Test Bench - Amplified Piezoelectric Actuator

The stack parameters taken from the data-sheet are shown in Table 1.

Let’s compute the generated force

The stroke is \(L_{\max} = 20\mu m\) for a voltage range of \(V_{\max} = 170 V\). Furthermore, the stiffness is \(k_a = 235 \cdot 10^6 N/m\).

The relation between the applied voltage and the generated force can be estimated as follows:

ka = 235e6; % [N/m]
Lmax = 20e-6; % [m]
Vmax = 170; % [V]

ga = ka*Lmax/Vmax; % [N/V]

ga = 27.6 [N/V]

From the parameters of the stack, it seems not possible to estimate the relation between the strain and the generated voltage.

In order to make the conversion from applied voltage to generated force or from the strain to the generated voltage, we need to defined some parameters corresponding to the piezoelectric material:

d33 = 300e-12; % Strain constant [m/V]
n   = 80;      % Number of layers per stack
ka  = 235e6;   % Stack stiffness [N/m]

The ratio of the developed force to applied voltage is:

where:

• \(F_a\): developed force in [N]
• \(n\): number of layers of the actuator stack
• \(d_{33}\): strain constant in [m/V]
• \(k_a\): actuator stack stiffness in [N/m]
• \(V_a\): applied voltage in [V]

If we take the numerical values, we obtain:

ga = d33*n*ka; % [N/V]

ga = 5.6 [N/V]

From (Fleming and Leang 2014) (page 123), the relation between relative displacement of the sensor stack and generated voltage is:

where:

• \(V_s\): measured voltage in [V]
• \(d_{33}\): strain constant in [m/V]
• \(\epsilon^T\): permittivity under constant stress in [F/m]
• \(s^D\): elastic compliance under constant electric displacement in [m^2/N]
• \(n\): number of layers of the sensor stack
• \(\Delta h\): relative displacement in [m]

If we take the numerical values, we obtain:

d33 = 300e-12; % Strain constant [m/V]
n   = 80;      % Number of layers per stack
eT  = 5.3e-9;  % Permittivity under constant stress [F/m]
sD  = 2e-11;   % Compliance under constant electric displacement [m2/N]

gs = d33/(eT*sD*n); % [V/m]

gs = 35.4 [V/um]

The idea here is to obtain the parameters \(g_a\) and \(g_s\) from the comparison of an experimental identification and the identification using Simscape.

Using the experimental identification, we can easily obtain the gain from the applied voltage to the generated displacement, but not to the generated force. However, from the Simscape model, we can easily have the link from the generated force to the displacement, them we can computed \(g_a\).

Similarly, it is fairly easy to experimentally obtain the gain from the stack displacement to the generated voltage across the stack. To link that to the strain of the sensor stack, the simscape model is used.

The obtained parameters \(g_a\) and \(g_s\) are not consistent between the different methods.

The one using the experimental data are saved and further used.

save('./matlab/mat/apa95ml_params.mat', 'ga', 'gs');

We first extract the stiffness and mass matrices.

K = readmatrix('APA95ML_K.CSV');
M = readmatrix('APA95ML_M.CSV');

Then, we extract the coordinates of the interface nodes.

[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA95ML_out_nodes_3D.txt');

The interface nodes are shown in Figure 9 and their coordinates are listed in Table 4.

Using K, M and int_xyz, we can use the Reduced Order Flexible Solid simscape block.

The flexible element is imported using the Reduced Order Flexible Solid Simscape block.

To model the actuator, an Internal Force block is added between the nodes 3 and 12. A Relative Motion Sensor block is added between the nodes 1 and 2 to measure the displacement and the amplified piezo.

One mass is fixed at one end of the piezo-electric stack actuator, the other end is fixed to the world frame.

m = 5.5;

load('apa95ml_params.mat', 'ga', 'gs');

The identified dynamics is shown in Figure 10.

%% Name of the Simulink File
mdl = 'piezo_amplified_3d';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'], 1, 'openinput');  io_i = io_i + 1; % Actuator Voltage [V]
io(io_i) = linio([mdl, '/y'],  1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]

Ghm = linearize(mdl, io);

The obtained dynamics is shown in Figure 11.

%% Name of the Simulink File
mdl = 'piezo_amplified_3d';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'], 1, 'openinput');  io_i = io_i + 1; % Voltage Actuator [V]
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage [V]

Gfm = linearize(mdl, io);

The data from the “noise test” and the identification test are loaded.

ht = load('huddle_test.mat', 't', 'u', 'y');
load('apa95ml_5kg_Amp_E505.mat', 't', 'um', 'y');

Any offset value is removed:

um = detrend(um, 0); % Input Voltage [V]
y  = detrend(y , 0); % Mass displacement [m]

ht.u  = detrend(ht.u, 0);
ht.y  = detrend(ht.y, 0);

Now we add a factor 10 to take into account the gain of the voltage amplifier.

um   = 10*um;

[tf_est, f] = tfestimate(um, -y, win, [], [], 1/Ts);
[co_est, ~] = mscohere(  um, -y, win, [], [], 1/Ts);

Comparison with the FEM model

load('mat/fem_simscape_models.mat', 'Ghm');

From the identified plant, a model of the transfer function from the actuator stack voltage to the force sensor generated voltage is developed.

w_z = 2*pi*111; % Zeros frequency [rad/s]
w_p = 2*pi*255; % Pole frequency [rad/s]
xi_z = 0.05;
xi_p = 0.015;
G_inf = 0.1;

Gi = G_inf*(s^2 + 2*xi_z*w_z*s + w_z^2)/(s^2 + 2*xi_p*w_p*s + w_p^2);

Its bode plot is shown in Figure 21.

The controller used in the Integral Force Feedback Architecture is:

where \(g\) is a gain that can be tuned.

Above 2 Hz the controller is basically an integrator, whereas an high pass filter is added at 0.5Hz to further reduce the low frequency gain.

In the frequency band of interest, this controller should mostly act as a pure integrator.

The Root Locus corresponding to this controller is shown in Figure 22.

The controller is now implemented in practice, and few controller gains are tested: \(g = 0\), \(g = 10\) and \(g = 100\).

For each controller gain, the identification shown in Figure 20 is performed.

iff_g10  = load('apa95ml_iff_g10_res.mat',  'u', 't', 'y', 'v');
iff_g100 = load('apa95ml_iff_g100_res.mat', 'u', 't', 'y', 'v');
iff_of   = load('apa95ml_iff_off_res.mat',  'u', 't', 'y', 'v');

Ts = 1e-4;
win = hann(ceil(10/Ts));

[tf_iff_g10, f] = tfestimate(iff_g10.u, iff_g10.y, win, [], [], 1/Ts);
[co_iff_g10, ~] = mscohere(iff_g10.u, iff_g10.y, win, [], [], 1/Ts);

[tf_iff_g100, ~] = tfestimate(iff_g100.u, iff_g100.y, win, [], [], 1/Ts);
[co_iff_g100, ~] = mscohere(iff_g100.u, iff_g100.y, win, [], [], 1/Ts);

[tf_iff_of, ~] = tfestimate(iff_of.u, iff_of.y, win, [], [], 1/Ts);
[co_iff_of, ~] = mscohere(iff_of.u, iff_of.y, win, [], [], 1/Ts);

The coherence between the excitation signal and the mass displacement as measured by the interferometer is shown in Figure 23.

The obtained transfer functions are shown in Figure 24. It is clear that the IFF architecture can actively damp the main resonance of the system.

Then, the same identification test is performed for many more gains.

load('apa95ml_iff_test.mat', 'results');

Ts = 1e-4;
win = hann(ceil(10/Ts));

tf_iff = {zeros(1, length(results))};
co_iff = {zeros(1, length(results))};
g_iff = [0, 1, 5, 10, 50, 100];

for i=1:length(results)
    [tf_est, f] = tfestimate(results{i}.u, results{i}.y, win, [], [], 1/Ts);
    [co_est, ~] = mscohere(results{i}.u, results{i}.y, win, [], [], 1/Ts);

    tf_iff(i) = {tf_est};
    co_iff(i) = {co_est};
end

The obtained dynamics are shown in Figure 25.

For each gain, the parameters of a second order resonant system that best fits the data are estimated and are compared with the data in Figure 26.

G_id = {zeros(1,length(results))};

f_start = 70; % [Hz]
f_end = 500; % [Hz]

for i = 1:length(results)
    tf_id = tf_iff{i}(sum(f<f_start):length(f)-sum(f>f_end));
    f_id = f(sum(f<f_start):length(f)-sum(f>f_end));

    gfr = idfrd(tf_id, 2*pi*f_id, Ts);
    G_id(i) = {procest(gfr,'P2UDZ')};
end

Finally, we can represent the position of the poles of the 2nd order systems on the Root Locus plot (Figure 27).

Bibliography