## Contents

## CSS2TH

Converts a continuous-time LTI state-space model into a parameter vector.

## Syntax

`[theta,params,T] = css2th(A,C,partype)`

`[theta,params,T] = css2th(A,B,C,partype)`

`[theta,params,T] = css2th(A,B,C,D,partype)`

`[theta,params,T] = css2th(A,B,C,D,x0,partype)`

`[theta,params,T] = css2th(A,B,C,D,x0,K,partype)`

## Description

This function converts a continuous-time LTI state-space model into a parameter vector that describes the model. Model structure:

## Inputs

`A,B,C,D` are system matrices describing the state space system. The `B` and `D` matrices are optional and can be left out or passed as an empty matrix to indicate it is not part of the model.

`x0` is the (optional) initial state.

`K` is the (optional) Kalman gain.

`partype` is string which specifies the type of parameterization that is used to parameterize the state space model. Three types of parameterization are supported:

`'on'`for output Normal parametrization.`'tr'`for tridiagonal parametrization.`'fl'`for full parametrization.

Rules for input parameters:

- The final parameter should always be the parametrization type. The order for the parameters prior to
`partype`is`A,B,C,D,x0,K`. The only exception is`A,C`, when only those are to be parametrized. - All parameters after
`A,B,C`and before`partype`are optional. If the last one is not to be parametrized it can be omitted. If any other is not to be parametrized, an empty matrix should be passed. `(A,B,C,partype)`is thus equivalent to`(A,B,C,[],[],[],partype)`However,`(A,B,C,[],x0,partype)`cannot be abbreviated.

## Outputs

`theta` is the parameter vector describing the system.

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

`T` is the transformation matrix between the input state space system and the state space system in the form described by `theta`.

## Remarks

This function is based on the SMI Toolbox 2.0 function `css2th`, copyright 1996 Johan Bruls. Support for the omission of `D`, `x0` and/or `K` has been added, as well as support for the full parametrization.

## Algorithm

The model parametrization for the output normal form and the tridiagonal parametrization is carried out according to [1]. The full model parametrization is a simple vectorization of the system matrices. In its most general form, the parameter vector is given by

## Used By

## See Also

## References

[1] B. Haverkamp, *Subspace Method Identification, Theory and Practice.* PhD thesis, Delft University of Technology, Delft, The Netherlands, 2000.