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lft
linear fractional transformation
Calling Sequence
[P1]=lft(P,K) [P1]=lft(P,r,K) [P1,r1]=lft(P,r,Ps,rs)
Arguments
- P
linear system (
syslin
list), the ``augmented'' plant, implicitly partitioned into four blocks (two input ports and two output ports).- K
linear system (
syslin
list), the controller (possibly an ordinary gain).- r
1x2 row vector, dimension of
P22
- Ps
linear system (
syslin
list), implicitly partitioned into four blocks (two input ports and two output ports).- rs
1x2 row vector, dimension of
Ps22
Description
Linear fractional transform between two standard plants
P
and Ps
in state space form or in
transfer form (syslin
lists).
r= size(P22) rs=size(P22s)
lft(P,r, K)
is the linear fractional transform
between P
and a controller K
(K
may be a gain or a controller in state space form
or in transfer form);
lft(P,K)
is lft(P,r,K)
with
r
=size of K
transpose;
P1= P11+P12*K* (I-P22*K)^-1 *P21
[P1,r1]=lft(P,r,Ps,rs)
returns the generalized (2
ports) lft of P
and Ps
.
P1
is the pair two-port interconnected plant and the
partition of P1
into 4 blocks in given by
r1
which is the dimension of the 22
block of P1
.
P
and R
can be PSSDs i.e. may admit a
polynomial D
matrix.
Examples
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)))
See Also
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