# lft

linear fractional transformation

### Syntax

S = lft(P, R) [S, s] = lft(P, p, R) [S, s] = lft(P, p, R, r)

### Arguments

- P
linear system (in state space or transfer function representation) or a simple gain, the ``augmented'' plant, implicitly partitioned into four blocks (two input ports and two output ports).

- p
1x2 row vector, the dimensions of the

`P_22`

block (see below).- R
llinear system (in state space or transfer function representation) or a simple gain, implicitly partitioned into four blocks (two input ports and two output ports).

- r
1x2 row vector, dimension of the

`R_22`

block . This argument should not be used. It is retained for compatibility with previous versions.- S
linear system, the linear fractional transform.

- s
1x2 row vector, dimension of the

`S_22`

block (see below).

### Description

Linear fractional transform between two standard plants in state space form or in transfer form:

- Syntax
`S=lft(P,R)`

Computes the linear fractional transform between the systems

`P`

and a controller`R`

. The system`S`

corresponds to the transferif

`ny`

and`nu`

are respectively the number of inputs and outputs of`R`

, one must have`size(P)>=[ny nu]`

. The system returned is formally equivalent toUsingi1 = 1:($-ny);j1=1:($-nu); i2 = ($-ny+1):$;j1=($-nu+1):$; S = P(i1,j1) + P(i1,j2) * R * (eye() - P(i2,j2) * R) \P(i2,j1)

`lft`

instead of the code above avoids numerical problems and non minimal realization.- Syntax
`[S,s]=lft(P,p,R)`

with

`p= [ny,nu]`

Forms the generalized (2 ports) lft of`P`

and`R`

.`S`

is the two-port interconnected plant, which correspond to the transfer:`s`

is the dimension of the`22`

block of`S`

.

`P`

and `R`

can be PSSDs i.e. may admit a
polynomial `D`

matrix.

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