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Estimate process model using time-domain or frequency-domain data


Estimate Process Model


sys = procest(data,type) estimates a process model sys using time-domain or frequency-domain data data. type defines the structure of sys.

A simple SISO process model has a gain, a time constant, and a delay:


Kp is a proportional gain. Tp1 is the time constant of the real pole, and Td is the transport delay (dead time). More complex process models can include zeroes, additional time constants, complex poles, and integration. For more information on process models, see idproc.


sys = procest(data,type,'InputDelay',InputDelay) specifies the input delay InputDelay.

Configure Initial Parameters


sys = procest(data,init_sys) uses the process model init_sys to configure the initial parameterization.

Specify Additional Options


sys = procest(___,opt) specifies additional model estimation options. Use opt with any of the input argument combinations in the previous syntaxes.

Return Estimated Offset and Initial Conditions


[sys,offset] = procest(___) returns the estimated value of the offset in input signal. procest automatically estimates the input offset when the model contains an integrator or when you set the InputOffset estimation option to 'estimate' using procestOptions.


[sys,offset,ic] = procest(___) returns the estimated initial conditions as an initialCondition object. Use this syntax if you plan to simulate or predict the model response using the same estimation input data and then compare the response with the same estimation output data. Incorporating the initial conditions yields a better match during the first part of the simulation.


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Estimate a process model and compare its response with the measured output.

Load the input/output data, which is stored in an iddata object z1.

load iddata1 z1

Estimate a first-order process model sys that contains one pole and no zeroes or delays. This model structure has type P1.

sys = procest(z1,'P1');

Compare the simulated model response with the measured output.


Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent z1 (y1), sys: 33.65%.

The fit percentage for the model is low. Add a delay to the model and compare the simulated and measured outputs.

sys = procest(z1,'P1D');

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent z1 (y1), sys: 44.87%.

The fit percentage has improved, but is still below 50%. The plot shows that the model output peaks do not attain the height of the measured output peaks, which indicates that the model needs to include more dynamics.

Create a second-order process model with complex (underdamped) poles.

sys = procest(z1,'P2U');

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent z1 (y1), sys: 70.44%.

The fit now exceeds 70%.

You can view more information about the estimation by exploring the idproc property sys.Report.

ans = 
              Status: 'Estimated using PROCEST'
              Method: 'PROCEST'
    InitialCondition: 'zero'
                 Fit: [1x1 struct]
          Parameters: [1x1 struct]
         OptionsUsed: [1x1 idoptions.procest]
           RandState: []
            DataUsed: [1x1 struct]
         Termination: [1x1 struct]

View the estimated gain Kp.

Kp = sys.Kp
Kp = 7.6818

Estimate a process model after specifying initial guesses for parameter values and bounding them.

Obtain input/output data.

data = idfrd(idtf([10 2],[1 1.3 1.2],'iod',0.45),logspace(-2,2,256));

Specify the parameters of the estimation initialization model.

type = 'P2UZD';
init_sys = idproc(type);

init_sys.Structure.Kp.Value = 1; 
init_sys.Structure.Tw.Value = 2; 
init_sys.Structure.Zeta.Value = 0.1; 
init_sys.Structure.Td.Value = 0; 
init_sys.Structure.Tz.Value = 1; 
init_sys.Structure.Kp.Minimum = 0.1;
init_sys.Structure.Kp.Maximum = 10;
init_sys.Structure.Td.Maximum = 1;
init_sys.Structure.Tz.Maximum = 10;

Specify the estimation options.

opt = procestOptions('Display','full','InitialCondition','Zero');
opt.SearchMethod = 'lm'; 
opt.SearchOptions.MaxIterations = 100;

Estimate the process model.

sys = procest(data,init_sys,opt);

Since the 'Display' option is specified as 'full', the estimation progress is displayed in a separate Plant Identification Progress window.

Compare the data to the estimated model.


Figure contains 2 axes objects. Axes object 1 with title From: u1 contains 2 objects of type line. These objects represent Data, sys: 100%. Axes object 2 contains 2 objects of type line. These objects represent Data, sys: 100%.

load iddata1
[sys,offset] = procest(z1,'P1DI');
offset = 0.0412

Load the data.

load iddata1ic z1i

Estimate a first-order plus dead time process model sys and return the initial conditions in ic. First specify 'estimate' for 'InitialCondition' to force the software to estimate ic. The default 'auto' setting uses the 'estimate' method only when the influence of the initial conditions on the overall model error exceed a threshold. When the initial conditions have a negligible effect on the overall estimation-error minimization process, the 'auto' setting uses 'zero'.

opt = procestOptions('InitialCondition','estimate');
[sys,offset,ic] = procest(z1i,'P1D',opt);
ic = 
  initialCondition with properties:

     A: -3.8997
    X0: -1.0871
     C: 4.5652
    Ts: 0

ic is an initialCondition object that encapsulates the free response of sys, in state-space form, to the initial state vector in X0. You can incorporate ic when you simulate sys with the z1i input signal and compare the response with the z1i output signal.

Obtain input/output data.

load iddata1 z1
load iddata2 z2
data = [z1 z2(1:300)];

data is a data set with 2 inputs and 2 outputs. The first input affects only the first output. Similarly, the second input affects only the second output.

In the estimated process model, the cross terms, which model the effect of the first input on the second output and vice versa, should be negligible. If the estimation process instead assigns higher orders to the cross dynamics, the degrees of estimation uncertainty for those terms should be high.

Estimate the process model.

type = 'P2UZ'; 
sys = procest(data,type);

The type variable denotes a model with complex-conjugate pair of poles, a zero, and a delay.

To evaluate the uncertainties, plot the frequency response.

w = linspace(0,20*pi,100);
h = bodeplot(sys,w);

The responses from the cross pairs show larger uncertainty, indicating that using a single type for each input/output pair results in too much energy in the cross pairs.

Use regularization to estimate parameters of an overparameterized process model.

Load the data.

load iddata1 z1;

Construct an initial system sysi by specifying parameter values for a model that includes three poles, one zero, and underdamped modes. Assume that gain Kp is known with a higher degree of confidence than the other model parameters.

sysi = idproc('P3UZ','Kp',7.5,'Tw',0.25,'Zeta',0.3,'Tp3',20,'Tz',0.02);

Estimate an unregularized process model sys1 using sysi to initialize the estimation model.

sys1 = procest(z1,sysi);

Estimate a regularized process model sys2 from sysi. Because K has a higher level of confidence, set the regularization constant R higher than for the other model parameters. This setting causes the estimation process to place more emphasis on maintaining the initial value of K.

opt = procestOptions;
opt.Regularization.Nominal = 'model';
opt.Regularization.R = [100;1;1;1;1];
opt.Regularization.Lambda = 0.1;
sys2 = procest(z1,sysi,opt);

Compare the model outputs with data.


Figure contains an axes object. The axes object contains 3 objects of type line. These objects represent z1 (y1), sys1: 26.74%, sys2: 70.16%.

Regularization helps steer the estimation process towards the correct parameter values, as the better fit for sys2 shows.

Compare the estimated gain values for sys1 and sys2.

g1 = sys1.Kp
g1 = -0.2320
g2 = sys2.Kp
g2 = 6.6236

The Kp value for the regularized system is much closer to the initial value than for the unregularized system.

Obtain the measured input-output data.

load iddemo_heatexchanger_data;
data = iddata(pt,ct,Ts);  
data.InputName  = '\Delta CTemp';
data.InputUnit  = 'C';
data.OutputName = '\Delta PTemp';
data.OutputUnit = 'C';
data.TimeUnit   = 'minutes';

Estimate a first-order plus dead time process model.

type = 'P1D';
sysP1D = procest(data,type);

Compare the model with the data.


Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent data (\Delta PTemp), sysP1D: 70.76%.

Plot the model residuals.


Figure contains 2 axes objects. Axes object 1 with title AutoCorr contains 2 objects of type line. This object represents sysP1D. Axes object 2 with title X C o r r blank ( Delta blank C T e m p ) contains 2 objects of type line. This object represents sysP1D.

The figure shows that the residuals are correlated. To account for that, add a first order ARMA disturbance component to the process model.

opt = procestOptions('DisturbanceModel','ARMA1');
sysP1D_noise = procest(data,'p1d',opt);

Compare the models.


Figure contains an axes object. The axes object contains 3 objects of type line. These objects represent data (\Delta PTemp), sysP1D: 70.76%, sysP1D\_noise: 24.43%.

Plot the model residuals.


Figure contains 2 axes objects. Axes object 1 with title AutoCorr contains 2 objects of type line. This object represents sysP1D\_noise. Axes object 2 with title X C o r r blank ( Delta blank C T e m p ) contains 2 objects of type line. This object represents sysP1D\_noise.

The residues of sysP1D_noise are uncorrelated.

Input Arguments

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Estimation data, specified as an iddata object, an frd object, or an idfrd object.

For time-domain estimation, data must be an iddata object containing the input and output signal values.

Time series models, which are time-domain models that contain no measured inputs, cannot be estimated using procest. Use ar, arx, or armax for time series models instead.

For frequency-domain estimation, data can be one of the following:

  • Recorded frequency response data (frd (Control System Toolbox) or idfrd)

  • iddata object with properties specified as follows:

    • InputData — Fourier transform of the input signal

    • OutputData — Fourier transform of the output signal

    • Domain'Frequency'

Estimation data must be uniformly sampled.

For multiexperiment data, the sample times and intersample behavior of all the experiments must match.

You can estimate both continuous-time and discrete-time models (of sample time matching that of data) using time-domain data and discrete-time frequency-domain data. You can estimate only continuous-time models using continuous-time frequency-domain data.

Process model structure, specified for SISO models as a string or character vector that represents an acronym for the model structure, such as 'P1D' or 'P2DZ'. The acronym starts with P and can contain any combination of the other following components:

  • P — Poles. All 'Type' acronyms start with P, because all process modes must have at least one pole.

  • 0, 1, 2, or 3 — Number of time constants (poles) to be modeled. This number does not include possible integrations (poles in the origin).

  • I — Integration is enforced (self-regulating process).

  • D — Time delay (dead time).

  • Z — Extra numerator term, a zero.

  • U — Underdamped modes (complex-valued poles) permitted. If U is not included in type, all poles must be real. The number of poles must be 2 or 3.

For MIMO models, specify typeas an Ny-by-Nu cell array of character vectors or string array, with one entry for each input-output pair. Here Ny is the number of inputs and Nu is the number of outputs.

For information regarding how type affects the structure of a process model, see idproc.

Input delays, specified as a numeric vector specifying a time delay for each input channel. Specify input delays in the time unit stored in the TimeUnit property.

For a system with Nu inputs, set InputDelay to an Nu-by-1 vector. Each entry of this vector is a numerical value that represents the input delay for the corresponding input channel. You can also set InputDelay to a scalar value to apply the same delay to all channels.

The software treats InputDelay as a fixed delay that is separate from any transport delay that the Td property of the model introduces.

Process model that configures initial parameterization of sys, specified as an idproc object. You obtain init_sys by either performing an estimation using measured data or by direct construction using idproc. The software uses the parameters and constraints defined in init_sys as the initial guess for estimating sys.

Use the Structure property of init_sys to configure initial guesses and constraints for Kp, Tp1, Tp2, Tp3, Tw, ζ, Td, and Tz. For example:

  • To specify an initial guess for the Tp1 parameter of init_sys, set init_sys.Structure.Tp1.Value as the initial guess.

  • To specify constraints for the Tp2 parameter of init_sys:

    • Set init_sys.Structure.Tp2.Minimum to the minimum Tp2 value.

    • Set init_sys.Structure.Tp2.Maximum to the maximum Tp2 value.

    • Set init_sys.Structure.Tp2.Free to indicate if Tp2 is a free parameter for estimation.

If you do not specify opt, and init_sys was obtained by estimation rather than construction, then the software uses estimation options from init_sys.Report.OptionsUsed

Estimation options, specified as an procestOptions option set. The estimation options include:

  • Estimation objective

  • Handling on initial conditions and disturbance component

  • Numerical search method to be used in estimation

Output Arguments

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Identified process model, returned as an idproc model of a structure defined by type.

Information about the estimation results and options used is stored in the model's Report property. Report has the following fields:

Report FieldDescription

Summary of the model status, which indicates whether the model was created by construction or obtained by estimation.


Estimation command used.


Handling of initial conditions during model estimation, returned as one of the following values:

  • 'zero' — The initial conditions were set to zero.

  • 'estimate' — The initial conditions were treated as independent estimation parameters.

  • 'backcast' — The initial conditions were estimated using the best least squares fit.

This field is especially useful to view how the initial conditions were handled when the InitialCondition option in the estimation option set is 'auto'.


Quantitative assessment of the estimation, returned as a structure. See Loss Function and Model Quality Metrics for more information on these quality metrics. The structure has the following fields:


Normalized root mean squared error (NRMSE) measure of how well the response of the model fits the estimation data, expressed as the percentage fitpercent = 100(1-NRMSE).


Value of the loss function when the estimation completes.


Mean squared error (MSE) measure of how well the response of the model fits the estimation data.


Final prediction error for the model.


Raw Akaike Information Criteria (AIC) measure of model quality.


Small-sample-size corrected AIC.


Normalized AIC.


Bayesian Information Criteria (BIC).


Estimated values of model parameters.


Option set used for estimation. If no custom options were configured, this is a set of default options. See procestOptions for more information.


State of the random number stream at the start of estimation. Empty, [], if randomization was not used during estimation. For more information, see rng.


Attributes of the data used for estimation. Structure with the following fields:


Name of the data set.


Data type. For idnlarx models, this is set to 'Time domain data'.


Number of data samples.


Sample time. This is equivalent to Data.Ts.


Input intersample behavior. One of the following values:

  • 'zoh' — Zero-order hold maintains a piecewise-constant input signal between samples.

  • 'foh' — First-order hold maintains a piecewise-linear input signal between samples.

  • 'bl' — Band-limited behavior specifies that the continuous-time input signal has zero power above the Nyquist frequency.

The value of Intersample has no effect on estimation results for discrete-time models.


Empty, [], for nonlinear estimation methods.


Empty, [], for nonlinear estimation methods.


Termination conditions for the iterative search used for prediction error minimization, returned as a structure with the following fields:


Reason for terminating the numerical search.


Number of search iterations performed by the estimation algorithm.


-norm of the gradient search vector when the search algorithm terminates.


Number of times the objective function was called.


Norm of the gradient search vector in the last iteration. Omitted when the search method is 'lsqnonlin' or 'fmincon'.


Criterion improvement in the last iteration, expressed as a percentage. Omitted when the search method is 'lsqnonlin' or 'fmincon'.


Algorithm used by 'lsqnonlin' or 'fmincon' search method. Omitted when other search methods are used.

For estimation methods that do not require numerical search optimization, the Termination field is omitted.

For more information on using Report, see Estimation Report.

Estimated value of input offset, returned as a vector. When data has multiple experiments, offset is a matrix where each column corresponds to an experiment.

Estimated initial conditions, returned as an initialCondition object or an object array of initialCondition values.

  • For a single-experiment data set, ic represents, in state-space form, the free response of the transfer function model (A and C matrices) to the estimated initial states (x0).

  • For a multiple-experiment data set with Ne experiments, ic is an object array of length Ne that contains one set of initialCondition values for each experiment.

If procest returns ic values of 0 and the you know that you have non-zero initial conditions, set the 'InitialCondition' option in procestOptions to 'estimate' and pass the updated option set to procest. For example:

opt = procestOptions('InitialCondition','estimate')
[sys,offset,ic] = procest(data,type,opt)
The default 'auto' setting of 'InitialCondition' uses the 'zero' method when the initial conditions have a negligible effect on the overall estimation-error minimization process. Specifying 'estimate' ensures that the software estimates values for ic.

For more information, see initialCondition. For an example of using this argument, see Obtain Initial Conditions.

Extended Capabilities

Version History

Introduced in R2012a