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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.
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 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.
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.
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. opt contains the options for configuring the estimation of the state-space models. These options include the handling of the initial conditions, input offset, estimation focus and search algorithm options. opt can be followed by name and 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 estimator n4sid:
m = n4sid(data,n,opt,Name,Value)
Unless you specify the sample time as a name-value pair input argument, ssest estimates a continuous-time model, while n4sid estimates a model whose sample time matches that of data.
Note: ssest uses n4sid to initialize the state-space matrices, and takes longer than n4sid to estimate a model but typically provides better fit to data.
For information about validating your model, see Validating Models After Estimation
By default, all entries of the A, B, and C state-space matrices are treated as free parameters. Using the Form name and value pair input argument of ssest , you can choose various canonical forms, such as the companion and modal forms, that use fewer parameters.
For more information about estimating a specific state-space parameterization, see:
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 and 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 estimates a model whose sample time matches that of the data. Thus, for time-domain data, n4sid delivers a discrete-time model. You can estimate a continuous-time model by using:
model = n4sid(data,nx,'Ts',0);
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 of 'Feedthrough' name and value pair input argument.
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 and 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 output, 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 represents the noise matrix of the model, such that the noise component of the model is:.
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.
Black box estimation: Use the DisturbanceModel name and 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;
The initial state vector X0 is obtained as the by-product of model estimation. The n4sid and ssest 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) and ssestOptions (for ssest) 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.