100 KiB
100 KiB
Simscape Uniaxial Model
- Introduction
- Simscape Model
- Undamped System
- Integral Force Feedback
- Relative Motion Control
- Direct Velocity Feedback
- With Cedrat Piezo-electric Actuators
- Comparison of Active Damping Techniques
Introduction ignore
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.
Simscape Model
A schematic of the uniaxial model used for simulations is represented in figure fig:uniaxial-model-nass-flexible.
The perturbations $w$ are:
- $F_s$: direct forces applied to the sample such as inertia forces and cable forces
- $F_{rz}$: parasitic forces due to the rotation of the spindle
- $F_{ty}$: parasitic forces due to scans with the translation stage
- $D_w$: ground motion
The quantity to $z$ to control is:
- $D$: the position of the sample with respect to the granite
The measured quantities $v$ are:
- $D$: the position of the sample with respect to the granite
We study the use of an additional sensor:
- $F_n$: a force sensor located in the nano-hexapod
- $v_n$: an absolute velocity sensor located on the top platform of the nano-hexapod
- $d_r$: a relative motion sensor located in the nano-hexapod
The control signal $u$ is:
- $F$ the force applied by the nano-hexapod actuator
Few active damping techniques will be compared in order to decide which sensor is to be included in the system. Schematics of the active damping techniques are displayed in figure fig:uniaxial-model-nass-flexible-active-damping.
Undamped System
Introduction ignore
Let's start by study the undamped system.
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.
All the controllers are set to 0 (Open Loop).
Identification
We identify the dynamics of the system.
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'sim_nano_station_uniaxial';The inputs and outputs are defined below and corresponds to the name of simulink blocks.
%% 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
Finally, we use the linearize Matlab function to extract a state space model from the simscape model.
%% 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]Finally, we save the identified system dynamics for further analysis.
save('./uniaxial/mat/plants.mat', 'G');Sensitivity to Disturbances
We show several plots representing the sensitivity to disturbances:
- in figure fig:uniaxial-sensitivity-disturbances the transfer functions from ground motion $D_w$ to the sample position $D$ and the transfer function from direct force on the sample $F_s$ to the sample position $D$ are shown
- in figure fig:uniaxial-sensitivity-force-dist, it is the effect of parasitic forces of the positioning stages ($F_{ty}$ and $F_{rz}$) on the position $D$ of the sample that are shown
<<plt-matlab>>
<<plt-matlab>>
Plant
The transfer function from the force $F$ applied by the nano-hexapod to the position of the sample $D$ is shown in figure fig:uniaxial-plant. It corresponds to the plant to control.
<<plt-matlab>>
Integral Force Feedback
<<sec:iff>>
Introduction ignore
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.
<<plt-matlab>>
The controller for each pair of actuator/sensor is:
K_iff = -1000/s; <<plt-matlab>>
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] save('./uniaxial/mat/plants.mat', 'G_iff', '-append');Sensitivity to Disturbance
Conclusion
Integral Force Feedback:
Relative Motion Control
<<sec:rmc>>
Introduction ignore
In the Relative Motion Control (RMC), a derivative feedback is applied between the measured actuator displacement to the actuator force input.
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.
<<plt-matlab>>
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); <<plt-matlab>>
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] save('./uniaxial/mat/plants.mat', 'G_rmc', '-append');Sensitivity to Disturbance
Conclusion
Relative Motion Control:
Direct Velocity Feedback
<<sec:dvf>>
Introduction ignore
In the Relative Motion Control (RMC), a feedback is applied between the measured velocity of the platform to the actuator force input.
Control Design
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] save('./uniaxial/mat/plants.mat', 'G_dvf', '-append');Sensitivity to Disturbance
Conclusion
Direct Velocity Feedback:
With Cedrat Piezo-electric Actuators
Identification
We identify the dynamics of the system.
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'sim_nano_station_uniaxial_cedrat';The inputs and outputs are defined below and corresponds to the name of simulink blocks.
%% 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
Finally, we use the linearize Matlab function to extract a state space model from the simscape model.
%% 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]Control Design
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.
<<plt-matlab>>
The controller for each pair of actuator/sensor is:
K_cedrat = 1000/s; <<plt-matlab>>
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_cedrat;
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_cedrat'; %% 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_cedrat = linearize(mdl, io, options);
G_cedrat.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_cedrat.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] save('./uniaxial/mat/plants.mat', 'G_cedrat', '-append');Sensitivity to Disturbance
Conclusion
This gives similar results than with a classical force sensor.
Comparison of Active Damping Techniques
<<sec:comparison>>
Load the plants
load('./uniaxial/mat/plants.mat', 'G', 'G_iff', 'G_rmc', 'G_dvf');Sensitivity to Disturbance
Conclusion
#name: tab:active_damping_comparison
| IFF | RMC | DVF | |
|---|---|---|---|
| Sensor Type | Force sensor | Relative Motion | Inertial |
| Guaranteed Stability | + | + | - |
| Sensitivity ($D_w$) | - | + | - |
| Sensitivity ($F_s$) | - (at low freq) | + | + |
| Sensitivity ($F_{ty,rz}$) | + | - | + |
The next step is to take into account the power spectral density of each disturbance.