Updated 06 Dec 2019
The present submission is an algorithm used to automatically identify the eigenfrequencies, mode shapes and damping ratios of a line-like structure using ambient vibrations only. The covariance driven stochastic subspace identification method (SSI-COV) is used in combination with a clustering algorithm to automatically analyse the stabilization diagrams. The algorithm is inspired by the one used by Magalhaes et al. . It has been applied for ambient vibration monitoring of the Lysefjord Bridge  and was compared to the frequency domain decomposition technique . Finally, the algorithm was found accurate enough to visualise the evolution of the bridge eigenfrequencies with the temperature .
The submission file contains:
- A data file BridgeData.mat
- A Matlab Live Script Example1.mlx that illustrates the application of the algorithm.
- The function SSICOV which is the automated SSI-COV algorithm.
- The function plotStabDiag.m, which plot the stabilization diagram.
Any question, suggestion or comment is welcomed.
 Magalhaes, F., Cunha, A., & Caetano, E. (2009). Online automatic identification of the modal parameters of a long span arch bridge. Mechanical Systems and Signal Processing, 23(2), 316-329.
 Cheynet, E., Jakobsen, J. B., & Snæbjörnsson, J. (2016).Buffeting response of a suspension bridge in complex terrain. Engineering Structures, 128, 474-487.
 Cheynet, E., Jakobsen, J. B., & Snæbjörnsson, J. (2017).Damping estimation of large wind-sensitive structures.Procedia Engineering, 199, 2047-2053.
 Cheynet, E., Snæbjörnsson, J., & Jakobsen, J. B. (2017).Temperature Effects on the Modal Properties of a Suspension Bridge.In Dynamics of Civil Structures, Volume 2 (pp. 87-93). Springer.
E. Cheynet (2020). Operational modal analysis with automated SSI-COV algorithm (https://www.mathworks.com/matlabcentral/fileexchange/69030-operational-modal-analysis-with-automated-ssi-cov-algorithm), MATLAB Central File Exchange. Retrieved .
Note: The problem with Itachi was solved in private discussion. The main conclusions were: (1) the input must be the dynamic acceleration or displacement response, not the free decay response (2) The response along the line-like structure must be recorded in at least two different positions.
I get this error, I have an 8 second signal with dt= 0.005. Please, how can I get rid of this problem?
Error using SSICOV/blockHankel (line 196)
the IRF must be a 3D matrix with dimensions <M x M x N>
Error in SSICOV (line 95)
[U,S,V] = blockHankel(IRF);
Error in ex (line 59)
[fn0,zeta0,phi0,paraPlot] = SSICOV(Y,t(1));
Requires "linkage" and "cluster" from the Statistics and Machine Learning Toolbox.
The best method I have used ever.
Sometimes measured data is recorded as a column vector. How about putting these lines on line 82?
if Nyy > N
y = y';
thanks for replying .sir i am using matlab 2015 version.
Hi mati ullah,
I cannot reproduce the error you get. The compilation is running without crash in Matlab 2018b. I do not know which Matlab version you use but a too old version may be one possible reason for the error message.
i have run the example1.mlx with the default data and got the following errors, E. Cheynet sir kindly help me to eliminate these errors as i am new to matlab.
Error in SSICOV (line 149)
[fn3,zeta3,phi3] = myClusterFun(fnS,zetaS,phiS,MACS);
Error in Example1 (line 34)
[fn0,zeta0,phi0,paraPlot] = SSICOV(rz,dt,'Nmin',2,'Nmax',50,'Ts',15);
Thank you for the answer
the code is work, if i reduce sampling frequency about 100 times (Fs=1e-4)
By setting a maximal lag of 0.1 sec with a time step of 1e-6 sec, you create a covariance matrix that is [ M x M x 1e5], where M is the number of sensors. The resulting Block Toeplitz matrix is way too large to be handled by Matlab. To avoid crashing Matlab, you need to have a number much lower than 1e5.
I have a signal with length 0.1 sec and time step 1e-6.
when i put Ts=0.1 sec then there is a error:
"Error using zeros
Out of memory. Type HELP MEMORY for your options.
Error in SSICOV/blockHankel (line 205)
T1 = zeros(M*N1);
Error in SSICOV (line 97)
[U,S,V] = blockHankel(IRF);"
and when i put Ts=1 sec or greater, there is another error:
"Index exceeds matrix dimensions.
Error in SSICOV/NExT (line 270)
IRF(oo,jj,:) = h0(1:M);
Error in SSICOV (line 95)
[IRF,~] = NExT(y,dt,p.Results.Ts,p.Results.methodCOV);"
What is the problem and how i can get true results.
I see, thanks for explanation,
In the article "Cardoso R., Cury A., Barbosa F., 2015 - On the advances of automatic modal identification for SHM" they attempt a similar non-normalized distance between mode estimates, but now including MAC that it scaled with arbitrary constant with [Hz] units, maybe you can take a look at that for inspiration. Maybe, it would be cool for the average user to stick to one of the chosen distance metrics, but for the advanced user to be able to choose among a few or even better, load his own distance function as an optional parameter. Similar to MATLAB's own implementation of "pdist".
That's a good remark! I will try to see if I can implement more rigorously the MAC number for pos = fn0(:)+1-MAC0(:); in a further version of the submission.
If the MAC number is not included, the clusters are selected based on the criterion 'eps_cluster'. I do not know until which point the cluster analysis will work properly without using the MAC number as an additional criterion. So far, I have been able to distinguish modes separated by only 0.009 Hz for a 1.3 km long span bridge using a low value for 'eps_cluster'.
Thank you for the answer,
Another question, if MAC is not used in the distance, is it then possible to detect closely spaced modes. I see that due to their closeness in frequency they will be clustered together and MAC could be useful to avoid it?
That is a very good point! I initially defined the variable "pos" as a quantity with the same dimension as a frequency because it was more intuitive to use the criterion "eps_cluster" this way. I had forgotten to remove the "1-MAC0(:)" in the previous version of the function SSICOV. The function still worked well because the dataset investigated in the nested function corresponds already to stables poles, such that 1-MAC0 is almost zero.
In the paper from Magalhaes et al , the variable pos is indeed without dimension. In the updated version, I have used the definition used in Ref. . However, because the relative distance is now used for the cluster analysis, the criterion "eps_cluster" is much larger in the updated function. I think that the choice of pos = fn0(:) is still acceptable since the MAC number has a limited influence on the cluster analysis. Another argument for using "pos = fn0(:)" is to avoid introducing a dependency between the criteria "eps_cluster" and "eps_MAC".
Hi, thank you very much for your contribution,
I have a question regarding line 450 in SSICOV function,
when you cluster, you firstly calculate positions with:
pos = fn0(:)+1-MAC0(:);
Magalhaes in paper  uses the same distance, but the frequency is normalized, so is it a mistake in your implementation or you meant to use this distance?
Anyways, really appreciate your work, it helps to develop Structural Health Monitoring toolbox for my university
Hi Xinzhe Yuan,
The error is probably triggered because the cross-covariance matrix is not built properly. I will update the function to provide a more robust definition of T1. However, it will probably not solve your problem, but simply trigger a new error. I suggest you to check whether the time lag you use as input (see the optional arguments) is appropriate.
Hi there, I got an error:
Undefined function or variable 'T1'.
Error in SSICOV/blockHankel (line 207)
Much appreicated if you can have a look at it.
typos corrected + minor modification to improve the performances
Set the default value of Ts as 500*dt instead of 20 s to avoid the crashing problem of Matlab if a very high sampling frequency is used. Ideally, Ts should be between two and six times the value of the lowest eigenfrequency of the system.
The distance parameter "pos" for the clustering algorithm is now properly defined using both the difference in terms of frequencies and the MAC number (thank Mihhail Samusev for the help!). Finally, the description of the submission is updated.
Updated the definition for the variable "pos" (Cluster analysis) + The default value for the time lag (cross-covariance) is more robustly defined + the variable T1 is preallocated