'e' — The initial states are estimated such that the norm of prediction error is minimized.

For nonlinear grey-box models, only those initial states i that are designated as free in the model (sys.InitialStates(i).Fixed = false) are estimated. To estimate all the states of the model, first specify all the Nx states of the idnlgrey model sys as free.

for i = 1:Nx
sys.InitialStates(i).Fixed = false;
end 

Similarly, to fix all the initial states to values specified in sys.InitialStates, first specify all the states as fixed in the sys.InitialStates property of the nonlinear grey-box model.

'd' — Similar to 'e', but absorbs nonzero delays into the model coefficients. The delays are first converted to explicit model states, and the initial values of those states are also estimated and returned.

Use this option for discrete-time linear models only.

Vector or Matrix — Initial guess for state values, when using nonlinear models. Specify a column vector of length equal to the number of states. For multi-experiment data, use a matrix with Ne columns, where Ne is the number of experiments.

Use this option for nonlinear models only.

x0obj — Specification object created using idpar. Use x0obj to impose constraints on the initial states by fixing their value or specifying minimum or maximum bounds.

Use x0obj only for nonlinear grey-box models and linear state-space models (idss or idgrey). This option is applicable only for prediction horizon equal to 1 or Inf. See findstates for more details about the prediction horizon.

A column vector of positive integers of length Nu, where Nu is the number of inputs.

[] — Indicates no offset.

Nu-by-Ne matrix — For multi-experiment data, specify InputOffset as an Nu-by-Ne matrix. Nu is the number of inputs, and Ne is the number of experiments.

A column vector of length Ny, where Ny is the number of outputs.

[] — Indicates no offset.

Ny-by-Ne matrix — For multi-experiment data, specify OutputOffset as a Ny-by-Ne matrix. Ny is the number of outputs, and Ne is the number of experiments.

[] — No weighting is used. Specifying as [] is the same as eye(Ny), where Ny is the number of outputs.

'noise' — Inverse of the noise variance stored with the model is used for weighting during estimation of initial states.

Positive semidefinite matrix, W, of size Ny-by-Ny — This weighting minimizes trace(E'*E*W) for estimation of initial states, where E is the matrix of prediction errors.

'auto' — A combination of the line search algorithms, 'gn', 'lm', 'gna', and 'grad' methods is tried in sequence at each iteration. The first descent direction leading to a reduction in estimation cost is used.

'gn' — Subspace Gauss-Newton least squares search. Singular values of the Jacobian matrix less than GnPinvConstant*eps*max(size(J))*norm(J) are discarded when computing the search direction. J is the Jacobian matrix. The Hessian matrix is approximated as J^TJ. If there is no improvement in this direction, the function tries the gradient direction.

'gna' — Adaptive subspace Gauss-Newton search. Eigenvalues less than gamma*max(sv) of the Hessian are ignored, where sv contains the singular values of the Hessian. The Gauss-Newton direction is computed in the remaining subspace. gamma has the initial value InitialGnaTolerance (see Advanced in 'SearchOptions' for more information). This value is increased by the factor LMStep each time the search fails to find a lower value of the criterion in fewer than five bisections. This value is decreased by the factor 2*LMStep each time a search is successful without any bisections.

'lm' — Levenberg-Marquardt least squares search, where the next parameter value is -pinv(H+d*I)*grad from the previous one. H is the Hessian, I is the identity matrix, and grad is the gradient. d is a number that is increased until a lower value of the criterion is found.

'grad' — Steepest descent least squares search.

'lsqnonlin' — Trust-region-reflective algorithm of lsqnonlin (Optimization Toolbox). Requires Optimization Toolbox™ software.

'fmincon' — Constrained nonlinear solvers. You can use the sequential quadratic programming (SQP) and trust-region-reflective algorithms of the fmincon (Optimization Toolbox) solver. If you have Optimization Toolbox software, you can also use the interior-point and active-set algorithms of the fmincon solver. Specify the algorithm in the SearchOptions.Algorithm option. The fmincon algorithms may result in improved estimation results in the following scenarios: