| | | Scilab Reference Manual | |
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lft — linear fractional transformation
[P1]=LFT(P,K) [P1]=LFT(P,r,K) [P1,r1]=LFT(P,r,P#,r#)
| 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 |
| P# | : linear system (syslin list), implicitly partitioned into four blocks (two input ports and two output ports). |
| r# | : 1x2 row vector, dimension of P#22 |
Linear fractional transform between two standard plants P and P# in state space form or in transfer form (syslin lists).
r= size(P22) r#=size(P22#)
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,P#,r#) returns the generalized (2 ports) lft of P and P#.
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)))
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