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files /dev/null and b/figs/iff_variability_plant_ws.pdf differ diff --git a/figs/iff_variability_plant_ws.png b/figs/iff_variability_plant_ws.png new file mode 100644 index 0000000..0cb3398 Binary files /dev/null and b/figs/iff_variability_plant_ws.png differ diff --git a/figs/poles_cl_system.pdf b/figs/poles_cl_system.pdf index 69955bf..5aa4f12 100644 Binary files a/figs/poles_cl_system.pdf and b/figs/poles_cl_system.pdf differ diff --git a/figs/poles_cl_system.png b/figs/poles_cl_system.png index ec94d2b..152e715 100644 Binary files a/figs/poles_cl_system.png and b/figs/poles_cl_system.png differ diff --git a/index.html b/index.html index 1b16e63..0aa542d 100644 Binary files a/index.html and b/index.html differ diff --git a/index.org b/index.org index 833fcce..cbb6dbd 100644 --- a/index.org +++ b/index.org @@ -23,6 +23,7 @@ #+PROPERTY: header-args:shell :eval no-export :END: +* Introduction :ignore: The objective of this note it to highlight some control problems that arises when controlling the position of an object using actuators that are rotating with respect to a fixed reference frame. In section [[sec:system]], a simple system composed of a spindle and a translation stage is defined and the equations of motion are written. @@ -38,10 +39,10 @@ In section [[sec:simscape]], the analytical study will be validated using a mult Finally, in section [[sec:control]], the control strategies are implemented using Simulink and Simscape and compared. * System Description and Analysis - :PROPERTIES: - :HEADER-ARGS:matlab+: :tangle matlab/system_numerical_analysis.m - :END: - <> +:PROPERTIES: +:HEADER-ARGS:matlab+: :tangle matlab/system_numerical_analysis.m +:END: +<> ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> @@ -437,7 +438,7 @@ Then we compare the result between voice coil and piezoelectric actuators. freqs = logspace(-2, 1, 1000); figure; - ax1 = subaxis(2,1,1); + ax1 = subplot(2,1,1); hold on; for i = 1:length(ws) plot(freqs, abs(squeeze(freqresp(Gs{i}, freqs, 'Hz')))); @@ -448,7 +449,7 @@ Then we compare the result between voice coil and piezoelectric actuators. set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]'); - ax2 = subaxis(2,1,2); + ax2 = subplot(2,1,2); hold on; for i = 1:length(ws) plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}, freqs, 'Hz'))), 'DisplayName', sprintf('w = %.0f [rpm]', ws(i)*60/2/pi)); @@ -477,7 +478,7 @@ Then we compare the result between voice coil and piezoelectric actuators. freqs = logspace(-2, 1, 1000); figure; - ax1 = subaxis(2,1,1); + ax1 = subplot(2,1,1); hold on; for i = 1:length(ws) plot(freqs, abs(squeeze(freqresp(Gcs{i}, freqs, 'Hz')))); @@ -488,7 +489,7 @@ Then we compare the result between voice coil and piezoelectric actuators. set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]'); - ax2 = subaxis(2,1,2); + ax2 = subplot(2,1,2); hold on; for i = 1:length(ws) plot(freqs, 180/pi*angle(squeeze(freqresp(Gcs{i}, freqs, 'Hz'))), 'DisplayName', sprintf('w = %.0f [rpm]', ws(i)*60/2/pi)); @@ -533,7 +534,7 @@ Then we compare the result between voice coil and piezoelectric actuators. freqs = logspace(2, 3, 1000); figure; - ax1 = subaxis(2,1,1); + ax1 = subplot(2,1,1); hold on; for i = 1:length(ws) plot(freqs, abs(squeeze(freqresp(Gs{i}, freqs, 'Hz')))); @@ -544,7 +545,7 @@ Then we compare the result between voice coil and piezoelectric actuators. set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]'); - ax2 = subaxis(2,1,2); + ax2 = subplot(2,1,2); hold on; for i = 1:length(ws) plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}, freqs, 'Hz'))), 'DisplayName', sprintf('w = %.0f [rpm]', ws(i)*60/2/pi)); @@ -570,7 +571,7 @@ Then we compare the result between voice coil and piezoelectric actuators. #+begin_src matlab :exports none figure; - ax1 = subaxis(2,1,1); + ax1 = subplot(2,1,1); hold on; for i = 1:length(ws) plot(freqs, abs(squeeze(freqresp(Gcs{i}, freqs, 'Hz')))); @@ -581,7 +582,7 @@ Then we compare the result between voice coil and piezoelectric actuators. set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]'); - ax2 = subaxis(2,1,2); + ax2 = subplot(2,1,2); hold on; for i = 1:length(ws) plot(freqs, 180/pi*angle(squeeze(freqresp(Gcs{i}, freqs, 'Hz'))), 'DisplayName', sprintf('w = %.0f [rpm]', ws(i)*60/2/pi)); @@ -728,6 +729,512 @@ For the voice coil (figure [[fig:poles_w_vc]]), the system is unstable when the #+CAPTION: Real and Imaginary part of the poles of the system as a function of the rotation speed - Piezoelectric actuator and light sample [[file:figs/poles_w_pz.png]] +* Integral Force Feedback +<> +** Introduction :ignore: +In this section, we study the use of Decentralized Integral Force Feedback (IFF) to damp the resonance of the positioning system. + +We thus suppose there is a force sensor in series with the actuator in the $u$ and $v$ directions. + +** Analytical Derivation of the transfer function for IFF control +The sensed forces are equal to: +\begin{align} + F_{um} &= F_u - k d_u \\ + F_{vm} &= F_v - k d_v +\end{align} + +If we consider a constant rotating speed $\dot{\theta} = \omega$, we have: +\begin{equation} +\begin{bmatrix} d_u \\ d_v \end{bmatrix} = +\frac{1}{(m s^2 + (k - m{\omega}^2))^2 + (2 m {\omega} s)^2} +\begin{bmatrix} + ms^2 + (k-m{\omega}^2) & 2 m \omega s \\ + -2 m \omega s & ms^2 + (k-m{\omega}^2) \\ +\end{bmatrix} +\begin{bmatrix} F_u \\ F_v \end{bmatrix} +\end{equation} + +Which then give: +\begin{align} + F_{um} &= \frac{(m s^2 + (k - m\omega^2))^2 + (2 m \omega s)^2 - k(m s^2 + (k - m\omega^2))}{(m s^2 + (k - m\omega^2))^2 + (2 m \omega s)^2} F_u - \frac{2 m \omega s}{(m s^2 + (k - m\omega^2))^2 + (2 m \omega s)^2} F_v\\ + F_{vm} &= \frac{2 m \omega s}{(m s^2 + (k - m\omega^2))^2 + (2 m \omega s)^2} F_u + \frac{(m s^2 + (k - m\omega^2))^2 + (2 m \omega s)^2 - k(m s^2 + (k - m\omega^2))}{(m s^2 + (k - m\omega^2))^2 + (2 m \omega s)^2} F_v +\end{align} + +If we note $\omega_0 = \sqrt{\frac{k}{m}}$, which corresponds to the resonance of the positioning system when not rotating, we obtain: +#+begin_important +\begin{align} + F_{um} &= \frac{s^4 + (\omega_0^2 + 2\omega^2) s^2 + \omega^2(\omega^2 - \omega_0^2)}{s^4 + 2 (\omega_0^2 + \omega^2) s^2 + (\omega_0^2 - \omega^2)^2} F_u - \frac{2 \omega_0^2 \omega s}{s^4 + 2 (\omega_0^2 + \omega^2) s^2 + (\omega_0^2 - \omega^2)^2} F_v\\ + F_{vm} &= \frac{2 \omega_0^2 \omega s}{s^4 + 2 (\omega_0^2 + \omega^2) s^2 + (\omega_0^2 - \omega^2)^2} F_u + \frac{s^4 + (\omega_0^2 + 2\omega^2) s^2 + \omega^2(\omega^2 - \omega_0^2)}{s^4 + 2 (\omega_0^2 + \omega^2) s^2 + (\omega_0^2 - \omega^2)^2} F_v +\end{align} +#+end_important + +** Low and High frequency Behavior +At high frequency, the force sensors give: +\begin{align} + F_{um} &= F_u \\ + F_{vm} &= F_v +\end{align} +which is usual for IFF control. + + +At low frequency: +\begin{align} + F_{um} &= \frac{\omega^2}{\omega_0^2 - \omega^2} F_u \\ + F_{vm} &= \frac{\omega^2}{\omega_0^2 - \omega^2} F_v +\end{align} + +The gain at low frequency is no more zero as it is usually the case. +Thus, a constant force will be sensed by the force sensor. + + +This could be explained by the fact that a constant force will induce some displacement of the mass. +The mass being more off-center with the rotation axis, it is experiencing a larger centrifugal force and this (constant) centrifugal force is sensed by the force sensor. +It is like a *negative stiffness is in parallel with both the stiffness and the force sensor*. + +** Poles and Zeros +Let's use Matlab Symbolic toolbox to find the analytical formula for the poles and zeros of the transfer function from the force actuators to the force sensors. + +Let's define the parameters and suppose that we have $\omega > \omega_0$ (stable poles). +#+begin_src matlab + syms w w0 positive + assume(w > w0) + syms x +#+end_src + +The characteristic equations to find the poles and zeros are: +#+begin_src matlab + z = x^4 + (w0^2 + 2*w^2)*x^2 + w^2*(w^2 - w0^2) == 0 + p = x^4 + 2*(w0^2 + w^2)*x^2 +(w^2 - w0^2)^2 == 0 +#+end_src + +#+begin_src matlab + solve(p, x) +#+end_src + +The obtained poles are: +\begin{align} + p_1 &= \pm j (\omega_0 - \omega) \\ + p_2 &= \pm j (\omega_0 + \omega) +\end{align} + +We obtain two pairs of complex conjugate poles: +- $p_1$ is going to lower frequencies as the rotating speed $\omega$ is increased +- $p_2$ is going to higher frequencies as the rotating speed $\omega$ is increased + +#+begin_src matlab + solve(z, x) +#+end_src + +The obtained zeros are: +\begin{align} + z_1 &= \pm j \sqrt{ \frac{1}{2} \omega_0 \sqrt{(8 \omega^2 + \omega_0^2)} + \omega^2 + \frac{1}{2} \omega_0^2 } \\ + z_2 &= \pm \sqrt{ \frac{1}{2} \omega_0 \sqrt{(8 \omega^2 + \omega_0^2)} - \omega^2 - \frac{1}{2} \omega_0^2 } +\end{align} + +Thus, there are *two complex conjugate zeros* and *two real zeros*. + +We can easily show that the frequency of the two complex conjugate zeros is between the frequency of $p_1$ and $p_2$. + +The effect of the two real zeros is an increase of +2 of the slope without change of phase (non-minimum phase behavior). + +#+begin_important +Also, as one zero has a positive real part, the *IFF control is no more unconditionally stable*. + +This is due to the fact that the zeros of the plant are the poles of the closed loop system with an infinite gain. + +Thus, for some finite IFF gain, one pole will have a positive real part and the system will be unstable. +#+end_important + +** IFF Plant and Root Locus +#+begin_src matlab :exports none + w0 = 2*pi; % [rad/s] + m = 1; % [kg] + k = m*w0^2; % [N/m] + + ws = linspace(0, 0.9*w0, 4); + + G_iff = {zeros(2, 2, length(ws))}; + + for i = 1:length(ws) + w = ws(i); + G_iff(:, :, i) = {[((m*s^2 + k-m*w^2)^2 + (2*m*w*s)^2 - k*(m*s^2 + k - m*w^2))/((m*s^2 + (k - m*w^2))^2 + (2*m*w*s)^2), -(2*k*m*w*s)/((m*s^2 + (k - m*w^2))^2 + (2*m*w*s)^2); + (2*k*m*w*s)/((m*s^2 + (k - m*w^2))^2 + (2*m*w*s)^2), ((m*s^2 + k-m*w^2)^2 + (2*m*w*s)^2 - k*(m*s^2 + k - m*w^2))/((m*s^2 + (k - m*w^2))^2 + (2*m*w*s)^2)]}; + end +#+end_src + +The bode plot for the diagonal and off-diagonal elements are shown in Figure [[fig:iff_plant_and_coupling]]. + +The change of dynamics from $F_u$ to $F_{u,m}$ due to the change in rotation speed is shown in Figure [[fig:iff_variability_plant_ws]]. +It is shown that the rotation speed does change the low frequency gain (explained in previous section). + +The root locus for Decentralized Direct Velocity Feedback is shown in Figure [[fig:iff_root_locus_ws]]. + +It is shown that: +- the high frequency pair of complex conjugate poles cannot be critically damped. + There is an optimal gain that gives maximum damping of this resonance. +- the low frequency resonance can be critically damped with sufficient gain +- a closed loop unstable pole is present whatever the gain of the IFF (if the rotation speed in not null) + +#+begin_src matlab :exports none + freqs = logspace(-2, 1, 1000); + + i = 2; + + 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') + 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 $', ws(i)/w0)]); + + ax3 = subplot(2, 2, 3); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff{i}(1, 1), freqs, 'Hz')))); + hold off; + yticks(-180:90:180); + ylim([-180 180]); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + + ax2 = subplot(2, 2, 2); + hold on; + plot(freqs, abs(squeeze(freqresp(G_iff{i}(1,2), freqs, 'Hz')))) + plot((sqrt(k/m) + ws(i))/2/pi, 1, 'kx') + plot((sqrt(k/m) - ws(i))/2/pi, 1, 'kx') + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + title(['$\frac{F_{v,m}}{F_u}$' sprintf(', $\\omega = %.2f \\omega_0 $', ws(i)/w0)]); + + ax4 = subplot(2, 2, 4); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff{i}(1, 2), freqs, 'Hz')))); + hold off; + yticks(-180:90:180); + ylim([-180 180]); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + xlabel('Frequency [Hz]'); + + linkaxes([ax1,ax2,ax3,ax4],'x'); + linkaxes([ax1,ax2],'y'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/iff_plant_and_coupling.pdf', 'width', 'full', 'height', 'full') +#+end_src + +#+name: fig:iff_plant_and_coupling +#+caption: Transfer Function from $F_u$ to $F_{u,m}$ and from $F_u$ to $F_{v,m}$ for $\omega = 0.3 \omega_0$ +#+RESULTS: +[[file:figs/iff_plant_and_coupling.png]] + +#+begin_src matlab :exports none + freqs = logspace(-2, 1, 1000); + + figure; + hold on; + for i = 1:length(ws) + plot(freqs, abs(squeeze(freqresp(G_iff{i}(1,1), freqs, 'Hz'))), ... + 'DisplayName', sprintf('$\\omega = %.2f \\omega_0 $', ws(i)/w0)) + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Amplitude [N/N]'); + legend('location', 'northwest'); + ylim([0, 1e3]); + title('$\frac{F_{u,m}}{F_u}$'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/iff_variability_plant_ws.pdf', 'width', 'full', 'height', 'tall') +#+end_src + +#+name: fig:iff_variability_plant_ws +#+caption: Transfer Function from $F_u$ to $F_{u,m}$ for multiple rotation speed +#+RESULTS: +[[file:figs/iff_variability_plant_ws.png]] + +Let's see how the root locus is changing with the rotation speed. +#+begin_src matlab :exports none + figure; + + gains = logspace(0, 3, 100); + + hold on; + 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)); + set(gca,'ColorOrderIndex',i); + plot(real(tzero(G_iff{i})), imag(tzero(G_iff{i})), 'o', ... + 'HandleVisibility', 'off'); + for k = 1:length(gains) + set(gca,'ColorOrderIndex',i); + cl_poles = pole(feedback(G_iff{i}, gains(k)/s*eye(2))); + plot(real(cl_poles), imag(cl_poles), '.', ... + 'HandleVisibility', 'off'); + end + end + hold off; + axis square; + xlim([-13, 2]); ylim([0, 15]); + + xlabel('Real Part'); ylabel('Imaginary Part'); + legend('location', 'northwest'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/iff_root_locus_ws.pdf', 'width', 'wide', 'height', 'tall') +#+end_src + +#+name: fig:iff_root_locus_ws +#+caption: Change of the Root Locus plot for different rotating speed +#+RESULTS: +[[file:figs/iff_root_locus_ws.png]] + +** Conclusion +#+begin_important +The centrifugal forces can be modeled as a negative stiffness in parallel with the actuators and force sensors. +This changes the low frequency gain of the IFF plant. +It also add zeros with positive real part which removes the typical unconditional stability property of the IFF control. + +From the Root Locus plot in Figure [[fig:iff_root_locus_ws]], it is shown that when rotating, unstable poles will be introduced due to IFF control. +Moreover, only one of the two pairs of complex conjugate poles can be critically damped. + + +Stiffness can be added in parallel to the force sensor to counteract the negative stiffness due to centrifugal forces. +If the added stiffness is higher than the maximum negative stiffness, then the poles of the IFF damped system will stay in the (stable) right half-plane. + + +Thus, IFF should not be used for rotating positioning platforms unless some stiffness is added in parallel with the force sensor. +If no parallel stiffness is added, an unstable pole is added in the system. +The frequency of this pole can be made small if the frequency of the position system resonance is much larger than the frequency of the rotation. +#+end_important + +* Direct Velocity Feedback +<> +** Introduction :ignore: +In this section, we study the use of Direct Velocity Feedback to damp the resonances of the positioning station. +We thus suppose that we are able to measure to relative displacement (or velocity) $d_u$ and $d_v$. + +** Analytical Study +The equation of motion are: +\begin{equation} +\begin{bmatrix} d_u \\ d_v \end{bmatrix} = +\frac{1}{(m s^2 + (k - m{\omega}^2))^2 + (2 m {\omega} s)^2} +\begin{bmatrix} + ms^2 + (k-m{\omega}^2) & 2 m \omega s \\ + -2 m \omega s & ms^2 + (k-m{\omega}^2) \\ +\end{bmatrix} +\begin{bmatrix} F_u \\ F_v \end{bmatrix} +\end{equation} + +If we note $\omega_0 = \sqrt{\frac{k}{m}}$, we obtain +\begin{align} + d_u &= \frac{1}{m}\frac{s^2 + (\omega_0^2 - \omega^2)}{s^4 + 2 (\omega_0^2 + \omega^2) s^2 + (\omega_0^2 - \omega^2)^2} F_u + \frac{1}{m} \frac{2 \omega s}{s^4 + 2 (\omega_0^2 + \omega^2) s^2 + (\omega_0^2 - \omega^2)^2} F_v\\ + d_v &= - \frac{1}{m} \frac{2 \omega s}{s^4 + 2 (\omega_0^2 + \omega^2) s^2 + (\omega_0^2 - \omega^2)^2} F_u + \frac{1}{m}\frac{s^2 + (\omega_0^2 - \omega^2)}{s^4 + 2 (\omega_0^2 + \omega^2) s^2 + (\omega_0^2 - \omega^2)^2} F_v +\end{align} + +** High and Low frequency behavior +At low frequency: +\begin{align} + d_u &= \frac{1}{m}\frac{1}{\omega_0^2 - \omega^2} F_u \\ + d_v &= \frac{1}{m}\frac{1}{\omega_0^2 - \omega^2} F_v +\end{align} + +Which can be re-written: +\begin{align} + d_u &= \frac{1}{k} \frac{1}{1 - \frac{\omega^2}{\omega_0^2}} F_u \\ + d_v &= \frac{1}{k} \frac{1}{1 - \frac{\omega^2}{\omega_0^2}} F_v +\end{align} + +The usual low frequency gain of $\frac{1}{k}$ is modified by a factor $\frac{1}{1 - \frac{\omega^2}{\omega_0^2}}$. + +At high frequency: +\begin{align} + d_u &= \frac{1}{m s^2} F_u + \frac{2 \omega}{m s^3} F_v\\ + d_v &= -\frac{2 \omega}{m s^3} F_u + \frac{1}{m s^2} F_v +\end{align} + +Thus the rotating speed does not change the high frequency behavior of the diagonal terms but change the coupling. + +** Poles and Zeros +The direct terms (diagonal terms of the 2x2 transfer function matrix) have two complex conjugate zeros at: +\begin{equation} + z = \pm j \sqrt{\omega_0^2 - \omega^2} +\end{equation} + +Which are between the two pairs of complex conjugate poles at: +\begin{align} + p_1 &= \pm j (\omega_0 - \omega) \\ + p_2 &= \pm j (\omega_0 + \omega) +\end{align} + +** DVF Plant and Root Locus +#+begin_src matlab :exports none + w0 = 2*pi; % [rad/s] + m = 1; % [kg] + k = m*w0^2; % [N/m] + + ws = linspace(0, 0.9*w0, 4); + + G_dvf = {zeros(2, 2, length(ws))}; + + for i = 1:length(ws) + w = ws(i); + G_dvf(:, :, i) = {[ (m*s^2 + k-m*w^2)/((m*s^2 + (k - m*w^2))^2 + (2*m*w*s)^2), (2*m*w*s)/((m*s^2 + (k - m*w^2))^2 + (2*m*w*s)^2); + -(2*m*w*s)/((m*s^2 + (k - m*w^2))^2 + (2*m*w*s)^2), (m*s^2 + k-m*w^2)/((m*s^2 + (k - m*w^2))^2 + (2*m*w*s)^2) ]}; + end +#+end_src + +The bode plot for the diagonal and off-diagonal elements are shown in Figure [[fig:dvf_plant_and_coupling]]. + +The change of dynamics from $F_u$ to $d_u$ due to the change in rotation speed is shown in Figure [[fig:dvf_variability_plant_ws]]. +It is shown that the rotation speed does not change much the low frequency and high frequency gains. +It only modifies the location of the two pair of complex conjugate pairs. + +The root locus for Decentralized Direct Velocity Feedback is shown in Figure [[fig:dvf_root_locus_ws]]. +It is shown that the closed-loop poles are in the left-half of the complex plane, whatever the gain. +Moreover, has much damping can be added to both pairs of complex conjugate poles. + +#+begin_src matlab :exports none + freqs = logspace(-2, 1, 1000); + + i = 2; + + figure; + ax1 = subplot(2, 2, 1); + hold on; + plot(freqs, abs(squeeze(freqresp(G_dvf{i}(1,1), freqs, 'Hz')))) + plot(freqs, ones(size(freqs))*1/(k * (1 - ws(i)^2/(k/m))), 'k--') + plot(freqs, abs(squeeze(freqresp(1/m/s^2, freqs, 'Hz'))), 'k--') + plot(sqrt(k/m - ws(i))/2/pi, 1, 'ko') + plot((sqrt(k/m) + ws(i))/2/pi, 1, 'kx') + plot((sqrt(k/m) - ws(i))/2/pi, 1, 'kx') + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); + title(['$\frac{d_u}{F_u}$' sprintf(', $\\omega = %.2f \\omega_0 $', ws(i)/w0)]); + + ax3 = subplot(2, 2, 3); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf{i}(1, 1), freqs, 'Hz')))); + hold off; + yticks(-180:90:180); + ylim([-180 180]); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + + ax2 = subplot(2, 2, 2); + hold on; + plot(freqs, abs(squeeze(freqresp(G_dvf{i}(1,2), freqs, 'Hz')))) + plot((sqrt(k/m) + ws(i))/2/pi, 1, 'kx') + plot((sqrt(k/m) - ws(i))/2/pi, 1, 'kx') + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + title(['$\frac{d_v}{F_u}$' sprintf(', $\\omega = %.2f \\omega_0 $', ws(i)/w0)]); + + ax4 = subplot(2, 2, 4); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf{i}(1, 2), freqs, 'Hz')))); + hold off; + yticks(-180:90:180); + ylim([-180 180]); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + xlabel('Frequency [Hz]'); + + linkaxes([ax1,ax2,ax3,ax4],'x'); + linkaxes([ax1,ax2],'y'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/dvf_plant_and_coupling.pdf', 'width', 'full', 'height', 'full') +#+end_src + +#+name: fig:dvf_plant_and_coupling +#+caption: Transfer Function from $F_u$ to $d_u$ and from $F_u$ to $d_v$ for $\omega = 0.3 \omega_0$ +#+RESULTS: +[[file:figs/dvf_plant_and_coupling.png]] + +#+begin_src matlab :exports none + freqs = logspace(-2, 1, 1000); + + figure; + hold on; + for i = 1:length(ws) + plot(freqs, abs(squeeze(freqresp(G_dvf{i}(1,1), freqs, 'Hz'))), ... + 'DisplayName', sprintf('$\\omega = %.2f \\omega_0 $', ws(i)/w0)) + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]'); + legend('location', 'northwest'); + title('$\frac{d_u}{F_u}$'); + ylim([0, 1e1]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/dvf_variability_plant_ws.pdf', 'width', 'full', 'height', 'tall') +#+end_src + +#+name: fig:dvf_variability_plant_ws +#+caption: Transfer Function from $F_u$ to $d_u$ for multiple rotation speed +#+RESULTS: +[[file:figs/dvf_variability_plant_ws.png]] + +#+begin_src matlab :exports none + figure; + + gains = logspace(-1, 3, 200); + + 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', ... + 'HandleVisibility', 'off'); + for k = 1:length(gains) + set(gca,'ColorOrderIndex',i); + cl_poles = pole(feedback(G_dvf{i}, s*gains(k)*eye(2))); + plot(real(cl_poles), imag(cl_poles), '.', ... + 'HandleVisibility', 'off'); + end + end + hold off; + axis square; + xlim([-14, 1]); ylim([0, 15]); + + xlabel('Real Part'); ylabel('Imaginary Part'); + legend('location', 'northwest'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/dvf_root_locus_ws.pdf', 'width', 'wide', 'height', 'tall') +#+end_src + +#+name: fig:dvf_root_locus_ws +#+caption: Change of the Root Locus plot for multiple rotating speed +#+RESULTS: +[[file:figs/dvf_root_locus_ws.png]] + +** Conclusion +#+begin_important +(Relative) Direct Velocity Feedback can be used to damp the resonance of a rotating positioning stage. + +As shown by the Root Locus in Figure [[fig:dvf_root_locus_ws]]: +- DVF permits to critically damp the vibrations of the positioning stage +- DVF is unconditionally stage +#+end_important + + * Control Strategies <> ** Measurement in the fixed reference frame @@ -913,6 +1420,62 @@ We obtain the same result than the analytical case (figures [[fig:coupling_light #+CAPTION: Coupling ratio obtained with the Simscape model [[file:figs/coupling_ratio_light_heavy.png]] +** Transfer function from force to force sensor (IFF plant) +#+begin_src matlab :exports code + %% Name of the Simulink File + mdl = 'rotating_frame'; + + %% Input/Output definition + io(1) = linio([mdl, '/fu'], 1, 'input'); + io(2) = linio([mdl, '/fv'], 1, 'input'); + + io(3) = linio([mdl, '/fum'], 1, 'output'); + io(4) = linio([mdl, '/fvm'], 1, 'output'); +#+end_src + +We start we identify the transfer functions at high speed with the light sample. +#+begin_src matlab :exports code + w = wlight; % Rotation speed [rad/s] + m = mlight; % mass of the sample [kg] + + kTuv = kpz; + Gpz_light = linearize(mdl, io, 0.1); + Gpz_light.InputName = {'Fu', 'Fv'}; + Gpz_light.OutputName = {'Fum', 'Fvm'}; + + kTuv = kvc; + Gvc_light = linearize(mdl, io, 0.1); + Gvc_light.InputName = {'Fu', 'Fv'}; + Gvc_light.OutputName = {'Fum', 'Fvm'}; +#+end_src + +Then we identify the system with an heavy mass and low speed. +#+begin_src matlab :exports code + w = wheavy; % Rotation speed [rad/s] + m = mheavy; % mass of the sample [kg] + + kTuv = kpz; + Gpz_heavy = linearize(mdl, io, 0.1); + Gpz_heavy.InputName = {'Fu', 'Fv'}; + Gpz_heavy.OutputName = {'Fum', 'Fvm'}; + + kTuv = kvc; + Gvc_heavy = linearize(mdl, io, 0.1); + Gvc_heavy.InputName = {'Fu', 'Fv'}; + Gvc_heavy.OutputName = {'Fum', 'Fvm'}; +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + plot(freqs, abs(squeeze(freqresp(Gvc_heavy('Fum', 'Fu'), freqs, 'Hz')))); + hold off; + xlim([freqs(1), freqs(end)]); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Coupling ratio'); + legend({'light - VC', 'light - PZ', 'heavy - VC', 'heavy - PZ'}, 'Location', 'northeast') +#+end_src + ** Plant Control - SISO approach *** Plant identification The goal is to study the control problems due to the coupling that appears because of the rotation. @@ -963,7 +1526,7 @@ The bode plot of the system not rotating and rotating at 60rpm is shown figure [ #+begin_src matlab :exports none figure; - ax1 = subaxis(2,2,1); + ax1 = subplot(2,2,1); title('From $F_u$ to $D_u$'); hold on; plot(freqs, abs(squeeze(freqresp(Gvc(1, 1), freqs, 'Hz')))); @@ -974,7 +1537,7 @@ The bode plot of the system not rotating and rotating at 60rpm is shown figure [ set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]'); - ax2 = subaxis(2,2,3); + ax2 = subplot(2,2,3); hold on; plot(freqs, 180/pi*angle(squeeze(freqresp(Gvc(1, 1), freqs, 'Hz'))), 'DisplayName', 'Gvc - $\omega = 0$'); plot(freqs, 180/pi*angle(squeeze(freqresp(Gtvc(1, 1), freqs, 'Hz'))), 'DisplayName', 'Gvc - $\omega = 60$rpm'); @@ -987,7 +1550,7 @@ The bode plot of the system not rotating and rotating at 60rpm is shown figure [ legend('Location', 'northeast'); linkaxes([ax1,ax2],'x'); - ax1 = subaxis(2,2,2); + ax1 = subplot(2,2,2); title('From $F_u$ to $D_v$'); hold on; plot(freqs, abs(squeeze(freqresp(Gvc(1, 2), freqs, 'Hz')))); @@ -997,7 +1560,7 @@ The bode plot of the system not rotating and rotating at 60rpm is shown figure [ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); - ax2 = subaxis(2,2,4); + ax2 = subplot(2,2,4); hold on; plot(freqs, 180/pi*angle(squeeze(freqresp(Gvc(1, 2), freqs, 'Hz'))), 'DisplayName', 'Gvc - $\omega = 0$'); plot(freqs, 180/pi*angle(squeeze(freqresp(Gtvc(1, 2), freqs, 'Hz'))), 'DisplayName', 'Gvc - $\omega = 60$rpm'); @@ -1052,7 +1615,7 @@ To stabilize the unstable pole, we need a control bandwidth of at least twice of freqs = logspace(-2, 2, 1000); figure; - ax1 = subaxis(2,2,1); + ax1 = subplot(2,2,1); title('$D_u/F_u$'); hold on; for i = 1:length(ws) @@ -1065,7 +1628,7 @@ To stabilize the unstable pole, we need a control bandwidth of at least twice of set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]'); - ax2 = subaxis(2,2,3); + ax2 = subplot(2,2,3); hold on; for i = 1:length(ws) plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_vc{i}(1, 1), freqs, 'Hz')))); @@ -1078,7 +1641,7 @@ To stabilize the unstable pole, we need a control bandwidth of at least twice of xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); linkaxes([ax1,ax2],'x'); - ax1 = subaxis(2,2,2); + ax1 = subplot(2,2,2); title('$D_v/F_u$'); hold on; for i = 1:length(ws) @@ -1091,7 +1654,7 @@ To stabilize the unstable pole, we need a control bandwidth of at least twice of set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); - ax2 = subaxis(2,2,4); + ax2 = subplot(2,2,4); hold on; for i = 1:length(ws) plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_vc{i}(1, 2), freqs, 'Hz'))), 'DisplayName', sprintf('w = %.0f [rpm]', ws(i)*60/2/pi)); @@ -1139,7 +1702,7 @@ Then, we can look at the same plots for the piezoelectric actuator (figure [[fig freqs = logspace(2, 3, 1000); figure; - ax1 = subaxis(2,1,1); + ax1 = subplot(2,1,1); hold on; for i = 1:length(ws) plot(freqs, abs(squeeze(freqresp(Gs_pz{i}(1, 1), freqs, 'Hz')))); @@ -1150,7 +1713,7 @@ Then, we can look at the same plots for the piezoelectric actuator (figure [[fig set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]'); - ax2 = subaxis(2,1,2); + ax2 = subplot(2,1,2); hold on; for i = 1:length(ws) plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_pz{i}(1, 1), freqs, 'Hz'))), 'DisplayName', sprintf('w = %.0f [rpm]', ws(i)*60/2/pi)); @@ -1194,7 +1757,7 @@ The loop gain is displayed figure [[fig:Gvc_loop_gain]]. figure; % Amplitude - ax1 = subaxis(2,1,1); + ax1 = subplot(2,1,1); hold on; plot(freqs, abs(squeeze(freqresp(Gvc('Du', 'fu')*Kll, freqs, 'Hz'))), '-'); set(gca,'xscale','log'); set(gca,'yscale','log'); @@ -1202,7 +1765,7 @@ The loop gain is displayed figure [[fig:Gvc_loop_gain]]. set(gca, 'XTickLabel',[]); hold off; % Phase - ax2 = subaxis(2,1,2); + ax2 = subplot(2,1,2); hold on; plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gvc('Du', 'fu')*Kll, freqs, 'Hz')))), '-'); set(gca,'xscale','log'); @@ -1233,7 +1796,7 @@ Let's first plot the SISO loop gain. figure; % Amplitude - ax1 = subaxis(2,1,1); + ax1 = subplot(2,1,1); hold on; plot(freqs, abs(squeeze(freqresp(Gs_vc{1}(1, 1)*Kll, freqs, 'Hz'))), '-'); plot(freqs, abs(squeeze(freqresp(Gs_vc{2}(1, 1)*Kll, freqs, 'Hz'))), '-'); @@ -1244,7 +1807,7 @@ Let's first plot the SISO loop gain. hold off; % Phase - ax2 = subaxis(2,1,2); + ax2 = subplot(2,1,2); hold on; plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_vc{1}(1, 1)*Kll, freqs, 'Hz'))), 'DisplayName', sprintf('%.0f rpm', ws(1)./(2*pi).*60)); plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_vc{2}(1, 1)*Kll, freqs, 'Hz'))), 'DisplayName', sprintf('%.0f rpm', ws(2)./(2*pi).*60)); @@ -1283,7 +1846,7 @@ The system then goes unstable at some point (here for $\omega=60rpm$). hold on; for i = 1:length(ws) sys = feedback(Gs_vc{i}(1, 1), K(1, 1), 'name'); - plot(real(pole(sys)), imag(pole(sys)), 'o', 'DisplayName', sprintf('$\\omega = %.0f rpm$', ws(i)/(2*pi)*60)); + plot(real(pole(sys)), imag(pole(sys)), 'x', 'DisplayName', sprintf('$\\omega = %.0f rpm$', ws(i)/(2*pi)*60)); end hold off; xlim([-80, 10]); @@ -1306,7 +1869,7 @@ If we look at the poles of the closed loop-system for multiple rotating speed (f figure; hold on; for i = 1:length(ws) - plot(real(pole(Gvc_cl{i})), imag(pole(Gvc_cl{i})), 'o', 'DisplayName', sprintf('$\\omega = %.0f rpm$', ws(i)/(2*pi)*60)); + plot(real(pole(Gvc_cl{i})), imag(pole(Gvc_cl{i})), 'x', 'DisplayName', sprintf('$\\omega = %.0f rpm$', ws(i)/(2*pi)*60)); end hold off; xlim([-80, 0]); @@ -1323,6 +1886,12 @@ If we look at the poles of the closed loop-system for multiple rotating speed (f #+CAPTION: Evolution of the poles of the closed-loop system [[file:figs/poles_cl_system.png]] +*** TODO Neglected coupling +Take a diagonal system and apply the diagonal controller. + +I expect that the system will be unstable +=> in some way the coupling makes the system stable + *** Close loop performance First, we create the closed loop systems. Then, we plot the transfer function from the reference signals $[r_u, r_v]$ to the output $[d_u, d_v]$ (figure [[fig:perfcomp]]). @@ -1369,6 +1938,59 @@ First, we create the closed loop systems. Then, we plot the transfer function fr #+CAPTION: Close loop performance for $\omega = 0$ and $\omega = 60 rpm$ [[file:figs/perfcomp.png]] +*** Campbell Diagram for the close loop system + +#+begin_src matlab :exports code + m = mlight; + k = kvc; + c = 0.1*sqrt(k*m); + + wsvc = linspace(0, 10, 100); % [rad/s] + + polesvc = zeros(8, length(wsvc)); + + for i = 1:length(wsvc) + Gs = (m*s^2 + (k-m*w^2))/((m*s^2 + (k - m*w^2))^2 + (2*m*w*s)^2); + Gcs = (2*m*w*s)/((m*s^2 + (k - m*w^2))^2 + (2*m*w*s)^2); + + G = [Gs, Gcs; Gcs, Gs]; + G.InputName = {'Fu', 'Fv'}; + G.OutputName = {'Du', 'Dv'}; + + polei = pole(feedback(G, K, 'name')); + polesvc(:, i) = sort(polei(imag(polei) > 0)); + + polei = pole(feedback(G, 10*K, 'name')); + polesvcb(:, i) = sort(polei(imag(polei) > 0)); + end +#+end_src + +#+begin_src matlab :exports none + figure; + % Amplitude + ax1 = subplot(1,2,1); + hold on; + for i = 1:8 + plot(wsvc, real(polesvc(i, :)), 'b') + plot(wsvc, real(polesvcb(i, :)), 'r') + end + plot(wsvc, zeros(size(wsvc)), 'k--') + hold off; + xlabel('Rotation Frequency [rad/s]'); + ylabel('Pole Real Part'); + + ax2 = subplot(1,2,2); + hold on; + for i = 1:8 + plot(wsvc, imag(polesvc(i, :)), 'b') + plot(wsvc, imag(polesvcb(i, :)), 'r') + end + hold off; + xlabel('Rotation Frequency [rad/s]'); + ylabel('Pole Imaginary Part'); +#+end_src + + *** Conclusion #+begin_important Even though considering one input and output at a time would results in an unstable system (figure [[fig:evolution_poles_u]]), when using the diagonal MIMO controller, the system stays stable (figure [[fig:poles_cl_system]]). This could be understood by saying that when controlling both directions at the same time, the coupling effect should be much lower than when controlling only one direction. @@ -1417,5 +2039,6 @@ We first look at the evolution of the singular values as a function of frequency * Control Implementation <> -* Bibliography :ignore: -# #+BIBLIOGRAPHY: /home/tdehaeze/MEGA/These/Ressources/references.bib plain option:-a option:-noabstract option:-nokeywords option:-noheader option:-nofooter option:-nobibsource limit:t +* Bibliography :ignore: +# bibliographystyle:unsrt +# bibliography:refs.bib diff --git a/refs.bib b/refs.bib new file mode 100644 index 0000000..e69de29 diff --git a/rotating_frame.slx b/rotating_frame.slx index 50e1bb8..6d553e9 100644 Binary files a/rotating_frame.slx and b/rotating_frame.slx differ