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Active Damping of Rotating Platforms using Integral Force Feedback - Matlab Computation

Table of Contents

This document gathers the Matlab code used to for the paper (Dehaeze and Collette 2020).

It is structured in several sections:

The matlab code is accessible on Zonodo and Github (Dehaeze 2020).

To run the Matlab code, go in the matlab directory and run the following Matlab files corresponding to each section.

Table 1: Paper’s sections and corresponding Matlab files
Sections Matlab File
Section 1 s1_system_description.m
Section 2 s2_iff_pure_int.m
Section 3 s3_iff_hpf.m
Section 4 s4_iff_kp.m
Section 5 s5_act_damp_comparison.m

1 System Description and Analysis

1.1 System description

The system consists of one 2 degree of freedom translation stage on top of a spindle (figure 1).

system.png

Figure 1: Schematic of the studied system

The control inputs are the forces applied by the actuators of the translation stage (\(F_u\) and \(F_v\)). As the translation stage is rotating around the Z axis due to the spindle, the forces are applied along \(\vec{i}_u\) and \(\vec{i}_v\).

1.2 Equations

Based on the Figure 1, the equations of motions are:

\begin{equation} \begin{bmatrix} d_u \\ d_v \end{bmatrix} = \bm{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix} \end{equation}

Where \(\bm{G}_d\) is a \(2 \times 2\) transfer function matrix.

\begin{equation} \bm{G}_d = \frac{1}{k} \frac{1}{G_{dp}} \begin{bmatrix} G_{dz} & G_{dc} \\ -G_{dc} & G_{dz} \end{bmatrix} \end{equation}

With:

\begin{align} G_{dp} &= \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\ G_{dz} &= \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \\ G_{dc} &= 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \end{align}

1.3 Numerical Values

Let’s define initial values for the model.

k = 1;    % Actuator Stiffness [N/m]
c = 0.05; % Actuator Damping [N/(m/s)]
m = 1;    % Payload mass [kg]
xi = c/(2*sqrt(k*m));
w0 = sqrt(k/m); % [rad/s]

1.4 Campbell Diagram

The Campbell Diagram displays the evolution of the real and imaginary parts of the system as a function of the rotating speed.

It is shown in Figure 2, and one can see that the system becomes unstable for \(\Omega > \omega_0\) (the real part of one of the poles becomes positive).

campbell_diagram.png

Figure 2: Campbell Diagram

1.5 Simscape Model

In order to validate all the equations of motion, a Simscape model of the same system has been developed. The dynamics of the system can be identified from the Simscape model and compare with the analytical model.

The rotating speed for the Simscape Model is defined.

W = 0.1; % Rotation Speed [rad/s]
open('rotating_frame.slx');

The transfer function from \([F_u, F_v]\) to \([d_u, d_v]\) is identified from the Simscape model.

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

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/K'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/G'], 2, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io, 0);

%% Input/Output definition
G.InputName  = {'Fu', 'Fv'};
G.OutputName = {'du', 'dv'};

The same transfer function from \([F_u, F_v]\) to \([d_u, d_v]\) is written down from the analytical model.

Gth = (1/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
      [(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ...
       -2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)];

Both transfer functions are compared in Figure 3 and are found to perfectly match.

plant_simscape_analytical.png

Figure 3: Bode plot of the transfer function from \([F_u, F_v]\) to \([d_u, d_v]\) as identified from the Simscape model and from an analytical model

1.6 Effect of the rotation speed

The transfer functions from \([F_u, F_v]\) to \([d_u, d_v]\) are identified for the following rotating speeds.

Ws = [0, 0.2, 0.7, 1.1]*w0; % Rotating Speeds [rad/s]
Gs = {zeros(2, 2, length(Ws))};

for W_i = 1:length(Ws)
    W = Ws(W_i);

    Gs(:, :, W_i) = {(1/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
                     [(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ...
                      -2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)]};
end

They are compared in Figure 4.

plant_compare_rotating_speed.png

Figure 4: Comparison of the transfer functions from \([F_u, F_v]\) to \([d_u, d_v]\) for several rotating speed

2 Problem with pure Integral Force Feedback

Force sensors are added in series with the two actuators (Figure 5).

Two identical controllers \(K_F\) are used to feedback each of the sensed force to its associated actuator.

system_iff.png

Figure 5: System with added Force Sensor in series with the actuators

2.1 Plant Parameters

Let’s define initial values for the model.

k = 1;    % Actuator Stiffness [N/m]
c = 0.05; % Actuator Damping [N/(m/s)]
m = 1;    % Payload mass [kg]
xi = c/(2*sqrt(k*m));
w0 = sqrt(k/m); % [rad/s]

2.2 Equations

The sensed forces are equal to:

\begin{equation} \begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} F_u \\ F_v \end{bmatrix} - (c s + k) \begin{bmatrix} d_u \\ d_v \end{bmatrix} \end{equation}

Which then gives:

\begin{equation} \begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \bm{G}_{f} \begin{bmatrix} F_u \\ F_v \end{bmatrix} \end{equation} \begin{equation} \begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \frac{1}{G_{fp}} \begin{bmatrix} G_{fz} & -G_{fc} \\ G_{fc} & G_{fz} \end{bmatrix} \begin{bmatrix} F_u \\ F_v \end{bmatrix} \end{equation} \begin{align} G_{fp} &= \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\ G_{fz} &= \left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\ G_{fc} &= \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right) \end{align}

2.3 Comparison of the Analytical Model and the Simscape Model

The rotation speed is set to \(\Omega = 0.1 \omega_0\).

W = 0.1*w0; % [rad/s]
open('rotating_frame.slx');

And the transfer function from \([F_u, F_v]\) to \([f_u, f_v]\) is identified using the Simscape model.

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

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/K'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/G'], 1, 'openoutput'); io_i = io_i + 1;
Giff = linearize(mdl, io, 0);

%% Input/Output definition
Giff.InputName  = {'Fu', 'Fv'};
Giff.OutputName = {'fu', 'fv'};

The same transfer function from \([F_u, F_v]\) to \([f_u, f_v]\) is written down from the analytical model.

Giff_th = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
          [(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
           (2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2];

The two are compared in Figure 6 and found to perfectly match.

plant_iff_comp_simscape_analytical.png

Figure 6: Comparison of the transfer functions from \([F_u, F_v]\) to \([f_u, f_v]\) between the Simscape model and the analytical one

2.4 Effect of the rotation speed

The transfer functions from \([F_u, F_v]\) to \([f_u, f_v]\) are identified for the following rotating speeds.

Ws = [0, 0.2, 0.7, 1.1]*w0; % Rotating Speeds [rad/s]
Gsiff = {zeros(2, 2, length(Ws))};

for W_i = 1:length(Ws)
    W = Ws(W_i);

    Gsiff(:, :, W_i) = {1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
                      [(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
                       (2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2]};
end

The obtained transfer functions are shown in Figure 7.

plant_iff_compare_rotating_speed.png

Figure 7: Comparison of the transfer functions from \([F_u, F_v]\) to \([f_u, f_v]\) for several rotating speed

2.5 Decentralized Integral Force Feedback

The decentralized IFF controller consists of pure integrators:

\begin{equation} \bm{K}_{\text{IFF}}(s) = \frac{g}{s} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \end{equation}

The Root Locus (evolution of the poles of the closed loop system in the complex plane as a function of \(g\)) is shown in Figure 8. It is shown that for non-null rotating speed, one pole is bound to the right-half plane, and thus the closed loop system is unstable.

root_locus_pure_iff.png

Figure 8: Root Locus for the Decentralized Integral Force Feedback controller. Several rotating speed are shown.

3 Integral Force Feedback with an High Pass Filter

3.1 Plant Parameters

Let’s define initial values for the model.

k = 1;    % Actuator Stiffness [N/m]
c = 0.05; % Actuator Damping [N/(m/s)]
m = 1;    % Payload mass [kg]
xi = c/(2*sqrt(k*m));
w0 = sqrt(k/m); % [rad/s]

3.2 Modified Integral Force Feedback Controller

Let’s modify the initial Integral Force Feedback Controller ; instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used:

\begin{equation} K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \end{equation}

where \(\omega_i\) characterize down to which frequency the signal is integrated.

Let’s arbitrary choose the following control parameters:

g = 2;
wi = 0.1*w0;

And the following rotating speed.

Giff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
        [(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
         (2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2];

The obtained Loop Gain is shown in Figure 9.

loop_gain_modified_iff.png

Figure 9: Loop Gain for the modified IFF controller

3.3 Root Locus

As shown in the Root Locus plot (Figure 10), for some value of the gain, the system remains stable.

root_locus_modified_iff.png

Figure 10: Root Locus for the modified IFF controller

3.4 What is the optimal \(\omega_i\) and \(g\)?

In order to visualize the effect of \(\omega_i\) on the attainable damping, the Root Locus is displayed in Figure 11 for the following \(\omega_i\):

wis = [0.01, 0.1, 0.5, 1]*w0; % [rad/s]

root_locus_wi_modified_iff.png

Figure 11: Root Locus for the modified IFF controller (zoomed plot on the left)

For the controller

\begin{equation} K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \end{equation}

The gain at which the system becomes unstable is

\begin{equation} g_\text{max} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right) \label{eq:iff_gmax} \end{equation}

While it seems that small \(\omega_i\) do allow more damping to be added to the system (Figure 11), the control gains may be limited to small values due to \eqref{eq:iff_gmax} thus reducing the attainable damping.

There must be an optimum for \(\omega_i\). To find the optimum, the gain that maximize the simultaneous damping of the mode is identified for a wide range of \(\omega_i\) (Figure 12).

wis = logspace(-2, 1, 100)*w0; % [rad/s]

opt_xi = zeros(1, length(wis)); % Optimal simultaneous damping
opt_gain = zeros(1, length(wis)); % Corresponding optimal gain

for wi_i = 1:length(wis)
    wi = wis(wi_i);
    Kiff = 1/(s + wi)*eye(2);

    fun = @(g)computeSimultaneousDamping(g, Giff, Kiff);

    [g_opt, xi_opt] = fminsearch(fun, 0.5*wi*((w0/W)^2 - 1));
    opt_xi(wi_i) = 1/xi_opt;
    opt_gain(wi_i) = g_opt;
end

mod_iff_damping_wi.png

Figure 12: Simultaneous attainable damping of the closed loop poles as a function of \(\omega_i\)

4 IFF with a stiffness in parallel with the force sensor

4.1 Schematic

In this section additional springs in parallel with the force sensors are added to counteract the negative stiffness induced by the rotation.

system_parallel_springs.png

Figure 13: Studied system with additional springs in parallel with the actuators and force sensors

In order to keep the overall stiffness \(k = k_a + k_p\) constant, a scalar parameter \(\alpha\) (\(0 \le \alpha < 1\)) is defined to describe the fraction of the total stiffness in parallel with the actuator and force sensor

\begin{equation} k_p = \alpha k, \quad k_a = (1 - \alpha) k \end{equation}

4.2 Equations

\begin{equation} \begin{bmatrix} f_u \\ f_v \end{bmatrix} = \bm{G}_k \begin{bmatrix} F_u \\ F_v \end{bmatrix} \end{equation} \begin{equation} \begin{bmatrix} f_u \\ f_v \end{bmatrix} = \frac{1}{G_{kp}} \begin{bmatrix} G_{kz} & -G_{kc} \\ G_{kc} & G_{kz} \end{bmatrix} \begin{bmatrix} F_u \\ F_v \end{bmatrix} \end{equation}

With:

\begin{align} G_{kp} &= \left( \frac{s^2}{{\omega_0}^2} + 2\xi \frac{s}{{\omega_0}^2} + 1 - \frac{\Omega^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0}\frac{s}{\omega_0} \right)^2 \\ G_{kz} &= \left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} + \alpha \right) \left( \frac{s^2}{{\omega_0}^2} + 2\xi \frac{s}{{\omega_0}^2} + 1 - \frac{\Omega^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0}\frac{s}{\omega_0} \right)^2 \\ G_{kc} &= \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0}\frac{s}{\omega_0} \right) \end{align}

If we compare \(G_{kz}\) and \(G_{fz}\), we see that the spring in parallel adds a term \(\alpha\). In order to have two complex conjugate zeros (instead of real zeros):

\begin{equation} \alpha > \frac{\Omega^2}{{\omega_0}^2} \quad \Leftrightarrow \quad k_p > m \Omega^2 \end{equation}

4.3 Plant Parameters

Let’s define initial values for the model.

k = 1;    % Actuator Stiffness [N/m]
c = 0.05; % Actuator Damping [N/(m/s)]
m = 1;    % Payload mass [kg]
xi = c/(2*sqrt(k*m));
w0 = sqrt(k/m); % [rad/s]

4.4 Comparison of the Analytical Model and the Simscape Model

The same transfer function from \([F_u, F_v]\) to \([f_u, f_v]\) is written down from the analytical model.

W = 0.1*w0; % [rad/s]

kp = 1.5*m*W^2;
cp = 0;
open('rotating_frame.slx');
%% Name of the Simulink File
mdl = 'rotating_frame';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/K'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/G'], 1, 'openoutput'); io_i = io_i + 1;

Giff = linearize(mdl, io, 0);

%% Input/Output definition
Giff.InputName  = {'Fu', 'Fv'};
Giff.OutputName = {'fu', 'fv'};
w0p = sqrt((k + kp)/m);
xip = c/(2*sqrt((k+kp)*m));

Giff_th = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
                   (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
                   (2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2 ];
Giff_th.InputName  = {'Fu', 'Fv'};
Giff_th.OutputName = {'fu', 'fv'};

plant_iff_kp_comp_simscape_analytical.png

Figure 14: Comparison of the transfer functions from \([F_u, F_v]\) to \([f_u, f_v]\) between the Simscape model and the analytical one

4.5 Effect of the parallel stiffness on the IFF plant

The rotation speed is set to \(\Omega = 0.1 \omega_0\).

W = 0.1*w0; % [rad/s]

And the IFF plant (transfer function from \([F_u, F_v]\) to \([f_u, f_v]\)) is identified in three different cases:

  • without parallel stiffness
  • with a small parallel stiffness \(k_p < m \Omega^2\)
  • with a large parallel stiffness \(k_p > m \Omega^2\)

The results are shown in Figure 15.

One can see that for \(k_p > m \Omega^2\), the systems shows alternating complex conjugate poles and zeros.

kp = 0;

w0p = sqrt((k + kp)/m);
xip = c/(2*sqrt((k+kp)*m));

Giff = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
    (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
    (2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
kp = 0.5*m*W^2;
k = 1 - kp;

w0p = sqrt((k + kp)/m);
xip = c/(2*sqrt((k+kp)*m));

Giff_s = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
    (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
    (2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
kp = 1.5*m*W^2;
k = 1 - kp;

w0p = sqrt((k + kp)/m);
xip = c/(2*sqrt((k+kp)*m));

Giff_l = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
    (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
    (2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];

plant_iff_kp.png

Figure 15: Transfer function from \([F_u, F_v]\) to \([f_u, f_v]\) for \(k_p = 0\), \(k_p < m \Omega^2\) and \(k_p > m \Omega^2\)

4.6 IFF when adding a spring in parallel

In Figure 16 is displayed the Root Locus in the three considered cases with

\begin{equation} K_{\text{IFF}} = \frac{g}{s} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \end{equation}

One can see that for \(k_p > m \Omega^2\), the root locus stays in the left half of the complex plane and thus the control system is unconditionally stable.

Thus, decentralized IFF controller with pure integrators can be used if:

\begin{equation} k_{p} > m \Omega^2 \end{equation}

root_locus_iff_kp.png

Figure 16: Root Locus

4.7 Effect of \(k_p\) on the attainable damping

However, having large values of \(k_p\) may decrease the attainable damping.

To study the second point, Root Locus plots for the following values of \(k_p\) are shown in Figure 17.

kps = [2, 20, 40]*m*W^2;

It is shown that large values of \(k_p\) decreases the attainable damping.

root_locus_iff_kps.png

Figure 17: Root Locus plot

alphas = logspace(-2, 0, 100);

opt_xi = zeros(1, length(alphas)); % Optimal simultaneous damping
opt_gain = zeros(1, length(alphas)); % Corresponding optimal gain

Kiff = 1/s*eye(2);

for alpha_i = 1:length(alphas)
    kp = alphas(alpha_i);
    k = 1 - alphas(alpha_i);

    w0p = sqrt((k + kp)/m);
    xip = c/(2*sqrt((k+kp)*m));

    Giff = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
        (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
        (2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];

    fun = @(g)computeSimultaneousDamping(g, Giff, Kiff);

    [g_opt, xi_opt] = fminsearch(fun, 2);
    opt_xi(alpha_i) = 1/xi_opt;
    opt_gain(alpha_i) = g_opt;
end

opt_damp_alpha.png

5 Comparison

Two modifications to adapt the IFF control strategy to rotating platforms have been proposed. These two methods are now compared in terms of added damping, closed-loop compliance and transmissibility.

5.1 Plant Parameters

Let’s define initial values for the model.

k = 1;    % Actuator Stiffness [N/m]
c = 0.05; % Actuator Damping [N/(m/s)]
m = 1;    % Payload mass [kg]
xi = c/(2*sqrt(k*m));
w0 = sqrt(k/m); % [rad/s]

The rotating speed is set to \(\Omega = 0.1 \omega_0\).

W = 0.1*w0;

5.2 Root Locus

IFF with High Pass Filter

wi = 0.1*w0; % [rad/s]

Giff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
        [(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
         (2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2];

IFF With parallel Stiffness

kp = 5*m*W^2;
k = k - kp;

w0p = sqrt((k + kp)/m);
xip = c/(2*sqrt((k+kp)*m));

Giff_kp = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
                   (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
                   (2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2 ];

k = k + kp;

comp_root_locus.png

Figure 19: Root Locus plot - Comparison of IFF with additional high pass filter, IFF with additional parallel stiffness

5.3 Controllers - Optimal Gains

In order to compare to three considered Active Damping techniques, gains that yield maximum damping of all the modes are computed for each case.

The obtained damping ratio and control are shown below.

  Obtained \(\xi\) Control Gain
Modified IFF 0.83 1.99
IFF with \(k_p\) 0.83 2.02

5.4 Passive Damping - Critical Damping

\begin{equation} \xi = \frac{c}{2 \sqrt{km}} \end{equation}

Critical Damping corresponds to to \(\xi = 1\), and thus:

\begin{equation} c_{\text{crit}} = 2 \sqrt{km} \end{equation}
c_opt = 2*sqrt(k*m);

5.5 Transmissibility And Compliance

open('rotating_frame.slx');
%% Name of the Simulink File
mdl = 'rotating_frame';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/dw'], 1, 'input');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/fd'], 1, 'input');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/Meas'], 1, 'output');  io_i = io_i + 1;
G_ol = linearize(mdl, io, 0);

%% Input/Output definition
G_ol.InputName  = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
G_ol.OutputName = {'Dx', 'Dy'};

5.5.1 Passive Damping

kp = 0;
cp = 0;
c_old = c;
c = c_opt;
G_pas = linearize(mdl, io, 0);

%% Input/Output definition
G_pas.InputName  = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
G_pas.OutputName = {'Dx', 'Dy'};
c = c_old;
Kiff = opt_gain_iff/(wi + s)*tf(eye(2));
G_iff = linearize(mdl, io, 0);

%% Input/Output definition
G_iff.InputName  = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
G_iff.OutputName = {'Dx', 'Dy'};
kp = 5*m*W^2;
cp = 0.01;
Kiff = opt_gain_kp/s*tf(eye(2));
G_kp = linearize(mdl, io, 0);

%% Input/Output definition
G_kp.InputName  = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
G_kp.OutputName = {'Dx', 'Dy'};

comp_transmissibility.png

Figure 20: Comparison of the transmissibility

comp_compliance.png

Figure 21: Comparison of the obtained Compliance

6 Notations

  Mathematical Notation Matlab Unit
Actuator Stiffness \(k\) k N/m
Actuator Damping \(c\) c N/(m/s)
Payload Mass \(m\) m kg
Damping Ratio \(\xi = \frac{c}{2\sqrt{km}}\) xi  
Actuator Force \(\bm{F}, F_u, F_v\) F Fu Fv N
Force Sensor signal \(\bm{f}, f_u, f_v\) f fu fv N
Relative Displacement \(\bm{d}, d_u, d_v\) d du dv m
Resonance freq. when \(\Omega = 0\) \(\omega_0\) w0 rad/s
Rotation Speed \(\Omega = \dot{\theta}\) W rad/s
Low Pass Filter corner frequency \(\omega_i\) wi rad/s
  Mathematical Notation Matlab Unit
Laplace variable \(s\) s  
Complex number \(j\) j  
Frequency \(\omega\) w [rad/s]

Bibliography

Dehaeze, T., and C. Collette. 2020. “Active Damping of Rotating Platforms Using Integral Force Feedback.” In Proceedings of the International Conference on Modal Analysis Noise and Vibration Engineering (ISMA).
Dehaeze, Thomas. 2020. “Active Damping of Rotating Positioning Platforms.” https://doi.org/10.5281/zenodo.3894342.

Author: Thomas Dehaeze

Created: 2020-07-09 jeu. 10:19