Active Damping applied on the Simscape Model

1. Undamped System
- 1.1. Identification of the dynamics for Active Damping
- 1.2. Tomography Experiment
  - 1.2.1. Simulation
  - 1.2.2. Results
2. Integral Force Feedback
3. Direct Velocity Feedback
4. Inertial Control
5. Comparison
6. Useful Functions
- 6.1. prepareTomographyExperiment

First, in section 1, we will looked at the undamped system.

Then, we will compare three active damping techniques:

In section 2: the integral force feedback is used
In section 3: the direct velocity feedback is used
In section 4: inertial control is used

For each of the active damping technique, we will:

Look at the damped plant
Simulate tomography experiments
Compare the sensitivity from disturbances

The disturbances are:

Ground motion
Motion errors of all the stages

1 Undamped System

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

We first look at the undamped system. The performance of this undamped system will be compared with the damped system using various techniques.

1.1 Identification of the dynamics for Active Damping

1.1.1 Initialize the Simulation

We initialize all the stages with the default parameters.

initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();

The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.

initializeNanoHexapod('actuator', 'piezo');
initializeSample('mass', 50);

We set the references to zero.

initializeReferences();

And all the controllers are set to 0.

K = tf(zeros(6));
save('./mat/controllers.mat', 'K', '-append');
K_ine = tf(zeros(6));
save('./mat/controllers.mat', 'K_ine', '-append');
K_iff = tf(zeros(6));
save('./mat/controllers.mat', 'K_iff', '-append');
K_dvf = tf(zeros(6));
save('./mat/controllers.mat', 'K_dvf', '-append');

1.1.2 Identification

First, we identify the dynamics of the system using the linearize function.

%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'sim_nass_active_damping';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fnl'],           1, 'openinput');              io_i = io_i + 1;
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Vlm');  io_i = io_i + 1;

%% Run the linearization
G = linearize(mdl, io, options);
G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ...
                'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6', ...
                'Vnlm1', 'Vnlm2', 'Vnlm3', 'Vnlm4', 'Vnlm5', 'Vnlm6'};

We then create transfer functions corresponding to the active damping plants.

G_iff = minreal(G({'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}));
G_dvf = minreal(G({'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}));
G_ine = minreal(G({'Vnlm1', 'Vnlm2', 'Vnlm3', 'Vnlm4', 'Vnlm5', 'Vnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}));

And we save them for further analysis.

save('./active_damping/mat/undamped_plants.mat', 'G_iff', 'G_dvf', 'G_ine');

1.1.3 Obtained Plants for Active Damping

Figure 1: G_iff: IFF Plant (png, pdf)

Figure 2: G_dvf: Plant for Direct Velocity Feedback (png, pdf)

Figure 3: Inertial Feedback Plant (png, pdf)

1.2 Tomography Experiment

1.2.1 Simulation

We initialize elements for the tomography experiment.

prepareTomographyExperiment();

We change the simulation stop time.

load('mat/conf_simscape.mat');
set_param(conf_simscape, 'StopTime', '3');

And we simulate the system.

sim('sim_nass_active_damping');

Finally, we save the simulation results for further analysis

save('./active_damping/mat/tomo_exp.mat', 'En', 'Eg', '-append');

1.2.2 Results

We load the results of tomography experiments.

load('./active_damping/mat/tomo_exp.mat', 'En');
t = linspace(0, 3, length(En(:,1)));

Figure 4: Position Error during tomography experiment - Translations (png, pdf)

Figure 5: Position Error during tomography experiment - Rotations (png, pdf)

2 Integral Force Feedback

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

Integral Force Feedback is applied on the simscape model.

2.1 Control Design

2.1.1 Plant

Let’s load the previously indentified undamped plant:

load('./active_damping/mat/undamped_plants.mat', 'G_iff');

Let’s look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor (figure 6).

Figure 6: Transfer function from forces applied in the legs to force sensor (png, pdf)

2.1.2 Control Design

The controller for each pair of actuator/sensor is:

K_iff = 1000/s;

The corresponding loop gains are shown in figure 7.

Figure 7: Loop Gain for the Integral Force Feedback (png, pdf)

2.1.3 Diagonal Controller

We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.

K_iff = -K_iff*eye(6);

We save the controller for further analysis.

save('./active_damping/mat/K_iff.mat', 'K_iff');

2.1.4 IFF with High Pass Filter

w_hpf = 2*pi*10; % Cut-off frequency for the high pass filter [rad/s]
w_lpf = 2*pi*200; % Cut-off frequency for the low pass filter [rad/s]

K_iff = 2*pi*200/s * (s/w_hpf)/(s/w_hpf + 1) * 1/(s/w_lpf + 1);

The corresponding loop gains are shown in figure 8.

Figure 8: Loop Gain for the Integral Force Feedback with an High pass filter (png, pdf)

We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.

K_iff = -K_iff*eye(6);

We save the controller for further analysis.

save('./active_damping/mat/K_iff_hpf.mat', 'K_iff');

2.2 Tomography Experiment

2.2.1 Simulation with IFF Controller

We initialize elements for the tomography experiment.

prepareTomographyExperiment();

We set the IFF controller.

load('./active_damping/mat/K_iff.mat', 'K_iff');
save('./mat/controllers.mat', 'K_iff', '-append');

We change the simulation stop time.

load('mat/conf_simscape.mat');
set_param(conf_simscape, 'StopTime', '3');

And we simulate the system.

sim('sim_nass_active_damping');

Finally, we save the simulation results for further analysis

En_iff = En;
Eg_iff = Eg;
save('./active_damping/mat/tomo_exp.mat', 'En_iff', 'Eg_iff', '-append');

2.2.2 Simulation with IFF Controller with added High Pass Filter

We initialize elements for the tomography experiment.

prepareTomographyExperiment();

We set the IFF controller with the High Pass Filter.

load('./active_damping/mat/K_iff_hpf.mat', 'K_iff');
save('./mat/controllers.mat', 'K_iff', '-append');

We change the simulation stop time.

load('mat/conf_simscape.mat');
set_param(conf_simscape, 'StopTime', '3');

And we simulate the system.

sim('sim_nass_active_damping');

Finally, we save the simulation results for further analysis

En_iff_hpf = En;
Eg_iff_hpf = Eg;
save('./active_damping/mat/tomo_exp.mat', 'En_iff_hpf', 'Eg_iff_hpf', '-append');

2.2.3 Compare with Undamped system

We load the results of tomography experiments.

load('./active_damping/mat/tomo_exp.mat', 'En', 'En_iff', 'En_iff_hpf');
t = linspace(0, 3, length(En(:,1)));

Figure 9: Position Error during tomography experiment - XY Motion (png, pdf)

Figure 10: Position Error during tomography experiment - Translations (png, pdf)

Figure 11: Position Error during tomography experiment - Rotations (png, pdf)

2.3 Conclusion

Integral Force Feedback:

Robust (guaranteed stability)
Acceptable Damping
Increase the sensitivity to disturbances at low frequencies

3 Direct Velocity Feedback

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

In the Direct Velocity Feedback (DVF), a derivative feedback is applied between the measured actuator displacement to the actuator force input. The actuator displacement can be measured with a capacitive sensor for instance.

3.1 Control Design

3.1.1 Plant

Let’s load the undamped plant:

load('./active_damping/mat/undamped_plants.mat', 'G_dvf');

Let’s look at the transfer function from actuator forces in the nano-hexapod to the measured displacement of the actuator for all 6 pairs of actuator/sensor (figure 12).

Figure 12: Transfer function from forces applied in the legs to leg displacement sensor (png, pdf)

3.1.2 Control Design

The Direct Velocity Feedback is defined below. A Low pass Filter is added to make the controller transfer function proper.

K_dvf = s*20000/(1 + s/2/pi/10000);

The obtained loop gains are shown in figure 13.

Figure 13: Loop Gain for the Integral Force Feedback (png, pdf)

3.1.3 Diagonal Controller

We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.

K_dvf = -K_dvf*eye(6);

We save the controller for further analysis.

save('./active_damping/mat/K_dvf.mat', 'K_dvf');

3.2 Tomography Experiment

3.2.1 Initialize the Simulation

We initialize elements for the tomography experiment.

prepareTomographyExperiment();

We set the DVF controller.

load('./active_damping/mat/K_dvf.mat', 'K_dvf');
save('./mat/controllers.mat', 'K_dvf', '-append');

3.2.2 Simulation

We change the simulation stop time.

load('mat/conf_simscape.mat');
set_param(conf_simscape, 'StopTime', '3');

And we simulate the system.

sim('sim_nass_active_damping');

Finally, we save the simulation results for further analysis

En_dvf = En;
Eg_dvf = Eg;
save('./active_damping/mat/tomo_exp.mat', 'En_dvf', 'Eg_dvf', '-append');

3.2.3 Compare with Undamped system

We load the results of tomography experiments.

load('./active_damping/mat/tomo_exp.mat', 'En', 'En_dvf');
t = linspace(0, 3, length(En(:,1)));

Figure 14: Position Error during tomography experiment - XY Motion (png, pdf)

Figure 15: Position Error during tomography experiment - Translations (png, pdf)

Figure 16: Position Error during tomography experiment - Rotations (png, pdf)

3.3 Conclusion

Direct Velocity Feedback:

4 Inertial Control

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

In Inertial Control, a feedback is applied between the measured absolute motion (velocity or acceleration) of the platform to the actuator force input.

4.1 Control Design

4.1.1 Plant

Let’s load the undamped plant:

load('./active_damping/mat/undamped_plants.mat', 'G_ine');

Let’s look at the transfer function from actuator forces in the nano-hexapod to the measured velocity of the nano-hexapod platform in the direction of the corresponding actuator for all 6 pairs of actuator/sensor (figure 17).

Figure 17: Transfer function from forces applied in the legs to leg velocity sensor (png, pdf)

4.1.2 Control Design

The controller is defined below and the obtained loop gain is shown in figure 18.

K_ine = 1e4/(1+s/(2*pi*100));

Figure 18: Loop Gain for Inertial Control (png, pdf)

4.1.3 Diagonal Controller

We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.

K_ine = -K_ine*eye(6);

We save the controller for further analysis.

save('./active_damping/mat/K_ine.mat', 'K_ine');

4.2 Tomography Experiment

4.2.1 Initialize the Simulation

We initialize elements for the tomography experiment.

prepareTomographyExperiment();

We set the Inertial controller.

load('./active_damping/mat/K_ine.mat', 'K_ine');
save('./mat/controllers.mat', 'K_ine', '-append');

4.2.2 Simulation

We change the simulation stop time.

load('mat/conf_simscape.mat');
set_param(conf_simscape, 'StopTime', '3');

And we simulate the system.

sim('sim_nass_active_damping');

Finally, we save the simulation results for further analysis

En_ine = En;
Eg_ine = Eg;
save('./active_damping/mat/tomo_exp.mat', 'En_ine', 'Eg_ine', '-append');

4.2.3 Compare with Undamped system

We load the results of tomography experiments.

load('./active_damping/mat/tomo_exp.mat', 'En', 'En_ine');
t = linspace(0, 3, length(En_ine(:,1)));

Figure 19: Position Error during tomography experiment - XY Motion (png, pdf)

Figure 20: Position Error during tomography experiment - Translations (png, pdf)

Figure 21: Position Error during tomography experiment - Rotations (png, pdf)

4.3 Conclusion

Inertial Control:

5 Comparison

5.1 Load the plants

load('./active_damping/mat/plants.mat', 'G', 'G_iff', 'G_ine', 'G_dvf');

5.2 Sensitivity to Disturbance

Figure 22: caption (png, pdf)

Figure 23: caption (png, pdf)

Figure 24: caption (png, pdf)

Figure 25: caption (png, pdf)

Figure 26: caption (png, pdf)

5.3 Damped Plant

Figure 27: Plant for the \(z\) direction for different active damping technique used (png, pdf)

Figure 28: Plant for the \(x\) direction for different active damping technique used (png, pdf)

Figure 29: Comparison of one off-diagonal plant for different damping technique applied (png, pdf)

5.4 Tomography Experiment

load('./active_damping/mat/tomo_exp.mat', 'En', 'En_iff', 'En_dvf', 'En_ine');
t = linspace(0, 3, length(En(:,1)));

rms(sqrt(En(:, 1).^2     + En(:, 2).^2     + En(:, 3).^2))
rms(sqrt(En_ine(:, 1).^2 + En_ine(:, 2).^2 + En_ine(:, 3).^2))
rms(sqrt(En_dvf(:, 1).^2 + En_dvf(:, 2).^2 + En_dvf(:, 3).^2))
rms(sqrt(En_iff(:, 1).^2 + En_iff(:, 2).^2 + En_iff(:, 3).^2))

5.4.1 Frequency Domain

Ts = t(2)-t(1); % Sample Time for the Data [s]

n_av = 8;
han_win = hanning(ceil(length(En(:, 1))/n_av));

[pxx,     f] = pwelch(En(:,     1), han_win, [], [], 1/Ts);
[pxx_ine, ~] = pwelch(En_ine(:, 1), han_win, [], [], 1/Ts);
[pxx_dvf, ~] = pwelch(En_dvf(:, 1), han_win, [], [], 1/Ts);
[pxx_iff, ~] = pwelch(En_iff(:, 1), han_win, [], [], 1/Ts);

[pyy,     ~] = pwelch(En(:,     2), han_win, [], [], 1/Ts);
[pyy_ine, ~] = pwelch(En_ine(:, 2), han_win, [], [], 1/Ts);
[pyy_dvf, ~] = pwelch(En_dvf(:, 2), han_win, [], [], 1/Ts);
[pyy_iff, ~] = pwelch(En_iff(:, 2), han_win, [], [], 1/Ts);

[pzz,     ~] = pwelch(En(:,     3), han_win, [], [], 1/Ts);
[pzz_ine, ~] = pwelch(En_ine(:, 3), han_win, [], [], 1/Ts);
[pzz_dvf, ~] = pwelch(En_dvf(:, 3), han_win, [], [], 1/Ts);
[pzz_iff, ~] = pwelch(En_iff(:, 3), han_win, [], [], 1/Ts);

[prx,     ~] = pwelch(En(:,     4), han_win, [], [], 1/Ts);
[prx_ine, ~] = pwelch(En_ine(:, 4), han_win, [], [], 1/Ts);
[prx_dvf, ~] = pwelch(En_dvf(:, 4), han_win, [], [], 1/Ts);
[prx_iff, ~] = pwelch(En_iff(:, 4), han_win, [], [], 1/Ts);

[pry,     ~] = pwelch(En(:,     5), han_win, [], [], 1/Ts);
[pry_ine, ~] = pwelch(En_ine(:, 5), han_win, [], [], 1/Ts);
[pry_dvf, ~] = pwelch(En_dvf(:, 5), han_win, [], [], 1/Ts);
[pry_iff, ~] = pwelch(En_iff(:, 5), han_win, [], [], 1/Ts);

[prz,     ~] = pwelch(En(:,     6), han_win, [], [], 1/Ts);
[prz_ine, ~] = pwelch(En_ine(:, 6), han_win, [], [], 1/Ts);
[prz_dvf, ~] = pwelch(En_dvf(:, 6), han_win, [], [], 1/Ts);
[prz_iff, ~] = pwelch(En_iff(:, 6), han_win, [], [], 1/Ts);

figure;
hold on;
plot(f, prx_ine,    'DisplayName', 'Inertial')
plot(f, prx_dvf,    'DisplayName', 'DVF')
plot(f, prx_iff,    'DisplayName', 'IFF')
plot(f, prx, 'k--', 'DisplayName', 'Undamped')
hold off;
xlabel('Frequency [Hz]');
ylabel('Power Spectral Density [$m^2/Hz$]');
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
legend('location', 'northeast');
xlim([1, 500]);

figure;
hold on;
plot(f, pxx_ine,    'DisplayName', 'Inertial')
plot(f, pxx_dvf,    'DisplayName', 'DVF')
plot(f, pxx_iff,    'DisplayName', 'IFF')
plot(f, pxx, 'k--', 'DisplayName', 'Undamped')
hold off;
xlabel('Frequency [Hz]');
ylabel('Power Spectral Density [$m^2/Hz$]');
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
legend('location', 'northeast');
xlim([1, 500]);

6 Useful Functions

6.1 prepareTomographyExperiment

This Matlab function is accessible here.

Function Description

function [] = prepareTomographyExperiment(args)

Optional Parameters

arguments
    args.nass_actuator       char   {mustBeMember(args.nass_actuator,{'piezo', 'lorentz'})} = 'piezo'
    args.sample_mass   (1,1) double {mustBeNumeric, mustBePositive} = 50
    args.Ry_period     (1,1) double {mustBeNumeric, mustBePositive} = 1
end

Initialize the Simulation

We initialize all the stages with the default parameters.

initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();

The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.

initializeNanoHexapod('actuator', args.nass_actuator);
initializeSample('mass', args.sample_mass);

We set the references to zero.

initializeReferences('Rz_type', 'rotating', 'Rz_period', args.Ry_period);

And all the controllers are set to 0.

K = tf(zeros(6));
save('./mat/controllers.mat', 'K', '-append');
K_ine = tf(zeros(6));
save('./mat/controllers.mat', 'K_ine', '-append');
K_iff = tf(zeros(6));
save('./mat/controllers.mat', 'K_iff', '-append');
K_dvf = tf(zeros(6));
save('./mat/controllers.mat', 'K_dvf', '-append');

Active Damping applied on the Simscape Model

Table of Contents

1 Undamped System

1.1 Identification of the dynamics for Active Damping

1.1.1 Initialize the Simulation

1.1.2 Identification

1.1.3 Obtained Plants for Active Damping

1.2 Tomography Experiment

1.2.1 Simulation

1.2.2 Results

2 Integral Force Feedback

2.1 Control Design

2.1.1 Plant

2.1.2 Control Design

2.1.3 Diagonal Controller

2.1.4 IFF with High Pass Filter

2.2 Tomography Experiment

2.2.1 Simulation with IFF Controller

2.2.2 Simulation with IFF Controller with added High Pass Filter

2.2.3 Compare with Undamped system

2.3 Conclusion

3 Direct Velocity Feedback

3.1 Control Design

3.1.1 Plant

3.1.2 Control Design

3.1.3 Diagonal Controller

3.2 Tomography Experiment

3.2.1 Initialize the Simulation

3.2.2 Simulation

3.2.3 Compare with Undamped system

3.3 Conclusion

4 Inertial Control

4.1 Control Design

4.1.1 Plant

4.1.2 Control Design

4.1.3 Diagonal Controller

4.2 Tomography Experiment

4.2.1 Initialize the Simulation

4.2.2 Simulation

4.2.3 Compare with Undamped system

4.3 Conclusion

5 Comparison

5.1 Load the plants

5.2 Sensitivity to Disturbance

5.3 Damped Plant

5.4 Tomography Experiment

5.4.1 Frequency Domain

6 Useful Functions

6.1 prepareTomographyExperiment

Function Description

Optional Parameters

Initialize the Simulation