Embed Size (px)
Citation preview
Revista Mexicana de Ingeniería Química
ISSN: 1665-2738
Universidad Autónoma Metropolitana Unidad
Iztapalapa
México
Mata-Machuca, J. L.; Martínez-Guerra, R.; Aguilar-López, R.
DIFFERENTIAL ALGEBRAIC ESTIMATOR FOR THE MONITORING OF A CLASS OF PARTIALLY
KNOWN BIOREACTOR MODELS
Revista Mexicana de Ingeniería Química, vol. 10, núm. 2, agosto, 2011, pp. 313-320
Universidad Autónoma Metropolitana Unidad Iztapalapa
Distrito Federal, México
Available in: http://www.redalyc.org/articulo.oa?id=62020825014
How to cite
Complete issue
More information about this article
Journal's homepage in redalyc.org
Scientific Information System
Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal
Non-profit academic project, developed under the open access initiative
Revista Mexicana de Ingeniería Química
CONTENIDO
Volumen 8, número 3, 2009 / Volume 8, number 3, 2009
213 Derivation and application of the Stefan-Maxwell equations
(Desarrollo y aplicación de las ecuaciones de Stefan-Maxwell)
Stephen Whitaker
Biotecnología / Biotechnology
245 Modelado de la biodegradación en biorreactores de lodos de hidrocarburos totales del petróleo
intemperizados en suelos y sedimentos
(Biodegradation modeling of sludge bioreactors of total petroleum hydrocarbons weathering in soil
and sediments)
S.A. Medina-Moreno, S. Huerta-Ochoa, C.A. Lucho-Constantino, L. Aguilera-Vázquez, A. Jiménez-
González y M. Gutiérrez-Rojas
259 Crecimiento, sobrevivencia y adaptación de Bifidobacterium infantis a condiciones ácidas
(Growth, survival and adaptation of Bifidobacterium infantis to acidic conditions)
L. Mayorga-Reyes, P. Bustamante-Camilo, A. Gutiérrez-Nava, E. Barranco-Florido y A. Azaola-
Espinosa
265 Statistical approach to optimization of ethanol fermentation by Saccharomyces cerevisiae in the
presence of Valfor® zeolite NaA
(Optimización estadística de la fermentación etanólica de Saccharomyces cerevisiae en presencia de
zeolita Valfor® zeolite NaA)
G. Inei-Shizukawa, H. A. Velasco-Bedrán, G. F. Gutiérrez-López and H. Hernández-Sánchez
Ingeniería de procesos / Process engineering
271 Localización de una planta industrial: Revisión crítica y adecuación de los criterios empleados en
esta decisión
(Plant site selection: Critical review and adequation criteria used in this decision)
J.R. Medina, R.L. Romero y G.A. Pérez
Revista Mexicanade Ingenierıa Quımica
1
Academia Mexicana de Investigacion y Docencia en Ingenierıa Quımica, A.C.
Volumen 10, Numero 2, Agosto 2011
ISSN 1665-2738
1
Vol. 10, No. 2 (2011) 313-320
DIFFERENTIAL ALGEBRAIC ESTIMATOR FOR THE MONITORING OF A CLASSOF PARTIALLY KNOWN BIOREACTOR MODELS
ESTIMADOR ALGEBRAICO DIFERENCIAL PARA EL MONITOREO DE UNACLASE DE BIORREACTORES CON MODELOS PARCIALMENTE CONOCIDOS
J. L. Mata-Machuca1, R. Martınez-Guerra1 and R. Aguilar-Lopez2∗
1Departamento de Control Automatico-CINVESTAV IPN.2Departamento de Biotecnologıa y Bioingenierıa CINVESTAV-IPN. Av. Instituto Politecnico Nacional No. 2508,
San Pedro Zacatenco, Mexico, D.F. C.P. 07360
Received 10 of March 2010; Accepted 13 of May 2011
AbstractThe problem of monitoring in a common class of partially known bioreactor models is addressed. A reduced
order observer namely differential algebraic estimator is proposed. The biomass is estimated by means of substrateconcentration measurements. The estimation methodology is based on a suitable change of variable whichallows generating artificial variables to infer the remaining mass concentrations constructing a differential-algebraicstructure. The proposed methodology is applied to a class of Haldane unstructured kinetic model with success.Stability analysis in a Lyapunov sense for the estimation error is performed. Some remarks about the convergencecharacteristics of the proposed estimator are given and numerical simulations show its satisfactory performance.Finally, for comparison purposes, a high gain observer is presented: the convergence is possible only when themodel is perfectly known.
Keywords: differential-algebraic estimator, state variable estimation, continuous bioreactor, Haldane kinetics.
ResumenEn el presente trabajo se considera el problema del monitoreo de una clase de biorreactores con modelos
parcialmente conocidos. Se propone un tipo de observador de orden reducido denominado estimador diferencialalgebraico. La metodologıa de estimacion se basa en un cambio de variables que permite generar variablesartificiales para inferir las concentraciones no medibles. La metodologıa propuesta es aplicada a un modelo cineticono estructurado de Haldane con exito. Se efectua un analisis de Lyapunov para demostrar la estabilidad de lametodologıa considerada. Algunos comentarios sobre las caracterısticas de la convergencia del estimador sonproporcionados y simulaciones numericas muestran un desempeno satisfactorio. Finalmente, con propositos decomparacion, un observador de alta ganancia se presenta en donde su convergencia se garantiza solo cuando seconoce perfectamente el modelo del sistema bajo estudio.
Palabras clave: estimator algebraico-diferencial, estimacion de variables de estado, biorreactor continuo, cineticade Haldane.
∗Corresponding author. E-mail: [email protected]. + 52 55 5747 3800
Publicado por la Academia Mexicana de Investigacion y Docencia en Ingenierıa Quımica A.C. 313
J. L. Mata-Machuca et al./ Revista Mexicana de Ingenierıa Quımica Vol. 10, No. 2 (2011) 313-320
1 Introduction
Operating a bioreactor is not a simple task, asduring a bioreacting process, variables such asconcentrations are generally determined by off-linelaboratory analysis, making this set of variablesof limited use for control purposes and on-linemonitoring. However, these variables can be on-lineestimated using soft sensors.
Over the last few years, the importance ofon-line monitoring of biotechnological processeshas increased. A first step to efficient bioreactoroperation is the adequate implementation of onlinemeasurements of essential variables such as substrateand biomass concentrations. Advantages ofcontinuous monitoring of key variables includegaining knowledge about the state of the process andthe possibility of detecting and isolating abnormalprocess developments at early stages. This reducesprocess costs, contributes to process safety and helpsin trouble-shooting and process accommodation. Themain problem in fermentation monitoring and controlis the fact that process variables usually cannot bemeasured on-line. Monitoring and controlling theseprocesses can therefore be difficult because onlyindirect measurements are available online, whilecalculated values may be rather uncertain. This canbe due to uncertainty with respect to the equationsused, measurement errors or both. For automaticcontrol this may have serious consequences, especiallyas the actual variables of interest often cannot bedirectly controlled and related variables are controlledinstead. In fermentation processes, on-line and off-line measurements are the main source of informationabout the state of the process. In combination withmodel-based calculations, they are used to produceestimations for monitoring purposes as well as forautomatic and manual process control (Bastin andDochain, 1990), (Masoud, 1997).
Observation schemes are widely used forreconstructing states of dynamical systems (Aguilar-Lopez et al., 2006). Most of the contributionsare related to asymptotic observers for monitoring,fault detections and control issues whereas thereal necessities of industrial plants are related toa fast response of the monitoring and regulationmethodologies.
Special attention was given to filtering techniques,namely extended Kalman filter, adaptive observers,and artificial neural networks (ANN), (Davila andFridman, 2005), (Hu and Wang, 2002), (Levant,2001), however for these techniques the right tuning
of the estimators gains is difficult. It is shown thatsoftware based state estimation is a powerful techniquethat can be successfully used to enhance automaticcontrol performance of biological systems as well asin system monitoring and on-line optimization.
In this paper we consider the growth rate partiallyknown. Following this idea, the necessity to adaptan observation scheme to the available knowledgeof the growth rate immediately arises. The maincontribution in this work is to show a state estimatorwhich is a simplified version of the methodologygiven by (Lemesle and Gouze, 2005) where asimple linear change of variable given in a naturalmanner allows to develop a differential-algebraic stateestimator. Results show an adequate performanceof the considered methodology. The technique isnot the same as (Alvarez-Ramirez et al., 1999)since we do not have derivators. The proposedestimation methodology is applied to a kind ofunstructured kinetic model: the Haldane model, whichis considered for biological process with substrateinhibition. The above mentioned kinetic model isapplied to a class of continuous stirred bioreactors.
In what follows, the statement of the problemis presented; an observability condition is given inthe differential-algebraic setting. In section 3, thebounded error estimator is designed. Section 4 shows ahigh gain observer as a comparison with the proposedmethodology. Finally, we give some concludingremarks.
2 Problem statement
2.1 The model
Consider the following nonlinear system
x = f (x, u)
y = h (x)(1)
where, x ∈ Rn, u ∈ Rm, m ≤ n, y ∈ Rp.Let us recall the classical observer definition. An
observer for system (1) is a dynamical system ˙x =
f (x, u, y), whose task is state estimation. Usuallyis required at least that ‖x − x‖ → 0 as t → ∞.Although in some cases, exponentially convergence isalso required (Gauthier et al., 1992).Definition 1: an estimator is said to be bounded if theestimation error (‖x − x‖) belongs to an open ball withradius proportional to some value that depends on itsestimation error.
314 www.rmiq.org
J. L. Mata-Machuca et al./ Revista Mexicana de Ingenierıa Quımica Vol. 10, No. 2 (2011) 313-320
In all paper, we will consider a class of bioreactormodel. The simplified Haldane model taken from(Vargas et al., 2000), is described by
dSdt
= D (S in − S ) − µ (S )X
YS /X+ kdX (2a)
dXdt
= −DX + µ (S ) X − kdX (2b)
where µ (S ) = µmaxS/(δ + S + S 2
/ϕ)
is the specificgrowth rate and µmax is the maximum growth rate.
We assume that µ(S ) is partially known, whichis common in biology (Gouze and Lemesle, 2001).Generally, µ(S ) is between two bounds meaning thatwe know a function µ(S ) such that |µ (S ) − µ (S )| < a,where a ∈ R+, and µ (0) = µ(0) = 0. We introducean important lemma about lower bounded propertiesof µ(S ).Lema 1 (Hadj-Sadok, 1999): there exists a constantε ∈ 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 thatµ (S (t)) > µ (ε) = r, where r ∈ R+.
The state variables S , X are the substrate andbiomass concentrations, respectively, D = q/V is thedilution rate with V the volume of the bioreactor andq the constant flow passing through the bioreactor,S in is the input substrate concentration, YS/X is thecorresponding yield coefficient. Let us notice that theinputs D = u and S in are fixed. Moreover, we assumethat the measured output is,
y = S (3)
2.2 Algebraic Observability Condition(AOC)
Before proposing the bounded error estimator, adefinition concerning on algebraic observabilitycondition is given, for more details see (Diop andMartınez-Guerra, 2001).Definition 2: consider the system described by (1),where x =
(x1 x2 . . . xn
)T. A state xi, i =
{1, 2, · · · , n}, is said to be algebraically observablewith respect to {u, y} if it satisfies a differentialpolynomial in terms of u, y and some of their timederivatives, 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) +
(µmaxϕ y
δϕ + ϕ y + y2
1YS /X
− kd
)X = 0 (4)
From Eq. (4), it is clear that the state variable Xsatisfies the AOC thus, X is algebraically observable.
3 Bounded error estimator
3.1 Estimator design
In what follows, the corresponding estimatedconcentration is denoted by , and we assume thatS is measured exactly, i.e., S = S . Then, we onlyreconstruct the biomass variable X.
Consider the Haldane’s model given by system (2),and make the change of variable
z = X + k S (5)
where k ∈ R is fixed.The dynamics of z is
z = −
[D + kd − k kd +
(k
YS /X− 1
)µ (S )
]z+
+ (1 − k) k kdS +
(k
YS /X− 1
)k µ (S ) S + k D S in
(6)
Proposition 1: if we choose the estimator’s gain suchthat YS/X < k 6 1 + D/kd and |µ (S ) − µ (S )| < a,a ∈ R+. Then, the system (7) is a bounded errorestimator of (6).
˙z = −
[D + kd − k kd +
(k
YS /X− 1
)µ (S )
]z+
+ (1 − k) k kdS +
(k
YS /X− 1
)k µ (S ) S + k D S in
(7)
For the proof, define the estimation error,
e = z − z (8)
Then, using eqs. (6) and (7) the estimation errordynamic is obtained as
e = −
[D + kd − k kd +
(k
YS /X− 1
)µ (S )
]e+
+
(k
YS /X− 1
) [µ (S ) − µ (S )
]k S −
(k
YS /X− 1
) [µ (S ) − µ (S )
]z
(9)To analyze the stability of Eq. (9) we consider thefollowing Lyapunov function candidate
V =12
e2 (10)
www.rmiq.org 315
J. L. Mata-Machuca et al./ Revista Mexicana de Ingenierıa Quımica Vol. 10, No. 2 (2011) 313-320
The time derivative of Eq. (10) is
V = e e (11)
Replacing (9) into (11) yields
V = −
[D + kd − k kd +
(k
YS /X− 1
)µ (S )
]e2+
+
(k
YS /X− 1
) [µ (S ) − µ (S )
]k S e −
(k
YS /X− 1
)[µ (S ) − µ (S )
]z e
(12)Equation (12) is written alternatively as
V = −
[D + kd − k kd +
(k
YS /X− 1
)µ (S )
]e2−
−
(k
YS /X− 1
) [µ (S ) − µ (S )
]X e
(13)
Now, from lemma 1 and taking into account thatYS/X < k 6 1 + D/kd, |µ (S ) − µ (S )| < a, and X isbounded, Eq. (13) leads to,
V 6 −[D + kd − kkd +
(k
YS /X− 1
)r]
e2
+
(k
YS /X− 1
)aXmax |e| = −λ e2 + w | e |
where,
λ = D + kd − kkd +
(k
YS /X− 1
)r
and
w =
(k
YS /X− 1
)aXmax
The right-hand side of the foregoing inequality is notnegative since near the origin, the positive linear termw | 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 onthe 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 Vwill decrease monotonically until the solutions entersthe set {V 6 ε}. From that time on, the solutioncannot leave the set {V 6 ε} since V is negative onthe boundary V = ε. According to (Khalil, 2002),the solution is uniformly ultimately bounded with the
ultimate bound | e | 6√
2ε. For instance, defining cand ε as follows
c =
(k
YS /X
aXmax
λ
)2
, ε =
(k
aXmax
λ
)2
the ultimate bound is, | e | 6√
2 k aXmaxλ
.Corollary 1: if the growth rate is perfectly known, i.e.,µ (S ) = µ (S ), and we choose the estimator’s gain suchthat YS/X < k 6 1 + D/kd. Then, the system (14) is anasymptotic estimator of (6).
˙z = −
[D + kd − k kd +
(k
YS /X− 1
)µ (S )
]z+
+ (1 − k) k kdS +
(k
YS /X− 1
)k µ (S ) S + k D S in
(14)Indeed, the dynamics of the error in this case is
e = −
[D + kd − k kd +
(k
YS /X− 1
)µ (S )
]e
and the corresponding time derivative of Lyapunovfunction candidate (10) is
V = −
[D + kd − k kd +
(k
YS /X− 1
)µ (S )
]e2 < 0
Moreover, X can be reconstructed considering
X = z − k S (15)
3.2 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 conditionsS (0) = 60, X(0) = 40, X (0) = 30, z (0) = 90, withappropriate units, these values are taken from Vargaset al., (2000). The estimator’s gain is k = 1. Thegrowth rates are chosen as
µ (S ) =S
140 + S + S 2/81.25
and
µ (S ) =0.8 S
140 + S + S 2/81.25
when the model is well known for the asymptoticestimator and when the model is partially knownfor the bounded error estimator, respectively. Thesimulations results were carried out with the help ofMatlab 7.1 Software with Simulink 6.3 as the toolbox.
316 www.rmiq.org
J. L. Mata-Machuca et al./ Revista Mexicana de Ingenierıa Quımica Vol. 10, No. 2 (2011) 313-320
The performance index of the correspondingestimation process is calculated as (Martınez-Guerra,et al., 2000)
J =1
t + 0.001
t∫0
‖e (τ)‖2dτ (16)
where e(t) is the corresponding state estimation error(the difference between the actual observed signal andits estimate).
First, in Fig. 1 we show the simulation resultsfor the bounded error estimator given by proposition1, and the corresponding results for the asymptoticestimator given by corollary 1 (without any noise inthe system output). Furthermore, in Fig. 2 is shownthe effect of noise in the estimation process. A whitenoise is added in the measurement (σ = 0.1, ±10%around the current value of the measured output)this is considering the corresponding measurementerror of the corresponding sensor and/or experimentalmeasurement technique. We can observe that thebounded error estimator is robust against noisymeasurement. Finally, in Fig. 3 is illustrated theperformance index given by (16) for the correspondingestimation process. It should be noted that thequadratic estimation error (performance index) isbounded on average and has a tendency to decrease.
The performance index of the corresponding estimation process is calculated as (Martínez-1
Guerra, {\it et al.}, 2000) 2
3
t
det
J0
2
001.0
1 (16) 4
where te is the corresponding state estimation error (the difference between the actual 5
observed signal and its estimate). 6
First, in Fig. 1 we show the simulation results for the bounded error estimator given by 7
proposition 1, and the corresponding results for the asymptotic estimator given by corollary 1 8
(without any noise in the system output). Furthermore, in Fig. 2 is shown the effect of noise in 9
the estimation process. A white noise is added in the measurement ( 1.0 , %10 around the 10
current value of the measured output) this is considering the corresponding measurement error of 11
the corresponding sensor and/or experimental measurement technique. We can observe that the 12
bounded error estimator is robust against noisy measurement. Finally, in Fig. 3 is illustrated the 13
performance index given by (16) for the corresponding estimation process. It should be noted that 14
the quadratic estimation error (performance index) is bounded on average and has a tendency to 15
decrease. 16
17
0 10 20 30 40 50 600
10
20
30
40
50
60
time
S u
b s
t r a t
e
18
0 10 20 30 40 50 600
10
20
30
40
50
60
time
B i o m
a s
s
X realasymptotic estimatorbounded error estimator
19
Figure 1. State Variables. 20
21
Fig. 1. State Variables.
0 10 20 30 40 50 600
10
20
30
40
50
60
time
S u
b s
t r a t
e
0 10 20 30 40 50 60
0
10
20
30
40
50
60
time
B i o m
a s
s
X realasymptotic estimatorbounded error estimator
1
Figure 2. State Variables (with noise in the system output). 2
3
4
0 10 20 30 40 50 600
10
20
30
40
50
60
70
time
J
asymptotic estimatorbounded error estimator
5
(a) 6
0 10 20 30 40 50 600
10
20
30
40
50
60
70
time
J
asymptotic estimatorbounded error estimator
7
(b) 8
Figure 3. Quadratic estimation error. (a) Without any noise, (b) with white noise; in the system 9
output. 10
11
4. A NOTE ON FULL-ORDER OBSERVERS: THE HIGH GAIN OBSERVER 12
A. Observer design 13
Consider that system (1) satisfies the AOC. In this case to estimate the state-space vector $x$, 14
we can suggest a nonlinear high gain observer (Gauthier {\it et al.}, 1992), (Martínez-Guerra {\it 15
et al.}, 2000) with the following structure, 16
17
xCyKuxfx ˆ,ˆˆ (17) 18
0 10 20 30 40 50 600
10
20
30
40
50
60
time
S u
b s
t r a t
e
0 10 20 30 40 50 60
0
10
20
30
40
50
60
timeB i o m
a s
s
X realasymptotic estimatorbounded error estimator
1
Figure 2. State Variables (with noise in the system output). 2
3
4
0 10 20 30 40 50 600
10
20
30
40
50
60
70
time
J
asymptotic estimatorbounded error estimator
5
(a) 6
0 10 20 30 40 50 600
10
20
30
40
50
60
70
time
J
asymptotic estimatorbounded error estimator
7
(b) 8
Figure 3. Quadratic estimation error. (a) Without any noise, (b) with white noise; in the system 9
output. 10
11
4. A NOTE ON FULL-ORDER OBSERVERS: THE HIGH GAIN OBSERVER 12
A. Observer design 13
Consider that system (1) satisfies the AOC. In this case to estimate the state-space vector $x$, 14
we can suggest a nonlinear high gain observer (Gauthier {\it et al.}, 1992), (Martínez-Guerra {\it 15
et al.}, 2000) with the following structure, 16
17
xCyKuxfx ˆ,ˆˆ (17) 18
Fig. 2. State Variables (with noise in the systemoutput).
0 10 20 30 40 50 600
10
20
30
40
50
60
time
S u
b s
t r a t
e
0 10 20 30 40 50 60
0
10
20
30
40
50
60
time
B i o m
a s
s
X realasymptotic estimatorbounded error estimator
1
Figure 2. State Variables (with noise in the system output). 2
3
4
0 10 20 30 40 50 600
10
20
30
40
50
60
70
time
J
asymptotic estimatorbounded error estimator
5
(a) 6
0 10 20 30 40 50 600
10
20
30
40
50
60
70
time
J
asymptotic estimatorbounded error estimator
7
(b) 8
Figure 3. Quadratic estimation error. (a) Without any noise, (b) with white noise; in the system 9
output. 10
11
4. A NOTE ON FULL-ORDER OBSERVERS: THE HIGH GAIN OBSERVER 12
A. Observer design 13
Consider that system (1) satisfies the AOC. In this case to estimate the state-space vector $x$, 14
we can suggest a nonlinear high gain observer (Gauthier {\it et al.}, 1992), (Martínez-Guerra {\it 15
et al.}, 2000) with the following structure, 16
17
xCyKuxfx ˆ,ˆˆ (17) 18
a)
0 10 20 30 40 50 600
10
20
30
40
50
60
time
S u
b s
t r a t
e
0 10 20 30 40 50 60
0
10
20
30
40
50
60
time
B i o m
a s
s
X realasymptotic estimatorbounded error estimator
1
Figure 2. State Variables (with noise in the system output). 2
3
4
0 10 20 30 40 50 600
10
20
30
40
50
60
70
time
J
asymptotic estimatorbounded error estimator
5
(a) 6
0 10 20 30 40 50 600
10
20
30
40
50
60
70
time
J
asymptotic estimatorbounded error estimator
7
(b) 8
Figure 3. Quadratic estimation error. (a) Without any noise, (b) with white noise; in the system 9
output. 10
11
4. A NOTE ON FULL-ORDER OBSERVERS: THE HIGH GAIN OBSERVER 12
A. Observer design 13
Consider that system (1) satisfies the AOC. In this case to estimate the state-space vector $x$, 14
we can suggest a nonlinear high gain observer (Gauthier {\it et al.}, 1992), (Martínez-Guerra {\it 15
et al.}, 2000) with the following structure, 16
17
xCyKuxfx ˆ,ˆˆ (17) 18
b)
Fig. 3. Quadratic estimation error. (a) Without anynoise, (b) with white noise; in the system output.
www.rmiq.org 317
J. L. Mata-Machuca et al./ Revista Mexicana de Ingenierıa Quımica Vol. 10, No. 2 (2011) 313-320
4 A note on full-order observers:The high gain observer
4.1 Observer design
Consider that system (1) satisfies the AOC. In this caseto estimate the state-space vector x, we can suggest anonlinear high gain observer (Gauthier et al., 1992),(Martınez-Guerra et al., 2000) with the followingstructure,
˙x = f (x, u) + K (y −Cx) , x ∈ Rn , x0 = x (t0)(17)
where the observer’s gain matrix is given by,
K = S −1θ CT , S θ =
(1
θi+ j−1 S i j
)i, j=1,...,n
and the positive parameter θ determines the desiredconvergence velocity. Moreover, S θ > 0, S θ = S T
θ
should be a solution of the algebraic equation,
S θ
(E +
θ
2I)+
(ET +
θ
2I)
S θ = CT C, E =
(0 In−1,n−10 0
)As shown by (Gauthier et al., 1992), (Martınez Guerraand de Leon-Morales, 1996), under certain technicala assumptions (Lipschitz conditions for nonlinearfunctions under consideration) this nonlinear observerhas an arbitrary exponential decay for any initialconditions. We obtain the following high orderobserver for the system (2) applying the observationscheme (17),
˙S = D(S in − S
)−
µmaxS
δ + S + S 2/ϕ
XYS/X
+kd X−2θ(S − y
)˙X = −DX +
µmaxS
δ + S + S 2/ϕ
X − kd X−
−1
−µmaxS + YS /X
(δ + S + S 2
ϕ
)kd2θ
µmaxX(δ − S 2
/ϕ)(
δ + S + S 2/ϕ) +
+ θ2YS/X
(δ + S + S 2
/ϕ) } (
S − y)
4.2 Simulations
In the same way, we show two simulations: when themodel is well known and when the model is partiallyknown. The initial conditions for the observer are
S (0) = 40, X (0) = 30, with appropriate units. Theestimator’s gain is θ = 2. The simulations results ofhigh gain observer are presented in figs. 4 and 5. InFig. 4, without any noise in the system output, whenthe model is perfectly known the rate of convergenceis fast, on the other hand, when the model is partiallyknown the observer does not reconstruct the statevariables. In Fig. 5, we studied the effect of noisein the measurement (white noise with σ = 0.1, ±5%around the current value of the measured output), wecan see that the high gain observer is very sensitiveto the noise in the system output. Fig. 6 showsthe performance index. It should be noted that thisobserver only reconstruct the state variables when themodel is well known.
system output. Fig. 6 shows the performance index. It should be noted that this observer only 1
reconstruct the state variables when the model is well known. 2
3
0 10 20 30 40 50 600
10
20
30
40
50
60
time
S u
b s
t r a t
e
S realS (perfect knowdledge of the growth rate)S (partial knowdledge of the growth rate)
4
0 10 20 30 40 50 600
10
20
30
40
50
60
70
time
B i o m
a s
s
X realX (perfect knowdledge of the growth rate)X (partial knowdledge of the growth rate)
5
Fig. 4. State Variables 6
7
0 10 20 30 40 50 600
10
20
30
40
50
60
time
S u
b s
t r a t
e
S realS (perfect knowdledge of the growth rate)S (partial knowdledge of the growth rate)
8
0 10 20 30 40 50 600
10
20
30
40
50
60
70
80
time
B i o m
a s
s
X realX (perfect knowdledge of the growth rate)X (partial knowdledge of the growth rate)
9
Fig. 5. State Variables (with noise in the system output). 10
11
12
Fig. 4. State variables.
Conclusions
In this paper we have presented a bounded errorestimator for bioprocess with unstructured growthmodels. We have proven the stability of thecorresponding estimation error in a Lyapunov sense.By means of a linear change of variable given in anatural manner and with some algebraic manipulationshave been constructed the state estimator, whichconverges to the current states of the reference modelgiven. We have demonstrated that the bounded errorestimator under consideration provides good enough
318 www.rmiq.org
J. L. Mata-Machuca et al./ Revista Mexicana de Ingenierıa Quımica Vol. 10, No. 2 (2011) 313-320
system output. Fig. 6 shows the performance index. It should be noted that this observer only 1
reconstruct the state variables when the model is well known. 2
3
0 10 20 30 40 50 600
10
20
30
40
50
60
time
S u
b s
t r a t
e
S realS (perfect knowdledge of the growth rate)S (partial knowdledge of the growth rate)
4
0 10 20 30 40 50 600
10
20
30
40
50
60
70
time
B i o m
a s
s
X realX (perfect knowdledge of the growth rate)X (partial knowdledge of the growth rate)
5
Fig. 4. State Variables 6
7
0 10 20 30 40 50 600
10
20
30
40
50
60
time
S u
b s
t r a t
e
S realS (perfect knowdledge of the growth rate)S (partial knowdledge of the growth rate)
8
0 10 20 30 40 50 600
10
20
30
40
50
60
70
80
time
B i o m
a s
s
X realX (perfect knowdledge of the growth rate)X (partial knowdledge of the growth rate)
9
Fig. 5. State Variables (with noise in the system output). 10
11
12
Fig. 5. State variables (with noise in the systemoutput).
1
2
3
4
5
6
7
8
0 10 20 30 40 50 600
500
1000
1500
time
J
J (perfect knowdledge of the growth rate)J (partial knowdledge of the growth rate)
9
(a) 10
11
0 10 20 30 40 50 600
500
1000
1500
time
J
J (perfect knowdledge of the growth rate)J (partial knowdledge of the growth rate)
12
(b) 13
Fig. 6. Quadratic estimation error. (a) Without any noise, (b) with white noise; in the system 14
output. 15
16
5. CONCLUSIONS. 17
In this paper we have presented a bounded error estimator for bioprocess with unstructured 18
growth models. We have proven the stability of the corresponding estimation error in a Lyapunov 19
sense. By means of a linear change of variable given in a natural manner and with some algebraic 20
manipulations have been constructed the state estimator, which converges to the current states of 21
the reference model given. We have demonstrated that the bounded error estimator under 22
consideration provides good enough state-space estimates which were bounded on average, 23
besides the proposed state estimator do not depend of a particular set of initial conditions or 24
a)
1
2
3
4
5
6
7
8
0 10 20 30 40 50 600
500
1000
1500
time
J
J (perfect knowdledge of the growth rate)J (partial knowdledge of the growth rate)
9
(a) 10
11
0 10 20 30 40 50 600
500
1000
1500
time
J
J (perfect knowdledge of the growth rate)J (partial knowdledge of the growth rate)
12
(b) 13
Fig. 6. Quadratic estimation error. (a) Without any noise, (b) with white noise; in the system 14
output. 15
16
5. CONCLUSIONS. 17
In this paper we have presented a bounded error estimator for bioprocess with unstructured 18
growth models. We have proven the stability of the corresponding estimation error in a Lyapunov 19
sense. By means of a linear change of variable given in a natural manner and with some algebraic 20
manipulations have been constructed the state estimator, which converges to the current states of 21
the reference model given. We have demonstrated that the bounded error estimator under 22
consideration provides good enough state-space estimates which were bounded on average, 23
besides the proposed state estimator do not depend of a particular set of initial conditions or 24
b)
Fig. 6. Quadratic estimation error. (a) Without anynoise, (b) with white noise; in the system output.
state-space estimates which were bounded on average,besides the proposed state estimator does not dependof a particular set of initial conditions or specificmodel structure. Moreover, we have constructeda high gain observer in which the convergence isfast only if the model is well known, but does notexists convergence if the model is partially known.Finally, we have presented some simulations toillustrate the effectiveness of the suggested approach,which shows some robustness properties against noisymeasurements.
ReferencesAguilar-Lopez, R., Martınez-Guerra, R., Mendoza-
Camargo, J., and M. Neria-Gonzalez (2006).Monitoring of and industrial wastewaterplant employing finite-time convergenceobserver. Journal of Chemical Technology andBiotechnology 81, 851-857.
Alvarez-Ramirez, J (1999). Robust PI stabilizationof a class of continuously stirred-tank reactors.AIChE Journal 45, 1992-2000.
Bastin, G. and D. Dochain(1990). On-lineestimation and adaptative control of bioreactors1. Elsevier, Amsterdam.
Davila, J., Fridman, L. and A. Levant (2005). Secondorder sliding-mode observer for mechanicalsystems. IEEE Transactions on AutomaticControl 50, 1785-1789.
Diop, S. and R. Martınez-Guerra (2001). Analgebraic and data derivative informationapproach to nonlinear system diagnosis.Proceedings of the European ControlConference (ECC), Porto, Portugal, 2334-2339.
Farza, M., Busawon, K. and H. Hammouri(1998). Simple nonlinear observers for on-line estimation of kinetics rates in bioreactors.Automatica 34, 301-318.
Gauthier, J., Hammouri H. and S. Othman (1992).A simple observer for nonlinear systems.Applications to bioreactors. IEEE Transactionson Automatic Control 37, 875-880.
Gouze, J. and V. Lemesle (2001). A boundederror observer for a class of bioreactormodels. Proceedings of the European ControlConference (ECC), Porto, Portugal.
www.rmiq.org 319
J. L. Mata-Machuca et al./ Revista Mexicana de Ingenierıa Quımica Vol. 10, No. 2 (2011) 313-320
Hadj-Sadok, Z. (1999). Modelisation et estimationdans les bioreacteurs; prise en compte desincertudes: application au traitement de l’eau.PhD thesis. Nice-Sophia Antipolis University.Nice.
Hu, S. and J. Wang (2002). Global asymptoticstability and global exponential stability ofcontinuous time recurrent neural netwoks. IEEETransactions on Automatic Control 47, 802-807.
Keller, H (1987). Non-linear observer designby transformation into a generalized observercanonical form. International Journal ofControl 46, 1915-1930.
Khalil, H (2002). Nonlinear systems. Third edition.Prentice Hall, New Jersey.
Lemesle, V. and J. Gouze (2005). Hybrid boundederror observers for uncertain bioreactor models.Bioprocess & Biosystems Engineering 27, 311-318.
Levant, A (2001). Universal single-input-single-output (SISO) sliding-mode controllers withfinite-time convergence. IEEE Transactions onAutomatic Control 46, 1447-1451.
Luenberger, D (1979). Introduction to DynamicSystems. Theory Models and Applications.Wiley, New York.
Martınez-Guerra, R. and J. de Leon-Morales (1996).Nonlinear estimators: a differential algebraicapproach. Journal of Mathematics andComputer Modelling 20, 125-132.
Martınez-Guerra, R., Poznyak, A. and V. Dıaz(2000). Robustness of high-gain observersfor closed-loop nonlinear systems: theoreticalstudy and robotics control application.International Journal of Systems Science 31,1519-1529.
Masoud, S (1997). Nonlinear state-observerdesign with application to reactors. ChemicalEngineering Science 52, 387-404.
Vargas, A., Soto, G., Moreno, J. and G. Buitron(2000), Observer based time-optimal control ofan aerobic SBR for chemical and petrochemicalwastewater treatment. Water Science andTechnology 42, 163-170.
320 www.rmiq.org
