Build a State Space model from identified Modal Parameters

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Alan
Alan 2022-12-5
评论： Star Strider 2022-12-6
I have run the modalfit example, "ModalParametersUsingLeastSquaresRationalFunctionMethodExample.mlx" I get the modal frequencies, damping rates and mode shapes. I couldn't find a function for turning that data into a State Space model.
How do I do that?

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Star Strider
Star Strider 2022-12-5
编辑：Star Strider 2022-12-6
Ran in:
That may be possible using the Signal Processing Toolbox invfreqz and tf2ss functions. I put the idfrd and ssest calls at the end, for comaprison.
Make appropriate changes to get the correct tranfer function order and state space realisation —
load modaldata SpaceStationFRF
frf = SpaceStationFRF.FRF;
f = SpaceStationFRF.f;
fs = SpaceStationFRF.Fs;
% nf = max(f)/fs*2*pi
Sz1 = size(frf)
Sz1 = 1×3
1000 3 3
figure
semilogy(f, abs(frf(:,1,1)))
grid
[b,a] = invfreqz(frf(:,1,1),f/fs*2*pi,4,4) % Transfer Function From Frequency Response
b = 1×5
1.0e-04 * 0.1052 0.0348 -0.2793 0.0478 0.0948
a = 1×5
1.0000 -2.0241 0.1291 1.8385 -0.9435
[A,B,C,D] = tf2ss(b,a) % State Space Realisation
A = 4×4
2.0241 -0.1291 -1.8385 0.9435 1.0000 0 0 0 0 1.0000 0 0 0 0 1.0000 0
B = 4×1
1 0 0 0
C = 1×4
1.0e-04 * 0.2478 -0.2929 -0.1457 0.1941
D = 1.0524e-05
% Extract the modal parameters of the lowest 24 modes using the least-squares rational function method.
[fn,dr,ms,ofrf] = modalfit(frf,f,fs,24,'FitMethod','lsrf');
% Compare the reconstructed FRF array to the measured one.
figure
for ij = 1:3
for ji = 1:3
subplot(3,3,3*(ij-1)+ji)
loglog(f,abs(frf(:,ji,ij)))
hold on
loglog(f,abs(ofrf(:,ji,ij)))
hold off
axis tight
title(sprintf('In%d -> Out%d',ij,ji))
if ij==3
xlabel('Frequency (Hz)')
end
end
end
% whos
FN = fieldnames(SpaceStationFRF)
FN = 3×1 cell array
{'FRF'} {'f' } {'Fs' }
FRF11_Data = idfrd(frf(:,1,1), f, 1/fs)
FRF11_Data = IDFRD model. Contains Frequency Response Data for 1 output(s) and 1 input(s). Response data is available at 1000 frequency points, ranging from 0.01592 rad/s to 159.2 rad/s. Sample time: 0.003125 seconds Status: Created by direct construction or transformation. Not estimated.
ss_sys = ssest(FRF11_Data)
ss_sys = 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 x1 -0.001584 0.1563 1.977e-07 -3.976e-05 x2 -0.6433 -0.001584 -0.0008179 0.1629 x3 -1.083e-10 -6.133e-10 -0.0006166 0.1563 x4 9.888e-10 -8.261e-10 -0.09739 -0.0006166 B = u1 x1 2.587e-08 x2 -7.953e-05 x3 -7.629e-05 x4 0.3125 C = x1 x2 x3 x4 y1 -0.001145 -6.471e-06 -4.126e-06 0.0007976 D = u1 y1 0 K = y1 x1 0 x2 0 x3 0 x4 0 Parameterization: FREE form (all coefficients in A, B, C free). Feedthrough: none Disturbance component: none Number of free coefficients: 24 Use "idssdata", "getpvec", "getcov" for parameters and their uncertainties. Status: Estimated using SSEST on frequency response data "FRF11_Data". Fit to estimation data: 90.97% FPE: 2.13e-07, MSE: 2.096e-07
EDIT — (6 Dec 2022 at 00:04)
Minor code change.
.
  2 个评论
Alan
Alan 2022-12-6
Thanks for the answer. But this seems to involve two "fitting" steps. The first to get the modal parameter from modalfit and the second using invfreqz. I think we could have skipped a step.
I was looking for a function that would create the state space matrices from the modal parameters and vectors directly. I think I would know how to do that; I was hoping MATLAB had a function already (which I hadn't been able to find.)
Star Strider
Star Strider 2022-12-6
My pleasure!
The System Identification Toolbox has two functions that can be used for this, those being idfrd and ssest, coded here as:
FRF11_Data = idfrd(frf(:,1,1), f, 1/fs)
ss_sys = ssest(FRF11_Data)
It’s possible to combine those into one function call:
ss_sys = ssest(idfrd(frf(:,1,1), f, 1/fs))
and perhaps even create an anonymous function to do that:
ss_frd = @(frd,f,ts) ssest(idfrd(frd, f, ts))
ss_sys = ss_frd(frf(:,1,1), f, 1/fs)
There otherwsie does not appear to be an existing function for that. If it were necessary to include other arguments to the functions within ‘ss_frd’, they would need to be added to its argument list as well, and then referenced in the appropriate functions.
.

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