Diagonal control using the SVD and the Jacobian Matrix

In this part, diagonal control using both the SVD and the Jacobian matrices are applied on a gravimeter model:

• Section 1.2: the model is described and its parameters are defined.
• Section 1.3: the plant dynamics from the actuators to the sensors is computed from a Simscape model.
• Section 1.4: the plant is decoupled using the Jacobian matrices.
• Section 1.5: the Singular Value Decomposition is performed on a real approximation of the plant transfer matrix and further use to decouple the system.
• Section 1.6: the effectiveness of the decoupling is computed using the Gershorin radii
• Section 1.7: the effectiveness of the decoupling is computed using the Relative Gain Array
• Section 1.8: the obtained decoupled plants are compared
• Section 1.9: the diagonal controller is developed
• Section 1.10: the obtained closed-loop performances for the two methods are compared

The model of the gravimeter is schematically shown in Figure 1.

The parameters used for the simulation are the following:

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 = 2e1; % 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', 'Ay1', 'Ax2', 'Ay2'};

The inputs and outputs of the plant are shown in Figure 3.

More precisely there are three inputs (the three actuator forces):

And 4 outputs (the two 2-DoF accelerometers):

We can check the poles of the plant:

As expected, the plant as 6 states (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 4.

Consider the control architecture shown in Figure 5.

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 5. \[ \bm{G}_x(s) = J_a^{-1} \bm{G}(s) J_{\tau}^{-T} \]

\(\bm{G}_x(s)\) correspond to the \(3 \times 3\) transfer function matrix from forces and torques applied to the gravimeter at its center of mass to the absolute acceleration of the gravimeter’s center of mass (Figure 5).

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];

And the plant \(\bm{G}_x\) is computed:

Gx = pinv(Ja)*G*pinv(Jt');
Gx.InputName  = {'Fx', 'Fy', 'Mz'};
Gx.OutputName  = {'Dx', 'Dy', 'Rz'};

size(Gx)
State-space model with 3 outputs, 3 inputs, and 6 states.

The diagonal and off-diagonal elements of \(G_x\) are shown in Figure 6.

It is shown at the system is:

• decoupled at high frequency thanks to a diagonal mass matrix (the Jacobian being evaluated at the center of mass of the payload)
• coupled at low frequency due to the non-diagonal terms in the stiffness matrix, especially the term corresponding to a coupling between a force in the x direction to a rotation around z (due to the torque applied by the stiffness 1).

The choice of the frame in this the Jacobian is evaluated is discussed in Section 1.12.

In order to decouple the plant using the SVD, first a real approximation of the plant transfer function matrix as the crossover frequency is required.

Let’s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G(j\omega_c)\) 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 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));

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

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

The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 7.

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

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

size(Gsvd)
State-space model with 4 outputs, 3 inputs, and 6 states.

The 4th output (corresponding to the null singular value) is discarded, and we only keep the \(3 \times 3\) plant:

Gsvd = Gsvd(1:3, 1:3);

The diagonal and off-diagonal elements of the “SVD” plant are shown in Figure 8.

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: \[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \]

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 10.

The RGA-number is also a measure of diagonal dominance:

The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 12.

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 13.

The control diagram for the centralized control is shown in Figure 14.

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 15. 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*10;  % 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(Jt')*K_cen*pinv(Ja));

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;
U_inv = inv(U);
G_svd = feedback(G, inv(V')*K_svd*U_inv(1:3, :));

The obtained diagonal elements of the loop gains are shown in Figure 16.

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 17.

Let say we change the position of the actuators:

la = l/2*0.7; % Position of Act. [m]
ha = h/2*0.7; % Position of Act. [m]

The new plant is computed, and the centralized and SVD control architectures are applied using the previsouly computed Jacobian matrices and \(U\) and \(V\) matrices.

The closed-loop system are still stable, and their

If we want to decouple the system at low frequency (determined by the stiffness matrix), we have to compute the Jacobian at a point where the stiffness matrix is diagonal. A displacement (resp. rotation) of the mass at this particular point should induce a pure force (resp. torque) on the same point due to stiffnesses in the system. This can be verified by geometrical computations.

If we want to decouple the system at high frequency (determined by the mass matrix), we have tot compute the Jacobians at the Center of Mass of the suspended solid. Similarly to the stiffness analysis, when considering only the inertia effects (neglecting the stiffnesses), a force (resp. torque) applied at this point (the center of mass) should induce a pure acceleration (resp. angular acceleration).

Ideally, we would like to have a decoupled mass matrix and stiffness matrix at the same time. To do so, the actuators (springs) should be positioned such that the stiffness matrix is diagonal when evaluated at the CoM of the solid.

As the SVD is applied on a real approximation of the plant dynamics at a frequency \(\omega_0\), it is foreseen that the effectiveness of the decoupling depends on the validity of the real approximation.

Let’s do the SVD decoupling on a plant that is mostly real (low damping) and one with a large imaginary part (larger damping).

Start with small damping, the obtained diagonal and off-diagonal terms are shown in Figure 26.

c = 2e1; % Actuator Damping [N/(m/s)]

Now take a larger damping, the obtained diagonal and off-diagonal terms are shown in Figure 27.

c = 5e2; % Actuator Damping [N/(m/s)]

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 31 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 32.

One can easily see that the system is strongly coupled.

Consider the control architecture shown in Figure 33. 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'};

In order to decouple the plant using the SVD, first a real approximation of the plant transfer function matrix as the crossover frequency is required.

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

Now, 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 34.

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 36.

The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 37.

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 38.

The control diagram for the centralized control is shown in Figure 39.

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 40. 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 41.

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 42.