## Estimate State-Space Models at the Command Line

### Black Box vs. Structured State-Space Model Estimation

You can estimate state-space models in two ways at the command line, depending upon your prior knowledge of the nature of the system and your requirements.

#### Black Box Estimation

In this approach, you specify the model order, and, optionally, additional model structure attributes that configure the overall structure of the state-space matrices. You call ssest, ssregest or n4sid with data and model order as primary input arguments, and use name-value pairs to specify any additional attributes, such as model sample time, presence of feedthrough, absence of noise component, etc. You do not work directly with the coefficients of the A, B, C, D, K, and X0 matrices.

#### Structured Estimation

In this approach, you create and configure an idss model that contains the initial values for all the system matrices. You use the Structure property of the idss model to specify all the parameter constraints. For example, you can designate certain coefficients of system matrices as fixed and impose minimum/maximum bounds on the values of the others. For quick configuration of the parameterization and whether to estimate feedthrough and disturbance dynamics, use ssform.

After configuring the idss model with desired constraints, you specify this model as an input argument to the ssest command. You cannot use n4sid or ssregest for structured estimation.

Note

• The structured estimation approach is also referred to as grey-box modeling. However, in this toolbox, the “grey box modeling” terminology is used only when referring to idgrey and idnlgrey models.

• Using the structured estimation approach, you cannot specify relationships among state-space coefficients. Each coefficient is essentially considered to be independent of others. For imposing dependencies, or to use more complex forms of parameterization, use the idgrey model and greyest estimator.

### Estimating State-Space Models Using ssest, ssregest and n4sid

You can estimate continuous-time and discrete-time state-space models using the iterative estimation command ssest that minimizes the prediction errors to obtain maximum-likelihood values.

Use the following general syntax to both configure and estimate state-space models:

m = ssest(data,n,opt,Name,Value)

where data is the estimation data, n is the model order, and opt contains options for configuring the estimation of the state-space models. These options include the handling of the initial conditions, input and output offsets, estimation focus and search algorithm options. opt can be followed by name-value pair input arguments that specify optional model structure attributes such as the presence of feedthrough, the canonical form of the model, and input delay.

As an alternative to ssest, you can use the noniterative subspace estimators n4sid or ssregest:

m = n4sid(data,n,opt,Name,Value)
m = ssregest(data,n,opt,Name,Value)

Unless you specify the sample time as a name-value pair input argument, n4sid and ssregest estimate a discrete-time model, while ssest estimates a continuous-time model.

Note

ssest uses n4sid to initialize the state-space matrices, and takes longer than n4sid to estimate a model but typically provides a better fit to data.

### Choosing the Structure of A, B, C Matrices

By default, all entries of the A, B, and C state-space matrices are treated as free parameters. Using the Form name-value pair input argument of ssest , you can choose various canonical forms, such as the companion and modal forms, that use fewer parameters.

### Choosing Between Continuous-Time and Discrete-Time Representations

For estimation of state-space models, you have the option of switching the model sample time between zero and that of the estimation data. You can do this using the Ts name-value pair input argument.

• By default, ssest estimates a continuous-time model. If you are using data set with nonzero sample time, data, which includes all time domain data, you can also estimate a discrete-time model by using:

model = ssest(data,nx,'Ts',data.Ts);

If you are using continuous-time frequency-domain data, you cannot estimate a discrete-time model.

• By default, n4sid and ssregest estimate a model whose sample time matches that of the data. Thus, for time-domain data, n4sid and ssregest deliver a discrete-time model. You can estimate a continuous-time model by using:

model = n4sid(data,nx,'Ts',0);

or

model = ssregest(data,nx,'Ts',0);

### Choosing to Estimate D, K, and X0 Matrices

For state-space models with any parameterization, you can specify whether to estimate the D, K and X0 matrices, which represent the input-to-output feedthrough, noise model and the initial states, respectively.

For state-space models with structured parameterization, you can also specify to estimate the D matrix. However, for free and canonical forms, the structure of the D matrix is set based on your choice for the 'Feedthrough' name-value pair input argument.

#### D Matrix

By default, the D matrix is not estimated and its value is fixed to zero, except for static models.

• Black box estimation: Use the Feedthrough name-value pair input argument to denote the presence or absence of feedthrough from individual inputs. For example, in case of a two input model such that there is feedthrough from only the second input, use:

model = n4sid(data,n,'Feedthrough',[false true]);
• Structured estimation: Configure the values of the init_sys.Structure.D, where init_sys is an idss model that represents the desired model structure. To force no feedthrough for the i-th input, set:

init_sys.Structure.D.Value(:,i) = 0;
init_sys.Structure.D.Free = true;
init_sys.Structure.D.Free(:,i) = false;

The first line specifies the value of the i-th column of D as zero. The next line specifies all the elements of D as free, estimable parameters. The last line specifies that the i-th column of the D matrix is fixed for estimation.

Alternatively, use ssform with 'Feedthrough' name-value pair.

#### K Matrix

K represents the noise matrix of the model, such that the noise component of the model is:.

$\begin{array}{l}\stackrel{˙}{x}=Ax+Ke\\ {y}_{n}=Cx+e\end{array}$

For frequency-domain data, no noise model is estimated and K is set to 0. For time-domain data, K is estimated by default in the black box estimation setup. yn is the contribution of the disturbances to the model output.

• Black box estimation: Use the DisturbanceModel name-value pair input argument to indicate if the disturbance component is fixed to zero (specify Value = 'none') or estimated as a free parameter (specify Value = 'estimate'). For example, use :

model = n4sid(data,n,'DisturbanceModel','none');
• Structured estimation: Configure the value of the init_sys.Structure.K parameter, where init_sys is an idss model that represents the desired model structure. You can fix some K matrix coefficients to known values and prescribe minimum/maximum bounds for free coefficients. For example, to estimate only the first column of the K matrix for a two output model:

kpar = init_sys.Structure.K;
kpar.Free(:,1) = true;
kpar.Free(:,2) = false;
kpar.Value(:,2) = 0; % second column value is fixed to zero
init_sys.Structure.K = kpar;

Alternatively, use ssform.

When not sure how to easily fix or free all coefficients of K, initially you can omit estimating the noise parameters in K to focus on achieving a reasonable model for the system dynamics. After estimating the dynamic model, you can use ssest to refine the model while configuring the K parameters to be free. For example:

init_sys = ssest(data, n,'DisturbanceModel','none');
init_sys.Structure.K.Free = true;
sys = ssest(data,init_sys);

where init_sys is the dynamic model without noise.

To set K to zero in an existing model, you can set its Value to 0 and Free flag to false:

m.Structure.K.Value = 0;
m.Structure.K.Free = false;

#### X0 Matrices

The initial state vector X0 is obtained as the by-product of model estimation. The n4sid, ssest and ssregest commands return the value of X0 as their second output arguments. You can choose how to handle initial conditions during model estimation by using the InitialState estimation option. Use n4sidOptions (for n4sid), ssestOptions (for ssest) or ssregestOptions (for ssregest) to create the estimation option set. For example, in order to hold the initial states to zero during estimation using n4sid:

opt = n4sidOptions;
opt.InitialState = 'zero';
[m,X0] = n4sid(data,n,opt);

The returned X0 variable is a zero vector of length n.

When you estimate models using multiexperiment data, the X0 matrix contains as many columns as data experiments.

For a complete list of values for the InitialStates option, see Specifying Initial States for Iterative Estimation Algorithms.

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