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45400-AC9
Interval Operability as a Design Tool for Model Predictive Controllers
During the first year of the project substantial progress was achieved and one PhD thesis was completed. During this period we examined, from a steady state point of view, linear, non-square systems with more outputs than inputs. These are common in industrial chemical processes. Such systems are particularly challenging because it is impossible to control all the outputs at specific set-points when there are fewer degrees of freedom available than the number of controlled variables. Thus, interval control is needed for at least some of the output variables. These intervals must be wide enough to guarantee feasible operation in the presence of process disturbances, but still achieve satisfactory tightness of control of the outputs. To address this challenge, the concept of steady-state Interval Operability is introduced for multivariable non-square systems with fewer inputs than output variables. This concept is of great value in the design of model-based constrained controllers, such as the Model Predictive Controller (MPC) or the Dynamic Matrix Controller (DMC).
Two approaches were developed that systematically calculate the tightest feasible set of output constraints (represented by the Achievable Output Interval Set (AOIS)) that can be achieved. The first one uses an iterative numerical approach that is computationally intensive as it requires the repeated calculation of the intersection of high dimensional polytopes. The second approach uses a Linear Programming (LP) based framework and is very efficient computationally and can be easily performed online. The main difference between these algorithms is the AOIS calculation in Rn, where n is the dimensionality of the output set. The results of each method are used in the design of high-dimensional and non-square Model Predictive Controllers (MPC). Both approaches enable the offline design of output constraints before the MPC controller is deployed. Because of its high-speed of computations, the LP approach also enables the online design of such constraints making real-time adaptation of the control objectives possible.
The real world applicability of the developed methodologies is illustrated with several industrial-scale chemical processes provided by Air Products and Chemicals and DuPont. For the operable cases, it is shown that the constrained region of operation can be reduced, without causing infeasibilities, by a factor of 103 – 107 for systems with an output dimensionality of 6 – 15. The calculated new limits are validated by running DMCplusTM (from AspenTech, Inc.) simulations for the extreme values of the disturbances. Furthermore, for an initially inoperable process, the amount of relaxation necessary for each of the output constraints, to make the control problem feasible, was calculated.
In a more detailed application study, the LP-based Interval Operability approach was applied to calculate operable output constraints for the Sheet Forming Control Problem (SFCP) from DuPont. Here one wishes to control the sheet thickness at 15 different measurement points, which represent the 15 output variables, using 9 manipulated variables in the presence of 3 disturbances. Thus, this problem represents a computationally complex, high-dimensional non-square system with quite a few more outputs than inputs. This problem was addressed under two study cases: 1) the original non-square model, where all the 15 outputs are controlled independently of each other; 2) an approximate square problem, where 6 outputs are combined in pairs, or zone variables, and are controlled within their corresponding zone. Results show that significant reduction of the constrained region of process operation can be achieved for different output targets specified. Specifically, the hyper-volume ratio of the initial to the designed constrained regions range between 103 and 105. The calculated constraints are validated by running DMCplusTM simulations for the extreme values of the disturbances. The solution of the approximate square problem, where some of the outputs are not controlled individually but in zones, provides that these outputs can be controlled more tightly than they could if the solution of the exact non-square problem was used. This revealed that such approximations could lead to more optimistic results than can be achieved in the real non-square process. A study was also completed in collaboration with Professor M. Ierapetritou at Rutgers University comparing the concepts of operability and flexibility and their application to process design and control. In a joint paper recently submitted for publication, the operability and flexibility methodologies are summarized. The application of the operability framework to steady-state and dynamic systems is then illustrated through the examination of several categories of examples, such as linear and non-linear, square and non-square systems. The flexibility approach based on the active set strategy is employed to study the same examples from the flexibility point of view. The results discussed show that the operability and flexibility approaches examine a process from different perspectives and provide valuable complementary information.
