diff --git a/docs/amplified_piezoelectric_stack.html b/docs/amplified_piezoelectric_stack.html new file mode 100644 index 0000000..556ee47 --- /dev/null +++ b/docs/amplified_piezoelectric_stack.html @@ -0,0 +1,377 @@ + + + + + + +Amplified Piezoelectric Stack Actuator + + + + + + + + + + + + +
+ UP + | + HOME +
+

Amplified Piezoelectric Stack Actuator

+
+

Table of Contents

+
+ +
+
+ +

+The presented model is based on souleille18_concep_activ_mount_space_applic. +

+ +

+The model represents the amplified piezo APA100M from Cedrat-Technologies (Figure 1). +The parameters are shown in the table below. +

+ + +
+

souleille18_model_piezo.png +

+

Figure 1: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator

+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 1: Parameters used for the model of the APA 100M
 ValueMeaning
\(m\)\(1\,[kg]\)Payload mass
\(k_e\)\(4.8\,[N/\mu m]\)Stiffness used to adjust the pole of the isolator
\(k_1\)\(0.96\,[N/\mu m]\)Stiffness of the metallic suspension when the stack is removed
\(k_a\)\(65\,[N/\mu m]\)Stiffness of the actuator
\(c_1\)\(10\,[N/(m/s)]\)Added viscous damping
+ +
+

1 Simplified Model

+
+
+
+

1.1 Parameters

+
+
+
m = 1; % [kg]
+
+ke = 4.8e6; % [N/m]
+ce = 5; % [N/(m/s)]
+me = 0.001; % [kg]
+
+k1 = 0.96e6; % [N/m]
+c1 = 10; % [N/(m/s)]
+
+ka = 65e6; % [N/m]
+ca = 5; % [N/(m/s)]
+ma = 0.001; % [kg]
+
+h = 0.2; % [m]
+
+
+ +

+IFF Controller: +

+
+
Kiff = -8000/s;
+
+
+
+
+ +
+

1.2 Identification

+
+

+Identification in open-loop. +

+
+
%% Name of the Simulink File
+mdl = 'amplified_piezo_model';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/w'],  1, 'openinput');  io_i = io_i + 1; % Base Motion
+io(io_i) = linio([mdl, '/f'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1; % External Force
+
+io(io_i) = linio([mdl, '/Fs'], 3, 'openoutput'); io_i = io_i + 1; % Force Sensors
+io(io_i) = linio([mdl, '/x1'], 1, 'openoutput'); io_i = io_i + 1; % Mass displacement
+
+G = linearize(mdl, io, 0);
+G.InputName  = {'w', 'f', 'F'};
+G.OutputName = {'Fs', 'x1'};
+
+
+ +

+Identification in closed-loop. +

+
+
%% Name of the Simulink File
+mdl = 'amplified_piezo_model';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/w'],  1, 'input');  io_i = io_i + 1; % Base Motion
+io(io_i) = linio([mdl, '/f'],  1, 'input');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/F'],  1, 'input');  io_i = io_i + 1; % External Force
+
+io(io_i) = linio([mdl, '/Fs'], 3, 'output'); io_i = io_i + 1; % Force Sensors
+io(io_i) = linio([mdl, '/x1'], 1, 'output'); io_i = io_i + 1; % Mass displacement
+
+Giff = linearize(mdl, io, 0);
+Giff.InputName  = {'w', 'f', 'F'};
+Giff.OutputName = {'Fs', 'x1'};
+
+
+ + +
+

amplified_piezo_tf_ol_and_cl.png +

+

Figure 2: Matrix of transfer functions from input to output in open loop (blue) and closed loop (red)

+
+
+
+ +
+

1.3 Root Locus

+
+ +
+

amplified_piezo_root_locus.png +

+

Figure 3: Root Locus

+
+
+
+
+ +
+

2 Rotating X-Y platform

+
+
+
+

2.1 Parameters

+
+
+
m = 1; % [kg]
+
+ke = 4.8e6; % [N/m]
+ce = 5; % [N/(m/s)]
+me = 0.001; % [kg]
+
+k1 = 0.96e6; % [N/m]
+c1 = 10; % [N/(m/s)]
+
+ka = 65e6; % [N/m]
+ca = 5; % [N/(m/s)]
+ma = 0.001; % [kg]
+
+h = 0.2; % [m]
+
+
+ +
+
Kiff = tf(0);
+
+
+
+
+ +
+

2.2 Identification

+
+

+Rotating speed in rad/s: +

+
+
Ws = 2*pi*[0, 1, 10, 100];
+
+
+ +
+
Gs = {zeros(length(Ws), 1)};
+
+
+ +

+Identification in open-loop. +

+
+
%% 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
+
+
+ + +
+

amplitifed_piezo_xy_rotation_plant_iff.png +

+

Figure 4: Transfer function matrix from forces to force sensors for multiple rotation speed

+
+
+
+ +
+

2.3 Root Locus

+
+ +
+

amplified_piezo_xy_rotation_root_locus.png +

+

Figure 5: Root locus for 3 rotating speed

+
+
+
+ +
+

2.4 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. +

+
+
Ws = 2*pi*[140, 160];
+
+
+ +

+Identification +

+
+
%% 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
+
+
+ + +
+

amplified_piezo_xy_rotating_unstable_root_locus.png +

+

Figure 6: Root Locus for the two considered rotation speed. For the red curve, the system is unstable.

+
+
+
+
+
+
+

Author: Dehaeze Thomas

+

Created: 2020-05-20 mer. 15:49

+
+ + diff --git a/docs/figs/amplified_piezo_root_locus.pdf b/docs/figs/amplified_piezo_root_locus.pdf new file mode 100644 index 0000000..6298c98 Binary files /dev/null and b/docs/figs/amplified_piezo_root_locus.pdf differ diff --git a/docs/figs/amplified_piezo_root_locus.png b/docs/figs/amplified_piezo_root_locus.png new file mode 100644 index 0000000..9886bec Binary files /dev/null and b/docs/figs/amplified_piezo_root_locus.png differ diff --git a/docs/figs/amplified_piezo_tf_ol_and_cl.pdf b/docs/figs/amplified_piezo_tf_ol_and_cl.pdf new file mode 100644 index 0000000..b973524 --- /dev/null +++ b/docs/figs/amplified_piezo_tf_ol_and_cl.pdf @@ -0,0 +1,3065 @@ +%PDF-1.4 +% +1 0 obj +<< +/Producer (Apache FOP Version 2.4.0-SNAPSHOT: PDFDocumentGraphics2D) +/CreationDate (D:20200520154541+02'00') +>> +endobj +2 0 obj +<< +/Type /ExtGState +/CA 0.2509804 +>> +endobj +3 0 obj +<< +/Type /ExtGState +/CA 0.2509804 +>> +endobj +4 0 obj +<< +/Type /ExtGState +/CA 0.2509804 +>> +endobj +5 0 obj 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diff --git a/org/amplified_piezoelectric_stack.org b/org/amplified_piezoelectric_stack.org new file mode 100644 index 0000000..16c97a6 --- /dev/null +++ b/org/amplified_piezoelectric_stack.org @@ -0,0 +1,433 @@ +#+TITLE: Amplified Piezoelectric Stack Actuator +#+SETUPFILE: ./setup/org-setup-file.org + +* Introduction :ignore: +The presented model is based on cite:souleille18_concep_activ_mount_space_applic. + +The model represents the amplified piezo APA100M from Cedrat-Technologies (Figure [[fig:souleille18_model_piezo]]). +The parameters are shown in the table below. + +#+name: fig:souleille18_model_piezo +#+caption: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator +[[file:./figs/souleille18_model_piezo.png]] + +#+caption: Parameters used for the model of the APA 100M +| | Value | Meaning | +|-------+-------------------+----------------------------------------------------------------| +| $m$ | $1\,[kg]$ | Payload mass | +| $k_e$ | $4.8\,[N/\mu m]$ | Stiffness used to adjust the pole of the isolator | +| $k_1$ | $0.96\,[N/\mu m]$ | Stiffness of the metallic suspension when the stack is removed | +| $k_a$ | $65\,[N/\mu m]$ | Stiffness of the actuator | +| $c_1$ | $10\,[N/(m/s)]$ | Added viscous damping | + +* Simplified Model +** 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 + simulinkproject('../'); +#+END_SRC + +#+begin_src matlab + open 'amplified_piezo_model.slx' +#+end_src + +** Parameters +#+begin_src matlab + m = 1; % [kg] + + ke = 4.8e6; % [N/m] + ce = 5; % [N/(m/s)] + me = 0.001; % [kg] + + k1 = 0.96e6; % [N/m] + c1 = 10; % [N/(m/s)] + + ka = 65e6; % [N/m] + ca = 5; % [N/(m/s)] + ma = 0.001; % [kg] + + h = 0.2; % [m] +#+end_src + +IFF Controller: +#+begin_src matlab + Kiff = -8000/s; +#+end_src + +** Identification +Identification in open-loop. +#+begin_src matlab + %% Name of the Simulink File + mdl = 'amplified_piezo_model'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion + io(io_i) = linio([mdl, '/f'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs + io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % External Force + + io(io_i) = linio([mdl, '/Fs'], 3, 'openoutput'); io_i = io_i + 1; % Force Sensors + io(io_i) = linio([mdl, '/x1'], 1, 'openoutput'); io_i = io_i + 1; % Mass displacement + + G = linearize(mdl, io, 0); + G.InputName = {'w', 'f', 'F'}; + G.OutputName = {'Fs', 'x1'}; +#+end_src + +Identification in closed-loop. +#+begin_src matlab + %% Name of the Simulink File + mdl = 'amplified_piezo_model'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/w'], 1, 'input'); io_i = io_i + 1; % Base Motion + io(io_i) = linio([mdl, '/f'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs + io(io_i) = linio([mdl, '/F'], 1, 'input'); io_i = io_i + 1; % External Force + + io(io_i) = linio([mdl, '/Fs'], 3, 'output'); io_i = io_i + 1; % Force Sensors + io(io_i) = linio([mdl, '/x1'], 1, 'output'); io_i = io_i + 1; % Mass displacement + + Giff = linearize(mdl, io, 0); + Giff.InputName = {'w', 'f', 'F'}; + Giff.OutputName = {'Fs', 'x1'}; +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(1, 3, 1000); + + figure; + + ax1 = subplot(2, 3, 1); + title('$\displaystyle \frac{x_1}{w}$') + hold on; + plot(freqs, abs(squeeze(freqresp(G('x1', 'w'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Giff('x1', 'w'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]'); + + ax2 = subplot(2, 3, 2); + title('$\displaystyle \frac{x_1}{f}$') + hold on; + plot(freqs, abs(squeeze(freqresp(G('x1', 'f'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Giff('x1', 'f'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]'); + + ax3 = subplot(2, 3, 3); + title('$\displaystyle \frac{x_1}{F}$') + hold on; + plot(freqs, abs(squeeze(freqresp(G('x1', 'F'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Giff('x1', 'F'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]'); + + ax4 = subplot(2, 3, 4); + title('$\displaystyle \frac{F_s}{w}$') + hold on; + plot(freqs, abs(squeeze(freqresp(G('Fs', 'w'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'w'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]'); + + ax5 = subplot(2, 3, 5); + title('$\displaystyle \frac{F_s}{f}$') + hold on; + plot(freqs, abs(squeeze(freqresp(G('Fs', 'f'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'f'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]'); + + ax6 = subplot(2, 3, 6); + title('$\displaystyle \frac{F_s}{F}$') + hold on; + plot(freqs, abs(squeeze(freqresp(G('Fs', 'F'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'F'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + + linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/amplified_piezo_tf_ol_and_cl.pdf', 'width', 'full', 'height', 'full'); +#+end_src + +#+name: fig:amplified_piezo_tf_ol_and_cl +#+caption: Matrix of transfer functions from input to output in open loop (blue) and closed loop (red) +#+RESULTS: +[[file:figs/amplified_piezo_tf_ol_and_cl.png]] + +** Root Locus +#+begin_src matlab :exports none :post + figure; + + gains = logspace(1, 6, 500); + + hold on; + plot(real(pole(G('Fs', 'f'))), imag(pole(G('Fs', 'f'))), 'kx'); + plot(real(tzero(G('Fs', 'f'))), imag(tzero(G('Fs', 'f'))), 'ko'); + for k = 1:length(gains) + cl_poles = pole(feedback(G('Fs', 'f'), -gains(k)/s)); + plot(real(cl_poles), imag(cl_poles), 'k.'); + end + hold off; + axis square; + xlim([-2500, 100]); ylim([0, 2600]); + + xlabel('Real Part'); ylabel('Imaginary Part'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/amplified_piezo_root_locus.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:amplified_piezo_root_locus +#+caption: Root Locus +#+RESULTS: +[[file:figs/amplified_piezo_root_locus.png]] + +* Rotating X-Y platform +** 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 + simulinkproject('../'); +#+END_SRC + +#+begin_src matlab + open 'amplified_piezo_xy_rotating_stage.slx' +#+end_src + +** Parameters +#+begin_src matlab + m = 1; % [kg] + + ke = 4.8e6; % [N/m] + ce = 5; % [N/(m/s)] + me = 0.001; % [kg] + + k1 = 0.96e6; % [N/m] + c1 = 10; % [N/(m/s)] + + ka = 65e6; % [N/m] + ca = 5; % [N/(m/s)] + ma = 0.001; % [kg] + + h = 0.2; % [m] +#+end_src + +#+begin_src matlab + Kiff = tf(0); +#+end_src + +** Identification +Rotating speed in rad/s: +#+begin_src matlab + Ws = 2*pi*[0, 1, 10, 100]; +#+end_src + +#+begin_src matlab + Gs = {zeros(length(Ws), 1)}; +#+end_src + +Identification in open-loop. +#+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 + freqs = logspace(1, 3, 1000); + + figure; + + ax1 = subplot(2, 2, 1); + title('$\displaystyle \frac{F_{s,x}}{f_x}$') + hold on; + for i = 1:length(Ws) + plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsx', 'fx'), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]'); + + ax2 = subplot(2, 2, 2); + title('$\displaystyle \frac{F_{s,y}}{f_x}$') + hold on; + for i = 1:length(Ws) + plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsy', 'fx'), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]'); + + ax3 = subplot(2, 2, 3); + title('$\displaystyle \frac{F_{s,x}}{f_y}$') + hold on; + for i = 1:length(Ws) + plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsx', 'fy'), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]'); + + ax4 = subplot(2, 2, 4); + title('$\displaystyle \frac{F_{s,y}}{f_y}$') + hold on; + for i = 1:length(Ws) + plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsy', 'fy'), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]'); + + linkaxes([ax1,ax2,ax3,ax4],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/amplitifed_piezo_xy_rotation_plant_iff.pdf', 'width', 'full', 'height', 'full'); +#+end_src + +#+name: fig:amplitifed_piezo_xy_rotation_plant_iff +#+caption: Transfer function matrix from forces to force sensors for multiple rotation speed +#+RESULTS: +[[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]]