I NTERNATIONAL J OURNAL OF C HEMICAL R EACTOR E NGINEERING Volume 6
2008
Article A86
A Robust Nonlinear Observer for State Variables Estimation in Multi-Input Multi-Output Chemical Reactors
∗
Bachir Daaou∗
Abdellah Mansouri†
Mohamed Bouhamida‡
Mohammed Chenafa∗∗
Ecole Normale Sup´erieure d’Enseignement Technique, daaou
[email protected] Ecole Normale Sup´erieure d’Enseignement Technique,
[email protected] ‡ Universit´e des Sciences et de la Technologie Mohamed Boudiaf, m
[email protected] ∗∗ Ecole Normale Sup´erieure d’Enseignement Technique,
[email protected] ISSN 1542-6580 c Copyright 2008 The Berkeley Electronic Press. All rights reserved. †
A Robust Nonlinear Observer for State Variables Estimation in Multi-Input Multi-Output Chemical Reactors Bachir Daaou, Abdellah Mansouri, Mohamed Bouhamida, and Mohammed Chenafa
Abstract In this paper we tackle the on-line estimation of state variables in MIMO continuous stirred chemical reactor (CSTR) using a nonlinear observer. We prove the asymptotic stability of the resulting error system. Moreover, this observer has robust performance in the presence of model uncertainty and measurement noise. Finally, computer simulations are developed for showing the performance of the proposed nonlinear observer. KEYWORDS: nonlinear observers, state estimation, MIMO chemical reactors
Daaou et al.: A Robust Nonlinear Observer for MIMO Chemical Reactors
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1. INTRODUCTION Among of the most frequent and important challenges in the control of chemical processes is to find an adequate and a reliable sensors to measure all the important state variables of the plant. However, if a number of on-line sensors providing state information are available today at the industrial scale, they are still very expensive and their maintenance is usually consuming time. This is especially true in the field of chemical processes. One way to estimate the states and parameters of a system is by means of observers, also called software sensors. The estimated states and parameters can be then used only for supervising the process (Kazantzis and Kravaris, 2000), or they can be incorporated in a loop control (Michalska and Mayne, 1991). State estimation methods have been initiated in the 1960s. The Kalman filter and the Luenberger observer were the first ones to be introduced (Kalman, 1960; Luenberger, 1964; 1966). The extensions of these two methods, known as extended Kalman filter (EKF) and extended Luenberger observer (Jo and Bankoff, 1976; Schuler and Suzhen, 1985; Bellgardt et al., 1986; Ponnuswamy et al., 1986; Ramirez, 1987; Ellis et al., 1988; Choi and Khan, 1988; Adebekun and Schork, 1989; Bastin and Dochain4, 1990; Kim and Choi, 1991; QuinteroMarmol et al., 1991; Kozub and MacGregor, 1992; Ogunnaike, 1994; Robertson et al., 1995; Liotta et al., 1997). However, several studies show the inadequacy of these methods for highly non-linear processes (Kantor, 1989; Kozub and MacGregor ; Ogunnaike, 1995; Gudi et al., 1995; Valluri and Soroush, 1996; Tatiraju and Soroush, 1997), because these methods use linear approximation of the nonlinear process model (Soroush, 1997; Zambare, 2001). Gauthier et al. (1981; 1992) stated a canonical form and necessary and sufficient observability conditions for a class of nonlinear systems that are linear with respect to inputs. Farza et al. (1998; 1999) developed a simple nonlinear observer for on-line estimation of the reaction rates in chemical and biochemical reactors. The principal advantage of this observer lies in the simplicity of its design and implementation. The sliding mode observers based on the theory of the system with variable structure was proposed by (Slotine et al., 1987; Canudas and Slotine, 1991; Wang et al., 1997, Ahmed-Ali and Lamnabhi-Lagarrigue 1999). The idea in this approach is to force the error in estimation to join a stabilizing variety. The difficulty is to find a variety attainable and having this property. Using the differential geometry tools, Aguilar et al. (2002; 2003; 2005) developed a high gain observer for uncertainty estimation in nonlinear systems with observable and unobservable uncertainties. In a recent contribution, another estimation approach also based on the differential geometry technique includes
Published by The Berkeley Electronic Press, 2008
International Journal of Chemical Reactor Engineering
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Vol. 6 [2008], Article A86
the sliding-mode observer is provided to the estimation of reaction heat in continuous chemical reactors (Aguilar, 2007). The majority of the previous approaches have the common properties to provide an asymptotically converging estimate. Engel and Kreisselmeier (2002) have proposed finite-time converging observer. In this technique, a state observer converges exactly to the state after a predefined time delay. Frédéric et al. (2007) proposed a nonlinear finite-time converging observer for a class of nonlinear systems. The principal advantage of this method is that the observer design does not require computing the inverse coordinates transformation but only its Jacobian. Taking into account the characteristics of the observers discussed above, the main objective of this work is to present a nonlinear efficient observer for Multi-Input/Multi-Output continuous chemical reactors in which the series
reactions, , take place. This nonlinear observer is developed to simultaneously estimate the concentration of the reactant and using continuous measurements of temperature and the concentration of the product . The stability of the observer is analytically treated using the Lyapunov theory in order to show the conditions under which exponential convergence can be achieved. The observer design is simple and it requires small computational effort. As result, its implementation is greatly facilitated. The performance of the proposed multi-output observer (MOO) is compared with a high gain observer in the single output case (SOO) i.e., only the reactor temperature is measured. The work is organized as follows. In Section 2, we present the dynamical model under consideration. The design of the corresponding nonlinear observer is presented in section 3. In section 4, the computer simulations were developed to illustrate the performance of the proposed nonlinear observer. Finally, we will close the paper with some concluding remarks. 2. SYSTEM DESCRIPTION AND PROBLEM STATEMENT
On-line estimation of state variables, concentrations, mainly, in chemical reactors is very important. Concentrations are related to process productivity; however their direct measurement is often expensive or even impossible considering the current sensor technology. In order to show the improved features of the observer proposed, a continuous stirred chemical reactor is chosen as study case. This type of reactor has been perhaps one of the most widely studied unit operation, from both dynamic analysis and control perspectives. The typical diagram of this
reactor is shown in figure 1 in which the series reactions, , take place in the liquid phase.
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Daaou et al.: A Robust Nonlinear Observer for MIMO Chemical Reactors
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, # , ,
Figure 01: Schematic of the multivariable non-isothermal chemical reactor. In this paper some assumptions have been introduced in the attempt to reduce the model complexity. In particular, we will assume the following: 1. There is a complete uniformity of concentration and temperature within the reactor and a complete uniformity of temperature inside the jacket. 2. The reactor and jacket volumes are constant. 3. The thermal exchange between the reactor and the jacket is expressed by a constant global heat exchange coefficient and the amount of heat retained in the reactor walls is negligible. 4. The reaction enthalpies ∆ are independent of temperature. 5. The heat capacities of the process fluids are constant. Under the above assumptions, the material and energy balances applied to the jacket reactor give the mathematical model (Gibon-Fargeot et al., 2000):
∆
! , # ! " # # ,
" # $! $"
with:
(
%& ' +, , - 1,2 )*
!
∆
" # ,
(1) (2) (3) (4) (5) (6)
where , # and are the outlet concentrations of the reactant , and respectively, inlet concentration of the reactant , reactor outlet
Published by The Berkeley Electronic Press, 2008
International Journal of Chemical Reactor Engineering
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Vol. 6 [2008], Article A86
temperature, reactor inlet temperature, jacket temperature, jacket inlet temperature, 0 feed flow rate to reactor, overall heat-transfer coefficient, heat transfer surface area, 1 heat capacity of feed and product, 1 heat capacity of jacket fluid, 2 - 1,2 activation energy, 3 universal gas constant, %& - 1,2 pre-exponential factor, ∆ - 1,2 heat of reaction 4 density of mixture in reactor and 4 density of jacket fluid. Here 5 6, , # , 78 , 9 60, 78 and $ 6, 78 , then the model (16) can be rewritten as: ! (7) 5 ! : 5! 9" ;! ! 5! 5" ;" " 5! 5< 5! 9!
5 " ! 5! 5" 5" 9!
5 <
5 =
! 5! 5"
" 5! 5
