493 lines
14 KiB
Mathematica
493 lines
14 KiB
Mathematica
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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
|
||
|
|
% 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');
|
||
|
|
|
||
|
|
% 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);
|