Non-square systems with fewer degrees of freedom available than the controlled outputs are particularly challenging to control because, for such systems, it is not possible to achieve set-point control for all the outputs and thus interval control is needed. These intervals must be wide enough to guarantee feasible operation in the presence of process disturbances, but still achieve satisfactory tightness of control for some of the outputs. The Operability methodology originally introduced for square systems¹ and extended for non-square systems^2,3 provides a method for selecting such output constraints systematically, so that they are as tight as possible and do not render the controller infeasible.

Based on the concepts previously published, here we present an optimization-based approach for the design of output constraints for non-square model-based controllers (MPC, DMC). This new approach substantially removes or lessens computational constraints on the size of the problem that can be addressed. Specifically, an algorithm to calculate the Achievable Output Interval Set (AOIS) in Rⁿ is developed, where n is the number of outputs. This set represents the tightest feasible set of output constraints that can be achieved with the input ranges within an Available Input Set (AIS), and when the disturbances remain within their Expected Disturbance Set (EDS). Thus, the calculated AOIS ranges correspond to the intervals that will enable the MPC (or DMC) controller to operate without infeasibilities around the steady-state.

The applicability of the developed methodology is illustrated with high-dimensional industrial chemical processes provided by Air Products and Chemicals and DuPont. Preliminary results for these industrial problems show that very significant reduction of the constrained region can be achieved for specified conditions of process steady-state and relative output weights on the tightness of the desired control of each output. This reduction is quantified by the Output Constrained Hyper-volume Reduction Factor (OCHRF) defined as the ratio between the hyper-volumes of the original constrained region, represented by the Desired Output Set (DOS), and the designed constrained region, represented by the AOIS ranges. These results imply that in many cases the new high-dimensional space within which the process can be effectively operated is significantly tighter than the original constrained space. Several examples have been characterized by OCHRF values of 10³ – 10⁷ in systems that have output dimensionality of 6 – 15. The designed new limits are validated by running DMCplus^TM (AspenTech) simulations for the extreme values of the disturbances.

Finally, the proposed framework enables the design of model-based controllers which calculate online the tightness with which each of the outputs can be controlled. This enables the selection of operating points that are closer to the actual process constraints with significant economic benefits.

References:

1. Vinson DR, Georgakis C. A new measure of process output controllability. J Process Control. 2000;10:185-194.

2. Lima FV, Georgakis C. Issues on the operability of multivariable non-square systems. Presented at the 2005 AIChE Annual Meeting, Paper 9c. 2005.

3. Lima FV, Georgakis C. An operability-based methodology for the feasible output ranges in the control of non-square systems. Presented at the 2006 AIChE Annual Meeting, Paper 359a. 2006.