Adaptive extremum seeking control of a non-isothermal tubular reactor with unknown kinetics
Introduction
The study of non-isothermal plug-flow reactors (PFR) is of great importance in the applied distributed parameter systems (DPS) literature (see, e.g., Christofides, 2001, and references therein). Modeling of such reactors is well known (Varma & Aris, 1977, Chapter 2, pp. 79–155). Typically, convection-reaction tubular reactors are described by a set of first-order hyperbolic partial differential equations (PDEs). These reactors have been studied for control, optimization and design.
Several control design techniques of DPS have been applied to tubular reactors in the last few decades (see, for example, the review paper by Ray (1978)). Ray (1981) divided DPS control design into two main classes: early lumping and late lumping. In the first case, the system is first discretized in his space coordinate to obtain a set of ordinary differential equations (ODEs). The control design is then developed based on this lumped version of the model. The late lumping design approach addresses the control problem on the original distributed model and then the model and the controller are discretized in the space coordinate.
The advantage of using distributed parameter models over lumped models is to use the maximum of the available information about the profiles of the state variables in the reactor. These profiles are of great importance in some application in order to guarantee the product quality at the end of the reactor and to respect constraints along the reactor. For example, in Kraft pulping, the temperature magnitude in the digester must stay within bounds in order to avoid chip breakage (see, for example, Kayihan, 1997). In most process control applications, the main concern is to avoid the formation of hot spots and reactor temperature runaway (e.g., Karafyllis & Daoutidis, 2002; Wu, Morbidelli, & Varma, 1998). It is also often desired to optimize a given function of the exit components concentrations. Studies about profiles in tubular reactor date back to Bilous and Amundson (1956a). Optimal profiles studies using closed form solution of the PDEs have been presented in Bilous and Amundson, 1956b, Bilous and Amundson, 1956c. Numerical methods to solve the optimal temperature profile for tubular reactor were given in (Ray, 1981). Smets, Dochain, and van Impe (2002) used a Pontryagin maximum principle approach to determine the optimal temperature profile along the tubular reactor profile.
Various extensions of nonlinear control techniques can be found in the literature. For convection-reaction tubular reactors, Hanczyc and Palazoglu (1995) used sliding mode control based on an exact representation of nonlinear first-order partial differential equations obtained by the method of characteristics. Christofides and Daoutidis (1996) used concepts from geometric control to develop an output feedback control methodology for hyperbolic PDE systems where the manipulated input, the measured output and the controlled output are distributed. The resulting controller was applied to a non-isothermal plug-flow reactor. Christofides and Daoutidis (1998) presented robust control design for first-order hyperbolic PDE systems with unmodeled dynamics. Orlov and Dochain (2002) applied discontinuous feedback to plug-flow reactor and axial dispersion reactor model. Moreover, trajectory analysis of nonlinear plug-flow models has been investigated by Laabissi, Achhab, Winkin, and Dochain (2001) and Winkin, Dochain, and Ligarius (2000). Most of the control strategies are applied through variations of the inlet concentration or the inlet temperature (e.g., Bošković & Krstić, 2002; Fliess, Mounier, Rouchon, & Rudolph, 1998). In this study, we follow the example presented in Christofides (2001) and consider the jacket temperature profile as the manipulated control input of the tubular reactor.
A very few DPS adaptive control schemes are reported in the literature. Baumeister, Scondo, Demetriou, and Rosen (1997) developed a model-reference adaptive control (MRAC) scheme and persistency of excitation conditions for hyperbolic and parabolic models (Demetriou & Rosen, 1994). Dochain (2001) presented the application of state observers and adaptive linearizing control to tubular control without any assumed knowledge of the reaction kinetics. Bošković and Krstić (2002) applied backstepping to a parabolic model of a convection-diffusion reaction tubular reactor.
In this study, we assume no knowledge of the reaction kinetics. Thus, the optimal profiles are unknown. In order to find the optimal profile of the jacket temperature for a given objective function of the exit components concentration vector, we extend results of adaptive extremum seeking developed by Guay, Dochain, and Perrier (2003) based on a Lyapunov objective function. Many successful applications of extremum control approaches have been reported in the 1950s and the 1960s (see, for example, Sternby, 1980; Ariyur & Krstić, 2003, for an overview). To the authors’ knowledge, the proposed application of adaptive extremum seeking as a late lumping control design to DPS is not found in the literature.
We will use an extension of Lyapunov stability theory to DPS, mainly via Lyapunov functionals. Theory about the stability of DPS described by PDEs are presented in Berger and Lapidus (1968), Gordon (1987), Romicki (1977), Sirazetdinov (1972) and Zubov (1964). Some applications to reactors can be found in Clough and Ramirez (1971) and Liou, Lim, and Weigand (1974).
The paper is structured as follows. Section 2 presents the model under consideration and some preliminary analysis of the problem. In Section 3, the parameter estimation algorithm and the controller are developed based on a Lyapunov functional. In Section 4, we present a numerical application of the algorithm to the Williams–Otto reaction. Brief conclusions are given in Section 5.
Section snippets
Class of systems
In this paper, we consider the following hyperbolic PDE model of a non-isothermal tubular reactor:for , with L , the length of the reactor. Here, denotes the concentration of chemical components in the reactor. The temperature is denoted and takes value in , a subset of . is the matrix of the known stoichiometric coefficients for each n components on r
Adaptive extremum seeking design
In this section, we design a profile control strategy that tracks the unknown optimum. To avoid the problem of hot spots, constraints are imposed on the range of admissible operating temperatures. The optimization problem of interest is stated as follows:Thus, the objective of the control strategy is to steer the system to the optimum equilibrium value of a linear combination of the product concentrations in the outlet stream of the plug-flow reactor, . In order to
Simulation study
In this paper, we apply the method to the Williams–Otto reaction (Roberts, 1979; Williams & Otto, 1960) in a plug-flow reactor without recirculation. The reaction scheme is the following:The component state vector is:The objective is to maximize the concentration of the product P , hence H is given by:The stoichiometric matrix is given by:The pseudo-inverse of M , is given by the following:
Conclusions
We have solved a class of extremum seeking profile control problems for plug-flow reactors with unknown reaction kinetics subject to reactor temperature constraints. It has been shown that if the persistency of excitation condition is satisfied, then the proposed adaptive extremum seeking controller guarantees the convergence to a small neighborhood of its optimum. Numerical application shows the performance and interest of the method.
Acknowledgement
The collaboration with Dr. I. Smets (KUL, Belgium) during portions of this work is gratefully acknowledged.
References (40)
- et al.
Optimum temperature gradients in tubular reactor. I. General theory and methods
Chemical Engineering Science
(1956) - et al.
Optimum temperature gradients in tubular reactor. II. Numerical study
Chemical Engineering Science
(1956) - et al.
Backstepping control of chemical tubular reactors
Computers and Chemical Engineering
(2002) - et al.
Robust control of hyperbolic PDE systems
Chemical Engineering Science
(1998) - et al.
Control of hot spots in plug flow reactors
Computers and Chemical Engineering
(2002) - et al.
Trajectory analysis of nonisothermal tubular reactor nonlinear models
Systems and Control Letters
(2001) - et al.
Lyapunov functional method for analysis of nonadiabatic tubular reactors with axial mixing and recycle
Chemical Engineering Science
(1974) Some recent applications of distributed parameter systems theory—a survey
Automatica
(1978)Stability conditions and design of control using Lyapunov–Zubov functionals
- et al.
Dynamical analysis of distributed parameter tubular reactors
Automatica
(2000)