## Contents

## LTIFRF

Calculates an LTI Frequency Response Function

## Syntax

`H = ltifrf(A,B,C,[],[],w,outopt)`

`H = ltifrf(A,B,C,D,[],w,outopt)`

`H = ltifrf([],[],[],D,[],w,outopt)`

`H = ltifrf(A,B,C,[],dA,w,outopt)`

## Description

`ltifrf` will return the Frequency Response Function (FRF) of a linear time-invariant state-space model, evaluated at the complex frequencies provided in *w*:

This function is used internally by `ffunlti`, `fac2b` and `fac2bd`. It is not meant for stand-alone use.

## Inputs

`A` is the state-space model matrix *A*.

`B` is the state-space model matrix *B*.

`C` is the state-space model matrix *C*.

`D` is the (optional) state-space model matrix *D*.

`dA` (optional) calculates the change in FRF given the deviation *dA* in *A*. *D* and *dA* are mutually exclusive.

`w` is the vector of complex frequencies. For discrete-time systems:

and for continuous-time systems.

`outopt` controls how *H* will be returned (see below).

## Outputs

`H` is the FRF. Usually a 3D-array of size *l* x *m* x *N*. However, if `outopt` is non-empty and *1*, *H* will be a vector of size *lmN* x 1. If `outopt` is non-empty and *2*, *H* will be a matrix of size *l* x *mN*.

## Algorithm

The state-space model is first transformed such that its state-transistion matrix *A* is in upper-Hessenberg form. The matrix

is subsequently solved by an efficient upper-Hessenberg solver in SLICOT, after which premultiplication by *C* and addition of *D* yields the FRF. This approach follows [1].

If a deviation *deltaA* in *A* is given, the FRF deviation is given by:

Again, the model is transformed so that *A* has upper-Hessenberg form, after which the SLICOT Hessenberg solver is used to obtain

and

Multiplication then yeilds the FRF deviation.

## Used By

## Uses Functions

SLICOT-functions `MB02RZ`, `MB02SZ`, `TB05AD`

LAPACK-functions `DGEHRD` and `DORMHR`

(All built into the executable)

## See Also

## References

[1] A.J. Laub, "Efficient multivariable frequency response calculations", *IEEE Transactions on Automatic Control*, vol. 26, pp. 407-408, Apr. 1981.