Fuzzy Model Based Iterative Learning Control for Phenol Biodegradation

Fuzzy Model Based Iterative Learning Control for Phenol Biodegradation Marco M´ arquez1, , Julio Waissman1 , and Olivia Gut´ u2 1

Centro de Investigaci´ on en Tecnolog´ıas de la Informaci´ on y Sistemas 2 Centro de Investigaci´ on en Matem´ aticas Unversidad Aut´ onoma del Estado de Hidalgo Carr. Pachuca-Tulancingo Km. 4.5, Pachuca Hgo. 42084, Mexico [email protected], {julio,olivia}@uaeh.edu.mx

Abstract. In a fedbatch process the operational strategy can consist on controling the influent substrate concentration in the reactor, by means of the input flow manipulation. Due to the repetitive characteristic of the Sequencing Batch Reactor processes, it opens the possibility to explore the information generated in previous cycles to improve the process operation, without having on-line sensors and/or a very precise analytical model. In this work an iterative learning control strategy based on a fuzzy model is proposed. It is assumed that the measurements are analytical and only a few number of them can be obtained. So, an interpolation technique is used to improve the control performance. Simulation results for a phenol biodegradation process are presented. Keywords: Fuzzy Iterative Learning Control, Biotechnological Process.

1

Introduction

The Sequencing Batch Reactor (SBR) process operates in a true batch mode with aeration and sludge settling both occurring in the same tank. The major difference between a SBR and a typical activated sludge system is that the SBR tank carries out the functions of equalization, aeration and sedimentation in a time sequence rather than in the conventional space sequence. The SBR presents some advantages with respect to the continuous activated sludge process. Since SBR is a batch process, the effluent can be held in the reactor until it is treated if the influent can be stored far away. This can minimize the deterioration of effluent quality associated with influent spikes. Also, biomass will not be washed out of a SBR because of flow surges. In addition, settling occurs when there is no inflow or outflow. However, the SBR systems have also some disadvantages, generally related to a higher level of control sophistication. Identification and real-time control of SBR processes still represent a challenging area of endeavor for control engineers. In particular, the control design is difficult by at least two well-known factors [1]. Firstly, the processes involving

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P. Melin et al. (Eds.): IFSA 2007, LNAI 4529, pp. 328–337, 2007. c Springer-Verlag Berlin Heidelberg 2007

Fuzzy Model Based Iterative Learning Control for Phenol Biodegradation

329

living microorganisms exhibit large nonlinearities, strongly coupled variables and often poorly understood dynamics. Secondly, the real-time monitoring of many key process variables, which are needed by advanced control algorithms, is hampered by the lack of reliable on-line sensors. Different trends for the control of fedbatch biotechnological processes have emerged, principally optimal and adaptive approaches [2]. These methods can be applied to the SBR control, since it can be considered as a biotechnological fedbatch process. The main drawback of the model-based optimal control methods, which provides a theorically realizable optimum, is the assumption of a perfectly known model. On the other hand, model-independent adaptive controllers do not guarantee a priori optimality of the control policy results. Finally, the approach based on the concept of minimal modeling of the kinetics has emerged, in order to fill the gap between modeling accuracy and control needs [2]. In this approach, the optimal control of fedbatch processes can be replaced by a common nearly optimal regulation control case. According to this, the control objective is stated in terms of a substrate concentration set point tracking to fix the influent flow rate. An iterative learning control (ILC) algorithm [3] allows the output-tracking task to be carried out. The choice of an ILC technique is justified by the repetitive nature of the fedbatch cultures. Since ILC uses information from previous executions of the task in order to improve the tracking performance from trial to trial, it does not require any on-line measurement. The ILC differs from the majority of the control methods, as it employs all the possibilities of incorporation of control information from the past process operation, such as the error and input signals, to construct the actual control action. The ILC control has been proposed for the control of systems that can perform the same task repetitively [4]. Since fedbatch reactors are permanently in a transient regime, the tracking behaviour of the conventional ILC approach deteriorates as the number of off-line measurement samples decreases. The use of ILC for discrete-time systems is a common case, but a very small time period is needed to guarantee the algorithm convergence [5]. In this work, the measurements are considered to be done by a process operator, and so it is desirable to have a sampling period as larger as possible. The effect of a larger sampling period on the ILC control of a SBR process was studied in [6]. The proposed solution consists in generating an estimated output sequence in small intervals, to reduce the tracking error between samples. In this work, a fuzzy model of the process was developed. The learning law is based in the fuzzy model to improve the convergence rate of the learning algorithm, based in ideas proposed in [7] and [8]. In this work a fuzzy model of the growth rates are coupled with a Takagi-Sugeno type fuzzy model of the process in order to reduce de complexity of the model and reduce the computational effort of the learning law algorithm. The study is carried out on an analytical model of a SBR system for phenol biodegradation, whose kinetics are characterized by the production and later consumption of an inhibitory metabolic intermediate. The process under study

330

M. M´ arquez, J. Waissman, and O. Gut´ u

is a pilot plant of the CIQ-UAEH [9]. From this process, an analytical model was previously experimentally validated for fedbatch cultures [10].

2

Phenol Biodegradation Model

The mass balance equation for the various constituents of phenol biodegradation is given by the following first order system of differential equations [10]: Qin (t) ˙ X(t) = μ(t)X(t) − X(t) + d1 (t) V (t) Qin (t) in (S − S1 (t)) + d2 (t) S˙ 1 (t) = −qS1 (t)X(t) + V (t) 1 Qin (t) S2 (t) S˙ 2 (t) = vS2 (t)X(t) − qS2 (t)X(t) − V (t) V˙ (t) = Qin (t) − Qout (t)

(1)

y(t) = S1 (t) where S1 (t) is the phenol concentration, S2 (t) is the main metabolic intermediate concentration, X(t) is the total microbial concentration and V (t) is the volume; μ(t) is the specific biomass growth rate, qS1 (t) and qS2 (t) are, respectively, the specific consumption rate of phenol and the intermediate; vS2 (t) is the specific intermediate production rate. We consider that the reactor is a fedbatch process and in this case Qout (t) = 0, d1 (t) and d2 (t) are the external disturbances. The specific biomass growth rate is calculated by μ(t) = μ1 (t) + μ2 (t)

(2)

where μ1 (t) is a modified Haldane type equation and μ2 (t) is a Monod type [10], i.e: K2 μmax1 S1 (t) KS1 + S1 + S12 /Ki1 K2 + S2 (t) μmax2 S2 (t) K1 μ2 (t) = KS2 + S2 (t) K1 + S1 (t)

μ1 (t) =

(3) (4)

where, μmax1 is the maximus growth value due to the phenol concentration, μmax2 is the maximus growth value due to the intermediary concentration. The specific growth and consumption rates are correlated with the constants, biomass to phenol Y1 and biomass to intermediate Y2 as follows: qS1 (t) =

μ1 (t) Y1

qS2 (t) =

μ2 (t) . Y2

(5)

The specific production rate of intermediate is linearly correlated to the specific growth rate of biomass on phenol [10] as vS2 (t) = αμ1 (t).

(6)


331

The parameters values are described in [10] in base to a sensitivity analysis, the values used for simulation are given in the Table 1. Table 1. Parameter values Symbol Value SI units μmax1 Y1

0.39

0.57 mg/mg

KS1

30

mg/l

Ki1

170

mg/l

K2

160

mg/l

μmax2 0.028 Y2

3

1/h

1/h

0.67 mg/mg

KS2

350

mg/l

K1

66

mg/l

α

1.6

mg/l

Fuzzy Model of Microbial Growth Rate

For developing the learning algorithm, it is needed a fuzzy model of the process. In order to reduce the number of possible fuzzy rules a fuzzy model of the microbial growth rate coupled with a fuzzy model of the process is proposed. This approach allows to obtain a reduced number of rules. So, less sub-models are required for the learning law. There were used five membership functions to evaluate the phenol concentration μ1 (t), and another five for the metabolic intermediate concentration μ2 (t) to model both growth rates. So only 25 fuzzy rules were generated. The consequent parameters of the linear functions can be estimated by least squares from the available data. The condition is that the consequent functions were linear and the number of data items is much greater than the dimension of the regression vector. A diagonal matrix Γi ∈ RN ×N is formed with the normalized membership values λi (xk ) as the k-th diagonal element, where k is in [1, N ] and N is the number of data items. A Xe matrix was formed by S1 , S2 as its columns and a column of ones to determine the parameters bi . I it is so called the extended regressor matrix Xe = [S1 , S2 , 1] [11]. Another matrix ¯ ∈ RN ×nN is formed from the matrices Γi and Xe : X ¯ = [Γ1 Xe , Γ2 Xe , . . . , Γn Xe ]. X

332


If the parameters are defined as a vector γ, it is computed in the following way: T −1 T ¯ X ¯ ¯ y X γ= X

(7)

where y is the fuzzy estimation of microbial growth rate μ1 (t) or μ2 (t). In this way, γ is obtained as the vector: γ = [aT1 b1 , aT2 b2 , . . . , aTn bn ]. The fuzzy estimated values of μ1 (t) and μ2 (t) are presented in Fig. 1 and Fig. 2 respectively. Thus, the microbial growth rate is a function of the phenol S1 and the metabolic intermediate S2 , with a linguistic interpretation.

Fig. 1. Microbial growth rate due the phenol

Once the fuzzy model of the microbial growth rate is obtained, the number of rules of the process model will be reduced. If there is considered a Takagi-Sugeno fuzzy inference system, the model can be represented by a fuzzy combination of linear systems: (i)

R(i) : If x1 (t) is F1

and · · · and xn (t) is Fn(i) ,

(8)

Then x(t + 1) is Ai (x(t)) + Bi u(t) (i)

where Fj is a fuzzy set, x(t) = [X(t), S1 (t), S2 (t), V (t)] denotes the state vector, u(t) denotes the control input, Ai ∈ R4×4 , Bi4×1 are matrices that represent the dynamic of each linear subsystem. The output y(t) is the phenol concentration. A representative prototype of the ith rule is denoted by xi = [X i , S1i , S2i , V i ]


333

Fig. 2. Microbial growth rate due the metabolic intermediate

A normalized membership value can be obtained as: x(t + 1) =

n

λi (x(t))(Ai x(t) + Bi u(t)).

(9)

i

where

n

j μF (xj ) λi (x) = m n j . i j μFj (xj )

(10)

The matrices Ai and Bi for each rule are defined as a linearization of the process model: ⎛ ⎜ ⎜ ⎜ Ai = ⎜ ⎜ ⎜ ⎝

μi1 + μi2 −

Qi Vi

μi1 Y1

αμi1 −

0

Qi X i (V i )2

− VQi

0

−Qi (S1in −S1i ) Vi

0

− VQi

Qi S2i (V i )2

0

0

0

0 i

μi2 Y2

0 ⎛

i

i

−X Vi

⎜ in ⎜ (S1 − S1i ) 1i V ⎜ Bi = ⎜ i S ⎜ − V 2i ⎝ 1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(11)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(12)

334


where μi1 and μi2 are the fuzzy estimation of the growth rates for the prototype of the ith rule. This allows to use a reduced number of rules to represent the process.

4

Model Based Fuzzy ILC

The Iterative Learning Control (ILC) is a technique generating an input sequence for a working cycle of a system, by using a learning law and the information of the input and output sequences of the last cycle, as well as the desired output sequence. The learning law has to assure the convergence of the output sequence to the desired one after several working cycles. A desired output sequence {yd (0), yd (Δt), . . . , yd (N Δt))} is imposed and it is defined on the interval [0, T ]. The error is obtained at each point as: ei (kΔt) = yd (kΔt) − yi (kΔt) where {yi (0), yi (Δt), . . . , yi (N Δt))} is the output sequence of the system for the i iteration. A commonly assumption for ILC is that the initial condition is bounded for each working cycle. The proposed learning law is: ui+1 (kΔt) = ui (kΔt) + βi (kΔt)ei (kΔt)

(13)

where {ui+1 (0), ui+1 (Δt), . . . , ui+1 (N Δt))} is the input sequence for the next iteration. The control gain βi (kΔt) is obtained by the fuzzy model of the process by: n λj (x(kΔt))βij (14) βi (kΔt) = j=0

βij is the learning gain of the j th rule of the model in the ith iteration. The learning gains βij are obtained minimizing a quadratic performance index [12]. In the iteration i, the sequence gain is computed for the j-th rule as: βij = (GTj QGj + R)−1 GTj Q

(15)

where Q and R are symmetric positive definite matrices. Gj is obtained from the super-vector notation of the system. This notation is in discrete time and is formulated as yj (kΔt) = Gj u(kΔt), where ⎛ ⎜ ⎜ Gj = ⎜ ⎝

CBj CAj Bj .. .

0 CBj .. .

··· ··· .. .

0 0 .. .

−1 −2 CAN Bj CAN Bj · · · CBj j j

⎞ ⎟ ⎟ ⎟ ⎠

(16)

where Aj and Bj are the model matrices for the j th rule of the fuzzy system.


5

335

Simulation Results

An application example was developed for the phenol biodegradation process. The main goal was to follow a phenol concentration profile. It was assumed that only a reduced number of analytical samples can be made. To guarantee convergence an interpolation is needed. As it is shown in [5], to ensure a reduced tracking error for a sampled data non linear system, it is needed a small sample time. Due to the system measurement is assumed to be by the way of operator analysis, the sample time can not be small. In the other hand, a small sampling period implies that the matrices Gj (kΔt) will be too big, and the control signal computation can requires a considerable computational effort. The input signal was computed with a bigger sampling period, and then a finer input sequence was generated using a correct interpolation technique. In [6] is shown that a good choice of the interpolation technique allows to have a bigger sampling period for the output sequence. The system was simulated for working cycles of 15 hours. In order to ensure a bounded initial condition error for biomass and volume, a purge at the end of each cycle is considered has a realistic assumption. The initial conditions of biomass concentration and volume are X(0) = 450 mg/l ± 3% and V (0) = 1 l. Nevertheless, it is not possible to reduce the phenol and the intermediate concentrations from one cycle to other. Then, both substances have to be completely consumed at the end of each working cycle. For this reason, S1 (0) = S2 (0) = 0 for the first iteration and for the other iterations, S1 (0) and S2 (0) take the values of the final concentration of the last working cycle.

Fig. 3. Simulation of the phenol concentration

336


The signal reference is the concentration profile defined by:

t − 6.5 −5

. yd (t) = 70

2.5

(17)

A white noise was added to states X and S1 in order to simulate the analytical measurement error. The sample time was 20 minutes, so 46 values are used for each state in each iteration. A cubic spline technique was used to interpolate the control signal in order to have a 1.2 minutes as virtual sample time, and apply a sequence of 751 values to the process.

Fig. 4. Simulation of the metabolic intermediate

For the fuzzy process model, three membership functions were defined for the phenol and the metabolic intermediate concentration, two functions for the volume, the inlet flow, and the biomass concentration. In Fig. 3, the simulation of 10 iterations are presented for the phenol concentration. A good performance is presented after 10 iterations. Nevertheless, an important tracking error is presented at the begining of the phenol concentration profile, due to the large sampled time used. In order to avoid an effect of bioaccumulation of intermediate, it is important to ensure that the intermediate is completely consumed at the end of each working cycle. In Fig. 4 the simulation shows that it is consumed before 15 hours in all iterations.


6

337

Conclusion

The use of a fuzzy model whose consequents are linear systems is a good alternative to control a bioreactor when the measurement can not be made on-line. The number of measurements can be reduced if an interpolation technique is used to have an small virtual sample period with a cost in the performance.

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