## Contents

## DFUNLTI

Calculates the cost-function information for `doptlti`.

## Syntax

`[epsilon] = dfunlti(th,u,y,params)`

`[epsilon,psi] = dfunlti(th,u,y,params)`

`[epsilon,psi,U2] = dfunlti(th,u,y,params)`

`[epsilon] = dfunlti(th,u,y,params,options,OptType,sigman,filtera,CorrD)`

`[epsilon,psi] = dfunlti(th,u,y,params,options,... OptType,sigman,filtera,CorrD)`

`[epsilon,psi,U2] = dfunlti(th,u,y,params,options,... OptType,sigman,filtera,CorrD)`

## Description

This function implements the cost-fuction for `doptlti`. It is not meant for standalone use.

## Inputs

`th` is the parameter vector describing the system.

`u,y` is the input and output data of the system to be optimized.

`params` is a structure that contains the dimension parameters of the system, such as the order, the number of inputs, whether `D`, `x0` or `K` is present in the model.

`options` is an (optional) `optimset` compatible options-structure. The fields `options.RFactor`, `options.LargeScale`, `options.Manifold` and `options.BlockSize` should have been added by `doptlti`.

`OptType` (optional) indicates what kind of weighted least squares or maximum likelihood optimization is being performed:

`'no_mle'`implies a nonlinear (weighted) least squares optimization.`'uncorr'`implies a maximum likelihood optimization without correlation among the output perturbances [1].`'flcorr'`implies a maximum likelihood optimization with correlated output perturbances [2].

`sigman` (optional) If `OptType` is `'no_mle'`, this can be a vector of size *1* x *l* that indicates the standard deviation of the perturbance of each of the outputs. If `OptType` is `'uncorr'`, this should be a vector of size *1* x *l* that indicates the standard deviation of the white noise innovations for the output perturbance AR model. If `OptType` is `'flcorr'`, this should be a Cholesky factor of the AR process' inverse covariance matrix, as obtained by `cholicm`.

`filtera` (optional) If `OptType` is `'uncorr'`, this should be the *A*-polynomial of a *d* th order AR noise model. The first element should be *1*, and the other elements should be *d* filter coefficients. In the multi-output case *filtera* should be a matrix having *max(di)+1* rows and *l* columns. If a certain output noise model has a lower order, then the coefficient vector should be padded with |NaN|s.

`CorrD` (optional) If `OptType` is `'uncorr'`, this should be a correction matrix for the lower-right part of the ICM's Cholesky-factor. No details will be provided here.

## Outputs

`epsilon` is the output of the cost function, which is the square of the error between the output and the estimated output.

`psi` is the Jacobian

of epsilon, or

if the full parametrization is used.

`U2` is the left null-space of Manifold matrix for the full parametrization [3].

## Algorithm

The formation of the error-vector is done by simple simulation of the current model:

The error-vector

is build up such that its *i* th blockrow consists of

Note that this example corresponds to the error-vector of an output error model in which no output weighting is applied. For innovation models and maximum likelihood corrections, the error-vector is different from the one shown above.

The Jacobian is formed by simulation as well [4]. This is a special case of the Jacobian for LPV systems that has been described in [3]. A QR-factorization is used to obtain its left null-space.

## Used By

## Uses Functions

## See Also

## References

[1] B. David and G. Bastin, "An estimator of the inverse covariance matrix aqnd its application to ML parameter estimation in dynamical systems", *Automatica*, vol. 37, no. 1, pp. 99-106, 2001.

[2] B. Davis, *Parameter Estimation in Nonlinear Dynamical Systems with Correlated Noise.* PhD thesis, Universite Catholique de Louvain-La-Neuve, Belgium, 2001.

[3] L.H. Lee and K. Poolla, "Identification of linear parameter varying systems using nonlinear programming", *Journal of Dynamic Systems*, Measurement and Control, col. 121, pp. 71-78, Mar 1999.

[4] N. Bergboer, V. Verdult, and M. Verhaegen, "An effcient implementation of maximum likelihood identification of LTI state-space models by local gradient search", in *Proceedings of the 41st IEEE Conference on Decision and Control*, Las Vegas, Nevada, Dec. 2002.