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@ -1,15 +0,0 @@
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function [A] = align(V)
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%A!ALIGN(V) returns a constat matrix A which is the real alignment of the
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%INVERSE of the complex input matrix V
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%from Mohit slides
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if (nargin ==0) || (nargin > 1)
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disp('usage: mat_inv_real = align(mat)')
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return
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end
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D = pinv(real(V'*V));
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A = D*real(V'*diag(exp(1i * angle(diag(V*D*V.'))/2)));
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end
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@ -1,34 +0,0 @@
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function [] = pzmap_testCL(system,H,gain,feedin,feedout)
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% evaluate and plot the pole-zero map for the closed loop system for
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% different values of the gain
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[~, n] = size(gain);
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[m1, n1, ~] = size(H);
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[~,n2] = size(feedin);
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figure
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for i = 1:n
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% if n1 == n2
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system_CL = feedback(system,gain(i)*H,feedin,feedout);
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[P,Z] = pzmap(system_CL);
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plot(real(P(:)),imag(P(:)),'x',real(Z(:)),imag(Z(:)),'o');hold on
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xlabel('Real axis (s^{-1})');ylabel('Imaginary Axis (s^{-1})');
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% clear P Z
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% else
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% system_CL = feedback(system,gain(i)*H(:,1+(i-1)*m1:m1+(i-1)*m1),feedin,feedout);
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%
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% [P,Z] = pzmap(system_CL);
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% plot(real(P(:)),imag(P(:)),'x',real(Z(:)),imag(Z(:)),'o');hold on
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% xlabel('Real axis (s^{-1})');ylabel('Imaginary Axis (s^{-1})');
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% clear P Z
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% end
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end
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str = {strcat('gain = ' , num2str(gain(1)))}; % at the end of first loop, z being loop output
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str = [str , strcat('gain = ' , num2str(gain(1)))]; % after 2nd loop
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for i = 2:n
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str = [str , strcat('gain = ' , num2str(gain(i)))]; % after 2nd loop
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str = [str , strcat('gain = ' , num2str(gain(i)))]; % after 2nd loop
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end
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legend(str{:})
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end
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@ -1,492 +0,0 @@
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%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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addpath('gravimeter');
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% Simscape Model - Parameters
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open('gravimeter.slx')
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% Parameters
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l = 0.5; % Length of the mass [m]
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la = 0.5; % Position of Act. [m]
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h = 1.7; % Height of the mass [m]
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ha = 1.7; % Position of Act. [m]
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m = 400; % Mass [kg]
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I = 115; % Inertia [kg m^2]
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k = 15e3; % Actuator Stiffness [N/m]
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c = 0.03; % Actuator Damping [N/(m/s)]
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deq = 0.2; % Length of the actuators [m]
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g = 0; % Gravity [m/s2]
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% System Identification - Without Gravity
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%% Name of the Simulink File
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mdl = 'gravimeter';
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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, '/F1'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
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G = linearize(mdl, io);
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G.InputName = {'F1', 'F2', 'F3'};
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G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
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pole(G)
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% #+RESULTS:
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% #+begin_example
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% pole(G)
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% ans =
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% -0.000473481142385801 + 21.7596190728632i
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% -0.000473481142385801 - 21.7596190728632i
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% -7.49842879459177e-05 + 8.6593576906982i
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% -7.49842879459177e-05 - 8.6593576906982i
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% -5.15386867925747e-06 + 2.27025295182755i
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% -5.15386867925747e-06 - 2.27025295182755i
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% #+end_example
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% The plant as 6 states as expected (2 translations + 1 rotation)
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size(G)
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% #+RESULTS:
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% : State-space model with 4 outputs, 3 inputs, and 6 states.
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freqs = logspace(-2, 2, 1000);
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figure;
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for in_i = 1:3
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for out_i = 1:4
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subplot(4, 3, 3*(out_i-1)+in_i);
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plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), '-');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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end
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end
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% System Identification - With Gravity
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g = 9.80665; % Gravity [m/s2]
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Gg = linearize(mdl, io);
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Gg.InputName = {'F1', 'F2', 'F3'};
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Gg.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
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% We can now see that the system is unstable due to gravity.
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pole(Gg)
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% #+RESULTS:
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% #+begin_example
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% pole(G)
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% ans =
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% -10.9848275341276 + 0i
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% 10.9838836405193 + 0i
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% -7.49855396089326e-05 + 8.65962885769976i
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% -7.49855396089326e-05 - 8.65962885769976i
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% -6.68819341967921e-06 + 0.83296042226902i
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% -6.68819341967921e-06 - 0.83296042226902i
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% #+end_example
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freqs = logspace(-2, 2, 1000);
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figure;
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for in_i = 1:3
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for out_i = 1:4
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subplot(4, 3, 3*(out_i-1)+in_i);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), '-');
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plot(freqs, abs(squeeze(freqresp(Gg(out_i,in_i), freqs, 'Hz'))), '-');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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end
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end
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% Parameters
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% Control parameters
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g = 1e5;
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g_svd = 1e5;
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% System parameters
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w0 = 2*pi*.5; % MinusK BM1 tablle
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l = 0.8; % [m]
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la = l; % [m]
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h = 1.7; % [m]
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ha = h; % [m]
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m = 70; % [kg]
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k = 3e3; % [N/m]
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I = 10; % [kg m^2]
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% Bode options.
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P = bodeoptions;
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P.FreqUnits = 'Hz';
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P.MagUnits = 'abs';
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P.MagScale = 'log';
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P.Grid = 'on';
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P.PhaseWrapping = 'on';
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P.Title.FontSize = 14;
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P.XLabel.FontSize = 14;
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P.YLabel.FontSize = 14;
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P.TickLabel.FontSize = 12;
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P.Xlim = [1e-1,1e2];
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P.MagLowerLimMode = 'manual';
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P.MagLowerLim= 1e-3;
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%P.PhaseVisible = 'off';
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% Frequency vector.
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w = 2*pi*logspace(-1,2,1000); % [rad/s]
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% generation of the state space model
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M = [m 0 0
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0 m 0
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0 0 I];
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%Jacobian of the bottom sensor
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Js1 = [1 0 h/2
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0 1 -l/2];
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%Jacobian of the top sensor
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Js2 = [1 0 -h/2
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0 1 0];
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%Jacobian of the actuators
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Ja = [1 0 ha/2 %Left horizontal actuator
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%1 0 h/2 %Right horizontal actuator
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0 1 -la/2 %Left vertical actuator
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0 1 la/2]; %Right vertical actuator
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Jta = Ja';
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K = k*Jta*Ja;
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C = 0.06*k*Jta*Ja;
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E = [1 0 0
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0 1 0
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0 0 1]; %projecting ground motion in the directions of the legs
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AA = [zeros(3) eye(3)
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-M\K -M\C];
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BB = [zeros(3,6)
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M\Jta M\(k*Jta*E)];
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% BB = [zeros(3,3)
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% M\Jta ];
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%
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% CC = [Ja zeros(3)];
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% DD = zeros(3,3);
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CC = [[Js1;Js2] zeros(4,3);
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zeros(2,6)
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(Js1+Js2)./2 zeros(2,3)
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(Js1-Js2)./2 zeros(2,3)
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(Js1-Js2)./(2*h) zeros(2,3)];
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DD = [zeros(4,6)
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zeros(2,3) eye(2,3)
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zeros(6,6)];
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system_dec = ss(AA,BB,CC,DD);
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% - Input = three actuators and three ground motions
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% - Output = the bottom sensor; the top sensor; the ground motion; the half sum; the half difference; the rotation
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size(system_dec)
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% Comparison with the Simscape Model
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freqs = logspace(-2, 2, 1000);
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figure;
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for in_i = 1:3
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for out_i = 1:4
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subplot(4, 3, 3*(out_i-1)+in_i);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), '-');
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plot(freqs, abs(squeeze(freqresp(system_dec(out_i,in_i)*s^2, freqs, 'Hz'))), '-');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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end
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end
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% Analysis
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% figure
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% bode(system_dec,P);
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% return
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%% svd decomposition
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% system_dec_freq = freqresp(system_dec,w);
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% S = zeros(3,length(w));
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% for m = 1:length(w)
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% S(:,m) = svd(system_dec_freq(1:4,1:3,m));
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% end
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% figure
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% loglog(w./(2*pi), S);hold on;
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% % loglog(w./(2*pi), abs(Val(1,:)),w./(2*pi), abs(Val(2,:)),w./(2*pi), abs(Val(3,:)));
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% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');
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% legend('\sigma_1','\sigma_2','\sigma_3');%,'\sigma_4','\sigma_5','\sigma_6');
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% ylim([1e-8 1e-2]);
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%
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% %condition number
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% figure
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% loglog(w./(2*pi), S(1,:)./S(3,:));hold on;
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% % loglog(w./(2*pi), abs(Val(1,:)),w./(2*pi), abs(Val(2,:)),w./(2*pi), abs(Val(3,:)));
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% xlabel('Frequency [Hz]');ylabel('Condition number [-]');
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% % legend('\sigma_1','\sigma_2','\sigma_3');%,'\sigma_4','\sigma_5','\sigma_6');
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%
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% %performance indicator
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% system_dec_svd = freqresp(system_dec(1:4,1:3),2*pi*10);
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% [U,S,V] = svd(system_dec_svd);
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% H_svd_OL = -eye(3,4);%-[zpk(-2*pi*10,-2*pi*40,40/10) 0 0 0; 0 10*zpk(-2*pi*40,-2*pi*200,40/200) 0 0; 0 0 zpk(-2*pi*2,-2*pi*10,10/2) 0];% - eye(3,4);%
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% H_svd = pinv(V')*H_svd_OL*pinv(U);
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% % system_dec_control_svd_ = feedback(system_dec,g*pinv(V')*H*pinv(U));
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%
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% OL_dec = g_svd*H_svd*system_dec(1:4,1:3);
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% OL_freq = freqresp(OL_dec,w); % OL = G*H
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% CL_system = feedback(eye(3),-g_svd*H_svd*system_dec(1:4,1:3));
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% CL_freq = freqresp(CL_system,w); % CL = (1+G*H)^-1
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% % CL_system_2 = feedback(system_dec,H);
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% % CL_freq_2 = freqresp(CL_system_2,w); % CL = G/(1+G*H)
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% for i = 1:size(w,2)
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% OL(:,i) = svd(OL_freq(:,:,i));
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% CL (:,i) = svd(CL_freq(:,:,i));
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% %CL2 (:,i) = svd(CL_freq_2(:,:,i));
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% end
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%
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% un = ones(1,length(w));
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% figure
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% loglog(w./(2*pi),OL(3,:)+1,'k',w./(2*pi),OL(3,:)-1,'b',w./(2*pi),1./CL(1,:),'r--',w./(2*pi),un,'k:');hold on;%
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% % loglog(w./(2*pi), 1./(CL(2,:)),w./(2*pi), 1./(CL(3,:)));
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% % semilogx(w./(2*pi), 1./(CL2(1,:)),w./(2*pi), 1./(CL2(2,:)),w./(2*pi), 1./(CL2(3,:)));
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% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');
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% legend('GH \sigma_{inf} +1 ','GH \sigma_{inf} -1','S 1/\sigma_{sup}');%,'\lambda_1','\lambda_2','\lambda_3');
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%
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% figure
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% loglog(w./(2*pi),OL(1,:)+1,'k',w./(2*pi),OL(1,:)-1,'b',w./(2*pi),1./CL(3,:),'r--',w./(2*pi),un,'k:');hold on;%
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% % loglog(w./(2*pi), 1./(CL(2,:)),w./(2*pi), 1./(CL(3,:)));
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% % semilogx(w./(2*pi), 1./(CL2(1,:)),w./(2*pi), 1./(CL2(2,:)),w./(2*pi), 1./(CL2(3,:)));
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% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');
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% legend('GH \sigma_{sup} +1 ','GH \sigma_{sup} -1','S 1/\sigma_{inf}');%,'\lambda_1','\lambda_2','\lambda_3');
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% Control Section
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system_dec_10Hz = freqresp(system_dec,2*pi*10);
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system_dec_0Hz = freqresp(system_dec,0);
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system_decReal_10Hz = pinv(align(system_dec_10Hz));
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[Ureal,Sreal,Vreal] = svd(system_decReal_10Hz(1:4,1:3));
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normalizationMatrixReal = abs(pinv(Ureal)*system_dec_0Hz(1:4,1:3)*pinv(Vreal'));
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[U,S,V] = svd(system_dec_10Hz(1:4,1:3));
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normalizationMatrix = abs(pinv(U)*system_dec_0Hz(1:4,1:3)*pinv(V'));
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H_dec = ([zpk(-2*pi*5,-2*pi*30,30/5) 0 0 0
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0 zpk(-2*pi*4,-2*pi*20,20/4) 0 0
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0 0 0 zpk(-2*pi,-2*pi*10,10)]);
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H_cen_OL = [zpk(-2*pi,-2*pi*10,10) 0 0; 0 zpk(-2*pi,-2*pi*10,10) 0;
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0 0 zpk(-2*pi*5,-2*pi*30,30/5)];
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H_cen = pinv(Jta)*H_cen_OL*pinv([Js1; Js2]);
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% H_svd_OL = -[1/normalizationMatrix(1,1) 0 0 0
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% 0 1/normalizationMatrix(2,2) 0 0
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% 0 0 1/normalizationMatrix(3,3) 0];
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% H_svd_OL_real = -[1/normalizationMatrixReal(1,1) 0 0 0
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% 0 1/normalizationMatrixReal(2,2) 0 0
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% 0 0 1/normalizationMatrixReal(3,3) 0];
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H_svd_OL = -[1/normalizationMatrix(1,1)*zpk(-2*pi*10,-2*pi*60,60/10) 0 0 0
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0 1/normalizationMatrix(2,2)*zpk(-2*pi*5,-2*pi*30,30/5) 0 0
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0 0 1/normalizationMatrix(3,3)*zpk(-2*pi*2,-2*pi*10,10/2) 0];
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H_svd_OL_real = -[1/normalizationMatrixReal(1,1)*zpk(-2*pi*10,-2*pi*60,60/10) 0 0 0
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0 1/normalizationMatrixReal(2,2)*zpk(-2*pi*5,-2*pi*30,30/5) 0 0
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0 0 1/normalizationMatrixReal(3,3)*zpk(-2*pi*2,-2*pi*10,10/2) 0];
|
||||
% H_svd_OL_real = -[zpk(-2*pi*10,-2*pi*40,40/10) 0 0 0; 0 10*zpk(-2*pi*10,-2*pi*100,100/10) 0 0; 0 0 zpk(-2*pi*2,-2*pi*10,10/2) 0];%-eye(3,4);
|
||||
% H_svd_OL = -[zpk(-2*pi*10,-2*pi*40,40/10) 0 0 0; 0 zpk(-2*pi*4,-2*pi*20,4/20) 0 0; 0 0 zpk(-2*pi*2,-2*pi*10,10/2) 0];% - eye(3,4);%
|
||||
H_svd = pinv(V')*H_svd_OL*pinv(U);
|
||||
H_svd_real = pinv(Vreal')*H_svd_OL_real*pinv(Ureal);
|
||||
|
||||
OL_dec = g*H_dec*system_dec(1:4,1:3);
|
||||
OL_cen = g*H_cen_OL*pinv([Js1; Js2])*system_dec(1:4,1:3)*pinv(Jta);
|
||||
OL_svd = 100*H_svd_OL*pinv(U)*system_dec(1:4,1:3)*pinv(V');
|
||||
OL_svd_real = 100*H_svd_OL_real*pinv(Ureal)*system_dec(1:4,1:3)*pinv(Vreal');
|
||||
|
||||
% figure
|
||||
% bode(OL_dec,w,P);title('OL Decentralized');
|
||||
% figure
|
||||
% bode(OL_cen,w,P);title('OL Centralized');
|
||||
|
||||
figure
|
||||
bode(g*system_dec(1:4,1:3),w,P);
|
||||
title('gain * Plant');
|
||||
|
||||
figure
|
||||
bode(OL_svd,OL_svd_real,w,P);
|
||||
title('OL SVD');
|
||||
legend('SVD of Complex plant','SVD of real approximation of the complex plant')
|
||||
|
||||
figure
|
||||
bode(system_dec(1:4,1:3),pinv(U)*system_dec(1:4,1:3)*pinv(V'),P);
|
||||
|
||||
CL_dec = feedback(system_dec,g*H_dec,[1 2 3],[1 2 3 4]);
|
||||
CL_cen = feedback(system_dec,g*H_cen,[1 2 3],[1 2 3 4]);
|
||||
CL_svd = feedback(system_dec,100*H_svd,[1 2 3],[1 2 3 4]);
|
||||
CL_svd_real = feedback(system_dec,100*H_svd_real,[1 2 3],[1 2 3 4]);
|
||||
|
||||
pzmap_testCL(system_dec,H_dec,g,[1 2 3],[1 2 3 4])
|
||||
title('Decentralized control');
|
||||
|
||||
pzmap_testCL(system_dec,H_cen,g,[1 2 3],[1 2 3 4])
|
||||
title('Centralized control');
|
||||
|
||||
pzmap_testCL(system_dec,H_svd,100,[1 2 3],[1 2 3 4])
|
||||
title('SVD control');
|
||||
|
||||
pzmap_testCL(system_dec,H_svd_real,100,[1 2 3],[1 2 3 4])
|
||||
title('Real approximation SVD control');
|
||||
|
||||
P.Ylim = [1e-8 1e-3];
|
||||
figure
|
||||
bodemag(system_dec(1:4,1:3),CL_dec(1:4,1:3),CL_cen(1:4,1:3),CL_svd(1:4,1:3),CL_svd_real(1:4,1:3),P);
|
||||
title('Motion/actuator')
|
||||
legend('Control OFF','Decentralized control','Centralized control','SVD control','SVD control real appr.');
|
||||
|
||||
P.Ylim = [1e-5 1e1];
|
||||
figure
|
||||
bodemag(system_dec(1:4,4:6),CL_dec(1:4,4:6),CL_cen(1:4,4:6),CL_svd(1:4,4:6),CL_svd_real(1:4,4:6),P);
|
||||
title('Transmissibility');
|
||||
legend('Control OFF','Decentralized control','Centralized control','SVD control','SVD control real appr.');
|
||||
|
||||
figure
|
||||
bodemag(system_dec([7 9],4:6),CL_dec([7 9],4:6),CL_cen([7 9],4:6),CL_svd([7 9],4:6),CL_svd_real([7 9],4:6),P);
|
||||
title('Transmissibility from half sum and half difference in the X direction');
|
||||
legend('Control OFF','Decentralized control','Centralized control','SVD control','SVD control real appr.');
|
||||
|
||||
figure
|
||||
bodemag(system_dec([8 10],4:6),CL_dec([8 10],4:6),CL_cen([8 10],4:6),CL_svd([8 10],4:6),CL_svd_real([8 10],4:6),P);
|
||||
title('Transmissibility from half sum and half difference in the Z direction');
|
||||
legend('Control OFF','Decentralized control','Centralized control','SVD control','SVD control real appr.');
|
||||
|
||||
% Greshgorin radius
|
||||
|
||||
system_dec_freq = freqresp(system_dec,w);
|
||||
x1 = zeros(1,length(w));
|
||||
z1 = zeros(1,length(w));
|
||||
x2 = zeros(1,length(w));
|
||||
S1 = zeros(1,length(w));
|
||||
S2 = zeros(1,length(w));
|
||||
S3 = zeros(1,length(w));
|
||||
|
||||
for t = 1:length(w)
|
||||
x1(t) = (abs(system_dec_freq(1,2,t))+abs(system_dec_freq(1,3,t)))/abs(system_dec_freq(1,1,t));
|
||||
z1(t) = (abs(system_dec_freq(2,1,t))+abs(system_dec_freq(2,3,t)))/abs(system_dec_freq(2,2,t));
|
||||
x2(t) = (abs(system_dec_freq(3,1,t))+abs(system_dec_freq(3,2,t)))/abs(system_dec_freq(3,3,t));
|
||||
system_svd = pinv(Ureal)*system_dec_freq(1:4,1:3,t)*pinv(Vreal');
|
||||
S1(t) = (abs(system_svd(1,2))+abs(system_svd(1,3)))/abs(system_svd(1,1));
|
||||
S2(t) = (abs(system_svd(2,1))+abs(system_svd(2,3)))/abs(system_svd(2,2));
|
||||
S2(t) = (abs(system_svd(3,1))+abs(system_svd(3,2)))/abs(system_svd(3,3));
|
||||
end
|
||||
|
||||
limit = 0.5*ones(1,length(w));
|
||||
|
||||
figure
|
||||
loglog(w./(2*pi),x1,w./(2*pi),z1,w./(2*pi),x2,w./(2*pi),limit,'--');
|
||||
legend('x_1','z_1','x_2','Limit');
|
||||
xlabel('Frequency [Hz]');
|
||||
ylabel('Greshgorin radius [-]');
|
||||
|
||||
figure
|
||||
loglog(w./(2*pi),S1,w./(2*pi),S2,w./(2*pi),S3,w./(2*pi),limit,'--');
|
||||
legend('S1','S2','S3','Limit');
|
||||
xlabel('Frequency [Hz]');
|
||||
ylabel('Greshgorin radius [-]');
|
||||
% set(gcf,'color','w')
|
||||
|
||||
% Injecting ground motion in the system to have the output
|
||||
|
||||
Fr = logspace(-2,3,1e3);
|
||||
w=2*pi*Fr*1i;
|
||||
%fit of the ground motion data in m/s^2/rtHz
|
||||
Fr_ground_x = [0.07 0.1 0.15 0.3 0.7 0.8 0.9 1.2 5 10];
|
||||
n_ground_x1 = [4e-7 4e-7 2e-6 1e-6 5e-7 5e-7 5e-7 1e-6 1e-5 3.5e-5];
|
||||
Fr_ground_v = [0.07 0.08 0.1 0.11 0.12 0.15 0.25 0.6 0.8 1 1.2 1.6 2 6 10];
|
||||
n_ground_v1 = [7e-7 7e-7 7e-7 1e-6 1.2e-6 1.5e-6 1e-6 9e-7 7e-7 7e-7 7e-7 1e-6 2e-6 1e-5 3e-5];
|
||||
|
||||
n_ground_x = interp1(Fr_ground_x,n_ground_x1,Fr,'linear');
|
||||
n_ground_v = interp1(Fr_ground_v,n_ground_v1,Fr,'linear');
|
||||
% figure
|
||||
% loglog(Fr,abs(n_ground_v),Fr_ground_v,n_ground_v1,'*');
|
||||
% xlabel('Frequency [Hz]');ylabel('ASD [m/s^2 /rtHz]');
|
||||
% return
|
||||
|
||||
%converting into PSD
|
||||
n_ground_x = (n_ground_x).^2;
|
||||
n_ground_v = (n_ground_v).^2;
|
||||
|
||||
%Injecting ground motion in the system and getting the outputs
|
||||
system_dec_f = (freqresp(system_dec,abs(w)));
|
||||
PHI = zeros(size(Fr,2),12,12);
|
||||
for p = 1:size(Fr,2)
|
||||
Sw=zeros(6,6);
|
||||
Iact = zeros(3,3);
|
||||
Sw(4,4) = n_ground_x(p);
|
||||
Sw(5,5) = n_ground_v(p);
|
||||
Sw(6,6) = n_ground_v(p);
|
||||
Sw(1:3,1:3) = Iact;
|
||||
PHI(p,:,:) = (system_dec_f(:,:,p))*Sw(:,:)*(system_dec_f(:,:,p))';
|
||||
end
|
||||
x1 = PHI(:,1,1);
|
||||
z1 = PHI(:,2,2);
|
||||
x2 = PHI(:,3,3);
|
||||
z2 = PHI(:,4,4);
|
||||
wx = PHI(:,5,5);
|
||||
wz = PHI(:,6,6);
|
||||
x12 = PHI(:,1,3);
|
||||
z12 = PHI(:,2,4);
|
||||
PHIwx = PHI(:,1,5);
|
||||
PHIwz = PHI(:,2,6);
|
||||
xsum = PHI(:,7,7);
|
||||
zsum = PHI(:,8,8);
|
||||
xdelta = PHI(:,9,9);
|
||||
zdelta = PHI(:,10,10);
|
||||
rot = PHI(:,11,11);
|
||||
1098
index.html
1098
index.html
File diff suppressed because it is too large
Load Diff
794
index.org
794
index.org
@ -44,9 +44,6 @@
|
||||
:END:
|
||||
|
||||
* Gravimeter - Simscape Model
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle gravimeter/script.m
|
||||
:END:
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
@ -56,35 +53,11 @@
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
addpath('gravimeter');
|
||||
#+end_src
|
||||
|
||||
** Simscape Model - Parameters
|
||||
** Simulink
|
||||
#+begin_src matlab
|
||||
open('gravimeter.slx')
|
||||
#+end_src
|
||||
|
||||
Parameters
|
||||
#+begin_src matlab
|
||||
l = 0.5; % Length of the mass [m]
|
||||
la = 0.5; % Position of Act. [m]
|
||||
|
||||
h = 1.7; % Height of the mass [m]
|
||||
ha = 1.7; % Position of Act. [m]
|
||||
|
||||
m = 400; % Mass [kg]
|
||||
I = 115; % Inertia [kg m^2]
|
||||
|
||||
k = 15e3; % Actuator Stiffness [N/m]
|
||||
c = 0.03; % Actuator Damping [N/(m/s)]
|
||||
|
||||
deq = 0.2; % Length of the actuators [m]
|
||||
|
||||
g = 0; % Gravity [m/s2]
|
||||
#+end_src
|
||||
|
||||
** System Identification - Without Gravity
|
||||
#+begin_src matlab
|
||||
%% Name of the Simulink File
|
||||
mdl = 'gravimeter';
|
||||
@ -104,23 +77,8 @@ Parameters
|
||||
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results output replace :exports results
|
||||
pole(G)
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
#+begin_example
|
||||
pole(G)
|
||||
ans =
|
||||
-0.000473481142385801 + 21.7596190728632i
|
||||
-0.000473481142385801 - 21.7596190728632i
|
||||
-7.49842879459177e-05 + 8.6593576906982i
|
||||
-7.49842879459177e-05 - 8.6593576906982i
|
||||
-5.15386867925747e-06 + 2.27025295182755i
|
||||
-5.15386867925747e-06 - 2.27025295182755i
|
||||
#+end_example
|
||||
|
||||
The plant as 6 states as expected (2 translations + 1 rotation)
|
||||
|
||||
#+begin_src matlab :results output replace
|
||||
size(G)
|
||||
#+end_src
|
||||
@ -150,544 +108,278 @@ The plant as 6 states as expected (2 translations + 1 rotation)
|
||||
#+RESULTS:
|
||||
[[file:figs/open_loop_tf.png]]
|
||||
|
||||
** System Identification - With Gravity
|
||||
** Matlab Code :noexport:
|
||||
#+begin_src matlab
|
||||
g = 9.80665; % Gravity [m/s2]
|
||||
#+end_src
|
||||
clc;
|
||||
% close all
|
||||
|
||||
#+begin_src matlab
|
||||
Gg = linearize(mdl, io);
|
||||
Gg.InputName = {'F1', 'F2', 'F3'};
|
||||
Gg.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
|
||||
#+end_src
|
||||
g = 100000;
|
||||
|
||||
We can now see that the system is unstable due to gravity.
|
||||
#+begin_src matlab :results output replace :exports results
|
||||
pole(Gg)
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
#+begin_example
|
||||
pole(G)
|
||||
ans =
|
||||
-10.9848275341276 + 0i
|
||||
10.9838836405193 + 0i
|
||||
-7.49855396089326e-05 + 8.65962885769976i
|
||||
-7.49855396089326e-05 - 8.65962885769976i
|
||||
-6.68819341967921e-06 + 0.83296042226902i
|
||||
-6.68819341967921e-06 - 0.83296042226902i
|
||||
#+end_example
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(-2, 2, 1000);
|
||||
|
||||
figure;
|
||||
for in_i = 1:3
|
||||
for out_i = 1:4
|
||||
subplot(4, 3, 3*(out_i-1)+in_i);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), '-');
|
||||
plot(freqs, abs(squeeze(freqresp(Gg(out_i,in_i), freqs, 'Hz'))), '-');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
end
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/open_loop_tf_g.pdf', 'width', 'full', 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:open_loop_tf_g
|
||||
#+caption: Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity
|
||||
#+RESULTS:
|
||||
[[file:figs/open_loop_tf_g.png]]
|
||||
|
||||
** Analytical Model
|
||||
*** Parameters
|
||||
Control parameters
|
||||
#+begin_src matlab
|
||||
g = 1e5;
|
||||
g_svd = 1e5;
|
||||
#+end_src
|
||||
|
||||
System parameters
|
||||
#+begin_src matlab
|
||||
w0 = 2*pi*.5; % MinusK BM1 tablle
|
||||
l = 0.5; %[m]
|
||||
la = 1; %[m]
|
||||
h = 1.7; %[m]
|
||||
ha = 1.7;% %[m]
|
||||
m = 400; %[kg]
|
||||
k = 15e3;%[N/m]
|
||||
kv = k;
|
||||
kh = 15e3;
|
||||
I = 115;%[kg m^2]
|
||||
% c = 0.06;
|
||||
% l = 0.4719; %[m]
|
||||
% la = .477; %[m]
|
||||
% h = 1.8973; %[m]
|
||||
% ha = 1.9060;% %[m]
|
||||
% m = 98.1421; %[kg]
|
||||
% k = 14512;%[N/m]
|
||||
% I = 28.5372;%[kg m^2]
|
||||
cv = 0.03;
|
||||
ch = 0.03;
|
||||
|
||||
l = 0.8; % [m]
|
||||
la = l; % [m]
|
||||
%% System definition
|
||||
[Fr, x1, z1, x2, z2, wx, wz, x12, z12, PHIwx, PHIwz,xsum,zsum,xdelta,zdelta,rot]...
|
||||
= modelGeneration(m,I,k,h,ha,l,la,cv,ch,kv,kh);
|
||||
|
||||
h = 1.7; % [m]
|
||||
ha = h; % [m]
|
||||
|
||||
m = 70; % [kg]
|
||||
|
||||
k = 3e3; % [N/m]
|
||||
I = 10; % [kg m^2]
|
||||
#+end_src
|
||||
|
||||
Bode options.
|
||||
#+begin_src matlab
|
||||
%% Bode options
|
||||
P = bodeoptions;
|
||||
P.FreqUnits = 'Hz';
|
||||
P.MagUnits = 'abs';
|
||||
P.MagScale = 'log';
|
||||
P.Grid = 'on';
|
||||
P.PhaseWrapping = 'on';
|
||||
P.Title.FontSize = 14;
|
||||
P.XLabel.FontSize = 14;
|
||||
P.YLabel.FontSize = 14;
|
||||
P.TickLabel.FontSize = 12;
|
||||
P.Xlim = [1e-1,1e2];
|
||||
P.MagLowerLimMode = 'manual';
|
||||
P.MagLowerLim= 1e-3;
|
||||
%P.PhaseVisible = 'off';
|
||||
#+end_src
|
||||
w = 2*pi*logspace(-1,2,1000);
|
||||
|
||||
Frequency vector.
|
||||
#+begin_src matlab
|
||||
w = 2*pi*logspace(-1,2,1000); % [rad/s]
|
||||
#+end_src
|
||||
%% curves points
|
||||
% slide 4
|
||||
F_sl4 = [2e-1 4e-1 7e-1 1 2 3 5];
|
||||
Amp_sl4 = [ 1 2 4 2.5 1 7e-1 7e-1];
|
||||
F_sl4_phase = [2e-1 4e-1 7e-1 1 ];
|
||||
Phase_sl4 = (180/pi).*[0 0 -0.5 -1.7];
|
||||
|
||||
*** generation of the state space model
|
||||
#+begin_src matlab
|
||||
M = [m 0 0
|
||||
0 m 0
|
||||
0 0 I];
|
||||
%slide 6
|
||||
F_sl6 = [2e-1 4e-1 1 2 3 5];
|
||||
Amp_sl6 = [1 1 6e-1 2e-1 3e-1 3e-1];
|
||||
F_sl6_phase = [2e-1 4e-1 1 ];
|
||||
Phase_sl6 = (180/pi).*[0 0 0 ];
|
||||
|
||||
%Jacobian of the bottom sensor
|
||||
Js1 = [1 0 h/2
|
||||
0 1 -l/2];
|
||||
%Jacobian of the top sensor
|
||||
Js2 = [1 0 -h/2
|
||||
0 1 0];
|
||||
%slide 9
|
||||
F_sl9 = [2.5e-1 4e-1 6e-1 1 1.7 2.2 3 5 10];
|
||||
Amp_sl9 = [3 6 1 5e-1 1 2 7e-1 2.5e-1 7e-2];
|
||||
Phase_sl9 = (180/pi)*[0 -1 -pi 0 -1 -1.5 -pi -pi -pi];
|
||||
|
||||
%Jacobian of the actuators
|
||||
Ja = [1 0 ha/2 %Left horizontal actuator
|
||||
%1 0 h/2 %Right horizontal actuator
|
||||
0 1 -la/2 %Left vertical actuator
|
||||
0 1 la/2]; %Right vertical actuator
|
||||
Jta = Ja';
|
||||
K = k*Jta*Ja;
|
||||
C = 0.06*k*Jta*Ja;
|
||||
% slide 14
|
||||
F_sl14 = [ 2e-1 4e-1 6e-1 8e-1 1 2 3 5 10];
|
||||
Amp_sl14 = [9e-1 1.5 1.2 0.35 .3 1.2 .3 .1 5e-2];
|
||||
F_sl14_phase = [ 2e-1 4e-1 6e-1 8e-1 ];
|
||||
Phase_sl14 = (180/pi).*[0 0 -1.7 -2];
|
||||
|
||||
E = [1 0 0
|
||||
0 1 0
|
||||
0 0 1]; %projecting ground motion in the directions of the legs
|
||||
%rotation
|
||||
F_rot = [1e-1 2e-1 4e-1 5e-1 7e-1 1 2 3 6.5 10 20];
|
||||
Amp_rot = [7e-8 2.2e-7 3e-7 1e-7 2e-8 9e-9 3e-8 9e-9 1e-9 4e-10 8e-11];
|
||||
|
||||
AA = [zeros(3) eye(3)
|
||||
-M\K -M\C];
|
||||
%% Plots
|
||||
% %slide 3
|
||||
% figure
|
||||
% loglog(Fr,abs(x2).^.5,Fr,abs(x1).^.5,Fr,abs(xsum).^.5,Fr,abs(xdelta).^.5)
|
||||
% xlabel('Frequency [Hz]');ylabel('Acceleration [m/s^2/rtHz]')
|
||||
% legend('Top sensor','Bottom sensor','Half sum','Half difference');
|
||||
% title('Horizontal')
|
||||
% xlim([7e-2 1e1]);
|
||||
|
||||
BB = [zeros(3,6)
|
||||
M\Jta M\(k*Jta*E)];
|
||||
%slide 4
|
||||
figure
|
||||
subplot 211
|
||||
loglog(Fr, abs(x12)./abs(x1));hold on;
|
||||
loglog(F_sl4,Amp_sl4,'*');
|
||||
xlabel('Frequency [Hz]');ylabel('Amplitude [-]');
|
||||
title('X direction Top/bottom sensor');
|
||||
xlim([7e-2 1e1]);
|
||||
subplot 212
|
||||
semilogx(Fr, (180/pi).*angle(x12./abs(x1)));hold on;
|
||||
loglog(F_sl4_phase,Phase_sl4,'*');
|
||||
xlabel('Frequency [Hz]');ylabel('Phase [deg]');
|
||||
xlim([7e-2 1e1]);
|
||||
|
||||
% BB = [zeros(3,3)
|
||||
% M\Jta ];
|
||||
%slide 6
|
||||
figure
|
||||
subplot 211
|
||||
loglog(Fr, abs(z12)./abs(z1));hold on;
|
||||
loglog(F_sl6,Amp_sl6,'*');
|
||||
xlabel('Frequency [Hz]');ylabel('Amplitude [-]');
|
||||
title('Z direction Top/bottom sensor');
|
||||
xlim([7e-2 1e1]);
|
||||
subplot 212
|
||||
semilogx(Fr, (180/pi).*angle(z12./abs(z1)));hold on;
|
||||
loglog(F_sl6_phase,Phase_sl6,'*');
|
||||
xlabel('Frequency [Hz]');ylabel('Phase [deg]');
|
||||
xlim([7e-2 1e1]);ylim([-180 180]);
|
||||
|
||||
% %slide 6
|
||||
% figure
|
||||
% loglog(Fr,abs(z2).^.5,Fr,abs(z1).^.5,Fr,abs(zsum).^.5,Fr,abs(zdelta).^.5)
|
||||
% xlabel('Frequency [Hz]');ylabel('Acceleration [m/s^2/rtHz]')
|
||||
% legend('Top sensor','Bottom sensor','Half sum','Half difference');
|
||||
% title('Vertical')
|
||||
% xlim([7e-2 1e1]);
|
||||
|
||||
%slide 9
|
||||
figure
|
||||
subplot 211
|
||||
loglog(Fr, abs(PHIwx)./abs(wx));hold on;
|
||||
loglog(F_sl9,Amp_sl9,'*');
|
||||
xlabel('Frequency [Hz]');ylabel('Amplitude [-]');
|
||||
title('X direction bottom/ground sensor');
|
||||
xlim([7e-2 1e1]);
|
||||
ylim([0.01 10]);
|
||||
subplot 212
|
||||
semilogx(Fr, (180/pi).*angle(PHIwx./abs(wx)));hold on;
|
||||
loglog(F_sl9,Phase_sl9,'*');
|
||||
xlabel('Frequency [Hz]');ylabel('Phase [deg]');
|
||||
xlim([7e-2 1e1]);
|
||||
|
||||
% %slide 8
|
||||
% figure
|
||||
% loglog(Fr,abs(wx).^.5,Fr,abs(x1).^.5,'-.',Fr,abs(x2).^.5,'.');
|
||||
% grid on;xlabel('Frequency [Hz]');
|
||||
% ylabel('ASD [m/s^2/rtHz]');
|
||||
% xlim([7e-2 1e1]);
|
||||
% legend('Ground','Bottom sensor','Top sensor');
|
||||
% title('Horizontal');
|
||||
%
|
||||
% CC = [Ja zeros(3)];
|
||||
% DD = zeros(3,3);
|
||||
% %slide 13
|
||||
% figure
|
||||
% loglog(Fr,abs(wz).^.5,Fr,abs(z1).^.5,'-.',Fr,abs(z2).^.5,'.');
|
||||
% grid on;xlabel('Frequency [Hz]');
|
||||
% ylabel('ASD [m/s^2/rtHz]');
|
||||
% xlim([7e-2 1e1]);
|
||||
% legend('Ground','Bottom sensor','Top sensor');
|
||||
% title('Vertical');
|
||||
|
||||
CC = [[Js1;Js2] zeros(4,3);
|
||||
zeros(2,6)
|
||||
(Js1+Js2)./2 zeros(2,3)
|
||||
(Js1-Js2)./2 zeros(2,3)
|
||||
(Js1-Js2)./(2*h) zeros(2,3)];
|
||||
%slide 14
|
||||
figure
|
||||
subplot 211
|
||||
loglog(Fr, abs(PHIwz)./abs(wz));hold on;
|
||||
loglog(F_sl14,Amp_sl14,'*');
|
||||
xlabel('Frequency [Hz]');ylabel('Amplitude [-]');
|
||||
title('Z direction bottom/ground sensor');
|
||||
xlim([7e-2 1e1]);
|
||||
ylim([0.01 10]);
|
||||
subplot 212
|
||||
semilogx(Fr, (180/pi).*angle(PHIwz./abs(wz)));hold on;
|
||||
loglog(F_sl14_phase,Phase_sl14,'*');
|
||||
xlabel('Frequency [Hz]');ylabel('Phase [deg]');
|
||||
xlim([7e-2 1e1]);
|
||||
|
||||
DD = [zeros(4,6)
|
||||
zeros(2,3) eye(2,3)
|
||||
zeros(6,6)];
|
||||
|
||||
system_dec = ss(AA,BB,CC,DD);
|
||||
%rotation
|
||||
figure
|
||||
loglog(Fr,abs(rot).^.5./((2*pi*Fr').^2),F_rot,Amp_rot,'*');
|
||||
xlabel('Frequency [Hz]');ylabel('Rotation [rad/rtHz]')
|
||||
xlim([7e-2 1e1]);
|
||||
#+end_src
|
||||
|
||||
- Input = three actuators and three ground motions
|
||||
- Output = the bottom sensor; the top sensor; the ground motion; the half sum; the half difference; the rotation
|
||||
** Model Generation :noexport:
|
||||
#+begin_src matlab
|
||||
function [Fr, x1, z1, x2, z2, wx, wz, x12, z12, PHIwx, PHIwz,xsum,zsum,xdelta,zdelta,rot] = modelGeneration(m,I,k,h,ha,l,la,dampv,damph,kv,kh)
|
||||
%% generation of the state space model
|
||||
M = [m 0 0
|
||||
0 m 0
|
||||
0 0 I];
|
||||
|
||||
#+begin_src matlab :results output replace
|
||||
size(system_dec)
|
||||
#+end_src
|
||||
%Jacobian of the bottom sensor
|
||||
Js1 = [1 0 h/2
|
||||
0 1 -l/2];
|
||||
%Jacobian of the top sensor
|
||||
Js2 = [1 0 -h/2
|
||||
0 1 0];
|
||||
|
||||
#+RESULTS:
|
||||
: State-space model with 12 outputs, 6 inputs, and 6 states.
|
||||
%Jacobian of the actuators
|
||||
Ja = [1 0 ha/2 %Left horizontal actuator
|
||||
%1 0 h/2 %Right horizontal actuator
|
||||
0 1 -la/2 %Left vertical actuator
|
||||
0 1 la/2]; %Right vertical actuator
|
||||
Jah = [1 0 ha/2];
|
||||
Jav = [0 1 -la/2 %Left vertical actuator
|
||||
0 1 la/2]; %Right vertical actuator
|
||||
Jta = Ja';
|
||||
Jtah = Jah';
|
||||
Jtav = Jav';
|
||||
K = kv*Jtav*Jav + kh*Jtah*Jah;
|
||||
C = dampv*kv*Jtav*Jav+damph*kh*Jtah*Jah;
|
||||
|
||||
*** Comparison with the Simscape Model
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(-2, 2, 1000);
|
||||
E = [1 0 0
|
||||
0 1 0
|
||||
0 0 1]; %projecting ground motion in the directions of the legs
|
||||
|
||||
figure;
|
||||
for in_i = 1:3
|
||||
for out_i = 1:4
|
||||
subplot(4, 3, 3*(out_i-1)+in_i);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), '-');
|
||||
plot(freqs, abs(squeeze(freqresp(system_dec(out_i,in_i)*s^2, freqs, 'Hz'))), '-');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
AA = [zeros(3) eye(3)
|
||||
-M\K -M\C];
|
||||
|
||||
BB = [zeros(3,6)
|
||||
M\Jta M\(k*Jta*E)];
|
||||
|
||||
CC = [[Js1;Js2] zeros(4,3);
|
||||
zeros(2,6)
|
||||
(Js1+Js2)./2 zeros(2,3)
|
||||
(Js1-Js2)./2 zeros(2,3)
|
||||
(Js1-Js2)./(2*h) zeros(2,3)];
|
||||
|
||||
DD = [zeros(4,6)
|
||||
zeros(2,3) eye(2,3)
|
||||
zeros(6,6)];
|
||||
|
||||
system_dec = ss(AA,BB,CC,DD);
|
||||
%input = three actuators and three ground motions
|
||||
%output = the bottom sensor; the top sensor; the ground motion; the half
|
||||
%sum; the half difference; the rotation
|
||||
|
||||
%% Injecting ground motion in the system to have the output
|
||||
Fr = logspace(-2,3,1e3);
|
||||
w=2*pi*Fr*1i;
|
||||
%fit of the ground motion data in m/s^2/rtHz
|
||||
Fr_ground_x = [0.07 0.1 0.15 0.3 0.7 0.8 0.9 1.2 5 10];
|
||||
n_ground_x1 = [4e-7 4e-7 2e-6 1e-6 5e-7 5e-7 5e-7 1e-6 1e-5 3.5e-5];
|
||||
Fr_ground_v = [0.07 0.08 0.1 0.11 0.12 0.15 0.25 0.6 0.8 1 1.2 1.6 2 6 10];
|
||||
n_ground_v1 = [7e-7 7e-7 7e-7 1e-6 1.2e-6 1.5e-6 1e-6 9e-7 7e-7 7e-7 7e-7 1e-6 2e-6 1e-5 3e-5];
|
||||
|
||||
n_ground_x = interp1(Fr_ground_x,n_ground_x1,Fr,'linear');
|
||||
n_ground_v = interp1(Fr_ground_v,n_ground_v1,Fr,'linear');
|
||||
% figure
|
||||
% loglog(Fr,abs(n_ground_v),Fr_ground_v,n_ground_v1,'*');
|
||||
% xlabel('Frequency [Hz]');ylabel('ASD [m/s^2 /rtHz]');
|
||||
% return
|
||||
|
||||
%converting into PSD
|
||||
n_ground_x = (n_ground_x).^2;
|
||||
n_ground_v = (n_ground_v).^2;
|
||||
|
||||
%Injecting ground motion in the system and getting the outputs
|
||||
system_dec_f = (freqresp(system_dec,abs(w)));
|
||||
PHI = zeros(size(Fr,2),12,12);
|
||||
for p = 1:size(Fr,2)
|
||||
Sw=zeros(6,6);
|
||||
Iact = zeros(3,3);
|
||||
Sw(4,4) = n_ground_x(p);
|
||||
Sw(5,5) = n_ground_v(p);
|
||||
Sw(6,6) = n_ground_v(p);
|
||||
Sw(1:3,1:3) = Iact;
|
||||
PHI(p,:,:) = (system_dec_f(:,:,p))*Sw(:,:)*(system_dec_f(:,:,p))';
|
||||
end
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/gravimeter_analytical_system_open_loop_models.pdf', 'width', 'full', 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:gravimeter_analytical_system_open_loop_models
|
||||
#+caption: Comparison of the analytical and the Simscape models
|
||||
#+RESULTS:
|
||||
[[file:figs/gravimeter_analytical_system_open_loop_models.png]]
|
||||
|
||||
*** Analysis
|
||||
#+begin_src matlab
|
||||
% figure
|
||||
% bode(system_dec,P);
|
||||
% return
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
%% svd decomposition
|
||||
% system_dec_freq = freqresp(system_dec,w);
|
||||
% S = zeros(3,length(w));
|
||||
% for m = 1:length(w)
|
||||
% S(:,m) = svd(system_dec_freq(1:4,1:3,m));
|
||||
% end
|
||||
% figure
|
||||
% loglog(w./(2*pi), S);hold on;
|
||||
% % loglog(w./(2*pi), abs(Val(1,:)),w./(2*pi), abs(Val(2,:)),w./(2*pi), abs(Val(3,:)));
|
||||
% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');
|
||||
% legend('\sigma_1','\sigma_2','\sigma_3');%,'\sigma_4','\sigma_5','\sigma_6');
|
||||
% ylim([1e-8 1e-2]);
|
||||
%
|
||||
% %condition number
|
||||
% figure
|
||||
% loglog(w./(2*pi), S(1,:)./S(3,:));hold on;
|
||||
% % loglog(w./(2*pi), abs(Val(1,:)),w./(2*pi), abs(Val(2,:)),w./(2*pi), abs(Val(3,:)));
|
||||
% xlabel('Frequency [Hz]');ylabel('Condition number [-]');
|
||||
% % legend('\sigma_1','\sigma_2','\sigma_3');%,'\sigma_4','\sigma_5','\sigma_6');
|
||||
%
|
||||
% %performance indicator
|
||||
% system_dec_svd = freqresp(system_dec(1:4,1:3),2*pi*10);
|
||||
% [U,S,V] = svd(system_dec_svd);
|
||||
% H_svd_OL = -eye(3,4);%-[zpk(-2*pi*10,-2*pi*40,40/10) 0 0 0; 0 10*zpk(-2*pi*40,-2*pi*200,40/200) 0 0; 0 0 zpk(-2*pi*2,-2*pi*10,10/2) 0];% - eye(3,4);%
|
||||
% H_svd = pinv(V')*H_svd_OL*pinv(U);
|
||||
% % system_dec_control_svd_ = feedback(system_dec,g*pinv(V')*H*pinv(U));
|
||||
%
|
||||
% OL_dec = g_svd*H_svd*system_dec(1:4,1:3);
|
||||
% OL_freq = freqresp(OL_dec,w); % OL = G*H
|
||||
% CL_system = feedback(eye(3),-g_svd*H_svd*system_dec(1:4,1:3));
|
||||
% CL_freq = freqresp(CL_system,w); % CL = (1+G*H)^-1
|
||||
% % CL_system_2 = feedback(system_dec,H);
|
||||
% % CL_freq_2 = freqresp(CL_system_2,w); % CL = G/(1+G*H)
|
||||
% for i = 1:size(w,2)
|
||||
% OL(:,i) = svd(OL_freq(:,:,i));
|
||||
% CL (:,i) = svd(CL_freq(:,:,i));
|
||||
% %CL2 (:,i) = svd(CL_freq_2(:,:,i));
|
||||
% end
|
||||
%
|
||||
% un = ones(1,length(w));
|
||||
% figure
|
||||
% loglog(w./(2*pi),OL(3,:)+1,'k',w./(2*pi),OL(3,:)-1,'b',w./(2*pi),1./CL(1,:),'r--',w./(2*pi),un,'k:');hold on;%
|
||||
% % loglog(w./(2*pi), 1./(CL(2,:)),w./(2*pi), 1./(CL(3,:)));
|
||||
% % semilogx(w./(2*pi), 1./(CL2(1,:)),w./(2*pi), 1./(CL2(2,:)),w./(2*pi), 1./(CL2(3,:)));
|
||||
% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');
|
||||
% legend('GH \sigma_{inf} +1 ','GH \sigma_{inf} -1','S 1/\sigma_{sup}');%,'\lambda_1','\lambda_2','\lambda_3');
|
||||
%
|
||||
% figure
|
||||
% loglog(w./(2*pi),OL(1,:)+1,'k',w./(2*pi),OL(1,:)-1,'b',w./(2*pi),1./CL(3,:),'r--',w./(2*pi),un,'k:');hold on;%
|
||||
% % loglog(w./(2*pi), 1./(CL(2,:)),w./(2*pi), 1./(CL(3,:)));
|
||||
% % semilogx(w./(2*pi), 1./(CL2(1,:)),w./(2*pi), 1./(CL2(2,:)),w./(2*pi), 1./(CL2(3,:)));
|
||||
% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');
|
||||
% legend('GH \sigma_{sup} +1 ','GH \sigma_{sup} -1','S 1/\sigma_{inf}');%,'\lambda_1','\lambda_2','\lambda_3');
|
||||
#+end_src
|
||||
|
||||
*** Control Section
|
||||
#+begin_src matlab
|
||||
system_dec_10Hz = freqresp(system_dec,2*pi*10);
|
||||
system_dec_0Hz = freqresp(system_dec,0);
|
||||
|
||||
system_decReal_10Hz = pinv(align(system_dec_10Hz));
|
||||
[Ureal,Sreal,Vreal] = svd(system_decReal_10Hz(1:4,1:3));
|
||||
normalizationMatrixReal = abs(pinv(Ureal)*system_dec_0Hz(1:4,1:3)*pinv(Vreal'));
|
||||
|
||||
[U,S,V] = svd(system_dec_10Hz(1:4,1:3));
|
||||
normalizationMatrix = abs(pinv(U)*system_dec_0Hz(1:4,1:3)*pinv(V'));
|
||||
|
||||
H_dec = ([zpk(-2*pi*5,-2*pi*30,30/5) 0 0 0
|
||||
0 zpk(-2*pi*4,-2*pi*20,20/4) 0 0
|
||||
0 0 0 zpk(-2*pi,-2*pi*10,10)]);
|
||||
H_cen_OL = [zpk(-2*pi,-2*pi*10,10) 0 0; 0 zpk(-2*pi,-2*pi*10,10) 0;
|
||||
0 0 zpk(-2*pi*5,-2*pi*30,30/5)];
|
||||
H_cen = pinv(Jta)*H_cen_OL*pinv([Js1; Js2]);
|
||||
% H_svd_OL = -[1/normalizationMatrix(1,1) 0 0 0
|
||||
% 0 1/normalizationMatrix(2,2) 0 0
|
||||
% 0 0 1/normalizationMatrix(3,3) 0];
|
||||
% H_svd_OL_real = -[1/normalizationMatrixReal(1,1) 0 0 0
|
||||
% 0 1/normalizationMatrixReal(2,2) 0 0
|
||||
% 0 0 1/normalizationMatrixReal(3,3) 0];
|
||||
H_svd_OL = -[1/normalizationMatrix(1,1)*zpk(-2*pi*10,-2*pi*60,60/10) 0 0 0
|
||||
0 1/normalizationMatrix(2,2)*zpk(-2*pi*5,-2*pi*30,30/5) 0 0
|
||||
0 0 1/normalizationMatrix(3,3)*zpk(-2*pi*2,-2*pi*10,10/2) 0];
|
||||
H_svd_OL_real = -[1/normalizationMatrixReal(1,1)*zpk(-2*pi*10,-2*pi*60,60/10) 0 0 0
|
||||
0 1/normalizationMatrixReal(2,2)*zpk(-2*pi*5,-2*pi*30,30/5) 0 0
|
||||
0 0 1/normalizationMatrixReal(3,3)*zpk(-2*pi*2,-2*pi*10,10/2) 0];
|
||||
% H_svd_OL_real = -[zpk(-2*pi*10,-2*pi*40,40/10) 0 0 0; 0 10*zpk(-2*pi*10,-2*pi*100,100/10) 0 0; 0 0 zpk(-2*pi*2,-2*pi*10,10/2) 0];%-eye(3,4);
|
||||
% H_svd_OL = -[zpk(-2*pi*10,-2*pi*40,40/10) 0 0 0; 0 zpk(-2*pi*4,-2*pi*20,4/20) 0 0; 0 0 zpk(-2*pi*2,-2*pi*10,10/2) 0];% - eye(3,4);%
|
||||
H_svd = pinv(V')*H_svd_OL*pinv(U);
|
||||
H_svd_real = pinv(Vreal')*H_svd_OL_real*pinv(Ureal);
|
||||
|
||||
OL_dec = g*H_dec*system_dec(1:4,1:3);
|
||||
OL_cen = g*H_cen_OL*pinv([Js1; Js2])*system_dec(1:4,1:3)*pinv(Jta);
|
||||
OL_svd = 100*H_svd_OL*pinv(U)*system_dec(1:4,1:3)*pinv(V');
|
||||
OL_svd_real = 100*H_svd_OL_real*pinv(Ureal)*system_dec(1:4,1:3)*pinv(Vreal');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
% figure
|
||||
% bode(OL_dec,w,P);title('OL Decentralized');
|
||||
% figure
|
||||
% bode(OL_cen,w,P);title('OL Centralized');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
figure
|
||||
bode(g*system_dec(1:4,1:3),w,P);
|
||||
title('gain * Plant');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
figure
|
||||
bode(OL_svd,OL_svd_real,w,P);
|
||||
title('OL SVD');
|
||||
legend('SVD of Complex plant','SVD of real approximation of the complex plant')
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
figure
|
||||
bode(system_dec(1:4,1:3),pinv(U)*system_dec(1:4,1:3)*pinv(V'),P);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
CL_dec = feedback(system_dec,g*H_dec,[1 2 3],[1 2 3 4]);
|
||||
CL_cen = feedback(system_dec,g*H_cen,[1 2 3],[1 2 3 4]);
|
||||
CL_svd = feedback(system_dec,100*H_svd,[1 2 3],[1 2 3 4]);
|
||||
CL_svd_real = feedback(system_dec,100*H_svd_real,[1 2 3],[1 2 3 4]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
pzmap_testCL(system_dec,H_dec,g,[1 2 3],[1 2 3 4])
|
||||
title('Decentralized control');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
pzmap_testCL(system_dec,H_cen,g,[1 2 3],[1 2 3 4])
|
||||
title('Centralized control');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
pzmap_testCL(system_dec,H_svd,100,[1 2 3],[1 2 3 4])
|
||||
title('SVD control');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
pzmap_testCL(system_dec,H_svd_real,100,[1 2 3],[1 2 3 4])
|
||||
title('Real approximation SVD control');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
P.Ylim = [1e-8 1e-3];
|
||||
figure
|
||||
bodemag(system_dec(1:4,1:3),CL_dec(1:4,1:3),CL_cen(1:4,1:3),CL_svd(1:4,1:3),CL_svd_real(1:4,1:3),P);
|
||||
title('Motion/actuator')
|
||||
legend('Control OFF','Decentralized control','Centralized control','SVD control','SVD control real appr.');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
P.Ylim = [1e-5 1e1];
|
||||
figure
|
||||
bodemag(system_dec(1:4,4:6),CL_dec(1:4,4:6),CL_cen(1:4,4:6),CL_svd(1:4,4:6),CL_svd_real(1:4,4:6),P);
|
||||
title('Transmissibility');
|
||||
legend('Control OFF','Decentralized control','Centralized control','SVD control','SVD control real appr.');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
figure
|
||||
bodemag(system_dec([7 9],4:6),CL_dec([7 9],4:6),CL_cen([7 9],4:6),CL_svd([7 9],4:6),CL_svd_real([7 9],4:6),P);
|
||||
title('Transmissibility from half sum and half difference in the X direction');
|
||||
legend('Control OFF','Decentralized control','Centralized control','SVD control','SVD control real appr.');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
figure
|
||||
bodemag(system_dec([8 10],4:6),CL_dec([8 10],4:6),CL_cen([8 10],4:6),CL_svd([8 10],4:6),CL_svd_real([8 10],4:6),P);
|
||||
title('Transmissibility from half sum and half difference in the Z direction');
|
||||
legend('Control OFF','Decentralized control','Centralized control','SVD control','SVD control real appr.');
|
||||
#+end_src
|
||||
|
||||
*** Greshgorin radius
|
||||
#+begin_src matlab
|
||||
system_dec_freq = freqresp(system_dec,w);
|
||||
x1 = zeros(1,length(w));
|
||||
z1 = zeros(1,length(w));
|
||||
x2 = zeros(1,length(w));
|
||||
S1 = zeros(1,length(w));
|
||||
S2 = zeros(1,length(w));
|
||||
S3 = zeros(1,length(w));
|
||||
|
||||
for t = 1:length(w)
|
||||
x1(t) = (abs(system_dec_freq(1,2,t))+abs(system_dec_freq(1,3,t)))/abs(system_dec_freq(1,1,t));
|
||||
z1(t) = (abs(system_dec_freq(2,1,t))+abs(system_dec_freq(2,3,t)))/abs(system_dec_freq(2,2,t));
|
||||
x2(t) = (abs(system_dec_freq(3,1,t))+abs(system_dec_freq(3,2,t)))/abs(system_dec_freq(3,3,t));
|
||||
system_svd = pinv(Ureal)*system_dec_freq(1:4,1:3,t)*pinv(Vreal');
|
||||
S1(t) = (abs(system_svd(1,2))+abs(system_svd(1,3)))/abs(system_svd(1,1));
|
||||
S2(t) = (abs(system_svd(2,1))+abs(system_svd(2,3)))/abs(system_svd(2,2));
|
||||
S2(t) = (abs(system_svd(3,1))+abs(system_svd(3,2)))/abs(system_svd(3,3));
|
||||
end
|
||||
|
||||
limit = 0.5*ones(1,length(w));
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
figure
|
||||
loglog(w./(2*pi),x1,w./(2*pi),z1,w./(2*pi),x2,w./(2*pi),limit,'--');
|
||||
legend('x_1','z_1','x_2','Limit');
|
||||
xlabel('Frequency [Hz]');
|
||||
ylabel('Greshgorin radius [-]');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
figure
|
||||
loglog(w./(2*pi),S1,w./(2*pi),S2,w./(2*pi),S3,w./(2*pi),limit,'--');
|
||||
legend('S1','S2','S3','Limit');
|
||||
xlabel('Frequency [Hz]');
|
||||
ylabel('Greshgorin radius [-]');
|
||||
% set(gcf,'color','w')
|
||||
#+end_src
|
||||
|
||||
*** Injecting ground motion in the system to have the output
|
||||
#+begin_src matlab
|
||||
Fr = logspace(-2,3,1e3);
|
||||
w=2*pi*Fr*1i;
|
||||
%fit of the ground motion data in m/s^2/rtHz
|
||||
Fr_ground_x = [0.07 0.1 0.15 0.3 0.7 0.8 0.9 1.2 5 10];
|
||||
n_ground_x1 = [4e-7 4e-7 2e-6 1e-6 5e-7 5e-7 5e-7 1e-6 1e-5 3.5e-5];
|
||||
Fr_ground_v = [0.07 0.08 0.1 0.11 0.12 0.15 0.25 0.6 0.8 1 1.2 1.6 2 6 10];
|
||||
n_ground_v1 = [7e-7 7e-7 7e-7 1e-6 1.2e-6 1.5e-6 1e-6 9e-7 7e-7 7e-7 7e-7 1e-6 2e-6 1e-5 3e-5];
|
||||
|
||||
n_ground_x = interp1(Fr_ground_x,n_ground_x1,Fr,'linear');
|
||||
n_ground_v = interp1(Fr_ground_v,n_ground_v1,Fr,'linear');
|
||||
% figure
|
||||
% loglog(Fr,abs(n_ground_v),Fr_ground_v,n_ground_v1,'*');
|
||||
% xlabel('Frequency [Hz]');ylabel('ASD [m/s^2 /rtHz]');
|
||||
% return
|
||||
|
||||
%converting into PSD
|
||||
n_ground_x = (n_ground_x).^2;
|
||||
n_ground_v = (n_ground_v).^2;
|
||||
|
||||
%Injecting ground motion in the system and getting the outputs
|
||||
system_dec_f = (freqresp(system_dec,abs(w)));
|
||||
PHI = zeros(size(Fr,2),12,12);
|
||||
for p = 1:size(Fr,2)
|
||||
Sw=zeros(6,6);
|
||||
Iact = zeros(3,3);
|
||||
Sw(4,4) = n_ground_x(p);
|
||||
Sw(5,5) = n_ground_v(p);
|
||||
Sw(6,6) = n_ground_v(p);
|
||||
Sw(1:3,1:3) = Iact;
|
||||
PHI(p,:,:) = (system_dec_f(:,:,p))*Sw(:,:)*(system_dec_f(:,:,p))';
|
||||
end
|
||||
x1 = PHI(:,1,1);
|
||||
z1 = PHI(:,2,2);
|
||||
x2 = PHI(:,3,3);
|
||||
z2 = PHI(:,4,4);
|
||||
wx = PHI(:,5,5);
|
||||
wz = PHI(:,6,6);
|
||||
x12 = PHI(:,1,3);
|
||||
z12 = PHI(:,2,4);
|
||||
PHIwx = PHI(:,1,5);
|
||||
PHIwz = PHI(:,2,6);
|
||||
xsum = PHI(:,7,7);
|
||||
zsum = PHI(:,8,8);
|
||||
xdelta = PHI(:,9,9);
|
||||
zdelta = PHI(:,10,10);
|
||||
rot = PHI(:,11,11);
|
||||
#+end_src
|
||||
|
||||
* Gravimeter - Functions
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
||||
:END:
|
||||
** =align=
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle gravimeter/align.m
|
||||
:END:
|
||||
<<sec:align>>
|
||||
|
||||
This Matlab function is accessible [[file:gravimeter/align.m][here]].
|
||||
|
||||
#+begin_src matlab
|
||||
function [A] = align(V)
|
||||
%A!ALIGN(V) returns a constat matrix A which is the real alignment of the
|
||||
%INVERSE of the complex input matrix V
|
||||
%from Mohit slides
|
||||
|
||||
if (nargin ==0) || (nargin > 1)
|
||||
disp('usage: mat_inv_real = align(mat)')
|
||||
return
|
||||
end
|
||||
|
||||
D = pinv(real(V'*V));
|
||||
A = D*real(V'*diag(exp(1i * angle(diag(V*D*V.'))/2)));
|
||||
|
||||
|
||||
end
|
||||
#+end_src
|
||||
|
||||
|
||||
** =pzmap_testCL=
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle gravimeter/pzmap_testCL.m
|
||||
:END:
|
||||
<<sec:pzmap_testCL>>
|
||||
|
||||
This Matlab function is accessible [[file:gravimeter/pzmap_testCL.m][here]].
|
||||
|
||||
#+begin_src matlab
|
||||
function [] = pzmap_testCL(system,H,gain,feedin,feedout)
|
||||
% evaluate and plot the pole-zero map for the closed loop system for
|
||||
% different values of the gain
|
||||
|
||||
[~, n] = size(gain);
|
||||
[m1, n1, ~] = size(H);
|
||||
[~,n2] = size(feedin);
|
||||
|
||||
figure
|
||||
for i = 1:n
|
||||
% if n1 == n2
|
||||
system_CL = feedback(system,gain(i)*H,feedin,feedout);
|
||||
|
||||
[P,Z] = pzmap(system_CL);
|
||||
plot(real(P(:)),imag(P(:)),'x',real(Z(:)),imag(Z(:)),'o');hold on
|
||||
xlabel('Real axis (s^{-1})');ylabel('Imaginary Axis (s^{-1})');
|
||||
% clear P Z
|
||||
% else
|
||||
% system_CL = feedback(system,gain(i)*H(:,1+(i-1)*m1:m1+(i-1)*m1),feedin,feedout);
|
||||
%
|
||||
% [P,Z] = pzmap(system_CL);
|
||||
% plot(real(P(:)),imag(P(:)),'x',real(Z(:)),imag(Z(:)),'o');hold on
|
||||
% xlabel('Real axis (s^{-1})');ylabel('Imaginary Axis (s^{-1})');
|
||||
% clear P Z
|
||||
% end
|
||||
end
|
||||
str = {strcat('gain = ' , num2str(gain(1)))}; % at the end of first loop, z being loop output
|
||||
str = [str , strcat('gain = ' , num2str(gain(1)))]; % after 2nd loop
|
||||
for i = 2:n
|
||||
str = [str , strcat('gain = ' , num2str(gain(i)))]; % after 2nd loop
|
||||
str = [str , strcat('gain = ' , num2str(gain(i)))]; % after 2nd loop
|
||||
end
|
||||
legend(str{:})
|
||||
end
|
||||
|
||||
x1 = PHI(:,1,1);
|
||||
z1 = PHI(:,2,2);
|
||||
x2 = PHI(:,3,3);
|
||||
z2 = PHI(:,4,4);
|
||||
wx = PHI(:,5,5);
|
||||
wz = PHI(:,6,6);
|
||||
x12 = PHI(:,1,3);
|
||||
z12 = PHI(:,2,4);
|
||||
PHIwx = PHI(:,1,5);
|
||||
PHIwz = PHI(:,2,6);
|
||||
xsum = PHI(:,7,7);
|
||||
zsum = PHI(:,8,8);
|
||||
xdelta = PHI(:,9,9);
|
||||
zdelta = PHI(:,10,10);
|
||||
rot = PHI(:,11,11);
|
||||
#+end_src
|
||||
|
||||
* Stewart Platform - Simscape Model
|
||||
@ -1283,7 +975,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
|
||||
#+RESULTS:
|
||||
[[file:figs/stewart_platform_simscape_cl_transmissibility.png]]
|
||||
|
||||
* Stewart Platform - Analytical Model :noexport:
|
||||
* Stewart Platform - Analytical 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)
|
||||
<<matlab-dir>>
|
||||
|
||||
Loading…
Reference in New Issue
Block a user