Non-square process control systems with fewer degrees of freedom than the controlled outputs are quite common in chemical processes. In these systems, it is impossible to control all output variables at specific set-points and many of the outputs are controlled within an interval. If these intervals are chosen to be very narrow, the controller might be infeasible. On the other hand, if the intervals for the output variables are quite wide, the tightness of control achieved might be unsatisfactory.

The objective of this presentation is the extension to higher-order systems of the multivariable non-square Operability methodology proposed by Lima and Georgakis (2006). This methodology aims to be used in the design of non-square controllers. This will be achieved by analyzing the general problem with n outputs, m inputs but only 1 disturbance affecting the process.

In this problem the servo Achievable Output Space (AOS) will be shifted by the maximum and minimum disturbance values in an n-dimensional manifold. The union of all shifted locations for the possible disturbance values will give us the space AOS(d) which is a subset of Rⁿ. In order to calculate the feasible output ranges we will use the definition of the Achievable Output Interval Space (AOIS) given by Lima and Georgakis (2006). The AOIS was defined as the smallest possible interval constraints for the outputs that can be achieved with the available range of the manipulated variables and when the disturbances remain within their expected values. The desired degree of tightness in the control of each of the outputs will affect the aspect ratio of the corresponding side of the initial estimate of the AOIS. The idea of this calculation is to enlarge the sizes of the initial AOIS keeping the pre-specified aspect ratio constant until the AOIS grows in size enough to touch or intersect the extreme spaces associated with the minimum and maximum disturbance values of AOS(d).

In the previous publication we have motivated the new concepts examining some simple non-square systems. In this presentation, higher-order systems will be addressed to demonstrate the effectiveness of the new methodology. The non-square systems examined are related to a Steam Methane Reformer process (SMR), which has 4 manipulated, 1 disturbance and 9 controlled variables.

Reference: Lima, F.; Georgakis, C. (2006). Operability of Multivariable Non-Square Systems. ADCHEM Proceedings, 989-994.