An MPC-based control structure selection approach for simultaneous process and control design

Abstract

An optimization framework that addresses the simultaneous process and control design of chemical systems including the selection of the control structure is presented. Different control structures composed of centralized and fully decentralized predictive controllers are considered in the analysis. The system's dynamic performance is quantified using a variability cost function that assigns a cost to the worst-case closed-loop variability, which is calculated using analytical bounds derived from tests used for robust control design. The selection of the controller structure is based on a communication cost term that penalizes pairings between the manipulated and the controlled variables based on the tuning parameters of the MPC controller and the process gains. Both NLP and MINLP formulations are proposed. The NLP formulation is shown to be faster and converges to a similar solution to that obtained with the MINLP formulation. The proposed methods were applied to a wastewater treatment industrial plant.

Introduction

Incorporating control decisions during the design phase of a process is recognized as an effective way to improve process profitability (Kookos and Perkins, 2001, Luyben, 2004, Mohideen et al., 1996, Sakizlis et al., 2004, Munoz et al., 2012, Ricardez Sandoval et al., 2008). This integrated approach involves the minimization of plant costs related to process design, e.g., capital and operating costs, and to process control, e.g., variability costs, dynamic feasibility, and controller implementation. Control decisions related to the design of a new plant, or retrofit of an existing plant, involves different aspects such as the selection of a suitable control structure, the specification of the control algorithms to be included in that control scheme and the calculation of the controllers’ tuning parameters. This paper presents an approach for the integration of design and control that combines process design-related costs with these different aspects of control decisions, and its associated costs, into a single optimization problem.

While the idea of adding control decisions at the design stage is relatively straightforward, the development of a mathematical framework that can simultaneously consider steady state and plant dynamics in a closed loop is a challenging task. Earlier approaches used for integrating design and control differed in the way that the closed loop performance was accounted for in the analysis. A first group of studies involved formulations where the capital and operating costs are minimized while considering a controllability index such as the RGA, Resiliency index (Lenhoff and Morari, 1982, Luyben and Floudas, 1994), condition number (Palazoglu and Arkun, 1986, Palazoglu and Arkun, 1987, Skogestad and Postlethwaite, 1996) and minimum square deviation (Molina et al., 2011, Zumoffen and Basualdo, 2013, Zumoffen et al., 2011). While the use of these indexes is computationally attractive, they may be inaccurate since they rely on steady state and/or dynamic linear process models, which may not capture the true process (nonlinear) dynamics. In a second group of approaches that consider the true process (nonlinear) dynamic model, a formal mixed-integer dynamic optimization problem (MIDO) is formulated to assess the optimal process design under uncertainty (Bahri et al., 1997, Bansal et al., 2002, Kookos and Perkins, 2001, Mohideen et al., 1996, Sakizlis et al., 2004). In those dynamic, optimization-based methods the closed loop variability is estimated using the mechanistic process model and a user-defined, time-dependent disturbance function with uncertain (critical) model parameters, e.g., a sinusoidal function with uncertain amplitude and frequency. Therefore, the designs obtained by these dynamic approaches may not be valid when the actual disturbance affecting the process deviates significantly from the disturbance function model assumed in the analysis. Systematic approaches based on process heuristics and dynamic simulations (Gani, Hytoft, Jaksland, & Jensen, 1997) and probabilistic-based disturbances (Ricardez-Sandoval, 2012) have also been proposed in the literature. A review on previous integration of design and control methodologies can be found elsewhere, e.g., Ricardez-Sandoval, Budman, & Douglas (2009a), Sakizlis et al. (2004) and Seferlis and Georgiadis (2004).

A recent approach proposed by two of the authors in this work, Ricardez-Sandoval, Budman, & Douglas (2009b), Ricardez-Sandoval, Budman, & Douglas (2010), Ricardez-Sandoval, Douglas, & Budman (2011), involves a computationally efficient methodology that made use of uncertain process models, which are identified from numerical simulations of the mechanistic process model and used to compute closed-loop variability bounds that were used to assign variability costs to the system's dynamic performance. However, these latter methodologies were limited since the control structure considered in the simultaneous design and control analysis remained fixed during the calculations, i.e., only the controller tuning parameters were included as decision variables for optimization. Also, only PI feedback controllers were considered in those studies. The current study expands upon those previous studies of Ricardez-Sandoval et al. by considering, in addition to the choice of controller parameters, optimal controller structure selection and the use of a model-based controller, Model Predictive Control (MPC), which is widely used in the process industry (Morari & Lee, 1999). It is important to recognize the specific challenges arising from the consideration of controller structure selection and the use of MPC. First, the optimal selection of a control configuration has been generally tackled by the use of binary variables within a mixed integer problem (Flores-Tlacuahuac and Biegler, 2007, Flores-Tlacuahuac and Biegler, 2008, Mohideen et al., 1997, Schweiger and Floudas, 1997) or by considering all possible combinations (Zumoffen & Basualdo, 2013), which adds numerical difficulty to an already computationally expensive and highly non-convex optimization problem. Likewise, the implementation of an MPC-based control strategy requires the identification of an internal process model. That internal model needs to be updated at each step in the optimization search since it depends on the optimization variables related to the plant's design such as the process units’ dimensions and their corresponding operating conditions. Thus, using model-based controllers such as MPC is more challenging as compared to PI or PID controllers for which explicit controller equations exist. Due to these challenges, it is imperative to use practical methods that can be efficiently used to calculate closed loop variability while using model-based control strategies. In that regard, the use of analytical bounds computed from uncertain models previously proposed by Ricardez-Sandoval and co-workers has also been adopted in this work to quantify process variability.

The selection of an optimal control structure is addressed in the current study by adding a communication cost function within the overall cost function. A key idea in the current work is that the distribution of the control effort among the different possible pairings between manipulated and controlled variables can be related to an index involving the tuning parameters of the MPC controller and the process gains. Using this index, two different formulations of the optimization problem are proposed which differ in the way that the communication costs are included within the analysis. In the first formulation, the communication costs are defined in terms of binary variables assigned to each possible pairing, resulting in a mixed integer nonlinear optimization (MINLP) problem. It will be shown that this MINLP problem, despite the savings achieved by using the robust bounds of Ricardez-Sandoval et al., is computationally intensive for large-scale problems, i.e., for problems with a large number of inputs and outputs. To reduce the computational costs, a second optimization formulation is proposed in this work where the communication costs are estimated by the aforementioned index but without binary variables, which eliminates the need for the MINLP formulation of the first approach. A case study involving the optimal design of an industrial-scale water treatment plant is used to compare the two optimization approaches.

The paper is organized as follows: the mathematical formulations of the two optimization problems explained above are presented in Section 2. Section 3 shows the application of the two methods to a wastewater treatment plant. A comparison between the two methodologies in terms of the optimal designs and computational times is discussed at the end of this section. Concluding remarks are given in Section 4.