To demonstrate the practical results analysis procedure for RTO system with trust-region constraints, a small analytical example is considered in this subsection. Consider the model-based optimization problem with trust-region constraints in the RTO system,
where: x is an independent variable, u is a dependent variable. The
inequality equations in Problem (
) represent
the trust-region constraints, i.e., , is the current operating point; and the
equality equation represents the plant model, which
for the ``true'' plant is:
In most RTO applications the rigorous process model is unknown and must be approximated. Then, for the purposes of this example, an approximate model is assumed for the RTO system:
where is an adjustable model parameter that will be estimated on-line.
Assume that the current operating point , the model updater
provides a new value of the adjustable model parameter with
the variance matrix , then by optimization
calculations, the new RTO prediction is , where the second
inequality constraint in Problem () is
active and the corresponding value of Lagrangian multiplier is . By parametric sensitivities analysis:
then
The covariance matrix can be calculated:
where
and
Consider the fundamental statistic hypothesis testing at the confidence level, i.e., , assume that 100 measurement data points are used to calculate the variance matrix , the control limit for the hypothesis test is given by Miletic and Marlin (1998), where p=2 and n=100,
Evaluate the statistic ,
Apparently, and consequently the null hypothesis is accepted at the 95% level of significance, that is the new RTO prediction is a plausible value for the current operating point. Making decision of not changing the setpoints for the manipulated variables solely depending on the fundamental statistic hypothesis testing may result in the poor RTO performance since the effect of the trust-region constraints on the RTO prediction is ignored.
Under the situation that fundamental statistic hypothesis testing is failed, the auxiliary Lagrangian multipliers test should be applied. Evaluate
Comparing the observed with the control limit where p=1 and n=100,
we see that and as a result, the new RTO prediction
should be implemented. The
illustrative example can be shown graphically in Figure ().
Figure: Illustrative Example: (a) RTO prediction without Trust-Region
Constraints; (b) RTO prediction with Trust-Region Constraints