ssest

Estimate state-space model using time-domain or frequency-domain data

Syntax

sys = ssest(tt,nx)

sys = ssest(u,y,nx,'Ts',Ts)

sys = ssest(data,nx)

sys = ssest(tt,nx)

sys = ssest(tt,nx,'OutputName',outputVariables,'InputName',[])

sys = ssest([],y,nx,'Ts',Ts)

sys = ssest(data,nx)

sys = ssest(___,Name,Value)

sys = ssest(tt,init_sys)

sys = ssest(u,y,init_sys)

sys = ssest(data,init_sys)

sys = ssest(___,opt)

[sys,x0] = ssest(___)

Description

Estimate State-Space Model

sys = ssest(tt,nx) estimates the continuous-time state-space model sys of order nx, using all the input and output signals in the timetable tt.

You can use this syntax for SISO and MISO systems. The function assumes that the last variable in the timetable is the single output signal.

sys is an idss model of the following form:

$$\begin{array}{l}\dot{x}(t)=Ax(t)+Bu(t)+Ke(t)\\ y(t)=Cx(t)+Du(t)+e(t)\end{array}$$

A, B, C, D, and K are state-space matrices. u(t) is the input, y(t) is the output, e(t) is the disturbance, and x(t) is the vector of nx states.

All entries of A, B, C, and K are free estimable parameters by default. D is fixed to zero by default, meaning that there is no feedthrough, except for static systems (nx = 0).

To estimate discrete-time models, set 'Ts' to the model sample time using name-value syntax.

To estimate MIMO models, use name-value syntax to specify the input and output channels using 'InputName' and 'OutputName' to the corresponding timetable variable names.

You can also use 'InputName' and 'OutputName' to specify specific channels when you do not want to use all the available channels in tt.

sys = ssest(u,y,nx,'Ts',Ts) estimates a discrete-time state-space model using the time-domain input and output signals in the comma-separated matrices u,y and the model sample time Ts.

The software assumes that the data sample time is Ts seconds. You can use this syntax for SISO, MISO, and MIMO systems.

Estimating a continuous-time model with u,y by not specifying Ts is not recommended. The software assumes that the data sample time is 1 second. You cannot change this sample time. If you want to estimate a continuous-time model from data with a sample time other than 1 second, you must first convert your matrix data to a timetable or iddata object. For an example of this conversion, see Convert SISO Matrix Data to Timetable or Convert MIMO Matrix Data to Timetable for Continuous-Time Model Estimation.

sys = ssest(data,nx) estimates a continuous-time state-space model using the time-domain or frequency-domain data in the data object data.

Use this syntax especially when you want to estimate a state-space model using frequency-domain or frequency-response data, or when you want to take advantage of the additional information, such as intersample behavior, data sample time, or experiment labeling, that data objects provide.

Estimate Time Series State-Space Model

sys = ssest(tt,nx) estimates the continuous time series model sys to fit the data in the timetable tt. tt must contain a single numeric variable. The function interprets the timetable variable data as a time series, which has no inputs and a single output.

For a time series model, the sys idss model has the following form:

$$\begin{array}{l}\dot{x}(t)=Ax(t)+Ke(t)\\ y(t)=Cx(t)+e(t)\end{array}$$

sys = ssest(tt,nx,'OutputName',outputVariables,'InputName',[]) estimates a multivariate time series model that uses the timetable output signals that have the variable names specified in outputVariables. The function interprets the specified variables as a multivariate time series. If you specify all the variables in tt in 'OutputName', you can omit the specification of 'InputName'.

sys = ssest([],y,nx,'Ts',Ts) estimates a discrete time series model with the sample time Ts from the output data matrix y. sys has as many outputs as there are columns in y.

sys = ssest(data,nx) estimates a time series model that uses the data in the iddata property data.OutputData. The property data.InputData must be empty.

Specify Additional Model Options

sys = ssest(___,Name,Value) incorporates additional options specified by one or more name-value pair arguments.

For example, specify a discrete-time system from matrix data that has a sample time of 0.1 using sys = ssest(um,ym,np,'Ts',0.1). Specify input and output signal variable names that correspond with the variables to use for MIMO timetable data using sys = ssest(data,nx,'InputName',["u1","u4"],'OutputName',["y1","y3"]). Use the 'Form', 'Feedthrough', and 'DisturbanceModel' name-value arguments to modify the default behavior of the A, B, C, D, and K matrices.

You can use this syntax with any of the previous input-argument combinations.

Configure Initial Parameters

sys = ssest(tt,init_sys) uses the linear system init_sys to configure the initial parameterization of sys for estimation using the timetable tt.

sys = ssest(u,y,init_sys) uses the matrix data u,y for estimation. If init_sys is a continuous-time model, using a timetable instead of matrices is recommended.

sys = ssest(data,init_sys) uses the data object data for estimation.

Specify Additional Estimation Options

sys = ssest(___,opt) incorporates an option set opt that specifies options such as estimation objective, handling of initial conditions, regularization, and numerical search method used for estimation. You can specify opt after any of the previous input-argument combinations.

Return Estimated Initial States

[sys,x0] = ssest(___) returns the value of initial states computed during estimation.

Examples

State-Space Model

Estimate a state-space model and compare its response with the measured output.

Load the input/output data, which is stored in a timetable.

load sdata1 tt1

Estimate a fourth-order state-space model.

nx = 4;
sys = ssest(tt1,nx);

Compare the simulated model response with the measured output.

compare(tt1,sys)

The plot shows that the fit percentage between the simulated model and the estimation data is greater than 70%.

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

sys.Report

ans = 
          Status: 'Estimated using SSEST with prediction focus'
          Method: 'SSEST'
    InitialState: 'zero'
        N4Weight: 'CVA'
       N4Horizon: [6 10 10]
             Fit: [1x1 struct]
      Parameters: [1x1 struct]
     OptionsUsed: [1x1 idoptions.ssest]
       RandState: []
        DataUsed: [1x1 struct]
     Termination: [1x1 struct]

For example, find out more information about the termination conditions.

sys.Report.Termination

ans = struct with fields:
                 WhyStop: 'No improvement along the search direction with line search.'
              Iterations: 7
    FirstOrderOptimality: 85.9759
                FcnCount: 123
              UpdateNorm: 7.7097
         LastImprovement: 0

The report includes information on the number of iterations and the reason the estimation stopped iterating.

Determine Optimal Estimated Model Order

Load the input/output data in matrices umat1 and ymat1, as well as the sample time Ts. These matrices store the same estimation data as the timetable data used to estimate a fourth-order model in State-Space Model.

load sdata1 umat1 ymat1 Ts

Determine the optimal model order by specifying argument nx as a range from 1:10.

nx = 1:10;
sys = ssest(umat1,ymat1,nx,'Ts',Ts);

An automatically generated plot shows the Hankel singular values for models of the orders specified by nx.

States with relatively small Hankel singular values can be safely discarded. The suggested default order choice is 2.

Select the model order in the Chosen Order list and click Apply. sys is a discrete-time model.

Identify State-Space Model with Input Delay

Load time-domain system response data in the matrix pair umat7 and ymat7.

load sdata7 umat7 ymat7 Ts;

Identify a fourth-order state-space model of the data. Specify a known delay of 2 seconds for the first input and 0 seconds for the second input.

nx = 4;
sys = ssest(umat7(1:300,:),ymat7(1:300,:),nx,'Ts',Ts,'InputDelay',[2;0]);

Modify Form, Feedthrough, and Disturbance-Model Matrices

Modify the canonical form of the A, B, and C matrices, include a feedthrough term in the D matrix, and eliminate disturbance-model estimation in the K matrix.

Load input-output data and estimate a fourth-order system using the ssest default options.

load iddata1 z1
sys1 = ssest(z1,4);

Specify the companion form and compare the A matrix with the default A matrix.

sys2 = ssest(z1,4,'Form','companion');
A1 = sys1.A

A1 = 4×4

   -0.5155    3.8483   -0.6657   -0.2666
   -5.8665   -2.7285    1.0649    1.4694
    0.4487    0.9308   -0.6235  -18.8148
   -0.4192   -0.5595   16.0688    0.5399

A2 = sys2.A

A2 = 4×4
103 ×

         0         0         0   -7.1122
    0.0010         0         0   -0.9547
         0    0.0010         0   -0.3263
         0         0    0.0010   -0.0033

Include a feedthrough term and compare D matrices.

sys3 = ssest(z1,4,'Feedthrough',1);
D1 = sys1.D

D1 = 
0

D3 = sys3.D

D3 = 
0.0339

Eliminate disturbance modeling and compare K matrices.

sys4 = ssest(z1,4,'DisturbanceModel','none');
K1 = sys1.K

K1 = 4×1

    0.0520
   -0.0973
   -0.0151
    0.0270

K4 = sys4.K

K4 = 4×1

     0
     0
     0
     0

Estimate Initial States as Independent Parameters

Specify ssest initial states as independent estimation parameters.

ssest can handle initial states using one of several methods. By default, ssest chooses the method automatically based on your estimation data. You can choose the method yourself by modifying the option set using ssestOptions.

Load the input/output data tt1 and estimate a second-order state-space model sys using the default options. Use the syntax that returns initial states x0.

load sdata1 tt1
[sys,x0] = ssest(tt1,2);
x0

x0 = 2×1

     0
     0

By default, the estimation is performed using the 'auto' setting for InitialState. Find out which method ssest applied by reviewing the value of InitialState in sys.Report.

sys.Report.InitialState

ans = 
'zero'

The software applied the 'zero' method, meaning that the software set the initial states to zero instead of estimating them. This selection is consistent with the 0 values returned for x0.

Specify that ssest estimate the initial states instead as independent parameters using the 'estimate' setting. Use ssestOptions to create a modified option set and specify that option set to estimate a new model.

opt = ssestOptions('InitialState','estimate');
[sys1,x0] = ssest(tt1,2,opt);
x0

x0 = 2×1

    0.0068
    0.0052

x0 now has estimated parameters with nonzero values.

Estimate State-Space Model Using Regularization

Obtain a regularized fifth-order state-space model for a second-order system from a narrow bandwidth signal.

Load estimation data.

load regularizationExampleData eData;

Create the transfer function model used for generating the estimation data (true system).

trueSys = idtf([0.02008 0.04017 0.02008],[1 -1.561 0.6414],1);

Estimate an unregularized state-space model.

opt = ssestOptions('SearchMethod','lm');
m = ssest(eData,5,'form','modal','DisturbanceModel','none','Ts',eData.Ts,opt);

Estimate a regularized state-space model.

opt.Regularization.Lambda = 10;
mr = ssest(eData,5,'form','modal','DisturbanceModel','none','Ts',eData.Ts,opt);

Compare the model outputs with the estimation data.

compare(eData,m,mr);

Compare the model impulse responses.

impulse(trueSys,m,mr,50);
legend('trueSys','m','mr');

Estimate Partially Known State-Space Model Using Structured Estimation

Configure the parameter constraints and initial values for state-space model estimation.

Create an idss model to specify the initial parameterization for estimation.

A = blkdiag([-0.1 0.4; -0.4 -0.1],[-1 5; -5 -1]);
B = [1; zeros(3,1)]; 
C = [1 1 1 1]; 
D = 0; 
K = zeros(4,1);
x0 = [0.1 0.1 0.1 0.1];
Ts = 0;
init_sys = idss(A,B,C,D,K,x0,Ts);

Setting all entries of K to 0 creates an idss model with no state disturbance element.

Use the Structure property to fix the values of some of the model parameters. Configure the model so that B and K are fixed, and only the nonzero entries of A are estimable.

init_sys.Structure.A.Free = (A~=0);
init_sys.Structure.B.Free = false;
init_sys.Structure.K.Free = false;

The entries in init_sys.Structure.A.Free determine whether the corresponding entries in init_sys.A are free (true) or fixed (false).

Load the measured data and estimate a state-space model using the parameter constraints and initial values specified by init_sys.

load sdata2 tt2;
sys = ssest(tt2,init_sys);

The estimated parameters of sys satisfy the constraints specified by init_sys.

Input Arguments

tt — Timetable-based estimation data

timetable | cell array of timetables.

Estimation data, specified as a uniformly sampled timetable that contains variables representing input and output channels or, for multiexperiment data, a cell array of timetables.

Use Entire Timetable

If you want to use all the variables in tt as input or output channels, and the variables are organized so that the set of input channel variables is followed by the set of output channel variables, then:

• For SISO systems, specify tt as an N_s-by-2 timetable, where N_s is the number of samples and the two timetable variables represent the measured input channel and output channel respectively.

• For MIMO systems, specify tt as an N_s-by-(N_u+N_y) timetable, where N_u is the number of inputs and N_y is the number of outputs. The first N_u variables must contain the input channels and the remaining N_y variables must contain the output channels.

  When you are estimating state space or transfer function models, you must also explicitly specify the input and output channels, as the following section describes.

• For multiexperiment data, specify data as an N_e-by-1 cell array of timetables, where N_e is the number of experiments. The sample times of all the experiments must match.

Use Selected Variables from Timetable

If you want to explicitly identify the input and output channels, such as when you want to use only a subset of the available channels, when the input and output channel variables are intermixed, or when you are estimating a MIMO state-space or transfer function model, use the 'InputName' and 'OutputName' name-value arguments to specify which variables to use as inputs and outputs.

For example, suppose that tt contains six channel variables: "u1", "u2", "u3", and "y1", "y2", "y3". For estimation, you want to use the variables "u1" and "u2" as the inputs and the variables "y1" and "y3" as the outputs. Use the following command to perform the estimation:

sys = ssest(tt,__,'InputName',["u1" "u2"],'OutputName',["y1" "y3"])

Use Timetable to Estimate Time Series Models

If you want to estimate a time series model rather than an input/output model, use only output variables from tt. You can either specify tt to contain only the output variables that you want, or extract the output variables from tt if tt also contains input variables. The specification approach is similar to that for input/output model estimation.

• For a single-output system, specify tt as an N_s-by-1 timetable.

• For a multivariate system, specify tt as an N_s-by-(N_y) timetable. Even if you plan to use all the variables in tt, you must specify all of them using the 'OutputName' name-value argument so that the software does not interpret them as input variables.

For a timetable tt that has variables beyond what you want to use, such as input variables or additional output variables, specify both the output variables you want to use and, in 'InputName', an empty array.

For example, suppose that tt contains six variables: "u1", "u2", "u3", and "y1", "y2", "y3". For time series estimation, you want to use the output variables "y1" and "y3". Use the following command to perform the estimation:

sys = ssest(tt,__,'OutputName',["y1" "y3"],'InputName',[])

For more information about working with estimation data types, see Data Domains and Data Types in System Identification Toolbox.

u, y — Matrix-based estimation data

matrices | cell array of matrices

Estimation data, specified for SISO systems as a comma-separated pair of N_s-by-1 real-valued matrices that contain uniformly sampled input and output time-domain signal values. Here, N_s is the number of samples.

For MIMO systems, specify u,y as an input/output matrix pair with the following dimensions:

• u — N_s-by-N_u, where N_u is the number of inputs.

• y — N_s-by-N_y, where N_y is the number of outputs.

For multiexperiment data, specify u,y as a pair of 1-by-N_e cell arrays, where N_e is the number of experiments. The sample times of all the experiments must match.

For time series data, which contains only outputs and no inputs, specify [],y.

Limitations

• Matrix-based data does not support estimation from frequency-domain data. You must use a data object such as an iddata object or idfrd object (see data).

• Using matrices for estimation data is not recommended for continuous-time estimation because the data does not provide the sample time. The software assumes that the data is sampled at 1 Hz. For continuous-time estimation, it is recommended that you convert each matrix to a timetable. For example, to convert the matrices um and ym to a timetable tt with a sample time of 0.5 minutes, use the following command.

  tt = timetable(um,ym,'rowtimes',minutes(0.5*(1:size(u,1))))

  For a more detailed example of converting matrix-based SISO data to a timetable, see Convert SISO Matrix Data to Timetable. For an example of converting a MIMO matrix pair to a timetable, see Convert MIMO Matrix Data to Timetable for Continuous-Time Model Estimation.

  For more information about working with estimation data types, see Data Domains and Data Types in System Identification Toolbox.

data — Estimation data object

iddata object | frd object | idfrd object

Estimation data object, specified as an iddata object, an frd object, or an idfrd object that contains uniformly sampled input and output values. By default, the software sets the sample time of the model to the sample time of the estimation data.

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

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

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'

Limitations

You cannot estimate continuous-time models using discrete-time frequency-domain data.

nx — Order of estimated model

1:10 (default) | positive integer scalar | positive integer vector | 0

Order of the estimated model, specified as a nonnegative integer or as a vector containing a range of positive integers.

• If you already know what order you want your estimated model to have, specify nx as a scalar.

• If you want to compare a range of potential orders to choose the most effective order for your estimated model, specify the range in nx. ssest creates a Hankel singular-value plot that shows the relative energy contributions of each state in the system. States with relatively small Hankel singular values contribute little to the accuracy of the model and can be discarded with little impact. The index of the highest state you retain is the model order. The plot window includes a suggestion for the order to use. You can accept this suggestion or enter a different order. For an example, see Determine Optimal Estimated Model Order.

  If you do not specify nx, or if you specify nx as best, the software automatically chooses nx from the range 1:10.

• If you are identifying a static system, set nx to 0.

opt — Estimation options

ssestOptions option set

Estimation options, specified as an ssestOptions option set. Options specified by opt include:

• Estimation objective

• Handling of initial conditions

• Regularization

• Numerical search method used for estimation

• Intersample behavior

For examples showing how to use opt, see Estimate Initial States as Independent Parameters and Estimate State-Space Model Using Regularization.

init_sys — Linear system that configures initial parameterization of sys

idss model | linear model | structure

Linear system that configures the initial parameterization of sys, specified as an idss model or as a structure. You obtain init_sys by either performing an estimation using measured data or by direct construction.

If init_sys is an idss model, ssest uses the parameter values of init_sys as the initial guess for estimating sys. For information on how to specify idss, see Estimate State-Space Models with Structured Parameterization. ssest honors constraints on the parameters of init_sys, such as fixed coefficients and minimum/maximum bounds.

Use the Structure property of init_sys to configure initial parameter values and constraints for the A, B, C, D, and K matrices. For example:

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

• To specify constraints for the B matrix of init_sys:

  • Set init_sys.Structure.B.Minimum to the minimum B matrix value

  • Set init_sys.Structure.B.Maximum to the maximum B matrix value

  • Set init_sys.Structure.B.Free to indicate if entries of the B matrix are free parameters for estimation

  To set more complex constraints, such as interdependence of coefficients, use grey-box estimation using greyest and idgrey.

You must assign finite initial values for all matrix parameters.

If init_sys is not a state-space (idss) model, the software first converts init_sys to an idss model. ssest uses the parameters of the resulting model as the initial guess for estimation.

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

For an example, see Estimate Partially Known State-Space Model Using Structured Estimation.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: sys = ssest(data,nx,'Ts',0.1)

InputName — Input channel names

string | character vector | string array | cell array of character vectors

Input channel names, specified as a string, character vector, string array, or cell array of character vectors.

If you are using a timetable for the data source, the names in InputName must be a subset of the timetable variables.

Example: sys = ssest(tt,__,'InputName',["u1" "u2"]) selects the variables u1 and u2 as the input channels from the timetable tt to use for the estimation.

OutputName — Output channel names

string | character vector | string array | cell array of character vectors

Output channel names, specified as a string, character vector, string array, or cell array of character vectors.

If you are using a timetable for the data source, the names in OutputName must be a subset of the timetable variables.

Example: sys = ssest(tt,__,'OutputName',["y1" "y3"]) selects the variables y1 and y3 as the output channels from the timetable tt to use for the estimation.

Ts — Sample time of estimated model

0 (continuous time) (default) | sample time of data | positive scalar

Sample time of the estimated model, specified as the comma-separated pair consisting of 'Ts' and either 0 or a positive scalar.

• For continuous-time models, specify 'Ts' as 0.

• For discrete-time models, specify 'Ts' as the data sample time in units defined by the following:

  • For timetable-based data — The timetable Time column

  • For matrix-based data — Seconds

  • For data objects, such as iddata objects — The data.TimeUnit property

To obtain the data sample time for a timetable tt, use the timetable property tt.Properties.Timestep.

InputDelay — Input delays

0 (default) | scalar | vector

Input delay for each input channel, specified as the comma-separated pair consisting of 'InputDelay' and a numeric vector.

• For continuous-time models, specify 'InputDelay' in the time units stored in the timetable, the data object TimeUnit property, or, for matrix data, in seconds.

• For discrete-time models, specify 'InputDelay' in integer multiples of the sample time Ts. For example, setting 'InputDelay' to 3 specifies a delay of three sampling periods.

For a system with N_u inputs, set InputDelay to an N_u-by-1 vector. Each entry of this vector is a numerical value that represents the input delay for the corresponding input channel. For an example, see Identify State-Space Model with Input Delay.

To apply the same delay to all channels, specify InputDelay as a scalar.

Form — Type of canonical form

'free' (default) | 'modal' | 'companion' | 'canonical'

Type of canonical form of sys, specified as the comma-separated pair consisting of 'Form' and one of the following values:

• 'free' — All entries of the matrices A, B, C, D, and K are treated as free.

• 'modal' — Obtain sys in modal form.

• 'companion' — Obtain sys in companion form.

• 'canonical' — Obtain sys in the observability canonical form.

For definitions of the canonical forms, see State-Space Realizations.

For more information, see Estimate State-Space Models with Canonical Parameterization. For an example, see Modify Form, Feedthrough, and Disturbance-Model Matrices.

Feedthrough — Direct feedthrough from input to output

0 (default) | 1 | logical vector

Direct feedthrough from input to output, specified as the comma-separated pair consisting of 'Feedthrough' and a logical vector of length N_u, where N_u is the number of inputs. If you specify Feedthrough as a logical scalar, that value is applied to all the inputs. For static systems, the software always assumes 'Feedthrough' is 1.

For an example, see Modify Form, Feedthrough, and Disturbance-Model Matrices.

DisturbanceModel — Option to estimate time-domain noise component parameters

'estimate' (default) | 'none'

Option to estimate time-domain noise component parameters in the K matrix, specified as the comma-separated pair consisting of 'DisturbanceModel' and one of the following values:

• 'estimate' — Estimate the noise component. The K matrix is treated as a free parameter.

• 'none' — Do not estimate the noise component. The elements of the K matrix are fixed at zero.

For frequency-domain data, the software assumes that 'DisturbanceModel' is 'none'.

For an example, see Modify Form, Feedthrough, and Disturbance-Model Matrices.

Output Arguments

sys — Identified state-space model

idss model

Identified state-space model, returned as an idss model. This model is created using the specified model orders, delays, and estimation options.

Information about the estimation results and options used is stored in the Report property of the model. 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
InitialState	How initial states were handled during estimation, returned as one of the following values: 'zero' — The initial state is set to zero. 'estimate' — The initial state is treated as an independent estimation parameter. 'backcast' — The initial state is estimated using the best least squares fit. Column vector of length Nx, where Nx is the number of states. For multi-experiment data, a matrix with Ne columns, where Ne is the number of experiments. Parametric initial condition object (x0obj) created using idpar. Only for discrete-time state-space models. This field is especially useful when the InitialState option in the estimation option set is 'auto'.
N4Weight	Weighting scheme used for singular-value decomposition by the N4SID algorithm, returned as one of the following values: 'MOESP' — Uses the MOESP algorithm. 'CVA' — Uses the Canonical Variate Algorithm. 'SSARX' — A subspace identification method that uses an ARX estimation-based algorithm to compute the weighting. This option is especially useful when the N4Weight option in the estimation option set is 'auto'.
N4Horizon	Forward and backward prediction horizons used by the N4SID algorithm, returned as a row vector with three elements —  [r sy su], where r is the maximum forward prediction horizon. sy is the number of past outputs, and su is the number of past inputs that are used for the predictions.
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 these fields. FitPercent — 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) LossFcn — Value of the loss function when the estimation completes MSE — Mean squared error (MSE) measure of how well the response of the model fits the estimation data FPE — Final prediction error for the model AIC — Raw Akaike Information Criteria (AIC) measure of model quality AICc — Small-sample-size corrected AIC nAIC — Normalized AIC BIC — Bayesian Information Criteria (BIC)
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 ssestOptions for more information.
RandState	State of the random number stream at the start of estimation. Empty, [], if randomization was not used during estimation. For more information, see rng.
DataUsed	Attributes of the data used for estimation. Structure with the following fields: Field Description Name Name of the data set. Type Data type. Length Number of data samples. Ts Sample time. This is equivalent to Data.Ts. InterSample 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. InputOffset Offset removed from time-domain input data during estimation. OutputOffset Offset removed from time-domain output data during estimation.
Termination	Termination conditions for the iterative search used for prediction error minimization, returned as a structure with these fields. WhyStop — Reason for terminating the numerical search Iterations — Number of search iterations performed by the estimation algorithm FirstOrderOptimality — $$\infty$$-norm of the gradient search vector when the search algorithm terminates FcnCount — Number of times the objective function was called UpdateNorm — Norm of the gradient search vector in the last iteration. Omitted when the search method is 'lsqnonlin' or 'fmincon'. LastImprovement — Criterion improvement in the last iteration, expressed as a percentage. Omitted when the search method is 'lsqnonlin' or 'fmincon'. Algorithm — 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.

x0 — Initial states computed during estimation

column vector | matrix

Initial states computed during the estimation, returned as an array containing a column vector corresponding to each experiment.

This array is also stored in the Parameters field of the model Report property.

For an example, see Estimate Initial States as Independent Parameters.

Algorithms

ssest initializes the parameter estimates using either a noniterative subspace approach or an iterative rational function estimation approach. It then refines the parameter values using the prediction error minimization approach. For more information, see pem and ssestOptions.

References

[1] Ljung, L. System Identification: Theory for the User, Second Edition. Upper Saddle River, NJ: Prentice Hall PTR, 1999.

Version History

Introduced in R2012a

R2022b: Time-domain estimation data is accepted in the form of timetables and matrices

Most estimation, validation, analysis, and utility functions now accept time-domain input/output data in the form of a single timetable that contains both input and output data or a pair of matrices that contain the input and output data separately. These functions continue to accept iddata objects as a data source as well, for both time-domain and frequency-domain data.

R2018a: Advanced Options are deprecated for SearchOptions when SearchMethod is 'lsqnonlin'

Specification of lsqnonlin- related advanced options are deprecated, including the option to invoke parallel processing when estimating using the lsqnonlin search method, or solver, in Optimization Toolbox™.