Inventory control of particulate processes

Abstract

In this work we address the problem of designing model-based controllers for particulate processes described by population balance (PB) models. We focus on PB models that are solved by numerical discretization, for which many standard control methodologies are not suitable due to the high order of these models. We interpret discretized PB models as chemical reaction networks and suggest to combine inventory control with techniques of stability of chemical reaction networks to design the controller. Inventory control is based on the idea of manipulating process flows so that certain extensive variables defining the system, called inventories, follow their setpoints. The whole system is stabilized by controlling the dominant inventories. The discretized PB is exploited in all aspects of controller design, from determining the controlled inventories to the final implementation of the control law. The methodology is illustrated with an industrial leaching reactor, the Silgrain^®process. We show that the discretized PB model takes the form of a Feinberg–Horn–Jackson zero-deficiency network, allowing us to prove stabilization of the whole system. The performance of standard inventory control and robust inventory control are investigated by simulation, with satisfactory results even in the presence of modeling errors.

Introduction

Particulate processes, i.e. processes involving a set of entities that differ from each other in the values of certain distributed properties, are encountered in almost any branch of the process industries (such as in the petrochemical, pharmaceutical, and metallurgical branches). Although the term “particulate processes” comprises unit operations that are different in their nature such as crystallization, emulsification, leaching, etc…, there are certain basic mechanisms that are shared by all particulate processes. Hence, a unified approach to build mechanistic models of particulate processes has been possible: the so-called population balance (PB) approach. The main ideas behind the PB can be traced back to Fisher’s work in statistics, and to the work by Flory in polymer growth modeling (Flory, 1953). However, in its modern and unified form, the PB equation was developed in the 1960s by two groups of researchers studying crystal nucleation and growth (Hulburt and Katz, 1964, Randolph, 1964). Since then, extensive research has been carried out on PB modeling of particulate processes. There are conferences and journal issues exclusively dedicated to particulate processes and to the PB. Moreover, detailed models for a considerable number of particulate processes are available in the literature, see for example the review article by Ramkrishna (1985), and the book by the same author (Ramkrishna, 2000).

Despite the rapid and remarkable advances in modeling, numerical solution, and simulation of PB, the field of automatic control of particulate processes has not developed as much as could be expected. Examples of advanced control strategies implemented in real industrial settings are scarce. Some of the reasons that explain the lack of advanced controllers for particulate processes are: the nonlinear and multivariable input-output behavior of such processes, the distributed nature of the PB models (i.e. infinite number of internal states), limited instrumentation (it remains difficult to measure the distribution of properties), insufficient degrees of freedom or manipulated variables, and batch or semibatch operation. Nonlinear and multivariable control approaches would thus be desirable for many particulate processes, but they are also harder to implement than linear single-input single-output approaches.

Some review papers on the status of certain branches of particulate processes include sections on the status of automatic control, such as the papers by Rawlings et al. (1993) and Braatz (2002) dealing with crystallization; the papers by Wang and Cameron (2002) and Cameron, Wang, Immanuel, and Stepanek (2005) dealing with granulation. As regards theoretical control-related issues, one of the first references is the controllability analysis suggested in Semino and Ray, 1995a, Semino and Ray, 1995b.

The most extensive work on design of nonlinear controllers for particulate processes is probably a series of papers coauthored by Christofides and a book by the same author (Christofides, 2002). Their approach, nonlinear output feedback control, was tested on a crystallization process (Chiu and Christofides, 1999, Chiu and Christofides, 2000), and an aerosol flow reactor (Chiu & Christofides, 2000). The effect of input constraints was discussed in El-Farra, Chiu, and Christofides (2001), and robustness issues were analyzed in Chiu and Christofides (2000). The approach consists in reducing the order of the PB model by combining the method of weighted residuals and the concept of approximate inertial manifold. A nonlinear low-order output feedback controller that enforce exponential stability of the closed loop is synthesized using geometric and Lyapunov-based techniques. The method has shown promising results in simulation, but require exhaustive model manipulation and the use of mathematical tools that are not common in the field of particulate processes. Other research groups have also used standard output feedback linearization, such as Kurtz, Zhu, Zamamiri, Henson, and Hjortsø (1998) and Mantzaris and Daoutidis (2004).

Model-based predictive control (MPC) is a control approach that has been widely used in the chemical process industry for decades, since it is intuitive. Linear MPC has been proposed for the stabilization of oscillating microbial cultures in bioreactors (Kurtz et al., 1998; Zhu, Zamamiri, Henson, & Hjortsø, 2000), and for the emulsion polymerization of styrene (Zeaiter, Romagnoli, Barton, & Gomes, 2002). Eaton and Rawlings (1990) applied nonlinear programming to solve the nonlinear model predictive control formulation of a batch crystallizer. Nonlinear model predictive control was also applied by Crowley, Meadows, Kostoulas, and Doyle (2000) and Immanuel and Doyle (2002) to optimize the performance of semibatch emulsion polymerization. Although MPC has a number of strengths, there are some obstacles to wider application of MPC to particulate processes: (a) the upper computational time required for convergence may be too large for online applications; (b) handling unfeasible solutions is not straightforward, (c) implementation costs increase as models get larger. Hence again, MPC is suited for PB models for which good reduced-order models, and preferably linearized models, are available.

In addition to nonlinear output feedback control and MPC, robust $H_{\infty }$ controllers have also been used (Galán et al., 2002, Vollmer and Raisch, 2002). Some practical implementations of process controllers at laboratory scale are presented in Patience and Rawlings (2001), Immanuel and Doyle (2002), and Zeaiter et al. (2002).

One common aspect to all model-based control design approaches of particulate processes reviewed above is that they start by reducing the order of the PB model before designing the controller. Otherwise, the controller design and implementation would become infeasible (such as in output feedback linearization involving Lie algebra) or the resulting controller would be limited by calculation speed (such as in MPC). Order reduction involves more or less cumbersome analytical manipulation. There are many PB models for which the common reduction methods (method of moments, Hulburt & Katz, 1964; method of weighted residuals, Christofides, 2002; or integral approximation, Motz, Mannal, & Gilles, 2004) are difficult to apply or can not be applied, such as PB models that are solved by numerical discretization.¹ The motivation of this paper is to use discretized PB models directly for controller design, without further reducing their order. We do indeed exploit the fact that for most particulate processes there is a low dimensional dynamic space in which a few independent variables dominate the overall dynamic behavior of the system, as recognized by several authors (Christofides, 2002; Kothare, Shinnar, Rinard, & Morari, 2000). However, we do not reduce the model, we just use it to identify these dominant variables and the nonlinear feedback laws required to control them. The control technique we suggest in this work is a nonlinear control technique called inventory control, developed by Farschman, Viswanath, and Ydstie (1998), and closely related to input–output passivity theory of nonlinear control and output feedback linearization. Inventory control is based on the idea of manipulating process flows so that the extensive variables defining the system, called inventories, follow their setpoints. Inventory control belongs to a framework developed by Ydstie and coworkers, linking thermodynamics, passivity based control, and transport phenomena. The connection between macroscopic thermodynamics of process systems and the input–output passivity theory of nonlinear control was established in Ydstie and Alonso (1997) and further analyzed in Ruszkowski, Garcia-Osorio, and Ydstie (2005), using the second law of thermodynamics to develop sufficient conditions for strict state passivity in the space of intensive variables, such that all the state variables (not only the controlled ones) converge to stationary variables. The control methodology was then developed for lumped process models (Farschman et al., 1998) and distributed process models (Alonso et al., 2000, Alonso and Ydstie, 2001; Ruszkowski et al., 2005, Ydstie, 2002), respectively. The approach presented in Farschman et al. (1998) exploits the structure and positivity of first principle models directly in the formulation of the control law, which has the form of an output feedback linearization law. The approach by Farschman et al. (1998) was mostly used for systems that do not undergo chemical transformations, and to small systems with few extensive variables. The approach by Farschman et al. (1998) was tested on particulate processes for the first time in Dueñas Díez, Ydstie, and Lie (2002) and Dueñas Díez (2004), illustrating the methodology with the model of an industrial leaching reactor, and the current paper continues the work on application of inventory control to discretized PB models. The way we use inventory control for particulate processes is novel in the sense that we interpret discretized PB models as networks of chemical reaction networks, being thus able to connect inventory control to concepts and tools from nonlinear chemical dynamics to demonstrate stabilization of all states. In particular, we use the important work on the existence and uniqueness of equilibria for reaction networks of mass action kinetics by Feinberg, Horn, and Jackson (Feinberg and Horn, 1974, Feinberg, 1995a, Feinberg, 1995b, Horn and Jackson, 1972, Feinberg, 1980, Feinberg, 1991) to demonstrate that for the chosen nonlinear control law, the internal dynamics of the case study are stable, and no oscillatory behavior or multiplicity of steady states are possible. The paper is organized as follows: Section 2 introduces the PB equation, model reduction and discretization methods. Section 3 describes inventory control. The Feinberg, Horn and Jackson framework is described in Section 4. Section 5 describes the Silgrain^® process, the PB model, how the disintegration kinetics can be viewed as a chemical reaction network of the type described by Feinberg, Horn, and Jackson, and the design and performance of inventory control in two variants: standard inventory control and robust inventory control. Finally, Section 6 summarizes the main conclusions of the paper.