Reconstruction of Dynamics of Aqueous Phenols and Their Products

Ind. Eng. Chem. Res. 2007, 46, 5855-5866

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Reconstruction of Dynamics of Aqueous Phenols and Their Products Formation in Ozonation Using Differential Neural Network Observers I. Chairez,† A. Poznyak,† and T. Poznyak*,‡ Department of Automatic Control, CINVESTAV-IPN, AV.Instituto Polite´ cnico Nacional, Col. San Pedro Zacatenco, C.P. 07360, Mexico D.F., Me´ xico, Superior School of Chemical Engineering, National Polytechnic Institute of Mexico (ESIQIE-IPN,), Edif. 7, UPALM, C.P. 07738, Mexico D.F., Mexico

A differential neural network (DNN) is used to estimate the state dynamics in the phenols-ozone-water system of the model solution of phenol (PH), 4-chlorophenol (4-CPH), and 2,4-dichlorophenol (2,4-DCPH). This new technique, which is based on a differential neural network observer (DNNO), is applied to estimate decomposition dynamics of phenols, byproducts accumulation and decomposition, and final products accumulation. It is considered to be a process with an uncertain model (“black-box”) and is affected by internal (variation of experimental conditions) and external perturbations (measurement noises). The monitored ozone concentration in the gas phase in the reactor outlet is used to obtain the summary characteristic curve (ozonogram) in ozonation. The proposed DNNO is trained using the variation of this parameter with the experimental data of the phenols decomposition, obtained using the HPLC technique, at pH 2 and pH 9. The trained DNNO then is applied to reconstruct the dynamics of the phenols decomposition, as well as the byproducts accumulation and the decomposition and the final product accumulation at pH 7 and pH 12. The proposed DNNO technique has been tested to compare estimated results to those experimentally obtained during semibatch ozonation of the model solution of phenols. A good correspondence between the experimental decomposition dynamics and those estimated by DNNO was obtained. 1. Introduction Phenolic compounds are known to be common pollutants in the wastewater of oil refinery, petrochemical, coke, and grease, due, steel and textile industries.1,2 The decomposition of these contaminants is difficult in wastewater;3,4 particularly, chlorophenols constitute the characteristics of toxicity and a refractory nature, as well as being difficult to remove via the traditional biological treatment, because the efficiency of this procedure is usually not satisfactory, because of the long reaction time required and limited initial concentration of pollutants.5-9 Ozonation and related chemical advanced oxidation processes (AOPs),10-16 and their combination with bioremediation,17-22 were used to eliminate contaminants from water. However, the most of the studies address only decomposition of the initial compound and does not consider the decomposition of byproducts. The importance of the byproducts characterize is due to their toxicity, which can be higher than that of the original contaminants. Several mathematical models of ozonation, depending on the experimental conditions and reactor models, have been proposed.12,23-28 However, in the case of residual water treatment by ozone, any mathematical model cannot be applied directly, because of the complex organics composition and the possibility to realize at same time two different reaction mechanisms: via molecular ozone and via indirect reactions. Evidently, the most complete mathematical model of ozonation that is under consideration belongs to the class of partial differential equations, because it describes such physicochemical processes as mass transfer, dissolution, absorption, bulking, etc.29 * To whom correspondence should be addressed. Tel.: +(525) 55 5061-37-41. Fax: +(525) 55 747-70-89. E-mail: apoznyak@ ctrl.cinvestav.mx (A.P.); Tel.: +(525) 55 729-60-00, ext. 55288. Fax: +(525) 55 586- 27-28. E-mail address: [email protected] (T.P.). † Department of Automatic Control. ‡ Superior School of Chemical Engineering.

This complex situation arises when we examine unknown parameters, as well as incomplete state observations. The Dynamic Neural Network (DNN) technique30 is suggested to be applied to provide adequate state estimates without either preliminary parameter identification or knowledge of the exact model structure (particularly in the case of the phenols decomposition in water by ozone).31 To estimate the current concentration of phenols, ozonation byproducts, and final compounds, a Dynamic Neuro Network Observer (DNNO)32,33 with some modification is suggested. This approach, which exploits DNN feedback properties, permits many problems related to global extreme search to be avoided, converting the learning process to an adequate feedback design to regulate and adjust the parameters (weights) using dynamic structures.32,34 Referring to our previous study of the reconstruction of the phenols decomposition in water by ozone without knowledge of an exact physical-model structure,31 the contribution of this study is the differential neural networks observer (DNNO) application to the numerical reconstruction of the decomposition of phenols, as well as their byproducts and final compounds formation in ozonation at different pH values (pH 7 and pH 12), using only the ozone concentration variation in the reactor output and the training of DNNO, using experimental data of phenols ozonation at pH 2 and pH 9. Three phenols (phenol (PH), 4-chlorophenol (4-CPH), and 2,4-dichlorophenol (2,4DCPH)) are chosen as the model pollutants. The experimental data of the phenols ozonation are used for the proposed DNNO approach verification. As a result, the validity of the DNNO technique is demonstrated by good agreement of the predicted results with the experimental data. 2. Theoretical Analysis 2.1. Modified DNNO for Nonlinear System Estimation. In the present study, a modification of the observer structure

10.1021/ie0705103 CCC: $37.00 © 2007 American Chemical Society Published on Web 08/09/2007

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Ind. Eng. Chem. Res., Vol. 46, No. 18, 2007

Figure 1. Comparison between DNNO-approach and experimental data for: the ozone concentration variation in the gas phase at pH 7 (1) and 12 (2): (a) phenol, (c) 4-chlorophenol, (e) and 2,4-dichlorophenol; (b) the decomposition of phenol, (d) 4-chlorophenol, and (f) 2,4-dichlorophenol.

[

Table 1. DNNO Parameters Obtained in Training Procedure initial compound 4-chlorophenol

-10 0 0 0 0.3* 0 0 0 0 0

A

0.1 -7 0 0 0 0 0 0 0

0.2 0 -4 0 0 0 0 0 0

0.4 0 0 -8 0 0 0 0 0

0.6 0 0 0 -5 0 0 0 0

0.9 0 0 0 0 -4 0 0 0

given in ref 31 and related by the variable structure term (K2) is introduced. The main modification on the DNNO structure is associated with the term et/|et|, which corresponds to the so-called unitary correction (here, |‚| represents the Euclidean norm). The new term is added to the DNNO scheme to improve the zone-convergent rate from the state estimator trajectories to those provided by the nonlinear system.35 The class of systems to be treated within this work is described by the vector nonlinear differential equations

x˘ t ) f(xt, ut) + ξ1,t

(1a)

yt ) Cxt + ξ2,t

(1b)

and

where xt ∈ Rn is the system state, yt ∈ Rp is the system output (p en), and ut ∈ Rm is the bounded control action (m en) belonging to the following admissible set, Uadm: 2 ) uTΛuu e ν0 < ∞} Uadm ) {u : |u|Λ u

] [] [] K1

Λu ) ΛTu > 0

The output matrix C ∈ Rp×n is assumed to be known a priori.

1.5 0 0 0 0 0 -3 0 0

2.2 0 0 0 0 0 0 -4 0

2.5 0 0 0 0 0 0 0 -9

K2

6 2 8 9 0.02 8 7 6 5 4

3 5 4 6 0.005 8 7 9 5 6

The nominal closed-loop dynamics is supposed to be quadratically stable for a fixed control u* ∈ Uadm; that is, there exists a Lyapunov function V h t such that

∂V h f(x,u*) e - λ1|x| 2 < 0 ∂x ∂V h | e λ2|x | < ∞ ∂x

(2a)

λ1, λ 2 > 0

(2b)

λ 2λmax(ΛV) + ˜f 1| Λ f ||Λ1f˜| < λ1

(2c)

|

Here, the positive definite matrices ΛVΛ˜f can be selected to be as small as we wish, but in such a manner that the Riccati matrix, given by eq 13 (see below), still has a solution. Therefore, the last inequality in eqs 2 is not so restrictive. The vectors ξ1,t and ξ2,t represent the state and output deterministic bounded (immeasurable) disturbances, i.e., 2 e γj, Λξj ) ΛξTj > 0 |ξj,t|Λ ξ

(for j ) 1, 2)

j

and do not violate the existence of the solution to the ordinary differential equation (ODE) described by eq 1. The nominal output system (without external perturbations, ξ2,t ) 0) is

Ind. Eng. Chem. Res., Vol. 46, No. 18, 2007 5857

The activation vector functions σi(‚) and φ(‚) are usually constructed by the sigmoid function components:

Table 2. Differential Neural Network Observer State Structure variable

treatment

xˆ 1,t xˆ 2,t xˆ 3,t

ozone concentration variation in the gas phase ozone concentration variation in the liquid phase decomposition dynamics for phenol concentration variation accumulation and decomposition dynamics for cathecol concentration variation accumulation and decomposition dynamics for hydroquinone concentration variation accumulation dynamics for formic acid concentration variation accumulation dynamics for oxalic acid concentration variation accumulation and decomposition dynamics for maleic acid concentration variation accumulation and decomposition dynamics for muconic acid concentration variation accumulation and decomposition dynamics for fumaric acid concentration variation

xˆ 4,t xˆ 5,t xˆ 6,t xˆ 7,t xˆ 8,t xˆ 9,t xˆ 10,t

n

σj(x) :) aj(1 + bj exp(-

2 |σ(x) - σ(x′)|Λ e lσ |x - x′|2Λ′σ, σ

(4a) (4b)

The last assumption automatically implies the following cone property:

(5)

which is valid for any x and u. Note that eq 1 always can be represented as

x˘ t ) f0(xt, ut | Θ) + ˜f t + ξ1,t

(6a)

˜f t: ) f(xt, ut) - f0(x,u | Θ)

(6b)

where f0(x,u | Θ) is treated as a possible “nominal dynamics”, which can be selected according to a designer desires, and ˜ft is a vector called the “DNN-modeling error”, which is commonly related to parametric uncertainties. Here, the parameters Θ are subjected to an adjustment to obtain a better matching between the nominal and nonlinear dynamics. In view of eq 6 and the corresponding boundedness property, the following upper bound for the unmodelled dynamics ˜ft is observed: 2 Λf, Λ1˜f > 0 |f˜t|2 e ˜f 0 + ˜f 1|xt|Λ 1, ˜f

According to DNN dynamics as

theory,5

(9b)

It is easy to prove that each component in the activation functions satisfies the following sector conditions:

Here, Lf (‚) is the Lie derivative operator.10 To ensure the uniqueness and the existence of the solution of the nonlinear ODE described by eq 1, here the class of nonlinear functions in eq 1 is supposed to satisfy the Lipschitz condition in both arguments; that is,

|f(x,u)|2 e C1 + C2|x|2

cklxl))-1 ∑ j)1

j ) 1, n, k ) 1,n, l ) (1,m)

O ˜ :) 3x [C T, [Lf (Cxt)]T, Lf2(Cxt)]T, ..., Ln-1 (Cx t )TT (3) f

|f(0,0)|2 e C1; x, y ∈R n; u, ν ∈ R m; 0 e L1, L2 < ∞

(9a)

n

φkl(x) :) akl(1 + bkl exp(-

2 |(φ(x) - φ(x′))ut|Λ e lφν0|x - x′|2Λ′φ (10) φ

uniformly observable, that is, the following (observability) matrix is nonsingular for any t g 0:

|f(x,u) - f(y,ν)| e L1| x - y| + L2| u -V |

cjxj))-1 ∑ j)1

(7)

we will define the nominal

The modified DNN state estimator is covered by the following nonlinear ODE: {et} d xˆ t ) Axˆ t + W1,tσ(xˆt) + W2,tφ(xˆt)ut + K1et + K2 dt {|et|}

yˆ t ) Cxˆt

(11a)

(11b)

Here, xˆ t ∈ Rn ∈Rn is the state vector of DNNO representing the current estimates of phenols concentration, ut ∈ Rm is the input vector for DNNO and, for the experimental system (in this case, it is the inflow concentration), is the output of DNN corresponding the estimates of the measurable ozone concentration in the gas phase; A ∈ Rn×n, K1 ∈ Rn×p, and K2 ∈ Rn×p are constant matrices adjusted during DNN training, σ(‚) ∈ Rn and φ(‚) ∈ Rn×m are standard vector fields constructed (each element) by sigmoid functions, C ∈ Rp×n is an output matrix, Wi ∈ Rn×n (i ) 1, 2) are the weights layer (the first one corresponds to the self-feedback adjustment and the second one is related with the input effect on the state estimation process) tuning by a special on-line learning procedure.32 The measurable data are the ozone concentration variation in the gas phase in the reactor output; that is, yt ) x1 ∈ R. Therefore, C ) (1, 0, ..., 0). The gain matrix K1 corresponds to a linear correction term,36 K2 ) -kP-1CT is a sliding mode-type correction term matrix.37 When et ) 0, eq 11a is treated as a differential inclusion in the Filippov’s sense (a differential inclusion in the Fillipov’s sense).34 The adequate learning of DNNO (eqs 11) provides a sufficiently small upper bound (in an average sense) for the state estimation error ∆t ) xˆ t - xt. The learning process is given by

1 ˜ j,t χj + Pxˆ t χtj W˙ j,t ) -kj,t ΛjPCTet + ΠjPW 2 (for j ) 1, 2) (12)

[

]

for

f0(x,u,t | Θ) ) A0x + W01σ(x) + W02φ(x)u Θ :) [W01, W02], A0 ∈ R n×n, W01, W02, ∈ R n×n, σ ∈ R1×1, φ ∈ R n×m (8)

χ1 ) σ(xˆ t), χ2 ) φ(xˆ t)ut, Πj ) Πtj > 0, Πj ∈ Rn×n, P ) PT > 0, P ∈Rn×n, Λj ) ΛTj > 0, Λj ∈Rn×n, W ˜ j,t ) Wj,t - W* j,t, kj > 0, kj ∈ R

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Figure 2. Comparison between the DNNO approach and experimental data for the byproducts and final compounds in the ozonation of phenol at pH 7: (a) hydroquinone (**) and catechol (++ ); (b) formic acid (**); (c) fumaric acid (**); and (d) maleic acid (**) and oxalic acid (++) and the DNNO reconstruction (s).

where the matrix P is the positive definite solution for the Riccati equation:

-1 -1/2 FQ :) ˜f 0|Λf| + ν0 + γ1[1 + λmax(Λ-1/2 V Λξ1 ΛV )] + 3γ2 +

PA ˜ * + (A ˜ *) P + PRP + Q ) 0 (0)

(0)

T

8λxηγ2, RQ :) λmin(P-1/2Q0P-1/2)

A h :) A * - K1C (0)

-1 Q :) Λ-1 1 + Λ2 + Λσ + ν0Λφ + Q0

R :) W10Λσ-1[W10]T + W20Λφ-1[W20]T + Λ˜f + Λξ1 + K1Λξ2KT1 + K2ΛKT2 (13) Here, it is important to notice that the observer (eqs 11) does not use any information on the model structure of the real process; that is, any a priori given dynamic model of the process is not required to be in use for the suggested DNN observers. In this sense, the suggested observer scheme may be treated as a “software sensor”. Let us consider the DNNO defined by eqs 11 with the corresponding learning law governed by eq 12. It is possible to ensure the following upper bound for the estimation process:

lim sup tf∞

1 t+

t |xˆ s - xs|Q ∫s)0

0

ds e

FQ RQ

for

(14)

Because the matrix Λf is a “free parameter”, it can be selected as one with a sufficiently small norm. Therefore, the influence of the constant ˜f0 on the upper bound (eq 14) can be made insignificant. The proof of the state estimation convergence and the discussions regarding how this upper bound has been derived are presented in the Appendix at the end of this article. The last inequality means that the reconstructed trajectories for the immeasurable variables that are involved in the ozonation are similar to their real values as small as the disturbance values are, as was expected. The details of the meaning of the upper limit for the averaged estimation error (eq 14), the convergence properties of the learning algorithm (eq 12), and any other special feature on the DNNO structure and its “philosophy” are discussed.32,38 2.2. Training of DNNO. To guarantee a sufficiently small (near to zero) state estimation error, the adequate DNNO parameters at eqs 11 should be selected. The off-line adjustable parameters A, K1, and K2 may be tuned during the “training” process, where the weights Wi,t (i ∈ 1, 2) are quickly adjusted


Figure 3. Comparison between the DNNO approach and experimental data for the byproducts and final compounds in the ozonation of phenol at pH 12: (a) hydroquinone (**); (b) fumaric acid (**); (c) maleic acid (**); and (d) oxalic acid (**) and the DNNO reconstruction (s).

on-line by the matrix differential learning law (eq 12). The training procedure has two ways of DNNO training: (1) Using any simplified mathematical model of phenols ozonation, including the byproducts dynamics, to generate the corresponding trainer input-output data sequences, which may serve as the DNNO parameters adjustment as well as for adequate selection of the initial conditions in the applied (Wj,0) learning procedure as described.31 (2) Using the experimental data of the phenols ozonation, a training data for the DNNO adjustment procedure may be obtained too. The data set, composed by the gas-phase ozone concentration variation and the phenols and byproducts decomposition dynamics, must have representative information about the phenols decomposition with ozone (an observability requirement) to apply any adaptive nonlinear observer (in this study, based on DNN approach). In the present study, the second method was used to train the DNNO observer. The phenols at different pH values (pH 2.0, 7.0, 9.0, and 12.0) were decomposed. Because of the difference between the ozonation mechanisms at acidic and basic pH, the DNNO training was performed under the following conditions: (1) The adjustable parameters on the DNNO structure were selected based on the data set generated at pH 2.0 and pH 9.0. (2) The state estimation, using the DNN approach, was tested with the data generated at pH 7.0 and pH 12.0.

(3) The trained DNNO at pH 2.0 was used to reconstruct the ozonation dynamics at pH 7.0, because the reaction is direct between the molecular ozone and the organics. (4) The ozonation at pH 12.0 was tested with the trained DNNO using the data set supplied at pH 9.0. At these two pH values, the reaction mechanism is similar, where the combination of free radicals and the direct reaction with molecular ozone happens. 2.3. State Estimation Technique. The estimation process (reconstruction of immeasurable variables) was conducted using special software that was designed to connect the acquisition board with a virtual instrument (computer-assisted sensor). This “artificial sensor” measures the ozone concentration in the gas phase in the reactor output. The generated information is input into the DNNO software that reconstructs (using the algorithm described by eqs 2 and 3) the unknown variables involved in the ozonation (phenols decomposition, byproducts formation and decomposition, final products formation). In fact, the suggested algorithm is running during the ozonation, so the reaction is realized simultaneously with the on-line estimation providing a current concentration value for each phenol and their byproducts. 3. Materials and Methods 3.1. Ozonation Procedure. The ozonation of phenols solution (with an initial concentration of 100 mg/L) was performed in a

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Figure 4. Comparison between the DNNO approach and experimental data for the byproducts and final compounds in the ozonation of 4-CPH at pH 7: (a) phenol (**); (b) hydroquinone (**) and catechol (++); (c) formic acid (**) and oxalic acid (++); and (d) muconic acid (**), maleic acid (++), and fumaric acid (]]) and the DNNO reconstruction (s).

Table 3. Intermediates and Final Species in the Ozonation of Phenol, 4-Chlorophenol, and 2,4-Dichlorophenol compound phenol 4-chlorophenol 2,4-dichlorophenol phenol 4-chlorophenol 2,4-dichlorophenol phenol 4-chlorophenol 2,4-dichlorophenol

hydroquinone × × × × × × traces traces traces

catechol traces × × × × ×

muconic acid

fumaric acid

maleic acid

oxalic acid

formic acid

phenol

4-chlorophenol

× × ×

× × ×

× × ×

traces traces

traces

pH 7 traces × traces

× × ×

× × ×

× × ×

traces ×

×

pH 12 traces traces traces

traces traces traces

× × ×

traces traces

traces

pH 2 ×

× traces traces

× traces

×

semibatch reactor (0.5 L) that had been provided with a diffuser plate at its bottom for feeding the ozonated gas, using the ozone generator (“AZCO”) with an initial ozone concentration of 32 mg/L and a gas flow of 0.5 L/min. All experiments were performed at ambient temperature. The measurements of ozone in gas phase at the reactor output was done with an ozone sensor (model BMT 930) that was connected to a personal computer (PC) (using an acquisition data board (NI-6024)), to reproduce the ozonation reaction experimental curve (ozonogram) in Matlab. The phenols ozonation at pH values of 2, 7, 9, and 12 was realized. The initial pH was controlled using H2SO4 and

0.05 N NaOH solutions, and this parameter was measured by a Conductronic pH meter (model PC 18 with a P100C-BNC electrode). 3.2. Analytical Methods. The aqueous samples were analyzed via high-performance liquid chromatography (HPLC). The HPLC analysis was performed using a liquid chromatograph (Perkin-Elmer series 200) that was coupled with the UV/VIS detector and a chromatographic column (Nova Pack C-18, 250 m × 4.6 mm). A mobile phase of water-acetonitrilephosphoric acid was used for the phenols (50:50:0.1) and for the acids (89.9:10:0.1) separation with the flow 0.8 mL/min.


Figure 5. Comparison between the DNNO approach and experimental data for the byproducts and final compounds in the ozonation of 4-CPH at pH 12: (a) phenol (**); (b) hydroquinone (**); (c) maleic acid (**); and (d) oxalic acid (**) and the DNNO reconstruction (s).

The identification of byproducts and final products of ozonation was realized by the comparison of the retention time and the ultraviolet (UV) spectrums of the patterns at λ ) 210 nm. The corresponding organic compound concentration was measured at different reaction times, for the purpose of comparison with the decomposition dynamic generated by DNNO. 4. Results and Discussion 4.1. Results of the DNNO Training. The first stage in the DNNO numerical estimation is the corresponding training. To train the DNNO, the experimental “trainer” data is used. By applying the suggested DNNO, the phenols decomposition dynamics and the byproducts and final product trajectories then were reconstructed. This method was used with the experimental data supplied by the ozonation of phenols at pH 2.0 and 9.0. The adjustable parameters were selected by the “try-to-test” technique. Obviously, the DNNO parameters involved in each reaction, changes slightly, because the computer algorithm, realizing the DNNO numerical algorithm (ODE-1), is strongly dependent on the ozonation variables. As an example, the parameters derived by the training method (A, K1, and K2) in the 4-chlorophenol ozonation at pH 2 are presented in Table 1. The training represents the most important part of the DNNO design, because this guarantees better possible correspondence between the experimental data and the reconstructed variables given by the state estimator. Figures 1a-f represent the

experimental ozonograms for three phenols and their numerical reconstructed decomposition in ozonation at pH 7 and pH 12. In Table 2, the physical sense of each component xˆ s,t (s ∈[1,l]) of the estimated state-vector is presented. Similar tables can be proposed for the reactions of ozone with other phenols. A reduced version of this table, which describes the corresponding byproducts and accumulated acid, will be presented below. 4.2. State Estimation Results. The main objective in the DNNO design is to reproduce the trajectories given by a partially unknown nonlinear system, without the complete and exact model usage. As mentioned, the quality of the training process guarantees the quality of the estimation technique. The previous section introduces the “best possible” (at least in a computational sense) DNN parameters to make the estimation error as small as possible. The observing (estimation) process is proved at pH values of 7.0 and 12.0, considering that three phenols (PH, 4-CPH, and 2,4-DCPH) were studied at four different pH values; however, the data at pH 2.0 and pH 9.0 were used to train the DNNO. Table 3 shows the byproducts and the final acids obtained in ozonation at different pH values, as identified via the HPLC technique. The basic byproducts in the phenols ozonation are catechol and hydroquinone, as well as muconic, fumaric, and maleic acids, whereas the final compounds are acids with low molecular weight (such as oxalic and formic acids). The DNNO application in the phenols ozonation allows

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Figure 6. Comparison between the DNNO approach and experimental data for the byproducts and final compounds in the ozonation of 2,4-DCPH at pH 7: (a) phenol and 4-chlorophenol (**); (b) hydroquinone (**) and catechol (++); (c) muconic acid (**), maleic acid (++), and fumaric acid (]]); and (d) oxalic acid (**) and the DNNO reconstruction (s).

all byproducts and accumulated acids to be reconstructed, using the corresponding on-line data (gas-phase ozone concentration) at pH 7 and pH 12 and a suitable training that generates the adequate static parameters described previously. (See Figures 2 and 3.) The estimation capability for the introduced DNNO approach is not only for specific reaction conditions. This fact can be demonstrated by phenol ozonation conducted at pH 12.0, where the combination of both mechanisms of ozonation (direct and indirect) is presented.39 In this case, the complete phenol decomposition is reached at 1000 s, compared with the previous examples at pH 7 that finished at 3000 s (see Figure 1b). Furthermore, the important differences between these two cases can be summarized as follows: (1) The reduction of the hydroquinone concentration is 10 times less than that at pH 7. (2) Muconic and fumaric acids are eliminated 10 times faster than at pH 2 and pH 7. (3) Oxalic acid is accumulated and formic acid was not formed at pH 12 (see Figure 3). The most important element on the DNN behavior is their capability to reconstruct a wide class of nonlinear system without previous exact knowledge of a system structure. Therefore, it is possible to apply the same observer for the ozonation of other phenols, but after a new training process based on different experimental data sets. For example, the decomposition of 4-CPH in ozonation is reconstructed using the corresponded ozonogram (see Figure 1c). In this case, the DNNO trained in

the phenol ozonation was used for the 4-CPH reconstructed destruction. It is possible to apply the same technique for the reconstruction of the intermediate and final products in 4-CPH ozonation at pH 7 and pH 12 (see Figures 4 and 5). In this case, the phenol formation during the 4-CPH dechloration with other intermediates as catechol, hydroquinone, muconic and malonic acids is observed. The final product is oxalic acid. In 4-CPH, ozonation at pH 12 reduces the decomposition time to 1000 s, in comparison to 3000 s at pH 7 (see Figure 1d). The mechanism of 4-CPH ozonation is changed, so catechol is not produced (see Figures 5a-d). The ozonation of 2,4-DCPH also was treated at four different pH values: pH 2, 7, 9, and 12. As previously stated, at pH 2.0 and pH 9, the DNNO is trained based on the ozonogram and the 2,4-DCPH decomposition dynamics. In these cases, the 4-CPH and phenol formation as products of the 2,4-DCPH dechloration also is observed. Furthermore, the similar intermediates and oxalic acid as a final product are formed (see Figure 6a-d). The pH increase has important effects on the decomposition of 2,4-DCPH: (1) The decomposition time is reduced from 4700 s at pH 2 up to 500 s at pH 12 (see Figure 1f). (2) The concentration of all byproducts is reduced. Therefore, the hydroquinone content decreases from 10-4 mol/L to 10-6 mol/L, and catechol is not identified at pH 12 (see Figures 6 and 7).


Figure 7. Comparison between the DNNO approach and experimental data for the byproducts and final compounds in the ozonation of 2,4-DCPH at pH 12: (a) phenol and 4-CPH (**); (b) hydroquinone (**); (c) fumaric acid (**) and maleic acid (++); and (d) oxalic acid and the DNNO reconstruction (s).

(3) The concentration of oxalic acid, as an accumulated product, increases by a factor of 5, in comparison with ozonation at pH 7 (see Figures 6d and 7d). All of these results show that the DNNO technique proposed in this study is extremely promising for the reconstruction of the phenols ozonation dynamics, as well as the byproducts formation and decomposition, and final product accumulation. These results are important, because the contaminant decomposition dynamics information may be used for a possible automatic algorithm design to control the ozonation process, using the DNNO in sequence of the toxic byproducts decomposition. This control could be applied to optimize the contaminants decomposition by means of a reduction in treatment time and reducing the ozone consumption. 4.3. Limits of the Proposed Reconstruction Method. Here, it should be emphasized that the estimation method cannot reproduce any dynamics that has not been considered during the training process, because the DNN state has its own constant order (i.e., the number of states that are involved in the observer description remains constant during the ozonation reaction).

decomposition, as well as the final compounds accumulation with DNNO training based on the experimental data of phenol decomposition. (2) The dynamics behavior, predicted by DNNO, shows sufficiently good correspondence with experimental data for byproducts, as well as for final compounds. (3) The phenol ozonation at the different pH values has been applied to demonstrate the robust (with respect to different experimental conditions) behavior of the suggested DNNO provided by the new learning laws, which includes the correction (switching) terms. (4) The effect of the pH increase on the decomposition dynamics of phenol, 4-chlorophenol, and 2,4-dichlorophenol is observed to be very significant, in that it reduces the total decomposition time by a factor of 10 (from 5000 s to 500 s). (5) The presence of the chloro species in the phenol molecules also reduces the degradation time in ozonation.

5. Conclusions

Appendix: Zone Convergence of the State Estimator

(1) The differential neural network observer (DNNO) approach shows good estimation of the phenols decomposition dynamics (the reconstruction), the byproducts accumulation and

Proof. Define the state estimation error as ∆t ) xt - xˆ t and the output error as et ) yt - yˆ t ) Cxt + ξ2,t - Cxˆ t ) C∆t + ξ2,t, for which the following identities hold:

5864


CTet ) CT[C∆t + ξ2,t] ) Nδ-1∆t + CTξ2,t - δ∆t

2∆Tt P1∆t e ∆tT[PA h +A h TP]∆t + ∆tTP[W02Λφ-1[W02]T + W01Λσ-1[W01]T]P∆t + ∆Tt P[Λf-1 + Λξ1-1 + Λ3-1] P∆t +

∆t ) Nδ(CTet - CTξ2,t + δ∆t)

∆Tt [Λ1 + Λ2] ∆t + σ˜ T(x, xˆ t)Λσσ˜ (x, xˆ t) +

The dynamics of ∆t is governed by the following ordinary differential equation (ODE):

T Λξ1ξ1,t + KT1 Λ3K1 + (φ(x, xˆ t)u)TΛφφ˜ (x,xt)u + ˜f tTΛf ˜f t + ξ1,t

(

T 3ξ2,t Λξ2ξ2,t - 2∆Tt PK2

dxˆ t ) A(0)∆t + W ∆˙ t ) x˘ t ˜ 1,tσ(xˆ t) + W01 σ˜ (x, xˆ t) + dt ˜ (x, xˆ t)u + ˜f t + ξ1,t - K1(yt - yˆ t) W ˜ 2,tφ(xˆ )ut -W02φ {et} K2 {|et|} where

˜ 2,t (xˆ )ut + et CNδPW T

)

C∆t + ξ2,t + etTCNδPW ˜ 1,tσ(xˆ t) + |C∆t + ξ2,t| T σ (xˆ t)W ˜ 1,t PNδΠ1NδPW ˜ 1,tσ(xˆ t) + T T ˜ 2,tPNδΠ2NδPW ˜ 2,tφ(xˆ )ut [φ(xˆ )ut] W T

where

∏j :) CT Λ-1 j

(for j ) 1, 2)

Selecting K2 ) -kP-1CT and applying the inequality described in expression 60 from ref 39, we derive

σ˜ t :) σ(xt) - σ(xˆ t)

∆Tt PK2

and

(C∆t + ξ2,t) |C∆t + ξ2,t|

) k(C∆t)T

C∆t + ξ2,t |C∆t + ξ2,t|

g

n

φ˜ (x, xˆ t) ) φ(x) - φ(xˆ t)

k(

|[C∆t]i| - 2xn|ξ2,t|) g k(xRP|∆t|P ∑ i)1

xδ∆ ∆ - 2xn|ξ T t

Define the energetic function as T ˜ 1,t W ˜ 1,t} + V :) V(∆,xˆ ,W ˜))V h (xt) + |∆|P2 + 2-1k1,t-1 tr{W T ˜ 2,t W ˜ 2,t} 2-1k2,t-1 tr{W

The matrix inequality introduced above allows the time derivative of the first term of the Lyapunov function to be represented as

lφν0Λ′φ + Q0]∆t + ∆Tt P[W02Λφ-1[W02]T + W01Λσ-1[W01]T]P∆t + ∆Tt P[Λf-1 + Λξ1-1 +

d -2 T V h (xt) + 2∆Tt P∆˙ t -2-1k1,t k˙ 1,t tr{W ˜ 1,t W ˜ 1,t} + dt -1 T -2 T tr{W˙ 1,t W˙ 1,t} - 2-1k2,t k˙ 2,t tr{W ˜ 2,t W ˜ 2,t} + k1,t

Λ3-1]P∆t - |∆t|Q02 + |∆f|(f˜0 + ˜f 1|xt|Λ˜f 1) + γ1 + KT1 Λ3K1 + 3γ2 - 2kxRP |∆t|P + 2kxδ∆Tt ∆t +

-1 T tr{W ˙ 2,t W˙ 2,t} ) 2∆Tt P1∆˙ t + k2,t

T ˜ 1,t PδCTetσT(xˆ t)} + 4xnk|ξ2,t| + tr{W

T [-2-1k1,t-1k˙ 1,tW ˜ 1,t + W˙ 1,t]} + tr{k1,t-1W˙ 1,t T [- 2-1k2,t-1k˙ 2,tW ˜ 2,t + W˙ 2,t]} tr{k2,t-1W˙ 2,t

Using the last relation and the corresponding state estimation error dynamics, the next identity could be stated:

˜ 1,tσ(xˆ t) + W01σ˜ (x, xˆ t)] + ∆Tt P∆˙ t ) ∆Tt P[A(0)∆t + W ˜ 2,tφ(xˆ )ut + W02φ ˜ (x, xˆ t)u + ˜f t] + ∆Tt P ∆Tt P[W ξ1,t - K1(yt - yˆ t) - K2(yt - yˆ t) |yt - yˆ t|

[

]

Notice that ∆Tt PA(0)∆˙ t ) 1/2∆Tt [PA(0) + (A(0))TP]∆t. The remaining terms in the last equation then can be estimated using the matrix inequality

XTY + (XTY)T e XTΛ-1 X + YTΛY X,Y ∈ R

2,t|)

2∆Tt P1∆t e ∆Tt [PA h +A h TP]∆t + ∆Tt [Λ1 + Λ2 + lσΛ′σ +

Its time derivative is

V˙ )

t

n×m

,

0

Reconstruction of Dynamics of Aqueous Phenols and Their Products

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