ssest

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

Syntax

sys = ssest(data,nx)

sys = ssest(data,nx,Name,Value)

sys = ssest(___,opt)

sys = ssest(data,init_sys)

sys = ssest(data,init_sys,opt)

[sys,x0] = ssest(___)

Description

Estimate a State-Space Model

sys = ssest(data,nx) estimates a continuous-time state-space model sys of order nx, using data data that can be in the time domain or the frequency domain. sys is a 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).

sys = ssest(data,nx,Name,Value) incorporates additional options specified by one or more name-value pair arguments. For example, estimate a discrete-time model by specifying the sample time 'Ts' name-value pair argument. Use the 'Form', 'Feedthrough', and 'DisturbanceModel' name-value pair arguments to modify the default behavior of the A, B, C, D, and K matrices.

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.

Configure Initial Parameters

sys = ssest(data,init_sys) uses the state-space model init_sys to configure the initial parameterization.

sys = ssest(data,init_sys,opt) estimates the model using an option set opt.

Return Estimated Initial States

[sys,x0] = ssest(___) returns the value of initial states computed during estimation. You can use this syntax with any of the previous input-argument combinations.

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 an iddata object.

load iddata1 z1

Estimate a fourth-order state-space model.

nx = 4;
sys = ssest(z1,nx);

Compare the simulated model response with the measured output.

compare(z1,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: 9.9156
         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 z1, which is stored in an iddata object. This is the same data used to estimate a fourth-order model in State-Space Model.

load iddata1 z1;

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

nx = 1:10;
sys = ssest(data,nx);

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.

Identify State-Space Model with Input Delay

Load time-domain system response data.

load iddata7 z7;

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(z7(1:300),nx,'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 estimate 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 z1 and estimate a second-order state-space model sys using the default options. Use the syntax that returns initial states x0.

load iddata1 z1
[sys,x0] = ssest(z1,2);
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(z1,2,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

Estimate a state-space model of measured input-output data. Configure the parameter constraints and initial values for estimation using a state-space model.

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 iddata2 z2;
sys = ssest(z2,init_sys);

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

Input Arguments

data — Estimation data

iddata object | frd object | idfrd object

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.

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

• Recorded frequency response data (frd 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'

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, 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

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 | 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 Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

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

'Ts' — Sample time of estimated model

0 (continuous time) (default) | sample time of data data.Ts) | 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 the units stored in the TimeUnit property.

'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 TimeUnit property.

• 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 [1].

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 noise component parameters

'estimate' (default) | 'none'

Option to estimate 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 more information, see Estimate State-Space Models with Canonical Parameterization. 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 Variable 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 the following fields: Field Description FitPercent Normalized root mean squared error (NRMSE) measure of how well the response of the model fits the estimation data, expressed as a percentage. 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 in the MATLAB® documentation.
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. Structure with the following fields: Field Description 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.

More About

Modal Form

In modal form, A is a block-diagonal matrix. The block size is typically 1-by-1 for real eigenvalues and 2-by-2 for complex eigenvalues. However, if there are repeated eigenvalues or clusters of nearby eigenvalues, the block size can be larger.

For example, for a system with eigenvalues $$({\lambda }_1,\sigma \pm j\omega ,{\lambda }_2)$$, the modal A matrix is of the form

$$\left[\begin{array}{cccc}{\lambda }_1& 0& 0& 0\\ 0& \sigma & \omega & 0\\ 0& -\omega & \sigma & 0\\ 0& 0& 0& {\lambda }_2\end{array}\right]$$

Companion Form

In the companion realization, the characteristic polynomial of the system appears explicitly in the rightmost column of the A matrix.

For a system with characteristic polynomial

$$P(s)=s^n+{\alpha }_1{s}^{n-1}+\dots +{\alpha }_{n-1}s+{\alpha }_n$$

the corresponding companion A matrix is

$$A=\left[\begin{array}{ccc}\begin{array}{l}0\\ 1\\ 0\\ 0\\ ⋮\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 1\\ 0\\ ⋮\\ 0\end{array}& \begin{array}{ccc}\begin{array}{cc}\begin{array}{l}0\\ 0\\ 0\\ 1\\ ⋮\\ 0\end{array}& \begin{array}{l}\dots \\ \dots \\ \dots \\ \dots \\ \ddots \\ \dots \end{array}\end{array}& \begin{array}{l}0\\ 0\\ 0\\ 0\\ ⋮\\ 1\end{array}& \begin{array}{l}-{\alpha }_n\\ -{\alpha }_{n-1}\\ -{\alpha }_{n-2}\\ -{\alpha }_{n-3}\\ ⋮\\ -{\alpha }_1\end{array}\end{array}\end{array}\right]$$

The companion transformation requires that the system be controllable from the first input. The companion form is poorly conditioned for most state-space computations; avoid using it when possible.

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.

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 ssestOptions, set SearchMethod to 'lsqnonlin', and set SearchOptions.Advanced.UseParallel to true.

For example:

opt = ssestOptions;
opt.SearchMethod = 'lsqnonlin';
opt.SearchOptions.Advanced.UseParallel = true;