n4sid

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

Syntax

sys = n4sid(data,nx)

sys = n4sid(data,nx,Name,Value)

sys = n4sid(___,opt)

[sys,x0] = n4sid(___)

Description

sys = n4sid(data,nx) estimates a discrete-time state-space model sys of order nx using data, which can be time-domain or frequency-domain data. 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. 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.

sys = n4sid(data,nx,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.

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.

[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

State-Space Model

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

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

load iddata1 z1

Estimate a fourth-order state-space model.

nx = 4;
sys = n4sid(z1,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 = 
              u1
   x1    0.02635
   x2   -0.03301
   x3  7.256e-05
   x4  0.0005861
 
  C = 
            x1       x2       x3       x4
   y1    69.08    26.64   -2.237  -0.5601
 
  D = 
       u1
   y1   0
 
  K = 
              y1
   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 "z1".  
Fit to estimation data: 76.33% (prediction focus)
FPE: 1.21, MSE: 1.087                            

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 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]

For example, find out more information about the estimated initial state.

sys.Report.Parameters.X0

ans = 4×1

   -0.0085
    0.0052
   -0.0193
    0.0282

Determine Optimal Estimated Model Order

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

load iddata1 z1

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 Apply.

Specify Estimation Options

Load estimation data.

load iddata2 z2

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'
           InputOffset: []
          OutputOffset: []
    EstimateCovariance: 1
          OutputWeight: []
                 Focus: 'prediction'
       WeightingFilter: []
      EnforceStability: 0
              Advanced: [1x1 struct]

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

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

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 n4sid default options.

load iddata1 z1
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

Continuous-Time Canonical-Form Model

Estimate a continuous-time canonical-form model.

Load estimation data.

load iddata1 z1

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 State-Space Model from Closed-Loop Data

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

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 (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

nx — Order of estimated model

1:10 (default) | positive integer scalar | positive integer vector | best | 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 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.

opt — Estimation options

n4sidOptions option set

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

Ts — Sample time of estimated model

sample time of data (data.Ts) (default) | 0 | 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. 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.

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.

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' — 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 Canonical State-Space Realizations.

For more information about using these forms for identification, 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' | '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. 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

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. 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: Field Description 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. Field Description 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 [].

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.

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.