As shown in Figure (
), an simplified RTO system is
built for simulation study. Figure () to () show the
RTO predictions trajectories for (1) System A: RTO system without
results analysis; (2) System B: RTO system with fundamental statistic
hypothesis testing; and (3) System C: RTO system with both
fundamental statistic hypothesis testing and auxiliary Lagrangian multipliers
testing (proposed method in this paper), respectively, starting from the same
operating point . Figure () shows that System A
originally tend to the true plant optima , where each step towards
to the true plant optima is constrained by the trust-region constraints,
however, when System A is close to the true plant optima, RTO
prediction randomly lies in a certain bounded region due to the propagation of
the measurement noises around RTO loop and as a result, such random setpoint
changes result in system instability. In order to avoid the unnecessary
setpoint changes, fundamental statistic hypothesis testing developed by
Miletic and Marlin (1998) is applied to this case study at the 98% level of
significance. Unfortunately, due to the presence of the trust-region
constraints, their method rejects any possible setpoint changes and the
operating point never moves during RTO executions, as shown in Figure
(). It happens because that the confidence region of the RTO
prediction is large enough so as to encompass the trust-region such that every
new RTO prediction, which is within the trust-region, is a plausible value for
the current operating point.
Figure: RTO Predictions without Results Analysis
Figure: RTO Predictions with Fundamental Statistic Hypothesis Testing
Figure () shows the simulation results obtained by using
practical results analysis procedure proposed in this paper. At the same
significance level, while RTO system converge to the true plant optima, the
practical results analysis procedure rejects a large amount of unnecessary
setpoint changes. For the purpose of comparison, each manipulated variable
trajectory with respect to RTO interval is shown in Figure (). It
is clear that System C can converge closely to the true plant optima,
and reduce the system instability by only implementing the meaningful setpoint changes.
Figure: RTO Predictions with both Fundamental Statistic Hypothesis Testing
and Auxiliary Lagrangian Multipliers Testing
Figure: Manipulated Variables Trajectory for RTO Systems (1) without Results
Analysis, (2) with Miletic and Marlin's Method, and (3) with Practical Results
Analysis Procedure