S. Skogestad, "Ratio control: Theoretical basis and practical implementation", JPC, Vol. 160 (2026)


---- COMMENT 1:  End of Section 3.1 "Flow controller": 

The starting point is that we want F2 = R F1, where F1 is measured and F2 can be manipulated (through the valve position z2). The simplest implementation is to set F2s = R F1 and use a flow controller (FC) for F2. This requires the FC for F2 to be reasonably fast; otherwise, F2 will “lag behind” F1.
One solution, provided that there is also an FC for F1, is to set F2s=R F1s. This is perfect (giving F2=R F1 dynamically) provided that the flow controllers for F1 and F2 have the same dynamics. The problem is that it does not work, or requires additional logic, if F1 saturates. Also, disturbances that temporarily change F1 will have no effect on F2.

An even "smarter" solution is to use a "blend" :
F2s = R* (g*F1s + (1-g)*F1)    (*)
where g (gamma) is an adjustable parameter, normally said to between 0 and 1. 
It is quite easy to show theoretically (see Visioli’s 2005 paper) that the ideal g which makes F2=R*F1 at all times (dynamically) is equal to the ratio of the two loop transfer function for the two flow controllers:

		g(s) = L_1(s)/L_2(s) = (C1 P1) / (C2 P2) 
	
This becomes dynamic, but if the dynamics are the same, it gives g=1 (as expected). And if the dynamics of the two loops are similar, it reduces to the ratio of the closed-loop time constants,

		g = tauc2 / tauc1 .

Thus, if FC2 is much faster than FC1, then tauc2/tauc1 is small and we get g=0, which gives F2s=R F1. If the two flow controllers are equally fast, then tauc2/tauc1=1, so g=1, which gives F2s=R F1s.
If FC1 is faster than FC2, then we actually get g>1. This may seem a little strange, but it's optimal as it "speeds up" the response for FC2.
The value g=0.5 is obtained when FC2 is “twice as fast” as FC1.

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