Effect of Input Intersample Behavior on Continuous-Time Models

The intersample behavior of the input signals influences the estimation, simulation and prediction of continuous-time models. A sampled signal is characterized only by its values at the sampling instants. However, when you apply a continuous-time input to a continuous-time system, the output values at the sampling instants depend on the inputs at the sampling instants and on the inputs between these points.

The iddata and idfrd objects have an InterSample property which stores how the input behaves between the sampling instants. You can specify the behavior between the samples to be piecewise constant (zero-order hold), linearly interpolated between the samples (first-order hold) or band-limited. A band-limited intersample behavior of the input signal means:

When the input signal is a band-limited discrete-time frequency-domain data (iddata with domain = 'frequency' or idfrd with sample time Ts≠0), the model estimation is performed by treating the data as continuous-time data (Ts = 0). For more information, see Pintelon, R. and J. Schoukens, System Identification. A Frequency Domain Approach, section 10.2, pp-352-356,Wiley-IEEE Press, New York, 2001.

The intersample behavior of the input data also affects the results of simulation and prediction of continuous-time models. sim and predict commands use the InterSample property to choose the right algorithm for computing model response.

The following example simulates a system using first-order hold ( foh ) intersample behavior for input signal.

sys = idtf([-1 -2],[1 2 1 0.5]);
rng('default')
u = idinput([100 1 5],'sine',[],[],[5 10 1]);
Ts = 2;
y = lsim(sys,u,(0:Ts:999)','foh');

Create an iddata object for the simulated input-output data.

data = iddata(y,u,Ts);

The default intersample behavior is zero-order hold (zoh).

data.InterSample

ans = 
'zoh'

Estimate a transfer function using this data.

np = 3; % number of poles
nz = 1; % number of zeros
opt = tfestOptions('InitializeMethod','all','Display','on');
opt.SearchOptions.MaxIterations = 100;	
modelZOH = tfest(data,np,nz,opt)

modelZOH =
 
  From input "u1" to output "y1":
          -217.2 s - 391.6
  ---------------------------------
  s^3 + 354.4 s^2 + 140.2 s + 112.4
 
Continuous-time identified transfer function.

Parameterization:
   Number of poles: 3   Number of zeros: 1
   Number of free coefficients: 5
   Use "tfdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:                                          
Estimated using TFEST on time domain data "data".
Fit to estimation data: 81.38%                   
FPE: 0.1146, MSE: 0.111                          
 

The model gives about 80% fit to data. The sample time of the data is large enough that intersample inaccuracy (using zoh rather than foh ) leads to significant modeling errors.

Re-estimate the model using foh intersample behavior.

data.InterSample = 'foh';
modelFOH = tfest(data,np,nz,opt)

modelFOH =
 
  From input "u1" to output "y1":
           -1.197 s - 0.06843
  -------------------------------------
  s^3 + 0.4824 s^2 + 0.3258 s + 0.01723
 
Continuous-time identified transfer function.

Parameterization:
   Number of poles: 3   Number of zeros: 1
   Number of free coefficients: 5
   Use "tfdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:                                          
Estimated using TFEST on time domain data "data".
Fit to estimation data: 97.7%                    
FPE: 0.001748, MSE: 0.001693                     
 

modelFOH is able to retrieve the original system correctly.

Compare the model outputs with data.

compare(data,modelZOH,modelFOH)

modelZOH is compared to data whose intersample behavior is foh. Therefore, its fit decreases to around 70%.

See Also

iddata | idfrd

Related Topics

• Frequency Domain Identification: Estimating Models Using Frequency Domain Data