## Contents

## SIMLNS

Calculates the left null-space of the basis of similarity transformations.

## Syntax

`U2 = simlns(A,B,C,[],[],[])`

`U2 = simlns(A,B,C,K,fD,fx)`

## Description

The function `simlns` calculates the left null-space of an LTI system's similarity map `Mtheta`. In the most general case, when *A*, *B*, *C*, *D*, *K* and *x0* are part of the parameter vector, this matrix is given by [1]:

A QR-factorization is used to obtain the left null-space.

This function is used internally by `dfunlti` and `ffunlti` and is not meant for stand-alone use.

## Inputs

`A,B,C` are the system matrices describing the LTI State Space system.

`K` is the (optional) Kalman gain, specify as empty matrix when not present.

`fD` (optional) specifies whether *D* is part of the parameter vector, specify as empty, *0* or *1*.

`fx` (optional) specifies whether *x0* is part of the parameter vector, specify as empty, *0* or *1*.

## Outputs

`U2` is the left null-space of the similarity map.

## Remarks

Specifying `fx=1` only causes an *n* x *n* identify-matrix to be appended to the lower right of the left null-space matrix; in a non-linear optimization, applying the left null-space ensures that the state-basis does not change. It thus does not have to be projected.

## Algorithm

The manifold matrix `Mtheta` is calculated according to [1]. A QR-factorization is used subsequently to obtain the left null-space

## Used By

## References

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