Box Level Constraints

Model h

4.4 Model Predictive Formulation

4.4.2 Box Level Constraints

Modifying the constant level constraints to include the fixed endpoint condition, h( t

+

Pit)

=

0, results in:

1 1

I 0

which we will refer to as "box constr.aints."

1 1

1 0

(4.34)

With these constraints the optimization finds, at each sample time, future outlet flow rates for which the predicted level reaches its nominal value in P time steps. The outlet flow given by solving (29 )-(30b )-(34) at each sample time is given by:

where:

l~q~I >

l~q;I l~q~I

~

l~q;I

~ _{0 _} 20(t) 2A[h(t) - h1 ]

q⁰ - P

+

I

+

^{T P( P}

+

I) and ~q; is as defined in ( 20).

(4.35)

( 4.36)

As in the constant level constraint case, this solution has a straightforward in- terpretation.

~q;

is the minimum magnitude change in outlet flow which prevents constraint violation for times less than t

+

k*; ~q~ is the minimum magnitude change in outlet flow which satisfies the fixed endpoint condition. The best feasible solution is then clearly the larger of these flow changes. For a particular choice of P, large flow imbalances result in l~q~I <

l~q;j

and the solution qo(t)

=

qo(t - l}

+

~q; is implemented. Since this recovers the discrete time infinite horizon MRCO optimal solution (22), the fixed endpoint condition has no effect on filtering performance. For small imbalances, l~q~I >

l~q;I

and the solution qo(t)

=

^{q0(t - 1)}

+

~q~ is imple- mented. In this situation the fixed endpoint condition causes an increase in MRCO.

Theorem 2 provides a condition on the horizon length which insures that the fixed endpoint condition does not interfere with flow filtering.

Theorem 4.2 For step inlet flow disturbances, the sequence q0 (

t +

k), k E K deter- rnined by {95) satisfies the conditions of Theorem 1 if:

Proof See Appendix B. ■

Thus whenever P

>

Pcrit, the fixed_ endpoint condition has no impact on filtering performance as measured by MRCO.

It is easily verified tha.t Pcrit decreases as the fl.ow imbalance increases and as the level approaches its nominal value. This observation allows us to choose P to

· guarantee optimal fl.ow filtering for all disturbances above a particular magnitude which occur while the level is within some range about its nominal value.

As stated (37) is a sufficient condition. However, as we discuss in the appendix, it is only conservative when h(t)

#

h, and a fl.ow imbalance O(t) occurs which is in the direction which tends to return the level to its nominal value. For example this is the case when the tank level is above nominal and the inlet flow rate drops. In any other situation, the condition (37) is necessary as well as sufficient.

The fixed endpoint condition is not sufficient to guarantee the realization of zero offset in P time steps. Since the on-line optimization is resolved at each sample time, the complete solution ~( t), determined at time t, which provides zero offset in P steps, is not implemented. Instead the "moving horizon" approach of implementing only the first element of q~(t), results in the realization of the sequence {q;₁(t),

q;

1(t+

1),

q;

1(t

+

2), ... } (given by (35)) which need not provide zero offset in P steps. This condition does however insure that there is no steady state level offset.

Theorem 4.3 The moving horizon model predictive controller defined by (29)-(30b)-

(34} achieves zero steady state level offset for constant inlet flow disturbances.

Proof See Appendix C. ■

Simulation experience has shown that in general the level returns to within 5% of its nominal value in between 2P and 2.5P sampling times for step inlet disturbances.

For small inlet disturbances the settling time is often smaller.

The significance of these results is that for any given fl.ow imbalance,

n,

there exists a finite P for which the moving horizon model predictive controller with box constraints achieves the minimum possible MRCO and integral action. It follows that by selecting P adequately large, optimal flow filtering and integral action can be achieved for disturbances of arbitrarily small magnitude. Suboptimal filtering of small disturbances (as determined by the selection of P) is not a practical concern since these disturbances pose the least trouble for downstream equipment.

What we have achieved by introducing box constraints is to assure satisfaction of the secondary objective of integral action with no adverse impact on the primary objective of fl.ow filtering for large disturbances. The price we pay for integral action is suboptimal filtering of small disturbances, but as we have argued, this is not significant in practice. In contrast, the addition of an integral term to an otherwise optimal controller, as in the OPC, results in suboptimal performance whenever the integral term is non-zero ( essentially always). Interaction and in some cases competition between the integral and optimal terms can significantly impact filtering performance and settling time as we will see in the examples below.

The single "tuning parameter" of this algorithm is the horizon length, P, which directly determines the trade-off between the incompatible objectives of good flow filtering ( requiring P large) _and rapid integral action ( requiring P small). The ap- propriate value of P is determined by the characteristics of a specific implementation.

The operating conditions of the upstream equipment will dictate the magnitude and frequency of expected inlet flow disturbances. The sensitivity of downstream equip- ment will dictate the filtering performance required for the expected disturbances.

Ideally P is selected equal to or greater than P crit for the smallest disturbance for which optimal filtering is required. In general if rapid integral action is not required ( disturbances are infrequent) P should be large. If large disturbances occur frequently it may be advantageous to reduce P so that tank volume is recovered rapidly to be used to filter subsequent disturbances.

The effect of the horizon length is demonstrated in Figures 2a and b. The single tank system used in this example is described in Table 1. Figure 2a shows the level response to a 50% step change in inlet flow rate (for which Pcrit = 14) for P = 5, 8, 14, 25, and oo. Figure 2b shows the corresponding outlet flow rates. For P < Pcrit increasing P improves flow filtering from M RCO

=

1.00 for P

=

5 to M RCO

=

^0.36

for P

=

14, at the expense of settling time. For P > Pcrit no improvement in flow filtering as measured by MRCO is possible and settling time increases.

Note that as P is increased, relaxing the desired settling time, the maximum peak in outlet flow is reduced. For P infinite, the maximum peak is equal to the

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