lft — linear fractional transformation
[P1]=lft(P,K) [P1]=lft(P,r,K) [P1,r1]=lft(P,r,Ps,rs)
linear system (syslin list), the ``augmented'' plant, implicitly partitioned into four blocks (two input ports and two output ports).
linear system (syslin list), the controller (possibly an ordinary gain).
1x2 row vector, dimension of P22
linear system (syslin list), implicitly partitioned into four blocks (two input ports and two output ports).
1x2 row vector, dimension of Ps22
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.
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)))