Auxiliary Lagrangian Multipliers Testing

It is clear from Equation () that the fundamental statistic hypothesis testing only consider the RTO problems with equality constraints. In other words, the concerned RTO system is essentially an unconstrained optimization problem in the reduced space (Fiacco, 1983). However, there exist lots of inequality constraints, such as operating constraints, equipments constraints, safety constraints, etc., in the realistic process industries and as a result, the practical application of the fundamental statistic hypothesis testing is possibly limited. In this paper, a practical results analysis procedure is developed for dealing with inequality constraints in RTO systems, where some certain of setpoints moving limits, namely trust-region constraints, are particularly addressed since the trust-region constraints are used in many RTO/APC systems for system safety and stability. Apparently, the fundamental statistic hypothesis test will fail when it is applied to such a RTO system in the presence of trust-region constraints. In order to satisfy the industrial requirements, an alternative method based on the fundamental statistic hypothesis testing is to run the RTO systems twice: once the new RTO prediction from the first run () is a plausible value for the current operating point (), then the trust-region constraints will be released, and the RTO system will be re-run to predict another RTO prediction, denotes . If is significantly different from depending on the hypothesis test, then the RTO prediction from the first run () should be implemented. Although this method can solve the problem of the trust-region constraints, it is time consuming and low effective. The proposed practical results analysis procedure in this paper will introduce a auxiliary Lagrangian multipliers testing to represent the level of activity of the trust-region constraints.

Consider the model-based optimization problem,

where P is the profit function, and are manipulated variables and dependent variables, is a set of inequality constraints, is a set of equality constraints (process model equations), and are the model parameters. By first order Karush-Kuhn-Tucker Optimality Conditions, it is known that:

where L is the Lagrangian function given by:

and are the vectors of Lagrangian multipliers for the inequality constraints () and equality constraints (h), respectively.

The inequality constraints set () can be divided into two subsets: active inequality constraints () and inactive inequality constraints (), then as in Ganesh and Biegler (1987),

Rearranging these equations in the linear matrix form:

 

It is clear that can be solved by Equation () if

where is a vector providing information about the sensitivity of the Lagrangian multipliers for the inequality constraints with respect to the model parameters.

Let , a new covariance matrix which can be formed including information about the inequality constraints is given by:

Then a similar hypothesis testing is built for testing the Lagrangian multipliers for the active inequality constraints:

The Rejection Region is decided by:

In the case of insignificant set point movement, the activity levels of the trust-region constraints can be compared with a zero value. If the trust-region constraints can be shown to be active to a certain confidence level, it can be assumed that those active inequality constraints are restricting the plant from moving towards the optimum, and therefore the new RTO predictions should be implemented actually, see Figure ().

 

In summary, the practical results analysis procedure for RTO systems with trust-region constraints can be stated as:

1)

Run RTO systems and get the new RTO predictions and the value of Lagrangian multipliers for the active trust-region constraints;

2)

Sensitivities calculations;

3)

Calculation of the covariance matrix of the RTO prediction with Lagrangian multipliers;

4)

Applying fundamental statistic hypothesis testing;

5)

If fundamental statistic hypothesis testing pass, then implement the new RTO prediction, and the results analysis procedure end; otherwise, applying auxiliary Lagrangian multipliers testing;

6)

If auxiliary Lagrangian multipliers testing pass, then Implement the new RTO prediction, and results analysis end; otherwise, the new RTO prediction will be rejected, and no setpoints change.