DAE-EKF-Based Nonlinear Predictive Control of Reactive Distillation Systems Exhibiting Input and Output Multiplicities Jalesh L. Purohit, Sachin C. Patwardhan,* and Sanjay M. Mahajani Department of Chemical Engineering, Indian Institute of Technology, Bombay, Mumbai 400076, India ABSTRACT: Reactive distillation (RD) has become one of the most important hybrid separation processes in recent times because of its economic and operational advantages. Reactive distillation systems, however, can exhibit complex input and output multiplicity behavior simultaneously in the desired operating region. Moreover, such a system is generally stiff and is typically modeled as a set of differential-algebraic equations. These two factors render the control of RD systems a challenging problem, particularly when the desirable operating point is unstable. In this work, an observer error feedback-based NMPC scheme has been developed for achieving offset-free control of RD systems modeled as DAEs. A recently developed version of the EKF for DAE systems (Mandela et al. Chem. Eng. Sci. 2010, 65, 4548-4556) was used to carry out state estimation. Because direct use of a DAE solver in the NMPC formulation can prove to be prohibitively computationally intensive and unsuitable for online implementation, a successive-linearization-based NMPC scheme (SLNMPC) was also developed. The effectiveness of the proposed control schemes is demonstrated by simulating servo and regulatory control problems associated with a hypothetical ideal RD column that exhibits input and output multiplicity behavior simultaneously at an unstable but economically desirable operating point. The servo and regulatory performances of the proposed SLNMPC scheme were also studied by simulating an industrial RD system involving MTBE synthesis. Analysis of the simulation results indicates that the proposed SLNMPC formulation provides an effective approach for handling control problems involving moderately large-magnitude servo and regulatory changes in the operation of RD systems. Moreover, the average computation time for the SLNMPC formulation was found to be quite small when compared with the sampling interval, which establishes the feasibility of implementing the SLNMPC scheme in real time. 1. INTRODUCTION Reactive distillation (RD) has become one of the most important hybrid unit operations because of its advantages, such as complete conversion in equilibrium-limited reactions, heat integration through the utilization of the exothermic heat of reaction for volatilization, minimum waste generation, and economic benefits over the conventional reaction-separation sequence in the chemical process industries. 1 The interaction of reaction and separation in RD systems leads to a highly nonlinear behavior such as the occurrence of steady-state input and output multiplicities. 2 Process intensification often leads to extrema of the critical process parameters (such as conversion) inside the operating region, 3 which can give rise to input multiplicity behavior. Input multiplicities result if competing effects are present in nonlinear chemical processes and the steady-state gain matrix of the process becomes singular in the desired operating region. 4 Several authors have reported the existence of input and output multiplicities in RD systems. 5-7 A phenomenon associated with input multiplicity is change(s) in the sign(s) of the steady-state gain(s) in the desired operating region. As a consequence, the robustness of the linear controller with integral action is lost in the presence of input multiplicity. 8 Moreover, the input multiplicity behavior can pose significant difficulties even for a nonlinear control scheme. 9 The presence of output multiplicity behavior, on the other hand, leads to open-loop unstable dynamics at some of the multiple steady states. Al-Arfaj and Luyben 10 established that output multiplicity exists at constant reflux flow and constant reflux ratio for both low- and high-conversion designs of methyl acetate RD columns. The behavior of the process in the presence of output multiplicities depends on the process history, and the process output can be different for the same input move. 11 If it is desired to control the system at one of the unstable operating points, then designing a controller that stabilizes the system and achieves desired closed-loop perform- ance is not an easy task. In fact, input and output multiplicities can occur simultaneously in some reactive distillation systems. Monroy-Loperena et al. 12 reported the simultaneous occur- rence of input and output multiplicity behavior in the RD process for ethylene glycol synthesis. Controlling such an RD system is a challenging task. Difficulties in controller synthesis are further compounded by the fact that RD systems are often modeled as sets of coupled differential-algebraic equations (DAEs). As a consequence, most nonlinear state estimation and control approaches, which are typically developed for systems of ordinary differential equations (ODEs), have to be further modified to handle nonlinear algebraic constraints. The implications of the occurrence of input multiplicity on the closed-loop performance of an ideal RD column regulated using multiloop proportional-integral-derivative (PID) con- trollers were investigated by Kumar and Kaishtha. 13 Whereas a conventional multiloop PID or linear multivariable control scheme, such as the robust multivariable controller developed by Volker et al., 14 might be sufficient for carrying out regulation Received: February 3, 2013 Revised: July 6, 2013 Accepted: August 6, 2013 Published: August 6, 2013 Article pubs.acs.org/IECR © 2013 American Chemical Society 13699 dx.doi.org/10.1021/ie4004128 | Ind. Eng. Chem. Res. 2013, 52, 13699-13716