procest
Estimate process model using time-domain or frequency-domain data
Syntax
Description
Estimate Process Model
estimates a process model sys
= procest(data
,type
)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
.
specifies the input delay sys
= procest(data
,type
,'InputDelay',InputDelay
)InputDelay
.
Configure Initial Parameters
Specify Additional Options
Return Estimated Offset and Initial Conditions
[
returns the estimated value of the offset in input signal. sys
,offset
] = procest(___)procest
automatically estimates the input offset when the model contains an integrator or when you
set the InputOffset
estimation option to 'estimate'
using procestOptions
.
[
returns the estimated initial conditions as an sys
,offset
,ic
] = procest(___)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.
Examples
Estimate and Refine Process Model
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.
compare(z1,sys)
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');
compare(z1,sys)
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');
compare(z1,sys)
The fit now exceeds 70%.
You can view more information about the estimation by exploring the idproc
property sys.Report
.
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
Specify Parameter Initial Values for Estimated Process Model
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.
compare(data,sys);
Return Input Offsets Estimated During Process Model Estimation
Obtain Initial Conditions
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
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.
Detect Overparameterization of Estimated Model
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); showConfidence(h);
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.
Estimate Overparameterized Process Model Using Regularization
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.
compare(z1,sys1,sys2);
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.
Estimate a First Order Plus Dead Time Model
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.
compare(data,sysP1D)
Plot the model residuals.
figure resid(data,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.
compare(data,sysP1D,sysP1D_noise)
Plot the model residuals.
figure resid(data,sysP1D_noise);
The residues of sysP1D_noise
are uncorrelated.
Input Arguments
data
— Estimation data
iddata
| idfrd
| frd
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:
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.
type
— Process model structure
character vector | string | cell array of character vectors | string array
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 withP
, because all process modes must have at least one pole.0
,1
,2
, or3
— 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. IfU
is not included intype
, all poles must be real. The number of poles must be 2 or 3.
For MIMO models, specify type
as 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
.
InputDelay
— Input delays
0
for all input channels (default) | numeric vector
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.
init_sys
— Process model that configures initial parameterization of sys
idproc
object
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
, setinit_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
opt
— Estimation options
procestOptions
option set
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
sys
— Identified process model
idproc
model
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 Field | Description | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Status | Summary of the model status, which indicates whether the model was created by construction or obtained by estimation. | ||||||||||||||||||
Method | Estimation command used. | ||||||||||||||||||
InitialCondition | Handling of initial conditions during model estimation, returned as one of the following values:
This field is especially useful to view
how the initial conditions were handled when the | ||||||||||||||||||
Fit | 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:
| ||||||||||||||||||
Parameters | Estimated values of model parameters. | ||||||||||||||||||
OptionsUsed | Option set used for estimation. If no custom options were configured,
this is a set of default options. See | ||||||||||||||||||
RandState | State of the random number stream at the start of estimation. Empty,
| ||||||||||||||||||
DataUsed | Attributes of the data used for estimation. Structure with the following fields:
| ||||||||||||||||||
Termination | Termination conditions for the iterative search used for prediction error minimization, returned as a structure with the following fields:
For estimation methods that do not require numerical search optimization,
the |
For more information on using Report
, see Estimation Report.
offset
— Estimated value of input offset
vector
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.
ic
— Initial conditions
initialCondition
object | object array of initialCondition
values
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 ofinitialCondition
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)
'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
Automatic Parallel Support
Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.
Parallel computing support is available for estimation using the
lsqnonlin
search method (requires Optimization Toolbox™). To enable parallel computing, use procestOptions
, set SearchMethod
to
'lsqnonlin'
, and set
SearchOptions.Advanced.UseParallel
to true
.
For example:
opt = procestOptions;
opt.SearchMethod = 'lsqnonlin';
opt.SearchOptions.Advanced.UseParallel = true;
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