434 lines
13 KiB
Org Mode
434 lines
13 KiB
Org Mode
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#+TITLE: Amplified Piezoelectric Stack Actuator
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#+SETUPFILE: ./setup/org-setup-file.org
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* Introduction :ignore:
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The presented model is based on cite:souleille18_concep_activ_mount_space_applic.
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The model represents the amplified piezo APA100M from Cedrat-Technologies (Figure [[fig:souleille18_model_piezo]]).
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The parameters are shown in the table below.
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#+name: fig:souleille18_model_piezo
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#+caption: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator
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[[file:./figs/souleille18_model_piezo.png]]
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#+caption: Parameters used for the model of the APA 100M
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| | Value | Meaning |
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|-------+-------------------+----------------------------------------------------------------|
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| $m$ | $1\,[kg]$ | Payload mass |
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| $k_e$ | $4.8\,[N/\mu m]$ | Stiffness used to adjust the pole of the isolator |
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| $k_1$ | $0.96\,[N/\mu m]$ | Stiffness of the metallic suspension when the stack is removed |
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| $k_a$ | $65\,[N/\mu m]$ | Stiffness of the actuator |
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| $c_1$ | $10\,[N/(m/s)]$ | Added viscous damping |
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* Simplified Model
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+BEGIN_SRC matlab
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simulinkproject('../');
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#+END_SRC
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#+begin_src matlab
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open 'amplified_piezo_model.slx'
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#+end_src
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** Parameters
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#+begin_src matlab
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m = 1; % [kg]
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ke = 4.8e6; % [N/m]
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ce = 5; % [N/(m/s)]
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me = 0.001; % [kg]
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k1 = 0.96e6; % [N/m]
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c1 = 10; % [N/(m/s)]
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ka = 65e6; % [N/m]
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ca = 5; % [N/(m/s)]
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ma = 0.001; % [kg]
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h = 0.2; % [m]
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#+end_src
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IFF Controller:
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#+begin_src matlab
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Kiff = -8000/s;
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#+end_src
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** Identification
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Identification in open-loop.
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#+begin_src matlab
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%% Name of the Simulink File
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mdl = 'amplified_piezo_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion
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io(io_i) = linio([mdl, '/f'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
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io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % External Force
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io(io_i) = linio([mdl, '/Fs'], 3, 'openoutput'); io_i = io_i + 1; % Force Sensors
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io(io_i) = linio([mdl, '/x1'], 1, 'openoutput'); io_i = io_i + 1; % Mass displacement
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G = linearize(mdl, io, 0);
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G.InputName = {'w', 'f', 'F'};
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G.OutputName = {'Fs', 'x1'};
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#+end_src
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Identification in closed-loop.
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#+begin_src matlab
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%% Name of the Simulink File
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mdl = 'amplified_piezo_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/w'], 1, 'input'); io_i = io_i + 1; % Base Motion
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io(io_i) = linio([mdl, '/f'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs
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io(io_i) = linio([mdl, '/F'], 1, 'input'); io_i = io_i + 1; % External Force
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io(io_i) = linio([mdl, '/Fs'], 3, 'output'); io_i = io_i + 1; % Force Sensors
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io(io_i) = linio([mdl, '/x1'], 1, 'output'); io_i = io_i + 1; % Mass displacement
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Giff = linearize(mdl, io, 0);
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Giff.InputName = {'w', 'f', 'F'};
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Giff.OutputName = {'Fs', 'x1'};
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#+end_src
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#+begin_src matlab :exports none
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freqs = logspace(1, 3, 1000);
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figure;
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ax1 = subplot(2, 3, 1);
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title('$\displaystyle \frac{x_1}{w}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('x1', 'w'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Giff('x1', 'w'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
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ax2 = subplot(2, 3, 2);
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title('$\displaystyle \frac{x_1}{f}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('x1', 'f'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Giff('x1', 'f'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
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ax3 = subplot(2, 3, 3);
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title('$\displaystyle \frac{x_1}{F}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('x1', 'F'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Giff('x1', 'F'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
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ax4 = subplot(2, 3, 4);
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title('$\displaystyle \frac{F_s}{w}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('Fs', 'w'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'w'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
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ax5 = subplot(2, 3, 5);
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title('$\displaystyle \frac{F_s}{f}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('Fs', 'f'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'f'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
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ax6 = subplot(2, 3, 6);
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title('$\displaystyle \frac{F_s}{F}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('Fs', 'F'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'F'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
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linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/amplified_piezo_tf_ol_and_cl.pdf', 'width', 'full', 'height', 'full');
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#+end_src
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#+name: fig:amplified_piezo_tf_ol_and_cl
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#+caption: Matrix of transfer functions from input to output in open loop (blue) and closed loop (red)
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#+RESULTS:
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[[file:figs/amplified_piezo_tf_ol_and_cl.png]]
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** Root Locus
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#+begin_src matlab :exports none :post
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figure;
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gains = logspace(1, 6, 500);
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hold on;
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plot(real(pole(G('Fs', 'f'))), imag(pole(G('Fs', 'f'))), 'kx');
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plot(real(tzero(G('Fs', 'f'))), imag(tzero(G('Fs', 'f'))), 'ko');
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for k = 1:length(gains)
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cl_poles = pole(feedback(G('Fs', 'f'), -gains(k)/s));
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plot(real(cl_poles), imag(cl_poles), 'k.');
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end
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hold off;
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axis square;
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xlim([-2500, 100]); ylim([0, 2600]);
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xlabel('Real Part'); ylabel('Imaginary Part');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/amplified_piezo_root_locus.pdf', 'width', 'wide', 'height', 'tall');
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#+end_src
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#+name: fig:amplified_piezo_root_locus
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#+caption: Root Locus
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#+RESULTS:
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[[file:figs/amplified_piezo_root_locus.png]]
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* Rotating X-Y platform
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+BEGIN_SRC matlab
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simulinkproject('../');
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#+END_SRC
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#+begin_src matlab
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open 'amplified_piezo_xy_rotating_stage.slx'
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#+end_src
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** Parameters
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#+begin_src matlab
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m = 1; % [kg]
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ke = 4.8e6; % [N/m]
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ce = 5; % [N/(m/s)]
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me = 0.001; % [kg]
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k1 = 0.96e6; % [N/m]
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c1 = 10; % [N/(m/s)]
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ka = 65e6; % [N/m]
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ca = 5; % [N/(m/s)]
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ma = 0.001; % [kg]
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h = 0.2; % [m]
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#+end_src
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#+begin_src matlab
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Kiff = tf(0);
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#+end_src
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** Identification
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Rotating speed in rad/s:
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#+begin_src matlab
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Ws = 2*pi*[0, 1, 10, 100];
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#+end_src
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#+begin_src matlab
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Gs = {zeros(length(Ws), 1)};
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#+end_src
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Identification in open-loop.
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#+begin_src matlab
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%% Name of the Simulink File
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mdl = 'amplified_piezo_xy_rotating_stage';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/fx'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/fy'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Fs'], 2, 'openoutput'); io_i = io_i + 1;
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for i = 1:length(Ws)
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ws = Ws(i);
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G = linearize(mdl, io, 0);
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G.InputName = {'fx', 'fy'};
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G.OutputName = {'Fsx', 'Fsy'};
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Gs(i) = {G};
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end
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#+end_src
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#+begin_src matlab :exports none
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freqs = logspace(1, 3, 1000);
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figure;
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ax1 = subplot(2, 2, 1);
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title('$\displaystyle \frac{F_{s,x}}{f_x}$')
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hold on;
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for i = 1:length(Ws)
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plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsx', 'fx'), freqs, 'Hz'))));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
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ax2 = subplot(2, 2, 2);
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title('$\displaystyle \frac{F_{s,y}}{f_x}$')
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hold on;
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for i = 1:length(Ws)
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plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsy', 'fx'), freqs, 'Hz'))));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
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ax3 = subplot(2, 2, 3);
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title('$\displaystyle \frac{F_{s,x}}{f_y}$')
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hold on;
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for i = 1:length(Ws)
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plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsx', 'fy'), freqs, 'Hz'))));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
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ax4 = subplot(2, 2, 4);
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title('$\displaystyle \frac{F_{s,y}}{f_y}$')
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hold on;
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for i = 1:length(Ws)
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plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsy', 'fy'), freqs, 'Hz'))));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
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linkaxes([ax1,ax2,ax3,ax4],'x');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/amplitifed_piezo_xy_rotation_plant_iff.pdf', 'width', 'full', 'height', 'full');
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#+end_src
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#+name: fig:amplitifed_piezo_xy_rotation_plant_iff
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#+caption: Transfer function matrix from forces to force sensors for multiple rotation speed
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#+RESULTS:
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[[file:figs/amplitifed_piezo_xy_rotation_plant_iff.png]]
|
||
|
|
|
||
|
|
** Root Locus
|
||
|
|
#+begin_src matlab :exports none :post
|
||
|
|
figure;
|
||
|
|
|
||
|
|
gains = logspace(1, 6, 500);
|
||
|
|
|
||
|
|
hold on;
|
||
|
|
for i = 1:length(Ws)
|
||
|
|
set(gca,'ColorOrderIndex',i);
|
||
|
|
plot(real(pole(Gs{i})), imag(pole(Gs{i})), 'x');
|
||
|
|
set(gca,'ColorOrderIndex',i);
|
||
|
|
plot(real(tzero(Gs{i})), imag(tzero(Gs{i})), 'o');
|
||
|
|
for k = 1:length(gains)
|
||
|
|
set(gca,'ColorOrderIndex',i);
|
||
|
|
cl_poles = pole(feedback(Gs{i}, -gains(k)/s*eye(2)));
|
||
|
|
plot(real(cl_poles), imag(cl_poles), '.');
|
||
|
|
end
|
||
|
|
end
|
||
|
|
hold off;
|
||
|
|
axis square;
|
||
|
|
xlim([-2900, 100]); ylim([0, 3000]);
|
||
|
|
|
||
|
|
xlabel('Real Part'); ylabel('Imaginary Part');
|
||
|
|
#+end_src
|
||
|
|
|
||
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
||
|
|
exportFig('figs/amplified_piezo_xy_rotation_root_locus.pdf', 'width', 'tall', 'height', 'wide');
|
||
|
|
#+end_src
|
||
|
|
|
||
|
|
#+name: fig:amplified_piezo_xy_rotation_root_locus
|
||
|
|
#+caption: Root locus for 3 rotating speed
|
||
|
|
#+RESULTS:
|
||
|
|
[[file:figs/amplified_piezo_xy_rotation_root_locus.png]]
|
||
|
|
|
||
|
|
** Analysis
|
||
|
|
The negative stiffness induced by the rotation is equal to $m \omega_0^2$.
|
||
|
|
Thus, the maximum rotation speed where IFF can be applied is:
|
||
|
|
\[ \omega_\text{max} = \sqrt{\frac{k_1}{m}} \approx 156\,[Hz] \]
|
||
|
|
|
||
|
|
Let's verify that.
|
||
|
|
#+begin_src matlab
|
||
|
|
Ws = 2*pi*[140, 160];
|
||
|
|
#+end_src
|
||
|
|
|
||
|
|
#+begin_src matlab :exports none
|
||
|
|
Gs = {zeros(length(Ws), 1)};
|
||
|
|
#+end_src
|
||
|
|
|
||
|
|
Identification
|
||
|
|
#+begin_src matlab
|
||
|
|
%% Name of the Simulink File
|
||
|
|
mdl = 'amplified_piezo_xy_rotating_stage';
|
||
|
|
|
||
|
|
%% Input/Output definition
|
||
|
|
clear io; io_i = 1;
|
||
|
|
io(io_i) = linio([mdl, '/fx'], 1, 'openinput'); io_i = io_i + 1;
|
||
|
|
io(io_i) = linio([mdl, '/fy'], 1, 'openinput'); io_i = io_i + 1;
|
||
|
|
|
||
|
|
io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
|
||
|
|
io(io_i) = linio([mdl, '/Fs'], 2, 'openoutput'); io_i = io_i + 1;
|
||
|
|
|
||
|
|
for i = 1:length(Ws)
|
||
|
|
ws = Ws(i);
|
||
|
|
G = linearize(mdl, io, 0);
|
||
|
|
G.InputName = {'fx', 'fy'};
|
||
|
|
G.OutputName = {'Fsx', 'Fsy'};
|
||
|
|
Gs(i) = {G};
|
||
|
|
end
|
||
|
|
#+end_src
|
||
|
|
|
||
|
|
#+begin_src matlab :exports none
|
||
|
|
figure;
|
||
|
|
|
||
|
|
gains = logspace(1, 6, 500);
|
||
|
|
|
||
|
|
hold on;
|
||
|
|
for i = 1:length(Ws)
|
||
|
|
set(gca,'ColorOrderIndex',i);
|
||
|
|
plot(real(pole(Gs{i})), imag(pole(Gs{i})), 'x');
|
||
|
|
set(gca,'ColorOrderIndex',i);
|
||
|
|
plot(real(tzero(Gs{i})), imag(tzero(Gs{i})), 'o');
|
||
|
|
for k = 1:length(gains)
|
||
|
|
set(gca,'ColorOrderIndex',i);
|
||
|
|
cl_poles = pole(feedback(Gs{i}, -gains(k)/s*eye(2)));
|
||
|
|
plot(real(cl_poles), imag(cl_poles), '.');
|
||
|
|
end
|
||
|
|
end
|
||
|
|
hold off;
|
||
|
|
axis square;
|
||
|
|
xlim([-100, 50]); ylim([0, 150]);
|
||
|
|
|
||
|
|
xlabel('Real Part'); ylabel('Imaginary Part');
|
||
|
|
#+end_src
|
||
|
|
|
||
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
||
|
|
exportFig('figs/amplified_piezo_xy_rotating_unstable_root_locus.pdf', 'width', 'wide', 'height', 'tall');
|
||
|
|
#+end_src
|
||
|
|
|
||
|
|
#+name: fig:amplified_piezo_xy_rotating_unstable_root_locus
|
||
|
|
#+caption: Root Locus for the two considered rotation speed. For the red curve, the system is unstable.
|
||
|
|
#+RESULTS:
|
||
|
|
[[file:figs/amplified_piezo_xy_rotating_unstable_root_locus.png]]
|