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.TH lft 5 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
lft - linear fractional transformation
.SH CALLING SEQUENCE
.nf
[P1]=LFT(P,K)
[P1]=LFT(P,r,K)
[P1,r1]=LFT(P,r,P#,r#)
.fi
.SH PARAMETERS
.TP
P
: linear system (\fVsyslin\fR list), the ``augmented'' plant, implicitly
partitioned into four blocks (two input ports and two output ports).
.TP
K
: linear system (\fVsyslin\fR list), the controller (possibly
an ordinary gain).
.TP
r
: 1x2 row vector, dimension of \fVP22\fR
.TP
P#
: linear system (\fVsyslin\fR list), implicitly
partitioned into four blocks (two input ports and two output ports).
.TP
r#
: 1x2 row vector, dimension of \fVP#22\fR
.SH DESCRIPTION
Linear fractional transform between two standard plants
\fVP\fR and \fVP#\fR in state space form or in transfer form
(\fVsyslin\fR lists).
.LP
\fVr= size(P22) r#=size(P22#)\fR
.LP
\fVLFT(P,r, K)\fR is the linear fractional transform between \fVP\fR and
a controller \fVK\fR (\fVK\fR may be a gain or a controller in
state space form or in transfer form);
.LP
\fVLFT(P,K)\fR is \fVLFT(P,r,K)\fR with \fVr\fR=size of \fVK\fR transpose;
.LP
\fVP1= P11+P12*K* (I-P22*K)^-1 *P21\fR
.LP
.Vb [P1,r1]=LFT(P,r,P#,r#)
returns the generalized (2 ports) lft of \fVP\fR and \fVP#\fR.
.LP
\fVP1\fR is the pair two-port interconnected plant and the partition
of \fVP1\fR into 4 blocks in given by \fVr1\fR which is the dimension
of the \fV22\fR block of \fVP1\fR.
.LP
\fVP\fR and \fVR\fR can be PSSDs i.e. may admit a polynomial \fVD\fR matrix.
.SH EXAMPLE
.nf
s=poly(0,'s');
P=[1/s, 1/(s+1); 1/(s+2),2/s]; K= 1/(s-1);
lft(P,K)
lft(P,[1,1],K)
P(1,1)+P(1,2)*K*inv(1-P(2,2)*K)*P(2,1) //Numerically dangerous!
ss2tf(lft(tf2ss(P),tf2ss(K)))
lft(P,-1)
f=[0,0;0,1];w=P/.f; w(1,1)
//Improper plant (PID control)
W=[1,1;1,1/(s^2+0.1*s)];K=1+1/s+s
lft(W,[1,1],K); ss2tf(lft(tf2ss(W),[1,1],tf2ss(K)))
.fi
.SH SEE ALSO
sensi, augment, feedback, sysdiag
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