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Better Matlab notations

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This commit is contained in:
Thomas Dehaeze
2020-11-09 14:37:04 +01:00
parent 292ba73fb1
commit e97a3d58ab
25 changed files with 1786 additions and 1666 deletions
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552
stewart_platform/simscape_model.m
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@@ -6,42 +6,14 @@ s = zpk('s');
addpath('STEP');
% Jacobian
% First, the position of the "joints" (points of force application) are estimated and the Jacobian computed.
open('drone_platform_jacobian.slx');
sim('drone_platform_jacobian');
Aa = [a1.Data(1,:);
a2.Data(1,:);
a3.Data(1,:);
a4.Data(1,:);
a5.Data(1,:);
a6.Data(1,:)]';
Ab = [b1.Data(1,:);
b2.Data(1,:);
b3.Data(1,:);
b4.Data(1,:);
b5.Data(1,:);
b6.Data(1,:)]';
As = (Ab - Aa)./vecnorm(Ab - Aa);
l = vecnorm(Ab - Aa)';
J = [As' , cross(Ab, As)'];
save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
% Simscape Model
% Simscape Model - Parameters
% <<sec:stewart_simscape>>
open('drone_platform.slx');
% Definition of spring parameters
% Definition of spring parameters:
kx = 0.5*1e3/3; % [N/m]
ky = 0.5*1e3/3;
@@ -51,31 +23,53 @@ cx = 0.025; % [Nm/rad]
cy = 0.025;
cz = 0.025;
% Gravity:
g = 0;
% We load the Jacobian.
% We load the Jacobian (previously computed from the geometry):
load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
% Identification of the plant
% The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
% We initialize other parameters:
U = eye(6);
V = eye(6);
Kc = tf(zeros(6));
% #+name: fig:stewart_platform_plant
% #+caption: Considered plant $\bm{G} = \begin{bmatrix}G_d\\G_u\end{bmatrix}$. $D_w$ is the translation/rotation of the support, $\tau$ the actuator forces, $a$ the acceleration/angular acceleration of the top platform
% #+RESULTS:
% [[file:figs/stewart_platform_plant.png]]
%% Name of the Simulink File
mdl = 'drone_platform';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1; % Ground Motion
io(io_i) = linio([mdl, '/V-T'], 1, 'openinput'); io_i = io_i + 1; % Actuator Forces
io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Top platform acceleration
G = linearize(mdl, io);
G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
% Plant
Gu = G(:, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
% Disturbance dynamics
Gd = G(:, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'});
% There are 24 states (6dof for the bottom platform + 6dof for the top platform).
@@ -87,176 +81,59 @@ size(G)
% #+RESULTS:
% : State-space model with 6 outputs, 12 inputs, and 24 states.
% The elements of the transfer matrix $\bm{G}$ corresponding to the transfer function from actuator forces $\tau$ to the measured acceleration $a$ are shown in Figure [[fig:stewart_platform_coupled_plant]].
% G = G*blkdiag(inv(J), eye(6));
% G.InputName = {'Dw1', 'Dw2', 'Dw3', 'Dw4', 'Dw5', 'Dw6', ...
% 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
% Thanks to the Jacobian, we compute the transfer functions in the frame of the legs and in an inertial frame.
Gx = G*blkdiag(eye(6), inv(J'));
Gx.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
% Gl = J*G;
% Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
% Obtained Dynamics
freqs = logspace(-1, 2, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$A_x/F_x$');
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$A_y/F_y$');
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$A_z/F_z$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast');
ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-360:90:360]);
linkaxes([ax1,ax2],'x');
% #+name: fig:stewart_platform_translations
% #+caption: Stewart Platform Plant from forces applied by the legs to the acceleration of the platform
% #+RESULTS:
% [[file:figs/stewart_platform_translations.png]]
% One can easily see that the system is strongly coupled.
freqs = logspace(-1, 2, 1000);
figure;
ax1 = subplot(2, 1, 1);
% Magnitude
hold on;
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$A_{R_x}/M_x$');
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$A_{R_y}/M_y$');
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$A_{R_z}/M_z$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [rad/(Nm)]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast');
ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-360:90:360]);
linkaxes([ax1,ax2],'x');
% #+name: fig:stewart_platform_rotations
% #+caption: Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform
% #+RESULTS:
% [[file:figs/stewart_platform_rotations.png]]
freqs = logspace(-1, 2, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for out_i = 1:5
for in_i = i+1:6
plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', out_i), sprintf('F%i', in_i)), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
end
for i_in = 1:6
for i_out = [1:i_in-1, i_in+1:6]
plot(freqs, abs(squeeze(freqresp(Gu(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off');
end
end
for ch_i = 1:6
plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', ch_i), sprintf('F%i', ch_i)), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gu(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'DisplayName', '$G_u(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:6
plot(freqs, abs(squeeze(freqresp(Gu(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_u(%d,%d)$', i_in_out, i_in_out));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
for ch_i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(sprintf('A%i', ch_i), sprintf('F%i', ch_i)), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-360:90:360]);
linkaxes([ax1,ax2],'x');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
ylim([1e-2, 1e5]);
legend('location', 'northwest');
% #+name: fig:stewart_platform_legs
% #+caption: Stewart Platform Plant from forces applied by the legs to displacement of the legs
% #+name: fig:plant_decouple_jacobian
% #+caption: Decoupled plant $\bm{G}_x$ using the Jacobian matrix $J$
% #+RESULTS:
% [[file:figs/stewart_platform_legs.png]]
% [[file:figs/plant_decouple_jacobian.png]]
% We define a new plant:
% \[ G_x(s) = G(s) J^{-T} \]
% $G_x(s)$ correspond to the transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform.
freqs = logspace(-1, 2, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_x/D_{w,x}$');
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_y/D_{w,y}$');
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Dwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_z/D_{w,z}$');
% set(gca,'ColorOrderIndex',1)
% plot(freqs, abs(squeeze(freqresp(TR(1,1), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
% plot(freqs, abs(squeeze(freqresp(TR(2,2), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
% plot(freqs, abs(squeeze(freqresp(TR(3,3), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - Translations'); xlabel('Frequency [Hz]');
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Rwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_x/R_{w,x}$');
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'Rwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_y/R_{w,y}$');
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Rwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_z/R_{w,z}$');
% set(gca,'ColorOrderIndex',1)
% plot(freqs, abs(squeeze(freqresp(TR(4,4), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
% plot(freqs, abs(squeeze(freqresp(TR(5,5), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
% plot(freqs, abs(squeeze(freqresp(TR(6,6), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - Rotations'); xlabel('Frequency [Hz]');
legend('location', 'northeast');
linkaxes([ax1,ax2],'x');
Gx = Gu*inv(J');
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
% Real Approximation of $G$ at the decoupling frequency
% Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_c(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
% <<sec:stewart_real_approx>>
% Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_u(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
wc = 2*pi*30; % Decoupling frequency [rad/s]
Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, ...
{'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
H1 = evalfr(Gc, j*wc);
H1 = evalfr(Gu, j*wc);
@@ -265,55 +142,58 @@ H1 = evalfr(Gc, j*wc);
D = pinv(real(H1'*H1));
H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
% Verification of the decoupling using the "Gershgorin Radii"
% SVD Decoupling
% <<sec:stewart_svd_decoupling>>
% First, the Singular Value Decomposition of $H_1$ is performed:
% \[ H_1 = U \Sigma V^H \]
[U,S,V] = svd(H1);
[U,~,V] = svd(H1);
% Then, the "Gershgorin Radii" is computed for the plant $G_c(s)$ and the "SVD Decoupled Plant" $G_d(s)$:
% \[ G_d(s) = U^T G_c(s) V \]
% #+name: fig:plant_decouple_svd
% #+caption: Decoupled plant $\bm{G}_{SVD}$ using the Singular Value Decomposition
% #+RESULTS:
% [[file:figs/plant_decouple_svd.png]]
% The decoupled plant is then:
% \[ G_{SVD}(s) = U^{-1} G_u(s) V^{-H} \]
Gsvd = inv(U)*Gu*inv(V');
% Verification of the decoupling using the "Gershgorin Radii"
% <<sec:comp_decoupling>>
% The "Gershgorin Radii" is computed for the coupled plant $G(s)$, for the "Jacobian plant" $G_x(s)$ and the "SVD Decoupled Plant" $G_{SVD}(s)$:
% The "Gershgorin Radii" of a matrix $S$ is defined by:
% \[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \]
% This is computed over the following frequencies.
freqs = logspace(-2, 2, 1000); % [Hz]
% Gershgorin Radii for the coupled plant:
Gr_coupled = zeros(length(freqs), size(Gc,2));
H = abs(squeeze(freqresp(Gc, freqs, 'Hz')));
for out_i = 1:size(Gc,2)
Gr_coupled = zeros(length(freqs), size(Gu,2));
H = abs(squeeze(freqresp(Gu, freqs, 'Hz')));
for out_i = 1:size(Gu,2)
Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end
% Gershgorin Radii for the decoupled plant using SVD:
Gd = U'*Gc*V;
Gr_decoupled = zeros(length(freqs), size(Gd,2));
H = abs(squeeze(freqresp(Gd, freqs, 'Hz')));
for out_i = 1:size(Gd,2)
Gr_decoupled = zeros(length(freqs), size(Gsvd,2));
H = abs(squeeze(freqresp(Gsvd, freqs, 'Hz')));
for out_i = 1:size(Gsvd,2)
Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end
% Gershgorin Radii for the decoupled plant using the Jacobian:
Gj = Gc*inv(J');
Gr_jacobian = zeros(length(freqs), size(Gj,2));
H = abs(squeeze(freqresp(Gj, freqs, 'Hz')));
for out_i = 1:size(Gj,2)
Gr_jacobian = zeros(length(freqs), size(Gx,2));
H = abs(squeeze(freqresp(Gx, freqs, 'Hz')));
for out_i = 1:size(Gx,2)
Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end
@@ -334,31 +214,55 @@ plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
hold off;
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
legend('location', 'northeast');
legend('location', 'northwest');
ylim([1e-3, 1e3]);
% Decoupled Plant
% Let's see the bode plot of the decoupled plant $G_d(s)$.
% \[ G_d(s) = U^T G_c(s) V \]
% Obtained Decoupled Plants
% <<sec:stewart_decoupled_plant>>
% The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown in Figure [[fig:simscape_model_decoupled_plant_svd]].
freqs = logspace(-1, 2, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on;
for ch_i = 1:6
plot(freqs, abs(squeeze(freqresp(Gd(ch_i, ch_i), freqs, 'Hz'))), ...
'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
for i_in = 1:6
for i_out = [1:i_in-1, i_in+1:6]
plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off');
end
end
for in_i = 1:5
for out_i = in_i+1:6
plot(freqs, abs(squeeze(freqresp(Gd(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(Gsvd(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
'DisplayName', '$G_{SVD}(i,j),\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for ch_i = 1:6
plot(freqs, abs(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))), ...
'DisplayName', sprintf('$G_{SVD}(%i,%i)$', ch_i, ch_i));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); xlabel('Frequency [Hz]');
legend('location', 'southeast');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
legend('location', 'northwest');
ylim([1e-1, 1e5])
% Phase
ax2 = nexttile;
hold on;
for ch_i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180:90:360]);
linkaxes([ax1,ax2],'x');
@@ -367,53 +271,135 @@ legend('location', 'southeast');
% #+RESULTS:
% [[file:figs/simscape_model_decoupled_plant_svd.png]]
% Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant $G_x(s)$ using the Jacobian are shown in Figure [[fig:simscape_model_decoupled_plant_jacobian]].
freqs = logspace(-1, 2, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on;
for ch_i = 1:6
plot(freqs, abs(squeeze(freqresp(Gj(ch_i, ch_i), freqs, 'Hz'))), ...
'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
end
for in_i = 1:5
for out_i = in_i+1:6
plot(freqs, abs(squeeze(freqresp(Gj(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
for i_in = 1:6
for i_out = [1:i_in-1, i_in+1:6]
plot(freqs, abs(squeeze(freqresp(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(Gx(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
'DisplayName', '$G_x(i,j),\ i \neq j$');
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$G_x(1,1) = A_x/F_x$');
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$G_x(2,2) = A_y/F_y$');
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = A_z/F_z$');
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$G_x(4,4) = A_{R_x}/M_x$');
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$G_x(5,5) = A_{R_y}/M_y$');
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$G_x(6,6) = A_{R_z}/M_z$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); xlabel('Frequency [Hz]');
legend('location', 'southeast');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
legend('location', 'northwest');
ylim([1e-2, 2e6])
% Diagonal Controller
% The controller $K$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$.
% Phase
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([0, 180]);
yticks([0:45:360]);
wc = 2*pi*0.1; % Crossover Frequency [rad/s]
C_g = 50; % DC Gain
K = eye(6)*C_g/(s+wc);
% #+RESULTS:
% [[file:figs/centralized_control.png]]
G_cen = feedback(G, inv(J')*K, [7:12], [1:6]);
linkaxes([ax1,ax2],'x');
% #+name: fig:svd_control
% #+caption: Control Diagram for the SVD control
% #+RESULTS:
% [[file:figs/svd_control.png]]
% SVD Control
G_svd = feedback(G, pinv(V')*K*pinv(U), [7:12], [1:6]);
% We choose the controller to be a low pass filter:
% \[ K_c(s) = \frac{G_0}{1 + \frac{s}{\omega_0}} \]
% $G_0$ is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to $\omega_c$
wc = 2*pi*80; % Crossover Frequency [rad/s]
w0 = 2*pi*0.1; % Controller Pole [rad/s]
K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
L_cen = K_cen*Gx;
G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
L_svd = K_svd*Gsvd;
G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
% The obtained diagonal elements of the loop gains are shown in Figure [[fig:stewart_comp_loop_gain_diagonal]].
freqs = logspace(-1, 2, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on;
plot(freqs, abs(squeeze(freqresp(L_svd(1, 1), freqs, 'Hz'))), 'DisplayName', '$L_{SVD}(i,i)$');
for i_in_out = 2:6
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(L_cen(1, 1), freqs, 'Hz'))), ...
'DisplayName', '$L_{J}(i,i)$');
for i_in_out = 2:6
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
legend('location', 'northwest');
ylim([5e-2, 2e3])
% Phase
ax2 = nexttile;
hold on;
for i_in_out = 1:6
set(gca,'ColorOrderIndex',1)
plot(freqs, 180/pi*angle(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))));
end
set(gca,'ColorOrderIndex',2)
for i_in_out = 1:6
set(gca,'ColorOrderIndex',2)
plot(freqs, 180/pi*angle(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180:90:360]);
linkaxes([ax1,ax2],'x');
% Closed-Loop system Performances
% <<sec:stewart_closed_loop_results>>
% Results
% Let's first verify the stability of the closed-loop systems:
isstable(G_cen)
@@ -433,69 +419,61 @@ isstable(G_svd)
% #+RESULTS:
% : ans =
% : logical
% : 0
% : 1
% The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:stewart_platform_simscape_cl_transmissibility]].
freqs = logspace(-3, 3, 1000);
freqs = logspace(-2, 2, 1000);
figure
figure;
tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = subplot(2, 3, 1);
ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop');
plot(freqs, abs(squeeze(freqresp(G_cen('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized');
plot(freqs, abs(squeeze(freqresp(G_svd('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'SVD');
plot(freqs, abs(squeeze(freqresp(G_svd('Ax', 'Dwx')/s^2, freqs, 'Hz'))), '--', 'DisplayName', 'SVD');
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $D_x/D_{w,x}$'); xlabel('Frequency [Hz]');
ylabel('$D_x/D_{w,x}$, $D_y/D_{w, y}$'); set(gca, 'XTickLabel',[]);
legend('location', 'southwest');
ax2 = subplot(2, 3, 2);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $D_y/D_{w,y}$'); xlabel('Frequency [Hz]');
ax3 = subplot(2, 3, 3);
ax2 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Az', 'Dwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Az', 'Dwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Az', 'Dwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Az', 'Dwz')/s^2, freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $D_z/D_{w,z}$'); xlabel('Frequency [Hz]');
ylabel('$D_z/D_{w,z}$'); set(gca, 'XTickLabel',[]);
ax4 = subplot(2, 3, 4);
ax3 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Arx', 'Rwx')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $R_x/R_{w,x}$'); xlabel('Frequency [Hz]');
ax5 = subplot(2, 3, 5);
hold on;
plot(freqs, abs(squeeze(freqresp(G_svd('Arx', 'Rwx')/s^2, freqs, 'Hz'))), '--');
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(G( 'Ary', 'Rwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
ylabel('$R_x/R_{w,x}$, $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
ax6 = subplot(2, 3, 6);
ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Arz', 'Rwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Arz', 'Rwz')/s^2, freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $R_z/R_{w,z}$'); xlabel('Frequency [Hz]');
ylabel('$R_z/R_{w,z}$'); xlabel('Frequency [Hz]');
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
linkaxes([ax1,ax2,ax3,ax4],'xy');
xlim([freqs(1), freqs(end)]);
ylim([1e-3, 1e2]);