lft

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

• sensi — sensitivity functions
• augment — augmented plant
• feedback — feedback operation
• sysdiag — block diagonal system connection