SVD Control

open('gravimeter.slx')

Parameters

l  = 1.0; % Length of the mass [m]
h  = 1.7; % Height of the mass [m]

la = l/2; % Position of Act. [m]
ha = h/2; % 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]

%% Name of the Simulink File
mdl = 'gravimeter';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;

G = linearize(mdl, io);
G.InputName  = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};

The inputs and outputs of the plant are shown in Figure 2.

We can check the poles of the plant:

-0.000183495485977108 +       13.546056874877i
-0.000183495485977108 -       13.546056874877i
-7.49842878906757e-05 +      8.65934902322567i
-7.49842878906757e-05 -      8.65934902322567i
-1.33171230256362e-05 +      3.64924169037897i
-1.33171230256362e-05 -      3.64924169037897i

The plant as 6 states as expected (2 translations + 1 rotation)

size(G)

State-space model with 4 outputs, 3 inputs, and 6 states.

The bode plot of all elements of the plant are shown in Figure 3.

Consider the control architecture shown in Figure 4.

The Jacobian matrix \(J_{\tau}\) is used to transform forces applied by the three actuators into forces/torques applied on the gravimeter at its center of mass. The Jacobian matrix \(J_{a}\) is used to compute the vertical acceleration, horizontal acceleration and rotational acceleration of the mass with respect to its center of mass.

We thus define a new plant as defined in Figure 4. \[ G_x(s) = J_a G(s) J_{\tau}^{-T} \]

\(G_x(s)\) correspond to the transfer function from forces and torques applied to the gravimeter at its center of mass to the absolute acceleration of the gravimeter’s center of mass.

The jacobian corresponding to the sensors and actuators are defined below.

Ja = [1 0  h/2
      0 1 -l/2
      1 0 -h/2
      0 1  0];

Jt = [1 0  ha
      0 1 -la
      0 1  la];

Gx = pinv(Ja)*G*pinv(Jt');
Gx.InputName  = {'Fx', 'Fz', 'My'};
Gx.OutputName  = {'Dx', 'Dz', 'Ry'};

The diagonal and off-diagonal elements of \(G_x\) are shown in Figure 5.

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*10; % Decoupling frequency [rad/s]

H1 = evalfr(G, j*wc);

The real approximation is computed as follows:

D = pinv(real(H1'*H1));
H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));

First, the Singular Value Decomposition of \(H_1\) is performed: \[ H_1 = U \Sigma V^H \]

[U,~,V] = svd(H1);

The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 6.

The decoupled plant is then: \[ G_{SVD}(s) = U^{-1} G_u(s) V^{-H} \]

Gsvd = inv(U)*G*inv(V');

The diagonal and off-diagonal elements of the “SVD” plant are shown in Figure 7.

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]

The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 9.

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 10.

The control diagram for the centralized control is shown in Figure 11.

The controller \(K_c\) is “working” in an cartesian frame. The Jacobian is used to convert forces in the cartesian frame to forces applied by the actuators.

The SVD control architecture is shown in Figure 12. The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).

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 13.

Let’s first verify the stability of the closed-loop systems:

isstable(G_cen)

ans =
  logical
   1

isstable(G_svd)

ans =
  logical
   1

The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 14.

open('drone_platform.slx');

Definition of spring parameters:

kx = 0.5*1e3/3; % [N/m]
ky = 0.5*1e3/3;
kz = 1e3/3;

cx = 0.025; % [Nm/rad]
cy = 0.025;
cz = 0.025;

We suppose the sensor is perfectly positioned.

sens_pos_error = zeros(3,1);

Gravity:

g = 0;

We load the Jacobian (previously computed from the geometry):

load('jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');

We initialize other parameters:

U = eye(6);
V = eye(6);
Kc = tf(zeros(6));

The plant shown in Figure 18 is identified from the Simscape model.

The inputs are:

• \(D_w\) translation and rotation of the bottom platform (with respect to the center of mass of the top platform)
• \(\tau\) the 6 forces applied by the voice coils

The outputs are the 6 accelerations measured by the inertial unit.

%% 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; % 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).

size(G)

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 19.

One can easily see that the system is strongly coupled.

Consider the control architecture shown in Figure 20. The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator.

The Jacobian matrix is computed from the geometry of the platform (position and orientation of the actuators).

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.

Gx = Gu*inv(J');
Gx.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};

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]

H1 = evalfr(Gu, j*wc);

The real approximation is computed as follows:

D = pinv(real(H1'*H1));
H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));

Note that the plant \(G_u\) at \(\omega_c\) is already an almost real matrix. This can be seen on the Bode plots where the phase is close to 1. This can be verified below where only the real value of \(G_u(\omega_c)\) is shown

First, the Singular Value Decomposition of \(H_1\) is performed: \[ H_1 = U \Sigma V^H \]

[U,~,V] = svd(H1);

The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 21.

The decoupled plant is then: \[ G_{SVD}(s) = U^{-1} G_u(s) V^{-H} \]

Gsvd = inv(U)*Gu*inv(V');

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.

The relative gain array (RGA) is defined as:

where \(\times\) denotes an element by element multiplication and \(G(s)\) is an \(n \times n\) square transfer matrix.

The obtained RGA elements are shown in Figure 23.

The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 9.

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 10.

The control diagram for the centralized control is shown in Figure 11.

The controller \(K_c\) is “working” in an cartesian frame. The Jacobian is used to convert forces in the cartesian frame to forces applied by the actuators.

The SVD control architecture is shown in Figure 12. The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).

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 28.

Let’s first verify the stability of the closed-loop systems:

isstable(G_cen)

ans =
  logical
   1

isstable(G_svd)

ans =
  logical
   1

The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 29.

Let’s now consider a small position error of the sensor:

sens_pos_error = [105 5 -1]*1e-3; % [m]

The system is identified again:

Gx = Gu*inv(J');
Gx.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};

Gsvd = inv(U)*Gu*inv(V');

L_cen = K_cen*Gx;
G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);

L_svd = K_svd*Gsvd;
G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);

isstable(G_cen)

isstable(G_svd)