[Luyben] Process Mod.. - Student subdomain for University of Bath

ANALYSIS OF MULTIYARIABLE SYSTEMS 575
One important aspect of the MRI calculations should be emphasized at this
point. The singular values depend on the scaling use in the steadystate gains of
the transfer functions. If different engineering units are used for the gains, different
singular values will be calculated. For example, as we change from time units
of minutes to hours or change from temperature units in Fahrenheit to Celsius,
the values of the singular values will change. Thus the comparison of alternative
control structures and processes can be obscured by this effect.
The practical solution to the problem is to always use dimensionless gains
in the transfer functions. The gains with engineering units should be divided by
the appropriate transmitter spans and multiplied by the appropriate valve gains.
Typical transmitter spans are 100”F, 50 gpm, 250 psig, 2 mol % benzene, and the
like. Typical valve gains are perhaps twice the normal steadystate flow rate of the
manipulated variable. The slope of the installed characteristics of the control
valve at the steadystate operating conditions should be used. Thus the gains that
should be used are those that the control system will see. It gets inputs in mA or
psig from transmitters and puts out mA or psig to valves or to flow controller
setpoints.
16.3 INTERACTION
Interaction among control loops in a multivariable system has been the subject of
much research over the last 20 years. Various types of decouplcrs were explored
to separate the loops. Rosenbrock presented the inverse Nyquist array (INA) to
quantify the amount of interaction. Bristol, Shinskey, and McAvoy developed the
relative gain array (RGA) as an index of loop interaction.
All of this work was based on the premise that interaction was undesirable.
This is true for setpoint disturbances. One would like to be able to change a
setpoint in one loop without affecting the other loops. And if the loops do not
interact, each individual loop can be tuned by itself and the whole system should
be stable if each individual loop is stable.
Unfortunately much of this interaction analysis work has clouded the issue
of how to design an effective control system for a multivariable process. In most
process control applications the problem is not setpoint responses but load
responses. We want a system that holds the process at the desired values in the
face of load disturbances. Interaction is therefore not necessarily bad, and in fact
in some systems it helps in rejecting the effects of load disturbances. Niederlinski
(AZChE J 1971, Vol. 17, p. 1261) showed in an early paper that the use of decouplers
made the load rejection worse.
Therefore the discussions of the RGA, INA, and decoupling techniques will
be quite brief. I include them, not because they are all that useful, but because
they are part of the history of multivariable control. You should be aware of
what they are and of their limitations.