A robust modifier adaptation method via Hessian augmentation using model uncertainties

Simultaneous online model identification and production optimization using modifier adaptation

“…The drawback is the need to estimate second order information of the plant based on plant measurements. More recently, [12] used an aggregate model, which combined several cision variables are referred to, in this paper, as "controls". models of the plant.…”

“…Since the plant optimum is unknown beforehand, this could only be guaranteed a priori by enforcing that all models are convex over the whole feasible input space or by convexifying them (see, e.g. [12]). Despite no proof of guaranteed convergence, the case studies in the following section illustrate the benefits of our algorithms when an appropriated model set is chosen.…”

Robust Modifier Adaptation via Worst-Case and Probabilistic Approaches

“…Using this assumption of parametrically bounded second derivatives, an approach known as directional Hessian modifier adaptation (DHMA) was developed . By setting the second directional derivative of the modified model constraint equal to the maximum second directional derivative of the set of models, where refers to the upper-bounded second directional with respect to θ .…”

“…Proposition 2 (WCMA−KKT matching upon convergence). Consider the WCMA problem, given by (31), with no measurement noise and perfect estimates of the gradients using either a constant filter or the new filter given by ( 26), let u p * be a KKT point of the plant [given by ( 1)], and assume that (31) has converged to the fixed point u ∞ = lim k→∞ u k . Then, at u p *, the KKT conditions of (31) match those of the plant.…”

“…Then, at u p *, the KKT conditions of (31) match those of the plant. Furthermore, u ∞ is a fixed point of (31) if and only if it satisfies the KKT conditions of the plant. Thus, u ∞ * = u p * and plant optimality is guaranteed upon convergence.…”