Optimality Analysis of Self-Optimising Control Systems

The goal of self-optimising control is to select a set of controlled variables, which when kept at constant setpoints, indirectly lead to suboptimal operation. Based on the necessary condition of unconstrained optimisation, Cao (2004) has identified that the steady-state optimizing control for unconstrained degrees of freedom is to keep a dimension-reduced gradient at zero. Alternatively, two sensitivity measures of the gradient to disturbances and to implementation errors have been introduced as criteria to select other controlled variables (Cao 2003). In this work, the local optimality and dynamic optimality of the dimension-reduced gradient have been analysed. It is shown that the local optimality is guaranteed when the gradient control system is closed-loop stable. Furthermore, the dynamic gradient is derived based on the necessary condition of dynamic optimality. The dynamic form of gradient has two parts, a static part corresponding to the static optimality, and the dynamic part represented as a time derivative. Therefore, the static gradient is dynamically optimal or suboptimal if the dynamic part is zero or near zero. Direct control using the dynamic gradient as controlled variables is discussed and demonstrated in the evaporator case study.