[414b] - Controllability of processes with large gains and valve
stiction
- Sigurd
Skogestad (speaker)
- Norwegian University of Science
and Technology (NTNU)
- Sem Sealands vei
4
- Trondheim, N7491
- Norway
- Phone:
+4773594154
- Fax:
- Email: skoge@chemeng.ntnu.no
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- A R B de Araujo
- Norwegian University of Science
and Technology (NTNU), Chemical Engineering
- Sem Sealands vei
4
- Trondheim, N7491
- Norway
- Phone:
+4774594030
- Fax:
- Email:
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Abstract:
Controllability of processes with large
gains and valve stiction
S. Skogestad and A.C.B. de
Araujo
NTNU, Trondheim, Norway
In a recent publication
McAvoy and Braatz (2003) state that for control purposes the
magnitude of steady-state process gain (maximum singular value)
should not exceed about 50. Otherwise, they claim, the process will
be prone exhibit transient oscillations because of valve resolution
problems, e.g. caused by valve stiction or hysteresis (a nonlinear
issue). If these claims are correct, then this has important
implications for the design of many processes, so the objective of
this work was to study this in more detail.
Intuitively, the
claims seem to be incorrect. Consider, for example, a holdup tank
where we need to control the liquid level. This is an integrating
process, so the steady-state gain is infinite, and the process
should be very difficult to control according McAvoy and Braatz
(2003). However, in practice control of liquid level is known to be
quite straightforward, and a simple proportional feedback controller
usually suffices.
We find that the claims by McAvoy and
Braatz (2003) about the magnitude of the steady-state gain hold if
we restrict ourselves to feedforward control. This is because
feedforward control is sensitive to unmeasured disturbances and
uncertainty at steady state. A valve resolution problem may be
viewed as an input disturbance. For example, let v denote the valve
resolution (e.g. v=0.01 or 1 % of the input range) and G the steady
state gain (e.g. G=50). Then the output error with feedforward
control is e = G v. (e.g. e = 0.5 with the numbers given). If we
want e to remain well below 1 (say 0.5) and we assume that a typical
valve resolution error is 1 %, then this simple analysis supports
the above claim that the steady-state gain should be less than about
50 when the inputs and outputs are scaled to be of the order 1.
However, this analysis only holds for feedforward
control.
McAvoy and Braatz present simulation examples for
feedback control, but a closer analysis of these reveals several
misinterpretations. First, they confuse the issue by considering a
2x2 example, whereas a SISO example would be sufficient to
illustrate the points. Also, the 2x2 example they study is very
interactive with an off-diagonal element 10 times larger then the
diagonal elements, and this kind of plant gives strong interactions
and control problems irrespective of valve stiction. Furthermore, in
one case (Figure 3) the simulation time is too short such that they
incorrectly conclude that the sensitivity to uncertainty decreases
when we decrease the feedback controller gain. Actually, it can be
shown that with integral action the controller gain should have no
effect on the magnitude of the oscillations. This is because around
the steady-state the system will be oscillating in a manner
corresponding to an on/off-valve with zero deadband.
With
feedback control we can easily control systems even with an on/off
valve, which has a valve resolution error of 100% (v=1) provided the
process provides some natural damping at the closed-loop bandwidth
of the system. For example, consider a thermostat used for heating
your home or car. The main issue here is that the delay in the loop
is sufficiently small, so that switching is fast. Then we can even
have infinite gain, e.g. consider an on/off-valve used for level
control. Oscillations are unavoidable if we have limited valve
resolution, because the steady-state can only be achieved by cycling
the input between two (or more) finite input values, like with an
on/off-valve.
Based on a more detailed controllability
analysis, based on simple nonlinear describing function theory
similar to that used for the relay tuning method), we conclude that
large process gains may indeed pose a fundamental problem in terms
of control, but only if the gain is large at the frequency w180
corresponding to the natural oscillations of the system. More
precisely, for an appropriately scaled model we must approximately
require that
| G(jw180) | * v < 1
where G is the
process, w180 is the frequency where the phase lag of the system
(process + controller) is 180 degrees, and v is the relative valve
resolution (e.g. v=1 for an on/off-valve). Note that the time scale
of the oscillations (duration of the pulse time) depends on the loop
dynamics (w180), whereas the magnitude depends on both w180 and the
valve accuracy.
Thus, large steady-state gains are by
themselves not a problem. The analysis is backed up by some simple
case studies which indeed show that systems with a large
high-frequency gain may be sensitive to limited valve resolution.
The magnitude of the resulting oscillations may be reduced by (1)
improving the accuracy of the valve, or (2) increasing w180 by
reducing the effective delay around the loop. Note that the latter
also reduces the pulse time, but if the resulting pulse time is till
too large, then another possibility is to (3) introduce pulse
modulation. Here the pulse time is set to a desired value (limited
by the dynamics of the valve only and not of the whole control
loop), and the controller adjusts the relative time the controller
spends in the various positions. This approach is mainly used if we
have an on/off valve.
T.A. McAvoy and R.D. Braatz, 2003,
“Controllability of processes with large singular values”,
Ind.Eng.Chem.Res., 42, 6155-6165.
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