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lft

linear fractional transformation

Syntax

S = lft(P, R)
[S, s] = lft(P, p, R)
[S, s] = lft(P, p, R, r)

Arguments

P

linear system (in state space or transfer function representation) or a simple gain, the ``augmented'' plant, implicitly partitioned into four blocks (two input ports and two output ports).

p

1x2 row vector, the dimensions of the P_22 block (see below).

R

linear system (in state space or transfer function representation) or a simple gain, implicitly partitioned into four blocks (two input ports and two output ports).

r

1x2 row vector, dimension of the R_22 block . This argument should not be used. It is retained for compatibility with previous versions.

S

linear system, the linear fractional transform.

s

1x2 row vector, dimension of the S_22 block (see below).

Description

Linear fractional transform between two standard plants in state space form or in transfer form:

Syntax S=lft(P,R)

Computes the linear fractional transform between the systems P and a controller R. The system S corresponds to the transfer

if ny and nu are respectively the number of inputs and outputs of R, one must have size(P)>=[ny nu]. The system returned is formally equivalent to

i1 = 1:($-ny);j1=1:($-nu);
i2 = ($-ny+1):$;j1=($-nu+1):$;
S = P(i1,j1) + P(i1,j2) * R * (eye() - P(i2,j2) * R) \P(i2,j1)
Using lft instead of the code above avoids numerical problems and non minimal realization.

Syntax [S,s]=lft(P,p,R)

with p= [ny,nu] Forms the generalized (2 ports) lft of P and R.

S is the two-port interconnected plant, which correspond to the transfer: s is the dimension of the 22 block of S.

P and R can be PSSDs i.e. may admit a polynomial D matrix.

Examples

//simple feeback case

P=[1/%s, 1/(%s+1); 1/(%s+2),2/%s];
R= 1/(%s-1);
lft(P,R)

P(1,1)+P(1,2)*R*inv(1-P(2,2)*R)*P(2,1)   //Numerically dangerous!

lft(P,-1)

//Improper systems
W=[1,1;1,1/(%s^2+0.1*%s)];
K=tf2ss(1+1/%s+%s);////Improper (PID control)
lft(W,[1,1],K)

References

"Review of LFTs, LMIs, and μ",John Doyle, Andy Packard and Kemin Zhou, CDC december 1991

See also

History

VersionDescription
2025.0.0 lft(P, p, R, r) is obsolete and will be removed in 2026.0.0.

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