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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)))

• sensi — sensitivity functions
• augment — augmented plant
• feedback — feedback operation
• sysdiag — соединение блочно-диагональной системы