A fundamental results analysis for RTO system is based on statistic hypothesis test, which was developed by Miletic and Marlin (1998).
Consider the problem of determining if a given vector is a plausible value for the mean of a multivariate normal distribution, there are two hypothesis that are specified in the statistical testing procedure: the null hypothesis and the alternative hypothesis. In the context of this paper, the null hypothesis, denoted , defines that is a plausible value for the normal population mean; the alternative hypothesis, denoted , are against , that is:
Let be a random sample from an population, i.e.,
then the statistic is defined as:
where
and n is the number of samples, p is the number of dimension of the vector , denotes a random variable with an F-distribution with p and degrees of freedom, is a constant associated with p and n which depends on the specific form of the hypothesis test (Miletic and Marlin, 1998). From the view point of hypothesis test, at the level of significance, reject in favor of if
In an RTO system, as given by Equation (
),
assume that is zero-centered, normal distributed and have no
correlation in time, that is:
then by multivariate normal theory (Johnson and Wichern, 1996),
where is the covariance matrix of , given by,
Therefore, the fundamental statistic hypothesis testing for RTO results analysis can be built as follows:
The Rejection Region is decided by:
Apparently, if the statistic is too large, then is ``too far'' from , i.e., is significantly different from , and should be downloaded to low-level MCC for implementation; otherwise, is a plausible value for , i.e., the setpoints change is unnecessary and should be rejected. Note that in fact there is always a set of plausible values for , that means do not reject at confidence level is equivalent to lies in the confidence region, denotes . Beginning at the center , the axes of the ellipsoid are
where
In other words, if , then the new RTO prediction is
significantly different from the current operating point, and should be implemented for tracking the changed optimal operating
point of the plant, see Figure ((a)); on the
contrary, if , then
is a plausible value for the current operating point,
and the unnecessary setpoint change should be rejected to reduce the RTO loop
instability, see Figure ((b)).
Figure: Fundamental Statistic Hypothesis Testing
The fundamental statistic hypothesis testing requires the calculation of
which is given by Equation (). is the
sensitivity of the model updater, that represents the variance of parameter
estimates due to the perturbation of the process measurements. It can be
calculated numerically or given by analytical solution (Miletic and Marlin,
1998),
where is the objective function of parameter estimation problem. is the sensitivity of the model-based optimizer, that represents the variance of RTO prediction due to the perturbation of the adjustable model parameters. Similarly, besides numerical calculation, it can be given by solving the following sensitivity equations (Ganesh and Biegler, 1987):
where L is Lagrangian function for model-based optimization problem, and is the vector of Lagrangian multipliers for the equality constraints (). is covariance matrix of process measurements. It can be either estimated from the available plant data, or chosen carefully to accurately reflect current sensor capabilities, for example, the actual value for each measurement are chosen such that the standard deviation of the flow, temperature and composition measurements are approximately and of the nominal value, respectively.