From 9addb4bbbd265daeb03d2ac7d32afdcbc184f426 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Mon, 6 Jul 2020 23:25:41 +0200 Subject: [PATCH] Add analysis about DC compliance gain --- matlab/figs/compliance_dc_gain_alpha.pdf | Bin 0 -> 29762 bytes matlab/figs/compliance_dc_gain_alpha.png | Bin 0 -> 85426 bytes matlab/figs/compliance_dc_gain_kp.pdf | Bin 0 -> 30841 bytes matlab/figs/compliance_dc_gain_kp.png | Bin 0 -> 93581 bytes matlab/figs/compliance_dc_gain_wi.pdf | Bin 0 -> 29762 bytes matlab/figs/compliance_dc_gain_wi.png | Bin 0 -> 85426 bytes matlab/figs/mod_iff_damping_wi.pdf | Bin 27845 -> 26365 bytes matlab/figs/mod_iff_damping_wi.png | Bin 91459 -> 74101 bytes matlab/figs/opt_damp_alpha.pdf | Bin 0 -> 24104 bytes matlab/figs/opt_damp_alpha.png | Bin 0 -> 59694 bytes matlab/figs/opt_damp_vs_dc_comp.pdf | Bin 0 -> 19534 bytes matlab/figs/opt_damp_vs_dc_comp.png | Bin 0 -> 41192 bytes matlab/index.html | 944 ++++++++++------------- matlab/src/computeSimultaneousDamping.m | 10 + 14 files changed, 419 insertions(+), 535 deletions(-) create mode 100644 matlab/figs/compliance_dc_gain_alpha.pdf create mode 100644 matlab/figs/compliance_dc_gain_alpha.png create mode 100644 matlab/figs/compliance_dc_gain_kp.pdf create mode 100644 matlab/figs/compliance_dc_gain_kp.png create mode 100644 matlab/figs/compliance_dc_gain_wi.pdf create mode 100644 matlab/figs/compliance_dc_gain_wi.png create mode 100644 matlab/figs/opt_damp_alpha.pdf create mode 100644 matlab/figs/opt_damp_alpha.png create mode 100644 matlab/figs/opt_damp_vs_dc_comp.pdf create mode 100644 matlab/figs/opt_damp_vs_dc_comp.png create mode 100644 matlab/src/computeSimultaneousDamping.m diff --git a/matlab/figs/compliance_dc_gain_alpha.pdf b/matlab/figs/compliance_dc_gain_alpha.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f06402f6db95101cf0c4915d8535843af5699482 GIT binary patch literal 29762 zcmd43WmFy8wk-@H5Fog_ySoKA29ElMc6elb zHoE#IhIqmnM=2CpkK 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Table of Contents

    -
  • Section 1
  • -
  • Section 2
  • -
  • Section 3
  • -
  • Section 4
  • -
  • Section 5
  • -
  • Section 6
  • -
  • Section 7
  • +
  • Section 1
  • +
  • Section 2
  • +
  • Section 3
  • +
  • Section 4
  • +
  • Section 5
  • +
  • Section 6
-
-

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).

-
+

system.png

Figure 1: Schematic of the studied system

@@ -142,11 +136,11 @@ The measurement is either the \(x-y\) displacement of the object located on top
-
-

1.2 Equations

+
+

1.2 Equations

-Based on the Figure 1, the equations of motions are: +Based on the Figure 1, the equations of motions are:

\begin{equation} @@ -183,8 +177,8 @@ Explain Coriolis and Centrifugal Forces (negative Stiffness)
-
-

1.3 Numerical Values

+
+

1.3 Numerical Values

Let’s define initial values for the model. @@ -204,19 +198,19 @@ w0 = sqrt(k/m); % [rad/s]

-
-

1.4 Campbell Diagram

+
+

1.4 Campbell Diagram

The Campbell Diagram displays the evolution of the real and imaginary parts of the system as a function of the rotating speed.

-It is shown in Figure 2, and one can see that the system becomes unstable for \(\Omega > \omega_0\) (the real part of one of the poles becomes positive). +It is shown in Figure 2, and one can see that the system becomes unstable for \(\Omega > \omega_0\) (the real part of one of the poles becomes positive).

-
+

campbell_diagram.png

Figure 2: Campbell Diagram

@@ -224,8 +218,8 @@ It is shown in Figure 2, and one can see that the syst
-
-

1.5 Simscape Model

+
+

1.5 Simscape Model

Define the rotating speed for the Simscape Model. @@ -275,11 +269,11 @@ The same transfer function from \([F_u, F_v]\) to \([d_u, d_v]\) is written down

-Both transfer functions are compared in Figure 3 and are found to perfectly match. +Both transfer functions are compared in Figure 3 and are found to perfectly match.

-
+

plant_simscape_analytical.png

Figure 3: Bode plot of the transfer function from \([F_u, F_v]\) to \([d_u, d_v]\) as identified from the Simscape model and from an analytical model

@@ -287,8 +281,8 @@ Both transfer functions are compared in Figure 3 and a
-
-

1.6 Effect of the rotation speed

+
+

1.6 Effect of the rotation speed

The transfer functions from \([F_u, F_v]\) to \([d_u, d_v]\) are identified for the following rotating speeds. @@ -312,11 +306,11 @@ end

-They are compared in Figure 4. +They are compared in Figure 4.

-
+

plant_compare_rotating_speed.png

Figure 4: Comparison of the transfer functions from \([F_u, F_v]\) to \([d_u, d_v]\) for several rotating speed

@@ -325,11 +319,11 @@ They are compared in Figure 4.
-
-

2 Problem with pure Integral Force Feedback

+
+

2 Problem with pure Integral Force Feedback

- +

  • Diagram with the controller
  • @@ -337,8 +331,8 @@ They are compared in Figure 4.
-
-

2.1 Schematic

+
+

2.1 Schematic

figs-tikz/system_iff.pdf @@ -346,8 +340,8 @@ They are compared in Figure 4.

-
-

2.2 Plant Parameters

+
+

2.2 Plant Parameters

Let’s define initial values for the model. @@ -367,8 +361,8 @@ w0 = sqrt(k/m); % [rad/s]

-
-

2.3 Equations

+
+

2.3 Equations

The sensed forces are equal to: @@ -413,8 +407,8 @@ Which then gives:

-
-

2.4 Simscape Model

+
+

2.4 Comparison of the Analytical Model and the Simscape Model

The rotation speed is set to \(\Omega = 0.1 \omega_0\). @@ -451,12 +445,7 @@ Giff.InputName = {'Fu', 'Fv'}; Giff.OutputName = {'fu', 'fv'};

-
-
-
-

2.5 Comparison of the Analytical Model and the Simscape Model

-

The same transfer function from \([F_u, F_v]\) to \([f_u, f_v]\) is written down from the analytical model.

@@ -468,11 +457,10 @@ The same transfer function from \([F_u, F_v]\) to \([f_u, f_v]\) is written down

-The two are compared in Figure 5 and found to perfectly match. +The two are compared in Figure 5 and found to perfectly match.

- -
+

plant_iff_comp_simscape_analytical.png

Figure 5: Comparison of the transfer functions from \([F_u, F_v]\) to \([f_u, f_v]\) between the Simscape model and the analytical one

@@ -480,9 +468,9 @@ The two are compared in Figure 5 and found to perfectl
-
-

2.6 Effect of the rotation speed

-
+
+

2.5 Effect of the rotation speed

+

The transfer functions from \([F_u, F_v]\) to \([f_u, f_v]\) are identified for the following rotating speeds.

@@ -505,10 +493,10 @@ end

-The obtained transfer functions are shown in Figure 6. +The obtained transfer functions are shown in Figure 6.

-
+

plant_iff_compare_rotating_speed.png

Figure 6: Comparison of the transfer functions from \([F_u, F_v]\) to \([f_u, f_v]\) for several rotating speed

@@ -516,9 +504,9 @@ The obtained transfer functions are shown in Figure 6.
-
-

2.7 Decentralized Integral Force Feedback

-
+
+

2.6 Decentralized Integral Force Feedback

+

The decentralized IFF controller consists of pure integrators:

@@ -530,12 +518,12 @@ The decentralized IFF controller consists of pure integrators: \end{equation}

-The Root Locus (evolution of the poles of the closed loop system in the complex plane as a function of \(g\)) is shown in Figure 7. +The Root Locus (evolution of the poles of the closed loop system in the complex plane as a function of \(g\)) is shown in Figure 7. It is shown that for non-null rotating speed, one pole is bound to the right-half plane, and thus the closed loop system is unstable.

-
+

root_locus_pure_iff.png

Figure 7: Root Locus for the Decentralized Integral Force Feedback controller. Several rotating speed are shown.

@@ -544,19 +532,19 @@ It is shown that for non-null rotating speed, one pole is bound to the right-hal
-
-

3 Integral Force Feedback with an High Pass Filter

+
+

3 Integral Force Feedback with an High Pass Filter

- +

  • Classical modification of the IFF
-
-

3.1 Plant Parameters

+
+

3.1 Plant Parameters

Let’s define initial values for the model. @@ -576,8 +564,8 @@ w0 = sqrt(k/m); % [rad/s]

-
-

3.2 Modified Integral Force Feedback Controller

+
+

3.2 Modified Integral Force Feedback Controller

Let’s modify the initial Integral Force Feedback Controller ; instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used: @@ -612,10 +600,10 @@ And the following rotating speed.

-The obtained Loop Gain is shown in Figure 8. +The obtained Loop Gain is shown in Figure 8.

-
+

loop_gain_modified_iff.png

Figure 8: Loop Gain for the modified IFF controller

@@ -623,15 +611,15 @@ The obtained Loop Gain is shown in Figure 8.
-
-

3.3 Root Locus

+
+

3.3 Root Locus

-As shown in the Root Locus plot (Figure 9), for some value of the gain, the system remains stable. +As shown in the Root Locus plot (Figure 9), for some value of the gain, the system remains stable.

-
+

root_locus_modified_iff.png

Figure 9: Root Locus for the modified IFF controller

@@ -639,11 +627,11 @@ As shown in the Root Locus plot (Figure 9), for some v
-
-

3.4 What is the optimal \(\omega_i\) and \(g\)?

+
+

3.4 What is the optimal \(\omega_i\) and \(g\)?

-In order to visualize the effect of \(\omega_i\) on the attainable damping, the Root Locus is displayed in Figure 10 for the following \(\omega_i\): +In order to visualize the effect of \(\omega_i\) on the attainable damping, the Root Locus is displayed in Figure 10 for the following \(\omega_i\):

wis = [0.01, 0.1, 0.5, 1]*w0; % [rad/s]
@@ -651,7 +639,7 @@ In order to visualize the effect of \(\omega_i\) on the attainable damping, the
 
-
+

root_locus_wi_modified_iff.png

Figure 10: Root Locus for the modified IFF controller (zoomed plot on the left)

@@ -674,40 +662,36 @@ The gain at which the system becomes unstable is \end{equation}

-While it seems that small \(\omega_i\) do allow more damping to be added to the system (Figure 10), the control gains may be limited to small values due to \eqref{eq:iff_gmax} thus reducing the attainable damping. +While it seems that small \(\omega_i\) do allow more damping to be added to the system (Figure 10), the control gains may be limited to small values due to \eqref{eq:iff_gmax} thus reducing the attainable damping.

There must be an optimum for \(\omega_i\). -To find the optimum, the gain that maximize the simultaneous damping of the mode is identified for a wide range of \(\omega_i\) (Figure 11). +To find the optimum, the gain that maximize the simultaneous damping of the mode is identified for a wide range of \(\omega_i\) (Figure 11).

+
-
wis = logspace(-2, 1, 31)*w0; % [rad/s]
+
wis = logspace(-2, 1, 100)*w0; % [rad/s]
 
 opt_zeta = zeros(1, length(wis)); % Optimal simultaneous damping
 opt_gain = zeros(1, length(wis)); % Corresponding optimal gain
 
 for wi_i = 1:length(wis)
     wi = wis(wi_i);
-    gains = linspace(0, (w0^2/W^2 - 1)*wi, 100);
+    Kiff = 1/(s + wi)*eye(2);
 
-    for g = gains
-        Kiff = (g/(wi+s))*eye(2);
+    fun = @(g)computeSimultaneousDamping(g, Giff, Kiff);
 
-        [w, zeta] = damp(minreal(feedback(Giff, Kiff)));
-
-        if min(zeta) > opt_zeta(wi_i) && all(zeta > 0)
-            opt_zeta(wi_i) = min(zeta);
-            opt_gain(wi_i) = g;
-        end
-    end
+    [g_opt, xi_opt] = fminsearch(fun, 0.5*wi*((w0/W)^2 - 1));
+    opt_zeta(wi_i) = 1/xi_opt;
+    opt_gain(wi_i) = g_opt;
 end
 
-
+

mod_iff_damping_wi.png

Figure 11: Simultaneous attainable damping of the closed loop poles as a function of \(\omega_i\)

@@ -716,19 +700,19 @@ end
-
-

4 IFF with a stiffness in parallel with the force sensor

+
+

4 IFF with a stiffness in parallel with the force sensor

- +

-
-

4.1 Schematic

+
+

4.1 Schematic

-
+

system_parallel_springs.png

Figure 12: Figure caption

@@ -736,8 +720,8 @@ end
-
-

4.2 Equations

+
+

4.2 Equations

\begin{equation} @@ -789,8 +773,8 @@ Which is equivalent to
-
-

4.3 Physical Explanation

+
+

4.3 Physical Explanation

  • Negative stiffness induced by gyroscopic effects
  • @@ -816,8 +800,8 @@ And thus, the stiffness in parallel should be such that:
-
-

4.4 Plant Parameters

+
+

4.4 Plant Parameters

Let’s define initial values for the model. @@ -837,8 +821,8 @@ w0 = sqrt(k/m); % [rad/s]

-
-

4.5 Comparison of the Analytical Model and the Simscape Model

+
+

4.5 Comparison of the Analytical Model and the Simscape Model

The same transfer function from \([F_u, F_v]\) to \([f_u, f_v]\) is written down from the analytical model. @@ -886,7 +870,7 @@ Giff_th.OutputName = {'fu', 'fv'};

-
+

plant_iff_kp_comp_simscape_analytical.png

Figure 13: Comparison of the transfer functions from \([F_u, F_v]\) to \([f_u, f_v]\) between the Simscape model and the analytical one

@@ -894,8 +878,8 @@ Giff_th.OutputName = {'fu', 'fv'};
-
-

4.6 Effect of the parallel stiffness on the IFF plant

+
+

4.6 Effect of the parallel stiffness on the IFF plant

The rotation speed is set to \(\Omega = 0.1 \omega_0\). @@ -915,7 +899,7 @@ And the IFF plant (transfer function from \([F_u, F_v]\) to \([f_u, f_v]\)) is i

-The results are shown in Figure 14. +The results are shown in Figure 14.

@@ -924,7 +908,6 @@ One can see that for \(k_p > m \Omega^2\), the systems shows alternating complex

kp = 0;
-cp = 0;
 
 w0p = sqrt((k + kp)/m);
 xip = c/(2*sqrt((k+kp)*m));
@@ -937,7 +920,7 @@ Giff = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 )
 
 
kp = 0.5*m*W^2;
-cp = 0;
+k = 1 - kp;
 
 w0p = sqrt((k + kp)/m);
 xip = c/(2*sqrt((k+kp)*m));
@@ -950,7 +933,7 @@ Giff_s = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2
 
 
kp = 1.5*m*W^2;
-cp = 0;
+k = 1 - kp;
 
 w0p = sqrt((k + kp)/m);
 xip = c/(2*sqrt((k+kp)*m));
@@ -962,7 +945,7 @@ Giff_l = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2
 
-
+

plant_iff_kp.png

Figure 14: Transfer function from \([F_u, F_v]\) to \([f_u, f_v]\) for \(k_p = 0\), \(k_p < m \Omega^2\) and \(k_p > m \Omega^2\)

@@ -970,11 +953,11 @@ Giff_l = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2
-
-

4.7 IFF when adding a spring in parallel

+
+

4.7 IFF when adding a spring in parallel

-In Figure 15 is displayed the Root Locus in the three considered cases with +In Figure 15 is displayed the Root Locus in the three considered cases with

\begin{equation} K_{\text{IFF}} = \frac{g}{s} \begin{bmatrix} @@ -995,7 +978,7 @@ Thus, decentralized IFF controller with pure integrators can be used if: \end{equation} -
+

root_locus_iff_kp.png

Figure 15: Root Locus

@@ -1003,8 +986,8 @@ Thus, decentralized IFF controller with pure integrators can be used if:
-
-

4.8 Effect of \(k_p\) on the attainable damping

+
+

4.8 Effect of \(k_p\) on the attainable damping

However, having large values of \(k_p\) may: @@ -1015,7 +998,7 @@ However, having large values of \(k_p\) may:

-To study the second point, Root Locus plots for the following values of \(k_p\) are shown in Figure 16. +To study the second point, Root Locus plots for the following values of \(k_p\) are shown in Figure 16.

kps = [2, 20, 40]*m*W^2;
@@ -1027,16 +1010,50 @@ It is shown that large values of \(k_p\) decreases the attainable damping.
 

-
+

root_locus_iff_kps.png

Figure 16: Root Locus plot

+ +
+
alphas = logspace(-2, 0, 100);
+
+opt_zeta = zeros(1, length(alphas)); % Optimal simultaneous damping
+opt_gain = zeros(1, length(alphas)); % Corresponding optimal gain
+
+Kiff = 1/s*eye(2);
+
+for alpha_i = 1:length(alphas)
+    kp = alphas(alpha_i);
+    k = 1 - alphas(alpha_i);
+
+    w0p = sqrt((k + kp)/m);
+    xip = c/(2*sqrt((k+kp)*m));
+
+    Giff = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
+        (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
+        (2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
+
+    fun = @(g)computeSimultaneousDamping(g, Giff, Kiff);
+
+    [g_opt, xi_opt] = fminsearch(fun, 2);
+    opt_zeta(alpha_i) = 1/xi_opt;
+    opt_gain(alpha_i) = g_opt;
+end
+
+
+ + +
+

opt_damp_alpha.png +

+
-
-

4.9 Optimal Gain

+
+

4.9 Optimal Gain

Let’s take \(k_p = 5 m \Omega^2\) and find the optimal IFF control gain \(g\) such that maximum damping are added to the poles of the closed loop system. @@ -1044,7 +1061,7 @@ Let’s take \(k_p = 5 m \Omega^2\) and find the optimal IFF control gain \(

kp = 5*m*W^2;
-cp = 0.01;
+k = 1 - kp;
 
 w0p = sqrt((k + kp)/m);
 xip = c/(2*sqrt((k+kp)*m));
@@ -1075,231 +1092,56 @@ end
 
-
+

root_locus_opt_gain_iff_kp.png

-

Figure 17: Root Locus for \(k_p = 5 m \Omega^2\) and the poles corresponding to the identified optimal gain

+

Figure 18: Root Locus for \(k_p = 5 m \Omega^2\) and the poles corresponding to the identified optimal gain

-
-

5 Direct Velocity Feedback

+
+

5 Comparison

- +

-
-

5.1 Schematic

+
+

5.1 Plant Parameters

- -
-

system_dvf.png +

+Let’s define initial values for the model.

-

Figure 18: Figure caption

+
+
k = 1;    % Actuator Stiffness [N/m]
+c = 0.05; % Actuator Damping [N/(m/s)]
+m = 1;    % Payload mass [kg]
+
+
+ +
+
xi = c/(2*sqrt(k*m));
+w0 = sqrt(k/m); % [rad/s]
+
+
+ +

+The rotating speed is set to \(\Omega = 0.1 \omega_0\). +

+
+
W = 0.1*w0;
+
-
-

5.2 Equations

+
+

5.2 Root Locus

-The sensed relative velocity are equal to: -

-
-\begin{equation} -\begin{bmatrix} \dot{d}_u \\ \dot{d}_v \end{bmatrix} = -\bm{G}_v -\begin{bmatrix} F_u \\ F_v \end{bmatrix} -\end{equation} - -\begin{equation} -\begin{bmatrix} \dot{d}_u \\ \dot{d}_v \end{bmatrix} = -\frac{s}{k} \frac{1}{G_{vp}} -\begin{bmatrix} - G_{vz} & G_{vc} \\ - -G_{vc} & G_{vz} -\end{bmatrix} -\begin{bmatrix} F_u \\ F_v \end{bmatrix} -\end{equation} -

-With: -

-\begin{align} - G_{vp} &= \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\ - G_{vz} &= \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \\ - G_{vc} &= 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} -\end{align} - -
-
-
- -
-

5.3 Plant Parameters

-
-

-Let’s define initial values for the model. -

-
-
k = 1;    % Actuator Stiffness [N/m]
-c = 0.05; % Actuator Damping [N/(m/s)]
-m = 1;    % Payload mass [kg]
-
-
- -
-
xi = c/(2*sqrt(k*m));
-w0 = sqrt(k/m); % [rad/s]
-
-
-
-
- -
-

5.4 Comparison of the Analytical Model and the Simscape Model

-
-

-The rotating speed is set to \(\Omega = 0.1 \omega_0\). -

-
-
W = 0.1*w0;
-
-
- -
-
open('rotating_frame.slx');
-
-
- -

-And the transfer function from \([F_u, F_v]\) to \([v_u, v_v]\) is identified using the Simscape model. -

-
-
%% Name of the Simulink File
-mdl = 'rotating_frame';
-
-%% Input/Output definition
-clear io; io_i = 1;
-io(io_i) = linio([mdl, '/K'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/G'], 1, 'openoutput'); io_i = io_i + 1;
-
-
- -
-
Gdvf = linearize(mdl, io, 0);
-
-%% Input/Output definition
-Gdvf.InputName  = {'Fu', 'Fv'};
-Gdvf.OutputName = {'Vu', 'Vv'};
-
-
- -

-The same transfer function from \([F_u, F_v]\) to \([v_u, v_v]\) is written down from the analytical model. -

-
-
Gdvf_th = (s/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
-          [(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ...
-           -2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)];
-
-Gdvf_th.InputName  = {'Fu', 'Fv'};
-Gdvf_th.OutputName = {'vu', 'vv'};
-
-
- -

-The two are compared in Figure 5 and found to perfectly match. -

- - -
-

plant_dvf_comp_simscape_analytical.png -

-

Figure 19: Comparison of the transfer functions from \([F_u, F_v]\) to \([v_u, v_v]\) between the Simscape model and the analytical one

-
-
-
- -
-

5.5 Root Locus

-
-

-The Decentralized Direct Velocity Feedback controller consist of a pure gain on the diagonal: -

-\begin{equation} - K_{\text{DVF}}(s) = g \begin{bmatrix} - 1 & 0 \\ - 0 & 1 -\end{bmatrix} -\end{equation} - -

-The corresponding Root Locus plots for the following rotating speeds are shown in Figure 20. -

-
-
Ws = [0, 0.2, 0.7, 1.1]*w0; % Rotating Speeds [rad/s]
-
-
- -

-It is shown that for rotating speed \(\Omega < \omega_0\), the closed loop system is unconditionally stable and arbitrary damping can be added to the poles. -

- -
-

root_locus_dvf.png -

-

Figure 20: Root Locus for the Decentralized Direct Velocity Feedback controller. Several rotating speed are shown.

-
-
-
-
- -
-

6 Comparison

-
-

- -

-
- -
-

6.1 Plant Parameters

-
-

-Let’s define initial values for the model. -

-
-
k = 1;    % Actuator Stiffness [N/m]
-c = 0.05; % Actuator Damping [N/(m/s)]
-m = 1;    % Payload mass [kg]
-
-
- -
-
xi = c/(2*sqrt(k*m));
-w0 = sqrt(k/m); % [rad/s]
-
-
- -

-The rotating speed is set to \(\Omega = 0.1 \omega_0\). -

-
-
W = 0.1*w0;
-
-
-
-
- -
-

6.2 Root Locus

-
-

IFF with High Pass Filter

@@ -1329,28 +1171,18 @@ k = k + kp;
-

-DVF -

-
-
Gdvf = (s/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
-       [(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ...
-        -2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)];
-
-
- -
+

comp_root_locus.png

-

Figure 21: Root Locus plot - Comparison of IFF with additional high pass filter, IFF with additional parallel stiffness and DVF

+

Figure 19: Root Locus plot - Comparison of IFF with additional high pass filter, IFF with additional parallel stiffness

-
-

6.3 Controllers - Optimal Gains

-
+
+

5.3 Controllers - Optimal Gains

+

In order to compare to three considered Active Damping techniques, gains that yield maximum damping of all the modes are computed for each case.

@@ -1388,51 +1220,43 @@ The obtained damping ratio and control are shown below. 0.83 2.01 - - -DVF -0.85 -1.67 -
-
-

6.4 Transmissibility

-
+
+

5.4 Passive Damping - Critical Damping

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

- +Critical Damping corresponds to to \(\xi = 1\), and thus: +

+\begin{equation} + c_{\text{crit}} = 2 \sqrt{km} +\end{equation} + +
+
c_opt = 2*sqrt(k*m);
+
+
+
+
+ +
+

5.5 Transmissibility And Compliance

+
+

+

open('rotating_frame.slx');
 
-
-
%% Name of the Simulink File
-mdl = 'rotating_frame';
-
-%% Input/Output definition
-clear io; io_i = 1;
-io(io_i) = linio([mdl, '/dw'], 1, 'input');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Meas'], 1, 'output');  io_i = io_i + 1;
-
-
- -
-
Tol = linearize(mdl, io, 0);
-
-%% Input/Output definition
-Tol.InputName  = {'Dwx', 'Dwy'};
-Tol.OutputName = {'Dx', 'Dy'};
-
-
-
-
Kiff = opt_gain_iff/(wi + s)*tf(eye(2));
-
-
%% Name of the Simulink File
@@ -1441,119 +1265,68 @@ mdl = 'rotating_frame';
 %% Input/Output definition
 clear io; io_i = 1;
 io(io_i) = linio([mdl, '/dw'], 1, 'input');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Meas'], 1, 'output');  io_i = io_i + 1;
-
-
- -
-
Tiff = linearize(mdl, io, 0);
-
-%% Input/Output definition
-Tiff.InputName  = {'Dwx', 'Dwy'};
-Tiff.OutputName = {'Dx', 'Dy'};
-
-
-
-
kp = 5*m*W^2;
-cp = 0.01;
-
-
- -
-
Kiff = opt_gain_kp/s*tf(eye(2));
-
-
- -
-
%% Name of the Simulink File
-mdl = 'rotating_frame';
-
-%% Input/Output definition
-clear io; io_i = 1;
-io(io_i) = linio([mdl, '/dw'], 1, 'input');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Meas'], 1, 'output');  io_i = io_i + 1;
-
-
- -
-
Tiff_kp = linearize(mdl, io, 0);
-
-%% Input/Output definition
-Tiff_kp.InputName  = {'Dwx', 'Dwy'};
-Tiff_kp.OutputName = {'Dx', 'Dy'};
-
-
-
-
Kdvf = opt_gain_kp*tf(eye(2));
-
-
- -
-
%% Name of the Simulink File
-mdl = 'rotating_frame';
-
-%% Input/Output definition
-clear io; io_i = 1;
-io(io_i) = linio([mdl, '/dw'], 1, 'input');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Meas'], 1, 'output');  io_i = io_i + 1;
-
-
- -
-
Tdvf = linearize(mdl, io, 0);
-
-%% Input/Output definition
-Tdvf.InputName  = {'Dwx', 'Dwy'};
-Tdvf.OutputName = {'Dx', 'Dy'};
-
-
- -
-

comp_transmissibility.png -

-

Figure 22: Comparison of the transmissibility

-
-
-
- -
-

6.5 Compliance

-
-

- -

-
-
%% Name of the Simulink File
-mdl = 'rotating_frame';
-
-%% Input/Output definition
-clear io; io_i = 1;
 io(io_i) = linio([mdl, '/fd'], 1, 'input');  io_i = io_i + 1;
 io(io_i) = linio([mdl, '/Meas'], 1, 'output');  io_i = io_i + 1;
 
-
-
Col = linearize(mdl, io, 0);
+
G_ol = linearize(mdl, io, 0);
 
 %% Input/Output definition
-Col.InputName  = {'Fdx', 'Fdy'};
-Col.OutputName = {'Dx', 'Dy'};
+G_ol.InputName  = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
+G_ol.OutputName = {'Dx', 'Dy'};
 
+
+ +
+

5.5.1 Passive Damping

+
+
+
kp = 0;
+cp = 0;
+
+
+ +
+
c_old = c;
+c = c_opt;
+
+
+ +
+
G_pas = linearize(mdl, io, 0);
+
+%% Input/Output definition
+G_pas.InputName  = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
+G_pas.OutputName = {'Dx', 'Dy'};
+
+
+ +
+
c = c_old;
+
+
+
+
+ +
Kiff = opt_gain_iff/(wi + s)*tf(eye(2));
 
-
Ciff = linearize(mdl, io, 0);
+
G_iff = linearize(mdl, io, 0);
 
 %% Input/Output definition
-Ciff.InputName  = {'Fdx', 'Fdy'};
-Ciff.OutputName = {'Dx', 'Dy'};
+G_iff.InputName  = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
+G_iff.OutputName = {'Dx', 'Dy'};
 
+
+ +
kp = 5*m*W^2;
 cp = 0.01;
@@ -1566,41 +1339,142 @@ cp = 0.01;
 
-
Ciff_kp = linearize(mdl, io, 0);
+
G_kp = linearize(mdl, io, 0);
 
 %% Input/Output definition
-Ciff_kp.InputName  = {'Fdx', 'Fdy'};
-Ciff_kp.OutputName = {'Dx', 'Dy'};
+G_kp.InputName  = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
+G_kp.OutputName = {'Dx', 'Dy'};
 
-
-
Kdvf = opt_gain_kp*tf(eye(2));
-
-
-
Cdvf = linearize(mdl, io, 0);
+
-%% Input/Output definition -Cdvf.InputName = {'Fdx', 'Fdy'}; -Cdvf.OutputName = {'Dx', 'Dy'}; -
+
+

comp_transmissibility.png +

+

Figure 20: Comparison of the transmissibility

+
-
+
+ +

comp_compliance.png

-

Figure 23: Comparison of the obtained Compliance

+

Figure 21: Comparison of the obtained Compliance

+
+
+
+ +
+

5.6 DC Compliance

+
+
+
wis = logspace(-2, 1, 100)*w0; % [rad/s]
+
+opt_zeta_wi  = zeros(1, length(wis)); % Optimal simultaneous damping
+opt_gain_wi  = zeros(1, length(wis)); % Corresponding optimal gain
+C_dc_gain_wi = zeros(1, length(wis)); % Compliance DC value
+
+for wi_i = 1:length(wis)
+    wi = wis(wi_i);
+    Kiff = 1/(s + wi)*eye(2);
+
+    fun = @(g)computeSimultaneousDamping(g, Gwi({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff);
+
+    [g_opt, xi_opt] = fminsearch(fun, 0.5*wi*((w0/W)^2 - 1));
+    opt_zeta_wi(wi_i) = 1/xi_opt;
+    opt_gain_wi(wi_i) = g_opt;
+
+    G_dc_gain = dcgain(lft(Gwi, -g_opt/(s + wi)*eye(2), 2, 2));
+    C_dc_gain_wi(wi_i) = G_dc_gain(1,1);
+end
+
+
+
+
alphas = logspace(-2, 0, 100);
+
+opt_zeta_kp = zeros(1, length(alphas)); % Optimal simultaneous damping
+opt_gain_kp = zeros(1, length(alphas)); % Corresponding optimal gain
+C_dc_gain_kp = zeros(1, length(alphas)); % DC gain of the compliance
+
+Kiff = 1/s*eye(2);
+
+for alpha_i = 1:length(alphas)
+    kp = alphas(alpha_i);
+    k = 1 - alphas(alpha_i);
+
+    w0p = sqrt((k + kp)/m);
+    xip = c/(2*sqrt((k+kp)*m));
+
+    Gkp = tf(zeros(4,4));
+    Gkp.InputName  = {'Fx', 'Fy', 'Fu', 'Fv'};
+    Gkp.OutputName = {'dx', 'dy', 'fu', 'fv'};
+
+    Gp = ((s^2)/(w0p^2) + 2*xip*s/w0p + 1 - (W^2)/(w0p^2))^2 + (2*W*s/(w0p^2))^2;
+
+    Gkp('dx', 'Fu') = (1/(k + kp))*((s^2)/(w0p^2) + 2*xip*s/w0p + 1 - (W^2)/(w0p^2))/Gp;
+    Gkp('dy', 'Fv') = (1/(k + kp))*((s^2)/(w0p^2) + 2*xip*s/w0p + 1 - (W^2)/(w0p^2))/Gp;
+
+    Gkp('dx', 'Fx') = (1/(k + kp))*((s^2)/(w0p^2) + 2*xip*s/w0p + 1 - (W^2)/(w0p^2))/Gp;
+    Gkp('dy', 'Fy') = (1/(k + kp))*((s^2)/(w0p^2) + 2*xip*s/w0p + 1 - (W^2)/(w0p^2))/Gp;
+
+    Gkp('dx', 'Fv') =  (1/(k + kp))*(2*W*s/(w0p^2))/Gp;
+    Gkp('dy', 'Fu') = -(1/(k + kp))*(2*W*s/(w0p^2))/Gp;
+
+    Gkp('dx', 'Fy') =  (1/(k + kp))*(2*W*s/(w0p^2))/Gp;
+    Gkp('dy', 'Fx') = -(1/(k + kp))*(2*W*s/(w0p^2))/Gp;
+
+    Gkp('fu', 'Fu') = ((s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2)/Gp;
+    Gkp('fv', 'Fv') = ((s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2)/Gp;
+
+    Gkp('fu', 'Fv') = -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p))/Gp;
+    Gkp('fv', 'Fu') =  (2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p))/Gp;
+
+    Gkp('fu', 'Fx') = -(c*s + k)*(1/(k + kp))*((s^2)/(w0p^2) + 2*xip*s/w0p + 1 - (W^2)/(w0p^2))/Gp;
+    Gkp('fv', 'Fy') = -(c*s + k)*(1/(k + kp))*((s^2)/(w0p^2) + 2*xip*s/w0p + 1 - (W^2)/(w0p^2))/Gp;
+
+    Gkp('fu', 'Fy') = -(c*s + k)*(1/(k + kp))*(2*W*s/(w0p^2))/Gp;
+    Gkp('fv', 'Fx') =  (c*s + k)*(1/(k + kp))*(2*W*s/(w0p^2))/Gp;
+
+    fun = @(g)computeSimultaneousDamping(g, Gkp({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff);
+
+    [g_opt, xi_opt] = fminsearch(fun, 2);
+    opt_zeta_kp(alpha_i) = 1/xi_opt;
+    opt_gain_kp(alpha_i) = g_opt;
+
+    G_dc_gain = dcgain(lft(Gkp, -g_opt/s*eye(2), 2, 2));
+    C_dc_gain_kp(alpha_i) = G_dc_gain(1,1);
+end
+
+
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compliance_dc_gain_wi.png +

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compliance_dc_gain_kp.png +

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opt_damp_vs_dc_comp.png +

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7 Notations

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6 Notations

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@@ -1802,7 +1676,7 @@ Cdvf.OutputName = {'Dx', 'Dy'};

Author: Thomas Dehaeze

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Created: 2020-06-24 mer. 08:44

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Created: 2020-07-06 lun. 23:23

diff --git a/matlab/src/computeSimultaneousDamping.m b/matlab/src/computeSimultaneousDamping.m new file mode 100644 index 0000000..2cfb68d --- /dev/null +++ b/matlab/src/computeSimultaneousDamping.m @@ -0,0 +1,10 @@ +% Compute Damping + +function [xi_min] = computeSimultaneousDamping(g, G, K) + [w, xi] = damp(minreal(feedback(G, g*K))); + xi_min = 1/min(xi); + + if xi_min < 0 + xi_min = 1e8; + end +end