A Robust Extended State Observer for the Estimation of Concentration and Kinetics in a CSTR

Transcription

1 Instituto Tecnológico y de Estudios Superiores de Occidente Repositorio Institucional del ITESO rei.iteso.mx Departamento de Matemáticas y Física DMAF - Artículos y ponencias con arbitrae 05- A Robust Extended State Observer for the Estimation of Concentration and Kinetics in a CSTR Sánchez-Torres, Juan D.; Jiménez-Rodríguez, Esteban; Jaramillo-Zuluaga, Óscar; Botero-Castro, Héctor; Loukianov, Alexander Sanchez-Torres, J.D.; Jiménez-Rodríguez, E.; Jaramillo-Zuluaga, Ó; Botero-Castro, H. and Loukianov, A. "A Robust Extended State Observer For The Estimation Of Concentration And Kinetics In A CSTR". International Journal of Chemical Reactor Engineering. Volume 4, Issue, Pages , ISSN (Online) , ISSN (Print) , DOI: December 05. Enlace directo al documento: Este documento obtenido del Repositorio Institucional del Instituto Tecnológico y de Estudios Superiores de Occidente se pone a disposición general bao los términos y condiciones de la siguiente licencia: (El documento empieza en la siguiente página)

2 Int. J. Chem. React. Eng. 05; aop Juan Diego Sánchez Torres, Héctor A. Botero, Esteban Jiménez*, Oscar Jaramillo and Alexander G. Loukianov A Robust Extended State Observer for the Estimation of Concentration and Kinetics in a CSTR DOI 0.55/icre Abstract: This paper presents a state estimation structure for a Continuous Stirred Tank Reactor (CSTR), by means of an Asymptotic Observer ointly with a disturbance high order sliding mode-based estimator. The proposed estimation scheme allows the asymptotic reconstruction of the concentration inside the reactor based on the measures of the temperature inside the reactor and the temperature inside the acket, in presence of changes in the global coefficient of heat transfer UA, the Arrhenius constant k 0 and the activation energy E. Additionally, the structure is able to estimate UA and the kinetics term k 0 e RT E. The properties of the proposed scheme are proved mathematically and verified through numerical simulations. Keywords: state estimator, state observer, sliding modes, nonlinear observers Introduction The continuous stirred-tank reactor (CSTR) is one of the most studied operation units due its wide application in several processes (Bequette 00). In addition, it is a representative case of the so-called reaction systems. Those systems are a class of nonlinear dynamical systems that is widely used in areas such as chemical engineering and biotechnology. For these chemical processes, it is well known that nonlinear state observers, parameter and *Corresponding author: Esteban Jiménez, Universidad Nacional de Colombia, Sede Medelín, Calle 59A No 63 0 Medellín, Colombia, esimenezro@unal.edu.co Juan Diego Sánchez Torres, CINVESTAV-IPN Unidad Guadalaara, Av. del Bosque 45, Guadalaara, México, dsanchez@gdl.cinvestav.mx Héctor A. Botero, Oscar Jaramillo, Universidad Nacional de Colombia, Sede Medelín, Calle 59A No 63 0 Medellín, Colombia, habotero@unal.edu.co, odaramilloz@unal.edu.co Alexander G. Loukianov, CINVESTAV-IPN Unidad Guadalaara, Av. del Bosque 45, Guadalaara, México, louk@gdl.cinvestav.mx unknown inputs estimators have been becoming of great interest due to they allow to reduce the quantity of installed sensors in a plant, since they only require some variables measurements and a dynamic model of the process (Walcott, Corless, and Zak 987; Bequette 00). The main uses for those methods are the design of observerbased controllers, monitoring schemes, the synthesis of fault detection and isolation methods, among other applications in a reliable and affordable way (Dochain 003a). In a chemical process, there are dynamic changes as a result from the normal operation of the plant, the aging of the materials, the conduct obstruction, the physical changes of the components and the changes on the properties of the reagents entering to the reactor. These changes make some parameters of the process to be modified, particularly those related to the reaction kinetics and the heat transfer. As a result, the process model used by a state observer become outdated. However, most of the observation methods are based on models that assume the parameters to be set to a perfectly known fixed value, while in the real system these parameters are unknown and only approximations can be obtained, moreover some of them are changing permanently. This issue makes the estimated variables to deviate from their actual values leading to undesirable situations as low performance controllers, insufficient quality products and operation regimes with unacceptable safety levels. Considering the mentioned uncertainties related to the mathematical model and limited availability of online sensors, the proposal of robust observation schemes for this class of systems is of fundamental importance (Bastin and Dochain 990; Soroush 998). There are several solutions to the observer design problem for these systems as the exposed in (Bastin and Dochain 990), including the use of the extended Kalman and Luenberger observers (Dochain and Vanrolleghem 00), adaptive observers (Bastin and Dochain 990; Dochain 003b; Farza et al. 009), the asymptotic observers which allow the estimation of some variables independently from the knowledge of the reaction kinetics (Bastin and Dochain 990; Dochain, Perrier, and Ydstie 99;

3 J. D. Sánchez Torres et al.: Estimation of Concentration and Kinetics in a CSTR Dochain and Vanrolleghem 00; Moreno and Dochain 005), Luenberger-asymptotic observers (Hulhoven, Vande Wouwer, and Bogaerts 006), interval observers (Gouzé, Rapaport, and Had-Sadok 000), dissipative observers (Moreno 008) and observer based estimators (OBE) for the case of parameter estimation (Oliveira, Ferreira, and Feyo de Azevedo 00; Perrier et al. 000). On the other hand, an important class of nonlinear observers are the sliding mode observers (SMO) (Utkin 99; S. V. Drakunov 99; Boukhobza and Barbot 998; Barbot, Boukhobza and Demai 996; Spurgeon 008) which have the main features of the sliding mode (SM) algorithms. Those algorithms are proposed with the idea to drive the dynamics of a system to a sliding manifold, that is an integral manifold with finite reaching time (Drakunov and Utkin 99), exhibiting very interesting features such as work with reduced observation error dynamics, the possibility of obtain a step by step design, robustness and insensitivity under parameter variations and external disturbances, and finite time stability (Utkin 99; Drakunov and Utkin 995). In addition, the last feature can be extended to uniform finite-time stability (Cruz-Zavala, Moreno, and Fridman 00) or fixed-time stability (Polyakov 0), allowing the controllers and estimators to converge with a settling-time independent to the initial conditions. For a SMO design case, the chattering reduces to a numerical problem (Slotine, Hedrick, and Misawa 986; Utkin 99). Hence, some SMO have attractive properties similar to those of the Kalman filter (i.e. noise resilience) but with simpler implementation (Drakunov 983). For chemical processes the SMO have been applied for several cases, some examples are shown in (Wang, Peng, and Huang Wang, Shiou Peng, and Ping Huang 997; Drakunov and Law 007; Martínez-Guerra, Aguilar, and Poznyak 004; Moreno and Mendoza 04) including oint schemes for parameter estimation based on the OBE design (Botero and Alvarez 0; Giraldo, Botero, and Sánchez-Torres 0; Botero et al. 04). As an extension of the mentioned approaches, the aim of this paper is to present a nonlinear estimation structure for a CSTR. In the previous works, asymptotic observers are used to estimate a variable, usually a concentration, by compensating the effect of the uncertainty caused by the kinetics. An estimate of the kinetics can be helpful to know the process traceability; however, in some cases this effect ofthekineticsisonlyeliminatedbutthekineticsitselfisnot estimated. In addition, the initial formulation of the asymptotic observer does not take into account the parametric change in the global coefficient of heat transfer, which can cause variations in the routine operation of the process. On the other hand, adaptive or SM observers are used to estimate the kinetics under the assumption of the concentration or the global coefficient of heat transfer availability. Consequently, a state estimation structure for a CSTR is proposed here, by means of an asymptotic observer ointly with a second order sliding mode-based estimator. By noting that the asymptotic observers use a state transformation in order to avoid using the kinetic model and this transformation results to be the regular form (Luk yanov and Utkin 98) it is presented a novel asymptotic observer that allows to obtain an estimate of the concentration compensating the effect of two uncertainty sources, namely the kinetics and the global coefficient of heat transfer. This structure is obtained by taking advantage of the regular form nesting properties. In addition, based on the observer presented in (Davila, Fridman, and Levant 005), the kinetics and the global coefficient of heat transfer are estimated with a second order SMO based on the generalized super twisting algorithm (GSTA) (Cruz-Zavala, Moreno, and Fridman 00), a fixed-time stable extension of the well known super twisting algorithm (Levant 993). The whole estimation scheme allows the exact and exponentially fast reconstruction of the concentration, the reaction kinetics termandtheglobalcoefficientofheattransfer. In the following, the Section presents the mathematical model of the CSTR. The estimation structure is presented in Section 3. The Section 4 presents results of numerical simulation. Finally, the conclusions of this paper are presented in the Section 5. Mathematical model for the CSTR This work uses a CSTR model, which is widespread in the process industry and its complexity has been evidenced (Bequette 00). In this case, it is assumed that the model has three state variables, which are: concentration C A, temperature inside the reactor T and temperature inside the acket T. It is also assumed that the tank level is perfectly controlled. With these assumptions, the model is still valid for lots of common situations in chemical processes and bioprocesses (Bequette 00; Dochain 003a). Inside the reactor, there is produced an exothermic reaction of the form A! B. The model equations, which are obtained from mass and energy balances at the reactive mass and at the flow inside the acket, are: _C A = F ð V C in C A Þ κc A _T = F ð V T in TÞ ΔH κc A + V T T () _T = F f T f T T T. V ρ C p V

4 J. D. Sánchez Torres et al.: Estimation of Concentration and Kinetics in a CSTR 3 Here, C A is the concentration of reactive inside the reactor, T is the temperature inside the reactor, T is the temperature inside the acket, F is the flow into the reactor, which is always positive and usually constant due to the continuous operation of the reactor, V is the volume of the reaction mass, C in is the reactive input concentration, T in is the inlet temperature of the reactant, ΔH is the heat of reaction, ρ is the density of the mixture in the reactor, C p is the heat capacity of food, F f is the feeding flow, V is the acket volume, T f is the inlet temperature to the acket, ρ is the density of acket flow and, C p is the heat capacity of the acket flow. The term κ = k 0 e RT E represents the kinetics, where k 0 is the Arrhenius kinetic constant, E is the activation energy and R is the universal gas constant. Finally, the parameter = UA is the global coefficient of heat transfer, where U is the overall coefficient of heat transfer and A is the heat transfer area (Bequette 00). It is assumed that the temperatures T and T are available for continuous measurement. The problem consists in the estimation of the concentration C A, the reaction kinetics κ and the global coefficient of heat transfer. 3 Observation scheme In this section, with the continuous measurements of the temperatures T and T, an observer for the variables C A, κ and is designed. 3. CSTR extended model Taking into account that in () the concentration C A is the unmeasured state variable, κ is an unknown function and, is an unknown, possibly, time-varying parameter, the plant model () is extended to the following system: _C A = F ð V C in C A Þ κc A _T = F ð V T in TÞ ΔH κc A + V T T _T = F f V _κ = f κ _ = f, T f T ρ C p V T T where f κ and f are unknown but bounded functions with f κ δ κ and f δ, and δ κ, δ > 0 are known constants. () 3. On the regular form and asymptotic observers Consider the following system: _x = a x + a x + b fðþ t _x = a x + a x + b fðþ t y = x, where x, x R are state variables, y is the measured output, fðþr t is an unknown input depending on the time t R + and a, a, a, a, b, b R are the system parameters. Here, the main obective is to find a state transformation such that the input fðþdoes t not appear in the first equation of (3). The transformed system, if existing, is called the regular form of (3) with respect to fðþ t (Luk yanov and Utkin 98). With this aim, define z = x b b x and z = x ; with this transformation the system (3) reduces to: _z = a z + a z (4) _z = a z + a z + b fðþ, t with a = a ab b b, a = a b + ð a a Þ b b + a, a = a and a = a + b b a. The expression (4), in addition to be a suitable representation of the system (3) for control design as shown in (Luk yanov and Utkin 98), is also very useful for observer design. To illustrate the observer design case, consider the system (3). Here, the main obective is to design an state observer to estimate the unmeasured variable x considering x to be available for continuous measurements and fðþas t an unknown input. Transforming the system (3) to the regular form with respect to the unknown input fðþ t (4), the following observer is presented: _^z = a ^z + a z ^x = ^z + b b x. The observer (5) is commonly known as the asymptotic observer (Bastin and Dochain 990; Dochain, Perrier, and Ydstie 99) and its main characteristic is its independence and therefore its robustness against the unknown input fðþ. t Defining the error variable ~z = z ^z, it follows _~z = a ~z ; where it is noted that the convergence of the observer depends on the parameter a which is not tunable. (3) (5)

5 4 J. D. Sánchez Torres et al.: Estimation of Concentration and Kinetics in a CSTR This procedure can be extended to the following third order system: _x = a x + a x + a 3 x 3 + b f ðþ t _x = a x + a x + a 3 x 3 + f ðþ+ t b f ðþ t _x 3 = a 3 x + a 3 x + a 33 x 3 + b 3 f ðþ t y = x y = x 3, where x, x, x 3 R are state variables, y and y are the measured outputs, f ðþ, t f ðþr t are unknown inputs depending on the time t R + and a, a, a 3, a, a, a 3, a 3, a 3, a 33, b,, b, b 3 R are the system parameters. To obtain the regular form with respect to the inputs f ðþand t f ðþ, t define the new nested variables z = x b x + b b b 3 x 3, z = x and z 3 = x 3. Using this transformation the system (6) becomes: (6) and third equations, as f ðþin t (6), the procedure exposed in (6)-(8) is applied. The variable z is defined as z = C A ΔH T + ρ C pv V T, (9) it can be considered as a transformation of the concentration C A. The dynamics of (9) is given by _z = F V C in z ρc p ΔH T + ρ C pv V T F ð VΔH T in TÞ ρ C pf f VΔH T f T, (0) where is observed the independence with respect to the unknown variables κ and, as exposed in (8), while the others variables remain untransformed. Thus, the system () is yields to regular form with respect to the unknown variables κ and. _z = a z + a z + a 3 z 3 _z = a z + a z + a 3 z 3 + f ðþ+ t b f ðþ t _z 3 = a 3 z + a 3 z + a 33 z 3 + b 3 f ðþ, t where a = a b a + bb b 3 a 3, a = b (7) a b a + bb b 3 a 3 + a b a + bb b 3 a 3, a 3 = a 3 b a 3 + bb b 3 a 33 bb b 3 ða b a + bb b 3 a 3 Þ, a = a, a = a + b a, a 3 = a 3 bb b 3 a, a 3 = a 3, a 3 = a 3 + b a 3, a 33 = a 33 bb b 3 a 3. Using the transformed system (7), an observer for the system (6) to estimate the unmeasured variable x, can be designed similarly to the observer (5), in the following form: _^z = a ^z + a z + a 3 z 3 ^x = ^z + b x b b b 3 x 3. The proposed observer (8) can be considered as an extension of the observer (5). Its main characteristic is its invariance with respect to the unknown inputs inputs f ðþand t f ðþ. t 3.3 A regular form for the CSTR In this subsection the main obective is to design an observer that does not depend on the kinetics κ and global coefficient of heat transfer. Taking advantage that for system () κ appears in the first and second equations, as f ðþin t (6), and appears in the second (8) 3.4 Observer structure Using (), (9) and (0), the following observer is proposed: ^C A = ^z + ρc p ΔH ^T + ρ C pv V ^T _^z = F V C in ^C A _^T = F V T in ^T _^T = F f T f V ^T F VΔH ΔH ^κ^c A + _^κ = k ϕ ~ ΔHδ T ρ C p V _^ = k ϕ ~ δ T _^κ f = ^κ ^κ f τ _^ f = ^ ^f, τ ^ ρ C p V T in ^T ^ V ^T ^T ρ C pf f VΔH ^T ^T + k ϕ ~ T T f ^T + k ϕ ~ T () where ^C A, ^z, ^T, ^T, ^κ and ^ are the estimates of C A, z, T, T, κ and, respectively. The observer input inections are given by ϕ ðþ= be 3 + μbe and ϕ ðþ= be0 +μ + 3 μ be with the gain μ 0. The function be α = α signðþ is defined for α 0, where signðxþ= for x >0,

6 J. D. Sánchez Torres et al.: Estimation of Concentration and Kinetics in a CSTR 5 signðxþ= for x < 0 and signð0þ ½, Š; T ~ = T ^T and ~T = T ^T are the observation error variables; ^κ f and ^ f are the filtered variables of ^κ and ^, respectively; k, k, k and k are the observer positive gains. The parameters τ and τ are the time constants of the filters. Finally, δ and δ are positive observer parameters which will be chosen thereafter. The filters for ^κ and ^, are added in order to reduce the noise effect on the variables estimation, taking advantage of the large time constants which are usually present on chemical processes. ~C A = ~z + ρc p ΔH _~z = F V ~z + ~ T + ρ C pv V ~ T F f VΔH _~T = F T V ~ + q k ϕ ~ T Fρ C pv V ΔH _~T = F f T ~ + q k ϕ ~ V T _q = k θ ϕ ~ T + Δq _q = k θ ϕ ~ T + Δq, ~T (5) 3.5 Convergence analysis Define the observation error variables ~z = z ^z, variables ~z = z ^z, ~ C A = C A ^C A, ~κ = κ ^κ and ~ = ^. Then, from (), (9), (0) and (), it follows ~C A = ~z + ΔH _~z = F V ~z + F f VΔH T ~ + ρ C pv T V ~ Fρ C pv V ΔH ~T _~T = F T V ~ ΔH ^CA ~κ ΔH CA ~ κ + V T T ^ V ^T ^T k ϕ ~ T _~T = F f V ~ T ~ ρ C p V _~κ = f κ + k ϕ ~ ΔHδ T ρ C p V _^ = f k ϕ ~ δ T ^T ^T Define now the new variables q and q as q = ΔH ^CA ~κ ΔH ~ CA κ + e q = ^T ρ C p V ^T Then, the system () yields T ~ T ρ C p V ~ k ϕ ~ T V T ^ T V ρ C p V () ^T ^T (3) T ~ T ~. (4) where θ = ^T ^T δ Δ q = ΔH ^CA f κ ΔH _^CA ~κ + d dt and θ = ^C A δ. The terms " ΔH CA ~ κ + V T ^ T V ^T ^T # and f Δ q = ^T ρ C p V ^T ~ _^T _^T ρ C p V " # d T ~ T dt ρ C p V ~, are considered in (5) as unknown but bounded disturbances with Δ q δq and Δ q δ q, and δ q, δ q > 0 are known positive constants. If the observer gains k, k, k and k are chosen as n qffiffiffiffiffiffiffiffi k = ðk, k, k, k Þ R 4 <0k δ q, 0 qffiffiffiffiffiffiffiffi < k δ q, k > k 4 + 4δ q k fðk, k, k, k ÞR 4 k > qffiffiffiffiffiffiffiffi > δ q, k >δ q, k >δ q g,, k > k 4 + 4δ q k q ffiffiffiffiffiffiffiffi δ q, and the parameters δ and δ are selected such that θ = ^T ^T δ > and θ = ^C A δ >, then a sliding mode appears in the system (5) on the manifold T, ~ T ~, q, q Þ = ð0,0,0,0þ in a fixed time t q > 0 (Cruz-Zavala, Moreno, and Fridman 00). From (4), it follows that ~ ðþ= t 0 for t > t q, and from (3) to (5), the motion of the system () is constrained to the sliding manifold T, ~ T ~, q, ~ = ð0,0,0,0þ, which means ^T, ^T, q, ^ = T, T,0,. In this manifold, the dynamics of () is described by the reduced order system k )

7 6 J. D. Sánchez Torres et al.: Estimation of Concentration and Kinetics in a CSTR _~z = F V ~z ~C A = ~z ~κ = κ ~z. ^C A (6) Therefore, from the model hypothesis F > 0, the observer error pair CA ~, ~κ converges exponentially to ð0, 0Þ. Finally, taking into account that _^κf = τ ^κ ^κ f and _^f = τ ^ ^f are linear first order low-pass filters for the estimates of κ and, respectively, the convergence analysis is completed. 4 Numerical simulation results In this section, the numerical simulations results of the proposed estimation scheme applied to a CSTR, are presented. All simulations presented here were conducted using the Euler integration method, with a fundamental step size of 0 3 ½minŠ. The CSTR parameters are shown on Table. It can be noticed that, the proposed observer () does not use the information relative to the function κ (i.e. the parameters E, k 0 and R) nor the parameter. To verify the robustness of the proposed observer, plant parameter variations were introduced in the simulation, see Table. Under normal operating conditions, this sort of variation is very common, and consequently, the simulation results (see Figures 6) show that the observer is able to deal with what happens in reality. Is worth to notice that this variation is unknown for the proposed estimation scheme. Table : Nominal parameters of the CSTR. Parameter Values [Unit] Parameter Values [Unit] F ½m 3 min Š C p 3, 57.3 ½ J kg Š V.4069 ½m 3 Š U h i J ðmin m KÞ C in, 4.5 ½gmol.m 3 Š A ½m Š k ½min Š F f ½m 3. min Š E ½J gmol Š V ½m 3 Š R ½J gmol K Š T f 79 ½KŠ T in 95. K ρ 000 ½kg m 3 Š ΔH ½ J g mol Š C p ½J kg Š ρ 000 ½kg m 3 Š Table : CSTR parametric variations. Parameter Variation time Variation magnitude, min %-negative step The initial conditions for the CSTR were selected to: C A ð0þ= 05.0½gmol m 3 Š, T ð0þ= ½KŠ and Tð0Þ= ½KŠ; and for the observer to: zð0þ= , ^T ð0þ=300½kš, ^T ð0þ=90½kš, ^κ ð0þ= ^κ f ð0þ= 0½min Š and ^ ð0þ= ^f ð0þ=. 0 5 ½J ðmin KÞ Š. Furthermore, the parameter values for the observer were adusted to: k =, k = 0, k =5, k = 0, δ = 0, δ =, μ = and τ = τ = 5 ½minŠ. In the following, this section is divided into two parts. In the first part, there is assumed that the temperature measurements are noiseless; and in the second part C A [gmol.m 3 ] 600 C Aest C A [gmol.m 3 ] 600 C Aest 400 C A 400 C A Figure : Concentration of the reactive inside the reactor C A (estimated and actual).

8 J. D. Sánchez Torres et al.: Estimation of Concentration and Kinetics in a CSTR T est T T est T T est T T est T T [K] T [K] T [K] 8 80 T [K] (a) Temperature inside the reactor (estimated and actual). (b) Temperature in the acket (estimated and actual). Figure : Temperature inside the reactor T and temperature in the acket T (estimated and actual). Kinetics κ f κ Kinetics κ f κ x f x (a) Reaction kinetics (estimated and actual). (b) Global coefficient of heat transfer (estimated and actual). Figure 3: Reaction kinetics kinetics κ and global coefficient of heat transfer (estimated and actual). f 00 C Aest 000 C A C A [gmol.m 3 ] Figure 4: Concentration of the reactive inside the reactor C A (estimated and actual). Noisy measurements.

9 8 J. D. Sánchez Torres et al.: Estimation of Concentration and Kinetics in a CSTR 345 T meas T est T 86 T [K] T [K] T meas T est T (a) Temperature inside the reactor (measured, estimated and actual). Noisy measurements t[min] (b) Temperature in the acket (measured, estimated and actual). Noisy measurements. Figure 5: Temperature inside the reactor T and temperature in the acket T (measured, estimated and actual). Noisy measurements. Kinetics (a) Reaction kinetics (estimated and actual). Noisy measurements. κ f κ x (b) Global coefficient of heat transfer (estimated and actual). Noisy measurements. f Figure 6: Reaction kinetics κ and global coefficient of heat transfer (estimated and actual). Noisy measurements. instead, there is included a normally distributed random signal as measurement noise in the temperature. 4. Noiseless measurements In this subsection, it is assumed no noise in the measurements. The following results were obtained: Figures 3 show the comparison between the actual and estimated variables corresponding to the concentration of reactive inside the reactor C A, the temperature inside the reactor T and the temperature inside the acket T, the reaction kinetics κ and the global coefficient of heat transfer, respectively, for noiseless measurements. It is observed the fast convergence of the variables T, T and, while the variables C A and κ converge exponentially with a time constant of F V. 4. Noisy measurements In this subsection, it is assumed that the temperature measurements were corrupted by a normally distributed random signal with zero mean and unitary variance; this assumed variance corresponds to temperature sensors with an accuracy of ± 3½KŠ. The following results were obtained: Figures 4 6 show the comparison between the actual and estimated variables corresponding to the concentration of reactive inside the C A, the temperature inside the reactor T and the temperature inside the acket T, the reaction kinetics κ and the global coefficient of heat transfer, respectively, for noisy measurements. Based on the presented simulations results, it can be observed a good performance of the proposed observer

10 J. D. Sánchez Torres et al.: Estimation of Concentration and Kinetics in a CSTR 9 scheme, also under noisy measurement conditions. The observer for C A presents a correct estimation of this variable due to its independence of the unknown terms and κ (Figures and 4). In addition, a correct simultaneous estimation of T, T, κ and using the high order sliding mode observer is achieved (Figures, 3, 5 and 6), making the proposed system suitable for observer-based control applications. Furthermore, under noisy conditions, the estimated temperatures are much closer to their actual values than their measurements (Figure 5). 5 Conclusions An estimation scheme based on an asymptotic observer working ointly with a second order sliding mode estimator, was proposed for a CSTR. Measuring the temperature inside the reactor and the temperature inside the acket, this scheme achieves the asymptotic estimation of the concentration, despite of changes in the global coefficient of heat transfer, the Arrhenius constant k 0 and the activation energy E. Additionally, the proposed scheme allows to estimate the terms and k 0 e RT E which are well-known as very difficult to measure, especially for real-time applications. These terms were considered as unknowns but with known bounds of their time derivatives. Hence, this configuration ensures an estimation of the concentration, which is robust with respect to variations of the mentioned above terms. With a suitable selection of the observer gains, the proposed scheme presents a fast observer error convergence with high accuracy. The performance of the proposed approach was verified via numerical simulations, showing characteristics as good estimation of the state variable and both terms, and robustness with respect to plant parameter variations under noisy measurements. Acknowledgements: Juan Diego Sánchez acknowledges to CONACyT, México for the PhD scholarship number 4080 and the support of Proyecto de Movilidad Internacional de la Diáspora Científica de Alto Reconocimiento, año 0 Colciencias Banco Mundial to his internship in Colombia. References. Barbot, J.-P., Boukhobza, T., Demai, M., 996. Sliding mode observer for triangular input form. In Proceedings of the 35th IEEE Conference on Decision and Control, doi: 0.09/CDC Bastin, G.P., Dochain, D., 990. On-line estimation and adaptative control of bioreactors. Laboratoire D Automatique. method=post&formato=&cantidad=&expresion=mfn= \npapers://78a e7-4c85-97-d9cb4b46b/paper/ p Bequette, B.W., 00. Behavior of a CSTR with a recirculating acket heat transfer system. Proceedings of the 00 American Control Conference (IEEE Cat. No.CH3730) 4. doi:0.09/acc Botero, H., Alvarez, H., 0. Non linear state and parameters estimation in chemical processes: analysis and improvement of three estimation structures applied to a CSTR. International Journal of Chemical Reactor Engineering 9, Botero, H., Sánchez-Torres, J.D., Jiménez, E., Jaramillo, O., 04. Estimación de Estado E Incertidumbre En Un CSTR Mediante Observador Asintótico Y Modos Deslizantes de Alto Orden. In XXVII Congreso Interamericano Y Colombiano de Ingeniería Química. Cartagena. 6. Boukhobza, T., Barbot, J.-P., 998. High order sliding modes observer. Proceedings of the 37th IEEE Conference on Decision and Control, 998. doi:0.09/cdc Cruz-Zavala, E., Moreno, J.A., Fridman, L., 00. Uniform second-order sliding mode observer for mechanical systems. In Proceedings of the 00 th International Workshop on Variable Structure Systems, VSS 00, 4 9. doi:0.09/vss Davila, J., Fridman, L., Levant, A., 005. Second-order sliding-mode observer for mechanical systems. IEEE Transactions on Automatic Control 50, doi:0.09/tac Dochain, D., 003a. State and parameter estimation in chemical and biochemical processes: a tutorial. Journal of Process Control 3, doi:0.06/s (03)0006 X. 0. Dochain, D., 003b. State observers for processes with uncertain kinetics. International Journal of Control. doi:0. 080/ Dochain, D., Perrier, M., Ydstie, B.E., 99. Asymptotic observers for stirred tank reactors. Chemical Engineering Science. doi:0.06/ (9) Dochain, D., Vanrolleghem, P. (eds.), 00. Dynamical Modelling and Estimation in Wastewater Treatment Processes. IWA Publishing. 3. Drakunov, S.V., 983. An adaptive quasioptimal filter with discontinuous parameters. Automation and Remote Control 44, Drakunov, S.V., 99. Sliding-mode observers based on equivalent control method. [99] Proceedings of the 3st IEEE Conference on Decision and Control. doi:0.09/cdc Drakunov, S.V., Law, V.J., 007. Parameter estimation using sliding mode observers: application to the Monod kinetic model. Chemical Product and Process Modeling, doi:0.0/ Drakunov, S.V., Utkin, V.I., 99. Sliding mode control in dynamic systems. International Journal of Control. doi:0. 080/ Drakunov, S.V., Utkin, V.I., 995. Sliding mode observers. Tutorial. Proceedings of th IEEE Conference on Decision and Control 4. doi:0.09/cdc

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