Build a State Space model from identified Modal Parameters

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?

Answer 1

[Star Strider]

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.

.

[Alan]

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]

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.

.