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System Identification Using Eigensystem Realization Algorithm (ERA)

This example considers a 3-degree-of-freedom (DOF) system that is excited by a series of hammer strikes. The resulting displacements are recorded by sensors. The system is proportionally damped such that the dampling matrix is a linear combination of the mass and stiffness matrices. This example requires Signal Processing Toolbox™ for the section on modal analysis.

3DOF.png

Preprocess Data

Import data for the first set of measurements. The data includes the excitation signals, reponse signals, time signals, and ground truth frequency-response functions. The response signal, denoted by Y1, gives the displacement of the first mass. The excitation signal consists of ten concatenated hammer impacts, and the response signal contains the corresponding displacement. The duration for each impact signal is 2.53 seconds. The excitation and response signals are corrupted with additive noise.

load modaldata XhammerMISO1 YhammerMISO1 fs; 
rng('default');

% Add noise to excitation and response signals
XhammerMISO1 = XhammerMISO1 + randn(size(YhammerMISO1))/1250;
YhammerMISO1 = YhammerMISO1 + randn(size(YhammerMISO1))/1e11;

% Define outputs from function
t = (0:size(XhammerMISO1,1)-1)/fs';
X1 = 1e2*XhammerMISO1;
Y1 = 1e2*YhammerMISO1;
X0 = X1(:,1);
Y0 = Y1(:,1);

Visualize the first excitation and the response channel of the measurement.

subplot(2,1,1)
plot(t,X0)
xlabel('Time (s)')
ylabel('Force (N)')
grid on;
title('Excitation and Response for a 3DOF System')
subplot(2,1,2)
plot(t,Y0)
xlabel('Time (s)')
ylabel('Displacement (m)')
grid on;

Figure contains 2 axes objects. Axes object 1 with title Excitation and Response for a 3DOF System contains an object of type line. Axes object 2 contains an object of type line.

Identify System Using ERA

Create iddata objects for estimation and validation data.

Ts = 1/fs; % sample time
estimationData = iddata(Y0(1:1000), X0(1:1000), 1/fs);
validationData = iddata(Y0(1001:2000), X0(1001:2000), 1/fs);

Visualize the estimation data.

figure
plot(estimationData)

Figure contains 2 axes objects. Axes object 1 with title y1 contains an object of type line. This object represents estimationData. Axes object 2 with title u1 contains an object of type line. This object represents estimationData.

The plot of the input data shows the presence of an input delay. Remove the delay from the data.

[~,inputDelay] = max(estimationData.InputData);
estimationData = estimationData(inputDelay:end);

The era function requires data to be passed as a timetable or numeric matrix. Convert the iddata object into a timetable.

L = length(estimationData.InputData);
t = seconds(estimationData.Tstart + (0:L-1)'*Ts);
y1 = estimationData.OutputData;
tt = timetable(t,y1);

Use era to estimate a state-space model using this estimation data.

order = 6;
sys = era(tt, order, 'Feedthrough', true)
sys =
  Discrete-time identified state-space model:
    x(t+Ts) = A x(t) + B u(t) + K e(t)
       y(t) = C x(t) + D u(t) + e(t)
 
  A = 
               x1          x2          x3          x4          x5          x6
   x1      0.8339     -0.5523   -0.001099    0.003053   -0.001497   0.0004633
   x2      0.5523      0.8323    -0.00714   -0.001638   -0.002676   0.0008873
   x3   -0.001099     0.00714      0.2293      0.9709  -0.0009399  -0.0006184
   x4   -0.003053   -0.001638     -0.9709      0.2289    0.006094   -0.002101
   x5   -0.001497    0.002676  -0.0009399   -0.006094     -0.5463     -0.8302
   x6  -0.0004633   0.0008873   0.0006184   -0.002101      0.8302     -0.5465
 
  B = 
               u1
   x1   -0.001403
   x2   0.0007461
   x3   -0.001515
   x4  -0.0001808
   x5   -0.000818
   x6  -0.0002384
 
  C = 
               x1          x2          x3          x4          x5          x6
   y1  -3.508e-07  -1.865e-07  -3.787e-07    4.52e-08  -2.045e-07   5.959e-08
 
  D = 
              u1
   y1  1.852e-10
 
  K = 
       y1
   x1   0
   x2   0
   x3   0
   x4   0
   x5   0
   x6   0
 
Sample time: 0.00025 seconds

Parameterization:
   FREE form (all coefficients in A, B, C free).
   Feedthrough: yes
   Disturbance component: none
   Number of free coefficients: 49
   Use "idssdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:                                              
Estimated using the Eigensystem Realization Algorithm
 
Model Properties

Compare the response of the estimated system with the validation data.

compare(validationData,sys);

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent Validation data (y1), sys: 99.88%.

Compare the performance of the estimated model against another state-space model estimated using ssest.

sys2 = ssest(estimationData, order, 'Feedthrough', true)
sys2 =
  Continuous-time identified state-space model:
      dx/dt = A x(t) + B u(t) + K e(t)
       y(t) = C x(t) + D u(t) + e(t)
 
  A = 
           x1      x2      x3      x4      x5      x6
   x1    4.99    2323   66.74  -241.3   35.31  -5.076
   x2   -2341  -7.761   300.1  -11.19    11.2   25.28
   x3  -49.75    -311  -5.595   -5360   30.91    15.2
   x4   270.4   6.758    5333  -15.04  -1.474  -64.71
   x5  -52.95  -16.58   14.17   34.15   3.021    8548
   x6  -41.44  -15.95    16.8   134.7   -8674  -51.74
 
  B = 
             u1
   x1  0.005996
   x2   0.03915
   x3   -0.2075
   x4   -0.1881
   x5     1.007
   x6   -0.9297
 
  C = 
               x1          x2          x3          x4          x5          x6
   y1   2.998e-05  -2.043e-05   1.586e-05   8.501e-06  -2.459e-06  -2.867e-06
 
  D = 
             u1
   y1  5.33e-09
 
  K = 
               y1
   x1   9.881e+06
   x2  -8.603e+06
   x3   4.435e+06
   x4   2.032e+07
   x5  -7.865e+07
   x6   4.392e+07
 
Parameterization:
   FREE form (all coefficients in A, B, C free).
   Feedthrough: yes
   Disturbance component: estimate
   Number of free coefficients: 55
   Use "idssdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:                                                    
Estimated using SSEST on time domain data "estimationData".
Fit to estimation data: 99.6% (prediction focus)           
FPE: 5.081e-17, MSE: 4.773e-17                             
 
Model Properties
compare(validationData,sys,sys2);
legend('Validation data','ERA','SSEST');

Figure contains an axes object. The axes object contains 3 objects of type line. These objects represent Validation data, ERA: 99.88%, SSEST: 99.83%.

The two models have a similar fit.

Modal Analysis

Perform a modal analysis of the state-space model estimated using era by using the modalfit function from Signal Processing Toolbox. modalfit returns the damping ratios zeta and the damped natural frequencies fd.

[~,f] = modalfrf(sys);
[fd, zeta] = modalfit(sys,f,3);

View zeta.

zeta
zeta = 3×1

    0.0008
    0.0018
    0.0028

View fd.

fd
fd = 3×1
103 ×

    0.3727
    0.8525
    1.3706

Compare these results with the modes obtained from the model estimated using ssest.

[frf,f] = modalfrf(sys2);
[fd, zeta] = modalfit(sys2,f,3)
fd = 3×1
103 ×

    0.3727
    0.8525
    1.3706

zeta = 3×1

    0.0008
    0.0018
    0.0029

There is a close match between the modes obtained from modalfit for the second estimated model and those obtained for the model estimated using era.

Reduced-order modeling comparison

Compare the performance of era and ssest when using a lower model order for system identification.

order = 4;
sys = era(tt, order, 'Feedthrough', true);
sys2 = ssest(estimationData, order, 'Feedthrough', true);

Compare the performance of both models as before.

compare(validationData,sys,sys2);
legend('Validation data','ERA','SSEST');

Figure contains an axes object. The axes object contains 3 objects of type line. These objects represent Validation data, ERA: 99.84%, SSEST: 87.57%.

For the reduced-order model, the ERA model has a significantly better fit than the SSEST model. This result demonstrates the capability of the ERA approach in obtaining state-space models from noisy impulse response data.

See Also

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