# n4sid

Estimate state-space model using subspace method with time-domain or frequency-domain data

## Description

### Estimate State-Space Model

example

sys = n4sid(tt,nx) estimates a discrete-time state-space model sys of order nx using all the input and output signals in the timetable tt.

sys is a model of the following form:

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

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. For dynamic systems, D is fixed to zero by default, meaning that the system has no feedthrough. For static systems (nx = 0), D is an estimable parameter by default.

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. You can also use this syntax to estimate a time-series model if tt contains a single variable that represents the sole output.

For MIMO systems and for timetables that contain more variables than you plan to use for estimation, you must also use name-value arguments to specify the names of the input and output channels you want. For more information, see tt.

To estimate a continuous-time model, set 'Ts' to 0 using name-value syntax.

sys = n4sid(u,y,nx,'Ts',Ts) uses the time-domain input and output signals in the comma-separated matrices u,y and the sample time Ts. You can use this syntax for SISO, MISO, and MIMO systems.

Estimating continuous-time models from matrix-based data is not recommended.

example

sys = n4sid(data,nx) uses 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 data sample time or experiment labeling, that data objects provide.

example

sys = n4sid(___,Name,Value) incorporates additional options specified by one or more name-value pair arguments. For example, to estimate a continuous-time model, specify the sample time 'Ts' as 0. Use the 'Form', 'Feedthrough', and 'DisturbanceModel' name-value pair 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 com

example

sys = n4sid(___,opt) specifies the estimation options opt. These options can include the initial states, estimation objective, and subspace algorithm related choices to be used for estimation. Specify opt after any of the previous input-argument combinations.

### Return Estimated Initial States

[sys,x0] = n4sid(___) returns the value of initial states computed during estimation. You can use this syntax with any of the previous input-argument combinations.

## Examples

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

Load the input-output data tt1, which is stored in a timetable.

Estimate a fourth-order state-space model.

nx = 4;
sys = n4sid(tt1,nx)
sys =
Discrete-time identified state-space model:
x(t+Ts) = A x(t) + B u(t) + K e(t)
y(t) = C x(t) + D u(t) + e(t)

A =
x1         x2         x3         x4
x1     0.8392    -0.3129    0.02105    0.03743
x2     0.4768     0.6671     0.1428   0.003757
x3   -0.01951    0.08374   -0.09761      1.046
x4  -0.003885   -0.02914    -0.8796   -0.03171

B =
u
x1    0.02635
x2   -0.03301
x3  7.256e-05
x4  0.0005861

C =
x1       x2       x3       x4
y    69.08    26.64   -2.237  -0.5601

D =
u
y  0

K =
y
x1   0.003282
x2   0.009339
x3  -0.003232
x4   0.003809

Sample time: 0.1 seconds

Parameterization:
FREE form (all coefficients in A, B, C free).
Feedthrough: none
Disturbance component: estimate
Number of free coefficients: 28
Use "idssdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:
Estimated using N4SID on time domain data "tt1".
Fit to estimation data: 76.33% (prediction focus)
FPE: 1.21, MSE: 1.087

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%.

sys.Report
ans =
Status: 'Estimated using N4SID with prediction focus'
Method: 'N4SID'
InitialState: 'estimate'
N4Weight: 'CVA'
N4Horizon: [6 10 10]
Fit: [1x1 struct]
Parameters: [1x1 struct]
OptionsUsed: [1x1 idoptions.n4sid]
RandState: [1x1 struct]
DataUsed: [1x1 struct]

sys.Report.Parameters.X0
ans = 4×1

-0.0085
0.0052
-0.0193
0.0282

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

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

nx = 1:10;
sys = n4sid(z1,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 .

Specify estimation options. Set the weighting scheme 'N4Weight' to 'SSARX' and estimation-status display option 'Display' to 'on'.

opt = n4sidOptions('N4Weight','SSARX','Display','on')
Option set for the n4sid command:

InitialState: 'estimate'
N4Weight: 'SSARX'
N4Horizon: 'auto'
Display: 'on'
InputInterSample: 'auto'
InputOffset: []
OutputOffset: []
EstimateCovariance: 1
OutputWeight: []
Focus: 'prediction'
WeightingFilter: []
EnforceStability: 0

Estimate a third-order state-space model using the updated option set.

nx = 3;
sys = n4sid(z2,nx,opt);

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 n4sid default options.

sys1 = n4sid(z1,4);

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

sys2 = n4sid(z1,4,'Form','modal');
A1 = sys1.A
A1 = 4×4

0.8392   -0.3129    0.0211    0.0374
0.4768    0.6671    0.1428    0.0038
-0.0195    0.0837   -0.0976    1.0462
-0.0039   -0.0291   -0.8796   -0.0317

A2 = sys2.A
A2 = 4×4

0.7554    0.3779         0         0
-0.3779    0.7554         0         0
0         0   -0.0669    0.9542
0         0   -0.9542   -0.0669

Include a feedthrough term and compare the D matrices.

sys3 = n4sid(z1,4,'Feedthrough',1);
D1 = sys1.D
D1 = 0
D3 = sys3.D
D3 = 0.0487

Eliminate disturbance modeling and compare the K matrices.

sys4 = n4sid(z1,4,'DisturbanceModel','none');
K1 = sys1.K
K1 = 4×1

0.0033
0.0093
-0.0032
0.0038

K4 = sys4.K
K4 = 4×1

0
0
0
0

Estimate a continuous-time canonical-form model.

Estimate the model. Set Ts to 0 to specify a continuous model.

nx = 2;
sys = n4sid(z1,nx,'Ts',0,'Form','canonical');

sys is a second-order continuous-time state-space model in the canonical form.

Estimate a state-space model from closed-loop data using the subspace algorithm SSARX. This algorithm is better at capturing feedback effects than other weighting algorithms.

Generate closed-loop estimation data for a second-order system corrupted by white noise.

N = 1000;
K = 0.5;
rng('default');
w = randn(N,1);
z = zeros(N,1);
u = zeros(N,1);
y = zeros(N,1);
e = randn(N,1);
v = filter([1 0.5],[1 1.5 0.7],e);
for k = 3:N
u(k-1) = -K*y(k-2) + w(k);
u(k-1) = -K*y(k-1) + w(k);
z(k) = 1.5*z(k-1) - 0.7*z(k-2) + u(k-1) + 0.5*u(k-2);
y(k) = z(k) + 0.8*v(k);
end
dat = iddata(y, u, 1);

Specify the weighting scheme 'N4weight' used by the N4SID algorithm. Create two option sets. For one option set, set 'N4weight' to 'CVA'. For the other option set, set the 'N4weight' to 'SSARX'.

optCVA = n4sidOptions('N4weight','CVA');
optSSARX = n4sidOptions('N4weight','SSARX');

Estimate state-space models using the option sets.

sysCVA = n4sid(dat,2,optCVA);
sysSSARX = n4sid(dat,2,optSSARX);

Compare the fit of the two models with the estimation data.

compare(dat,sysCVA,sysSSARX);

As the plot shows, the model estimated using the SSARX algorithm produces a better fit than the model estimated using the CVA algorithm.

## Input Arguments

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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 Ns-by-2 timetable, where Ns 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 Ns-by-(Nu+Ny) timetable, where Nu is the number of inputs and Ny is the number of outputs. The first Nu variables must contain the input channels and the remaining Ny 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 Ne-by-1 cell array of timetables, where Ne 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 = n4sid(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 Ns-by-1 timetable.

• For a multivariate system, specify tt as an Ns-by-(Ny) 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 = n4sid(tt,__,'OutputName',["y1" "y3"],'InputName',[])

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

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

• uNs-by-Nu, where Nu is the number of inputs.

• yNs-by-Ny, where Ny is the number of outputs.

For multiexperiment data, specify u,y as a pair of 1-by-Ne cell arrays, where Ne 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 only.

#### 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.

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 (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. 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.

The domain of your data determines the type of model you can estimate.

• Time-domain or discrete-time frequency-domain data — Continuous-time and discrete-time models

• Continuous-time frequency-domain data — Continuous-time models only

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 that range for nx. n4sid 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.

Estimation options, specified as an n4sidOptions option set. Options specified by opt include:

• Estimation objective

• Handling of initial conditions

• Subspace algorithm-related choices

For examples showing how to specify options, see Specify Estimation Options and Continuous-Time Canonical-Form Model.

### 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 = n4sid(data,nx,'Form','modal')

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. In the frequency domain, using continuous-time frequency-domain data results in a continuous-time model.

• For discrete-time models, the software sets 'Ts' to the sample time of the data in the units stored in the TimeUnit property.

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 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.

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

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

• 'free' — Treat all entries of the matrices A, B, C, D, and K 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 an example, see Modify Form, Feedthrough, and Disturbance-Model Matrices.

Direct feedthrough from input to output, specified as the comma-separated pair consisting of 'Feedthrough' and a logical vector of length Nu, where Nu 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.

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. For time-domain data, 'estimate' is the default.

• 'none' — Do not estimate the noise component. The elements of the K matrix are fixed at zero. For frequency-domain data, 'none' is the default and the only acceptable value.

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

## Output Arguments

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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 FieldDescription
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.

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 the following fields:

FieldDescription
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 you did not configure any custom options, OptionsUsed is the set of default options. See n4sidOptions 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, returned as a structure with the following fields.

FieldDescription
Name

Name of the data set.

Type

Data type.

Length

Number of data samples.

Ts

Sample time.

InterSample

Input intersample behavior, returned as 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.

InputOffset

Offset removed from time-domain input data during estimation. For nonlinear models, it is [].

OutputOffset

Offset removed from time-domain output data during estimation. For nonlinear models, it is [].

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.

## References

[1] Ljung, L. System Identification: Theory for the User, Appendix 4A, Second Edition, pp. 132–134. Upper Saddle River, NJ: Prentice Hall PTR, 1999.

[2] van Overschee, P., and B. De Moor. Subspace Identification of Linear Systems: Theory, Implementation, Applications. Springer Publishing: 1996.

[3] Verhaegen, M. "Identification of the deterministic part of MIMO state space models." Automatica, 1994, Vol. 30, pp. 61–74.

[4] Larimore, W.E. "Canonical variate analysis in identification, filtering and adaptive control." Proceedings of the 29th IEEE Conference on Decision and Control, 1990, pp. 596–604.

[5] McKelvey, T., H. Akcay, and L. Ljung. "Subspace-based multivariable system identification from frequency response data." IEEE Transactions on Automatic Control, 1996, Vol. 41, pp. 960–979.

## Version History

Introduced before R2006a

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