Estimate Process Models at the Command Line

Prerequisites

Before you can perform this task, you must have

Input-output data as an iddata object or frequency response data as frd or idfrd objects. See Representing Time- and Frequency-Domain Data Using iddata Objects. For supported data formats, see Data Supported by Process Models.
Performed any required data preprocessing operations. When working with time domain data, if you need to model nonzero offsets, such as when model contains integration behavior, do not detrend your data. In other cases, to improve the accuracy of your model, you should detrend your data. See Ways to Prepare Data for System Identification.

Using procest to Estimate Process Models

You can estimate process models using the iterative estimation method procest that minimizes the prediction errors to obtain maximum likelihood estimates. The resulting models are stored as idproc model objects.

You can use the following general syntax to both configure and estimate process models:

m = procest(data,mod_struc,opt)

data is the estimation data and mod_struc is one of the following:

A character vector that represents the process model structure, as described in Process Model Structure Specification.
A template idproc model. opt is an option set for configuring the estimation of the process model, such as handling of initial conditions, input offset and numerical search method.

Tip

You do not need to construct the model object using idproc before estimation unless you want to specify initial parameter guesses, minimum/maximum bounds, or fixed parameter values, as described in Estimate Process Models with Fixed Parameters.

For more information about validating a process model, see Validating Models After Estimation.

You can use procest to refine parameter estimates of an existing process model, as described in Refine Linear Parametric Models.

For detailed information, see procest and idproc.

Estimate Process Models with Free Parameters

Open Live Script

This example shows how to estimate the parameters of a first-order process model:

$G (s) = \frac{K_{p}}{1 + s T_{p 1}} e^{- s T_{d}}$ .

This process has two inputs and the response from each input is estimated by a first-order process model. All parameters are free to vary.

Load estimation data.

load co2data

Specify known sample time of 0.5 min.

Ts = 0.5;

Split data set into estimation data ze and validation data zv.

ze = iddata(Output_exp1,Input_exp1,Ts,...
                        'TimeUnit','min');
zv = iddata(Output_exp2,Input_exp2,Ts,...
                        'TimeUnit','min');

Estimate model with one pole, a delay, and a first-order disturbance component. The data contains known offsets. Specify them using the InputOffset and OutputOffset options.

opt = procestOptions;
opt.InputOffset = [170;50];
opt.OutputOffset = -45;
opt.Display = 'on';
opt.DisturbanceModel = 'arma1';
m = procest(ze,'p1d',opt)

m =
 
Process model with 2 inputs: y = G11(s)u1 + G12(s)u2        
  From input "u1" to output "y1":                           
             Kp                                             
  G11(s) = ---------- * exp(-Td*s)                          
            1+Tp1*s                                         
                                                            
        Kp = 2.6553                                         
       Tp1 = 0.15515                                        
        Td = 2.3175                                         
                                                            
  From input "u2" to output "y1":                           
             Kp                                             
  G12(s) = ---------- * exp(-Td*s)                          
            1+Tp1*s                                         
                                                            
        Kp = 9.9756                                         
       Tp1 = 2.0653                                         
        Td = 4.9195                                         
                                                            
  An additive ARMA disturbance model exists for output "y1":
      y = G u + (C/D)e                                      
                                                            
      C(s) = s + 2.676                                      
      D(s) = s + 0.6228                                     
                                                            
Parameterization:
    {'P1D'}    {'P1D'}
   Number of free coefficients: 8
   Use "getpvec", "getcov" for parameters and their uncertainties.

Status:                                          
Estimated using PROCEST on time domain data "ze".
Fit to estimation data: 91.07% (prediction focus)
FPE: 2.431, MSE: 2.412

Use dot notation to get the value of any model parameter. For example, get the value of dc gain parameter Kp.

m.Kp

ans = 1×2

    2.6553    9.9756

Estimate Process Models with Fixed Parameters

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This example shows how to estimate a process model with fixed parameters.

When you know the values of certain parameters in the model and want to estimate only the values you do not know, you must specify the fixed parameters after creating the idproc model object. Use the following commands to prepare the data and construct a process model with one pole and a delay:

Load estimation data.

load co2data

Specify known sample time is 0.5 minutes.

Ts = 0.5;

Split data set into estimation data ze and validation data zv.

ze = iddata(Output_exp1,Input_exp1,Ts,...
                        'TimeUnit','min');
zv = iddata(Output_exp2,Input_exp2,Ts,...
                        'TimeUnit','min');
mod = idproc({'p1d','p1d'},'TimeUnit','min')

mod =
 
Process model with 2 inputs: y = G11(s)u1 + G12(s)u2
  From input 1 to output 1:                         
             Kp                                     
  G11(s) = ---------- * exp(-Td*s)                  
            1+Tp1*s                                 
                                                    
        Kp = NaN                                    
       Tp1 = NaN                                    
        Td = NaN                                    
                                                    
  From input 2 to output 1:                         
             Kp                                     
  G12(s) = ---------- * exp(-Td*s)                  
            1+Tp1*s                                 
                                                    
        Kp = NaN                                    
       Tp1 = NaN                                    
        Td = NaN                                    
                                                    
Parameterization:
    {'P1D'}    {'P1D'}
   Number of free coefficients: 6
   Use "getpvec", "getcov" for parameters and their uncertainties.

Status:                                                         
Created by direct construction or transformation. Not estimated.

The model parameters Kp , Tp1 , and Td are assigned NaN values, which means that the parameters have not yet been estimated from the data.

Use the Structure model property to specify the initial guesses for unknown parameters, minimum/maximum parameter bounds and fix known parameters.

Set the value of Kp for the second transfer function to 10 and specify it as a fixed parameter. Initialize the delay values for the two transfer functions to 2 and 5 minutes, respectively. Specify them as free estimation parameters.

mod.Structure(2).Kp.Value = 10;
mod.Structure(2).Kp.Free = false;

mod.Structure(1).Tp1.Value = 2;
mod.Structure(2).Td.Value = 5;

Estimate Tp1 and Td only.

mod_proc = procest(ze,mod)

mod_proc =
 
Process model with 2 inputs: y = G11(s)u1 + G12(s)u2
  From input "u1" to output "y1":                   
             Kp                                     
  G11(s) = ---------- * exp(-Td*s)                  
            1+Tp1*s                                 
                                                    
        Kp = -3.2213                                
       Tp1 = 2.1707                                 
        Td = 4.44                                   
                                                    
  From input "u2" to output "y1":                   
             Kp                                     
  G12(s) = ---------- * exp(-Td*s)                  
            1+Tp1*s                                 
                                                    
        Kp = 10                                     
       Tp1 = 2.0764                                 
        Td = 4.5205                                 
                                                    
Parameterization:
    {'P1D'}    {'P1D'}
   Number of free coefficients: 5
   Use "getpvec", "getcov" for parameters and their uncertainties.

Status:                                          
Estimated using PROCEST on time domain data "ze".
Fit to estimation data: 77.44%                   
FPE: 15.5, MSE: 15.39

In this case, the value of Kp is fixed, but Tp1 and Td are estimated.