UP | HOME

Simscape Uniaxial Model

Table of Contents

The idea is to use the same model as the full Simscape Model but to restrict the motion only in the vertical direction.

This is done in order to more easily study the system and evaluate control techniques.

1 Undamped System

1.1 Init

We initialize all the stages with the default parameters. The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg. 

initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeNanoHexapod(struct('actuator', 'piezo'));
initializeSample(struct('mass', 50));

All the controllers are set to 0. 

K = tf(0);
save('./mat/controllers.mat', 'K', '-append');
K_iff = tf(0);
save('./mat/controllers.mat', 'K_iff', '-append');
K_rmc = tf(0);
save('./mat/controllers.mat', 'K_rmc', '-append');
K_dvf = tf(0);
save('./mat/controllers.mat', 'K_dvf', '-append');

1.2 Identification

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

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

%% Input/Output definition
io(1)  = linio([mdl, '/Dw'],    1, 'input');  % Ground Motion
io(2)  = linio([mdl, '/Fs'],    1, 'input');  % Force applied on the sample
io(3)  = linio([mdl, '/Fnl'],   1, 'input');  % Force applied by the NASS
io(4)  = linio([mdl, '/Fdty'],  1, 'input');  % Parasitic force Ty
io(5)  = linio([mdl, '/Fdrz'],  1, 'input');  % Parasitic force Rz

io(6)  = linio([mdl, '/Dsm'],  1, 'output'); % Displacement of the sample
io(7)  = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
io(8)  = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
io(9)  = linio([mdl, '/Dgm'],  1, 'output'); % Absolute displacement of the granite
io(10) = linio([mdl, '/Vlm'],  1, 'output'); % Measured absolute velocity of the top NASS platform

%% Run the linearization
G = linearize(mdl, io, options);
G.InputName  = {'Dw',   ... % Ground Motion [m]
                'Fs',   ... % Force Applied on Sample [N]
                'Fn',   ... % Force applied by NASS [N]
                'Fty',  ... % Parasitic Force Ty [N]
                'Frz'};     % Parasitic Force Rz [N]
G.OutputName = {'D',    ... % Measured sample displacement x.r.t. granite [m]
                'Fnm',  ... % Force Sensor in NASS [N]
                'Dnm',  ... % Displacement Sensor in NASS [m]
                'Dgm',  ... % Asbolute displacement of Granite [m]
                'Vlm'}; ... % Absolute Velocity of NASS [m/s]

1.3 Sensitivity to Disturbances

uniaxial-sensitivity-disturbances.png

Figure 1: Sensitivity to disturbances (png, pdf)

uniaxial-sensitivity-force-dist.png

Figure 2: Sensitivity to disturbances (png, pdf)

1.4 Plant

uniaxial-plant.png

Figure 3: Bode plot of the Plant (png, pdf)

1.5 Save

save('./uniaxial/mat/plants.mat', 'G');

2 Integral Force Feedback

2.1 Control Design

load('./uniaxial/mat/plants.mat', 'G');

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.

uniaxial_iff_plant.png

Figure 4: Transfer function from forces applied in the legs to force sensor (png, pdf)

The controller for each pair of actuator/sensor is: 

K_iff = -1000/s;

uniaxial_iff_open_loop.png

Figure 5: Loop Gain for the Integral Force Feedback (png, pdf)

2.2 Identification

Let's initialize the system prior to identification. 

initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeNanoHexapod(struct('actuator', 'piezo'));
initializeSample(struct('mass', 50));

All the controllers are set to 0. 

K = tf(0);
save('./mat/controllers.mat', 'K', '-append');
K_iff = -K_iff;
save('./mat/controllers.mat', 'K_iff', '-append');
K_rmc = tf(0);
save('./mat/controllers.mat', 'K_rmc', '-append');
K_dvf = tf(0);
save('./mat/controllers.mat', 'K_dvf', '-append');

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

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

%% Input/Output definition
io(1)  = linio([mdl, '/Dw'],    1, 'input');  % Ground Motion
io(2)  = linio([mdl, '/Fs'],    1, 'input');  % Force applied on the sample
io(3)  = linio([mdl, '/Fnl'],   1, 'input');  % Force applied by the NASS
io(4)  = linio([mdl, '/Fdty'],  1, 'input');  % Parasitic force Ty
io(5)  = linio([mdl, '/Fdrz'],  1, 'input');  % Parasitic force Rz

io(6)  = linio([mdl, '/Dsm'],  1, 'output'); % Displacement of the sample
io(7)  = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
io(8)  = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
io(9)  = linio([mdl, '/Dgm'],  1, 'output'); % Absolute displacement of the granite
io(10) = linio([mdl, '/Vlm'],  1, 'output'); % Measured absolute velocity of the top NASS platform

%% Run the linearization
G_iff = linearize(mdl, io, options);
G_iff.InputName  = {'Dw',   ... % Ground Motion [m]
                    'Fs',   ... % Force Applied on Sample [N]
                    'Fn',   ... % Force applied by NASS [N]
                    'Fty',  ... % Parasitic Force Ty [N]
                    'Frz'};     % Parasitic Force Rz [N]
G_iff.OutputName = {'D',    ... % Measured sample displacement x.r.t. granite [m]
                    'Fnm',  ... % Force Sensor in NASS [N]
                    'Dnm',  ... % Displacement Sensor in NASS [m]
                    'Dgm',  ... % Asbolute displacement of Granite [m]
                    'Vlm'}; ... % Absolute Velocity of NASS [m/s]

2.3 Sensitivity to Disturbance

uniaxial_sensitivity_dist_iff.png

Figure 6: Sensitivity to disturbance once the IFF controller is applied to the system (png, pdf)

uniaxial_sensitivity_dist_stages_iff.png

Figure 7: Sensitivity to force disturbances in various stages when IFF is applied (png, pdf)

2.4 Damped Plant

uniaxial_plant_iff_damped.png

Figure 8: Damped Plant after IFF is applied (png, pdf)

2.5 Save

save('./uniaxial/mat/plants.mat', 'G_iff', '-append');

2.6 Conclusion

Integral Force Feedback:

3 Relative Motion Control

In the Relative Motion Control (RMC), a derivative feedback is applied between the measured actuator displacement to the actuator force input.

3.1 Control Design

load('./uniaxial/mat/plants.mat', 'G');

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.

uniaxial_rmc_plant.png

Figure 9: Transfer function from forces applied in the legs to leg displacement sensor (png, pdf)

The Relative Motion Controller is defined below. A Low pass Filter is added to make the controller transfer function proper. 

K_rmc = s*50000/(1 + s/2/pi/10000);

uniaxial_rmc_open_loop.png

Figure 10: Loop Gain for the Integral Force Feedback (png, pdf)

3.2 Identification

Let's initialize the system prior to identification. 

initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeNanoHexapod(struct('actuator', 'piezo'));
initializeSample(struct('mass', 50));

And initialize the controllers. 

K = tf(0);
save('./mat/controllers.mat', 'K', '-append');
K_iff = tf(0);
save('./mat/controllers.mat', 'K_iff', '-append');
K_rmc = -K_rmc;
save('./mat/controllers.mat', 'K_rmc', '-append');
K_dvf = tf(0);
save('./mat/controllers.mat', 'K_dvf', '-append');

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

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

%% Input/Output definition
io(1)  = linio([mdl, '/Dw'],    1, 'input');  % Ground Motion
io(2)  = linio([mdl, '/Fs'],    1, 'input');  % Force applied on the sample
io(3)  = linio([mdl, '/Fnl'],   1, 'input');  % Force applied by the NASS
io(4)  = linio([mdl, '/Fdty'],  1, 'input');  % Parasitic force Ty
io(5)  = linio([mdl, '/Fdrz'],  1, 'input');  % Parasitic force Rz

io(6)  = linio([mdl, '/Dsm'],  1, 'output'); % Displacement of the sample
io(7)  = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
io(8)  = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
io(9)  = linio([mdl, '/Dgm'],  1, 'output'); % Absolute displacement of the granite
io(10) = linio([mdl, '/Vlm'],  1, 'output'); % Measured absolute velocity of the top NASS platform

%% Run the linearization
G_rmc = linearize(mdl, io, options);
G_rmc.InputName  = {'Dw',   ... % Ground Motion [m]
                    'Fs',   ... % Force Applied on Sample [N]
                    'Fn',   ... % Force applied by NASS [N]
                    'Fty',  ... % Parasitic Force Ty [N]
                    'Frz'};     % Parasitic Force Rz [N]
G_rmc.OutputName = {'D',    ... % Measured sample displacement x.r.t. granite [m]
                    'Fnm',  ... % Force Sensor in NASS [N]
                    'Dnm',  ... % Displacement Sensor in NASS [m]
                    'Dgm',  ... % Asbolute displacement of Granite [m]
                    'Vlm'}; ... % Absolute Velocity of NASS [m/s]

3.3 Sensitivity to Disturbance

uniaxial_sensitivity_dist_rmc.png

Figure 11: Sensitivity to disturbance once the RMC controller is applied to the system (png, pdf)

uniaxial_sensitivity_dist_stages_rmc.png

Figure 12: Sensitivity to force disturbances in various stages when RMC is applied (png, pdf)

3.4 Damped Plant

uniaxial_plant_rmc_damped.png

Figure 13: Damped Plant after RMC is applied (png, pdf)

3.5 Save

save('./uniaxial/mat/plants.mat', 'G_rmc', '-append');

3.6 Conclusion

Relative Motion Control:

4 Direct Velocity Feedback

In the Relative Motion Control (RMC), a feedback is applied between the measured velocity of the platform to the actuator force input.

4.1 Control Design

load('./uniaxial/mat/plants.mat', 'G');

uniaxial_dvf_plant.png

Figure 14: Transfer function from forces applied in the legs to leg velocity sensor (png, pdf)

K_dvf = tf(5e4);

uniaxial_dvf_loop_gain.png

Figure 15: Transfer function from forces applied in the legs to leg velocity sensor (png, pdf)

4.2 Identification

Let's initialize the system prior to identification. 

initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeNanoHexapod(struct('actuator', 'piezo'));
initializeSample(struct('mass', 50));

And initialize the controllers. 

K = tf(0);
save('./mat/controllers.mat', 'K', '-append');
K_iff = tf(0);
save('./mat/controllers.mat', 'K_iff', '-append');
K_rmc = tf(0);
save('./mat/controllers.mat', 'K_rmc', '-append');
K_dvf = -K_dvf;
save('./mat/controllers.mat', 'K_dvf', '-append');

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

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

%% Input/Output definition
io(1)  = linio([mdl, '/Dw'],    1, 'input');  % Ground Motion
io(2)  = linio([mdl, '/Fs'],    1, 'input');  % Force applied on the sample
io(3)  = linio([mdl, '/Fnl'],   1, 'input');  % Force applied by the NASS
io(4)  = linio([mdl, '/Fdty'],  1, 'input');  % Parasitic force Ty
io(5)  = linio([mdl, '/Fdrz'],  1, 'input');  % Parasitic force Rz

io(6)  = linio([mdl, '/Dsm'],  1, 'output'); % Displacement of the sample
io(7)  = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
io(8)  = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
io(9)  = linio([mdl, '/Dgm'],  1, 'output'); % Absolute displacement of the granite
io(10) = linio([mdl, '/Vlm'],  1, 'output'); % Measured absolute velocity of the top NASS platform

%% Run the linearization
G_dvf = linearize(mdl, io, options);
G_dvf.InputName  = {'Dw',   ... % Ground Motion [m]
                    'Fs',   ... % Force Applied on Sample [N]
                    'Fn',   ... % Force applied by NASS [N]
                    'Fty',  ... % Parasitic Force Ty [N]
                    'Frz'};     % Parasitic Force Rz [N]
G_dvf.OutputName = {'D',    ... % Measured sample displacement x.r.t. granite [m]
                    'Fnm',  ... % Force Sensor in NASS [N]
                    'Dnm',  ... % Displacement Sensor in NASS [m]
                    'Dgm',  ... % Asbolute displacement of Granite [m]
                    'Vlm'}; ... % Absolute Velocity of NASS [m/s]

4.3 Sensitivity to Disturbance

uniaxial_sensitivity_dist_dvf.png

Figure 16: Sensitivity to disturbance once the DVF controller is applied to the system (png, pdf)

uniaxial_sensitivity_dist_stages_dvf.png

Figure 17: Sensitivity to force disturbances in various stages when DVF is applied (png, pdf)

4.4 Damped Plant

uniaxial_plant_dvf_damped.png

Figure 18: Damped Plant after DVF is applied (png, pdf)

4.5 Save

save('./uniaxial/mat/plants.mat', 'G_dvf', '-append');

4.6 Conclusion

Direct Velocity Feedback:

5 Comparison of Active Damping Techniques

5.1 Load the plants

load('./uniaxial/mat/plants.mat', 'G', 'G_iff', 'G_rmc', 'G_dvf');

5.2 Sensitivity to Disturbance

uniaxial_sensitivity_dist_comp.png

Figure 19: Sensitivity to disturbance - Comparison (png, pdf)

uniaxial_sensitivity_dist_stages_comp.png

Figure 20: Sensitivity to force disturbances - Comparison (png, pdf)

5.3 Damped Plant

uniaxial_plant_damped_comp.png

Figure 21: Damped Plant - Comparison (png, pdf)

5.4 Conclusion

Author: Dehaeze Thomas

Created: 2019-10-24 jeu. 17:44

Validate