State feedback synthesis for on-line optimization in the presence of measurable disturbances

In this paper, state feedback laws are developed for online optimization of batch reactors in the presence of measurable disturbances. Pontryagin's minimum principle is used to derive the first order necessary conditions of optimality. Then the optimal state feedback laws are derived by eliminating the adjoint variables from the first order necessary conditions. System Lie brackets are defined to express the developed state feedback laws in a compact form. It is found that depending on the degree of singularity of manipulated input the state feedback laws are either static or dynamic. Finally, the efficacy of the online methodology is illustrated through a series reaction occurring in a semi-batch reactor where the objective is to maximize a product by manipulating batch temperature in presence of disturbance in the flow rate of the reactant.