lftf (Robust Control Toolbox)

Two-port or one-port state-space linear fractional transformation.

[a,b1,b2,c1,c2,d11,d12,d21,d22] = 
lftf(A,B1,B2,,a,b1,b2,) 
[aa,bb,cc,dd] = 
lftf(a,b1,b2,c1,c2,d11,d12,d21,d22,aw,bw,cw,dw) 
[aa,bb,cc,dd] = 
lftf(aw,bw,cw,dw,a,b1,b2,c1,c2,d11,d12,d21,d22) 
tss = lftf(tss1,tss2)
ss = lftf(tss1,ss2) 
ss = lftf(ss1,tss2)

Description
lftf computes a state-space closed loop transfer function from input u₁ to output (see Figure 1-11), given the open loop transfer function from u₁ to y₁

and the transfer function tss2 from u₂ to y₂

Either of the systems (tss1 or tss2) can be "one-port" state space or "two-port." lftf also handles the case when some of the A, B or C matrices are empty.

The output variables will be returned in state-space form or, if the inputs are in the optional mksys form, then the returned outputs will likewise be in mksys form.

Figure 1-11: Two-Port Linear Fractional Transformation.

Algorithm
lftf implements the formulae in [1] for (aa, bb, cc, dd), when the second input system is a "one-port" :

where

The formula for the other cases are similar.

See Also
interc, sectf

References

[1]: M. G. Safonov, E. A. Jonckheere, M. Verma and D. J. N. Limebeer, "Synthesis of Positive Real Multivariable Feedback Systems," Int. J. Control, vol. 45, no. 3, pp. 817-842, 1987.

iofr, iofc lqg