Approximating Necessary Conditions of Optimality as Controlled Variables

Abstract

Selection of controlled variables (CVs) has recently gained wide attention, because of its paramount importance in real-time optimization (RTO) of plant operation. The so-called self-optimizing control (SOC) strategy aims to select appropriate CVs so that when they are maintained at constant setpoints, the overall plant operation is optimal or near optimal, despite various disturbances and uncertainties. Recent progresses of the SOC methodology have focused on finding linear combinations of measurements as CVs via linearization of the process around its nominal operating point, which results in the plant operation being only locally optimal. In this work, the concept of necessary conditions of optimality (NCO) is incorporated into CV selection to overcome the "local" shortcoming of existing SOC methods. Theoretically, the NCO should be selected as the optimal CV, although it may not be practical because of the measurability of the NCO. To address this issue, in this work, CVs are selected to approximate unmeasured NCO over the entire operation region with zero setpoints to achieve near-optimal operation globally. The NCO approximation CVs can be obtained through any existing regression approaches. Among them, two particular regression methodsnamely, least-squares and neural networksare adopted in this work as an illustration of the proposed methodology. The effectiveness and advantages of the new approach are demonstrated through two case studies. Results are compared with those obtained by using existing SOC methods and an NCO tracking technique.

Data-driven self-optimizing control: constrained optimization problem

“…The optimization problem can be written as (5) For this type of problem, the NCO was split into two parts by Ye et al (2013a) which are, the active constraint, (constraint with strict equality) and reduced gradients . These are respectively given as (6) and (7) The reduced gradient has components which can be compressed to dimensions using singular value decomposition.…”

“…Other local methods include null space method (Alstad and Skogestad, 2007), and branch and bound methods Cao, 2010, 2009;Cao and Kariwala, 2008). Ye et al(2013a) have defined two expressions for loss in objective function that are used as criteria for CV selection. These are worst case and average losses for uniformly distributed disturbance given as follows (1) (2) where and are the maximum singular value and Frobenius norm of a matrix, respectively.…”

“…These are worst case and average losses for uniformly distributed disturbance given as follows (1) (2) where and are the maximum singular value and Frobenius norm of a matrix, respectively. The matrix M was defined as (3) where , and the Hessian matrices are given as (Ye et al, 2013a) (4) Equation (1) and Equation (2) are used to select the right CV candidate as a subset of measurements . The CV selection procedure involves minimizing the loss expressions with respect to H (Ye et al, 2013a).…”

“…The matrix M was defined as (3) where , and the Hessian matrices are given as (Ye et al, 2013a) (4) Equation (1) and Equation (2) are used to select the right CV candidate as a subset of measurements . The CV selection procedure involves minimizing the loss expressions with respect to H (Ye et al, 2013a).…”

“…In a related work by Cao (2004), chain rule differentiation was proposed to explicitly express the gradient as a function of system's Jacobian. Other methods were also developed for CV selection which approximate necessary condition of optimality (NCO) to achieve near optimal operation globally (Ye et al, 2013a(Ye et al, , 2012. However, global optimal operation was achievable with these methods, system model is still necessary for NCO evaluation, which is a short coming to systems with unknown or complicated model.…”