Dynamic optimization for batch processes with uncertainties via approximating invariant

The dynamic optimization problem for batch processes with uncertainties is considered in this paper. The invariant of optimality conditions is analytically derived. However, the invariant usually depends on the unmeasured states and disturbances; hence cannot be directly used for on-line control. To address this difficulty, we propose to approximate the invariant using available measurements, including manipulated variables, so that the optimal control can be derived as a function of available measurements. To this end, off-line experiments are conducted under various operating conditions and numerical regression method is applied to obtain an approximate expression of invariant in terms of manipulated variables and measurements. The approximated optimal control law is then derived by solving the manipulated variables as a function of measurements to keep the approximated invariant at zero, which can be used for on-line implementation. An illustrative example of fed batch reactor is provided to illustrate the proposed approach.