From 75b39626e302f8cb343a4f7f60b6549384f4aadc Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 17 Dec 2019 18:04:21 +0100 Subject: [PATCH] Create a new folder for active damping --- active_damping/index.org | 3 - active_damping_uniaxial/figs | 1 + active_damping_uniaxial/index.html | 1548 +++++++++++++ active_damping_uniaxial/index.org | 2063 +++++++++++++++++ .../mat/plants.mat | Bin .../matlab/dvf.m | 0 .../matlab/iff.m | 0 .../matlab/rmc.m | 0 .../matlab/sim_nano_station_id.slx | Bin index.html | 82 +- index.org | 7 +- 11 files changed, 3663 insertions(+), 41 deletions(-) create mode 120000 active_damping_uniaxial/figs create mode 100644 active_damping_uniaxial/index.html create mode 100644 active_damping_uniaxial/index.org rename {active_damping => active_damping_uniaxial}/mat/plants.mat (100%) rename {active_damping => active_damping_uniaxial}/matlab/dvf.m (100%) rename {active_damping => active_damping_uniaxial}/matlab/iff.m (100%) rename {active_damping => active_damping_uniaxial}/matlab/rmc.m (100%) rename {active_damping => active_damping_uniaxial}/matlab/sim_nano_station_id.slx (100%) diff --git a/active_damping/index.org b/active_damping/index.org index 37d3a3b..6e70a4a 100644 --- a/active_damping/index.org +++ b/active_damping/index.org @@ -2058,6 +2058,3 @@ Direct Velocity Feedback: #+NAME: fig:plant_comp_damping_coupling #+CAPTION: Comparison of one off-diagonal plant for different damping technique applied ([[./figs/plant_comp_damping_coupling.png][png]], [[./figs/plant_comp_damping_coupling.pdf][pdf]]) [[file:figs/plant_comp_damping_coupling.png]] - -* Conclusion -<> diff --git a/active_damping_uniaxial/figs b/active_damping_uniaxial/figs new file mode 120000 index 0000000..d440220 --- /dev/null +++ b/active_damping_uniaxial/figs @@ -0,0 +1 @@ +../figs/ \ No newline at end of file diff --git a/active_damping_uniaxial/index.html b/active_damping_uniaxial/index.html new file mode 100644 index 0000000..32bd328 --- /dev/null +++ b/active_damping_uniaxial/index.html @@ -0,0 +1,1548 @@ + + + + + + + +Active Damping + + + + + + + + + + + + + + + +
+ UP + | + HOME +
+

Active Damping

+ + +

+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 relative motion control is used
    • +
  • In section 4: the direct velocity feedback is used
    • +
+ +

+For each of the active damping technique, we will: +

+
    +
  • Compare the sensitivity from disturbances
    • +
  • Look at the damped plant
    • +
+ +

+The disturbances are: +

+
    +
  • Ground motion
    • +
  • Direct forces
    • +
  • 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 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. +

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

1.2 Identification

+
+

+We identify the various transfer functions of the system +

+
+
G = identifyPlant();
+
+
+ +

+And we save it for further analysis. +

+
+
save('./active_damping/mat/plants.mat', 'G', '-append');
+
+
+
+
+ +
+

1.3 Sensitivity to disturbances

+
+

+The sensitivity to disturbances are shown on figure 1. +

+ + +
+

sensitivity_dist_undamped.png +

+

Figure 1: Undamped sensitivity to disturbances (png, pdf)

+
+ + +
+

sensitivity_dist_stages.png +

+

Figure 2: Sensitivity to force disturbances in various stages (png, pdf)

+
+
+
+ +
+

1.4 Undamped Plant

+
+

+The "plant" (transfer function from forces applied by the nano-hexapod to the measured displacement of the sample with respect to the granite) bode plot is shown on figure 1. +

+ + +
+

plant_undamped.png +

+

Figure 3: Transfer Function from cartesian forces to displacement for the undamped plant (png, pdf)

+
+
+
+
+ +
+

2 Integral Force Feedback

+
+

+ +

+
+

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

+ +
+

+Integral Force Feedback is applied. +In section 2.1, IFF is applied on a uni-axial system to understand its behavior. +Then, it is applied on the simscape model. +

+
+ +
+

2.1 One degree-of-freedom example

+
+

+ +

+
+
+

2.1.1 Equations

+
+ +
+

iff_1dof.png +

+

Figure 4: Integral Force Feedback applied to a 1dof system

+
+ +

+The dynamic of the system is described by the following equation: +

+\begin{equation} + ms^2x = F_d - kx - csx + kw + csw + F +\end{equation} +

+The measured force \(F_m\) is: +

+\begin{align} + F_m &= F - kx - csx + kw + csw \\ + &= ms^2 x - F_d +\end{align} +

+The Integral Force Feedback controller is \(K = -\frac{g}{s}\), and thus the applied force by this controller is: +

+\begin{equation} + F_{\text{IFF}} = -\frac{g}{s} F_m = -\frac{g}{s} (ms^2 x - F_d) +\end{equation} +

+Once the IFF is applied, the new dynamics of the system is: +

+\begin{equation} + ms^2x = F_d + F - kx - csx + kw + csw - \frac{g}{s} (ms^2x - F_d) +\end{equation} + +

+And finally: +

+\begin{equation} + x = F_d \frac{1 + \frac{g}{s}}{ms^2 + (mg + c)s + k} + F \frac{1}{ms^2 + (mg + c)s + k} + w \frac{k + cs}{ms^2 + (mg + c)s + k} +\end{equation} + +

+We can see that this: +

+
    +
  • adds damping to the system by a value \(mg\)
    • +
  • lower the compliance as low frequency by a factor: \(1 + g/s\)
    • +
+ +

+If we want critical damping: +

+\begin{equation} + \xi = \frac{1}{2} \frac{c + gm}{\sqrt{km}} = \frac{1}{2} +\end{equation} + +

+This is attainable if we have: +

+\begin{equation} + g = \frac{\sqrt{km} - c}{m} +\end{equation} +
+
+ +
+

2.1.2 Matlab Example

+
+

+Let define the system parameters. +

+
+
m = 50; % [kg]
+k = 1e6; % [N/m]
+c = 1e3; % [N/(m/s)]
+
+
+ +

+The state space model of the system is defined below. +

+
+
A = [-c/m -k/m;
+     1     0];
+
+B = [1/m 1/m -1;
+     0   0    0];
+
+C = [ 0  1;
+     -c -k];
+
+D = [0 0 0;
+     1 0 0];
+
+sys = ss(A, B, C, D);
+sys.InputName = {'F', 'Fd', 'wddot'};
+sys.OutputName = {'d', 'Fm'};
+sys.StateName = {'ddot', 'd'};
+
+
+ +

+The controller \(K_\text{IFF}\) is: +

+
+
Kiff = -((sqrt(k*m)-c)/m)/s;
+Kiff.InputName = {'Fm'};
+Kiff.OutputName = {'F'};
+
+
+ +

+And the closed loop system is computed below. +

+
+
sys_iff = feedback(sys, Kiff, 'name', +1);
+
+
+ + +
+

iff_1dof_sensitivitiy.png +

+

Figure 5: Sensitivity to disturbance when IFF is applied on the 1dof system (png, pdf)

+
+
+
+
+ +
+

2.2 Control Design

+
+

+Let's load the undamped plant: +

+
+
load('./active_damping/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 (figure 6). +

+ + +
+

iff_plant.png +

+

Figure 6: 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;
+
+
+ +

+The corresponding loop gains are shown in figure 7. +

+ + +
+

iff_open_loop.png +

+

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

+
+
+
+ +
+

2.3 Identification of the damped plant

+
+

+Let's initialize the system prior to identification. +

+
+
initializeReferences();
+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(zeros(6));
+save('./mat/controllers.mat', 'K', '-append');
+K_iff = -K_iff*eye(6);
+save('./mat/controllers.mat', 'K_iff', '-append');
+K_rmc = tf(zeros(6));
+save('./mat/controllers.mat', 'K_rmc', '-append');
+K_dvf = tf(zeros(6));
+save('./mat/controllers.mat', 'K_dvf', '-append');
+
+
+ +

+We identify the system dynamics now that the IFF controller is ON. +

+
+
G_iff = identifyPlant();
+
+
+ +

+And we save the damped plant for further analysis +

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

2.4 Sensitivity to disturbances

+
+

+As shown on figure 8: +

+
    +
  • The top platform of the nano-hexapod how behaves as a "free-mass".
    • +
  • The transfer function from direct forces \(F_s\) to the relative displacement \(D\) is equivalent to the one of an isolated mass.
    • +
  • The transfer function from ground motion \(D_g\) to the relative displacement \(D\) tends to the transfer function from \(D_g\) to the displacement of the granite (the sample is being isolated thanks to IFF). +However, as the goal is to make the relative displacement \(D\) as small as possible (e.g. to make the sample motion follows the granite motion), this is not a good thing.
    • +
+ + +
+

sensitivity_dist_iff.png +

+

Figure 8: Sensitivity to disturbance once the IFF controller is applied to the system (png, pdf)

+
+ +
+

+The order of the models are very high and thus the plots may be wrong. +For instance, the plots are not the same when using minreal. +

+ +
+ + +
+

sensitivity_dist_stages_iff.png +

+

Figure 9: Sensitivity to force disturbances in various stages when IFF is applied (png, pdf)

+
+
+
+ +
+

2.5 Damped Plant

+
+

+Now, look at the new damped plant to control. +

+ +

+It damps the plant (resonance of the nano hexapod as well as other resonances) as shown in figure 10. +

+ + +
+

plant_iff_damped.png +

+

Figure 10: Damped Plant after IFF is applied (png, pdf)

+
+ +

+However, it increases coupling at low frequency (figure 11). +

+ +
+

plant_iff_coupling.png +

+

Figure 11: Coupling induced by IFF (png, pdf)

+
+
+
+ +
+

2.6 Conclusion

+
+
+

+Integral Force Feedback: +

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

3 Relative Motion Control

+
+

+ +

+
+

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

+ +
+

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

+
+ +
+

3.1 One degree-of-freedom example

+
+

+ +

+
+
+

3.1.1 Equations

+
+ +
+

rmc_1dof.png +

+

Figure 12: Relative Motion Control applied to a 1dof system

+
+ +

+The dynamic of the system is: +

+\begin{equation} + ms^2x = F_d - kx - csx + kw + csw + F +\end{equation} +

+In terms of the stage deformation \(d = x - w\): +

+\begin{equation} + (ms^2 + cs + k) d = -ms^2 w + F_d + F +\end{equation} +

+The relative motion control law is: +

+\begin{equation} + K = -g s +\end{equation} +

+Thus, the applied force is: +

+\begin{equation} + F = -g s d +\end{equation} +

+And the new dynamics will be: +

+\begin{equation} + d = w \frac{-ms^2}{ms^2 + (c + g)s + k} + F_d \frac{1}{ms^2 + (c + g)s + k} + F \frac{1}{ms^2 + (c + g)s + k} +\end{equation} + +

+And thus damping is added. +

+ +

+If critical damping is wanted: +

+\begin{equation} + \xi = \frac{1}{2}\frac{c + g}{\sqrt{km}} = \frac{1}{2} +\end{equation} +

+This corresponds to a gain: +

+\begin{equation} + g = \sqrt{km} - c +\end{equation} +
+
+ +
+

3.1.2 Matlab Example

+
+

+Let define the system parameters. +

+
+
m = 50; % [kg]
+k = 1e6; % [N/m]
+c = 1e3; % [N/(m/s)]
+
+
+ +

+The state space model of the system is defined below. +

+
+
A = [-c/m -k/m;
+     1     0];
+
+B = [1/m 1/m -1;
+     0   0    0];
+
+C = [ 0  1;
+     -c -k];
+
+D = [0 0 0;
+     1 0 0];
+
+sys = ss(A, B, C, D);
+sys.InputName = {'F', 'Fd', 'wddot'};
+sys.OutputName = {'d', 'Fm'};
+sys.StateName = {'ddot', 'd'};
+
+
+ +

+The controller \(K_\text{RMC}\) is: +

+
+
Krmc = -(sqrt(k*m)-c)*s;
+Krmc.InputName = {'d'};
+Krmc.OutputName = {'F'};
+
+
+ +

+And the closed loop system is computed below. +

+
+
sys_rmc = feedback(sys, Krmc, 'name', +1);
+
+
+ + +
+

rmc_1dof_sensitivitiy.png +

+

Figure 13: Sensitivity to disturbance when RMC is applied on the 1dof system (png, pdf)

+
+
+
+
+ +
+

3.2 Control Design

+
+

+Let's load the undamped plant: +

+
+
load('./active_damping/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 (figure 14). +

+ + +
+

rmc_plant.png +

+

Figure 14: 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);
+
+
+ +

+The obtained loop gains are shown in figure 15. +

+ + +
+

rmc_open_loop.png +

+

Figure 15: Loop Gain for the Integral Force Feedback (png, pdf)

+
+
+
+ +
+

3.3 Identification of the damped plant

+
+

+Let's initialize the system prior to identification. +

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

+And initialize the controllers. +

+
+
K = tf(zeros(6));
+save('./mat/controllers.mat', 'K', '-append');
+K_iff = tf(zeros(6));
+save('./mat/controllers.mat', 'K_iff', '-append');
+K_rmc = -K_rmc*eye(6);
+save('./mat/controllers.mat', 'K_rmc', '-append');
+K_dvf = tf(zeros(6));
+save('./mat/controllers.mat', 'K_dvf', '-append');
+
+
+ +

+We identify the system dynamics now that the RMC controller is ON. +

+
+
G_rmc = identifyPlant();
+
+
+ +

+And we save the damped plant for further analysis. +

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

3.4 Sensitivity to disturbances

+
+

+As shown in figure 16, RMC control succeed in lowering the sensitivity to disturbances near resonance of the system. +

+ + +
+

sensitivity_dist_rmc.png +

+

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

+
+ + +
+

sensitivity_dist_stages_rmc.png +

+

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

+
+
+
+ +
+

3.5 Damped Plant

+
+ +
+

plant_rmc_damped.png +

+

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

+
+
+
+ +
+

3.6 Conclusion

+
+
+

+Relative Motion Control: +

+
    +
  • +
+ +
+
+
+
+ +
+

4 Direct Velocity Feedback

+
+

+ +

+
+

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

+ +
+

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

+
+ +
+

4.1 One degree-of-freedom example

+
+

+ +

+
+
+

4.1.1 Equations

+
+ +
+

dvf_1dof.png +

+

Figure 19: Direct Velocity Feedback applied to a 1dof system

+
+ +

+The dynamic of the system is: +

+\begin{equation} + ms^2x = F_d - kx - csx + kw + csw + F +\end{equation} +

+In terms of the stage deformation \(d = x - w\): +

+\begin{equation} + (ms^2 + cs + k) d = -ms^2 w + F_d + F +\end{equation} +

+The direct velocity feedback law shown in figure 19 is: +

+\begin{equation} + K = -g +\end{equation} +

+Thus, the applied force is: +

+\begin{equation} + F = -g \dot{x} +\end{equation} +

+And the new dynamics will be: +

+\begin{equation} + d = w \frac{-ms^2 - gs}{ms^2 + (c + g)s + k} + F_d \frac{1}{ms^2 + (c + g)s + k} + F \frac{1}{ms^2 + (c + g)s + k} +\end{equation} + +

+And thus damping is added. +

+ +

+If critical damping is wanted: +

+\begin{equation} + \xi = \frac{1}{2}\frac{c + g}{\sqrt{km}} = \frac{1}{2} +\end{equation} +

+This corresponds to a gain: +

+\begin{equation} + g = \sqrt{km} - c +\end{equation} +
+
+ +
+

4.1.2 Matlab Example

+
+

+Let define the system parameters. +

+
+
m = 50; % [kg]
+k = 1e6; % [N/m]
+c = 1e3; % [N/(m/s)]
+
+
+ +

+The state space model of the system is defined below. +

+
+
A = [-c/m -k/m;
+     1     0];
+
+B = [1/m 1/m -1;
+     0   0    0];
+
+C = [1 0;
+     0 1;
+     0 0];
+
+D = [0 0 0;
+     0 0 0;
+     0 0 1];
+
+sys = ss(A, B, C, D);
+sys.InputName = {'F', 'Fd', 'wddot'};
+sys.OutputName = {'ddot', 'd', 'wddot'};
+sys.StateName = {'ddot', 'd'};
+
+
+ +

+Because we need \(\dot{x}\) for feedback, we compute it from the outputs +

+
+
G_xdot = [1, 0, 1/s;
+          0, 1, 0];
+G_xdot.InputName = {'ddot', 'd', 'wddot'};
+G_xdot.OutputName = {'xdot', 'd'};
+
+
+ +

+Finally, the system is described by sys as defined below. +

+
+
sys = series(sys, G_xdot, [1 2 3], [1 2 3]);
+
+
+ +

+The controller \(K_\text{DVF}\) is: +

+
+
Kdvf = tf(-(sqrt(k*m)-c));
+Kdvf.InputName = {'xdot'};
+Kdvf.OutputName = {'F'};
+
+
+ +

+And the closed loop system is computed below. +

+
+
sys_dvf = feedback(sys, Kdvf, 'name', +1);
+
+
+ +

+The obtained sensitivity to disturbances is shown in figure 20. +

+ +
+

dvf_1dof_sensitivitiy.png +

+

Figure 20: Sensitivity to disturbance when DVF is applied on the 1dof system (png, pdf)

+
+
+
+
+ +
+

4.2 Control Design

+
+

+Let's load the undamped plant: +

+
+
load('./active_damping/mat/plants.mat', 'G');
+
+
+ +

+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 21). +

+ + +
+

dvf_plant.png +

+

Figure 21: Transfer function from forces applied in the legs to leg velocity sensor (png, pdf)

+
+ +

+The controller is defined below and the obtained loop gain is shown in figure 22. +

+ +
+
K_dvf = tf(3e4);
+
+
+ + +
+

dvf_open_loop_gain.png +

+

Figure 22: Loop Gain for DVF (png, pdf)

+
+
+
+ +
+

4.3 Identification of the damped plant

+
+

+Let's initialize the system prior to identification. +

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

+And initialize the controllers. +

+
+
K = tf(zeros(6));
+save('./mat/controllers.mat', 'K', '-append');
+K_iff = tf(zeros(6));
+save('./mat/controllers.mat', 'K_iff', '-append');
+K_rmc = tf(zeros(6));
+save('./mat/controllers.mat', 'K_rmc', '-append');
+K_dvf = -K_dvf*eye(6);
+save('./mat/controllers.mat', 'K_dvf', '-append');
+
+
+ +

+We identify the system dynamics now that the RMC controller is ON. +

+
+
G_dvf = identifyPlant();
+
+
+ +

+And we save the damped plant for further analysis. +

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

4.4 Sensitivity to disturbances

+
+ +
+

sensitivity_dist_dvf.png +

+

Figure 23: Sensitivity to disturbance once the DVF controller is applied to the system (png, pdf)

+
+ + + +
+

sensitivity_dist_stages_dvf.png +

+

Figure 24: Sensitivity to force disturbances in various stages when DVF is applied (png, pdf)

+
+
+
+ +
+

4.5 Damped Plant

+
+ +
+

plant_dvf_damped.png +

+

Figure 25: Damped Plant after DVF is applied (png, pdf)

+
+
+
+ +
+

4.6 Conclusion

+
+
+

+Direct Velocity Feedback: +

+ +
+
+
+
+ +
+

5 Comparison

+
+

+ +

+
+
+

5.1 Load the plants

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

5.2 Sensitivity to Disturbance

+
+ +
+

sensitivity_comp_ground_motion_z.png +

+

Figure 26: caption (png, pdf)

+
+ + + +
+

sensitivity_comp_direct_forces_z.png +

+

Figure 27: caption (png, pdf)

+
+ + +
+

sensitivity_comp_spindle_z.png +

+

Figure 28: caption (png, pdf)

+
+ + +
+

sensitivity_comp_ty_z.png +

+

Figure 29: caption (png, pdf)

+
+ + + +
+

sensitivity_comp_ty_x.png +

+

Figure 30: caption (png, pdf)

+
+
+
+ +
+

5.3 Damped Plant

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+ +
+

plant_comp_damping_z.png +

+

Figure 31: Plant for the \(z\) direction for different active damping technique used (png, pdf)

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+ + +
+

plant_comp_damping_x.png +

+

Figure 32: Plant for the \(x\) direction for different active damping technique used (png, pdf)

+
+ + +
+

plant_comp_damping_coupling.png +

+

Figure 33: Comparison of one off-diagonal plant for different damping technique applied (png, pdf)

+
+
+
+
+ +
+

6 Conclusion

+
+

+ +

+
+
+
+
+

Author: Dehaeze Thomas

+

Created: 2019-12-11 mer. 16:55

+

Validate

+
+ + diff --git a/active_damping_uniaxial/index.org b/active_damping_uniaxial/index.org new file mode 100644 index 0000000..37d3a3b --- /dev/null +++ b/active_damping_uniaxial/index.org @@ -0,0 +1,2063 @@ +#+TITLE: Active Damping +:DRAWER: +#+STARTUP: overview + +#+LANGUAGE: en +#+EMAIL: dehaeze.thomas@gmail.com +#+AUTHOR: Dehaeze Thomas + +#+HTML_LINK_HOME: ../index.html +#+HTML_LINK_UP: ../index.html + +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: + +#+HTML_MATHJAX: align: center tagside: right font: TeX + +#+PROPERTY: header-args:matlab :session *MATLAB* +#+PROPERTY: header-args:matlab+ :comments org +#+PROPERTY: header-args:matlab+ :results none +#+PROPERTY: header-args:matlab+ :exports both +#+PROPERTY: header-args:matlab+ :eval no-export +#+PROPERTY: header-args:matlab+ :output-dir figs +#+PROPERTY: header-args:matlab+ :tangle no +#+PROPERTY: header-args:matlab+ :mkdirp yes + +#+PROPERTY: header-args:shell :eval no-export + +#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/}{config.tex}") +#+PROPERTY: header-args:latex+ :imagemagick t :fit yes +#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150 +#+PROPERTY: header-args:latex+ :imoutoptions -quality 100 +#+PROPERTY: header-args:latex+ :results raw replace :buffer no +#+PROPERTY: header-args:latex+ :eval no-export +#+PROPERTY: header-args:latex+ :exports both +#+PROPERTY: header-args:latex+ :mkdirp yes +#+PROPERTY: header-args:latex+ :output-dir figs +:END: + +* Introduction :ignore: +First, in section [[sec:undamped_system]], we will looked at the undamped system. + +Then, we will compare three active damping techniques: +- In section [[sec:iff]]: the integral force feedback is used +- In section [[sec:rmc]]: the relative motion control is used +- In section [[sec:dvf]]: the direct velocity feedback is used + +For each of the active damping technique, we will: +- Compare the sensitivity from disturbances +- Look at the damped plant + +The disturbances are: +- Ground motion +- Direct forces +- Motion errors of all the stages + +* Undamped System +:PROPERTIES: +:header-args:matlab+: :tangle matlab/undamped_system.m +:header-args:matlab+: :comments none :mkdirp yes +:END: +<> + +** ZIP file containing the data and matlab files :ignore: +#+begin_src bash :exports none :results none + if [ matlab/undamped_system.m -nt data/undamped_system.zip ]; then + cp matlab/undamped_system.m undamped_system.m; + zip data/undamped_system \ + undamped_system.m + rm undamped_system.m; + fi +#+end_src + +#+begin_note + All the files (data and Matlab scripts) are accessible [[file:data/undamped_system.zip][here]]. +#+end_note + +** Introduction :ignore: +We first look at the undamped system. +The performance of this undamped system will be compared with the damped system using various techniques. + +** 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 :tangle no + simulinkproject('../'); +#+end_src + +#+begin_src matlab + open('active_damping/matlab/sim_nano_station_id.slx') +#+end_src + +** 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. + +#+begin_src matlab + initializeReferences(); + initializeGround(); + initializeGranite(); + initializeTy(); + initializeRy(); + initializeRz(); + initializeMicroHexapod(); + initializeAxisc(); + initializeMirror(); + initializeNanoHexapod(struct('actuator', 'piezo')); + initializeSample(struct('mass', 50)); +#+end_src + +All the controllers are set to 0. +#+begin_src matlab + K = tf(zeros(6)); + save('./mat/controllers.mat', 'K', '-append'); + K_iff = tf(zeros(6)); + save('./mat/controllers.mat', 'K_iff', '-append'); + K_rmc = tf(zeros(6)); + save('./mat/controllers.mat', 'K_rmc', '-append'); + K_dvf = tf(zeros(6)); + save('./mat/controllers.mat', 'K_dvf', '-append'); +#+end_src + +** Identification +We identify the various transfer functions of the system +#+begin_src matlab + G = identifyPlant(); +#+end_src + +And we save it for further analysis. +#+begin_src matlab + save('./active_damping/mat/plants.mat', 'G', '-append'); +#+end_src + +** Sensitivity to disturbances +The sensitivity to disturbances are shown on figure [[fig:sensitivity_dist_undamped]]. + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + subplot(2, 1, 1); + title('$D_g$ to $D$'); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / D_{g,x}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / D_{g,y}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / D_{g,z}\right|$'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]'); + legend('location', 'southeast'); + + subplot(2, 1, 2); + title('$F_s$ to $D$'); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{s,x}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{s,y}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{s,z}\right|$'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/sensitivity_dist_undamped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:sensitivity_dist_undamped +#+CAPTION: Undamped sensitivity to disturbances ([[./figs/sensitivity_dist_undamped.png][png]], [[./figs/sensitivity_dist_undamped.pdf][pdf]]) +[[file:figs/sensitivity_dist_undamped.png]] + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{rz, z}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{ty, z}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{ty, x}\right|$'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/sensitivity_dist_stages.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:sensitivity_dist_stages +#+CAPTION: Sensitivity to force disturbances in various stages ([[./figs/sensitivity_dist_stages.png][png]], [[./figs/sensitivity_dist_stages.pdf][pdf]]) +[[file:figs/sensitivity_dist_stages.png]] + +** Undamped Plant +The "plant" (transfer function from forces applied by the nano-hexapod to the measured displacement of the sample with respect to the granite) bode plot is shown on figure [[fig:sensitivity_dist_undamped]]. + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'DisplayName', '$D_x / F_x$'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))), 'DisplayName', '$D_y / F_y$'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'DisplayName', '$D_z / F_z$'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'southwest'); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/plant_undamped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:plant_undamped +#+CAPTION: Transfer Function from cartesian forces to displacement for the undamped plant ([[./figs/plant_undamped.png][png]], [[./figs/plant_undamped.pdf][pdf]]) +[[file:figs/plant_undamped.png]] + +* Integral Force Feedback +:PROPERTIES: +:header-args:matlab+: :tangle matlab/iff.m +:header-args:matlab+: :comments none :mkdirp yes +:END: +<> + +** ZIP file containing the data and matlab files :ignore: +#+begin_src bash :exports none :results none + if [ matlab/iff.m -nt data/iff.zip ]; then + cp matlab/iff.m iff.m; + zip data/iff \ + mat/plant.mat \ + iff.m + rm iff.m; + fi +#+end_src + +#+begin_note + All the files (data and Matlab scripts) are accessible [[file:data/iff.zip][here]]. +#+end_note + +** Introduction :ignore: +Integral Force Feedback is applied. +In section [[sec:iff_1dof]], IFF is applied on a uni-axial system to understand its behavior. +Then, it is applied on the simscape model. + +** One degree-of-freedom example +:PROPERTIES: +:header-args:matlab+: :tangle no +:END: +<> +*** Equations +#+begin_src latex :file iff_1dof.pdf :post pdf2svg(file=*this*, ext="png") :exports results + \begin{tikzpicture} + % Ground + \draw (-1, 0) -- (1, 0); + + % Ground Displacement + \draw[dashed] (-1, 0) -- ++(-0.5, 0) coordinate(w); + \draw[->] (w) -- ++(0, 0.5) node[left]{$w$}; + + % Mass + \draw[fill=white] (-1, 1.4) rectangle ++(2, 0.8) node[pos=0.5]{$m$}; + \node[forcesensor={0.4}{0.4}] (fsensn) at (0, 1){}; + \draw[] (-0.8, 1) -- (0.8, 1); + \node[left] at (fsensn.west) {$F_m$}; + + % Displacement of the mass + \draw[dashed] (-1, 2.2) -- ++(-0.5, 0) coordinate(x); + \draw[->] (x) -- ++(0, 0.5) node[left]{$x$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.8, 0) -- (-0.8, 1) node[midway, left=0.1]{$k$}; + \draw[damper] (0, 0) -- (0, 1) node[midway, left=0.2]{$c$}; + \draw[actuator={0.4}{0.2}] (0.8, 0) -- (0.8, 1) coordinate[midway, right=0.1](F); + + % Displacements + \node[block={0.8cm}{0.6cm}, right=0.6 of F] (Kiff) {$K$}; + \draw[->] (Kiff.west) -- (F) node[above right]{$F$}; + \draw[<-] (Kiff.east) -- ++(0.5, 0) |- (fsensn.east); + \end{tikzpicture} +#+end_src + +#+name: fig:iff_1dof +#+caption: Integral Force Feedback applied to a 1dof system +#+RESULTS: +[[file:figs/iff_1dof.png]] + +The dynamic of the system is described by the following equation: +\begin{equation} + ms^2x = F_d - kx - csx + kw + csw + F +\end{equation} +The measured force $F_m$ is: +\begin{align} + F_m &= F - kx - csx + kw + csw \\ + &= ms^2 x - F_d +\end{align} +The Integral Force Feedback controller is $K = -\frac{g}{s}$, and thus the applied force by this controller is: +\begin{equation} + F_{\text{IFF}} = -\frac{g}{s} F_m = -\frac{g}{s} (ms^2 x - F_d) +\end{equation} +Once the IFF is applied, the new dynamics of the system is: +\begin{equation} + ms^2x = F_d + F - kx - csx + kw + csw - \frac{g}{s} (ms^2x - F_d) +\end{equation} + +And finally: +\begin{equation} + x = F_d \frac{1 + \frac{g}{s}}{ms^2 + (mg + c)s + k} + F \frac{1}{ms^2 + (mg + c)s + k} + w \frac{k + cs}{ms^2 + (mg + c)s + k} +\end{equation} + +We can see that this: +- adds damping to the system by a value $mg$ +- lower the compliance as low frequency by a factor: $1 + g/s$ + +If we want critical damping: +\begin{equation} + \xi = \frac{1}{2} \frac{c + gm}{\sqrt{km}} = \frac{1}{2} +\end{equation} + +This is attainable if we have: +\begin{equation} + g = \frac{\sqrt{km} - c}{m} +\end{equation} + +*** 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 + +*** Matlab Example +Let define the system parameters. +#+begin_src matlab + m = 50; % [kg] + k = 1e6; % [N/m] + c = 1e3; % [N/(m/s)] +#+end_src + +The state space model of the system is defined below. +#+begin_src matlab + A = [-c/m -k/m; + 1 0]; + + B = [1/m 1/m -1; + 0 0 0]; + + C = [ 0 1; + -c -k]; + + D = [0 0 0; + 1 0 0]; + + sys = ss(A, B, C, D); + sys.InputName = {'F', 'Fd', 'wddot'}; + sys.OutputName = {'d', 'Fm'}; + sys.StateName = {'ddot', 'd'}; +#+end_src + +The controller $K_\text{IFF}$ is: +#+begin_src matlab + Kiff = -((sqrt(k*m)-c)/m)/s; + Kiff.InputName = {'Fm'}; + Kiff.OutputName = {'F'}; +#+end_src + +And the closed loop system is computed below. +#+begin_src matlab + sys_iff = feedback(sys, Kiff, 'name', +1); +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + subplot(2, 2, 1); + title('Fd to d') + hold on; + plot(freqs, abs(squeeze(freqresp(sys('d', 'Fd'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(sys_iff('d', 'Fd'), freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + xlim([freqs(1), freqs(end)]); + + subplot(2, 2, 3); + title('Fd to x') + hold on; + plot(freqs, abs(squeeze(freqresp(sys('d', 'Fd'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(sys_iff('d', 'Fd'), freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + xlim([freqs(1), freqs(end)]); + + subplot(2, 2, 2); + title('w to d') + hold on; + plot(freqs, abs(squeeze(freqresp(sys('d', 'wddot')*s^2, freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(sys_iff('d', 'wddot')*s^2, freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]'); + xlim([freqs(1), freqs(end)]); + + subplot(2, 2, 4); + title('w to x') + hold on; + plot(freqs, abs(squeeze(freqresp(1+sys('d', 'wddot')*s^2, freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(1+sys_iff('d', 'wddot')*s^2, freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/iff_1dof_sensitivitiy.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:iff_1dof_sensitivitiy +#+CAPTION: Sensitivity to disturbance when IFF is applied on the 1dof system ([[./figs/iff_1dof_sensitivitiy.png][png]], [[./figs/iff_1dof_sensitivitiy.pdf][pdf]]) +[[file:figs/iff_1dof_sensitivitiy.png]] + +** 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 :tangle no + simulinkproject('../'); +#+end_src + +#+begin_src matlab + open('active_damping/matlab/sim_nano_station_id.slx') +#+end_src + +** Control Design +Let's load the undamped plant: +#+begin_src matlab + load('./active_damping/mat/plants.mat', 'G'); +#+end_src + +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 [[fig:iff_plant]]). + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i=1:6 + plot(freqs, abs(squeeze(freqresp(G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i=1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/iff_plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:iff_plant +#+CAPTION: Transfer function from forces applied in the legs to force sensor ([[./figs/iff_plant.png][png]], [[./figs/iff_plant.pdf][pdf]]) +[[file:figs/iff_plant.png]] + +The controller for each pair of actuator/sensor is: +#+begin_src matlab + K_iff = -1000/s; +#+end_src + +The corresponding loop gains are shown in figure [[fig:iff_open_loop]]. + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i=1:6 + plot(freqs, abs(squeeze(freqresp(K_iff*G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i=1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(K_iff*G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/iff_open_loop.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:iff_open_loop +#+CAPTION: Loop Gain for the Integral Force Feedback ([[./figs/iff_open_loop.png][png]], [[./figs/iff_open_loop.pdf][pdf]]) +[[file:figs/iff_open_loop.png]] + +** Identification of the damped plant +Let's initialize the system prior to identification. +#+begin_src matlab + initializeReferences(); + initializeGround(); + initializeGranite(); + initializeTy(); + initializeRy(); + initializeRz(); + initializeMicroHexapod(); + initializeAxisc(); + initializeMirror(); + initializeNanoHexapod(struct('actuator', 'piezo')); + initializeSample(struct('mass', 50)); +#+end_src + +All the controllers are set to 0. +#+begin_src matlab + K = tf(zeros(6)); + save('./mat/controllers.mat', 'K', '-append'); + K_iff = -K_iff*eye(6); + save('./mat/controllers.mat', 'K_iff', '-append'); + K_rmc = tf(zeros(6)); + save('./mat/controllers.mat', 'K_rmc', '-append'); + K_dvf = tf(zeros(6)); + save('./mat/controllers.mat', 'K_dvf', '-append'); +#+end_src + +We identify the system dynamics now that the IFF controller is ON. +#+begin_src matlab + G_iff = identifyPlant(); +#+end_src + +And we save the damped plant for further analysis +#+begin_src matlab + save('./active_damping/mat/plants.mat', 'G_iff', '-append'); +#+end_src + +** Sensitivity to disturbances +As shown on figure [[fig:sensitivity_dist_iff]]: +- The top platform of the nano-hexapod how behaves as a "free-mass". +- The transfer function from direct forces $F_s$ to the relative displacement $D$ is equivalent to the one of an isolated mass. +- The transfer function from ground motion $D_g$ to the relative displacement $D$ tends to the transfer function from $D_g$ to the displacement of the granite (the sample is being isolated thanks to IFF). + However, as the goal is to make the relative displacement $D$ as small as possible (e.g. to make the sample motion follows the granite motion), this is not a good thing. + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + subplot(2, 1, 1); + title('$D_g$ to $D$'); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / D_{g,x}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / D_{g,y}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / D_{g,z}\right|$'); + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(G_iff.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); + + subplot(2, 1, 2); + title('$F_s$ to $D$'); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{s,x}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{s,y}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{s,z}\right|$'); + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(G_iff.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/sensitivity_dist_iff.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:sensitivity_dist_iff +#+CAPTION: Sensitivity to disturbance once the IFF controller is applied to the system ([[./figs/sensitivity_dist_iff.png][png]], [[./figs/sensitivity_dist_iff.pdf][pdf]]) +[[file:figs/sensitivity_dist_iff.png]] + +#+begin_warning + The order of the models are very high and thus the plots may be wrong. + For instance, the plots are not the same when using =minreal=. +#+end_warning + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{rz, z}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{ty, z}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{ty, x}\right|$'); + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(minreal(prescale(G_iff.G_dist('Dz', 'Frzz'), {2*pi, 2*pi*1e3})), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(minreal(G_iff.G_dist('Dz', 'Ftyz')), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(minreal(G_iff.G_dist('Dx', 'Ftyx')), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/sensitivity_dist_stages_iff.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:sensitivity_dist_stages_iff +#+CAPTION: Sensitivity to force disturbances in various stages when IFF is applied ([[./figs/sensitivity_dist_stages_iff.png][png]], [[./figs/sensitivity_dist_stages_iff.pdf][pdf]]) +[[file:figs/sensitivity_dist_stages_iff.png]] + +** Damped Plant +Now, look at the new damped plant to control. + +It damps the plant (resonance of the nano hexapod as well as other resonances) as shown in figure [[fig:plant_iff_damped]]. + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + ax1 = subplot(2, 2, 1); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + + ax2 = subplot(2, 2, 2); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [rad/(Nm)]'); xlabel('Frequency [Hz]'); + + ax3 = subplot(2, 2, 3); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{n,x}\right|$'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{n,y}\right|$'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{n,z}\right|$'); + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northwest'); + + ax4 = subplot(2, 2, 4); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), 'DisplayName', '$\left|R_x / M_{n,x}\right|$'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz'))), 'DisplayName', '$\left|R_y / M_{n,y}\right|$'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), 'DisplayName', '$\left|R_z / M_{n,z}\right|$'); + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northwest'); + + linkaxes([ax1,ax2,ax3,ax4],'x'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/plant_iff_damped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:plant_iff_damped +#+CAPTION: Damped Plant after IFF is applied ([[./figs/plant_iff_damped.png][png]], [[./figs/plant_iff_damped.pdf][pdf]]) +[[file:figs/plant_iff_damped.png]] + +However, it increases coupling at low frequency (figure [[fig:plant_iff_coupling]]). +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + for ix = 1:6 + for iy = 1:6 + subplot(6, 6, (ix-1)*6 + iy); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_cart(ix, iy), freqs, 'Hz'))), 'k-'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_cart(ix, iy), freqs, 'Hz'))), 'k--'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylim([1e-12, 1e-5]); + end + end +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/plant_iff_coupling.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:plant_iff_coupling +#+CAPTION: Coupling induced by IFF ([[./figs/plant_iff_coupling.png][png]], [[./figs/plant_iff_coupling.pdf][pdf]]) +[[file:figs/plant_iff_coupling.png]] + +** Conclusion +#+begin_important +Integral Force Feedback: +- Robust (guaranteed stability) +- Acceptable Damping +- Increase the sensitivity to disturbances at low frequencies +#+end_important + +* Relative Motion Control +:PROPERTIES: +:header-args:matlab+: :tangle matlab/rmc.m +:header-args:matlab+: :comments none :mkdirp yes +:END: +<> + +** ZIP file containing the data and matlab files :ignore: +#+begin_src bash :exports none :results none + if [ matlab/rmc.m -nt data/rmc.zip ]; then + cp matlab/rmc.m rmc.m; + zip data/rmc \ + mat/plant.mat \ + rmc.m + rm rmc.m; + fi +#+end_src + +#+begin_note + All the files (data and Matlab scripts) are accessible [[file:data/rmc.zip][here]]. +#+end_note + +** Introduction :ignore: +In the Relative Motion Control (RMC), a derivative feedback is applied between the measured actuator displacement to the actuator force input. + +** One degree-of-freedom example +:PROPERTIES: +:header-args:matlab+: :tangle no +:END: +<> +*** Equations +#+begin_src latex :file rmc_1dof.pdf :post pdf2svg(file=*this*, ext="png") :exports results + \begin{tikzpicture} + % Ground + \draw (-1, 0) -- (1, 0); + + % Ground Displacement + \draw[dashed] (-1, 0) -- ++(-0.5, 0) coordinate(w); + \draw[->] (w) -- ++(0, 0.5) node[left]{$w$}; + + % Mass + \draw[fill=white] (-1, 1) rectangle ++(2, 0.8) node[pos=0.5]{$m$}; + + % Displacement of the mass + \draw[dashed] (-1, 1.8) -- ++(-0.5, 0) coordinate(x); + \draw[->] (x) -- ++(0, 0.5) node[left]{$x$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.8, 0) -- (-0.8, 1) node[midway, left=0.1]{$k$}; + \draw[damper] (0, 0) -- (0, 1) node[midway, left=0.2]{$c$}; + \draw[actuator={0.4}{0.2}] (0.8, 0) -- (0.8, 1) coordinate[midway, right=0.1](F); + + % Measured deformation + \draw[dashed] (1, 0) -- ++(2, 0) coordinate(d_bot); + \draw[dashed] (1, 1) -- ++(2, 0) coordinate(d_top); + \draw[<->] (d_bot) --coordinate[midway](d) (d_top); + + % Displacements + \node[block={0.8cm}{0.6cm}, right=0.6 of F] (Krmc) {$K$}; + \draw[->] (Krmc.west) -- (F) node[above right]{$F$}; + \draw[->] (d)node[above left]{$d$} -- (Krmc.east); + \end{tikzpicture} +#+end_src + +#+name: fig:rmc_1dof +#+caption: Relative Motion Control applied to a 1dof system +#+RESULTS: +[[file:figs/rmc_1dof.png]] + +The dynamic of the system is: +\begin{equation} + ms^2x = F_d - kx - csx + kw + csw + F +\end{equation} +In terms of the stage deformation $d = x - w$: +\begin{equation} + (ms^2 + cs + k) d = -ms^2 w + F_d + F +\end{equation} +The relative motion control law is: +\begin{equation} + K = -g s +\end{equation} +Thus, the applied force is: +\begin{equation} + F = -g s d +\end{equation} +And the new dynamics will be: +\begin{equation} + d = w \frac{-ms^2}{ms^2 + (c + g)s + k} + F_d \frac{1}{ms^2 + (c + g)s + k} + F \frac{1}{ms^2 + (c + g)s + k} +\end{equation} + +And thus damping is added. + +If critical damping is wanted: +\begin{equation} + \xi = \frac{1}{2}\frac{c + g}{\sqrt{km}} = \frac{1}{2} +\end{equation} +This corresponds to a gain: +\begin{equation} + g = \sqrt{km} - c +\end{equation} + +*** 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 + +*** Matlab Example +Let define the system parameters. +#+begin_src matlab + m = 50; % [kg] + k = 1e6; % [N/m] + c = 1e3; % [N/(m/s)] +#+end_src + +The state space model of the system is defined below. +#+begin_src matlab + A = [-c/m -k/m; + 1 0]; + + B = [1/m 1/m -1; + 0 0 0]; + + C = [ 0 1; + -c -k]; + + D = [0 0 0; + 1 0 0]; + + sys = ss(A, B, C, D); + sys.InputName = {'F', 'Fd', 'wddot'}; + sys.OutputName = {'d', 'Fm'}; + sys.StateName = {'ddot', 'd'}; +#+end_src + +The controller $K_\text{RMC}$ is: +#+begin_src matlab + Krmc = -(sqrt(k*m)-c)*s; + Krmc.InputName = {'d'}; + Krmc.OutputName = {'F'}; +#+end_src + +And the closed loop system is computed below. +#+begin_src matlab + sys_rmc = feedback(sys, Krmc, 'name', +1); +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); + + figure; + + subplot(2, 2, 1); + title('Fd to d') + hold on; + plot(freqs, abs(squeeze(freqresp(sys('d', 'Fd'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(sys_rmc('d', 'Fd'), freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + xlim([freqs(1), freqs(end)]); + + subplot(2, 2, 3); + title('Fd to x') + hold on; + plot(freqs, abs(squeeze(freqresp(sys('d', 'Fd'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(sys_rmc('d', 'Fd'), freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + xlim([freqs(1), freqs(end)]); + + subplot(2, 2, 2); + title('w to d') + hold on; + plot(freqs, abs(squeeze(freqresp(sys('d', 'wddot')*s^2, freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(sys_rmc('d', 'wddot')*s^2, freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]'); + xlim([freqs(1), freqs(end)]); + + subplot(2, 2, 4); + title('w to x') + hold on; + plot(freqs, abs(squeeze(freqresp(1+sys('d', 'wddot')*s^2, freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(1+sys_rmc('d', 'wddot')*s^2, freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/rmc_1dof_sensitivitiy.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:rmc_1dof_sensitivitiy +#+CAPTION: Sensitivity to disturbance when RMC is applied on the 1dof system ([[./figs/rmc_1dof_sensitivitiy.png][png]], [[./figs/rmc_1dof_sensitivitiy.pdf][pdf]]) +[[file:figs/rmc_1dof_sensitivitiy.png]] + +** 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 :tangle no + simulinkproject('../'); +#+end_src + +#+begin_src matlab + open('active_damping/matlab/sim_nano_station_id.slx') +#+end_src + +** Control Design +Let's load the undamped plant: +#+begin_src matlab + load('./active_damping/mat/plants.mat', 'G'); +#+end_src + +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 [[fig:rmc_plant]]). + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i=1:6 + plot(freqs, abs(squeeze(freqresp(G.G_dleg(['Dm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i=1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_dleg(['Dm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/rmc_plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:rmc_plant +#+CAPTION: Transfer function from forces applied in the legs to leg displacement sensor ([[./figs/rmc_plant.png][png]], [[./figs/rmc_plant.pdf][pdf]]) +[[file:figs/rmc_plant.png]] + +The Relative Motion Controller is defined below. +A Low pass Filter is added to make the controller transfer function proper. +#+begin_src matlab + K_rmc = s*50000/(1 + s/2/pi/10000); +#+end_src + +The obtained loop gains are shown in figure [[fig:rmc_open_loop]]. + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i=1:6 + plot(freqs, abs(squeeze(freqresp(K_rmc*G.G_dleg(['Dm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i=1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(K_rmc*G.G_dleg(['Dm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/rmc_open_loop.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:rmc_open_loop +#+CAPTION: Loop Gain for the Integral Force Feedback ([[./figs/rmc_open_loop.png][png]], [[./figs/rmc_open_loop.pdf][pdf]]) +[[file:figs/rmc_open_loop.png]] + +** Identification of the damped plant +Let's initialize the system prior to identification. +#+begin_src matlab + initializeReferences(); + initializeGround(); + initializeGranite(); + initializeTy(); + initializeRy(); + initializeRz(); + initializeMicroHexapod(); + initializeAxisc(); + initializeMirror(); + initializeNanoHexapod(struct('actuator', 'piezo')); + initializeSample(struct('mass', 50)); +#+end_src + +And initialize the controllers. +#+begin_src matlab + K = tf(zeros(6)); + save('./mat/controllers.mat', 'K', '-append'); + K_iff = tf(zeros(6)); + save('./mat/controllers.mat', 'K_iff', '-append'); + K_rmc = -K_rmc*eye(6); + save('./mat/controllers.mat', 'K_rmc', '-append'); + K_dvf = tf(zeros(6)); + save('./mat/controllers.mat', 'K_dvf', '-append'); +#+end_src + +We identify the system dynamics now that the RMC controller is ON. +#+begin_src matlab + G_rmc = identifyPlant(); +#+end_src + +And we save the damped plant for further analysis. +#+begin_src matlab + save('./active_damping/mat/plants.mat', 'G_rmc', '-append'); +#+end_src + +** Sensitivity to disturbances +As shown in figure [[fig:sensitivity_dist_rmc]], RMC control succeed in lowering the sensitivity to disturbances near resonance of the system. + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + subplot(2, 1, 1); + title('$D_g$ to $D$'); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / D_{g,x}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / D_{g,y}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / D_{g,z}\right|$'); + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]'); + legend('location', 'southeast'); + + subplot(2, 1, 2); + title('$F_s$ to $D$'); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{s,x}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{s,y}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{s,z}\right|$'); + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/sensitivity_dist_rmc.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:sensitivity_dist_rmc +#+CAPTION: Sensitivity to disturbance once the RMC controller is applied to the system ([[./figs/sensitivity_dist_rmc.png][png]], [[./figs/sensitivity_dist_rmc.pdf][pdf]]) +[[file:figs/sensitivity_dist_rmc.png]] + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{rz, z}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{ty, z}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{ty, x}\right|$'); + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/sensitivity_dist_stages_rmc.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:sensitivity_dist_stages_rmc +#+CAPTION: Sensitivity to force disturbances in various stages when RMC is applied ([[./figs/sensitivity_dist_stages_rmc.png][png]], [[./figs/sensitivity_dist_stages_rmc.pdf][pdf]]) +[[file:figs/sensitivity_dist_stages_rmc.png]] + +** Damped Plant +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + ax1 = subplot(2, 2, 1); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + + ax2 = subplot(2, 2, 2); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [rad/(Nm)]'); xlabel('Frequency [Hz]'); + + ax3 = subplot(2, 2, 3); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{n,x}\right|$'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{n,y}\right|$'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{n,z}\right|$'); + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northwest'); + + ax4 = subplot(2, 2, 4); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), 'DisplayName', '$\left|R_x / M_{n,x}\right|$'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz'))), 'DisplayName', '$\left|R_y / M_{n,y}\right|$'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), 'DisplayName', '$\left|R_z / M_{n,z}\right|$'); + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northwest'); + + linkaxes([ax1,ax2,ax3,ax4],'x'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/plant_rmc_damped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:plant_rmc_damped +#+CAPTION: Damped Plant after RMC is applied ([[./figs/plant_rmc_damped.png][png]], [[./figs/plant_rmc_damped.pdf][pdf]]) +[[file:figs/plant_rmc_damped.png]] + +** Conclusion +#+begin_important +Relative Motion Control: +- +#+end_important + +* Direct Velocity Feedback +:PROPERTIES: +:header-args:matlab+: :tangle matlab/dvf.m +:header-args:matlab+: :comments none :mkdirp yes +:END: +<> + +** ZIP file containing the data and matlab files :ignore: +#+begin_src bash :exports none :results none + if [ matlab/dvf.m -nt data/dvf.zip ]; then + cp matlab/dvf.m dvf.m; + zip data/dvf \ + mat/plant.mat \ + dvf.m + rm dvf.m; + fi +#+end_src + +#+begin_note + All the files (data and Matlab scripts) are accessible [[file:data/dvf.zip][here]]. +#+end_note + +** Introduction :ignore: +In the Relative Motion Control (RMC), a feedback is applied between the measured velocity of the platform to the actuator force input. + +** One degree-of-freedom example +:PROPERTIES: +:header-args:matlab+: :tangle no +:END: +<> +*** Equations +#+begin_src latex :file dvf_1dof.pdf :post pdf2svg(file=*this*, ext="png") :exports results + \begin{tikzpicture} + % Ground + \draw (-1, 0) -- (1, 0); + + % Ground Displacement + \draw[dashed] (-1, 0) -- ++(-0.5, 0) coordinate(w); + \draw[->] (w) -- ++(0, 0.5) node[left]{$w$}; + + % Mass + \draw[fill=white] (-1, 1) rectangle ++(2, 0.8) node[pos=0.5]{$m$}; + + % Velocity Sensor + \node[inertialsensor={0.3}] (velg) at (1, 1.8){}; + \node[above] at (velg.north) {$\dot{x}$}; + + % Displacement of the mass + \draw[dashed] (-1, 1.8) -- ++(-0.5, 0) coordinate(x); + \draw[->] (x) -- ++(0, 0.5) node[left]{$x$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.8, 0) -- (-0.8, 1) node[midway, left=0.1]{$k$}; + \draw[damper] (0, 0) -- (0, 1) node[midway, left=0.2]{$c$}; + \draw[actuator={0.4}{0.2}] (0.8, 0) -- (0.8, 1) coordinate[midway, right=0.1](F); + + % Control + \node[block={0.8cm}{0.6cm}, right=0.6 of F] (Kdvf) {$K$}; + \draw[->] (Kdvf.west) -- (F) node[above right]{$F$}; + \draw[<-] (Kdvf.east) -- ++(0.5, 0) |- (velg.east); + \end{tikzpicture} +#+end_src + +#+name: fig:dvf_1dof +#+caption: Direct Velocity Feedback applied to a 1dof system +#+RESULTS: +[[file:figs/dvf_1dof.png]] + +The dynamic of the system is: +\begin{equation} + ms^2x = F_d - kx - csx + kw + csw + F +\end{equation} +In terms of the stage deformation $d = x - w$: +\begin{equation} + (ms^2 + cs + k) d = -ms^2 w + F_d + F +\end{equation} +The direct velocity feedback law shown in figure [[fig:dvf_1dof]] is: +\begin{equation} + K = -g +\end{equation} +Thus, the applied force is: +\begin{equation} + F = -g \dot{x} +\end{equation} +And the new dynamics will be: +\begin{equation} + d = w \frac{-ms^2 - gs}{ms^2 + (c + g)s + k} + F_d \frac{1}{ms^2 + (c + g)s + k} + F \frac{1}{ms^2 + (c + g)s + k} +\end{equation} + +And thus damping is added. + +If critical damping is wanted: +\begin{equation} + \xi = \frac{1}{2}\frac{c + g}{\sqrt{km}} = \frac{1}{2} +\end{equation} +This corresponds to a gain: +\begin{equation} + g = \sqrt{km} - c +\end{equation} + +*** 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 + +*** Matlab Example +Let define the system parameters. +#+begin_src matlab + m = 50; % [kg] + k = 1e6; % [N/m] + c = 1e3; % [N/(m/s)] +#+end_src + +The state space model of the system is defined below. +#+begin_src matlab + A = [-c/m -k/m; + 1 0]; + + B = [1/m 1/m -1; + 0 0 0]; + + C = [1 0; + 0 1; + 0 0]; + + D = [0 0 0; + 0 0 0; + 0 0 1]; + + sys = ss(A, B, C, D); + sys.InputName = {'F', 'Fd', 'wddot'}; + sys.OutputName = {'ddot', 'd', 'wddot'}; + sys.StateName = {'ddot', 'd'}; +#+end_src + +Because we need $\dot{x}$ for feedback, we compute it from the outputs +#+begin_src matlab + G_xdot = [1, 0, 1/s; + 0, 1, 0]; + G_xdot.InputName = {'ddot', 'd', 'wddot'}; + G_xdot.OutputName = {'xdot', 'd'}; +#+end_src + +Finally, the system is described by =sys= as defined below. +#+begin_src matlab + sys = series(sys, G_xdot, [1 2 3], [1 2 3]); +#+end_src + +The controller $K_\text{DVF}$ is: +#+begin_src matlab + Kdvf = tf(-(sqrt(k*m)-c)); + Kdvf.InputName = {'xdot'}; + Kdvf.OutputName = {'F'}; +#+end_src + +And the closed loop system is computed below. +#+begin_src matlab + sys_dvf = feedback(sys, Kdvf, 'name', +1); +#+end_src + +The obtained sensitivity to disturbances is shown in figure [[fig:dvf_1dof_sensitivitiy]]. +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); + + figure; + + subplot(2, 2, 1); + title('Fd to d') + hold on; + plot(freqs, abs(squeeze(freqresp(sys('d', 'Fd'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(sys_dvf('d', 'Fd'), freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + xlim([freqs(1), freqs(end)]); + + subplot(2, 2, 3); + title('Fd to x') + hold on; + plot(freqs, abs(squeeze(freqresp(sys('xdot', 'Fd')/s, freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(sys_dvf('xdot', 'Fd')/s, freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + xlim([freqs(1), freqs(end)]); + + subplot(2, 2, 2); + title('w to d') + hold on; + plot(freqs, abs(squeeze(freqresp(sys('d', 'wddot')*s^2, freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(sys_dvf('d', 'wddot')*s^2, freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]'); + xlim([freqs(1), freqs(end)]); + + subplot(2, 2, 4); + title('w to x') + hold on; + plot(freqs, abs(squeeze(freqresp(1+sys('d', 'wddot')*s^2, freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(1+sys_dvf('d', 'wddot')*s^2, freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/dvf_1dof_sensitivitiy.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:dvf_1dof_sensitivitiy +#+CAPTION: Sensitivity to disturbance when DVF is applied on the 1dof system ([[./figs/dvf_1dof_sensitivitiy.png][png]], [[./figs/dvf_1dof_sensitivitiy.pdf][pdf]]) +[[file:figs/dvf_1dof_sensitivitiy.png]] + +** 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 :tangle no + simulinkproject('../'); +#+end_src + +#+begin_src matlab + open('active_damping/matlab/sim_nano_station_id.slx') +#+end_src + +** Control Design +Let's load the undamped plant: +#+begin_src matlab + load('./active_damping/mat/plants.mat', 'G'); +#+end_src + +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 [[fig:dvf_plant]]). + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i=1:6 + plot(freqs, abs(squeeze(freqresp(G.G_geoph(['Vm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i=1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_geoph(['Vm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/dvf_plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:dvf_plant +#+CAPTION: Transfer function from forces applied in the legs to leg velocity sensor ([[./figs/dvf_plant.png][png]], [[./figs/dvf_plant.pdf][pdf]]) +[[file:figs/dvf_plant.png]] + +The controller is defined below and the obtained loop gain is shown in figure [[fig:dvf_open_loop_gain]]. + +#+begin_src matlab + K_dvf = tf(3e4); +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i=1:6 + plot(freqs, abs(squeeze(freqresp(K_dvf*G.G_geoph(['Vm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i=1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(K_dvf*G.G_geoph(['Vm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/dvf_open_loop_gain.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:dvf_open_loop_gain +#+CAPTION: Loop Gain for DVF ([[./figs/dvf_open_loop_gain.png][png]], [[./figs/dvf_open_loop_gain.pdf][pdf]]) +[[file:figs/dvf_open_loop_gain.png]] + +** Identification of the damped plant +Let's initialize the system prior to identification. +#+begin_src matlab + initializeReferences(); + initializeGround(); + initializeGranite(); + initializeTy(); + initializeRy(); + initializeRz(); + initializeMicroHexapod(); + initializeAxisc(); + initializeMirror(); + initializeNanoHexapod(struct('actuator', 'piezo')); + initializeSample(struct('mass', 50)); +#+end_src + +And initialize the controllers. +#+begin_src matlab + K = tf(zeros(6)); + save('./mat/controllers.mat', 'K', '-append'); + K_iff = tf(zeros(6)); + save('./mat/controllers.mat', 'K_iff', '-append'); + K_rmc = tf(zeros(6)); + save('./mat/controllers.mat', 'K_rmc', '-append'); + K_dvf = -K_dvf*eye(6); + save('./mat/controllers.mat', 'K_dvf', '-append'); +#+end_src + +We identify the system dynamics now that the RMC controller is ON. +#+begin_src matlab + G_dvf = identifyPlant(); +#+end_src + +And we save the damped plant for further analysis. +#+begin_src matlab + save('./active_damping/mat/plants.mat', 'G_dvf', '-append'); +#+end_src + +** Sensitivity to disturbances + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + subplot(2, 1, 1); + title('$D_g$ to $D$'); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / D_{g,x}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / D_{g,y}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / D_{g,z}\right|$'); + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); + + subplot(2, 1, 2); + title('$F_s$ to $D$'); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{s,x}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{s,y}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{s,z}\right|$'); + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/sensitivity_dist_dvf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:sensitivity_dist_dvf +#+CAPTION: Sensitivity to disturbance once the DVF controller is applied to the system ([[./figs/sensitivity_dist_dvf.png][png]], [[./figs/sensitivity_dist_dvf.pdf][pdf]]) +[[file:figs/sensitivity_dist_dvf.png]] + + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{rz, z}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{ty, z}\right|$'); + plot(freqs, abs(squeeze(freqresp(G.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{ty, x}\right|$'); + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/sensitivity_dist_stages_dvf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:sensitivity_dist_stages_dvf +#+CAPTION: Sensitivity to force disturbances in various stages when DVF is applied ([[./figs/sensitivity_dist_stages_dvf.png][png]], [[./figs/sensitivity_dist_stages_dvf.pdf][pdf]]) +[[file:figs/sensitivity_dist_stages_dvf.png]] + +** Damped Plant +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + ax1 = subplot(2, 2, 1); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + + ax2 = subplot(2, 2, 2); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [rad/(Nm)]'); xlabel('Frequency [Hz]'); + + ax3 = subplot(2, 2, 3); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{n,x}\right|$'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{n,y}\right|$'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{n,z}\right|$'); + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northwest'); + + ax4 = subplot(2, 2, 4); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), 'DisplayName', '$\left|R_x / M_{n,x}\right|$'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz'))), 'DisplayName', '$\left|R_y / M_{n,y}\right|$'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), 'DisplayName', '$\left|R_z / M_{n,z}\right|$'); + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northwest'); + + linkaxes([ax1,ax2,ax3,ax4],'x'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/plant_dvf_damped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:plant_dvf_damped +#+CAPTION: Damped Plant after DVF is applied ([[./figs/plant_dvf_damped.png][png]], [[./figs/plant_dvf_damped.pdf][pdf]]) +[[file:figs/plant_dvf_damped.png]] + +** Conclusion +#+begin_important +Direct Velocity Feedback: +#+end_important + +* Comparison +<> +** Introduction :ignore: +** 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 + cd('../'); +#+end_src + +** Load the plants +#+begin_src matlab + load('./active_damping/mat/plants.mat', 'G', 'G_iff', 'G_rmc', 'G_dvf'); +#+end_src + +** Sensitivity to Disturbance +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + title('$D_{g,z}$ to $D_z$'); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_gm( 'Dz', 'Dgz'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'Undamped'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/sensitivity_comp_ground_motion_z.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:sensitivity_comp_ground_motion_z +#+CAPTION: caption ([[./figs/sensitivity_comp_ground_motion_z.png][png]], [[./figs/sensitivity_comp_ground_motion_z.pdf][pdf]]) +[[file:figs/sensitivity_comp_ground_motion_z.png]] + + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + title('$F_{s,z}$ to $D_z$'); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_fs( 'Dz', 'Fsz'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'Undamped'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/sensitivity_comp_direct_forces_z.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:sensitivity_comp_direct_forces_z +#+CAPTION: caption ([[./figs/sensitivity_comp_direct_forces_z.png][png]], [[./figs/sensitivity_comp_direct_forces_z.pdf][pdf]]) +[[file:figs/sensitivity_comp_direct_forces_z.png]] + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + title('$F_{rz,z}$ to $D_z$'); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_dist( 'Dz', 'Frzz'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'Undamped'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/sensitivity_comp_spindle_z.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:sensitivity_comp_spindle_z +#+CAPTION: caption ([[./figs/sensitivity_comp_spindle_z.png][png]], [[./figs/sensitivity_comp_spindle_z.pdf][pdf]]) +[[file:figs/sensitivity_comp_spindle_z.png]] + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + title('$F_{ty,z}$ to $D_z$'); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_dist( 'Dz', 'Ftyz'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'Undamped'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/sensitivity_comp_ty_z.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:sensitivity_comp_ty_z +#+CAPTION: caption ([[./figs/sensitivity_comp_ty_z.png][png]], [[./figs/sensitivity_comp_ty_z.pdf][pdf]]) +[[file:figs/sensitivity_comp_ty_z.png]] + + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + title('$F_{ty,x}$ to $D_x$'); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_dist( 'Dx', 'Ftyx'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'Undamped'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/sensitivity_comp_ty_x.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:sensitivity_comp_ty_x +#+CAPTION: caption ([[./figs/sensitivity_comp_ty_x.png][png]], [[./figs/sensitivity_comp_ty_x.pdf][pdf]]) +[[file:figs/sensitivity_comp_ty_x.png]] + +** Damped Plant +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + title('$F_{n,z}$ to $D_z$'); + ax1 = subplot(2, 1, 1); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_cart( 'Dz', 'Fnz'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'Undamped'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + legend('location', 'northeast'); + + ax2 = subplot(2, 1, 2); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart ('Dz', 'Fnz'), freqs, 'Hz'))), 'k-'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'k:'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'k--'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'k-.'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/plant_comp_damping_z.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:plant_comp_damping_z +#+CAPTION: Plant for the $z$ direction for different active damping technique used ([[./figs/plant_comp_damping_z.png][png]], [[./figs/plant_comp_damping_z.pdf][pdf]]) +[[file:figs/plant_comp_damping_z.png]] + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + title('$F_{n,z}$ to $D_z$'); + ax1 = subplot(2, 1, 1); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_cart( 'Dx', 'Fnx'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'Undamped'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + legend('location', 'northeast'); + + ax2 = subplot(2, 1, 2); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart ('Dx', 'Fnx'), freqs, 'Hz'))), 'k-'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'k:'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'k--'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'k-.'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/plant_comp_damping_x.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:plant_comp_damping_x +#+CAPTION: Plant for the $x$ direction for different active damping technique used ([[./figs/plant_comp_damping_x.png][png]], [[./figs/plant_comp_damping_x.pdf][pdf]]) +[[file:figs/plant_comp_damping_x.png]] + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + title('$F_{n,x}$ to $R_z$'); + ax1 = subplot(2, 1, 1); + hold on; + plot(freqs, abs(squeeze(freqresp(G.G_cart( 'Rz', 'Fnx'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'Undamped'); + plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Rz', 'Fnx'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF'); + plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Rz', 'Fnx'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC'); + plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Rz', 'Fnx'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + legend('location', 'northeast'); + + ax2 = subplot(2, 1, 2); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart ('Ry', 'Fnx'), freqs, 'Hz'))), 'k-'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Ry', 'Fnx'), freqs, 'Hz'))), 'k:'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Ry', 'Fnx'), freqs, 'Hz'))), 'k--'); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Ry', 'Fnx'), freqs, 'Hz'))), 'k-.'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/plant_comp_damping_coupling.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:plant_comp_damping_coupling +#+CAPTION: Comparison of one off-diagonal plant for different damping technique applied ([[./figs/plant_comp_damping_coupling.png][png]], [[./figs/plant_comp_damping_coupling.pdf][pdf]]) +[[file:figs/plant_comp_damping_coupling.png]] + +* Conclusion +<> diff --git a/active_damping/mat/plants.mat b/active_damping_uniaxial/mat/plants.mat similarity index 100% rename from active_damping/mat/plants.mat rename to active_damping_uniaxial/mat/plants.mat diff --git a/active_damping/matlab/dvf.m b/active_damping_uniaxial/matlab/dvf.m similarity index 100% rename from active_damping/matlab/dvf.m rename to active_damping_uniaxial/matlab/dvf.m diff --git a/active_damping/matlab/iff.m b/active_damping_uniaxial/matlab/iff.m similarity index 100% rename from active_damping/matlab/iff.m rename to active_damping_uniaxial/matlab/iff.m diff --git a/active_damping/matlab/rmc.m b/active_damping_uniaxial/matlab/rmc.m similarity index 100% rename from active_damping/matlab/rmc.m rename to active_damping_uniaxial/matlab/rmc.m diff --git a/active_damping/matlab/sim_nano_station_id.slx b/active_damping_uniaxial/matlab/sim_nano_station_id.slx similarity index 100% rename from active_damping/matlab/sim_nano_station_id.slx rename to active_damping_uniaxial/matlab/sim_nano_station_id.slx diff --git a/index.html b/index.html index cb88a3d..51c0e60 100644 --- a/index.html +++ b/index.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Simscape Model of the Nano-Active-Stabilization-System @@ -258,17 +258,18 @@ for the JavaScript code in this tag.

Table of Contents

@@ -277,8 +278,8 @@ for the JavaScript code in this tag. Here are links to the documents related to the Simscape model of the Nano-Active-Stabilization-System.

-
-

1 Simulink Project (link)

+
+

1 Simulink Project (link)

The project is managed with a Simulink Project. @@ -287,8 +288,8 @@ Such project is briefly presented here

-
-

2 Simscape Model (link)

+
+

2 Simscape Model (link)

The model of the NASS is based on Simulink and Simscape Multi-Body. @@ -297,8 +298,8 @@ Such toolbox is presented here.

-
-

3 Simscape Subsystems (link)

+
+

3 Simscape Subsystems (link)

The model is decomposed of multiple subsystems. @@ -312,8 +313,8 @@ All these subsystems are described he

-
-

4 Kinematics of the Station (link)

+
+

4 Kinematics of the Station (link)

First, we consider perfectly rigid elements and joints and we just study the kinematic of the station. @@ -323,8 +324,8 @@ This is detailed here.

-
-

5 Metrology (link)

+
+

5 Metrology (link)

In this document (accessible here), we discuss the measurement of the sample with respect to the granite. @@ -332,8 +333,8 @@ In this document (accessible here), we disc

-
-

6 Computation of the positioning error of the Sample (link)

+
+

6 Computation of the positioning error of the Sample (link)

From the measurement of the position of the sample with respect to the granite and from the wanted position of each stage, we can compute the positioning error of the sample with respect to the nano-hexapod. @@ -342,8 +343,8 @@ This is done here.

-
-

7 Tuning of the Dynamics of the Simscape model (link)

+
+

7 Tuning of the Dynamics of the Simscape model (link)

From dynamical measurements perform on the real positioning station, we tune the parameters of the simscape model to have similar dynamics. @@ -355,8 +356,8 @@ This is explained here.

-
-

8 Disturbances (link)

+
+

8 Disturbances (link)

The effect of disturbances on the position of the micro-station have been measured. @@ -373,8 +374,8 @@ We also discuss how the disturbances are implemented in the model.

-
-

9 Tomography Experiment (link)

+
+

9 Tomography Experiment (link)

Now that the dynamics of the Model have been tuned and the Disturbances have included, we can simulate experiments. @@ -386,19 +387,28 @@ Tomography experiments are simulated and the results are presented -

10 Active Damping Techniques (link)

+
+

10 Active Damping Techniques on the Uni-axial Model (link)

-Active damping techniques are applied to the Simscape model. +Active damping techniques are applied to the Uniaxial Simscape model.

-
-

11 Useful Matlab Functions (link)

+
+

11 Active Damping Techniques on the full Simscape Model (link)

+Active damping techniques are applied to the full Simscape model. +

+
+
+ +
+

12 Useful Matlab Functions (link)

+
+

Many matlab functions are shared among all the files of the projects.

@@ -410,7 +420,7 @@ These functions are all defined here.

Author: Dehaeze Thomas

-

Created: 2019-12-13 ven. 17:55

+

Created: 2019-12-17 mar. 10:15

Validate

diff --git a/index.org b/index.org index b68e6b0..8633f10 100644 --- a/index.org +++ b/index.org @@ -88,8 +88,11 @@ Now that the dynamics of the Model have been tuned and the Disturbances have inc Tomography experiments are simulated and the results are presented [[./experiment_tomography/index.org][here]]. -* Active Damping Techniques ([[./active_damping/index.org][link]]) -Active damping techniques are applied to the Simscape model. +* Active Damping Techniques on the Uni-axial Model ([[./active_damping_uniaxial/index.org][link]]) +Active damping techniques are applied to the Uniaxial Simscape model. + +* Active Damping Techniques on the full Simscape Model ([[./active_damping/index.org][link]]) +Active damping techniques are applied to the full Simscape model. * Useful Matlab Functions ([[./functions/index.org][link]]) Many matlab functions are shared among all the files of the projects.