} = \frac{V_{c,v}}{\tilde{V}_{c,v}}$') + hold off; + set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Amplitude [V/V]'); + legend('location', 'southwest'); + ylim([10, 1000]); + + linkaxes([ax1, ax2, ax3], 'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/current_amplifier_tf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:current_amplifier_tf +#+CAPTION: Transfer function for the current amplifier ([[./figs/current_amplifier_tf.png][png]], [[./figs/current_amplifier_tf.pdf][pdf]]) +[[file:figs/current_amplifier_tf.png]] + +#+begin_important +Over the frequency band of interest, the current amplifier transfer function $G_i$ can be considered as constant. +This is the same for the impedance $Z_c$. +#+end_important + +#+begin_src matlab + Gi = tf(blkdiag(1/(Rh + Rch), 1/(Rv + Rcv))); + Zc = tf(blkdiag(Rch, Rcv)); +#+end_src + +*** Identified Transfer Functions +Noise is generated using the DAC ($[U_{c,h}\ U_{c,v}]$) and we measure the output of the voltage amplifier $[V_{c,h}, V_{c,v}]$. +From that, we should be able to identify $G_a Z_c G_i$. + +The identification data is loaded. +#+begin_src matlab + uh = load('mat/data_uch.mat', 't', 'Uch', 'Vch'); + uv = load('mat/data_ucv.mat', 't', 'Ucv', 'Vcv'); +#+end_src + +We remove the first seconds where the Cercalo is turned on. +#+begin_src matlab :exports none + t0 = 1; + + uh.Uch(uh.t> +#+end_src + +#+NAME: fig:current_amplifier_comp_theory_id +#+CAPTION: Identified and Theoretical Transfer Function $G_a G_i$ ([[./figs/current_amplifier_comp_theory_id.png][png]], [[./figs/current_amplifier_comp_theory_id.pdf][pdf]]) +[[file:figs/current_amplifier_comp_theory_id.png]] + +There is a gain mismatch, that is probably due to bad identification of the inductance and resistance measurement of the cercalo inductors. +Thus, we suppose $G_a$ is perfectly known (the gain and cut-off frequency of the voltage amplifier is very accurate) and that $G_i$ is also well determined as it mainly depends on the resistor used in the amplifier that is well measured. +#+begin_src matlab + Gi_resp_h = abs(GaZcGi_h)./squeeze(abs(freqresp(Ga(1,1)*Zc(1,1), f, 'Hz'))); + Gi_resp_v = abs(GaZcGi_v)./squeeze(abs(freqresp(Ga(2,2)*Zc(2,2), f, 'Hz'))); + Gi = tf(blkdiag(mean(Gi_resp_h(f>20 & f<200)), mean(Gi_resp_v(f>20 & f<200)))); +#+end_src + +#+begin_src matlab :exports none + GaZcGi_resp = abs(squeeze(freqresp(Ga*Zc*Gi, f, 'Hz'))); + + figure; + + ax1 = subplot(1, 2, 1); + hold on; + plot(f, abs(GaZcGi_h), 'k-', 'DisplayName', 'Identified') + plot(f, squeeze(GaZcGi_resp(1,1,:)), 'k--', 'DisplayName', 'Theoretical') + title('FRF $G_{i,h} Z_{c,h} G_{a,h} = \frac{V_{c,h}}{U_{c,h}}$') + set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); + ylabel('Amplitude'); xlabel('Frequency [Hz]'); + legend(); + hold off; + + ax2 = subplot(1, 2, 2); + hold on; + plot(f, abs(GaZcGi_v), 'k-', 'DisplayName', 'Identified') + plot(f, squeeze(GaZcGi_resp(2,2,:)), 'k--', 'DisplayName', 'Theoretical') + title('FRF $G_{a,v} Z_{c,v} G_{i,v} = \frac{V_{c,v}}{U_{c,v}}$') + set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); + ylabel('Amplitude'); xlabel('Frequency [Hz]'); + legend(); + hold off; + + linkaxes([ax1,ax2],'xy'); + xlim([1, 2000]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/current_amplifier_comp_theory_id_bis.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:current_amplifier_comp_theory_id_bis +#+CAPTION: Identified and Theoretical Transfer Function $G_a G_i$ ([[./figs/current_amplifier_comp_theory_id_bis.png][png]], [[./figs/current_amplifier_comp_theory_id_bis.pdf][pdf]]) +[[file:figs/current_amplifier_comp_theory_id_bis.png]] + +Finally, we have the following transfer functions: +#+begin_src matlab :results output replace :exports results + Gi,Zc,Ga +#+end_src + +#+RESULTS: +#+begin_example +ans = filepath; +if ischar(ans), fid = fopen('/tmp/babel-ZKMGJu/matlab-FA7h5L', 'w'); fprintf(fid, '%s\n', ans); fclose(fid); +else, dlmwrite('/tmp/babel-ZKMGJu/matlab-FA7h5L', ans, '\t') +end +'org_babel_eoe' +Gi,Zc,Ga +'org_babel_eoe' +ans = filepath; +if ischar(ans), fid = fopen('/tmp/babel-ZKMGJu/matlab-FA7h5L', 'w'); fprintf(fid, '%s\n', ans); fclose(fid); +else, dlmwrite('/tmp/babel-ZKMGJu/matlab-FA7h5L', ans, '\t') +end +'org_babel_eoe' +ans = + 'org_babel_eoe' +Gi,Zc,Ga + +Gi = + + From input 1 to output... + 1: 0.01275 + + 2: 0 + + From input 2 to output... + 1: 0 + + 2: 0.01382 + +Static gain. + + +Zc = + + From input 1 to output... + 1: 9.3 + + 2: 0 + + From input 2 to output... + 1: 0 + + 2: 8.3 + +Static gain. + + +Ga = + + From input 1 to output... + 6.2832e+06 + 1: ---------- + (s+6283) + + 2: 0 + + From input 2 to output... + 1: 0 + + 6.2832e+06 + 2: ---------- + (s+6283) + +Continuous-time zero/pole/gain model. +#+end_example + +** Identification of the Cercalo Dynamics +*** Introduction :ignore: +We now wish to identify the dynamics of the Cercalo identified by $G_c$ on the block diagram in Fig. [[fig:block_diagram_simplify]]. + +To do so, we inject some noise at the input of the current amplifier $[U_{c,h},\ U_{c,v}]$ (one input after the other) and we measure simultaneously the output of the 4QD $[V_{p,h},\ V_{p,v}]$. + +The transfer function obtained will be $G_c G_i$, and because we have already identified $G_i$, we can obtain $G_c$ by multiplying the obtained transfer function matrix by ${G_i}^{-1}$. + +*** Input / Output data +The identification data is loaded +#+begin_src matlab + uh = load('mat/data_uch.mat', 't', 'Uch', 'Vph', 'Vpv'); + uv = load('mat/data_ucv.mat', 't', 'Ucv', 'Vph', 'Vpv'); +#+end_src + +We remove the first seconds where the Cercalo is turned on. +#+begin_src matlab + t0 = 1; + + uh.Uch(uh.t> +#+end_src + +#+NAME: fig:identification_uh +#+CAPTION: Identification signals when exciting the horizontal direction ([[./figs/identification_uh.png][png]], [[./figs/identification_uh.pdf][pdf]]) +[[file:figs/identification_uh.png]] + + +#+begin_src matlab :exports none + figure; + ax1 = subplot(1, 2, 1); + plot(uv.t, uv.Ucv, 'DisplayName', '$Uc_v$'); + xlabel('Time [s]'); + ylabel('Amplitude [V]'); + + ax2 = subplot(1, 2, 2); + hold on; + plot(uv.t, uv.Vpv, 'DisplayName', '$Vp_v$'); + plot(uv.t, uv.Vph, 'DisplayName', '$Vp_h$'); + hold off; + xlabel('Time [s]'); + ylabel('Amplitude [V]'); + legend(); + + linkaxes([ax1,ax2],'x'); + xlim([uv.t(1), uv.t(end)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/identification_uv.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:identification_uv +#+CAPTION: Identification signals when exciting in the vertical direction ([[./figs/identification_uv.png][png]], [[./figs/identification_uv.pdf][pdf]]) +[[file:figs/identification_uv.png]] + +*** Coherence +The window used for the spectral analysis is an =hanning= windows with temporal size equal to 1 second. +#+begin_src matlab + win = hanning(ceil(1*fs)); +#+end_src + +#+begin_src matlab + [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); +#+end_src + +#+begin_src matlab :exports none + figure; + ax11 = subplot(2, 2, 1); + hold on; + plot(f, coh_Uch_Vph) + set(gca, 'Xscale', 'log'); + title('Coherence $\frac{Vp_h}{Uc_h}$') + ylabel('Coherence') + hold off; + + ax12 = subplot(2, 2, 2); + hold on; + plot(f, coh_Ucv_Vph) + set(gca, 'Xscale', 'log'); + title('Coherence $\frac{Vp_h}{Uc_v}$') + hold off; + + ax21 = subplot(2, 2, 3); + hold on; + plot(f, coh_Uch_Vpv) + set(gca, 'Xscale', 'log'); + title('Coherence $\frac{Vp_v}{Uc_h}$') + ylabel('Coherence') + xlabel('Frequency [Hz]') + hold off; + + ax22 = subplot(2, 2, 4); + hold on; + plot(f, coh_Ucv_Vpv) + set(gca, 'Xscale', 'log'); + title('Coherence $\frac{Vp_v}{Uc_v}$') + xlabel('Frequency [Hz]') + hold off; + + linkaxes([ax11,ax12,ax21,ax22],'xy'); + xlim([10, 1000]); ylim([0, 1]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/coh_cercalo.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:coh_cercalo +#+CAPTION: Coherence ([[./figs/coh_cercalo.png][png]], [[./figs/coh_cercalo.pdf][pdf]]) +[[file:figs/coh_cercalo.png]] + +*** Estimation of the Frequency Response Function Matrix +We compute an estimate of the transfer functions. +#+begin_src matlab + [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); +#+end_src + +#+begin_src matlab :exports none + figure; + ax11 = subplot(2, 2, 1); + hold on; + plot(f, abs(tf_Uch_Vph)) + title('Frequency Response Function $\frac{Vp_h}{Uc_h}$') + set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); + ylabel('Amplitude [V/V]') + hold off; + + ax12 = subplot(2, 2, 2); + hold on; + plot(f, abs(tf_Ucv_Vph)) + title('Frequency Response Function $\frac{Vp_h}{Uc_v}$') + set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); + hold off; + + ax21 = subplot(2, 2, 3); + hold on; + plot(f, abs(tf_Uch_Vpv)) + title('Frequency Response Function $\frac{Vp_v}{Uc_h}$') + set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); + ylabel('Amplitude [V/V]') + xlabel('Frequency [Hz]') + hold off; + + ax22 = subplot(2, 2, 4); + hold on; + plot(f, abs(tf_Ucv_Vpv)) + title('Frequency Response Function $\frac{Vp_v}{Uc_v}$') + set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); + xlabel('Frequency [Hz]') + hold off; + + linkaxes([ax11,ax12,ax21,ax22],'xy'); + xlim([10, 1000]); ylim([1e-2, 1e3]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/frf_cercalo_gain.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:frf_cercalo_gain +#+CAPTION: Frequency Response Matrix ([[./figs/frf_cercalo_gain.png][png]], [[./figs/frf_cercalo_gain.pdf][pdf]]) +[[file:figs/frf_cercalo_gain.png]] + +#+begin_src matlab :exports none + figure; + ax11 = subplot(2, 2, 1); + hold on; + plot(f, 180/pi*unwrap(angle(tf_Uch_Vph))) + title('Frequency Response Function $\frac{Vp_h}{Uc_h}$') + set(gca, 'Xscale', 'log'); + ylabel('Phase [deg]') + yticks(-180:90:180); + hold off; + + ax12 = subplot(2, 2, 2); + hold on; + plot(f, 180/pi*unwrap(angle(tf_Ucv_Vph))) + title('Frequency Response Function $\frac{Vp_h}{Uc_v}$') + set(gca, 'Xscale', 'log'); + yticks(-180:90:180); + hold off; + + ax21 = subplot(2, 2, 3); + hold on; + plot(f, 180/pi*unwrap(angle(tf_Uch_Vpv))) + title('Frequency Response Function $\frac{Vp_v}{Uc_h}$') + set(gca, 'Xscale', 'log'); + ylabel('Phase [deg]') + xlabel('Frequency [Hz]') + yticks(-180:90:180); + hold off; + + ax22 = subplot(2, 2, 4); + hold on; + plot(f, 180/pi*unwrap(angle(tf_Ucv_Vpv))) + title('Frequency Response Function $\frac{Vp_v}{Uc_v}$') + set(gca, 'Xscale', 'log'); + xlabel('Frequency [Hz]') + yticks(-180:90:180); + hold off; + + linkaxes([ax11,ax12,ax21,ax22],'xy'); + xlim([10, 1000]); ylim([-200, 200]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/frf_cercalo_phase.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:frf_cercalo_phase +#+CAPTION: Frequency Response Matrix_Phase ([[./figs/frf_cercalo_phase.png][png]], [[./figs/frf_cercalo_phase.pdf][pdf]]) +[[file:figs/frf_cercalo_phase.png]] + +*** Time Delay +Now, we would like to remove the time delay included in the FRF prior to the model extraction. + +Estimation of the time delay: +#+begin_src matlab + Ts_delay = Ts; % [s] + + G_delay = tf(1, 1, 'InputDelay', Ts_delay); + + G_delay_resp = squeeze(freqresp(G_delay, f, 'Hz')); +#+end_src + +We then remove the time delay from the frequency response function. +#+begin_src matlab + tf_Uch_Vph = tf_Uch_Vph./G_delay_resp; + tf_Uch_Vpv = tf_Uch_Vpv./G_delay_resp; + tf_Ucv_Vph = tf_Ucv_Vph./G_delay_resp; + tf_Ucv_Vpv = tf_Ucv_Vpv./G_delay_resp; +#+end_src + +*** Extraction of a transfer function matrix +First we define the initial guess for the resonance frequencies and the weights associated. +#+begin_src matlab + freqs_res_uh = [410]; % [Hz] + freqs_res_uv = [250]; % [Hz] +#+end_src + +We then make an initial guess on the complex values of the poles. +#+begin_src matlab + 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)]; +#+end_src + +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. +Finally, we ignore data above some frequency. +#+begin_src matlab + 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_uh) + 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 + + weight_Uch_Vph(f>1000) = 0; + weight_Uch_Vpv(f>1000) = 0; + weight_Ucv_Vph(f>1000) = 0; + weight_Ucv_Vpv(f>1000) = 0; +#+end_src + +The weights are shown in Fig. [[fig:weights_cercalo]]. + +#+begin_src matlab :exports none + figure; + hold on; + plot(f, weight_Uch_Vph, 'DisplayName', '$W_{U_{ch},V_{ph}}$'); + plot(f, weight_Uch_Vpv, 'DisplayName', '$W_{U_{ch},V_{pv}}$'); + plot(f, weight_Ucv_Vph, 'DisplayName', '$W_{U_{cv},V_{ph}}$'); + plot(f, weight_Ucv_Vpv, 'DisplayName', '$W_{U_{cv},V_{pv}}$'); + hold off; + xlabel('Frequency [Hz]'); ylabel('Weight Amplitude'); + set(gca, 'xscale', 'log'); + xlim([f(1), f(end)]); + legend('location', 'northwest'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/weights_cercalo.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:weights_cercalo +#+CAPTION: Weights amplitude ([[./figs/weights_cercalo.png][png]], [[./figs/weights_cercalo.pdf][pdf]]) +[[file:figs/weights_cercalo.png]] + +When we set some options for =vfit3=. +#+begin_src matlab + 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) +#+end_src + +We define the number of iteration. +#+begin_src matlab + Niter = 5; +#+end_src + +An we run the =vectfit3= algorithm. +#+begin_src matlab + 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 +#+end_src + +#+begin_src matlab :exports none + figure; + ax11 = subplot(2, 2, 1); + hold on; + plot(f, abs(tf_Uch_Vph)) + plot(f, abs(fit_Uch_Vph)) + title('Frequency Response Function $\frac{Vp_h}{Uc_h}$') + set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); + ylabel('Amplitude [V/V]') + hold off; + + ax12 = subplot(2, 2, 2); + hold on; + plot(f, abs(tf_Ucv_Vph)) + plot(f, abs(fit_Ucv_Vph)) + title('Frequency Response Function $\frac{Vp_h}{Uc_v}$') + set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); + hold off; + + ax21 = subplot(2, 2, 3); + hold on; + plot(f, abs(tf_Uch_Vpv)) + plot(f, abs(fit_Uch_Vpv)) + title('Frequency Response Function $\frac{Vp_v}{Uc_h}$') + set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); + ylabel('Amplitude [V/V]') + xlabel('Frequency [Hz]') + hold off; + + ax22 = subplot(2, 2, 4); + hold on; + plot(f, abs(tf_Ucv_Vpv)) + plot(f, abs(fit_Ucv_Vpv)) + title('Frequency Response Function $\frac{Vp_v}{Uc_v}$') + set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); + xlabel('Frequency [Hz]') + hold off; + + linkaxes([ax11,ax12,ax21,ax22],'xy'); + xlim([10, 1000]); ylim([1e-2, 1e3]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/identification_matrix_fit.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:identification_matrix_fit +#+CAPTION: Transfer Function Extraction of the FRF matrix ([[./figs/identification_matrix_fit.png][png]], [[./figs/identification_matrix_fit.pdf][pdf]]) +[[file:figs/identification_matrix_fit.png]] + + +#+begin_src matlab :exports none + figure; + ax11 = subplot(2, 2, 1); + hold on; + plot(f, 180/pi*unwrap(angle(tf_Uch_Vph))) + plot(f, 180/pi*unwrap(angle(fit_Uch_Vph))) + title('Frequency Response Function $\frac{Vp_h}{Uc_h}$') + set(gca, 'Xscale', 'log'); + ylabel('Phase [deg]') + yticks(-180:90:180); + hold off; + + ax12 = subplot(2, 2, 2); + hold on; + plot(f, 180/pi*unwrap(angle(tf_Ucv_Vph))) + plot(f, 180/pi*unwrap(angle(fit_Ucv_Vph))) + title('Frequency Response Function $\frac{Vp_h}{Uc_v}$') + set(gca, 'Xscale', 'log'); + yticks(-180:90:180); + hold off; + + ax21 = subplot(2, 2, 3); + hold on; + plot(f, 180/pi*unwrap(angle(tf_Uch_Vpv))) + plot(f, 180/pi*unwrap(angle(fit_Uch_Vpv))) + title('Frequency Response Function $\frac{Vp_v}{Uc_h}$') + set(gca, 'Xscale', 'log'); + ylabel('Phase [deg]') + xlabel('Frequency [Hz]') + yticks(-180:90:180); + hold off; + + ax22 = subplot(2, 2, 4); + hold on; + plot(f, 180/pi*unwrap(angle(tf_Ucv_Vpv))) + plot(f, 180/pi*unwrap(angle(fit_Ucv_Vpv))) + title('Frequency Response Function $\frac{Vp_v}{Uc_v}$') + set(gca, 'Xscale', 'log'); + xlabel('Frequency [Hz]') + yticks(-180:90:180); + hold off; + + linkaxes([ax11,ax12,ax21,ax22],'xy'); + xlim([10, 1000]); ylim([-200, 200]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/identification_matrix_fit_phase.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:identification_matrix_fit_phase +#+CAPTION: Transfer Function Extraction of the FRF matrix ([[./figs/identification_matrix_fit_phase.png][png]], [[./figs/identification_matrix_fit_phase.pdf][pdf]]) +[[file:figs/identification_matrix_fit_phase.png]] + +And finally, we create the identified $G_c$ matrix by multiplying by ${G_i}^{-1}$. +#+begin_src matlab + G_Uch_Vph = tf(minreal(ss(full(SER_Uch_Vph.A),SER_Uch_Vph.B,SER_Uch_Vph.C,SER_Uch_Vph.D))); + G_Ucv_Vph = tf(minreal(ss(full(SER_Ucv_Vph.A),SER_Ucv_Vph.B,SER_Ucv_Vph.C,SER_Ucv_Vph.D))); + G_Uch_Vpv = tf(minreal(ss(full(SER_Uch_Vpv.A),SER_Uch_Vpv.B,SER_Uch_Vpv.C,SER_Uch_Vpv.D))); + G_Ucv_Vpv = tf(minreal(ss(full(SER_Ucv_Vpv.A),SER_Ucv_Vpv.B,SER_Ucv_Vpv.C,SER_Ucv_Vpv.D))); + + Gc = [G_Uch_Vph, G_Ucv_Vph; + G_Uch_Vpv, G_Ucv_Vpv]*inv(Gi); +#+end_src + +** Identification of the Newport Dynamics +*** Introduction :ignore: +We here identify the transfer function from a reference sent to the Newport $[U_{n,h},\ U_{n,v}]$ to the measurement made by the 4QD $[V_{p,h},\ V_{p,v}]$. + +To do so, we inject noise to the Newport $[U_{n,h},\ U_{n,v}]$ and we record the 4QD measurement $[V_{p,h},\ V_{p,v}]$. + +*** Input / Output data +The identification data is loaded +#+begin_src matlab + uh = load('mat/data_unh.mat', 't', 'Unh', 'Vph', 'Vpv'); + uv = load('mat/data_unv.mat', 't', 'Unv', 'Vph', 'Vpv'); +#+end_src + +We remove the first seconds where the Cercalo is turned on. +#+begin_src matlab + t0 = 3; + + uh.Unh(uh.t> +#+end_src + +#+NAME: fig:identification_unh +#+CAPTION: Identification signals when exciting the horizontal direction ([[./figs/identification_unh.png][png]], [[./figs/identification_unh.pdf][pdf]]) +[[file:figs/identification_unh.png]] + +#+begin_src matlab :exports none + figure; + ax1 = subplot(1, 2, 1); + plot(uv.t, uv.Unv, 'DisplayName', '$Un_v$'); + xlabel('Time [s]'); + ylabel('Amplitude [V]'); + + ax2 = subplot(1, 2, 2); + hold on; + plot(uv.t, uv.Vpv, 'DisplayName', '$Vp_v$'); + plot(uv.t, uv.Vph, 'DisplayName', '$Vp_h$'); + hold off; + xlabel('Time [s]'); + ylabel('Amplitude [V]'); + legend(); + + linkaxes([ax1,ax2],'x'); + xlim([uv.t(1), uv.t(end)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/identification_unv.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:identification_unv +#+CAPTION: Identification signals when exciting in the vertical direction ([[./figs/identification_unv.png][png]], [[./figs/identification_unv.pdf][pdf]]) +[[file:figs/identification_unv.png]] + +*** Coherence +The window used for the spectral analysis is an =hanning= windows with temporal size equal to 1 second. +#+begin_src matlab + win = hanning(ceil(1*fs)); +#+end_src + +#+begin_src matlab + [coh_Unh_Vph, f] = mscohere(uh.Unh, uh.Vph, win, [], [], fs); + [coh_Unh_Vpv, ~] = mscohere(uh.Unh, uh.Vpv, win, [], [], fs); + [coh_Unv_Vph, ~] = mscohere(uv.Unv, uv.Vph, win, [], [], fs); + [coh_Unv_Vpv, ~] = mscohere(uv.Unv, uv.Vpv, win, [], [], fs); +#+end_src + +#+begin_src matlab :exports none + figure; + ax11 = subplot(2, 2, 1); + hold on; + plot(f, coh_Unh_Vph) + set(gca, 'Xscale', 'log'); + title('Coherence $\frac{Vp_h}{Un_h}$') + ylabel('Coherence') + hold off; + + ax12 = subplot(2, 2, 2); + hold on; + plot(f, coh_Unv_Vph) + set(gca, 'Xscale', 'log'); + title('Coherence $\frac{Vp_h}{Un_v}$') + hold off; + + ax21 = subplot(2, 2, 3); + hold on; + plot(f, coh_Unh_Vpv) + set(gca, 'Xscale', 'log'); + title('Coherence $\frac{Vp_v}{Un_h}$') + ylabel('Coherence') + xlabel('Frequency [Hz]') + hold off; + + ax22 = subplot(2, 2, 4); + hold on; + plot(f, coh_Unv_Vpv) + set(gca, 'Xscale', 'log'); + title('Coherence $\frac{Vp_v}{Un_v}$') + xlabel('Frequency [Hz]') + hold off; + + linkaxes([ax11,ax12,ax21,ax22],'xy'); + xlim([10, 1000]); ylim([0, 1]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/id_newport_coherence.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:id_newport_coherence +#+CAPTION: Coherence ([[./figs/id_newport_coherence.png][png]], [[./figs/id_newport_coherence.pdf][pdf]]) +[[file:figs/id_newport_coherence.png]] + +*** Estimation of the Frequency Response Function Matrix +We compute an estimate of the transfer functions. +#+begin_src matlab + [tf_Unh_Vph, f] = tfestimate(uh.Unh, uh.Vph, win, [], [], fs); + [tf_Unh_Vpv, ~] = tfestimate(uh.Unh, uh.Vpv, win, [], [], fs); + [tf_Unv_Vph, ~] = tfestimate(uv.Unv, uv.Vph, win, [], [], fs); + [tf_Unv_Vpv, ~] = tfestimate(uv.Unv, uv.Vpv, win, [], [], fs); +#+end_src + +#+begin_src matlab :exports none + figure; + ax11 = subplot(2, 2, 1); + hold on; + plot(f, abs(tf_Unh_Vph)) + title('Frequency Response Function $\frac{Vp_h}{Un_h}$') + set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); + ylabel('Amplitude [V/V]') + hold off; + + ax12 = subplot(2, 2, 2); + hold on; + plot(f, abs(tf_Unv_Vph)) + title('Frequency Response Function $\frac{Vp_h}{Un_v}$') + set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); + hold off; + + ax21 = subplot(2, 2, 3); + hold on; + plot(f, abs(tf_Unh_Vpv)) + title('Frequency Response Function $\frac{Vp_v}{Un_h}$') + set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); + ylabel('Amplitude [V/V]') + xlabel('Frequency [Hz]') + hold off; + + ax22 = subplot(2, 2, 4); + hold on; + plot(f, abs(tf_Unv_Vpv)) + title('Frequency Response Function $\frac{Vp_v}{Un_v}$') + set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); + xlabel('Frequency [Hz]') + hold off; + + linkaxes([ax11,ax12,ax21,ax22],'xy'); + xlim([10, 1000]); ylim([1e-4, 1e1]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/frf_newport_gain.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:frf_newport_gain +#+CAPTION: Frequency Response Matrix ([[./figs/frf_newport_gain.png][png]], [[./figs/frf_newport_gain.pdf][pdf]]) +[[file:figs/frf_newport_gain.png]] + +#+begin_src matlab :exports none + figure; + ax11 = subplot(2, 2, 1); + hold on; + plot(f, 180/pi*unwrap(angle(tf_Unh_Vph))) + title('Frequency Response Function $\frac{Vp_h}{Un_h}$') + set(gca, 'Xscale', 'log'); + ylabel('Phase [deg]') + yticks(-180:90:180); + hold off; + + ax12 = subplot(2, 2, 2); + hold on; + plot(f, 180/pi*unwrap(angle(tf_Unv_Vph))) + title('Frequency Response Function $\frac{Vp_h}{Un_v}$') + set(gca, 'Xscale', 'log'); + yticks(-180:90:180); + hold off; + + ax21 = subplot(2, 2, 3); + hold on; + plot(f, 180/pi*unwrap(angle(tf_Unh_Vpv))) + title('Frequency Response Function $\frac{Vp_v}{Un_h}$') + set(gca, 'Xscale', 'log'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + yticks(-180:90:180); + hold off; + + ax22 = subplot(2, 2, 4); + hold on; + plot(f, 180/pi*unwrap(angle(tf_Unv_Vpv))) + title('Frequency Response Function $\frac{Vp_v}{Un_v}$') + set(gca, 'Xscale', 'log'); + xlabel('Frequency [Hz]'); + yticks(-180:90:180); + hold off; + + linkaxes([ax11,ax12,ax21,ax22],'xy'); + xlim([10, 1000]); ylim([-200, 200]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/frf_newport_phase.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:frf_newport_phase +#+CAPTION: Frequency Response Matrix Phase ([[./figs/frf_newport_phase.png][png]], [[./figs/frf_newport_phase.pdf][pdf]]) +[[file:figs/frf_newport_phase.png]] + +*** Time Delay +Now, we would like to remove the time delay included in the FRF prior to the model extraction. + +Estimation of the time delay: +#+begin_src matlab + Ts_delay = 0.0005; % [s] + + G_delay = tf(1, 1, 'InputDelay', Ts_delay); + + G_delay_resp = squeeze(freqresp(G_delay, f, 'Hz')); +#+end_src + +We then remove the time delay from the frequency response function. +#+begin_src matlab :exports none + figure; + ax11 = subplot(2, 2, 1); + hold on; + plot(f, 180/pi*unwrap(angle(tf_Unh_Vph))) + plot(f, 180/pi*unwrap(angle(tf_Unh_Vph./G_delay_resp))) + title('Frequency Response Function $\frac{Vp_h}{Un_h}$') + set(gca, 'Xscale', 'log'); + ylabel('Phase [deg]') + yticks(-180:90:180); + hold off; + + ax12 = subplot(2, 2, 2); + hold on; + plot(f, 180/pi*unwrap(angle(tf_Unv_Vph))) + plot(f, 180/pi*unwrap(angle(tf_Unv_Vph./G_delay_resp))) + title('Frequency Response Function $\frac{Vp_h}{Un_v}$') + set(gca, 'Xscale', 'log'); + yticks(-180:90:180); + hold off; + + ax21 = subplot(2, 2, 3); + hold on; + plot(f, 180/pi*unwrap(angle(tf_Unh_Vpv))) + plot(f, 180/pi*unwrap(angle(tf_Unh_Vpv./G_delay_resp))) + title('Frequency Response Function $\frac{Vp_v}{Un_h}$') + set(gca, 'Xscale', 'log'); + ylabel('Phase [deg]') + xlabel('Frequency [Hz]') + yticks(-180:90:180); + hold off; + + ax22 = subplot(2, 2, 4); + hold on; + plot(f, 180/pi*unwrap(angle(tf_Unv_Vpv))) + plot(f, 180/pi*unwrap(angle(tf_Unv_Vpv./G_delay_resp))) + title('Frequency Response Function $\frac{Vp_v}{Un_v}$') + set(gca, 'Xscale', 'log'); + xlabel('Frequency [Hz]') + yticks(-180:90:180); + hold off; + + linkaxes([ax11,ax12,ax21,ax22],'xy'); + xlim([10, 1000]); ylim([-200, 200]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/time_delay_newport.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:time_delay_newport +#+CAPTION: Phase change due to time-delay in the Newport dynamics ([[./figs/time_delay_newport.png][png]], [[./figs/time_delay_newport.pdf][pdf]]) +[[file:figs/time_delay_newport.png]] + +*** Extraction of a transfer function matrix +From Fig. [[fig:frf_newport_gain]], it seems reasonable to model the Newport dynamics as diagonal and constant. + +#+begin_src matlab + Gn = blkdiag(tf(mean(abs(tf_Unh_Vph(f>10 & f<100)))), tf(mean(abs(tf_Unv_Vpv(f>10 & f<100))))); +#+end_src + +** Full System +We now have identified: +- $G_i$ +- $G_a$ +- $G_c$ +- $G_n$ +- $G_d$ + +We name the input and output of each transfer function: +#+begin_src matlab + Gi.InputName = {'Uch', 'Ucv'}; + Gi.OutputName = {'Ich', 'Icv'}; + + Zc.InputName = {'Ich', 'Icv'}; + Zc.OutputName = {'Vtch', 'Vtcv'}; + + Ga.InputName = {'Vtch', 'Vtcv'}; + Ga.OutputName = {'Vch', 'Vcv'}; + + Gc.InputName = {'Ich', 'Icv'}; + Gc.OutputName = {'Vpch', 'Vpcv'}; + + Gn.InputName = {'Unh', 'Unv'}; + Gn.OutputName = {'Vpnh', 'Vpnv'}; + + Gd.InputName = {'Rh', 'Rv'}; + Gd.OutputName = {'Vph', 'Vpv'}; +#+end_src + +#+begin_src matlab + Sh = sumblk('Vph = Vpch + Vpnh'); + Sv = sumblk('Vpv = Vpcv + Vpnv'); +#+end_src + +#+begin_src matlab + inputs = {'Uch', 'Ucv', 'Unh', 'Unv'}; + outputs = {'Vch', 'Vcv', 'Ich', 'Icv', 'Rh', 'Rv', 'Vph', 'Vpv'}; + + sys = connect(Gi, Zc, Ga, Gc, Gn, inv(Gd), Sh, Sv, inputs, outputs); +#+end_src + +The file =mat/plant.mat= is accessible [[./mat/plant.mat][here]]. +#+begin_src matlab + save('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd'); +#+end_src + +* Active Damping +** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab + freqs = logspace(1, 3, 1000); +#+end_src + +** Load Plant +#+begin_src matlab + load('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd'); +#+end_src + +** Test +#+begin_src matlab + bode(sys({'Vch', 'Vcv'}, {'Uch', 'Ucv'})); +#+end_src + +#+begin_src matlab + Kppf = blkdiag(-10000/s, tf(0)); + + Kppf.InputName = {'Vch', 'Vcv'}; + Kppf.OutputName = {'Uch', 'Ucv'}; +#+end_src + +#+begin_src matlab :exports none + figure; + % Magnitude + ax1 = subaxis(2,1,1); + hold on; + plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), 'k-'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); + ylabel('Magnitude [dB]'); + hold off; + + % Phase + ax2 = subaxis(2,1,2); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G, freqs, 'Hz'))), 'k-'); + set(gca,'xscale','log'); + yticks(-360:90:180); + ylim([-360 0]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab + inputs = {'Uch', 'Ucv', 'Unh', 'Unv'}; + outputs = {'Ich', 'Icv', 'Rh', 'Rv', 'Vph', 'Vpv'}; + + sys_cl = connect(sys, Kppf, inputs, outputs); + + figure; bode(sys_cl({'Vph', 'Vpv'}, {'Uch', 'Ucv'}), sys({'Vph', 'Vpv'}, {'Uch', 'Ucv'})) +#+end_src + +* TODO Huddle Test We load the data taken during the Huddle Test. #+begin_src matlab load('mat/data_huddle_test.mat', ... @@ -598,1121 +1948,6 @@ We compute the Power Spectral Density of the voltage across the inductance used xlim([1, 1000]); #+end_src -** Input / Output data -The identification data is loaded -#+begin_src matlab - uh = load('mat/data_uch.mat', ... - 't', 'Uch', 'Ucv', ... - 'Unh', 'Unv', ... - 'Vph', 'Vpv', ... - 'Vch', 'Vcv', ... - 'Vnh', 'Vnv', ... - 'Va'); - - uv = load('mat/data_ucv.mat', ... - 't', 'Uch', 'Ucv', ... - 'Unh', 'Unv', ... - 'Vph', 'Vpv', ... - 'Vch', 'Vcv', ... - 'Vnh', 'Vnv', ... - 'Va'); -#+end_src - -We remove the first seconds where the Cercalo is turned on. -#+begin_src matlab - t0 = 1; - - uh.Uch(uh.t> -#+end_src - -#+NAME: fig:identification_uh -#+CAPTION: Identification signals when exciting the horizontal direction ([[./figs/identification_uh.png][png]], [[./figs/identification_uh.pdf][pdf]]) -[[file:figs/identification_uh.png]] - - -#+begin_src matlab :exports none - figure; - ax1 = subplot(1, 2, 1); - plot(uv.t, uv.Ucv, 'DisplayName', '$Uc_v$'); - xlabel('Time [s]'); - ylabel('Amplitude [V]'); - - ax2 = subplot(1, 2, 2); - hold on; - plot(uv.t, uv.Vpv, 'DisplayName', '$Vp_v$'); - plot(uv.t, uv.Vph, 'DisplayName', '$Vp_h$'); - hold off; - xlabel('Time [s]'); - ylabel('Amplitude [V]'); - legend(); - - linkaxes([ax1,ax2],'x'); - xlim([uv.t(1), uv.t(end)]); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/identification_uv.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:identification_uv -#+CAPTION: Identification signals when exciting in the vertical direction ([[./figs/identification_uv.png][png]], [[./figs/identification_uv.pdf][pdf]]) -[[file:figs/identification_uv.png]] - -** Coherence -The window used for the spectral analysis is an =hanning= windows with temporal size equal to 1 second. -#+begin_src matlab - win = hanning(ceil(1*fs)); -#+end_src - -#+begin_src matlab - [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); -#+end_src - -#+begin_src matlab :exports none - figure; - ax11 = subplot(2, 2, 1); - hold on; - plot(f, coh_Uch_Vph) - set(gca, 'Xscale', 'log'); - title('Coherence $\frac{Vp_h}{Uc_h}$') - ylabel('Coherence') - hold off; - - ax12 = subplot(2, 2, 2); - hold on; - plot(f, coh_Ucv_Vph) - set(gca, 'Xscale', 'log'); - title('Coherence $\frac{Vp_h}{Uc_v}$') - hold off; - - ax21 = subplot(2, 2, 3); - hold on; - plot(f, coh_Uch_Vpv) - set(gca, 'Xscale', 'log'); - title('Coherence $\frac{Vp_v}{Uc_h}$') - ylabel('Coherence') - xlabel('Frequency [Hz]') - hold off; - - ax22 = subplot(2, 2, 4); - hold on; - plot(f, coh_Ucv_Vpv) - set(gca, 'Xscale', 'log'); - title('Coherence $\frac{Vp_v}{Uc_v}$') - xlabel('Frequency [Hz]') - hold off; - - linkaxes([ax11,ax12,ax21,ax22],'xy'); - xlim([10, 1000]); ylim([0, 1]); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/identification_coherence.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:identification_coherence -#+CAPTION: Coherence ([[./figs/identification_coherence.png][png]], [[./figs/identification_coherence.pdf][pdf]]) -[[file:figs/identification_coherence.png]] - -** Estimation of the Frequency Response Function Matrix -We compute an estimate of the transfer functions. -#+begin_src matlab - [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); -#+end_src - -#+begin_src matlab :exports none - figure; - ax11 = subplot(2, 2, 1); - hold on; - plot(f, abs(tf_Uch_Vph)) - title('Frequency Response Function $\frac{Vp_h}{Uc_h}$') - set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); - ylabel('Amplitude') - hold off; - - ax12 = subplot(2, 2, 2); - hold on; - plot(f, abs(tf_Ucv_Vph)) - title('Frequency Response Function $\frac{Vp_h}{Uc_v}$') - set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); - hold off; - - ax21 = subplot(2, 2, 3); - hold on; - plot(f, abs(tf_Uch_Vpv)) - title('Frequency Response Function $\frac{Vp_v}{Uc_h}$') - set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); - ylabel('Amplitude') - xlabel('Frequency [Hz]') - hold off; - - ax22 = subplot(2, 2, 4); - hold on; - plot(f, abs(tf_Ucv_Vpv)) - title('Frequency Response Function $\frac{Vp_v}{Uc_v}$') - set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); - xlabel('Frequency [Hz]') - hold off; - - linkaxes([ax11,ax12,ax21,ax22],'xy'); - xlim([10, 1000]); ylim([1e-2, 1e3]); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/frequency_response_matrix.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:frequency_response_matrix -#+CAPTION: Frequency Response Matrix ([[./figs/frequency_response_matrix.png][png]], [[./figs/frequency_response_matrix.pdf][pdf]]) -[[file:figs/frequency_response_matrix.png]] - -#+begin_src matlab :exports none - figure; - ax11 = subplot(2, 2, 1); - hold on; - plot(f, 180/pi*unwrap(angle(tf_Uch_Vph))) - title('Frequency Response Function $\frac{Vp_h}{Uc_h}$') - set(gca, 'Xscale', 'log'); - ylabel('Phase [deg]') - yticks([-180:90:180]); - hold off; - - ax12 = subplot(2, 2, 2); - hold on; - plot(f, 180/pi*unwrap(angle(tf_Ucv_Vph))) - title('Frequency Response Function $\frac{Vp_h}{Uc_v}$') - set(gca, 'Xscale', 'log'); - yticks([-180:90:180]); - hold off; - - ax21 = subplot(2, 2, 3); - hold on; - plot(f, 180/pi*unwrap(angle(tf_Uch_Vpv))) - title('Frequency Response Function $\frac{Vp_v}{Uc_h}$') - set(gca, 'Xscale', 'log'); - ylabel('Phase [deg]') - xlabel('Frequency [Hz]') - yticks([-180:90:180]); - hold off; - - ax22 = subplot(2, 2, 4); - hold on; - plot(f, 180/pi*unwrap(angle(tf_Ucv_Vpv))) - title('Frequency Response Function $\frac{Vp_v}{Uc_v}$') - set(gca, 'Xscale', 'log'); - xlabel('Frequency [Hz]') - yticks([-180:90:180]); - hold off; - - linkaxes([ax11,ax12,ax21,ax22],'xy'); - xlim([10, 1000]); ylim([-200, 200]); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/frequency_response_matrix_phase.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:frequency_response_matrix_phase -#+CAPTION: Frequency Response Matrix_Phase ([[./figs/frequency_response_matrix_phase.png][png]], [[./figs/frequency_response_matrix_phase.pdf][pdf]]) -[[file:figs/frequency_response_matrix_phase.png]] - -** Time Delay -Now, we would like to remove the time delay included in the FRF prior to the model extraction. - -Estimation of the time delay: -#+begin_src matlab - Ts_delay = Ts; % [s] - - G_delay = tf(1, 1, 'InputDelay', Ts_delay); - - G_delay_resp = squeeze(freqresp(G_delay, f, 'Hz')); -#+end_src - -We then remove the time delay from the frequency response function. -#+begin_src matlab - tf_Uch_Vph = tf_Uch_Vph./G_delay_resp; - tf_Uch_Vpv = tf_Uch_Vpv./G_delay_resp; - tf_Ucv_Vph = tf_Ucv_Vph./G_delay_resp; - tf_Ucv_Vpv = tf_Ucv_Vpv./G_delay_resp; -#+end_src - -** Extraction of a transfer function matrix -First we define the initial guess for the resonance frequencies and the weights associated. -#+begin_src matlab - freqs_res_uh = [410]; % [Hz] - freqs_res_uv = [250]; % [Hz] -#+end_src - -From the number of resonance frequency we want to fit, we define the order =N= of the system we want to obtain. -#+begin_src matlab - N = 2; -#+end_src - -We then make an initial guess on the complex values of the poles. -#+begin_src matlab - 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)]; -#+end_src - -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. -#+begin_src matlab - 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 -#+end_src - -Ignore data above some frequency. -#+begin_src matlab - weight_Uch_Vph(f>1000) = 0; - weight_Uch_Vpv(f>1000) = 0; - weight_Ucv_Vph(f>1000) = 0; - weight_Ucv_Vpv(f>1000) = 0; -#+end_src - -#+begin_src matlab :exports none - figure; - hold on; - plot(f, weight_Uch_Vph); - plot(f, weight_Uch_Vpv); - plot(f, weight_Ucv_Vph); - plot(f, weight_Ucv_Vpv); - hold off; - xlabel('Frequency [Hz]'); - xlabel('Weight Amplitude'); - set(gca, 'xscale', 'log'); - xlim([f(1), f(end)]); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/weights.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:weights -#+CAPTION: Weights amplitude ([[./figs/weights.png][png]], [[./figs/weights.pdf][pdf]]) -[[file:figs/weights.png]] - -When we set some options for =vfit3=. -#+begin_src matlab - 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) -#+end_src - -We define the number of iteration. -#+begin_src matlab - Niter = 5; -#+end_src - -An we run the =vectfit3= algorithm. -#+begin_src matlab - 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 -#+end_src - -#+begin_src matlab :exports none - figure; - ax11 = subplot(2, 2, 1); - hold on; - plot(f, abs(tf_Uch_Vph)) - plot(f, abs(fit_Uch_Vph)) - title('Frequency Response Function $\frac{Vp_h}{Uc_h}$') - set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); - ylabel('Amplitude') - hold off; - - ax12 = subplot(2, 2, 2); - hold on; - plot(f, abs(tf_Ucv_Vph)) - plot(f, abs(fit_Ucv_Vph)) - title('Frequency Response Function $\frac{Vp_h}{Uc_v}$') - set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); - hold off; - - ax21 = subplot(2, 2, 3); - hold on; - plot(f, abs(tf_Uch_Vpv)) - plot(f, abs(fit_Uch_Vpv)) - title('Frequency Response Function $\frac{Vp_v}{Uc_h}$') - set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); - ylabel('Amplitude') - xlabel('Frequency [Hz]') - hold off; - - ax22 = subplot(2, 2, 4); - hold on; - plot(f, abs(tf_Ucv_Vpv)) - plot(f, abs(fit_Ucv_Vpv)) - title('Frequency Response Function $\frac{Vp_v}{Uc_v}$') - set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); - xlabel('Frequency [Hz]') - hold off; - - linkaxes([ax11,ax12,ax21,ax22],'xy'); - xlim([10, 1000]); ylim([1e-2, 1e3]); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/identification_matrix_fit.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:identification_matrix_fit -#+CAPTION: Transfer Function Extraction of the FRF matrix ([[./figs/identification_matrix_fit.png][png]], [[./figs/identification_matrix_fit.pdf][pdf]]) -[[file:figs/identification_matrix_fit.png]] - - -#+begin_src matlab :exports none - figure; - ax11 = subplot(2, 2, 1); - hold on; - plot(f, 180/pi*unwrap(angle(tf_Uch_Vph))) - plot(f, 180/pi*unwrap(angle(fit_Uch_Vph))) - title('Frequency Response Function $\frac{Vp_h}{Uc_h}$') - set(gca, 'Xscale', 'log'); - ylabel('Phase [deg]') - yticks([-180:90:180]); - hold off; - - ax12 = subplot(2, 2, 2); - hold on; - plot(f, 180/pi*unwrap(angle(tf_Ucv_Vph))) - plot(f, 180/pi*unwrap(angle(fit_Ucv_Vph))) - title('Frequency Response Function $\frac{Vp_h}{Uc_v}$') - set(gca, 'Xscale', 'log'); - yticks([-180:90:180]); - hold off; - - ax21 = subplot(2, 2, 3); - hold on; - plot(f, 180/pi*unwrap(angle(tf_Uch_Vpv))) - plot(f, 180/pi*unwrap(angle(fit_Uch_Vpv))) - title('Frequency Response Function $\frac{Vp_v}{Uc_h}$') - set(gca, 'Xscale', 'log'); - ylabel('Phase [deg]') - xlabel('Frequency [Hz]') - yticks([-180:90:180]); - hold off; - - ax22 = subplot(2, 2, 4); - hold on; - plot(f, 180/pi*unwrap(angle(tf_Ucv_Vpv))) - plot(f, 180/pi*unwrap(angle(fit_Ucv_Vpv))) - title('Frequency Response Function $\frac{Vp_v}{Uc_v}$') - set(gca, 'Xscale', 'log'); - xlabel('Frequency [Hz]') - yticks([-180:90:180]); - hold off; - - linkaxes([ax11,ax12,ax21,ax22],'xy'); - xlim([10, 1000]); ylim([-200, 200]); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/identification_matrix_fit_phase.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:identification_matrix_fit_phase -#+CAPTION: Transfer Function Extraction of the FRF matrix ([[./figs/identification_matrix_fit_phase.png][png]], [[./figs/identification_matrix_fit_phase.pdf][pdf]]) -[[file:figs/identification_matrix_fit_phase.png]] - -And finally, we create the identified state space model: -#+begin_src matlab - G_Uch_Vph = minreal(ss(full(SER_Uch_Vph.A),SER_Uch_Vph.B,SER_Uch_Vph.C,SER_Uch_Vph.D)); - G_Ucv_Vph = minreal(ss(full(SER_Ucv_Vph.A),SER_Ucv_Vph.B,SER_Ucv_Vph.C,SER_Ucv_Vph.D)); - G_Uch_Vpv = minreal(ss(full(SER_Uch_Vpv.A),SER_Uch_Vpv.B,SER_Uch_Vpv.C,SER_Uch_Vpv.D)); - G_Ucv_Vpv = minreal(ss(full(SER_Ucv_Vpv.A),SER_Ucv_Vpv.B,SER_Ucv_Vpv.C,SER_Ucv_Vpv.D)); - - Gc = [G_Uch_Vph, G_Ucv_Vph; - G_Uch_Vpv, G_Ucv_Vpv]; -#+end_src - -#+begin_src matlab - save('mat/plant.mat', 'Gc'); -#+end_src - -** Parameter extraction -#+begin_src matlab - [Gz, Gp, Gk] = zpkdata(G_Uch_Vph); - G_Uch_Vph = zpk([], Gp, -prod(Gz{:})*Gk); - [Gz, Gp, Gk] = zpkdata(G_Ucv_Vph); - G_Ucv_Vph = zpk([], Gp, -prod(Gz{:})*Gk); - [Gz, Gp, Gk] = zpkdata(G_Uch_Vpv); - G_Uch_Vpv = zpk([], Gp, -prod(Gz{:})*Gk); - [Gz, Gp, Gk] = zpkdata(G_Ucv_Vpv); - G_Ucv_Vpv = zpk([], Gp, -prod(Gz{:})*Gk); - - % Plant without the Zero - Gb = [G_Uch_Vph, G_Ucv_Vph; - G_Uch_Vpv, G_Ucv_Vpv]; -#+end_src - -* Identification of the Newport 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) - <> -#+end_src - -#+begin_src matlab :exports none :results silent :noweb yes - <> -#+end_src - -#+begin_src matlab - fs = 1e4; - Ts = 1/fs; - freqs = logspace(1, 3, 1000); -#+end_src - -** Input / Output data -The identification data is loaded -#+begin_src matlab - uh = load('mat/data_unh.mat', ... - 't', 'Uch', 'Ucv', ... - 'Unh', 'Unv', ... - 'Vph', 'Vpv', ... - 'Vch', 'Vcv', ... - 'Vnh', 'Vnv', ... - 'Va'); - - uv = load('mat/data_unv.mat', ... - 't', 'Uch', 'Ucv', ... - 'Unh', 'Unv', ... - 'Vph', 'Vpv', ... - 'Vch', 'Vcv', ... - 'Vnh', 'Vnv', ... - 'Va'); -#+end_src - -We remove the first seconds where the Cercalo is turned on. -#+begin_src matlab - t0 = 3; - - uh.Uch(uh.t> -#+end_src - -#+NAME: fig:identification_unh -#+CAPTION: Identification signals when exciting the horizontal direction ([[./figs/identification_unh.png][png]], [[./figs/identification_unh.pdf][pdf]]) -[[file:figs/identification_unh.png]] - - -#+begin_src matlab :exports none - figure; - ax1 = subplot(1, 2, 1); - plot(uv.t, uv.Unv, 'DisplayName', '$Uc_v$'); - xlabel('Time [s]'); - ylabel('Amplitude [V]'); - - ax2 = subplot(1, 2, 2); - hold on; - plot(uv.t, uv.Vpv, 'DisplayName', '$Vp_v$'); - plot(uv.t, uv.Vph, 'DisplayName', '$Vp_h$'); - hold off; - xlabel('Time [s]'); - ylabel('Amplitude [V]'); - legend(); - - linkaxes([ax1,ax2],'x'); - xlim([uv.t(1), uv.t(end)]); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/identification_unv.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:identification_unv -#+CAPTION: Identification signals when exciting in the vertical direction ([[./figs/identification_unv.png][png]], [[./figs/identification_unv.pdf][pdf]]) -[[file:figs/identification_unv.png]] - - -** Coherence -The window used for the spectral analysis is an =hanning= windows with temporal size equal to 1 second. -#+begin_src matlab - win = hanning(ceil(1*fs)); -#+end_src - -#+begin_src matlab - [coh_Unh_Vph, f] = mscohere(uh.Unh, uh.Vph, win, [], [], fs); - [coh_Unh_Vpv, ~] = mscohere(uh.Unh, uh.Vpv, win, [], [], fs); - [coh_Unv_Vph, ~] = mscohere(uv.Unv, uv.Vph, win, [], [], fs); - [coh_Unv_Vpv, ~] = mscohere(uv.Unv, uv.Vpv, win, [], [], fs); -#+end_src - -#+begin_src matlab :exports none - figure; - ax11 = subplot(2, 2, 1); - hold on; - plot(f, coh_Unh_Vph) - set(gca, 'Xscale', 'log'); - title('Coherence $\frac{Vp_h}{Un_h}$') - ylabel('Coherence') - hold off; - - ax12 = subplot(2, 2, 2); - hold on; - plot(f, coh_Unv_Vph) - set(gca, 'Xscale', 'log'); - title('Coherence $\frac{Vp_h}{Un_v}$') - hold off; - - ax21 = subplot(2, 2, 3); - hold on; - plot(f, coh_Unh_Vpv) - set(gca, 'Xscale', 'log'); - title('Coherence $\frac{Vp_v}{Un_h}$') - ylabel('Coherence') - xlabel('Frequency [Hz]') - hold off; - - ax22 = subplot(2, 2, 4); - hold on; - plot(f, coh_Unv_Vpv) - set(gca, 'Xscale', 'log'); - title('Coherence $\frac{Vp_v}{Un_v}$') - xlabel('Frequency [Hz]') - hold off; - - linkaxes([ax11,ax12,ax21,ax22],'xy'); - xlim([10, 1000]); ylim([0, 1]); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/id_Newport_coherence.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:id_Newport_coherence -#+CAPTION: Coherence ([[./figs/id_Newport_coherence.png][png]], [[./figs/id_Newport_coherence.pdf][pdf]]) -[[file:figs/id_Newport_coherence.png]] - -** Estimation of the Frequency Response Function Matrix -We compute an estimate of the transfer functions. -#+begin_src matlab - [tf_Unh_Vph, f] = tfestimate(uh.Unh, uh.Vph, win, [], [], fs); - [tf_Unh_Vpv, ~] = tfestimate(uh.Unh, uh.Vpv, win, [], [], fs); - [tf_Unv_Vph, ~] = tfestimate(uv.Unv, uv.Vph, win, [], [], fs); - [tf_Unv_Vpv, ~] = tfestimate(uv.Unv, uv.Vpv, win, [], [], fs); -#+end_src - -#+begin_src matlab :exports none - figure; - ax11 = subplot(2, 2, 1); - hold on; - plot(f, abs(tf_Unh_Vph)) - title('Frequency Response Function $\frac{Vp_h}{Un_h}$') - set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); - ylabel('Amplitude') - hold off; - - ax12 = subplot(2, 2, 2); - hold on; - plot(f, abs(tf_Unv_Vph)) - title('Frequency Response Function $\frac{Vp_h}{Un_v}$') - set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); - hold off; - - ax21 = subplot(2, 2, 3); - hold on; - plot(f, abs(tf_Unh_Vpv)) - title('Frequency Response Function $\frac{Vp_v}{Un_h}$') - set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); - ylabel('Amplitude') - xlabel('Frequency [Hz]') - hold off; - - ax22 = subplot(2, 2, 4); - hold on; - plot(f, abs(tf_Unv_Vpv)) - title('Frequency Response Function $\frac{Vp_v}{Un_v}$') - set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); - xlabel('Frequency [Hz]') - hold off; - - linkaxes([ax11,ax12,ax21,ax22],'xy'); - xlim([10, 1000]); ylim([1e-4, 1e1]); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/frequency_response_matrix.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:frequency_response_matrix -#+CAPTION: Frequency Response Matrix ([[./figs/frequency_response_matrix.png][png]], [[./figs/frequency_response_matrix.pdf][pdf]]) -[[file:figs/frequency_response_matrix.png]] - -#+begin_src matlab :exports none - figure; - ax11 = subplot(2, 2, 1); - hold on; - plot(f, 180/pi*unwrap(angle(tf_Unh_Vph))) - title('Frequency Response Function $\frac{Vp_h}{Un_h}$') - set(gca, 'Xscale', 'log'); - ylabel('Phase [deg]') - yticks([-180:90:180]); - hold off; - - ax12 = subplot(2, 2, 2); - hold on; - plot(f, 180/pi*unwrap(angle(tf_Unv_Vph))) - title('Frequency Response Function $\frac{Vp_h}{Un_v}$') - set(gca, 'Xscale', 'log'); - yticks([-180:90:180]); - hold off; - - ax21 = subplot(2, 2, 3); - hold on; - plot(f, 180/pi*unwrap(angle(tf_Unh_Vpv))) - title('Frequency Response Function $\frac{Vp_v}{Un_h}$') - set(gca, 'Xscale', 'log'); - ylabel('Phase [deg]') - xlabel('Frequency [Hz]') - yticks([-180:90:180]); - hold off; - - ax22 = subplot(2, 2, 4); - hold on; - plot(f, 180/pi*unwrap(angle(tf_Unv_Vpv))) - title('Frequency Response Function $\frac{Vp_v}{Un_v}$') - set(gca, 'Xscale', 'log'); - xlabel('Frequency [Hz]') - yticks([-180:90:180]); - hold off; - - linkaxes([ax11,ax12,ax21,ax22],'xy'); - xlim([10, 1000]); ylim([-200, 200]); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/id_Newport_phase.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:id_Newport_phase -#+CAPTION: Frequency Response Matrix_Phase ([[./figs/id_Newport_phase.png][png]], [[./figs/id_Newport_phase.pdf][pdf]]) -[[file:figs/id_Newport_phase.png]] - -** Time Delay -Now, we would like to remove the time delay included in the FRF prior to the model extraction. - -Estimation of the time delay: -#+begin_src matlab - Ts_delay = 0.0005; % [s] - - G_delay = tf(1, 1, 'InputDelay', Ts_delay); - - G_delay_resp = squeeze(freqresp(G_delay, f, 'Hz')); -#+end_src - -We then remove the time delay from the frequency response function. -#+begin_src matlab :exports none - figure; - ax11 = subplot(2, 2, 1); - hold on; - plot(f, 180/pi*unwrap(angle(tf_Unh_Vph))) - plot(f, 180/pi*unwrap(angle(tf_Unh_Vph./G_delay_resp))) - title('Frequency Response Function $\frac{Vp_h}{Un_h}$') - set(gca, 'Xscale', 'log'); - ylabel('Phase [deg]') - yticks([-180:90:180]); - hold off; - - ax12 = subplot(2, 2, 2); - hold on; - plot(f, 180/pi*unwrap(angle(tf_Unv_Vph))) - plot(f, 180/pi*unwrap(angle(tf_Unv_Vph./G_delay_resp))) - title('Frequency Response Function $\frac{Vp_h}{Un_v}$') - set(gca, 'Xscale', 'log'); - yticks([-180:90:180]); - hold off; - - ax21 = subplot(2, 2, 3); - hold on; - plot(f, 180/pi*unwrap(angle(tf_Unh_Vpv))) - plot(f, 180/pi*unwrap(angle(tf_Unh_Vpv./G_delay_resp))) - title('Frequency Response Function $\frac{Vp_v}{Un_h}$') - set(gca, 'Xscale', 'log'); - ylabel('Phase [deg]') - xlabel('Frequency [Hz]') - yticks([-180:90:180]); - hold off; - - ax22 = subplot(2, 2, 4); - hold on; - plot(f, 180/pi*unwrap(angle(tf_Unv_Vpv))) - plot(f, 180/pi*unwrap(angle(tf_Unv_Vpv./G_delay_resp))) - title('Frequency Response Function $\frac{Vp_v}{Un_v}$') - set(gca, 'Xscale', 'log'); - xlabel('Frequency [Hz]') - yticks([-180:90:180]); - hold off; - - linkaxes([ax11,ax12,ax21,ax22],'xy'); - xlim([10, 1000]); ylim([-200, 200]); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/time_delay_Newport.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:time_delay_Newport -#+CAPTION: Phase change due to time-delay in the Newport dynamics ([[./figs/time_delay_Newport.png][png]], [[./figs/time_delay_Newport.pdf][pdf]]) -[[file:figs/time_delay_Newport.png]] - -** Extraction of a transfer function matrix -From Fig. [[fig:frequency_response_matrix]], it seems reasonable to model the Newport dynamics as a constant. - -#+begin_src matlab - Gnh0 = tf(mean(abs(tf_Unh_Vph(f>10 & f<100)))); - Gnv0 = tf(mean(abs(tf_Unv_Vpv(f>10 & f<100)))); - - Gn = [Gnh0, 0; - 0, Gnv0]; -#+end_src - -* Calibration of the 4 Quadrant Diode -** Introduction :ignore: - -We know precisely the Newport angle thanks to its metrology included. -Thus, we can choose precisely the angle of the Newport mirror and see what is the value measured by the 4 Quadrant Diode. -The goal is to obtain the "gain" of the 4QD in [V/rad]. - -** Input / Output data -The identification data is loaded -#+begin_src matlab - uh = load('mat/data_cal_pd_h.mat', ... - 't', 'Uch', 'Ucv', ... - 'Unh', 'Unv', ... - 'Vph', 'Vpv', ... - 'Vch', 'Vcv', ... - 'Vnh', 'Vnv', ... - 'Va'); - - uv = load('mat/data_cal_pd_v.mat', ... - 't', 'Uch', 'Ucv', ... - 'Unh', 'Unv', ... - 'Vph', 'Vpv', ... - 'Vch', 'Vcv', ... - 'Vnh', 'Vnv', ... - 'Va'); -#+end_src - -We remove the first seconds where the Cercalo is turned on. -#+begin_src matlab - t0 = 1; - - uh.Uch(uh.t> -#+end_src - -#+NAME: fig:calib_4qd_h -#+CAPTION: Identification signals when exciting the horizontal direction ([[./figs/calib_4qd_h.png][png]], [[./figs/calib_4qd_h.pdf][pdf]]) -[[file:figs/calib_4qd_h.png]] - - -#+begin_src matlab :exports none - figure; - ax1 = subplot(1, 2, 1); - plot(uv.t, uv.Unv, 'DisplayName', '$Uc_v$'); - xlabel('Time [s]'); - ylabel('Amplitude [V]'); - - ax2 = subplot(1, 2, 2); - hold on; - plot(uv.t, uv.Vpv, 'DisplayName', '$Vp_v$'); - plot(uv.t, uv.Vph, 'DisplayName', '$Vp_h$'); - hold off; - xlabel('Time [s]'); - ylabel('Amplitude [V]'); - legend(); - - linkaxes([ax1,ax2],'x'); - xlim([uv.t(1), uv.t(end)]); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/calib_4qd_v.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:calib_4qd_v -#+CAPTION: Identification signals when exciting in the vertical direction ([[./figs/calib_4qd_v.png][png]], [[./figs/calib_4qd_v.pdf][pdf]]) -[[file:figs/calib_4qd_v.png]] - -** Linear Regression -#+begin_src matlab - bh = [ones(size(uh.Unh)) uh.Unh]\uh.Vph; - bv = [ones(size(uv.Unv)) uv.Unv]\uv.Vpv; -#+end_src - -#+begin_src matlab :exports none - figure; - ax1 = subplot(1, 2, 1); - hold on; - plot(uh.Unh, uh.Vph, 'o', 'DisplayName', 'Exp. data'); - plot([min(uh.Unh) max(uh.Unh)], [min(uh.Unh) max(uh.Unh)].*bh(2) + bh(1), 'k--', 'DisplayName', sprintf('%.1e x + %.1e', bh(2), bh(1))) - hold off; - xlabel('$Un_h$ [V]'); ylabel('$Vp_h$ [V]'); - legend(); - - ax2 = subplot(1, 2, 2); - hold on; - plot(uv.Unv, uv.Vpv, 'o', 'DisplayName', 'Exp. data'); - plot([min(uv.Unv) max(uv.Unv)], [min(uv.Unv) max(uv.Unv)].*bv(2) + bv(1), 'k--', 'DisplayName', sprintf('%.1e x + %.1e', bv(2), bv(1))) - hold off; - xlabel('$Un_v$ [V]'); - ylabel('$Vp_v$ [V]'); - legend(); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/4qd_linear_reg.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:4qd_linear_reg -#+CAPTION: Linear Regression ([[./figs/4qd_linear_reg.png][png]], [[./figs/4qd_linear_reg.pdf][pdf]]) -[[file:figs/4qd_linear_reg.png]] - -** Gain of the photodiodes -If we note: -- $\alpha_o$ the angular displacement of the beam -- $\alpha_m$ the angular displacement of the mirror -- $V_{qd}$ the voltage measured by the 4QD -- $V_n$ the voltage input of the Newport - -The angular displacement of the beam is twice the mechanical angular displacement of the Newport mirror (supposing here that the mirror is flat and not concave): -\[ \alpha_o = 2 \alpha_m \] - -We want to determine $\frac{V_{qd}}{\alpha_o}$: -\begin{align*} - \frac{V_{qd}}{\alpha_o} &= \frac{1}{2} \frac{V_{qd}}{\frac{1}{2} \alpha_o} \\ - &= \frac{1}{2} \frac{V_{qd}}{V_n} \frac{V_n}{\alpha_m} -\end{align*} -with: -- $\frac{V_n}{\alpha_m} = \frac{1}{2.62 \cdot 10^{-3}}\ \left[\frac{V}{rad}\right]$ is the inverse of the gain of the Newport -- $\frac{V_{qd}}{V_n} = 0.185\ \left[\frac{V}{V}\right]$ is the identified DC gain of the transfer function from a voltage applied to the Newport to the 4QD measurement - -Thus: -#+begin_src matlab - Gqd = 0.5 * mean(diag(dcgain(Gn))) * 1/(2.62*10^-3); -#+end_src - -We obtain: -\[ \frac{\alpha_0}{V_{qd}} \approx 35.2\ \left[ \frac{V}{rad} \right] \] - -\[ \frac{\alpha_0}{V_{qd}} \approx 0.35\ \left[ \frac{mV}{\mu rad} \right] \] - * Plant Scaling - measured noise - expected perturbations @@ -1785,5 +2020,131 @@ Thus, we require the sensitivity function at $\omega_0 = 1\text{ Hz}$: In terms of loop gain, this is equivalent to: \[ |L(j\omega_0)| > 5 \cdot 10^{3} \] -* Control Design +* Decentralized Control +:PROPERTIES: +:header-args:matlab+: :tangle matlab/decentralized_control.m +:header-args:matlab+: :comments org :mkdirp yes +:END: +<> + +** Introduction :ignore: +In this section, we try to implement a simple decentralized controller. + +** ZIP file containing the data and matlab files :ignore: +#+begin_src bash :exports none :results none + if [ matlab/decentralized_control.m -nt data/decentralized_control.zip ]; then + cp matlab/decentralized_control.m decentralized_control.m; + zip data/decentralized_control \ + mat/plant.mat \ + decentralized_control.m + rm decentralized_control.m; + fi +#+end_src + +#+begin_note + All the files (data and Matlab scripts) are accessible [[file:data/decentralized_control.zip][here]]. +#+end_note + +** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab + freqs = logspace(0, 3, 1000); +#+end_src + +** Load Plant +#+begin_src matlab + load('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd'); +#+end_src + +** Diagonal Controller +Using =SISOTOOL=, a diagonal controller is designed. +The two SISO loop gains are shown in Fig. [[fig:diag_contr_loop_gain]]. +#+begin_src matlab + Kh = -0.25598*(s+112)*(s^2 + 15.93*s + 6.686e06)/((s^2*(s+352.5)*(1+s/2/pi/2000))); + Kv = 10207*(s+55.15)*(s^2 + 17.45*s + 2.491e06)/(s^2*(s+491.2)*(s+7695)); + + K = blkdiag(Kh, Kv); + K.InputName = {'Rh', 'Rv'}; + K.OutputName = {'Uch', 'Ucv'}; +#+end_src + +#+begin_src matlab :exports none + figure; + % Magnitude + ax1 = subaxis(2,1,1); + hold on; + plot(freqs, abs(squeeze(freqresp(Kh*sys('Rh', 'Uch'), freqs, 'Hz'))), 'DisplayName', '$L_h = K_h G_{d,h}^{-1} G_{\frac{V_{p,h}}{\tilde{U}_{c,h}}} G_{i,h} $'); + plot(freqs, abs(squeeze(freqresp(Kv*sys('Rv', 'Ucv'), freqs, 'Hz'))), 'DisplayName', '$L_v = K_v G_{d,v}^{-1} G_{\frac{V_{p,v}}{\tilde{U}_{c,v}}} G_{i,v} $'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); + ylabel('Magnitude [dB]'); + hold off; + legend('location', 'northeast'); + + % Phase + ax2 = subaxis(2,1,2); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(Kh*sys('Rh', 'Uch'), freqs, 'Hz')))); + plot(freqs, 180/pi*angle(squeeze(freqresp(Kv*sys('Rv', 'Ucv'), freqs, 'Hz')))); + set(gca,'xscale','log'); + yticks(-180:90:180); + ylim([-180 180]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/diag_contr_loop_gain.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:diag_contr_loop_gain +#+CAPTION: Loop Gain using the Decentralized Diagonal Controller ([[./figs/diag_contr_loop_gain.png][png]], [[./figs/diag_contr_loop_gain.pdf][pdf]]) +[[file:figs/diag_contr_loop_gain.png]] + +We then close the loop and we look at the transfer function from the Newport rotation signal to the beam angle (Fig. [[fig:diag_contr_effect_newport]]). +#+begin_src matlab + inputs = {'Uch', 'Ucv', 'Unh', 'Unv'}; + outputs = {'Vch', 'Vcv', 'Ich', 'Icv', 'Rh', 'Rv', 'Vph', 'Vpv'}; + + sys_cl = connect(sys, -K, inputs, outputs); +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(sys('Rh', 'Unh'), freqs, 'Hz'))), '-', 'DisplayName', 'OL - $R_h/U_{n,h}$'); + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(sys_cl('Rh', 'Unh'), freqs, 'Hz'))), '--', 'DisplayName', 'CL - $R_h/U_{n,h}$'); + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(sys('Rv', 'Unv'), freqs, 'Hz'))), '-', 'DisplayName', 'OL - $R_v/U_{n,v}$'); + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(sys_cl('Rv', 'Unv'), freqs, 'Hz'))), '--', 'DisplayName', 'CL - $R_v/U_{n,v}$'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Magnitude [dB]'); + hold off; + 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/diag_contr_effect_newport.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:diag_contr_effect_newport +#+CAPTION: Effect of the Newport rotation on the beam position when the loop is closed using the Decentralized Diagonal Controller ([[./figs/diag_contr_effect_newport.png][png]], [[./figs/diag_contr_effect_newport.pdf][pdf]]) +[[file:figs/diag_contr_effect_newport.png]] + * Measurement of the non-repeatability diff --git a/mat/plant.mat b/mat/plant.mat index 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squeeze(GaZcGi_resp(1,1,:)), 'k--', 'DisplayName', 'Theoretical') +title('FRF $G_{i,h} Z_{c,h} G_{a,h} = \frac{V_{c,h}}{U_{c,h}}$') +set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); +ylabel('Amplitude'); xlabel('Frequency [Hz]'); +legend(); +hold off; + +ax2 = subplot(1, 2, 2); +hold on; +plot(f, abs(GaZcGi_v), 'k-', 'DisplayName', 'Identified') +plot(f, squeeze(GaZcGi_resp(2,2,:)), 'k--', 'DisplayName', 'Theoretical') +title('FRF $G_{a,v} Z_{c,v} G_{i,v} = \frac{V_{c,v}}{U_{c,v}}$') +set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); +ylabel('Amplitude'); xlabel('Frequency [Hz]'); +legend(); +hold off; + +linkaxes([ax1,ax2],'xy'); +xlim([1, 2000]); + + + +% #+NAME: fig:current_amplifier_comp_theory_id_bis +% #+CAPTION: Identified and Theoretical Transfer Function $G_a G_i$ ([[./figs/current_amplifier_comp_theory_id_bis.png][png]], [[./figs/current_amplifier_comp_theory_id_bis.pdf][pdf]]) +% [[file:figs/current_amplifier_comp_theory_id_bis.png]] + +% Finally, we have the following transfer functions: + +Gi,Zc,Ga + +% Input / Output data +% The identification data is loaded + +uh = load('mat/data_uch.mat', 't', 'Uch', 'Vph', 'Vpv'); +uv = load('mat/data_ucv.mat', 't', 'Ucv', 'Vph', 'Vpv'); + + + +% We remove the first seconds where the Cercalo is turned on. + +t0 = 1; + +uh.Uch(uh.t(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 + +weight_Uch_Vph(f>1000) = 0; +weight_Uch_Vpv(f>1000) = 0; +weight_Ucv_Vph(f>1000) = 0; +weight_Ucv_Vpv(f>1000) = 0; + + + +% The weights are shown in Fig. [[fig:weights_cercalo]]. + + +figure; +hold on; +plot(f, weight_Uch_Vph, 'DisplayName', '$W_{U_{ch},V_{ph}}$'); +plot(f, weight_Uch_Vpv, 'DisplayName', '$W_{U_{ch},V_{pv}}$'); +plot(f, weight_Ucv_Vph, 'DisplayName', '$W_{U_{cv},V_{ph}}$'); +plot(f, weight_Ucv_Vpv, 'DisplayName', '$W_{U_{cv},V_{pv}}$'); +hold off; +xlabel('Frequency [Hz]'); ylabel('Weight Amplitude'); +set(gca, 'xscale', 'log'); +xlim([f(1), f(end)]); +legend('location', 'northwest'); + + + +% #+NAME: fig:weights_cercalo +% #+CAPTION: Weights amplitude ([[./figs/weights_cercalo.png][png]], [[./figs/weights_cercalo.pdf][pdf]]) +% [[file:figs/weights_cercalo.png]] + +% 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 + +figure; +ax11 = subplot(2, 2, 1); +hold on; +plot(f, abs(tf_Uch_Vph)) +plot(f, abs(fit_Uch_Vph)) +title('Frequency Response Function $\frac{Vp_h}{Uc_h}$') +set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); +ylabel('Amplitude [V/V]') +hold off; + +ax12 = subplot(2, 2, 2); +hold on; +plot(f, abs(tf_Ucv_Vph)) +plot(f, abs(fit_Ucv_Vph)) +title('Frequency Response Function $\frac{Vp_h}{Uc_v}$') +set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); +hold off; + +ax21 = subplot(2, 2, 3); +hold on; +plot(f, abs(tf_Uch_Vpv)) +plot(f, abs(fit_Uch_Vpv)) +title('Frequency Response Function $\frac{Vp_v}{Uc_h}$') +set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); +ylabel('Amplitude [V/V]') +xlabel('Frequency [Hz]') +hold off; + +ax22 = subplot(2, 2, 4); +hold on; +plot(f, abs(tf_Ucv_Vpv)) +plot(f, abs(fit_Ucv_Vpv)) +title('Frequency Response Function $\frac{Vp_v}{Uc_v}$') +set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); +xlabel('Frequency [Hz]') +hold off; + +linkaxes([ax11,ax12,ax21,ax22],'xy'); +xlim([10, 1000]); ylim([1e-2, 1e3]); + + + +% #+NAME: fig:identification_matrix_fit +% #+CAPTION: Transfer Function Extraction of the FRF matrix ([[./figs/identification_matrix_fit.png][png]], [[./figs/identification_matrix_fit.pdf][pdf]]) +% [[file:figs/identification_matrix_fit.png]] + + + +figure; +ax11 = subplot(2, 2, 1); +hold on; +plot(f, 180/pi*unwrap(angle(tf_Uch_Vph))) +plot(f, 180/pi*unwrap(angle(fit_Uch_Vph))) +title('Frequency Response Function $\frac{Vp_h}{Uc_h}$') +set(gca, 'Xscale', 'log'); +ylabel('Phase [deg]') +yticks(-180:90:180); +hold off; + +ax12 = subplot(2, 2, 2); +hold on; +plot(f, 180/pi*unwrap(angle(tf_Ucv_Vph))) +plot(f, 180/pi*unwrap(angle(fit_Ucv_Vph))) +title('Frequency Response Function $\frac{Vp_h}{Uc_v}$') +set(gca, 'Xscale', 'log'); +yticks(-180:90:180); +hold off; + +ax21 = subplot(2, 2, 3); +hold on; +plot(f, 180/pi*unwrap(angle(tf_Uch_Vpv))) +plot(f, 180/pi*unwrap(angle(fit_Uch_Vpv))) +title('Frequency Response Function $\frac{Vp_v}{Uc_h}$') +set(gca, 'Xscale', 'log'); +ylabel('Phase [deg]') +xlabel('Frequency [Hz]') +yticks(-180:90:180); +hold off; + +ax22 = subplot(2, 2, 4); +hold on; +plot(f, 180/pi*unwrap(angle(tf_Ucv_Vpv))) +plot(f, 180/pi*unwrap(angle(fit_Ucv_Vpv))) +title('Frequency Response Function $\frac{Vp_v}{Uc_v}$') +set(gca, 'Xscale', 'log'); +xlabel('Frequency [Hz]') +yticks(-180:90:180); +hold off; + +linkaxes([ax11,ax12,ax21,ax22],'xy'); +xlim([10, 1000]); ylim([-200, 200]); + + + +% #+NAME: fig:identification_matrix_fit_phase +% #+CAPTION: Transfer Function Extraction of the FRF matrix ([[./figs/identification_matrix_fit_phase.png][png]], [[./figs/identification_matrix_fit_phase.pdf][pdf]]) +% [[file:figs/identification_matrix_fit_phase.png]] + +% And finally, we create the identified $G_c$ matrix by multiplying by ${G_i}^{-1}$. + +G_Uch_Vph = tf(minreal(ss(full(SER_Uch_Vph.A),SER_Uch_Vph.B,SER_Uch_Vph.C,SER_Uch_Vph.D))); +G_Ucv_Vph = tf(minreal(ss(full(SER_Ucv_Vph.A),SER_Ucv_Vph.B,SER_Ucv_Vph.C,SER_Ucv_Vph.D))); +G_Uch_Vpv = tf(minreal(ss(full(SER_Uch_Vpv.A),SER_Uch_Vpv.B,SER_Uch_Vpv.C,SER_Uch_Vpv.D))); +G_Ucv_Vpv = tf(minreal(ss(full(SER_Ucv_Vpv.A),SER_Ucv_Vpv.B,SER_Ucv_Vpv.C,SER_Ucv_Vpv.D))); + +Gc = [G_Uch_Vph, G_Ucv_Vph; + G_Uch_Vpv, G_Ucv_Vpv]*inv(Gi); + +% Input / Output data +% The identification data is loaded + +uh = load('mat/data_unh.mat', 't', 'Unh', 'Vph', 'Vpv'); +uv = load('mat/data_unv.mat', 't', 'Unv', 'Vph', 'Vpv'); + + + +% We remove the first seconds where the Cercalo is turned on. + +t0 = 3; + +uh.Unh(uh.t10 & f<100)))), tf(mean(abs(tf_Unv_Vpv(f>10 & f<100))))); + +% Full System +% We now have identified: +% - $G_i$ +% - $G_a$ +% - $G_c$ +% - $G_n$ +% - $G_d$ + +% The file =mat/plant.mat= is accessible [[./mat/plant.mat][here]]. + +save('mat/plant.mat', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd'); diff --git a/matlab/decentralized_control.m b/matlab/decentralized_control.m new file mode 100644 index 0000000..2d486fa --- /dev/null +++ b/matlab/decentralized_control.m @@ -0,0 +1,77 @@ +%% Clear Workspace and Close figures +clear; close all; clc; + +%% Intialize Laplace variable +s = zpk('s'); + +freqs = logspace(0, 3, 1000); + +% Load Plant + +load('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd'); + +% Diagonal Controller +% Using =SISOTOOL=, a diagonal controller is designed. +% The two SISO loop gains are shown in Fig. [[fig:diag_contr_loop_gain]]. + +Kh = -0.25598*(s+112)*(s^2 + 15.93*s + 6.686e06)/((s^2*(s+352.5)*(1+s/2/pi/2000))); +Kv = 10207*(s+55.15)*(s^2 + 17.45*s + 2.491e06)/(s^2*(s+491.2)*(s+7695)); + +K = blkdiag(Kh, Kv); +K.InputName = {'Rh', 'Rv'}; +K.OutputName = {'Uch', 'Ucv'}; + +figure; +% Magnitude +ax1 = subaxis(2,1,1); +hold on; +plot(freqs, abs(squeeze(freqresp(Kh*sys('Rh', 'Uch'), freqs, 'Hz'))), 'DisplayName', '$L_h = K_h G_{d,h}^{-1} G_{\frac{V_{p,h}}{\tilde{U}_{c,h}}} G_{i,h} $'); +plot(freqs, abs(squeeze(freqresp(Kv*sys('Rv', 'Ucv'), freqs, 'Hz'))), 'DisplayName', '$L_v = K_v G_{d,v}^{-1} G_{\frac{V_{p,v}}{\tilde{U}_{c,v}}} G_{i,v} $'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); +ylabel('Magnitude [dB]'); +hold off; +legend('location', 'northeast'); + +% Phase +ax2 = subaxis(2,1,2); +hold on; +plot(freqs, 180/pi*angle(squeeze(freqresp(Kh*sys('Rh', 'Uch'), freqs, 'Hz')))); +plot(freqs, 180/pi*angle(squeeze(freqresp(Kv*sys('Rv', 'Ucv'), freqs, 'Hz')))); +set(gca,'xscale','log'); +yticks(-180:90:180); +ylim([-180 180]); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); + + + +% #+NAME: fig:diag_contr_loop_gain +% #+CAPTION: Loop Gain using the Decentralized Diagonal Controller ([[./figs/diag_contr_loop_gain.png][png]], [[./figs/diag_contr_loop_gain.pdf][pdf]]) +% [[file:figs/diag_contr_loop_gain.png]] + +% We then close the loop and we look at the transfer function from the Newport rotation signal to the beam angle (Fig. [[fig:diag_contr_effect_newport]]). + +inputs = {'Uch', 'Ucv', 'Unh', 'Unv'}; +outputs = {'Vch', 'Vcv', 'Ich', 'Icv', 'Rh', 'Rv', 'Vph', 'Vpv'}; + +sys_cl = connect(sys, -K, inputs, outputs); + +figure; +hold on; +set(gca,'ColorOrderIndex',1); +plot(freqs, abs(squeeze(freqresp(sys('Rh', 'Unh'), freqs, 'Hz'))), '-', 'DisplayName', 'OL - $R_h/U_{n,h}$'); +set(gca,'ColorOrderIndex',1); +plot(freqs, abs(squeeze(freqresp(sys_cl('Rh', 'Unh'), freqs, 'Hz'))), '--', 'DisplayName', 'CL - $R_h/U_{n,h}$'); +set(gca,'ColorOrderIndex',2); +plot(freqs, abs(squeeze(freqresp(sys('Rv', 'Unv'), freqs, 'Hz'))), '-', 'DisplayName', 'OL - $R_v/U_{n,v}$'); +set(gca,'ColorOrderIndex',2); +plot(freqs, abs(squeeze(freqresp(sys_cl('Rv', 'Unv'), freqs, 'Hz'))), '--', 'DisplayName', 'CL - $R_v/U_{n,v}$'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Magnitude [dB]'); +hold off; +xlim([freqs(1), freqs(end)]); +legend('location', 'southeast'); diff --git a/matlab/frf_processing.m b/matlab/frf_processing.m new file mode 100644 index 0000000..1ef0221 --- /dev/null +++ b/matlab/frf_processing.m @@ -0,0 +1,141 @@ +%% Clear Workspace and Close figures +clear; close all; clc; + +%% Intialize Laplace variable +s = zpk('s'); + +% Load Plant + +load('mat/plant.mat', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd'); + +% Test + +bodeFig({Ga*Zc*Gi}, struct('phase', true)); + +% TODO Huddle Test +% We load the data taken during the Huddle Test. + +load('mat/data_huddle_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(1-alpha)*freqs_res(freq_i) & omega<(1 + alpha)*2*pi*freqs_res(freq_i)) = freqs_res_weights(freq_i); -% end - - - -% Ignore data above some frequency. - -weight(f>1000) = 0; - -figure; -hold on; -plot(f, weight); -plot(freqs_res, ones(size(freqs_res)), 'rx'); -hold off; -xlabel('Frequency [Hz]'); -xlabel('Weight Amplitude'); -set(gca, 'xscale', 'log'); -xlim([f(1), f(end)]); - - - -% #+NAME: fig:weights -% #+CAPTION: Weights amplitude ([[./figs/weights.png][png]], [[./figs/weights.pdf][pdf]]) -% [[file:figs/weights.png]] - -% 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_ux_yx, poles, ~, fit_ux_yx] = vectfit3(tf_ux_yx.', 1i*2*pi*f, poles, weight, opts); -end -for iter = 1:Niter - [SER_uy_yx, poles, ~, fit_uy_yx] = vectfit3(tf_uy_yx.', 1i*2*pi*f, poles, weight, opts); -end -for iter = 1:Niter - [SER_ux_yy, poles, ~, fit_ux_yy] = vectfit3(tf_ux_yy.', 1i*2*pi*f, poles, weight, opts); -end -for iter = 1:Niter - [SER_uy_yy, poles, ~, fit_uy_yy] = vectfit3(tf_uy_yy.', 1i*2*pi*f, poles, weight, opts); -end - -figure; -ax11 = subplot(2, 2, 1); -hold on; -plot(f, abs(tf_ux_yx)) -plot(f, abs(fit_ux_yx)) -title('Frequency Response Function $\frac{y_x}{u_x}$') -set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); -ylabel('Amplitude') -hold off; - -ax12 = subplot(2, 2, 2); -hold on; -plot(f, abs(tf_uy_yx)) -plot(f, abs(fit_uy_yx)) -title('Frequency Response Function $\frac{y_x}{u_y}$') -set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); -hold off; - -ax21 = subplot(2, 2, 3); -hold on; -plot(f, abs(tf_ux_yy)) -plot(f, abs(fit_ux_yy)) -title('Frequency Response Function $\frac{y_y}{u_x}$') -set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); -ylabel('Amplitude') -xlabel('Frequency [Hz]') -hold off; - -ax22 = subplot(2, 2, 4); -hold on; -plot(f, abs(tf_uy_yy)) -plot(f, abs(fit_uy_yy)) -title('Frequency Response Function $\frac{y_y}{u_y}$') -set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); -xlabel('Frequency [Hz]') -hold off; - -linkaxes([ax11,ax12,ax21,ax22],'x'); -xlim([10, 1000]); -linkaxes([ax11,ax12,ax21,ax22],'y'); -ylim([1e-2, 1e3]) - - - -% #+NAME: fig:identification_matrix_fit -% #+CAPTION: Transfer Function Extraction of the FRF matrix ([[./figs/identification_matrix_fit.png][png]], [[./figs/identification_matrix_fit.pdf][pdf]]) -% [[file:figs/identification_matrix_fit.png]] - -% And finally, we create the identified state space model: - -G_ux_yx = minreal(ss(full(SER_ux_yx.A),SER_ux_yx.B,SER_ux_yx.C,SER_ux_yx.D)); -G_uy_yx = minreal(ss(full(SER_uy_yx.A),SER_uy_yx.B,SER_uy_yx.C,SER_uy_yx.D)); -G_ux_yy = minreal(ss(full(SER_ux_yy.A),SER_ux_yy.B,SER_ux_yy.C,SER_ux_yy.D)); -G_uy_yy = minreal(ss(full(SER_uy_yy.A),SER_uy_yy.B,SER_uy_yy.C,SER_uy_yy.D)); - -G = [G_ux_yx, G_uy_yx; - G_ux_yy, G_uy_yy]; - -save('mat/plant.mat', 'G'); diff --git a/run_test.m b/matlab/run_test.m similarity index 100% rename from run_test.m rename to matlab/run_test.m diff --git a/setup.m b/matlab/setup.m similarity index 100% rename from setup.m rename to 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