How can I get a time response from FRD model

I have a model in FRD mode, I want to a) get a time response of this model, b) implement it in Simulink (both seems to be unavailable in Matlab)

Answer 1

[Sam Chak]

Hi @Bentolila Yedidia

There isn't a direct function called 'frd2ss()' for the conversion you're looking for. However, you can estimate a state-space model from available frequency response data (FRD) using the 'ssest()' function. This function will provide you with the estimated state-space model, which you can then use in Simulink for simulation. This will allow you to obtain the time response of the equivalent FRD model.

%% Dummy state-space for FRD generation
ssA    = [0 1; -1 -2];
ssB    = [0; 1];
ssC    = [1 0];
ssD    = 0*ssC*ssB;
ss_Sys = ss(ssA, ssB, ssC, ssD);

%% Obtain Frequency-Response Data model
w      = logspace(-2, 2, 50);
frdSys = frd(ss_Sys, w)

frdSys =
 
    Frequency(rad/s)            Response       
    ----------------            --------       
          0.0100         9.997e-01 - 2.000e-02i
          0.0121         9.996e-01 - 2.413e-02i
          0.0146         9.994e-01 - 2.911e-02i
          0.0176         9.991e-01 - 3.513e-02i
          0.0212         9.987e-01 - 4.238e-02i
          0.0256         9.980e-01 - 5.112e-02i
          0.0309         9.971e-01 - 6.166e-02i
          0.0373         9.958e-01 - 7.435e-02i
          0.0450         9.939e-01 - 8.961e-02i
          0.0543         9.912e-01 - 1.079e-01i
          0.0655         9.872e-01 - 1.299e-01i
          0.0791         9.814e-01 - 1.562e-01i
          0.0954         9.731e-01 - 1.874e-01i
          0.1151         9.611e-01 - 2.243e-01i
          0.1389         9.439e-01 - 2.675e-01i
          0.1677         9.194e-01 - 3.173e-01i
          0.2024         8.851e-01 - 3.735e-01i
          0.2442         8.375e-01 - 4.350e-01i
          0.2947         7.730e-01 - 4.990e-01i
          0.3556         6.884e-01 - 5.605e-01i
          0.4292         5.817e-01 - 6.121e-01i
          0.5179         4.549e-01 - 6.440e-01i
          0.6251         3.150e-01 - 6.464e-01i
          0.7543         1.751e-01 - 6.128e-01i
          0.9103         5.124e-02 - 5.444e-01i
          1.0985        -4.246e-02 - 4.512e-01i
          1.3257        -9.962e-02 - 3.487e-01i
          1.5999        -1.231e-01 - 2.525e-01i
          1.9307        -1.220e-01 - 1.728e-01i
          2.3300        -1.072e-01 - 1.128e-01i
          2.8118        -8.707e-02 - 7.090e-02i
          3.3932        -6.714e-02 - 4.334e-02i
          4.0949        -4.995e-02 - 2.594e-02i
          4.9417        -3.624e-02 - 1.529e-02i
          5.9636        -2.585e-02 - 8.921e-03i
          7.1969        -1.822e-02 - 5.164e-03i
          8.6851        -1.274e-02 - 2.973e-03i
         10.4811        -8.858e-03 - 1.706e-03i
         12.6486        -6.135e-03 - 9.761e-04i
         15.2642        -4.237e-03 - 5.576e-04i
         18.4207        -2.921e-03 - 3.181e-04i
         22.2300        -2.011e-03 - 1.813e-04i
         26.8270        -1.384e-03 - 1.033e-04i
         32.3746        -9.514e-04 - 5.883e-05i
         39.0694        -6.538e-04 - 3.349e-05i
         47.1487        -4.492e-04 - 1.906e-05i
         56.8987        -3.086e-04 - 1.085e-05i
         68.6649        -2.120e-04 - 6.175e-06i
         82.8643        -1.456e-04 - 3.514e-06i
        100.0000        -9.997e-05 - 2.000e-06i
 
Continuous-time frequency response.

bode(ss_Sys,'b', frdSys,'r--'), grid on

%% Convert Frequency-Response Data model to Linear Time-Invariant System
ltiSys = ssest(frdSys)

ltiSys =
  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
   x1          -1           1
   x2  -2.302e-14          -1
 
  B = 
       u1
   x1   0
   x2   1
 
  C = 
       x1  x2
   y1   1   0
 
  D = 
       u1
   y1   0
 
  K = 
       y1
   x1   0
   x2   0
 
Parameterization:
   FREE form (all coefficients in A, B, C free).
   Feedthrough: none
   Disturbance component: none
   Number of free coefficients: 8
   Use "idssdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:                                                   
Estimated using SSEST on frequency response data "frdSys".
Fit to estimation data: 100%                              
FPE: 9.314e-30, MSE: 7.935e-30                            
 

%% Extract matrices and store in Workspace for State-space block in Simulink 
A = ltiSys.A;   % state  matrix
B = ltiSys.B;   % input  matrix
C = ltiSys.C;   % output matrix
D = ltiSys.D;   % direct matrix

%% Time response of the equivalent FRD model
step(ltiSys), grid on