Optimal measurement-based cost gradient estimate for feedback real-time optimization

This work presents a simple and efficient way of estimating the steady-state cost gradient $J_u$ based on available uncertain measurements $y$. The main motivation is to control $J_u$ to zero in order to minimize the economic cost $J$. For this purpose, it is shown that the optimal cost gradient estimate for unconstrained operation is simply ${\hat{J}}_u=H(y_m-y^{\ast })$ where $H$ is a constant matrix, $y_m$ is the vector of measurements and $y^{\ast }$ is their nominally unconstrained optimal value. The derivation of the optimal $H$-matrix is based on existing methods for self-optimizing control and therefore the result is exact for a convex quadratic economic cost $J$ with linear constraints and measurements. The optimality holds locally in other cases. For the constrained case, the unconstrained gradient estimate ${\hat{J}}_u$ should be multiplied by the nullspace of the active constraints and the resulting “reduced gradient” controlled to zero.