**1. Introduction**

High-fidelity and computationally efficient optimisation models are key for profitable decision making in process industries and have been the focus of extensive research over the years [1]. In recent years, the need for exploiting and explicitly considering interdependencies throughout the different layers of decision making has been underpinned by the enterprise-wide optimisation (EWO) concept [2]. Stemming from the progressively volatile and competitive market conditions, it is imperative for process industries to operate with agility in order to maximise their profitability [3]. EWO is aiming at increased profitability and resilience in process operations through the integration and simultaneous optimisation of existing information streams. Nonetheless, it comes at a considerable cost. Because of the multiple scales considered, EWO leads to computational challenges, thus preventing practitioners from harnessing the potential benefits such wide integration has to offer. Particularly, incorporating control considerations in an EWO fashion results in (mixed integer) nonconvex problems which are hard to solve.

By the same token, control considerations are ubiquitous in EWO problems. Figure 1 showcases how real-time optimisation and production scheduling exchange information with the layer of APC because of their interdependent decisions.

Real-time optimisation is concerned with the manipulation of systems' dynamics in order to achieve optimised profitability and operations. On the other hand, production scheduling determines the optimal allocation of resources for the completion of competing tasks. As indicated by Figure 1, both RTO and process scheduling exchange information with the layer of APC so as to achieve optimal dynamic operations. To this end, the research community has proposed different methods for their integration.

A common shortfall when focusing on integrating RTO and APC is that two different models are employed for the optimisation of the same system. Typically, a locally linear model of the initial nonlinear dynamics is used at the APC because of the need for fast solution rates while RTO considers the original nonlinear model. This leads subsequently to issues related to suboptimal trajectories and non-reachable states [4].

**Figure 1.** Interaction of APC with different layers of decision making in process industries.

Darby et al. [5], through their literature review regarding the integration of RTO and MPC, suggested that for a successful integration, common issues such as model mismatch among the layers of RTO and APC should be eliminated. Nonetheless, in real industrial processes, model degradation and other factors can result in model mismatch, so the consideration of parameter estimation and data reconciliation functionalities is needed to integrate RTO and MPC, as indicated by Figure 2.

**Figure 2.** Interaction between advanced process control and real-time optimisation.

The interaction between real-time optimisation and model predictive control can be categorised broadly into three classes: (i) dynamic RTO (d-RTO), (ii) static RTO (s-RTO) and (iii) economic model predictive control (e-MPC). Both s-RTO and d-RTO are twolayer schemes where reference trajectories are passed to the layer of APC in the form of set-points [6]. While under the static real-time optimisation paradigm, the optimisation problem is solved at specific instances whenever new data become available or when steady state is achieved, in the d-RTO paradigm, the system's transient behaviour is explicitly considered, thus resulting in dynamic optimisation problems. e-MPC [7] refers to singlelayer strategies which are incorporated into the control structure economic considerations. In that spirit, De Souza et al. [8] proposed the inclusion of the gradient of the economic objective function into the MPC objective as a single-layer strategy. Considering uncertain systems, Chachuat et al. [9] examined alternative model adaptation strategies.

This article is motivated by the abovementioned issues and aims at introducing a method for designing multi set-point explicit controllers for nonlinear systems through recent advances in multi-parametric programming. Multi-parametric programming (mp-P) has received considerable attention from the process systems engineering community because of its unique ability to aid in the design of explicit model predictive controllers and thus shift the computational burden associated with offline control [10]. We examine a case of multi-parametric nonlinear programs (mp-NLPs) that involve both endogenous uncertainty, in the form of left-hand side parameters (LHS), as well as exogenous uncertainty in the cost coefficient of the objective function (OFC), and, on the right-hand side of the constraints (RHS), uncertain parameters on the right-hand side (RHS). In engineering problems, LHS uncertainty arises from variations in model coefficients, due to parameter estimation errors or model mismatch; OFC uncertainty arises due to fluctuation in market prices or control penalties while RHS uncertainty can be due to varying system exogenous factors. The contribution of the present work is a novel framework for the design of multi set-point explicit controllers for nonlinear process systems.

The remainder of the article is organised as follows: in Section 2, an overview of the field of multi-parametric programming and explicit MPC is given, and then, in Section 3, the proposed algorithm is detailed and a framework for multi set-point explicit controllers is introduced. In Section 4, two case studies are examined so as to illustrate the main computational steps of the proposed methodology and, lastly, in Section 5, concluding remarks are made.
