Estimador algebraico diferencial para el monitoreo de una clase de ...

Revista Mexicana de Ingenier´ıa Qu´ımica Academia Mexicana de Investigaci´ on y Docencia en Ingenier´ıa Qu´ımica, A.C.

Revista Mexicana de Ingeniería Química Volumen 10, N´ umero 2, Agosto 2011

ISSN 1665-2738

CONTENIDO

1

Vol. 10, No. 2 (2011) 313-320 Volumen 8, número 3, 2009 / Volume 8, number 3, 2009 1

DIFFERENTIAL ALGEBRAIC ESTIMATOR FOR THE MONITORING OF A CLASS 213 Derivation and application of the Stefan-Maxwell OF PARTIALLY KNOWN equations BIOREACTOR MODELS (Desarrollo y ALGEBRAICO aplicación de las ecuaciones de Stefan-Maxwell) ESTIMADOR DIFERENCIAL PARA EL MONITOREO DE UNA CLASE DE BIORREACTORES CON MODELOS PARCIALMENTE CONOCIDOS Stephen Whitaker

J. L. Mata-Machuca1 , R. Mart´ınez-Guerra1 and R. Aguilar-López2∗ 1 Biotecnología / Biotechnology Departamento de Control Automático-CINVESTAV IPN. 2

Departamento de Biotecnolog´ıa y Bioingenier´ıa CINVESTAV-IPN. Av. Instituto Politécnico Nacional No. 2508, 245 Modelado de la biodegradación en biorreactores de lodos de hidrocarburos totales del petróleo San Pedro Zacatenco, México, D.F. C.P. 07360 intemperizados en suelos y sedimentos Received 10 of March 2010; Accepted 13 of May 2011 (Biodegradation modeling of sludge bioreactors of total petroleum hydrocarbons weathering in soil

Abstract and sediments) S.A. of Medina-Moreno, Lucho-Constantino, L. Aguilera-Vázquez, A. Jiménez-A reduced The problem monitoring inS.a Huerta-Ochoa, common classC.A. of partially known bioreactor models is addressed. order observer namely algebraic estimator is proposed. The biomass is estimated by means of substrate González y M.differential Gutiérrez-Rojas concentration measurements. The estimation methodology is based on a suitable change of variable which 259 Crecimiento, sobrevivencia y adaptación de Bifidobacterium infantis a condiciones ácidas allows generating artificial variables to infer the remaining mass concentrations constructing a differential-algebraic structure. (Growth, The proposed methodology is of applied to a classinfantis of Haldane unstructured survival and adaptation Bifidobacterium to acidic conditions) kinetic model with success. Stability analysis in a Lyapunov sense for the estimation error is performed. Some remarks about the convergence L. Mayorga-Reyes, P. Bustamante-Camilo, A. Gutiérrez-Nava, E. Barranco-Florido y A. Azaolacharacteristics of the proposed estimator are given and numerical simulations show its satisfactory performance. Finally, forEspinosa comparison purposes, a high gain observer is presented: the convergence is possible only when the model 265 is perfectly known. Statistical approach to optimization of ethanol fermentation by Saccharomyces cerevisiae in the Keywords: presence differential-algebraic estimator, of Valfor® zeolite NaA state variable estimation, continuous bioreactor, Haldane kinetics. Resumen (Optimización estadística de la fermentación etanólica de Saccharomyces cerevisiae en presencia de En el presente trabajo zeolite se considera zeolita Valfor® NaA) el problema del monitoreo de una clase de biorreactores con modelos parcialmente conocidos. Se propone un tipo de observador de orden reducido denominado estimador diferencial G. Inei-Shizukawa, H. A. Velasco-Bedrán, G. F. Gutiérrez-López and H. Hernández-Sánchez algebraico. La metodolog´ıa de estimación se basa en un cambio de variables que permite generar variables artificiales para inferir las concentraciones no medibles. La metodolog´ıa propuesta es aplicada a un modelo cinético no estructurado Haldane/ con e´ xito. Se efectúa un análisis de Lyapunov para demostrar la estabilidad de la Ingeniería de de procesos Process engineering metodolog´ıa considerada. Algunos comentarios sobre las caracter´ısticas de la convergencia del estimador son 271 Localización de una planta industrial: Revisión crítica y adecuación de los criterios empleados en proporcionados y simulaciones numéricas muestran un desempeño satisfactorio. Finalmente, con propósitos de decisión comparacióesta n, un observador de alta ganancia se presenta en donde su convergencia se garantiza solo cuando se conoce perfectamente el modelo del sistema bajoadequation estudio. criteria used in this decision) (Plant site selection: Critical review and J.R. Medina, R.L.algebraico-diferencial, Romero y G.A. Pérez estimación de variables de estado, biorreactor continuo, cinética Palabras clave: estimator de Haldane.

∗ Corresponding author. E-mail: [email protected] Tel. + 52 55 5747 3800

Publicado por la Academia Mexicana de Investigación y Docencia en Ingenier´ıa Qu´ımica A.C.

313

J. L. Mata-Machuca et al./ Revista Mexicana de Ingenier´ıa Qu´ımica Vol. 10, No. 2 (2011) 313-320

1

Introduction

Operating a bioreactor is not a simple task, as during a bioreacting process, variables such as concentrations are generally determined by off-line laboratory analysis, making this set of variables of limited use for control purposes and on-line monitoring. However, these variables can be on-line estimated using soft sensors. Over the last few years, the importance of on-line monitoring of biotechnological processes has increased. A first step to efficient bioreactor operation is the adequate implementation of online measurements of essential variables such as substrate and biomass concentrations. Advantages of continuous monitoring of key variables include gaining knowledge about the state of the process and the possibility of detecting and isolating abnormal process developments at early stages. This reduces process costs, contributes to process safety and helps in trouble-shooting and process accommodation. The main problem in fermentation monitoring and control is the fact that process variables usually cannot be measured on-line. Monitoring and controlling these processes can therefore be difficult because only indirect measurements are available online, while calculated values may be rather uncertain. This can be due to uncertainty with respect to the equations used, measurement errors or both. For automatic control this may have serious consequences, especially as the actual variables of interest often cannot be directly controlled and related variables are controlled instead. In fermentation processes, on-line and offline measurements are the main source of information about the state of the process. In combination with model-based calculations, they are used to produce estimations for monitoring purposes as well as for automatic and manual process control (Bastin and Dochain, 1990), (Masoud, 1997). Observation schemes are widely used for reconstructing states of dynamical systems (AguilarLópez et al., 2006). Most of the contributions are related to asymptotic observers for monitoring, fault detections and control issues whereas the real necessities of industrial plants are related to a fast response of the monitoring and regulation methodologies. Special attention was given to filtering techniques, namely extended Kalman filter, adaptive observers, and artificial neural networks (ANN), (Dávila and Fridman, 2005), (Hu and Wang, 2002), (Levant, 2001), however for these techniques the right tuning 314

of the estimators gains is difficult. It is shown that software based state estimation is a powerful technique that can be successfully used to enhance automatic control performance of biological systems as well as in system monitoring and on-line optimization. In this paper we consider the growth rate partially known. Following this idea, the necessity to adapt an observation scheme to the available knowledge of the growth rate immediately arises. The main contribution in this work is to show a state estimator which is a simplified version of the methodology given by (Lemesle and Gouzé, 2005) where a simple linear change of variable given in a natural manner allows to develop a differential-algebraic state estimator. Results show an adequate performance of the considered methodology. The technique is not the same as (Alvarez-Ramirez et al., 1999) since we do not have derivators. The proposed estimation methodology is applied to a kind of unstructured kinetic model: the Haldane model, which is considered for biological process with substrate inhibition. The above mentioned kinetic model is applied to a class of continuous stirred bioreactors. In what follows, the statement of the problem is presented; an observability condition is given in the differential-algebraic setting. In section 3, the bounded error estimator is designed. Section 4 shows a high gain observer as a comparison with the proposed methodology. Finally, we give some concluding remarks.

2 2.1

Problem statement The model

Consider the following nonlinear system x˙ = f (x, u) y = h (x)

(1)

where, x ∈ Rn , u ∈ Rm , m ≤ n, y ∈ R p . Let us recall the classical observer definition. An observer for system (1) is a dynamical system x˙ˆ = fˆ ( xˆ, u, y), whose task is state estimation. Usually is required at least that k xˆ − xk → 0 as t → ∞. Although in some cases, exponentially convergence is also required (Gauthier et al., 1992). Definition 1: an estimator is said to be bounded if the estimation error (k xˆ − xk) belongs to an open ball with radius proportional to some value that depends on its estimation error.

www.rmiq.org

J. L. Mata-Machuca et al./ Revista Mexicana de Ingenier´ıa Qu´ımica Vol. 10, No. 2 (2011) 313-320 In all paper, we will consider a class of bioreactor model. The simplified Haldane model taken from (Vargas et al., 2000), is described by dS X + kd X = D (S in − S ) − µ (S ) dt YS /X

(2a)

dX = −DX + µ (S ) X − kd X (2b) dt . . where µ (S ) = µmax S δ + S + S 2 ϕ is the specific growth rate and µmax is the maximum growth rate. We assume that µ(S ) is partially known, which is common in biology (Gouzé and Lemesle, 2001). Generally, µ(S ) is between two bounds meaning that we know a function µ(S ˆ ) such that |µ (S ) − µˆ (S )| < a, where a ∈ R+ , and µ (0) = µ(0) ˆ = 0. We introduce an important lemma about lower bounded properties of µ(S ). Lema 1 (Hadj-Sadok, 1999): there exists a constant ε ∈ R, such that S (0) > ε implies S (t) > ε for all t. Thus, for any smooth function µ(S ), µ (S (t)) > µ (ε) for all t. From lemma 1, we could always choose ε such that µˆ (S (t)) > µˆ (ε) = r, where r ∈ R+ . The state variables S , X are the substrate and biomass concentrations, respectively, D = q/V is the dilution rate with V the volume of the bioreactor and q the constant flow passing through the bioreactor, S in is the input substrate concentration, YS /X is the corresponding yield coefficient. Let us notice that the inputs D = u and S in are fixed. Moreover, we assume that the measured output is, y=S

2.2

Algebraic (AOC)

Observability

(3)

Condition

Before proposing the bounded error estimator, a definition concerning on algebraic observability condition is given, for more details see (Diop and Mart´ınez-Guerra, 2001). Definition 2: consider the system described by (1), T x1 x2 . . . xn . A state xi , i = where x = {1, 2, · · · , n}, is said to be algebraically observable with respect to {u, y} if it satisfies a differential polynomial in terms of u, y and some of their time derivatives, i.e., P (xi , u, u˙ , . . . , y, y˙ , . . .) = 0, i = {1, 2, · · · , n}. Replacing y = S into Eq. (2a), the algebraic observability condition for Haldane model is

calculated as follows, y˙ − u (S in − y) +

! 1 µmax ϕ y − k d X = 0 (4) δϕ + ϕ y + y2 YS /X

From Eq. (4), it is clear that the state variable X satisfies the AOC thus, X is algebraically observable.

3 3.1

Bounded error estimator Estimator design

In what follows, the corresponding estimated concentration is denoted by ˆ, and we assume that S is measured exactly, i.e., S = Sˆ . Then, we only reconstruct the biomass variable X. Consider the Haldane’s model given by system (2), and make the change of variable z = X +kS where k ∈ R is fixed. The dynamics of z is " z˙ = − D + kd − k kd + + (1 − k) k kd S +

k

(5)

! # (S ) −1 µ z+

YS /X ! (6) k (S ) −1 k µ S + k D S in

YS /X

Proposition 1: if we choose the estimator’s gain such that YS /X < k 6 1 + D/kd and |µ (S ) − µˆ (S )| < a, a ∈ R+ . Then, the system (7) is a bounded error estimator of (6). " ! # ˙zˆ = − D + kd − k kd + k − 1 µˆ (S ) zˆ+ YS /X ! (7) k − 1 k µˆ (S ) S + k D S in + (1 − k) k kd S + YS /X For the proof, define the estimation error, e = z − zˆ

(8)

Then, using eqs. (6) and (7) the estimation error dynamic is obtained as ! # " k − 1 µˆ (S ) e+ e˙ = − D + kd − k kd + YS /X ! ! k k + − 1 µ (S ) − µˆ (S ) k S − − 1 µ (S ) − µˆ (S ) z YS /X YS /X (9) To analyze the stability of Eq. (9) we consider the following Lyapunov function candidate

www.rmiq.org

V=

1 2 e 2

(10) 315

J. L. Mata-Machuca et al./ Revista Mexicana de Ingenier´ıa Qu´ımica Vol. 10, No. 2 (2011) 313-320 ultimate bound | e | 6 and ε as follows

The time derivative of Eq. (10) is V˙ = e e˙

(11)

Replacing (9) into (11) yields " ! # k ˙ V = − D + kd − k kd + − 1 µˆ (S ) e2 + YS /X ! ! k k − 1 µ (S ) − µˆ (S ) k S e − −1 + YS /X YS /X µ (S ) − µˆ (S ) z e (12) Equation (12) is written alternatively as " ! # k V˙ = − D + kd − k kd + − 1 µˆ (S ) e2 − YS /X ! (13) k − 1 µ (S ) − µˆ (S ) X e − YS /X Now, from lemma 1 and taking into account that YS /X < k 6 1 + D/kd , |µ (S ) − µˆ (S )| < a, and X is bounded, Eq. (13) leads to, " ! # k V˙ 6 − D + kd − kkd + − 1 r e2 YS /X ! k − 1 aXmax |e| = −λ e2 + w | e | + YS /X

2ε. For instance, defining c

!2 k aXmax , YS /X λ

aX 2 max ε= k λ

√ the ultimate bound is, | e | 6 2 k aXλmax . Corollary 1: if the growth rate is perfectly known, i.e., µ (S ) = µˆ (S ), and we choose the estimator’s gain such that YS /X < k 6 1 + D/kd . Then, the system (14) is an asymptotic estimator of (6). " ! # ˙zˆ = − D + kd − k kd + k − 1 µ (S ) zˆ+ YS /X ! k + (1 − k) k kd S + − 1 k µ (S ) S + k D S in YS /X (14) Indeed, the dynamics of the error in this case is # ! " k e˙ = − D + kd − k kd + − 1 µ (S ) e YS /X and the corresponding time derivative of Lyapunov function candidate (10) is " # ! k V˙ = − D + kd − k kd + − 1 µ (S ) e2 < 0 YS /X Moreover, X can be reconstructed considering

where, k

λ = D + kd − kkd +

YS /X

Xˆ = zˆ − k S

! −1 r

3.2

and

!

w=

k − 1 aXmax YS /X

The right-hand side of the foregoing inequality is not negative since near the origin, the positive linear term w | e | dominates the negative quadratic term −λe2 . However, V˙ is negative outside the set { | e | 6 w/λ}. Let. c, ε be some upper bounds for V (e). With c > w2 2λ2 , solutions starting in the set {V (e) 6 c} will remain therein for all time because V˙ is negative on the boundary V = c. Hence, the solutions of Eq. (9) are. uniformly bounded (Khalil, 2002). Moreover, if w2 2λ2 < ε < c, then V˙ will be negative in the set {ε 6 V 6 c}, which shows that, in this set V will decrease monotonically until the solutions enters the set {V 6 ε}. From that time on, the solution cannot leave the set {V 6 ε} since V˙ is negative on the boundary V = ε. According to (Khalil, 2002), the solution is uniformly ultimately bounded with the 316

c=

√

(15)

Numerical simulations

For all simulations in this paper we take S in = 50, D = 0.1, YS /X = 0.9, kd = 0.01 and the initial conditions S (0) = 60, X(0) = 40, Xˆ (0) = 30, zˆ (0) = 90, with appropriate units, these values are taken from Vargas et al., (2000). The estimator’s gain is k = 1. The growth rates are chosen as µ (S ) = and µˆ (S ) =

S . 140 + S + S 2 81.25 0.8 S . 140 + S + S 2 81.25

when the model is well known for the asymptotic estimator and when the model is partially known for the bounded error estimator, respectively. The simulations results were carried out with the help of Matlab 7.1 Software with Simulink 6.3 as the toolbox.

www.rmiq.org


1  0.001

t



e  d

Zt ke (τ)k2 dτ

(16)

0

40 30 20 10

2

60 60

where e(t) is the corresponding state estimation error 50 (the difference between the actual 50 observed signal1and 60 its estimate). corresponding state estimation 40 error (the difference 2

0

SS uubbsst rt raat e te

is the 50 First, al and its estimate).

40

50 Bi o ma s s

1 t + 0.001

50 calculated as (Martínez-

0 0

20

30 20 10

(16) 10

40

60 60 30 time

40

50

60

50 50

X real 0 X real 0 10 asymptotic estim asymptotic estim bounded error e bounded error es

60 real 40 between theFigure actual2.X State Variables (with noise in the 40 asymptotic estimator

BB io aassss i om m

J=

60

60

Su b s t r a t e

mance index al.}, 2000)

The performance index of the corresponding estimation process is calculated as (Mart´ınez-Guerra, ofet al., the2000) corresponding estimation process is

JJ

J

Bi o ma s s

Su b s t r a t e

50 in Fig. 1 we show the30simulation results bounded error estimator 30 30 30 3 for the40bounded error estimator given by proposition 40 20 20 and simulation the corresponding results for the the asymptotic 20 g. 1 we show1,the results for bounded error estimator given by 20 4 30 30 estimator given by corollary 1 (without any noise in 10 10 10 and the corresponding resultsFurthermore, for the asymptotic estimator given by corollary 1 10 70 the system output). in Fig. 2 is shown 20 20 asymptotic estima 0 0 noise in the system output). in Fig. 2 is shown the effect of noise in bounded40 error est5 the effect of noise inFurthermore, the estimation0000process. A white 10 20 30 10 20 30 40 50 60 6000 10 20 30 40 50 10 20 30 40 50 60 time time 10 10 1the measurement (σ = 0.1, ±10% time time 1 noise is added in n process. A white noise is added in the measurement (   0.1 , 10% around the 50 around 0the current of the measured output) 0 2is value Figure 2. measurement State Variables (with noise in the the50system system output). 0 Variables 10 20(with 30 60 output). 0 10 2 20 30 40 corresponding 50 Figure 60 40 40 2. State noise in of the measured output) this considering the error oftime this is considering the corresponding measurement time 1 30 ding sensor and/or measurement technique. WeFig. can2.observe that the error ofexperimental the corresponding sensor and/or experimental State Variables (with noise in the system 33 Figure 2 2. State Variables (with noise in the system output). measurement technique. We can observe Finally, that the in Fig. estimator is robust against noisy measurement. 3 is illustrated the 20 output). estimator is robust against noisy 44 3 ndex given by bounded (16) forerror the corresponding estimation process. It should be noted that 10 measurement. Finally, in Fig. 3 is illustrated the 0 70 70 0 10 20 30 40 50 asymptotic estimator estimation error (performance index) isfor bounded on average and has a tendency to asymptotic estimator performance index given by (16) the corresponding time 4 bounded errorestimator estimator 5 bounded error 60 60 estimation process. It should be noted that the 70 50 asymptotic estimator quadratic estimation error (performance index) 50 6 is (a) bounded error estimator 60 bounded on average and has a tendency to decrease. 40 40 70

40 J

50

5

40

20

30

5

20 10

6

0 0

10

20

60

30

30

20

20

7

10

50

10 10

40

30 time

(a)

760 8 9 10

40

asymptotic estima bounded error es

10 10

50

20 20

60

30 30 time time

a) (a) (a)

3040 40

50 50

60 60

20 10

0 0 10 20 asymptotic estimator estimator asymptotic bounded error estimator bounded error estimator

70 70 60 60

30 time

40

50

(b)

asymptotic estimator 50 50 bounded error estimator 40 40

Figure 3. Quadratic estimation error. (a) Without any noise, (b 30 30 output. 20 20

11

10 10

12

0 0 0 0

0 0

12

10 13

20 A.

4. A NOTE ON -ORDER : THE HIG 20 30 FULL 40 50 OBSERVERS 60 20 30 40 50 60

10 10

time time

design 30 Observer 40 50 60 8 b) time(b) 0 0 10 20 30 40 50 60 14 Consider that system (1) satisfies the AOC. In this case to es time Fig. 3.error. Quadratic estimationany error.noise, (a) Without any white (a) Without any noise, (b) with with white noise; noise; in 9 Figure (b) 3. Quadratic estimation (a) Without (b) 15 we can suggest a nonlinear high gain observer (Gauthier {\it in et Fig. 1. State Variables. noise, (b) with white noise; in the system output. Figure101. State Variables. output. output. 16 Without et al.},any 2000) with following structure, Figure 3. Quadratic estimation error. (a) noise, (b)the with white noise; in the system www.rmiq.org 317 output. 11 17

11

FULL OBSERVERS THE HIGH HIGH GAIN GAIN OBSERVER OBSERVER 4. A18NOTE ONxˆFULL :: THE   K  y OBSERVERS  f xˆ-, ORDER uORDER Cxˆ 

4.13A NOTE FULL-ORDER OBSERVERS: THE HIGH GAIN OBSERVER A. ONObserver design

12 13

70

50

20

J

Bi o ma s s

9 10

40

40

40

10

8

30 time

10

X real 60 asymptotic estimator 50 bounded error estimator

50

7

0 0

20 20

0 0 0 0

10

6

60

JJ

Su b s t r a t e

30

30 30

J

50

60

A.

Observer 14 design Consider that system (1) satisfies the AOC. AOC. In In this this case case to to estimate estimate the the state-space state-space


4

A note on full-order observers: The high gain observer

Sˆ (0) = 40, Xˆ (0) = 30, with appropriate units. The estimator’s gain is θ = 2. The simulations results of high gain observer are presented in figs. 4 and 5. In Fig. 4, without any noise in the system output, when 4.1 Observer design the model is perfectly known the rate of convergence is fast, on the other hand, when the model is partially Consider that system (1) satisfies the AOC. In this case known the observer does not reconstruct the state to estimate the state-space vector x, we can suggest a variables. In Fig. 5, we studied the effect of noise nonlinear high gain observer (Gauthier et al., 1992), in the measurement (white noise with σ = 0.1, ±5% (Mart´ınez-Guerra et al., 2000) with the following around the current value of the measured output), we structure, can see that the high gain observer is very sensitive x˙ˆ = f ( xˆ, u) + K (y − C xˆ) , xˆ ∈ Rn , xˆ0 = xˆ (t0 ) to the noise in the system output. Fig. 6 shows thethe performance index. index. It should noted be that noted this 1 system output. Fig.(17) 6 shows performance It be should where the observer’s 2gain matrix is given by, observer only reconstruct the state variables when the reconstruct the state variables when the model is well known. ! model is well known. 1 K = S θ−1C T , S3θ = i+ j−1 S i j θ i, j=1,...,n 60

and the positive parameter θ determines the desired convergence velocity. Moreover, S θ > 0, S θ = S θT should be a solution of the algebraic equation, Sθ

θ θ E + I + E T + I S θ = C T C, 2 2

E=

0 0

In−1,n−1 0

!

Su b s t r a t e

S real S (perfect knowdledge of the growth rate) S (partial knowdledge of the growth rate)

50 40 30 20 10

S˙ˆ = D S in − Sˆ −

0 0

20

30 time

50

60

50 40 30 20

X real X (perfect knowdledge of the growth rate) X (partial knowdledge of the growth rate)

10 0 0

10

20

30 time

Fig. 4. StateFig. variables. 4. State

1 7 ˆ2 −µmax Sˆ + YS /X δ + Sˆ + Sϕ kd  .  ˆ δ − S2 ϕ  µ X  max  . + 2θ     δ + Sˆ + Sˆ 2 ϕ . ) + θ2 YS /X δ + Sˆ + Sˆ 2 ϕ Sˆ − y

40

60

Xˆ µmax Sˆ ˆ . +kd X−2θ Sˆ − y δ + Sˆ + Sˆ 2 ϕ YS /X

5 µmax Sˆ ˆ . Xˆ − kd X− X˙ˆ = −DXˆ + δ6+ Sˆ + Sˆ 2 ϕ

10

70

B i o ma s s

As shown by (Gauthier et al., 1992), (Mart´ınez Guerra and de Leon-Morales, 1996), under certain technical 4 a assumptions (Lipschitz conditions for nonlinear functions under consideration) this nonlinear observer has an arbitrary exponential decay for any initial conditions. We obtain the following high order observer for the system (2) applying the observation scheme (17),

40

50

60

Variables

−

4.2

Conclusions 60 50


Su b s t r a t e

In this paper we have presented a bounded error 40 estimator for bioprocess with unstructured growth models. We have proven the stability of the 30 corresponding estimation error in a Lyapunov sense. 20 By means of a linear change of variable given in a 10 manner and with some algebraic manipulations natural have 0 been constructed the state estimator, which 0 10 20 30 40 50 60 converges to the current states time of the reference model given. We have demonstrated that the bounded error 80 estimator under consideration provides good enough 70

Simulations

In the same way, we 8show two simulations: when the model is well known and when the model is partially known. The initial conditions for the observer are

60

www.rmiq.org B i o ma s s

318

50 40 30 20 10 0 0

X real X (perfect knowdledge of the growth rate) X (partial knowdledge of the growth rate) 10

20

30

40

50

60

that this obse

20

X real X (perfect knowdledge of the growth rate) X (partial knowdledge of the growth rate)

10 0 0

10

20

30 time

40

50

60

J. L.Fig. Mata-Machuca et al./ Revista Mexicana de Ingenier´ıa Qu´ımica Vol. 10, No. 2 (2011) 313-320 4. State Variables

60

Su b s t r a t e

50


40 30 20 10 0 0

10

20

30 time

40

50

60

state-space estimates which were bounded on average, besides the proposed state estimator does not depend of a particular set of initial conditions or specific model structure. Moreover, we have constructed a high gain observer in which the convergence is fast only if the model is well known, but does not exists convergence if the model is partially known. Finally, we have presented some simulations to illustrate the effectiveness of the suggested approach, which shows some robustness properties against noisy measurements.

80

References

70

B i o ma s s

60 50 40 30 20 10 0 0

X real X (perfect knowdledge of the growth rate) X (partial knowdledge of the growth rate) 10

20

30 time

40

50

60

Aguilar-López, R., Mart´ınez-Guerra, R., MendozaCamargo, J., and M. Neria-González (2006). Monitoring of and industrial wastewater plant employing finite-time convergence observer. Journal of Chemical Technology and Biotechnology 81, 851-857. Alvarez-Ramirez, J (1999). Robust PI stabilization of a class of continuously stirred-tank reactors. AIChE Journal 45, 1992-2000.

5. Variables State variables noise in the systemoutput). Fig. 5. Fig. State (with(with noise in the system output). 1500 J (perfect knowdledge of the growth rate) J (partial knowdledge of the growth rate)

Bastin, G. and D. Dochain(1990). On-line estimation and adaptative control of bioreactors 1. Elsevier, Amsterdam.

1500 J (perfect knowdledge of the growth rate) J (partial knowdledge of the growth rate)

J

J

1000

1000

500

500

0 0

10

20

0 0

10

20

30 time

(a)

30 time

40 40

50 50

60 60

(a) a) 1500 J (perfect knowdledge of the growth rate) J (partial knowdledge of the growth rate)

J

1000

J

500

J (perfect knowdledge of the growth rate) J (partial knowdledge of the growth rate)

500

0 0

10

20

30 time

40

0 0

10

20

30

40

50 50

Diop, S. and R. Mart´ınez-Guerra (2001). An algebraic and data derivative information approach to nonlinear system diagnosis. Proceedings of the European Control Conference (ECC), Porto, Portugal, 2334-2339. Farza, M., Busawon, K. and H. Hammouri (1998). Simple nonlinear observers for online estimation of kinetics rates in bioreactors. Automatica 34, 301-318.

1500

1000

Dávila, J., Fridman, L. and A. Levant (2005). Second order sliding-mode observer for mechanical systems. IEEE Transactions on Automatic Control 50, 1785-1789.

60 60

Gauthier, J., Hammouri H. and S. Othman (1992). A simple observer for nonlinear systems. Applications to bioreactors. IEEE Transactions on Automatic Control 37, 875-880. Gouzé, J. and V. Lemesle (2001).

A bounded

b)time (b) error observer for a class of bioreactor Fig. 6. Quadratic estimation error. (a) Without any models. Proceedings of the European Control (b) adratic estimation error. (a) Without any noise, (b) with white noise; in the system noise, (b) with white noise; in the system output.

Conference (ECC), Porto, Portugal.

output. adratic estimation error. (a) Without any noise, (b) with white noise; in the system www.rmiq.org output.

5. CONCLUSIONS. 5. CONCLUSIONS. we have presented a bounded error estimator for bioprocess with unstructured ls.we Wehave havepresented proven thea stability the corresponding a Lyapunov boundedoferror estimator for estimation bioprocesserror withinunstructured

319

J. L. Mata-Machuca et al./ Revista Mexicana de Ingenier´ıa Qu´ımica Vol. 10, No. 2 (2011) 313-320 Hadj-Sadok, Z. (1999). Modélisation et estimation dans les bioréacteurs; prise en compte des incertudes: application au traitement de l’eau. PhD thesis. Nice-Sophia Antipolis University. Nice. Hu, S. and J. Wang (2002). Global asymptotic stability and global exponential stability of continuous time recurrent neural netwoks. IEEE Transactions on Automatic Control 47, 802-807. Keller, H (1987). Non-linear observer design by transformation into a generalized observer canonical form. International Journal of Control 46, 1915-1930. Khalil, H (2002). Nonlinear systems. Third edition. Prentice Hall, New Jersey. Lemesle, V. and J. Gouzé (2005). Hybrid bounded error observers for uncertain bioreactor models. Bioprocess & Biosystems Engineering 27, 311318. Levant, A (2001). Universal single-input-singleoutput (SISO) sliding-mode controllers with finite-time convergence. IEEE Transactions on Automatic Control 46, 1447-1451.

320

Luenberger, D (1979). Introduction to Dynamic Systems. Theory Models and Applications. Wiley, New York. Mart´ınez-Guerra, R. and J. de Leon-Morales (1996). Nonlinear estimators: a differential algebraic approach. Journal of Mathematics and Computer Modelling 20, 125-132. Mart´ınez-Guerra, R., Poznyak, A. and V. D´ıaz (2000). Robustness of high-gain observers for closed-loop nonlinear systems: theoretical study and robotics control application. International Journal of Systems Science 31, 1519-1529. Masoud, S (1997). Nonlinear state-observer design with application to reactors. Chemical Engineering Science 52, 387-404. Vargas, A., Soto, G., Moreno, J. and G. Buitrón (2000), Observer based time-optimal control of an aerobic SBR for chemical and petrochemical wastewater treatment. Water Science and Technology 42, 163-170.

www.rmiq.org