From 0e5ea4de1559ceb77f263682230fde7fd17aa207 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Thu, 11 Jun 2020 14:17:56 +0200 Subject: [PATCH] Rework all the analysis - better variable names - better figures --- matlab/index.html | 1004 +++++++++++-- matlab/index.org | 1248 ++++++++++------- matlab/matlab/rotating_frame.slx | Bin 38329 -> 38795 bytes matlab/rotating_frame.slxc | Bin 5632 -> 5624 bytes .../rotating_frame/checksumOfCache.mat | Bin 392 -> 392 bytes .../tmwinternal/simulink_cache.xml | 4 +- .../sim/varcache/rotating_frame/varInfo.mat | Bin 3072 -> 3072 bytes tikz/figs/rotating_xy_platform.pdf | Bin 62237 -> 62289 bytes tikz/figs/rotating_xy_platform.png | Bin 72354 -> 85356 bytes tikz/figs/rotating_xy_platform.svg | Bin 66135 -> 66901 bytes tikz/figs/rotating_xy_platform_springs.pdf | Bin 60656 -> 62057 bytes tikz/figs/rotating_xy_platform_springs.png | Bin 69119 -> 72726 bytes tikz/figs/rotating_xy_platform_springs.svg | Bin 56531 -> 63269 bytes tikz/index.org | 97 +- 14 files changed, 1717 insertions(+), 636 deletions(-) diff --git a/matlab/index.html b/matlab/index.html index 1ced094..6ea41fa 100644 --- a/matlab/index.html +++ b/matlab/index.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Matlab Computation @@ -34,43 +34,92 @@

Table of Contents

@@ -125,28 +174,31 @@ Notations: -
-

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).

-

-figs/rotating_xy_platform.pdf + +

+

rotating_xy_platform.png

+

Figure 1: Figure caption

+

The control inputs are the forces applied by the actuators of the translation stage (\(F_u\) and \(F_v\)). -As the translation stage is rotating around the Z axis due to the spindle, the forces are applied along \(u\) and \(v\). +As the translation stage is rotating around the Z axis due to the spindle, the forces are applied along \(\vec{i}_u\) and \(\vec{i}_v\).

@@ -155,11 +207,11 @@ 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. +Based on the Figure 1.

\begin{equation} @@ -174,8 +226,8 @@ Based on the Figure 1.
-
-

1.3 Numerical Values

+
+

1.3 Numerical Values

k = 2;
@@ -189,8 +241,8 @@ w0 = sqrt(k/m);
-
-

1.4 Campbell Diagram

+
+

1.4 Campbell Diagram

Compute the poles @@ -211,8 +263,8 @@ end

-
-

1.5 Simscape Model

+
+

1.5 Simscape Model

open('rotating_frame.slx');
@@ -254,8 +306,8 @@ G.OutputName = {'du', 'dv'};
-
-

1.6 Comparison with the model

+
+

1.6 Comparison with the model

Ga = (1/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2))^2 + (2*wr*s/(w0^2))^2) * ...
@@ -272,13 +324,17 @@ G.OutputName = {'du', 'dv'};
-
-

2 Integral Force Feedback

+
+

2 Integral Force Feedback

-
-

2.1 Numerical Values

+
+

2.1 IFF with pure integrator

+
+
+

2.1.1 Numerical Values

+
k = 1;
 m = 1;
@@ -291,9 +347,9 @@ wr = 0.1*w0;
-
-

2.2 Equations

-
+
+

2.1.2 Equations

+

The sensed forces are equal to:

@@ -329,9 +385,9 @@ The sensed forces are equal to:
-
-

2.3 Poles and Zeros

-
+
+

2.1.3 Poles and Zeros

+
syms wr w0 xi positive
 assumealso(w0 > wr)
@@ -378,9 +434,9 @@ Zeros (without damping)
-
-

2.4 Simscape Model

-
+
+

2.1.4 Simscape Model

+
%% Name of the Simulink File
 mdl = 'rotating_frame';
@@ -410,12 +466,12 @@ Giff.OutputName = {'Fmu', 'Fmv'};
-
-

2.5 IFF Plant

+
+

2.1.5 IFF Plant

-
-

2.6 Loop Gain

-
+
+

2.1.6 Loop Gain

+

Let’s take \(\Omega = \frac{\omega_0}{10}\).

@@ -430,16 +486,17 @@ G_iff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (ws^2)/(w0^2))^2 + (2*ws*s/(w0^2))^2)
-
-

2.7 Root Locus

+
+

2.1.7 Root Locus

-
-

2.8 Modified IFF

-
-
-

2.8.1 Control Law

-
+
+

2.2 Modified IFF (pseudo integrator)

+
+
+
+

2.2.1 Control Law

+

Let’s take the integral feedback controller as a low pass filter (pseudo integrator):

@@ -472,15 +529,15 @@ K_iff = (g/(1+s/wi))*eye(2);
-
-

2.8.2 Loop Gain

+
+

2.2.2 Loop Gain

-
-

2.8.3 Root Locus

+
+

2.2.3 Root Locus

-
-

2.8.4 Optimal Gain

-
+
+

2.2.4 Optimal Gain

+

The DC gain for Giff is (for \(\Omega < \omega_0\)):

@@ -511,14 +568,248 @@ G_cl = feedback(G_iff, K_opt);
+ +
+

2.3 Stiffness in parallel with the force sensor

+
+
+
+

2.3.1 Schematic

-
-

3 Direct Velocity Feedback

+
+

2.3.2 Equations

+
+

+The equations should be the same as before by taking \(k = k^\prime + k_a\). +I just have to determine the measured force by the sensor +

+
+
+ +
+

2.3.3 Parameters

+
+
+
k = 1;
+m = 1;
+c = 0.05;
+xi = c/(2*sqrt(k*m));
+
+w0 = sqrt(k/m);
+wr = 0.1;
+
+
+
+
+ +
+

2.3.4 IFF Plant

+
+
+
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, '/Fu'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Fv'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Tuv'], 5, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Tuv'], 6, 'openoutput'); io_i = io_i + 1;
+
+
+ +
+
kp = 0;
+cp = 0;
+
+
+ +
+
Giff = linearize(mdl, io, 0);
+
+%% Input/Output definition
+Giff.InputName  = {'Fu', 'Fv'};
+Giff.OutputName = {'Fmu', 'Fmv'};
+
+
+ +
+
kp = 0.5*m*wr^2;
+cp = 0.01;
+
+
+ +
+
Giffa = linearize(mdl, io, 0);
+
+%% Input/Output definition
+Giffa.InputName  = {'Fu', 'Fv'};
+Giffa.OutputName = {'Fmu', 'Fmv'};
+
+
+ +
+
kp = 1.5*m*wr^2;
+cp = 0.01;
+
+
+ +
+
Giffb = linearize(mdl, io, 0);
+
+%% Input/Output definition
+Giffb.InputName  = {'Fu', 'Fv'};
+Giffb.OutputName = {'Fmu', 'Fmv'};
+
+
+ +

+Comparison with the model +

+
+
figure; bode(Giff, 'k-', Giffa, 'b--', Giffb, 'r--')
+
+
+
+
+ +
+

2.3.5 Parallel Stiffness effect

+
+

+Pure IFF controller can be used if: +

+\begin{equation} + k_{p} > m \Omega^2 +\end{equation} + +

+However, having large values of \(k_p\) may: +

+
    +
  • decrease the actuator stroke
    • +
  • decrease the attainable damping (section about optimal value)
    • +
+
+
+ +
+

2.3.6 Root locus

+
+
+

2.3.7 Optimal value of \(k_p\)

+
+
+
kps = [0, 0.5, 1, 2, 10]*m*wr^2;
+cp = 0.01;
+
+
+
+
+ +
+

2.3.8 Physical Explanation

+
+

+Negative stiffness +Zeros are for high loop gain: remove the force sensor +Thus the stiffness in parallel should be higher than the virtual negative stiffness added by the gyroscopic effects +

+
+
+
+ +
+

2.4 Comparison

+
+

+Fix the parameters, and see how the root locus change and the final damped system +

+
+
k = 1;
+m = 1;
+c = 0.05;
+
+xi = c/(2*sqrt(k*m));
+
+w0 = sqrt(k/m);
+wr = 0.1;
+
+
+ +
+
kp = 0;
+cp = 0;
+
+
+ +
+
%% Name of the Simulink File
+mdl = 'rotating_frame';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Fu'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Fv'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Tuv'], 5, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Tuv'], 6, 'openoutput'); io_i = io_i + 1;
+
+
+ +
+
Giff = linearize(mdl, io, 0);
+
+%% Input/Output definition
+Giff.InputName  = {'Fu', 'Fv'};
+Giff.OutputName = {'Fmu', 'Fmv'};
+
+
+ +

+Modified IFF +

+
+
wi = w0;
+
+
+ +

+With parallel Stiffness +

+
+
kp = 2*m*wr^2;
+cp = 0.01;
+
+
+ +
+
Giff_kp = linearize(mdl, io, 0);
+
+%% Input/Output definition
+Giff_kp.InputName  = {'Fu', 'Fv'};
+Giff_kp.OutputName = {'Fmu', 'Fmv'};
+
+
+ +

+With parallel stiffness => unconditional stability and more damping but have to add mechanical parts +

+
+
+
+ +
+

3 Direct Velocity Feedback

-
-

3.1 Equations

+
+

3.1 Equations

The sensed relative velocity are equal to: @@ -539,8 +830,8 @@ The sensed relative velocity are equal to:

-
-

3.2 Numerical Values

+
+

3.2 Numerical Values

k = 1;
@@ -561,8 +852,8 @@ w0 = sqrt(k/m);
-
-

3.3 Simscape Model

+
+

3.3 Simscape Model

%% Name of the Simulink File
@@ -588,8 +879,8 @@ G.OutputName = {'Vu', 'Vv'};
-
-

3.4 DVF Plant

+
+

3.4 DVF Plant

figure; bode(G, Ga)
@@ -598,8 +889,8 @@ G.OutputName = {'Vu', 'Vv'};
-
-

3.5 Loop Gain

+
+

3.5 Loop Gain

Let’s take \(\Omega = \frac{\omega_0}{10}\). @@ -615,14 +906,535 @@ G_dvf = (s/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2))^2 + (2*wr*s/(w0^2)

-
-

3.6 Root Locus

+
+

3.6 Root Locus

+
+
+
+

4 Comparison

+
+
+
+

4.1 Parameters

+
+
+
k = 1;
+m = 1;
+c = 0.05;
+
+xi = c/(2*sqrt(k*m));
+
+w0 = sqrt(k/m);
+wr = 0.1;
+
+
+
+
+ +
+

4.2 Root Locus

+
+
+
+

4.2.1 Pseudo Integrator IFF

+
+
+
kp = 0;
+cp = 0;
+
+
+ +
+
%% 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'], 2, 'openoutput'); io_i = io_i + 1;
+
+
+ +
+
Kiff = tf(zeros(2));
+Kdvf = tf(zeros(2));
+
+
+ +
+
Giff = linearize(mdl, io, 0);
+
+%% Input/Output definition
+Giff.InputName  = {'Fu', 'Fv'};
+Giff.OutputName = {'Fmu', 'Fmv'};
+
+
+
+
+ +
+

4.2.2 IFF With parallel Stiffness

+
+
+
kp = 2*m*wr^2;
+cp = 0.01;
+
+
+ +
+
%% 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'], 2, 'openoutput'); io_i = io_i + 1;
+
+
+ +
+
Giff_kp = linearize(mdl, io, 0);
+
+%% Input/Output definition
+Giff_kp.InputName  = {'Fu', 'Fv'};
+Giff_kp.OutputName = {'Fmu', 'Fmv'};
+
+
+
+
+ +
+

4.2.3 DVF

+
+
+
%% 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'};
+
+
+
+
+ +
+

4.2.4 Root Locus

+
+
+
wi = w0;
+
+
+
+
+
+ +
+

4.3 Controllers - Optimal Gains

+
+

+Estimate the controller gain that yields good damping in all cases. +

+ +

+Pseudo integrator IFF: +Parallel Stiffness +DVF: +

+
+
opt_zeta_iff, opt_zeta_kp, opt_zeta_dvf
+
+
+
+
+ +
+

4.4 Transmissibility

+
+
+
+

4.4.1 Open Loop

+
+
+
wi = w0;
+
+
+ +
+
Kdvf = tf(zeros(2));
+Kiff = tf(zeros(2));
+
+
+ +
+
kp = 0;
+cp = 0;
+
+
+ +
+
%% 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'};
+
+
+
+
+ +
+

4.4.2 Pseudo Integrator IFF

+
+
+
wi = w0;
+
+
+ +
+
Kdvf = tf(zeros(2));
+Kiff = opt_gain_iff/(1 + s/wi)*tf(eye(2));
+
+
+ +
+
kp = 0;
+cp = 0;
+
+
+ +
+
%% 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 = linearize(mdl, io, 0);
+
+%% Input/Output definition
+Tiff.InputName  = {'Dwx', 'Dwy'};
+Tiff.OutputName = {'Dx', 'Dy'};
+
+
+
+
+ +
+

4.4.3 IFF With parallel Stiffness

+
+
+
kp = 2*m*wr^2;
+cp = 0.01;
+
+
+ +
+
Kdvf = tf(zeros(2));
+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'};
+
+
+
+
+ +
+

4.4.4 DVF

+
+
+
kp = 0;
+cp = 0;
+
+
+ +
+
Kdvf = opt_gain_kp*tf(eye(2));
+Kiff = tf(zeros(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'};
+
+
+
+
+ +
+

4.4.5 Transmissibility

+
+
+
freqs = logspace(-2, 1, 1000);
+
+figure;
+hold on;
+plot(freqs, abs(squeeze(freqresp(Tiff(1,1), freqs))), ...
+     'DisplayName', 'IFF Pseudo int')
+plot(freqs, abs(squeeze(freqresp(Tiff_kp(1,1), freqs))), ...
+     'DisplayName', 'IFF Paral. stiff')
+plot(freqs, abs(squeeze(freqresp(Tdvf(1,1), freqs))), ...
+     'DisplayName', 'DVF')
+plot(freqs, abs(squeeze(freqresp(Tol(1,1), freqs))), 'k-', ...
+     'DisplayName', 'IFF Pseudo int')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Frequency [rad/s]'); ylabel('Amplitude [m/m]');
+legend('location', 'northwest');
+
+
+
+
+
+ +
+

4.5 Compliance

+
+
+
+

4.5.1 Open Loop

+
+
+
Kdvf = tf(zeros(2));
+Kiff = tf(zeros(2));
+
+
+ +
+
kp = 0;
+cp = 0;
+
+
+ +
+
%% 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);
+
+%% Input/Output definition
+Col.InputName  = {'Fdx', 'Fdy'};
+Col.OutputName = {'Dx', 'Dy'};
+
+
+
+
+ +
+

4.5.2 Pseudo Integrator IFF

+
+
+
wi = w0;
+
+
+ +
+
Kdvf = tf(zeros(2));
+Kiff = opt_gain_iff/(1 + s/wi)*tf(eye(2));
+
+
+ +
+
kp = 0;
+cp = 0;
+
+
+ +
+
%% 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;
+
+
+ +
+
Ciff = linearize(mdl, io, 0);
+
+%% Input/Output definition
+Ciff.InputName  = {'Fdx', 'Fdy'};
+Ciff.OutputName = {'Dx', 'Dy'};
+
+
+
+
+ +
+

4.5.3 IFF With parallel Stiffness

+
+
+
kp = 2*m*wr^2;
+cp = 0.01;
+
+
+ +
+
Kdvf = tf(zeros(2));
+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, '/fd'], 1, 'input');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Meas'], 1, 'output');  io_i = io_i + 1;
+
+
+ +
+
Ciff_kp = linearize(mdl, io, 0);
+
+%% Input/Output definition
+Ciff_kp.InputName  = {'Fdx', 'Fdy'};
+Ciff_kp.OutputName = {'Dx', 'Dy'};
+
+
+
+
+ +
+

4.5.4 DVF

+
+
+
kp = 0;
+cp = 0;
+
+
+ +
+
Kdvf = opt_gain_kp*tf(eye(2));
+Kiff = tf(zeros(2));
+
+
+ +
+
%% 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;
+
+
+ +
+
Cdvf = linearize(mdl, io, 0);
+
+%% Input/Output definition
+Cdvf.InputName  = {'Fdx', 'Fdy'};
+Cdvf.OutputName = {'Dx', 'Dy'};
+
+
+
+
+ +
+

4.5.5 Compliance

+
+
+
freqs = logspace(-2, 1, 1000);
+
+figure;
+hold on;
+plot(freqs, abs(squeeze(freqresp(Ciff(1,1), freqs))), ...
+     'DisplayName', 'IFF Pseudo int')
+plot(freqs, abs(squeeze(freqresp(Ciff_kp(1,1), freqs))), ...
+     'DisplayName', 'IFF Paral. stiff')
+plot(freqs, abs(squeeze(freqresp(Cdvf(1,1), freqs))), ...
+     'DisplayName', 'DVF')
+plot(freqs, abs(squeeze(freqresp(Col(1,1), freqs))), 'k-', ...
+     'DisplayName', 'IFF Pseudo int')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Frequency [rad/s]'); ylabel('Compliance [m/N]');
+legend('location', 'northwest');
+% ylim([0, 1e3]);
+% xlim([freqs(1), freqs(end)]);
+
+
+
+

Author: Thomas Dehaeze

-

Created: 2020-06-10 mer. 10:17

+

Created: 2020-06-10 mer. 17:21

diff --git a/matlab/index.org b/matlab/index.org index c8ea5c8..173b9aa 100644 --- a/matlab/index.org +++ b/matlab/index.org @@ -25,19 +25,37 @@ :END: * Introduction :ignore: -Notations: -- $k$: Actuator's Stiffness [N/m] -- $c$: Actuator's Damping [N/(m/s)] -- $m$: Payload's mass [kg] -- $\omega_0 = \sqrt{\frac{k}{m}}$: Resonance of the (non-rotating) mass-spring system [rad/s] -- $\Omega = \dot{\theta}$: rotation speed [rad/s] -| | Mathematical Notation | Matlab | Unit | -|--------------------+-----------------------+--------+---------| -| Actuator Stiffness | $k$ | =k= | N/m | -| Actuator Damping | $c$ | =c= | N/(m/s) | +| | Mathematical Notation | Matlab | Unit | +|-----------------------------------+------------------------------+---------------+---------| +| Actuator Stiffness | $k$ | =k= | N/m | +| Actuator Damping | $c$ | =c= | N/(m/s) | +| Payload Mass | $m$ | =m= | kg | +| Damping Ratio | $\xi = \frac{c}{2\sqrt{km}}$ | =xi= | | +| Actuator Force | $\bm{F}, F_u, F_v$ | =F= =Fu= =Fv= | N | +| Force Sensor signal | $\bm{f}, f_u, f_v$ | =f= =fu= =fv= | N | +| Relative Displacement | $\bm{d}, d_u, d_v$ | =d= =du= =dv= | m | +| Relative Velocity | $\bm{v}, v_u, v_v$ | =v= =vu= =vv= | m/s | +| Resonance freq. when $\Omega = 0$ | $\omega_0$ | =w0= | rad/s | +| Rotation Speed | $\Omega = \dot{\theta}$ | =W= | rad/s | -* Matlab Init :noexport:ignore: +| | Mathematical Notation | Matlab | Unit | +|------------------+-----------------------+--------+---------| +| Laplace variable | $s$ | =s= | | +| Complex number | $j$ | =j= | | +| Frequency | $\omega$ | =w= | [rad/s] | + +| | Mathematical Notation | Matlab | Unit | +|----------------+------------------------------------------------+--------+---------| +| IFF Plant | $\bm{G}_\text{IFF}(s) = \frac{\bm{f}}{\bm{F}}$ | =Giff= | N/N | +| DVF Plant | $\bm{G}_\text{DVF}(s) = \frac{\bm{v}}{\bm{F}}$ | =Gdvf= | (m/s)/N | +| IFF Controller | $\bm{K}_\text{IFF}(s)$ | =Kiff= | | +| DVF Controller | $\bm{K}_\text{DVF}(s)$ | =Kdvf= | | + +* System Description and Analysis +<> +** 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 @@ -50,19 +68,19 @@ Notations: addpath('./matlab/'); #+end_src -* System Description and Analysis -<> +#+begin_src matlab + open('rotating_frame.slx'); +#+end_src ** System description The system consists of one 2 degree of freedom translation stage on top of a spindle (figure [[fig:rotating_xy_platform]]). #+name: fig:rotating_xy_platform #+caption: Figure caption -#+attr_latex: :scale 1 -[[file:figs/rotating_xy_platform.pdf]] +[[file:figs-tikz/rotating_xy_platform.png]] The control inputs are the forces applied by the actuators of the translation stage ($F_u$ and $F_v$). -As the translation stage is rotating around the Z axis due to the spindle, the forces are applied along $u$ and $v$. +As the translation stage is rotating around the Z axis due to the spindle, the forces are applied along $\vec{i}_u$ and $\vec{i}_v$. The measurement is either the $x-y$ displacement of the object located on top of the translation stage or the $u-v$ displacement of the sample with respect to a fixed reference frame. @@ -79,66 +97,74 @@ Based on the Figure [[fig:rotating_xy_platform]]. \begin{bmatrix} F_u \\ F_v \end{bmatrix} \end{equation} -** Numerical Values -#+begin_src matlab - k = 2; - m = 1; - c = 0.05; - xi = c/(2*sqrt(k*m)); +Explain Coriolis and Centrifugal Forces (negative Stiffness) - w0 = sqrt(k/m); +** Numerical Values +Let's define initial values for the model. +#+begin_src matlab + k = 1; % [N/m] + m = 1; % [kg] + c = 0.05; % [N/(m/s)] + + xi = c/(2*sqrt(k*m)); + w0 = sqrt(k/m); % [rad/s] +#+end_src + +We set the additional stiffness and damping to zero (this will be used later). +#+begin_src matlab + kp = 0; % [N/m] + cp = 0; % [N/(m/s)] #+end_src ** Campbell Diagram -Compute the poles -#+begin_src matlab :exports code - wrs = linspace(0, 2, 51); % [rad/s] +The system becomes unstable for $\Omega > \omega_0$. - polesvc = zeros(4, length(wrs)); +#+begin_src matlab + Ws = linspace(0, 2, 51); % Vector of considered rotation speed [rad/s] - for i = 1:length(wrs) - wr = wrs(i); - polei = pole(1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wrs(i)^2)/(w0^2))^2 + (2*wrs(i)*s/(w0^2))^2)); + polesvc = zeros(4, length(Ws)); + + for i = 1:length(Ws) + W = Ws(i); + + polei = pole(1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2)); [~, i_sort] = sort(imag(polei)); polesvc(:, i) = polei(i_sort); end #+end_src -#+begin_src matlab :exports none +#+begin_src matlab figure; ax1 = subplot(1,2,1); hold on; for i = 1:size(polesvc, 1) - plot(wrs, real(polesvc(i, :)), 'k-') + plot(Ws, real(polesvc(i, :)), 'k-') end - plot(wrs, zeros(size(wrs)), 'k--') + plot(Ws, zeros(size(Ws)), 'k--') hold off; xlabel('Rotation Frequency [rad/s]'); ylabel('Pole Real Part'); ax2 = subplot(1,2,2); hold on; for i = 1:size(polesvc, 1) - plot(wrs, imag(polesvc(i, :)), 'k-') - plot(wrs, -imag(polesvc(i, :)), 'k-') + plot(Ws, imag(polesvc(i, :)), 'k-') + plot(Ws, -imag(polesvc(i, :)), 'k-') end hold off; xlabel('Rotation Frequency [rad/s]'); ylabel('Pole Imaginary Part'); #+end_src ** Simscape Model +Define the rotating speed for the Simscape Model. #+begin_src matlab - open('rotating_frame.slx'); + W = 0.1; % Rotation Speed [rad/s] #+end_src +No controller for now. #+begin_src matlab - k = 2; - m = 1; - c = 0.05; - xi = c/(2*sqrt(k*m)); - - w0 = sqrt(k/m); - wr = 0.1; + Kiff = tf(zeros(2)); + Kdvf = tf(zeros(2)); #+end_src #+begin_src matlab @@ -147,10 +173,8 @@ Compute the poles %% Input/Output definition clear io; io_i = 1; - io(io_i) = linio([mdl, '/Fu'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Fv'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Tuv'], 1, 'openoutput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Tuv'], 2, 'openoutput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/K'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/G'], 3, 'openoutput'); io_i = io_i + 1; #+end_src #+begin_src matlab @@ -163,31 +187,104 @@ Compute the poles ** Comparison with the model #+begin_src matlab - Ga = (1/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2))^2 + (2*wr*s/(w0^2))^2) * ... - [(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2), 2*wr*s/(w0^2) ; ... - -2*wr*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2)]; + G_th = (1/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)]; #+end_src -#+begin_src matlab - figure; bode(G, 'k-', Ga, 'r--') +#+begin_src matlab :exports none + freqs = logspace(-1, 1, 1000); + + figure; + ax1 = subplot(2, 2, 1); + hold on; + plot(freqs, abs(squeeze(freqresp(G(1,1), freqs))), '-') + plot(freqs, abs(squeeze(freqresp(G_th(1,1), freqs))), '--') + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/N]'); + + ax3 = subplot(2, 2, 3); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G(1,1), freqs))), '-') + plot(freqs, 180/pi*angle(squeeze(freqresp(G_th(1,1), freqs))), '--') + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); + yticks(-180:90:180); + ylim([-180 180]); + hold off; + + ax2 = subplot(2, 2, 2); + hold on; + plot(freqs, abs(squeeze(freqresp(G(1,2), freqs))), '-') + plot(freqs, abs(squeeze(freqresp(G_th(1,2), freqs))), '--') + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/N]'); + + ax4 = subplot(2, 2, 4); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G(1,2), freqs))), '-') + plot(freqs, 180/pi*angle(squeeze(freqresp(G_th(1,2), freqs))), '--') + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); + yticks(-180:90:180); + ylim([-180 180]); + hold off; + + linkaxes([ax1,ax2,ax3,ax4],'x'); + xlim([freqs(1), freqs(end)]); + + linkaxes([ax1,ax2],'y'); #+end_src * Integral Force Feedback -** IFF with pure integrator -*** Numerical Values +** Introduction :ignore: + +- Diagram with the controller +- Basic idea of IFF + +** 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 - k = 1; - m = 1; - xi = 0.01; - c = 2*xi*sqrt(k*m); - w0 = sqrt(k/m); - wr = 0.1*w0; + addpath('./matlab/'); +#+end_src + +#+begin_src matlab + open('rotating_frame.slx'); +#+end_src + +** Plant Parameters +Let's define initial values for the model. +#+begin_src matlab + k = 1; % [N/m] + m = 1; % [kg] + c = 0.05; % [N/(m/s)] + + xi = c/(2*sqrt(k*m)); + w0 = sqrt(k/m); % [rad/s] +#+end_src + +** IFF with pure integrator +<> +*** No parallel stiffness +We set the additional stiffness and damping to zero (this will be used later). +#+begin_src matlab + kp = 0; % [N/m] + cp = 0; % [N/(m/s)] #+end_src *** Equations The sensed forces are equal to: \begin{equation} -\begin{bmatrix} F_{um} \\ F_{vm} \end{bmatrix} = +\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 @@ -196,9 +293,10 @@ The sensed forces are equal to: \begin{bmatrix} d_u \\ d_v \end{bmatrix} \end{equation} +Which then gives: #+begin_important \begin{equation} -\begin{bmatrix} F_{um} \\ F_{vm} \end{bmatrix} = +\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \frac{1}{\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} \begin{bmatrix} (\frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2}) (\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2}) + (2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0})^2 & - (2 \xi \frac{s}{\omega_0} + 1) 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \\ @@ -208,22 +306,16 @@ The sensed forces are equal to: \end{equation} #+end_important +*** Poles and Zeros without damping :noexport: #+begin_src matlab - Giffa = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2))^2 + (2*wr*s/(w0^2))^2) * ... - [(s^2/w0^2 - wr^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2)) + (2*wr*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*wr*s/(w0^2) ; ... - (2*xi*s/w0 + 1)*2*wr*s/(w0^2), (s^2/w0^2 - wr^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2))+ (2*wr*s/(w0^2))^2]; -#+end_src - -*** Poles and Zeros -#+begin_src matlab - syms wr w0 xi positive - assumealso(w0 > wr) + syms W w0 xi positive + assumealso(w0 > W) syms x #+end_src #+begin_src matlab - z = (x^2/w0^2 - wr^2/w0^2)*((x^2)/(w0^2) + 1 - (wr^2)/(w0^2)) + (2*wr*x/(w0^2))^2 == 0 - p = ((x^2)/(w0^2) + 1 - (wr^2)/(w0^2))^2 + (2*wr*x/(w0^2))^2 == 0 + z = (x^2/w0^2 - W^2/w0^2)*((x^2)/(w0^2) + 1 - (W^2)/(w0^2)) + (2*W*x/(w0^2))^2 == 0 + p = ((x^2)/(w0^2) + 1 - (W^2)/(w0^2))^2 + (2*W*x/(w0^2))^2 == 0 #+end_src #+begin_src matlab @@ -241,25 +333,34 @@ The zeros are the roots of: Poles (without damping) \begin{equation} - \left(\begin{array}{c} -w_{0}\,1{}\mathrm{i}-\mathrm{wr}\,1{}\mathrm{i}\\ -w_{0}\,1{}\mathrm{i}+\mathrm{wr}\,1{}\mathrm{i}\\ w_{0}\,1{}\mathrm{i}-\mathrm{wr}\,1{}\mathrm{i}\\ w_{0}\,1{}\mathrm{i}+\mathrm{wr}\,1{}\mathrm{i} \end{array}\right) + \left(\begin{array}{c} -w_{0}\,1{}\mathrm{i}-\mathrm{W}\,1{}\mathrm{i}\\ -w_{0}\,1{}\mathrm{i}+\mathrm{W}\,1{}\mathrm{i}\\ w_{0}\,1{}\mathrm{i}-\mathrm{W}\,1{}\mathrm{i}\\ w_{0}\,1{}\mathrm{i}+\mathrm{W}\,1{}\mathrm{i} \end{array}\right) \end{equation} Zeros (without damping) \begin{equation} - \left(\begin{array}{c} -\sqrt{-\frac{w_{0}\,\sqrt{{w_{0}}^2+8\,{\mathrm{wr}}^2}}{2}-\frac{{w_{0}}^2}{2}-{\mathrm{wr}}^2}\\ -\sqrt{\frac{w_{0}\,\sqrt{{w_{0}}^2+8\,{\mathrm{wr}}^2}}{2}-\frac{{w_{0}}^2}{2}-{\mathrm{wr}}^2}\\ \sqrt{-\frac{w_{0}\,\sqrt{{w_{0}}^2+8\,{\mathrm{wr}}^2}}{2}-\frac{{w_{0}}^2}{2}-{\mathrm{wr}}^2}\\ \sqrt{\frac{w_{0}\,\sqrt{{w_{0}}^2+8\,{\mathrm{wr}}^2}}{2}-\frac{{w_{0}}^2}{2}-{\mathrm{wr}}^2} \end{array}\right) + \left(\begin{array}{c} -\sqrt{-\frac{w_{0}\,\sqrt{{w_{0}}^2+8\,{\mathrm{W}}^2}}{2}-\frac{{w_{0}}^2}{2}-{\mathrm{W}}^2}\\ -\sqrt{\frac{w_{0}\,\sqrt{{w_{0}}^2+8\,{\mathrm{W}}^2}}{2}-\frac{{w_{0}}^2}{2}-{\mathrm{W}}^2}\\ \sqrt{-\frac{w_{0}\,\sqrt{{w_{0}}^2+8\,{\mathrm{W}}^2}}{2}-\frac{{w_{0}}^2}{2}-{\mathrm{W}}^2}\\ \sqrt{\frac{w_{0}\,\sqrt{{w_{0}}^2+8\,{\mathrm{W}}^2}}{2}-\frac{{w_{0}}^2}{2}-{\mathrm{W}}^2} \end{array}\right) \end{equation} *** Simscape Model +The rotation speed is set to $\Omega = 0.1 \omega_0$. +#+begin_src matlab + W = 0.1*w0; % [rad/s] +#+end_src + +No controller for now. +#+begin_src matlab + Kiff = tf(zeros(2)); + Kdvf = tf(zeros(2)); +#+end_src + #+begin_src matlab %% Name of the Simulink File mdl = 'rotating_frame'; %% Input/Output definition clear io; io_i = 1; - io(io_i) = linio([mdl, '/Fu'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Fv'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Tuv'], 5, 'openoutput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Tuv'], 6, 'openoutput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/K'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/G'], 2, 'openoutput'); io_i = io_i + 1; #+end_src #+begin_src matlab @@ -267,78 +368,79 @@ Zeros (without damping) %% Input/Output definition Giff.InputName = {'Fu', 'Fv'}; - Giff.OutputName = {'Fmu', 'Fmv'}; + Giff.OutputName = {'fu', 'fv'}; +#+end_src + +*** Comparison with the model +#+begin_src matlab + Giff_th = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ... + [(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ... + (2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2]; #+end_src #+begin_src matlab - figure; bode(Giff, Giffa) -#+end_src - -*** IFF Plant -#+begin_src matlab :exports none - ws = [0, 0.1, 0.3, 0.8, 1.1]; - - G_iff = {zeros(2, 2, length(ws))}; - - for i = 1:length(ws) - W = ws(i); - - G_iff(:, :, i) = {1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ... - [(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ... - (2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2]}; - end -#+end_src - -#+begin_src matlab :exports none - freqs = logspace(-2, 1, 1000); - - i = 2; + freqs = logspace(-1, 1, 1000); figure; ax1 = subplot(2, 2, 1); hold on; - plot(freqs, abs(squeeze(freqresp(G_iff{i}(1,1), freqs, 'Hz')))) - plot(freqs, ones(size(freqs))*(ws(i)^2/(k/m - ws(i)^2)), 'k--') - plot(sqrt(0.5*sqrt(k/m)*sqrt(8*ws(i)^2 + k/m) + ws(i) + 0.5*k/m)/2/pi, 1, 'ko') - plot(sqrt(0.5*sqrt(k/m)*sqrt(8*ws(i)^2 + k/m) - ws(i) - 0.5*k/m)/2/pi, 1, 'ko') - plot((sqrt(k/m) + ws(i))/2/pi, 1, 'kx') - plot((sqrt(k/m) - ws(i))/2/pi, 1, 'kx') + plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs))), '-') + plot(freqs, abs(squeeze(freqresp(Giff_th(1,1), freqs))), '--') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude [N/N]'); - title(['$\frac{F_{u,m}}{F_u}$' sprintf(', $\\omega = %.2f \\omega_0 $', ws(i)/w0)]); + set(gca, 'XTickLabel',[]); ylabel('Amplitude [N/N]'); ax3 = subplot(2, 2, 3); hold on; - plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff{i}(1, 1), freqs, 'Hz')))); - hold off; + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,1), freqs))), '-') + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_th(1,1), freqs))), '--') + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); yticks(-180:90:180); ylim([-180 180]); - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); - xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; ax2 = subplot(2, 2, 2); hold on; - plot(freqs, abs(squeeze(freqresp(G_iff{i}(1,2), freqs, 'Hz')))) - plot((sqrt(k/m) + ws(i))/2/pi, 1, 'kx') - plot((sqrt(k/m) - ws(i))/2/pi, 1, 'kx') + plot(freqs, abs(squeeze(freqresp(Giff(1,2), freqs))), '-') + plot(freqs, abs(squeeze(freqresp(Giff_th(1,2), freqs))), '--') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - title(['$\frac{F_{v,m}}{F_u}$' sprintf(', $\\omega = %.2f \\omega_0 $', ws(i)/w0)]); + set(gca, 'XTickLabel',[]); ylabel('Amplitude [N/N]'); ax4 = subplot(2, 2, 4); hold on; - plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff{i}(1, 2), freqs, 'Hz')))); - hold off; + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,2), freqs))), '-') + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_th(1,2), freqs))), '--') + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); yticks(-180:90:180); ylim([-180 180]); - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); - xlabel('Frequency [Hz]'); + hold off; linkaxes([ax1,ax2,ax3,ax4],'x'); + xlim([freqs(1), freqs(end)]); + linkaxes([ax1,ax2],'y'); #+end_src +*** Influence of the rotation speed on the IFF Plant +#+begin_src matlab + Ws = [0, 0.1, 0.3, 0.8, 1.1]; % Rotating Speeds [rad/s] +#+end_src + +#+begin_src matlab + Gsiff = {zeros(2, 2, length(Ws))}; + + for i = 1:length(Ws) + W = Ws(i); + + Gsiff(:, :, i) = {1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ... + [(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ... + (2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2]}; + end +#+end_src + #+begin_src matlab :exports none freqs = logspace(-2, 1, 1000); @@ -346,9 +448,9 @@ Zeros (without damping) ax1 = subplot(2, 1, 1); hold on; - for i = 1:length(ws) - plot(freqs, abs(squeeze(freqresp(G_iff{i}(1,1), freqs))), ... - 'DisplayName', sprintf('$\\omega = %.2f \\omega_0 $', ws(i)/w0)) + for i = 1:length(Ws) + plot(freqs, abs(squeeze(freqresp(Gsiff{i}(1,1), freqs))), ... + 'DisplayName', sprintf('$\\omega = %.2f \\omega_0 $', Ws(i)/w0)) end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); @@ -358,9 +460,9 @@ Zeros (without damping) ax2 = subplot(2, 1, 2); hold on; - for i = 1:length(ws) - plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff{i}(1,1), freqs))), ... - 'DisplayName', sprintf('$\\omega = %.1f \\omega_0 $', ws(i)/w0)) + for i = 1:length(Ws) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gsiff{i}(1,1), freqs))), ... + 'DisplayName', sprintf('$\\omega = %.1f \\omega_0 $', Ws(i)/w0)) end set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); @@ -375,38 +477,42 @@ Zeros (without damping) *** Loop Gain Let's take $\Omega = \frac{\omega_0}{10}$. #+begin_src matlab - ws = 0.1*w0; - G_iff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (ws^2)/(w0^2))^2 + (2*ws*s/(w0^2))^2) * ... - [(s^2/w0^2 - ws^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (ws^2)/(w0^2)) + (2*ws*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*ws*s/(w0^2) ; ... - (2*xi*s/w0 + 1)*2*ws*s/(w0^2), (s^2/w0^2 - ws^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (ws^2)/(w0^2))+ (2*ws*s/(w0^2))^2]; + W = 0.1*w0; + Giff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ... + [(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ... + (2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2]; #+end_src -#+begin_src matlab :exports none - g = 1; +The decentralized controller contains pure integrators: +\begin{equation} + \bm{K}_{\text{IFF}}(s) = \frac{g}{s} \begin{bmatrix} + 1 & 0 \\ + 0 & 1 + \end{bmatrix} +\end{equation} +#+begin_src matlab + g = 2; + + Kiff = g/s*tf(eye(2)); +#+end_src + +#+begin_src matlab :exports none freqs = logspace(-2, 1, 1000); figure; ax1 = subplot(2, 1, 1); hold on; - for i = 1:length(ws) - plot(freqs, abs(squeeze(freqresp(G_iff(1,1)*g/s, freqs))), ... - 'DisplayName', sprintf('$\\omega = %.2f \\omega_0 $', ws(i)/w0)) - end + plot(freqs, abs(squeeze(freqresp(Giff(1,1)*Kiff(1,1), freqs)))) hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); ylabel('Amplitude [N/N]'); - legend('location', 'northwest'); - ylim([0, 1e3]); + set(gca, 'XTickLabel',[]); ylabel('Loop Gain'); ax2 = subplot(2, 1, 2); hold on; - for i = 1:length(ws) - plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff(1,1)*g/s, freqs))), ... - 'DisplayName', sprintf('$\\omega = %.1f \\omega_0 $', ws(i)/w0)) - end + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,1)*Kiff(1,1), freqs)))) set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); yticks(-180:90:180); @@ -418,13 +524,15 @@ Let's take $\Omega = \frac{\omega_0}{10}$. #+end_src *** Root Locus -#+begin_src matlab :exports none - ws = [0, 0.1, 0.3, 0.8, 1.1]; +#+begin_src matlab + Ws = [0, 0.1, 0.3, 0.8, 1.1]; +#+end_src - G_iff = {zeros(2, 2, length(ws))}; +#+begin_src matlab + G_iff = {zeros(2, 2, length(Ws))}; - for i = 1:length(ws) - W = ws(i); + for i = 1:length(Ws) + W = Ws(i); G_iff(:, :, i) = {1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ... [(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ... @@ -438,10 +546,10 @@ Let's take $\Omega = \frac{\omega_0}{10}$. gains = logspace(-2, 4, 100); hold on; - for i = 1:length(ws) + for i = 1:length(Ws) set(gca,'ColorOrderIndex',i); plot(real(pole(G_iff{i})), imag(pole(G_iff{i})), 'x', ... - 'DisplayName', sprintf('$\\omega = %.2f \\omega_0 $', ws(i)/w0)); + 'DisplayName', sprintf('$\\omega = %.1f \\omega_0 $', Ws(i)/w0)); set(gca,'ColorOrderIndex',i); plot(real(tzero(G_iff{i})), imag(tzero(G_iff{i})), 'o', ... 'HandleVisibility', 'off'); @@ -461,57 +569,58 @@ Let's take $\Omega = \frac{\omega_0}{10}$. #+end_src ** Modified IFF (pseudo integrator) +<> +*** No parallel stiffness +We set the additional stiffness and damping to zero (this will be used later). +#+begin_src matlab + kp = 0; % [N/m] + cp = 0; % [N/(m/s)] +#+end_src + +*** Plant parameters +#+begin_src matlab + W = 0.1*w0; + + Giff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ... + [(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ... + (2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2]; + +#+end_src + *** Control Law Let's take the integral feedback controller as a low pass filter (pseudo integrator): \begin{equation} - K_{\text{IFF}} = \begin{bmatrix} - g\frac{\omega_i}{\omega_i + s} & 0 \\ - 0 & g\frac{\omega_i}{\omega_i + s} + K_{\text{IFF}}(s) = g\frac{1}{1 + \frac{s}{\omega_i}} \begin{bmatrix} + 1 & 0 \\ + 0 & 1 \end{bmatrix} \end{equation} #+begin_src matlab - xi = 0.005; - w0 = 1; - ws = 0.1*w0; - - G_iff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (ws^2)/(w0^2))^2 + (2*ws*s/(w0^2))^2) * ... - [(s^2/w0^2 - ws^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (ws^2)/(w0^2)) + (2*ws*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*ws*s/(w0^2) ; ... - (2*xi*s/w0 + 1)*2*ws*s/(w0^2), (s^2/w0^2 - ws^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (ws^2)/(w0^2))+ (2*ws*s/(w0^2))^2]; - + g = 2; + wi = w0; #+end_src #+begin_src matlab - g = 100; - wi = ws; - - K_iff = (g/(1+s/wi))*eye(2); + Kiff = (g/(1+s/wi))*eye(2); #+end_src *** Loop Gain -#+begin_src matlab :exports none +#+begin_src matlab freqs = logspace(-2, 1, 1000); figure; ax1 = subplot(2, 1, 1); hold on; - for i = 1:length(ws) - plot(freqs, abs(squeeze(freqresp(G_iff(1,1)*K_iff(1,1), freqs))), ... - 'DisplayName', sprintf('$\\omega = %.2f \\omega_0 $', ws(i)/w0)) - end + plot(freqs, abs(squeeze(freqresp(Giff(1,1)*Kiff(1,1), freqs)))) hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Amplitude [N/N]'); - legend('location', 'northwest'); - ylim([0, 1e3]); ax2 = subplot(2, 1, 2); hold on; - for i = 1:length(ws) - plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff(1,1)*K_iff(1,1), freqs))), ... - 'DisplayName', sprintf('$\\omega = %.1f \\omega_0 $', ws(i)/w0)) - end + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,1)*Kiff(1,1), freqs)))) set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); yticks(-180:90:180); @@ -523,32 +632,30 @@ Let's take the integral feedback controller as a low pass filter (pseudo integra #+end_src *** Root Locus +As shown in the Root Locus plot, for some value of the gain, the system is stable. +(The system is however still not conditionally stable). + #+begin_src matlab :exports none figure; gains = logspace(-2, 4, 100); hold on; - plot(real(pole(G_iff)), imag(pole(G_iff)), 'kx', ... - 'DisplayName', sprintf('$\\Omega = %.2f \\omega_0 $', ws/w0)); - plot(real(tzero(G_iff)), imag(tzero(G_iff)), 'ko', ... - 'HandleVisibility', 'off'); - for g_i = 1:length(gains) - K_iff = (gains(g_i)/(1+s/wi))*eye(2); - cl_poles = pole(feedback(G_iff, K_iff)); - plot(real(cl_poles), imag(cl_poles), 'k.', ... - 'HandleVisibility', 'off'); + plot(real(pole(Giff)), imag(pole(Giff)), 'kx'); + plot(real(tzero(Giff)), imag(tzero(Giff)), 'ko'); + for gi = 1:length(gains) + clpoles = pole(feedback(Giff, (gains(gi)/(1+s/wi))*eye(2))); + plot(real(clpoles), imag(clpoles), 'k.'); end hold off; axis square; - xlim([-2, 0.5]); ylim([0, 2.5]); + xlim([-2, 0.5]); ylim([-1.25, 1.25]); xlabel('Real Part'); ylabel('Imaginary Part'); - legend('location', 'northwest'); #+end_src -*** Optimal Gain -The DC gain for Giff is (for $\Omega < \omega_0$): +*** How does $\omega_i$ influences the attainable damping? Optimal Gain +The DC gain for $G_\text{IFF}$ is (for $\Omega < \omega_0$): \begin{equation} G_{\text{IFF}}(\omega = 0) = \frac{1}{1 - \frac{{\omega_0}^2}{\Omega^2}} \begin{bmatrix} 1 & 0 \\ @@ -562,28 +669,104 @@ The maximum gain where is system is still stable is \end{equation} -Let's find the gain that maximize the simultaneous damping of the two modes. -#+begin_src matlab :exports none +Root Locus => Small $\omega_i$ seems to allow more damping but may limit the gain. +#+begin_src matlab + wis = [0.01, 0.1, 0.5, 1]*w0; +#+end_src + +#+begin_src matlab + figure; + gains = logspace(-2, 4, 100); - opt_zeta = 0; - opt_gain = 0; - for g_i = 1:length(gains) - K_iff = (gains(g_i)/(1+s/wi))*eye(2); + hold on; + plot(real(pole(Giff)), imag(pole(Giff)), 'kx'); + plot(real(tzero(Giff)), imag(tzero(Giff)), 'ko'); + for wi_i = 1:length(wis) + set(gca,'ColorOrderIndex',wi_i); + wi = wis(wi_i); + L(wi_i) = plot(nan, nan, '.', 'DisplayName', sprintf('$\\omega_i = %.1e \\omega_0$', wi./w0)); + for gi = 1:length(gains) + clpoles = pole(feedback(Giff, (gains(gi)/(1+s/wi))*eye(2))); + set(gca,'ColorOrderIndex',wi_i); + plot(real(clpoles), imag(clpoles), '.'); + end + end + hold off; + axis square; + xlim([-2, 0.5]); ylim([-1.25, 1.25]); + legend(L, 'location', 'northwest'); + xlabel('Real Part'); ylabel('Imaginary Part'); + + clear L +#+end_src - [w, zeta] = damp(minreal(feedback(G_iff, K_iff))); +Find wi that yields most damping => there is an optimum. +Limitation of small $\omega_i$: stability/gain margin +Limitation of large $\omega_i$: no damping attainable +#+begin_src matlab + opt_zeta = zeros(1, length(wis)); + opt_gain = zeros(1, length(wis)); - if min(zeta) > opt_zeta && all(zeta > 0) - opt_zeta = min(zeta); - opt_gain = min(gains(g_i)); + for wi_i = 1:length(wis) + gains = linspace(0, w0^2/W^2 - 1, 100); + + for g_i = 1:length(gains) + Kiff = (gains(g_i)/(1+s/wis(wi_i)))*eye(2); + + [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) = min(gains(g_i)); + end end end #+end_src +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) + data2orgtable([wis; opt_zeta; opt_gain]', {}, {'wis', 'zeta', 'gain'}, ' %.3f '); +#+end_src + +#+RESULTS: +| wis | zeta | gain | +|------+-------+------| +| 0.01 | 0.437 | 98.0 | +| 0.1 | 0.829 | 20.0 | +| 0.5 | 0.376 | 3.0 | +| 1.0 | 0.204 | 2.0 | + +Root Locus that shows the maximum damping attainable. #+begin_src matlab - K_opt = (opt_gain/(1+s/wi))*eye(2); + figure; - G_cl = feedback(G_iff, K_opt); + gains = logspace(-2, 4, 100); + + hold on; + plot(real(pole(Giff)), imag(pole(Giff)), 'kx', 'HandleVisibility', 'off'); + plot(real(tzero(Giff)), imag(tzero(Giff)), 'ko', 'HandleVisibility', 'off'); + for wi_i = 1:length(wis) + wi = wis(wi_i); + for gi = 1:length(gains) + clpoles = pole(feedback(Giff, (gains(gi)/(1+s/wi))*eye(2))); + set(gca,'ColorOrderIndex',wi_i); + plot(real(clpoles), imag(clpoles), '.', 'HandleVisibility', 'off'); + end + clpoles = pole(feedback(Giff, (opt_gain(wi_i)/(1+s/wi))*eye(2))); + set(gca,'ColorOrderIndex',wi_i); + plot(real(clpoles), imag(clpoles), 'x', 'DisplayName', sprintf('$\\omega_i = %.2f \\omega_0$', wi./w0)); + end + hold off; + axis square; + xlim([-2, 0.5]); ylim([-1.25, 1.25]); + legend('location', 'northwest'); + xlabel('Real Part'); ylabel('Imaginary Part'); +#+end_src + +#+begin_src matlab :exports none + [~, opt_i] = max(opt_zeta); + Kiff_opt = (opt_gain(opt_i)/(1 + s/wis(opt_i)))*eye(2); + Giff_cl = feedback(Giff, Kiff_opt); #+end_src #+begin_src matlab :exports none @@ -593,51 +776,16 @@ Let's find the gain that maximize the simultaneous damping of the two modes. ax1 = subplot(2, 1, 1); hold on; - for i = 1:length(ws) - plot(freqs, abs(squeeze(freqresp(G_iff(1,1)*K_opt(1,1), freqs))), ... - 'DisplayName', sprintf('$\\omega = %.2f \\omega_0 $', ws(i)/w0)) - end + plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs)))) + plot(freqs, abs(squeeze(freqresp(Giff_cl(1,1), freqs)))) hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Amplitude [N/N]'); - legend('location', 'northwest'); - ylim([0, 1e3]); ax2 = subplot(2, 1, 2); hold on; - for i = 1:length(ws) - plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff(1,1)*K_opt(1,1), freqs))), ... - 'DisplayName', sprintf('$\\omega = %.1f \\omega_0 $', ws(i)/w0)) - end - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); - xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); - yticks(-180:90:180); - ylim([-180 180]); - hold off; - - linkaxes([ax1,ax2],'x'); - xlim([freqs(1), freqs(end)]); -#+end_src - -#+begin_src matlab :exports none - freqs = logspace(-2, 1, 1000); - - figure; - - ax1 = subplot(2, 1, 1); - hold on; - plot(freqs, abs(squeeze(freqresp(G_iff(1,1), freqs)))) - plot(freqs, abs(squeeze(freqresp(G_cl(1,1), freqs)))) - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); ylabel('Amplitude [N/N]'); - legend('location', 'northwest'); - ylim([0, 1e3]); - - ax2 = subplot(2, 1, 2); - hold on; - plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff(1,1), freqs)))) - plot(freqs, 180/pi*angle(squeeze(freqresp(G_cl(1,1), freqs)))) + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,1), freqs)))) + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_cl(1,1), freqs)))) set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); yticks(-180:90:180); @@ -649,82 +797,125 @@ Let's find the gain that maximize the simultaneous damping of the two modes. #+end_src ** Stiffness in parallel with the force sensor +<> *** Schematic -*** Equations +#+name: fig:figure_name +#+caption: Figure caption +[[file:figs-tikz/rotating_xy_platform_springs.png]] + +*** Physical Explanation +- Negative stiffness induced by gyroscopic effects +- Zeros of the open-loop <=> Poles of the subsystem with the force sensors removes +- As the zeros are the poles of the closed loop system for high gains, we want them to be in the left-half plane +- Thus we want the zeros to be in the left half plant and thus the system with the force sensors stable +- This can be done by adding springs in parallel with the force sensors with a stiffness larger than the virtual negative stiffness added by the gyroscopic effects + +The negative stiffness induced by the rotation is: +\begin{equation} + k_{n} = - m \Omega^2 +\end{equation} + +And thus, the stiffness in parallel should be such that: +\begin{equation} + k_{p} > m \Omega^2 +\end{equation} + +*** TODO Equations The equations should be the same as before by taking $k = k^\prime + k_a$. I just have to determine the measured force by the sensor -*** Parameters +*** Effect of the parallel stiffness on the IFF plant +Let's fix the rotating speed: #+begin_src matlab - k = 1; - m = 1; - c = 0.05; - xi = c/(2*sqrt(k*m)); - - w0 = sqrt(k/m); - wr = 0.1; + W = 0.1*w0; #+end_src -*** IFF Plant +And no controller is used. #+begin_src matlab - open('rotating_frame.slx'); + Kiff = tf(zeros(2)); + Kdvf = tf(zeros(2)); #+end_src +And the IFF plant is identified in three different cases: +- without parallel stiffness +- with a small parallel stiffness $k_p < m \Omega^2$ +- with a large parallel stiffness $k_p > m \Omega^2$ + #+begin_src matlab %% Name of the Simulink File mdl = 'rotating_frame'; %% Input/Output definition clear io; io_i = 1; - io(io_i) = linio([mdl, '/Fu'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Fv'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Tuv'], 5, 'openoutput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Tuv'], 6, 'openoutput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/K'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/G'], 2, 'openoutput'); io_i = io_i + 1; #+end_src #+begin_src matlab kp = 0; cp = 0; -#+end_src -#+begin_src matlab Giff = linearize(mdl, io, 0); %% Input/Output definition Giff.InputName = {'Fu', 'Fv'}; - Giff.OutputName = {'Fmu', 'Fmv'}; + Giff.OutputName = {'fu', 'fv'}; #+end_src #+begin_src matlab - kp = 0.5*m*wr^2; - cp = 0.01; -#+end_src + kp = 0.5*m*W^2; + cp = 0.001; -#+begin_src matlab - Giffa = linearize(mdl, io, 0); + Giff_s = linearize(mdl, io, 0); %% Input/Output definition - Giffa.InputName = {'Fu', 'Fv'}; - Giffa.OutputName = {'Fmu', 'Fmv'}; + Giff_s.InputName = {'Fu', 'Fv'}; + Giff_s.OutputName = {'fu', 'fv'}; #+end_src #+begin_src matlab - kp = 1.5*m*wr^2; - cp = 0.01; -#+end_src + kp = 1.5*m*W^2; + cp = 0.001; -#+begin_src matlab - Giffb = linearize(mdl, io, 0); + Giff_l = linearize(mdl, io, 0); %% Input/Output definition - Giffb.InputName = {'Fu', 'Fv'}; - Giffb.OutputName = {'Fmu', 'Fmv'}; + Giff_l.InputName = {'Fu', 'Fv'}; + Giff_l.OutputName = {'fu', 'fv'}; #+end_src -Comparison with the model #+begin_src matlab - figure; bode(Giff, 'k-', Giffa, 'b--', Giffb, 'r--') + freqs = logspace(-2, 1, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs))), 'k-') + plot(freqs, abs(squeeze(freqresp(Giff_s(1,1), freqs))), 'k--') + plot(freqs, abs(squeeze(freqresp(Giff_l(1,1), freqs))), 'k:') + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); ylabel('Amplitude [N/N]'); + + ax2 = subplot(2, 1, 2); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,1), freqs))), 'k-', ... + 'DisplayName', '$k_p = 0$') + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_s(1,1), freqs))), 'k--', ... + 'DisplayName', '$k_p < m\Omega^2$') + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_l(1,1), freqs))), 'k:', ... + 'DisplayName', '$k_p > m\Omega^2$') + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); + yticks(-180:90:180); + ylim([-180 180]); + hold off; + legend('location', 'southwest'); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); #+end_src *** Parallel Stiffness effect @@ -741,32 +932,50 @@ However, having large values of $k_p$ may: #+begin_src matlab :exports none figure; - gains = logspace(-2, 4, 100); + gains = logspace(-2, 2, 100); hold on; - plot(real(pole(Giffb)), imag(pole(Giffb)), 'kx', ... - 'DisplayName', sprintf('$\\Omega = %.2f \\omega_0 $', wr/w0)); - plot(real(tzero(Giffb)), imag(tzero(Giffb)), 'ko', ... + set(gca,'ColorOrderIndex',1); + plot(real(pole(Giff)), imag(pole(Giff)), 'x', ... + 'DisplayName', '$k_p < m\Omega^2$'); + set(gca,'ColorOrderIndex',1); + plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', ... 'HandleVisibility', 'off'); for g_i = 1:length(gains) - K_iff = (gains(g_i)/s)*eye(2); - cl_poles = pole(feedback(Giffb, K_iff)); - plot(real(cl_poles), imag(cl_poles), 'k.', ... + cl_poles = pole(feedback(Giff, (gains(g_i)/s)*eye(2))); + set(gca,'ColorOrderIndex',1); + plot(real(cl_poles), imag(cl_poles), '.', ... 'HandleVisibility', 'off'); end - plot(real(pole(Giffa)), imag(pole(Giffa)), 'rx', ... - 'DisplayName', sprintf('$\\Omega = %.2f \\omega_0 $', wr/w0)); - plot(real(tzero(Giffa)), imag(tzero(Giffa)), 'ro', ... + + set(gca,'ColorOrderIndex',2); + plot(real(pole(Giff_s)), imag(pole(Giff_s)), 'x', ... + 'DisplayName', '$k_p < m\Omega^2$'); + set(gca,'ColorOrderIndex',2); + plot(real(tzero(Giff_s)), imag(tzero(Giff_s)), 'o', ... 'HandleVisibility', 'off'); for g_i = 1:length(gains) - K_iffa = (gains(g_i)/s)*eye(2); - cl_poles = pole(feedback(Giffa, K_iffa)); - plot(real(cl_poles), imag(cl_poles), 'r.', ... + cl_poles = pole(feedback(Giff_s, (gains(g_i)/s)*eye(2))); + set(gca,'ColorOrderIndex',2); + plot(real(cl_poles), imag(cl_poles), '.', ... + 'HandleVisibility', 'off'); + end + + set(gca,'ColorOrderIndex',3); + plot(real(pole(Giff_l)), imag(pole(Giff_l)), 'x', ... + 'DisplayName', '$k_p > m\Omega^2$'); + set(gca,'ColorOrderIndex',3); + plot(real(tzero(Giff_l)), imag(tzero(Giff_l)), 'o', ... + 'HandleVisibility', 'off'); + for g_i = 1:length(gains) + set(gca,'ColorOrderIndex',3); + cl_poles = pole(feedback(Giff_l, (gains(g_i)/s)*eye(2))); + plot(real(cl_poles), imag(cl_poles), '.', ... 'HandleVisibility', 'off'); end hold off; axis square; - xlim([-2, 0.5]); ylim([0, 2.5]); + xlim([-1, 0.2]); ylim([0, 1.2]); xlabel('Real Part'); ylabel('Imaginary Part'); legend('location', 'northwest'); @@ -774,7 +983,7 @@ However, having large values of $k_p$ may: *** Optimal value of $k_p$ #+begin_src matlab - kps = [0, 0.5, 1, 2, 10]*m*wr^2; + kps = [0, 0.5, 1, 2, 10]*m*W^2; cp = 0.01; #+end_src @@ -790,7 +999,7 @@ However, having large values of $k_p$ may: set(gca,'ColorOrderIndex',kp_i); plot(real(pole(Giff)), imag(pole(Giff)), 'x', ... - 'DisplayName', sprintf('$k_p = %.1f m \\Omega^2$', kp/(m*wr^2))); + 'DisplayName', sprintf('$k_p = %.1f m \\Omega^2$', kp/(m*W^2))); set(gca,'ColorOrderIndex',kp_i); plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', ... 'HandleVisibility', 'off'); @@ -810,115 +1019,116 @@ However, having large values of $k_p$ may: legend('location', 'northwest'); #+end_src -*** Physical Explanation -Negative stiffness -Zeros are for high loop gain: remove the force sensor -Thus the stiffness in parallel should be higher than the virtual negative stiffness added by the gyroscopic effects +To have unconditional stability: +\begin{equation} + k_{p} > m \Omega^2 +\end{equation} -** Comparison -Fix the parameters, and see how the root locus change and the final damped system -#+begin_src matlab - k = 1; - m = 1; - c = 0.05; +But if $k_p$ is too late, the attainable damping is decreasing and this may as well limit the actuator stroke/force. - xi = c/(2*sqrt(k*m)); - - w0 = sqrt(k/m); - wr = 0.1; -#+end_src +*** Optimal Gain +Let's take $k_p = 2 m \Omega^2$ and find the optimal IFF control gain. #+begin_src matlab - kp = 0; - cp = 0; -#+end_src - -#+begin_src matlab - %% Name of the Simulink File - mdl = 'rotating_frame'; - - %% Input/Output definition - clear io; io_i = 1; - io(io_i) = linio([mdl, '/Fu'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Fv'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Tuv'], 5, 'openoutput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Tuv'], 6, 'openoutput'); io_i = io_i + 1; -#+end_src - -#+begin_src matlab - Giff = linearize(mdl, io, 0); - - %% Input/Output definition - Giff.InputName = {'Fu', 'Fv'}; - Giff.OutputName = {'Fmu', 'Fmv'}; -#+end_src - -Modified IFF -#+begin_src matlab - wi = w0; -#+end_src - -With parallel Stiffness -#+begin_src matlab - kp = 2*m*wr^2; + kp = 2*m*W; cp = 0.01; + + Giff = linearize(mdl, io, 0); #+end_src #+begin_src matlab - Giff_kp = linearize(mdl, io, 0); + opt_zeta = 0; + opt_gain = 0; - %% Input/Output definition - Giff_kp.InputName = {'Fu', 'Fv'}; - Giff_kp.OutputName = {'Fmu', 'Fmv'}; + gains = logspace(-2, 4, 100); + + for g_i = 1:length(gains) + Kiff = (gains(g_i)/s)*eye(2); + + [w, zeta] = damp(minreal(feedback(Giff, Kiff))); + + if min(zeta) > opt_zeta && all(zeta > 0) + opt_zeta = min(zeta); + opt_gain = min(gains(g_i)); + end + end #+end_src -#+begin_src matlab :exports none +#+begin_src matlab figure; gains = logspace(-2, 4, 100); hold on; - set(gca,'ColorOrderIndex',1); - plot(real(pole(Giff)), imag(pole(Giff)), 'x', ... - 'DisplayName', 'Pseudo Integrator'); - set(gca,'ColorOrderIndex',1); - plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', ... - 'HandleVisibility', 'off'); - for g_i = 1:length(gains) - K_iff = (gains(g_i)/(1 + s/wi))*eye(2); - cl_poles = pole(feedback(Giff, K_iff)); - set(gca,'ColorOrderIndex',1); - plot(real(cl_poles), imag(cl_poles), '.', ... - 'HandleVisibility', 'off'); - end - - set(gca,'ColorOrderIndex',2); - plot(real(pole(Giff_kp)), imag(pole(Giff_kp)), 'x', ... - 'DisplayName', 'Parallel Stiffness'); - set(gca,'ColorOrderIndex',2); - plot(real(tzero(Giff_kp)), imag(tzero(Giff_kp)), 'o', ... - 'HandleVisibility', 'off'); - for g_i = 1:length(gains) - K_iffa = (gains(g_i)/s)*eye(2); - cl_poles = pole(feedback(Giff_kp, K_iffa)); - set(gca,'ColorOrderIndex',2); - plot(real(cl_poles), imag(cl_poles), '.', ... - 'HandleVisibility', 'off'); + plot(real(pole(Giff)), imag(pole(Giff)), 'kx', 'HandleVisibility', 'off'); + plot(real(tzero(Giff)), imag(tzero(Giff)), 'ko', 'HandleVisibility', 'off'); + for gi = 1:length(gains) + clpoles = pole(feedback(Giff, (gains(gi)/s)*eye(2))); + plot(real(clpoles), imag(clpoles), 'k.', 'HandleVisibility', 'off'); end + % Optimal Gain + clpoles = pole(feedback(Giff, (opt_gain/s)*eye(2))); + set(gca,'ColorOrderIndex',1); + plot(real(clpoles), imag(clpoles), 'x', 'HandleVisibility', 'off'); hold off; axis square; - xlim([-2, 0.5]); ylim([0, 2.5]); - + xlim([-1.2, 0.05]); ylim([0, 1.25]); xlabel('Real Part'); ylabel('Imaginary Part'); - legend('location', 'northwest'); #+end_src -With parallel stiffness => unconditional stability and more damping but have to add mechanical parts +#+begin_src matlab :exports none + Kiff_opt = (opt_gain(opt_i)/s)*eye(2); + Giff_cl = feedback(Giff, Kiff_opt); +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(-2, 1, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs)))) + plot(freqs, abs(squeeze(freqresp(Giff_cl(1,1), freqs)))) + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); ylabel('Amplitude [N/N]'); + + ax2 = subplot(2, 1, 2); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,1), freqs)))) + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_cl(1,1), freqs)))) + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); + yticks(-180:90:180); + ylim([-180 180]); + hold off; + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src * Direct Velocity Feedback +** 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 + addpath('./matlab/'); +#+end_src + +#+begin_src matlab + open('rotating_frame.slx'); +#+end_src + ** Equations The sensed relative velocity are equal to: - #+begin_important \begin{equation} \begin{bmatrix} \dot{d}_u \\ \dot{d}_v \end{bmatrix} = @@ -931,124 +1141,141 @@ The sensed relative velocity are equal to: \end{equation} #+end_important -** Numerical Values +** Plant Parameters +Let's define initial values for the model. #+begin_src matlab - k = 1; - m = 1; - c = 0.05; + k = 1; % [N/m] + m = 1; % [kg] + c = 0.05; % [N/(m/s)] + xi = c/(2*sqrt(k*m)); - - w0 = sqrt(k/m); + w0 = sqrt(k/m); % [rad/s] #+end_src +** Plant - Bode Plot +The controllers are set to zero. #+begin_src matlab - Ga = (s/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2))^2 + (2*wr*s/(w0^2))^2) * ... - [(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2), 2*wr*s/(w0^2) ; ... - -2*wr*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2)]; + Kiff = tf(zeros(2)); + Kdvf = tf(zeros(2)); #+end_src -** Simscape Model +No parallel stiffness and damper is used. +#+begin_src matlab + kp = 0; % [N/m] + cp = 0; % [N/(m/s)] +#+end_src + +The rotating speed is set to $\Omega = 0.1 \omega_0$. +#+begin_src matlab + W = 0.1*w0; +#+end_src + +The DVF plant is identified from the Simscape model and compared with the analytical one. + #+begin_src matlab %% Name of the Simulink File mdl = 'rotating_frame'; %% Input/Output definition clear io; io_i = 1; - io(io_i) = linio([mdl, '/Fu'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Fv'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Tuv'], 3, 'openoutput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Tuv'], 4, 'openoutput'); io_i = 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; #+end_src #+begin_src matlab - G = linearize(mdl, io, 0); + Gdvf = linearize(mdl, io, 0); %% Input/Output definition - G.InputName = {'Fu', 'Fv'}; - G.OutputName = {'Vu', 'Vv'}; + Gdvf.InputName = {'Fu', 'Fv'}; + Gdvf.OutputName = {'Vu', 'Vv'}; #+end_src -** DVF Plant #+begin_src matlab - figure; bode(G, Ga) + 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)]; #+end_src -** Loop Gain -Let's take $\Omega = \frac{\omega_0}{10}$. #+begin_src matlab - ws = 0.1*w0; - G_dvf = (s/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2))^2 + (2*wr*s/(w0^2))^2) * ... - [(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2), 2*wr*s/(w0^2) ; ... - -2*wr*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2)]; - -#+end_src - -#+begin_src matlab :exports none - g = 1; - - freqs = logspace(-2, 1, 1000); + freqs = logspace(-1, 1, 1000); figure; - - ax1 = subplot(2, 1, 1); + ax1 = subplot(2, 2, 1); hold on; - for i = 1:length(ws) - plot(freqs, abs(squeeze(freqresp(G_dvf(1,1)*g, freqs))), ... - 'DisplayName', sprintf('$\\omega = %.2f \\omega_0 $', ws(i)/w0)) - end + plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs))), '-') + plot(freqs, abs(squeeze(freqresp(Gdvf_th(1,1), freqs))), '--') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); ylabel('Amplitude [N/N]'); - legend('location', 'northwest'); - ylim([0, 1e3]); + set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/s/N]'); - ax2 = subplot(2, 1, 2); + ax3 = subplot(2, 2, 3); hold on; - for i = 1:length(ws) - plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf(1,1)*g, freqs))), ... - 'DisplayName', sprintf('$\\omega = %.1f \\omega_0 $', ws(i)/w0)) - end + plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(1,1), freqs))), '-') + plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf_th(1,1), freqs))), '--') set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); yticks(-180:90:180); ylim([-180 180]); hold off; - linkaxes([ax1,ax2],'x'); + ax2 = subplot(2, 2, 2); + hold on; + plot(freqs, abs(squeeze(freqresp(Gdvf(1,2), freqs))), '-') + plot(freqs, abs(squeeze(freqresp(Gdvf_th(1,2), freqs))), '--') + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/s/N]'); + + ax4 = subplot(2, 2, 4); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(1,2), freqs))), '-') + plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf_th(1,2), freqs))), '--') + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); + yticks(-180:90:180); + ylim([-180 180]); + hold off; + + linkaxes([ax1,ax2,ax3,ax4],'x'); xlim([freqs(1), freqs(end)]); + + linkaxes([ax1,ax2],'y'); #+end_src ** Root Locus -#+begin_src matlab :exports none - ws = [0, 0.1, 0.3, 0.8, 1.1]; +The controller is a pure gain:: +\begin{equation} + K_{\text{DVF}}(s) = g +\end{equation} - % G_dvf = {zeros(2, 2, length(ws))}; - G_dvf = {zeros(length(ws), 1)}; - - for i = 1:length(ws) - W = ws(i); - - G_dvf(i) = {(s/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ... - [(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ... - -2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)]}; - end +#+begin_src matlab + Ws = [0, 0.1, 0.3, 0.8, 1.1]; #+end_src -#+begin_src matlab :exports none - gains = logspace(-2, 4, 100); +#+begin_src matlab + gains = logspace(-2, 1, 100); figure; hold on; - for i = 1:length(ws) - set(gca,'ColorOrderIndex',i); - plot(real(pole(G_dvf{i})), imag(pole(G_dvf{i})), 'x', ... - 'DisplayName', sprintf('$\\omega = %.2f \\omega_0 $', ws(i)/w0)); - set(gca,'ColorOrderIndex',i); - plot(real(tzero(G_dvf{i})), imag(tzero(G_dvf{i})), 'o', ... + for W_i = 1:length(Ws) + W = Ws(W_i); + + Gdvf = (s/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ... + [(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ... + -2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)]; + + set(gca,'ColorOrderIndex',W_i); + plot(real(pole(Gdvf)), imag(pole(Gdvf)), 'x', ... + 'DisplayName', sprintf('$\\omega = %.2f \\omega_0 $', W/w0)); + + set(gca,'ColorOrderIndex',W_i); + plot(real(tzero(Gdvf)), imag(tzero(Gdvf)), 'o', ... 'HandleVisibility', 'off'); + for g_i = 1:length(gains) - set(gca,'ColorOrderIndex',i); - cl_poles = pole(feedback(G_dvf{i}, gains(g_i)*eye(2))); + set(gca,'ColorOrderIndex',W_i); + cl_poles = pole(feedback(Gdvf, gains(g_i)*eye(2))); + plot(real(cl_poles), imag(cl_poles), '.', ... 'HandleVisibility', 'off'); end @@ -1062,29 +1289,43 @@ Let's take $\Omega = \frac{\omega_0}{10}$. #+end_src * 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 addpath('./matlab/'); #+end_src -** Parameters #+begin_src matlab - k = 1; - m = 1; - c = 0.05; + open('rotating_frame.slx'); +#+end_src + +** Plant Parameters +Let's define initial values for the model. +#+begin_src matlab + k = 1; % [N/m] + m = 1; % [kg] + c = 0.05; % [N/(m/s)] xi = c/(2*sqrt(k*m)); + w0 = sqrt(k/m); % [rad/s] +#+end_src - w0 = sqrt(k/m); - wr = 0.1; +The rotation speed is set to $\Omega = 0.1 \omega_0$. +#+begin_src matlab + W = 0.1*w0; % [rad/s] +#+end_src + +#+begin_src matlab + Kiff = tf(zeros(2)); + Kdvf = tf(zeros(2)); #+end_src ** Root Locus @@ -1104,11 +1345,6 @@ Let's take $\Omega = \frac{\omega_0}{10}$. io(io_i) = linio([mdl, '/G'], 2, 'openoutput'); io_i = io_i + 1; #+end_src -#+begin_src matlab - Kiff = tf(zeros(2)); - Kdvf = tf(zeros(2)); -#+end_src - #+begin_src matlab Giff = linearize(mdl, io, 0); @@ -1119,7 +1355,7 @@ Let's take $\Omega = \frac{\omega_0}{10}$. *** IFF With parallel Stiffness #+begin_src matlab - kp = 2*m*wr^2; + kp = 2*m*W^2; cp = 0.01; #+end_src @@ -1142,6 +1378,11 @@ Let's take $\Omega = \frac{\omega_0}{10}$. #+end_src *** DVF +#+begin_src matlab + kp = 0; + cp = 0; +#+end_src + #+begin_src matlab %% Name of the Simulink File mdl = 'rotating_frame'; @@ -1162,13 +1403,13 @@ Let's take $\Omega = \frac{\omega_0}{10}$. *** Root Locus #+begin_src matlab - wi = w0; + wi = 0.1*w0; #+end_src #+begin_src matlab :exports none figure; - gains = logspace(-2, 4, 100); + gains = logspace(-2, 2, 100); hold on; set(gca,'ColorOrderIndex',1); @@ -1214,13 +1455,12 @@ Let's take $\Omega = \frac{\omega_0}{10}$. end hold off; axis square; - xlim([-2, 0.5]); ylim([0, 2.5]); + xlim([-1.2, 0.05]); ylim([0, 1.25]); xlabel('Real Part'); ylabel('Imaginary Part'); legend('location', 'northwest'); #+end_src - ** Controllers - Optimal Gains Estimate the controller gain that yields good damping in all cases. @@ -1231,9 +1471,9 @@ Pseudo integrator IFF: opt_gain_iff = 0; for g_i = 1:length(gains) - K_iff = (gains(g_i)/(1+s/wi))*eye(2); + Kiff = (gains(g_i)/(1+s/wi))*eye(2); - [w, zeta] = damp(minreal(feedback(Giff, K_iff))); + [w, zeta] = damp(minreal(feedback(Giff, Kiff))); if min(zeta) > opt_zeta_iff && all(zeta > 0) opt_zeta_iff = min(zeta); @@ -1249,9 +1489,9 @@ Parallel Stiffness opt_gain_kp = 0; for g_i = 1:length(gains) - K_iff = gains(g_i)/s*eye(2); + Kiff = gains(g_i)/s*eye(2); - [w, zeta] = damp(minreal(feedback(Giff_kp, K_iff))); + [w, zeta] = damp(minreal(feedback(Giff_kp, Kiff))); if min(zeta) > opt_zeta_kp && all(zeta > 0) opt_zeta_kp = min(zeta); @@ -1262,16 +1502,16 @@ Parallel Stiffness DVF: #+begin_src matlab :exports none - gains = logspace(-2, 4, 100); + gains = logspace(0, 4, 100); opt_zeta_dvf = 0; opt_gain_dvf = 0; for g_i = 1:length(gains) - K_dvf = gains(g_i)*eye(2); + Kdvf = gains(g_i)*eye(2); - [w, zeta] = damp(minreal(feedback(Gdvf, K_dvf))); + [w, zeta] = damp(minreal(feedback(Gdvf, Kdvf))); - if min(zeta) > opt_zeta_dvf && all(zeta > 0) + if min(zeta) > opt_zeta_dvf && all(zeta > 0) && min(zeta) < 0.85 opt_zeta_dvf = min(zeta); opt_gain_dvf = min(gains(g_i)); end @@ -1284,10 +1524,6 @@ DVF: ** Transmissibility *** Open Loop -#+begin_src matlab - wi = w0; -#+end_src - #+begin_src matlab Kdvf = tf(zeros(2)); Kiff = tf(zeros(2)); @@ -1317,10 +1553,6 @@ DVF: #+end_src *** Pseudo Integrator IFF -#+begin_src matlab - wi = w0; -#+end_src - #+begin_src matlab Kdvf = tf(zeros(2)); Kiff = opt_gain_iff/(1 + s/wi)*tf(eye(2)); @@ -1351,7 +1583,7 @@ DVF: *** IFF With parallel Stiffness #+begin_src matlab - kp = 2*m*wr^2; + kp = 2*m*W^2; cp = 0.01; #+end_src @@ -1429,16 +1661,6 @@ DVF: ** Compliance *** Open Loop -#+begin_src matlab - Kdvf = tf(zeros(2)); - Kiff = tf(zeros(2)); -#+end_src - -#+begin_src matlab - kp = 0; - cp = 0; -#+end_src - #+begin_src matlab %% Name of the Simulink File mdl = 'rotating_frame'; @@ -1449,6 +1671,16 @@ DVF: io(io_i) = linio([mdl, '/Meas'], 1, 'output'); io_i = io_i + 1; #+end_src +#+begin_src matlab + Kdvf = tf(zeros(2)); + Kiff = tf(zeros(2)); +#+end_src + +#+begin_src matlab + kp = 0; + cp = 0; +#+end_src + #+begin_src matlab Col = linearize(mdl, io, 0); @@ -1458,10 +1690,6 @@ DVF: #+end_src *** Pseudo Integrator IFF -#+begin_src matlab - wi = w0; -#+end_src - #+begin_src matlab Kdvf = tf(zeros(2)); Kiff = opt_gain_iff/(1 + s/wi)*tf(eye(2)); @@ -1472,16 +1700,6 @@ DVF: cp = 0; #+end_src -#+begin_src matlab - %% 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; -#+end_src - #+begin_src matlab Ciff = linearize(mdl, io, 0); @@ -1492,7 +1710,7 @@ DVF: *** IFF With parallel Stiffness #+begin_src matlab - kp = 2*m*wr^2; + kp = 2*m*W^2; cp = 0.01; #+end_src @@ -1501,16 +1719,6 @@ DVF: Kiff = opt_gain_kp/s*tf(eye(2)); #+end_src -#+begin_src matlab - %% 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; -#+end_src - #+begin_src matlab Ciff_kp = linearize(mdl, io, 0); @@ -1530,16 +1738,6 @@ DVF: Kiff = tf(zeros(2)); #+end_src -#+begin_src matlab - %% 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; -#+end_src - #+begin_src matlab Cdvf = linearize(mdl, io, 0); @@ 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(0,0) {$(x, y)$}; + + \draw[->] (3.5, 0) arc (0:40:3.5) node[midway, left]{$\Omega$}; + \end{tikzpicture} +#+end_src + +#+RESULTS: +[[file:figs/rotating_xy_platform_bis.png]] + * X-Y Rotating Positioning Platform with Springs in parallel #+begin_src latex :file rotating_xy_platform_springs.pdf \begin{tikzpicture} @@ -134,17 +208,13 @@ Configuration file is accessible [[file:config.org][here]]. \draw[] ($(av) + (-0.5, -0.2)$)coordinate(kv) -- ($(av) + ( 0.5, 0)$); % Spring and Actuator for U - \draw[actuator={0.6}{0.2}] (actu) -- (actu-|-2.6,0); - \draw[spring=0.2] (ku) -- (ku-|-2.6,0); - \draw[spring=0.2] (-1, 0.8) -- (-1, 0.8-|-2.6,0); + \draw[actuator={0.6}{0.2}] (actu) -- node[below=0.1, rotate=\thetau]{$F_u$} (actu-|-2.6,0); + \draw[spring=0.2] (ku) -- node[below=0.1, rotate=\thetau]{$k_u$} (ku-|-2.6,0); + \draw[spring=0.2] (-1, 0.8) -- node[above=0.1, rotate=\thetau]{$k_{pu}$} (-1, 0.8-|-2.6,0); - \draw[actuator={0.6}{0.2}] (actv) -- (actv|-0,-2.6); - \draw[spring=0.2] (kv) -- (kv|-0,-2.6); - \draw[spring=0.2] (-0.8, -1) -- (-0.8, -1|-0,-2.6); - - % Displacement measurement - \draw[<->, dashed] (-2.6, -0.8) --node[midway, below, rotate=\thetau]{$d_u$} (-1 , -0.8); - \draw[<->, dashed] ( 0.8, -2.6) --node[midway, right, rotate=\thetau]{$d_v$} ( 0.8, -1); + \draw[actuator={0.6}{0.2}] (actv) -- node[right=0.1, rotate=\thetau]{$F_v$} (actv|-0,-2.6); + \draw[spring=0.2] (kv) -- node[right=0.1, rotate=\thetau]{$k_v$} (kv|-0,-2.6); + \draw[spring=0.2] (-0.8, -1) -- node[left=0.1, rotate=\thetau]{$k_{pv}$} (-0.8, -1|-0,-2.6); \end{scope} % Inertial Frame @@ -164,6 +234,9 @@ Configuration file is accessible [[file:config.org][here]]. \end{tikzpicture} #+end_src +#+RESULTS: +[[file:figs/rotating_xy_platform_springs.png]] + #+begin_src latex :file rotating_xy_platform_springs.pdf \begin{tikzpicture} % Angle