(522c) A Tube-Based Strategy for Robust Control Synthesis of Periodic Processes

It is well known that periodic operation of certain processes can lead to improved performance compared with steady-state operation (Bailey, 1974). In order to reach optimality, these processes are typically required to operate near operational or safety constraints. Although open-loop policies have been widely investigated, they have the disadvantage that, in the presence of disturbances, these constraints may become violated -- a motivation for robust control strategies.

Herein, we address the robust feedback optimal control problems for periodic processes with economic objectives using set-based methods. The focus is on tube-based optimal control problems, whereby one

optimizes over robust forward invariant tubes instead of optimizing over feedback laws. A robust forward invariant tube (RFIT) is essentially a set-valued function containing all state trajectories of the system, under a given feedback law and under all possible uncertainty realizations. We present a formulation of tube optimal control problems based on a condition introduced by Villanueva et al. (2017) for the construction of robust forward invariant tubes. This condition is restricted to compact and convex tubes, and are given in the form of a differential inequality for the support function of the tube. By restricting the search to tubes with ellipsoidal cross-sections, one arrives at a standard optimal problem formulation with periodicity constraints that conservatively approximates exact tube-based optimal control problem. A by-product of this construction is an explicit expression for the feedback law associated to the RFIT. Thus, once the periodic tube problem has been solved off-line, this feedback law -- which guarantees robust constraint satisfaction for the closed-loop system -- may be directly implemented online without the need for re-optimization. We illustrate this approach with the synthesis of robust feedback controller for a periodic bioreactor with an economic objective.

References:

Villanueva, M.E., Quirynen, R., Diehl, M., Chachuat, B., and Houska, B. (2017). Robust MPC via min–max differential inequalities. Automatica, 77, 311–321.

Bailey, J. (1974). Periodic operation of chemical reactors: a review. Chemical Engineering Communications, 1(3), 111–124.