77 KiB
Active Damping with an uni-axial model
- Introduction
- Undamped System
- Integral Force Feedback
- Relative Motion Control
- Direct Velocity Feedback
- Comparison
- Conclusion
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
<<sec:undamped_system>>
ZIP file containing the data and matlab files ignore
All the files (data and Matlab scripts) are accessible here.
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.
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('actuator', 'piezo');
initializeSample('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');Identification
We identify the various transfer functions of the system
G = identifyPlant();And we save it for further analysis.
save('./active_damping_uniaxial/mat/plants.mat', 'G', '-append');Sensitivity to disturbances
The sensitivity to disturbances are shown on figure fig:sensitivity_dist_undamped.
<<plt-matlab>>
<<plt-matlab>>
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.
<<plt-matlab>>
Integral Force Feedback
<<sec:iff>>
ZIP file containing the data and matlab files ignore
All the files (data and Matlab scripts) are accessible here.
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
<<sec:iff_1dof>>
Equations
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 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); <<plt-matlab>>
Control Design
Let's load the undamped plant:
load('./active_damping_uniaxial/mat/plants.mat', 'G');Let's look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor (figure fig:iff_plant).
<<plt-matlab>>
The controller for each pair of actuator/sensor is:
K_iff = -1000/s;The corresponding loop gains are shown in figure fig:iff_open_loop.
<<plt-matlab>>
Identification of the damped plant
Let's initialize the system prior to identification.
initializeReferences();
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeNanoHexapod('actuator', 'piezo');
initializeSample('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_uniaxial/mat/plants.mat', 'G_iff', '-append');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.
<<plt-matlab>>
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.
<<plt-matlab>>
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.
<<plt-matlab>>
However, it increases coupling at low frequency (figure fig:plant_iff_coupling).
<<plt-matlab>>
Conclusion
Integral Force Feedback:
- Robust (guaranteed stability)
- Acceptable Damping
- Increase the sensitivity to disturbances at low frequencies
Relative Motion Control
<<sec:rmc>>
ZIP file containing the data and matlab files ignore
All the files (data and Matlab scripts) are accessible here.
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
<<sec:rmc_1dof>>
Equations
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 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); <<plt-matlab>>
Control Design
Let's load the undamped plant:
load('./active_damping_uniaxial/mat/plants.mat', 'G');Let's look at the transfer function from actuator forces in the nano-hexapod to the measured displacement of the actuator for all 6 pairs of actuator/sensor (figure fig:rmc_plant).
<<plt-matlab>>
The Relative Motion Controller is defined below. A Low pass Filter is added to make the controller transfer function proper.
K_rmc = s*50000/(1 + s/2/pi/10000);The obtained loop gains are shown in figure fig:rmc_open_loop.
<<plt-matlab>>
Identification of the damped plant
Let's initialize the system prior to identification.
initializeReferences();
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeNanoHexapod('actuator', 'piezo');
initializeSample('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_uniaxial/mat/plants.mat', 'G_rmc', '-append');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.
<<plt-matlab>>
<<plt-matlab>>
Conclusion
Relative Motion Control:
Direct Velocity Feedback
<<sec:dvf>>
ZIP file containing the data and matlab files ignore
All the files (data and Matlab scripts) are accessible here.
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
<<sec:dvf_1dof>>
Equations
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 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 fig:dvf_1dof_sensitivitiy.
<<plt-matlab>>
Control Design
Let's load the undamped plant:
load('./active_damping_uniaxial/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 fig:dvf_plant).
<<plt-matlab>>
The controller is defined below and the obtained loop gain is shown in figure fig:dvf_open_loop_gain.
K_dvf = tf(3e4); <<plt-matlab>>
Identification of the damped plant
Let's initialize the system prior to identification.
initializeReferences();
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeNanoHexapod('actuator', 'piezo');
initializeSample('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_uniaxial/mat/plants.mat', 'G_dvf', '-append');Sensitivity to disturbances
Conclusion
Direct Velocity Feedback:
Comparison
<<sec:comparison>>
Introduction ignore
Load the plants
load('./active_damping_uniaxial/mat/plants.mat', 'G', 'G_iff', 'G_rmc', 'G_dvf');Sensitivity to Disturbance
Damped Plant
Conclusion
<<sec:conclusion>>