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- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - diff --git a/figs/coupling_small_m.svg b/figs/coupling_small_m.svg deleted file mode 100644 index 46c228d..0000000 --- a/figs/coupling_small_m.svg +++ /dev/null @@ -1,343 +0,0 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 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+ Control in a rotating frame @@ -240,79 +240,73 @@

Table of Contents

@@ -322,13 +316,17 @@ The objective of this note it to highlight some control problems that arises whe

-In section 1, a simple system composed of a spindle and a translation stage is defined and the equations of motion are written. +In section 1, a simple system composed of a spindle and a translation stage is defined and the equations of motion are written. The rotation induces some coupling between the actuators and their displacement, and modifies the dynamics of the system. This is studied using the equations, and some numerical computations are used to compare the use of voice coil and piezoelectric actuators.

-Then, in section 4, two different control approach are compared where: +In sections 2 and 3, the use of Integral Force Feedback and Direct Velocity Feedback is studied for the case of rotating positioning platforms. +

+ +

+Then, in section 4, two different control approach are compared where:

-In section 5, the analytical study will be validated using a multi body model of the studied system. +In section 5, the analytical study will be validated using a multi body model of the studied system.

-

-Finally, in section 6, the control strategies are implemented using Simulink and Simscape and compared. -

- -
-

1 System Description and Analysis

+
+

1 System Description and Analysis

- +

-
-

1.1 System description

+
+

1.1 System description

-The system consists of one 2 degree of freedom translation stage on top of a spindle (figure 1). +The system consists of one 2 degree of freedom translation stage on top of a spindle (figure 1).

@@ -367,7 +361,7 @@ The measurement is either the \(x-y\) displacement of the object located on top

-
+

rotating_frame_2dof.png

Figure 1: Schematic of the mecanical system

@@ -401,19 +395,19 @@ Indices \(u\) and \(v\) corresponds to signals in the rotating reference frame (
-
-

1.2 Equations

+
+

1.2 Equations

- -Based on the figure 1, we can write the equations of motion of the system. + +Based on the figure 1, we can write the equations of motion of the system.

Let’s express the kinetic energy \(T\) and the potential energy \(V\) of the mass \(m\):

\begin{align} -\label{org0832be2} +\label{orged17d67} T & = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) \\ V & = \frac{1}{2} k \left( x^2 + y^2 \right) \end{align} @@ -422,7 +416,7 @@ V & = \frac{1}{2} k \left( x^2 + y^2 \right) The Lagrangian is the kinetic energy minus the potential energy.

\begin{equation} -\label{orgf1dd577} +\label{orgdad1a41} L = T-V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \left( x^2 + y^2 \right) \end{equation} @@ -431,7 +425,7 @@ L = T-V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \le The partial derivatives of the Lagrangian with respect to the variables \((x, y)\) are:

\begin{align*} -\label{org7098c72} +\label{org14a5fd1} \frac{\partial L}{\partial x} & = -kx \\ \frac{\partial L}{\partial y} & = -ky \\ \frac{d}{dt}\frac{\partial L}{\partial \dot{x}} & = m\ddot{x} \\ @@ -501,11 +495,11 @@ We can then subtract and add the previous equations to obtain the following equa

\begin{equation} -\label{org0b782a6} +\label{org913ac5b} m \ddot{d_u} + (k - m\dot{\theta}^2) d_u = F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta} \end{equation} \begin{equation} -\label{orgc41ba93} +\label{orgc9c0d55} m \ddot{d_v} + (k - m\dot{\theta}^2) d_v = F_v - 2 m\dot{d_u}\dot{\theta} - m d_u\ddot{\theta} \end{equation} @@ -531,8 +525,8 @@ The resulting effect of those forces should then be higher when using softer act
-
-

1.3 Numerical Values for the NASS

+
+

1.3 Numerical Values for the NASS

Let’s define the parameters for the NASS. @@ -595,8 +589,8 @@ Let’s define the parameters for the NASS.

-
-

1.4 Euler and Coriolis forces - Numerical Result

+
+

1.4 Euler and Coriolis forces - Numerical Result

First we will determine the value for Euler and Coriolis forces during regular experiment. @@ -607,10 +601,10 @@ First we will determine the value for Euler and Coriolis forces during regular e

-The obtained values are displayed in table 1. +The obtained values are displayed in table 1.

- +
@@ -644,22 +638,22 @@ The obtained values are displayed in table 1. -
-

1.5 Negative Spring Effect - Numerical Result

+
+

1.5 Negative Spring Effect - Numerical Result

The negative stiffness due to the rotation is equal to \(-m{\omega_0}^2\).

-The values for the negative spring effect are displayed in table 2. +The values for the negative spring effect are displayed in table 2.

This is definitely negligible when using piezoelectric actuators. It may not be the case when using voice coil actuators.

-
Table 1: Euler and Coriolis forces for the NASS
+
@@ -687,15 +681,15 @@ This is definitely negligible when using piezoelectric actuators. It may not be -
-

1.6 Limitations due to coupling

+
+

1.6 Limitations due to coupling

To simplify, we consider a constant rotating speed \(\dot{\theta} = \omega_0\) and thus \(\ddot{\theta} = 0\).

-From equations \eqref{org0b782a6} and \eqref{orgc41ba93}, we obtain: +From equations \eqref{org913ac5b} and \eqref{orgc9c0d55}, we obtain:

\begin{align*} (m s^2 + (k - m{\omega_0}^2)) d_u &= F_u + 2 m {\omega_0} s d_v \\ @@ -733,7 +727,7 @@ The two previous equations can be written in a matrix form:

\begin{equation} -\label{org90e92a6} +\label{org8cb65e4} \begin{bmatrix} d_u \\ d_v \end{bmatrix} = \frac{1}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2} \begin{bmatrix} @@ -750,26 +744,26 @@ Then, coupling is negligible if \(|-m \omega^2 + (k - m{\omega_0}^2)| \gg |2 m {

-
-

1.6.1 Numerical Analysis

+
+

1.6.1 Numerical Analysis

We plot on the same graph \(\frac{|-m \omega^2 + (k - m {\omega_0}^2)|}{|2 m \omega_0 \omega|}\) for the voice coil and the piezo:

    -
  • with the light sample (figure 2).
    • -
  • with the heavy sample (figure 3).
    • +
  • with the light sample (figure 2).
    • +
  • with the heavy sample (figure 3).
-
+

coupling_light.png

Figure 2: Relative Coupling for light mass and high rotation speed

-
+

coupling_heavy.png

Figure 3: Relative Coupling for heavy mass and low rotation speed

@@ -785,17 +779,17 @@ Coupling is higher for actuators with small stiffness.
-
-

1.7 Limitations due to negative stiffness effect

+
+

1.7 Limitations due to negative stiffness effect

If \(\max{\dot{\theta}} \ll \sqrt{\frac{k}{m}}\), then the negative spring effect is negligible and \(k - m\dot{\theta}^2 \approx k\).

-Let’s estimate what is the maximum rotation speed for which the negative stiffness effect is still negligible (\(\omega_\text{max} = 0.1 \sqrt{\frac{k}{m}}\)). Results are shown table 3. +Let’s estimate what is the maximum rotation speed for which the negative stiffness effect is still negligible (\(\omega_\text{max} = 0.1 \sqrt{\frac{k}{m}}\)). Results are shown table 3.

-
Table 2: Negative Spring effect
+
@@ -847,7 +841,7 @@ The system can even goes unstable when \(m \omega^2 > k\), that is when the cent From this analysis, we can determine the lowest practical stiffness that is possible to use: \(k_\text{min} = 10 m \omega^2\) (table sec:tab:mink)

-
Table 3: Maximum rotation speed at which negative stiffness is negligible (\(0.1\sqrt{\frac{k}{m}}\))
+
@@ -875,11 +869,11 @@ From this analysis, we can determine the lowest practical stiffness that is poss -
-

1.8 Effect of rotation speed on the plant

+
+

1.8 Effect of rotation speed on the plant

-As shown in equation \eqref{org90e92a6}, the plant changes with the rotation speed \(\omega_0\). +As shown in equation \eqref{org8cb65e4}, the plant changes with the rotation speed \(\omega_0\).

@@ -891,11 +885,11 @@ Then we compare the result between voice coil and piezoelectric actuators.

-
-

1.8.1 Voice coil actuator

+
+

1.8.1 Voice coil actuator

-
+

G_ws_vc.png

Figure 4: Bode plot of the direct transfer function term (from \(F_u\) to \(D_u\)) for multiple rotation speed - Voice coil

@@ -903,7 +897,7 @@ Then we compare the result between voice coil and piezoelectric actuators. -
+

Gc_ws_vc.png

Figure 5: caption

@@ -911,18 +905,18 @@ Then we compare the result between voice coil and piezoelectric actuators.
-
-

1.8.2 Piezoelectric actuator

+
+

1.8.2 Piezoelectric actuator

-
+

G_ws_pz.png

Figure 6: Bode plot of the direct transfer function term (from \(F_u\) to \(D_u\)) for multiple rotation speed - Piezoelectric actuator

-
+

Gc_ws_pz.png

Figure 7: Bode plot of the coupling transfer function term (from \(F_u\) to \(D_v\)) for multiple rotation speed - Piezoelectric actuator

@@ -930,8 +924,8 @@ Then we compare the result between voice coil and piezoelectric actuators.
-
-

1.8.3 Analysis

+
+

1.8.3 Analysis

When the rotation speed is null, the coupling terms are equal to zero and the diagonal terms corresponds to one degree of freedom mass spring system. @@ -958,8 +952,8 @@ As shown in the previous figures, the system with voice coil is much more sensit

-
-

1.8.4 Campbell diagram

+
+

1.8.4 Campbell diagram

The poles of the system are computed for multiple values of the rotation frequency. To simplify the computation of the poles, we add some damping to the system. @@ -998,7 +992,7 @@ polespz = zeros(2, length(wspz));

-We then plot the real and imaginary part of the poles as a function of the rotation frequency (figures 8 and 9). +We then plot the real and imaginary part of the poles as a function of the rotation frequency (figures 8 and 9).

@@ -1006,11 +1000,11 @@ When the real part of one pole becomes positive, the system goes unstable.

-For the voice coil (figure 8), the system is unstable when the rotation speed is above 5 rad/s. The real and imaginary part of the poles of the system with piezoelectric actuators are changing much less (figure 9). +For the voice coil (figure 8), the system is unstable when the rotation speed is above 5 rad/s. The real and imaginary part of the poles of the system with piezoelectric actuators are changing much less (figure 9).

-
+

poles_w_vc.png

Figure 8: Real and Imaginary part of the poles of the system as a function of the rotation speed - Voice Coil and light sample

@@ -1018,7 +1012,7 @@ For the voice coil (figure 8), the system is unstable -
+

poles_w_pz.png

Figure 9: Real and Imaginary part of the poles of the system as a function of the rotation speed - Piezoelectric actuator and light sample

@@ -1028,11 +1022,11 @@ For the voice coil (figure 8), the system is unstable
-
-

2 Integral Force Feedback

+
+

2 Integral Force Feedback

- +

In this section, we study the use of Decentralized Integral Force Feedback (IFF) to damp the resonance of the positioning system. @@ -1042,8 +1036,8 @@ In this section, we study the use of Decentralized Integral Force Feedback (IFF) We thus suppose there is a force sensor in series with the actuator in the \(u\) and \(v\) directions.

-
-

2.1 Analytical Derivation of the transfer function for IFF control

+
+

2.1 Analytical Derivation of the transfer function for IFF control

The sensed forces are equal to: @@ -1087,8 +1081,8 @@ If we note \(\omega_0 = \sqrt{\frac{k}{m}}\), which corresponds to the resonance

-
-

2.2 Low and High frequency Behavior

+
+

2.2 Low and High frequency Behavior

At high frequency, the force sensors give: @@ -1124,8 +1118,8 @@ It is like a negative stiffness is in parallel with both the stiffness and th

-
-

2.3 Poles and Zeros

+
+

2.3 Poles and Zeros

Let’s use Matlab Symbolic toolbox to find the analytical formula for the poles and zeros of the transfer function from the force actuators to the force sensors. @@ -1213,20 +1207,20 @@ Thus, for some finite IFF gain, one pole will have a positive real part and the

-
-

2.4 IFF Plant and Root Locus

+
+

2.4 IFF Plant and Root Locus

-The bode plot for the diagonal and off-diagonal elements are shown in Figure 10. +The bode plot for the diagonal and off-diagonal elements are shown in Figure 10.

-The change of dynamics from \(F_u\) to \(F_{u,m}\) due to the change in rotation speed is shown in Figure 11. +The change of dynamics from \(F_u\) to \(F_{u,m}\) due to the change in rotation speed is shown in Figure 11. It is shown that the rotation speed does change the low frequency gain (explained in previous section).

-The root locus for Decentralized Direct Velocity Feedback is shown in Figure 12. +The root locus for Decentralized Direct Velocity Feedback is shown in Figure 12.

@@ -1240,14 +1234,14 @@ There is an optimal gain that gives maximum damping of this resonance. -

+

iff_plant_and_coupling.png

Figure 10: Transfer Function from \(F_u\) to \(F_{u,m}\) and from \(F_u\) to \(F_{v,m}\) for \(\omega = 0.3 \omega_0\)

-
+

iff_variability_plant_ws.png

Figure 11: Transfer Function from \(F_u\) to \(F_{u,m}\) for multiple rotation speed

@@ -1257,7 +1251,7 @@ There is an optimal gain that gives maximum damping of this resonance. Let’s see how the root locus is changing with the rotation speed.

-
+

iff_root_locus_ws.png

Figure 12: Change of the Root Locus plot for different rotating speed

@@ -1265,8 +1259,8 @@ Let’s see how the root locus is changing with the rotation speed.
-
-

2.5 Conclusion

+
+

2.5 Conclusion

@@ -1276,7 +1270,7 @@ It also add zeros with positive real part which removes the typical unconditiona

-From the Root Locus plot in Figure 12, it is shown that when rotating, unstable poles will be introduced due to IFF control. +From the Root Locus plot in Figure 12, it is shown that when rotating, unstable poles will be introduced due to IFF control. Moreover, only one of the two pairs of complex conjugate poles can be critically damped.

@@ -1298,19 +1292,19 @@ The frequency of this pole can be made small if the frequency of the position sy
-
-

3 Direct Velocity Feedback

+
+

3 Direct Velocity Feedback

- +

In this section, we study the use of Direct Velocity Feedback to damp the resonances of the positioning station. We thus suppose that we are able to measure to relative displacement (or velocity) \(d_u\) and \(d_v\).

-
-

3.1 Analytical Study

+
+

3.1 Analytical Study

The equation of motion are: @@ -1335,8 +1329,8 @@ If we note \(\omega_0 = \sqrt{\frac{k}{m}}\), we obtain

-
-

3.2 High and Low frequency behavior

+
+

3.2 High and Low frequency behavior

At low frequency: @@ -1372,8 +1366,8 @@ Thus the rotating speed does not change the high frequency behavior of the diago

-
-

3.3 Poles and Zeros

+
+

3.3 Poles and Zeros

The direct terms (diagonal terms of the 2x2 transfer function matrix) have two complex conjugate zeros at: @@ -1392,41 +1386,41 @@ Which are between the two pairs of complex conjugate poles at:

-
-

3.4 DVF Plant and Root Locus

+
+

3.4 DVF Plant and Root Locus

-The bode plot for the diagonal and off-diagonal elements are shown in Figure 13. +The bode plot for the diagonal and off-diagonal elements are shown in Figure 13.

-The change of dynamics from \(F_u\) to \(d_u\) due to the change in rotation speed is shown in Figure 14. +The change of dynamics from \(F_u\) to \(d_u\) due to the change in rotation speed is shown in Figure 14. It is shown that the rotation speed does not change much the low frequency and high frequency gains. It only modifies the location of the two pair of complex conjugate pairs.

-The root locus for Decentralized Direct Velocity Feedback is shown in Figure 15. +The root locus for Decentralized Direct Velocity Feedback is shown in Figure 15. It is shown that the closed-loop poles are in the left-half of the complex plane, whatever the gain. Moreover, has much damping can be added to both pairs of complex conjugate poles.

-
+

dvf_plant_and_coupling.png

Figure 13: Transfer Function from \(F_u\) to \(d_u\) and from \(F_u\) to \(d_v\) for \(\omega = 0.3 \omega_0\)

-
+

dvf_variability_plant_ws.png

Figure 14: Transfer Function from \(F_u\) to \(d_u\) for multiple rotation speed

-
+

dvf_root_locus_ws.png

Figure 15: Change of the Root Locus plot for multiple rotating speed

@@ -1434,8 +1428,8 @@ Moreover, has much damping can be added to both pairs of complex conjugate poles
-
-

3.5 Conclusion

+
+

3.5 Conclusion

@@ -1443,7 +1437,7 @@ Moreover, has much damping can be added to both pairs of complex conjugate poles

-As shown by the Root Locus in Figure 15: +As shown by the Root Locus in Figure 15:

  • DVF permits to critically damp the vibrations of the positioning stage
    • @@ -1456,15 +1450,15 @@ As shown by the Root Locus in Figure 15:
-
-

4 Control Strategies

+
+

4 Control Strategies

- +

-
-

4.1 Measurement in the fixed reference frame

+
+

4.1 Measurement in the fixed reference frame

First, let’s consider a measurement in the fixed referenced frame. @@ -1487,11 +1481,11 @@ Finally, the control low \(K\) links the position errors \([\epsilon_u, \epsilon

-The block diagram is shown on figure 16. +The block diagram is shown on figure 16.

-
+

control_measure_fixed_2dof.png

Figure 16: Control with a measure from fixed frame

@@ -1507,19 +1501,19 @@ One question we wish to answer is: is \(G(\theta) J(\theta) = G(\theta_0) J(\the
-
-

4.2 Measurement in the rotating frame

+
+

4.2 Measurement in the rotating frame

Let’s consider that the measurement is made in the rotating reference frame.

-The corresponding block diagram is shown figure 17. +The corresponding block diagram is shown figure 17.

-
+

control_measure_rotating_2dof.png

Figure 17: Control with a measure from rotating frame

@@ -1532,16 +1526,16 @@ The loop gain is \(L = G K\).
-
-

5 Multi Body Model - Simscape

+
+

5 Multi Body Model - Simscape

- +

-
-

5.1 Initialization

+
+

5.1 Initialization

Let’s load the previously defined parameters for the model. @@ -1598,8 +1592,8 @@ freqs = logspace(-2, 3, 1000); -

5.2 Identification in the rotating referenced frame

+
+

5.2 Identification in the rotating referenced frame

We initialize the inputs and outputs of the system to identify: @@ -1666,22 +1660,22 @@ Gvc_heavy.OutputName = {'Du', -

5.3 Coupling ratio between \(f_{uv}\) and \(d_{uv}\)

+
+

5.3 Coupling ratio between \(f_{uv}\) and \(d_{uv}\)

In order to validate the equations written, we can compute the coupling ratio using the simscape model and compare with the equations.

-From the previous identification, we plot the coupling ratio in both case (figure 18). +From the previous identification, we plot the coupling ratio in both case (figure 18).

-We obtain the same result than the analytical case (figures 2 and 3). +We obtain the same result than the analytical case (figures 2 and 3).

-
+

coupling_ratio_light_heavy.png

Figure 18: Coupling ratio obtained with the Simscape model

@@ -1689,8 +1683,8 @@ We obtain the same result than the analytical case (figures -

5.4 Transfer function from force to force sensor (IFF plant)

+
+

5.4 Transfer function from force to force sensor (IFF plant)

%% Name of the Simulink File
@@ -1745,12 +1739,12 @@ Gvc_heavy.OutputName = {'Fum', 
-
5.5 Plant Control - SISO approach

+

+
5.5 Plant Control - SISO approach

 

 

-

-
5.5.1 Plant identification

+

+
5.5.1 Plant identification

 

 

 The goal is to study the control problems due to the coupling that appears because of the rotation.
@@ -1765,11 +1759,11 @@ The actuators are voice coil with some damping added.
 

 
 

-The bode plot of the system not rotating and rotating at 60rpm is shown figure 19.
+The bode plot of the system not rotating and rotating at 60rpm is shown figure 19.
 

 
 
-

+

 
Gvc_speed.png
 

 
Figure 19: Change of transfer functions due to rotating speed

@@ -1777,12 +1771,12 @@ The bode plot of the system not rotating and rotating at 60rpm is shown figure <
 

 

 
-

-
5.5.2 Effect of rotation speed

+

+
5.5.2 Effect of rotation speed

 

 

 We first identify the system (voice coil and light mass) for multiple rotation speed.
-Then we compute the bode plot of the system (figure 20).
+Then we compute the bode plot of the system (figure 20).
 

 
 

@@ -1799,18 +1793,18 @@ To stabilize the unstable pole, we need a control bandwidth of at least twice of
 

 
 
-

+

 
Guu_uv_ws.png
 

 
Figure 20: Diagonal term as a function of the rotation frequency

 

 
 

-Then, we can look at the same plots for the piezoelectric actuator (figure 21). The effect of the rotation frequency has very little effect on the dynamics of the system to control.
+Then, we can look at the same plots for the piezoelectric actuator (figure 21). The effect of the rotation frequency has very little effect on the dynamics of the system to control.
 

 
 
-

+

 
Guu_ws_pz.png
 

 
Figure 21: Diagonal term as a function of the rotation frequency

@@ -1818,8 +1812,8 @@ Then, we can look at the same plots for the piezoelectric actuator (figure 
 

 
-

-
5.5.3 Controller design

+

+
5.5.3 Controller design

 

 

 We design a controller based on the identification when the system is not rotating.
@@ -1839,11 +1833,11 @@ K.OutputName = {'Fu', '
 

 
 

-The loop gain is displayed figure 22.
+The loop gain is displayed figure 22.
 

 
 
-

+

 
Gvc_loop_gain.png
 

 
Figure 22: Loop gain obtained for a lead-lag controller on the system with a voice coil

@@ -1851,8 +1845,8 @@ The loop gain is displayed figure 22.
 

 

 
-

-
5.5.4 Controlling the rotating system

+

+
5.5.4 Controlling the rotating system

 

 

 We here want to see if the system is robust with respect to the rotation speed.
@@ -1864,7 +1858,7 @@ Let’s first plot the SISO loop gain.
 

 
 
-

+

 
loop_gain_turning.png
 

 
Figure 23: Loop Gain \(G_u * K\)

@@ -1882,7 +1876,7 @@ We can now compute the close-loop systems.
 

 
 

-Let’s now look on figure 24 at the evolution of the poles of the system when closing only one loop (controlling only one direction). We see that two complex conjugate poles are approaching the real axis and then they separate: one goes to positive real part and the other goes to negative real part.
+Let’s now look on figure 24 at the evolution of the poles of the system when closing only one loop (controlling only one direction). We see that two complex conjugate poles are approaching the real axis and then they separate: one goes to positive real part and the other goes to negative real part.
 The system then goes unstable at some point (here for \(\omega=60rpm\)).
 

 
@@ -1901,14 +1895,14 @@ legend('Location', 'nor
 

 
 
-

+

 
evolution_poles_u.png
 

 
Figure 24: Evolution of the poles of the closed-loop system when closing just one loop

 

 
 

-If we look at the poles of the closed loop-system for multiple rotating speed (figure 25) when closing the two loops (MIMO system), we see that they all have a negative real part (stable system), and their evolution on the complex plane is rather small compare to the open loop evolution.
+If we look at the poles of the closed loop-system for multiple rotating speed (figure 25) when closing the two loops (MIMO system), we see that they all have a negative real part (stable system), and their evolution on the complex plane is rather small compare to the open loop evolution.
 

 
 

@@ -1925,7 +1919,7 @@ legend('Location', 'nor
 

 
 
-

+

 
poles_cl_system.png
 

 
Figure 25: Evolution of the poles of the closed-loop system

@@ -1933,8 +1927,8 @@ legend('Location', 'nor
 

 

 
-

-
5.5.5 TODO Neglected coupling

+

+
5.5.5 TODO Neglected coupling

 

 

 Take a diagonal system and apply the diagonal controller.
@@ -1947,11 +1941,11 @@ I expect that the system will be unstable
 

 

 
-

-
5.5.6 Close loop performance

+

+
5.5.6 Close loop performance

 

 

-First, we create the closed loop systems. Then, we plot the transfer function from the reference signals \([r_u, r_v]\) to the output \([d_u, d_v]\) (figure 26).
+First, we create the closed loop systems. Then, we plot the transfer function from the reference signals \([r_u, r_v]\) to the output \([d_u, d_v]\) (figure 26).
 

 
 

@@ -1990,7 +1984,7 @@ linkaxes([ax1,ax2],'x');
 

 
 
-

+

 
perfcomp.png
 

 
Figure 26: Close loop performance for \(\omega = 0\) and \(\omega = 60 rpm\)

@@ -1998,8 +1992,8 @@ linkaxes([ax1,ax2],'x');
 

 

 
-

-
5.5.7 Campbell Diagram for the close loop system

+

+
5.5.7 Campbell Diagram for the close loop system

 

 

 
m = mlight;
@@ -2029,71 +2023,27 @@ polesvc = zeros(8, length(wsvc));
 

 

 
-

-
5.5.8 Conclusion

+

+
5.5.8 Conclusion

 

 

 

-Even though considering one input and output at a time would results in an unstable system (figure 24), when using the diagonal MIMO controller, the system stays stable (figure 25). This could be understood by saying that when controlling both directions at the same time, the coupling effect should be much lower than when controlling only one direction.
+Even though considering one input and output at a time would results in an unstable system (figure 24), when using the diagonal MIMO controller, the system stays stable (figure 25). This could be understood by saying that when controlling both directions at the same time, the coupling effect should be much lower than when controlling only one direction.
 

 
 

-The close-loop performance does not vary a lot with the rotating speed (figure 26) even tough the open loop system is varying quite a lot (figure 20).
+The close-loop performance does not vary a lot with the rotating speed (figure 26) even tough the open loop system is varying quite a lot (figure 20).
 

 
 

 

 

 

-
-

-
5.6 BKMK Plant Control - MIMO approach

-

-

-

-
5.6.1 Analysis - SVD

-

-

-The singular value decomposition of a MIMO system \(G\) is defined as follow:
-\[ G = U \Sigma V^H \]
-

-
-

-With:
-

-
    
-
  • \(\Sigma\) is an \(2 \times 2\) matrix with 2 non-negative singular values \(\sigma_i\), arranged in descending order along its main diagonal
    • 
-
  • \(U\) is an \(2 \times 2\) unitary matrix. The columns vectors of \(U\), denoted \(u_i\), represent the output directions of the plant. They are orthonomal
    • 
-
  • \(V\) is an \(2 \times 2\) unitary matrix. The columns vectors of \(V\), denoted \(v_i\), represent the input directions of the plant. They are orthonomal
    • 
-

-
-

-We first look at the evolution of the singular values as a function of frequency (figure 27).
-

-
-
-

-
G_sigma.png
-

-
Figure 27: Evolution of the singular values with frequency

-

-

-

-

-

-
-

-
6 Control Implementation

-

-

-
-

-

 

 

 

 
Author: Dehaeze Thomas

-
Created: 2020-04-09 jeu. 17:20

+
Created: 2020-04-09 jeu. 17:35

 

 
 
diff --git a/index.org b/index.org
index cbb6dbd..aa3cf9a 100644
--- a/index.org
+++ b/index.org
@@ -30,14 +30,14 @@ In section [[sec:system]], a simple system composed of a spindle and a translati
 The rotation induces some coupling between the actuators and their displacement, and modifies the dynamics of the system.
 This is studied using the equations, and some numerical computations are used to compare the use of voice coil and piezoelectric actuators.
 
+In sections [[sec:iff]] and [[sec:dvf]], the use of Integral Force Feedback and Direct Velocity Feedback is studied for the case of rotating positioning platforms.
+
 Then, in section [[sec:control_strategies]], two different control approach are compared where:
 - the measurement is made in the fixed frame
 - the measurement is made in the rotating frame
 
 In section [[sec:simscape]], the analytical study will be validated using a multi body model of the studied system.
 
-Finally, in section [[sec:control]], the control strategies are implemented using Simulink and Simscape and compared.
-
 * System Description and Analysis
 :PROPERTIES:
 :HEADER-ARGS:matlab+: :tangle matlab/system_numerical_analysis.m
@@ -1236,7 +1236,7 @@ As shown by the Root Locus in Figure [[fig:dvf_root_locus_ws]]:
 
 
 * Control Strategies
-  <>
+<>
 ** Measurement in the fixed reference frame
 First, let's consider a measurement in the fixed referenced frame.
 
@@ -1270,10 +1270,10 @@ The corresponding block diagram is shown figure [[fig:control_measure_rotating_2
 The loop gain is $L = G K$.
 
 * Multi Body Model - Simscape
-  :PROPERTIES:
-  :HEADER-ARGS:matlab+: :tangle matlab/simscape_analysis.m
-  :END:
-  <>
+:PROPERTIES:
+:HEADER-ARGS:matlab+: :tangle matlab/simscape_analysis.m
+:END:
+<>
 
 ** Matlab Init                                              :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -1998,47 +1998,3 @@ First, we create the closed loop systems. Then, we plot the transfer function fr
   The close-loop performance does not vary a lot with the rotating speed (figure [[fig:perfcomp]]) even tough the open loop system is varying quite a lot (figure [[fig:Guu_uv_ws]]).
 #+end_important
 
-** BKMK Plant Control - MIMO approach
-*** Analysis - SVD
-The singular value decomposition of a MIMO system $G$ is defined as follow:
-\[ G = U \Sigma V^H \]
-
-With:
-- $\Sigma$ is an $2 \times 2$ matrix with 2 non-negative *singular values* $\sigma_i$, arranged in descending order along its main diagonal
-- $U$ is an $2 \times 2$ unitary matrix. The columns vectors of $U$, denoted $u_i$, represent the *output directions* of the plant. They are orthonomal
-- $V$ is an $2 \times 2$ unitary matrix. The columns vectors of $V$, denoted $v_i$, represent the *input directions* of the plant. They are orthonomal
-
-We first look at the evolution of the singular values as a function of frequency (figure [[fig:G_sigma]]).
-
-#+begin_src matlab :exports none
-  freqs = logspace(-2, 1, 1000);
-
-  figure;
-  hold on;
-  for i = 1:length(ws)
-    sv = sigma(Gs_vc{i}, 2*pi*freqs);
-    set(gca,'ColorOrderIndex',i)
-    plot(freqs, sv(1, :), 'DisplayName', sprintf('w = %.0f rpm', ws(i)*60/2/pi));
-    set(gca,'ColorOrderIndex',i)
-    plot(freqs, sv(2, :), '--', 'HandleVisibility', 'off');
-  end
-  hold off;
-  set(gca,'xscale','log'); set(gca,'yscale','log');
-  legend('location', 'southwest');
-#+end_src
-
-#+HEADER: :tangle no :exports results :results none :noweb yes
-#+begin_src matlab :var filepath="figs/G_sigma.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
-  <>
-#+end_src
-
-#+LABEL: fig:G_sigma
-#+CAPTION: Evolution of the singular values with frequency
-[[file:figs/G_sigma.png]]
-
-* Control Implementation
-  <>
-
-* Bibliography                                                      :ignore:
-# bibliographystyle:unsrt
-# bibliography:refs.bib
diff --git a/references.html b/references.html
deleted file mode 100644
index f7cd043..0000000
--- a/references.html
+++ /dev/null
@@ -1,16 +0,0 @@
-
-
Table 4: Minimum possible stiffness
- - - - - -
-[1] - -Steven W. Smith. - The Scientist and Engineer's Guide to Digital Signal Processing - - Second Edition. - California Technical Publishing, 1999. - -
\ No newline at end of file diff --git a/refs.bib b/refs.bib deleted file mode 100644 index e69de29..0000000 diff --git a/rotating_frame.slxc b/rotating_frame.slxc deleted file mode 100644 index 330f6f5..0000000 Binary files a/rotating_frame.slxc and /dev/null differ diff --git a/rotating_frame_ctrl.slx b/rotating_frame_ctrl.slx deleted file mode 100644 index ee27159..0000000 Binary files a/rotating_frame_ctrl.slx and /dev/null differ diff --git a/rotating_frame_ctrl.slxc b/rotating_frame_ctrl.slxc deleted file mode 100644 index dd3a8cd..0000000 Binary files a/rotating_frame_ctrl.slxc and /dev/null differ