Detecting Bifurcations in an Electrochemical Cell Employing an

Article pubs.acs.org/JPCA

Detecting Bifurcations in an Electrochemical Cell Employing an Assisted Reference Model Strategy E. Ramírez-Á lvarez,*,† M. Calderón Ramírez,† R. Rico-Martínez,† C. González-Figueredo,‡ and P. Parmananda§ †

Departamento de Ingeniería Química, Instituto Tecnológico de Celaya, Av. Tecnológico y A. García Cubas s/n, 38010 Celaya, Guanajuato, México ‡ Departamento de Ingeniería Química, Universidad Jesuita de Guadalajara, Periférico Sur Manuel Gómez Morín 8585, Tlaquepaque, Jalisco 45604, México § Department of Physics, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India ABSTRACT: Bifurcations are detected in experiments involving the electrochemical oxidation of copper in phosphoric acid. Ideally, a bifurcation detection protocol requires a model with good qualitative and quantitative prediction capabilities that is also able to reproduce the drift inherent to the electrochemical experiment. A generic protocol for manipulation of nonlinear systems is proposed to achieve this goal. This protocol is based on a good qualitative predictor coupled to a modified Kalman filter that corrects the quantitative errors of the model point by point, adapting to the slowly varying conditions imposed by the electrochemical drift. for bifurcation detection have been proposed.4,6 Every instability has its own signature and devising methodologies to reveal and detect such signature has served as a way of marking its onset.7 In addition, a protocol for bifurcation detections assisted by a model with good predictive capabilities has been previously proposed.8−10 This protocol has been tested in microscopic simulations11 and in electronic circuits.9 The protocol has some limitations that hinder its widespread application, the principle drawback being its dependence on a robust model required to appropriately reject noise, and a starting point relatively close to the bifurcation, because it is based on mimicking the normal form of the desired bifurcation.12 These limitations contraindicate its application in systems with slow-moving time-dependent components, such as electrochemical systems. The characterization of a system dynamical response usually starts by efforts to model the occurrence of the underlying physicochemical phenomena based on first-principles. The resulting models, however, often lack the accuracy in describing the experiment observations, or may be too complex for realtime applications. For electrochemical systems, common mechanisms for the recurrent observation of certain types of nonlinear phenomena have been identified (see, e.g., ref 13 and references therein) but its translation to models with quantitative capabilities remains a scientific challenge. In parallel, the application of empirical data-driven modeling techniques has been proposed as a mechanism to gain

I. INTRODUCTION The location of bifurcations is one of the most important tasks associated to the study of the nonlinear dynamical response of real systems (see, e.g., refs 1−3). Manipulation, optimization, and control of nonlinear systems in all science areas require previous knowledge of the dynamic characteristics such as unstable states, bifurcations and bistability regions and different methods have been proposed to assist in this task. Bifurcation detection in nonlinear systems is usually carried out by fixing the parameter value and letting the systems stabilize at a fixed point or an oscillatory state. The process is repeated by slightly perturbing the operating parameter values until a qualitative change in the dynamics is observed. One says that a bifurcation occurs somewhere between the last two fixed parameter values, and a new window in parameter space is delimited to continue with the search for more bifurcations. This method has several drawbacks, the most obvious is the large amount of experimental time required in the vicinity of bifurcations, because one has to wait for the dynamics to settle to the final attractor. Furthermore, the presence of “catastrophic bifurcations”, which brings about the disappearance of stable states, such as saddle-node bifurcations, may require frequent resetting of the experiment. Moreover, the presence of inherent slow time-dependences may prove the pinpointing of a bifurcation an almost impossible task. However, because the location of a bifurcation is vital for controlling the systems in their vicinity, this remain a problem of interest (see, e.g., ref 4 and references therein). Among the different approaches, Langer and Parlitz5 proposed a method for bifurcation location for periodically driven systems. More recently, several model-free approaches © 2012 American Chemical Society

Received: October 13, 2012 Revised: December 27, 2012 Published: December 31, 2012 535

dx.doi.org/10.1021/jp310127a | J. Phys. Chem. A 2013, 117, 535−540

The Journal of Physical Chemistry A

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understanding on experimentally observed sequences of transitions. 14 Because these empirical models rely on experimental observations they quite often retain certain predictive capabilities. However, the complexity of bifurcations scenarios, and the lack of robust adaptive mechanism for improving their prediction online with the experiments, has hindered their use for nonlinear system manipulation. Previously,15 we had proposed an assisted reference model strategy and implemented it successfully for the induction of chaotic dynamics observed during the anodic oxidation of copper in phosphoric acid. These dynamics were described via next minimum maps and the system was perturbed in the vicinity of stable periodic orbits based on the reference model. Feedback and predictive control policies were successfully implemented. The promotion of the chaotic dynamics proved robust, even in the presence of a significant electrochemical drift. In this contribution, we build upon the previously proposed bifurcation detection protocol9 by incorporating this robust reference model strategy. This protocol is assisted by a model with good qualitative predictive capabilities, an artificial neural network model for this illustration, coupled with the modified Kalman filter, enhancing its noise-rejection and adaptation capabilities. The model serves as a reference to predict the bifurcation point location, improving the prediction as the bifurcation is reached, and assisting the protocol in keeping the dynamics nearby, once the system has been driven to the bifurcation.8,9 This strategy is implemented in the copper electro-oxidation in phosphoric acid in a parameter region where a forward and a reverse period doubling bifurcations are observed under potentiostatic conditions. The paper is organized as follows: First, the experimental system used in the illustration is described. In section III, the bifurcation detections protocol and the generic reference model, which constitute the methodological approach, are presented. In section IV, experimental results involving the pinpointing of the forward and reverse period doubling bifurcations are discussed. Finally, in section V, brief conclusions are outlined.

Figure 1. Schematic of the experimental setup for the anodic copper electro-oxidation in phosphoric acid.

current (C) using a data acquisition and a control card of 12 bits at 500 Hz sampling frequency. For the above-mentioned experimental conditions one observes periodic attractors undergoing a period doubling bifurcation closely followed by a similar reverse period doubling bifurcation as V (anodic potential) is continuously varied.

III. BIFURCATION DETECTION STRATEGY The previously proposed bifurcation detection protocol8,9 is modified to incorporate the use of the hybrid reference model. The protocol consists of four modules. The first module relies on a reference data-driven model. For our illustration, a static model with “good” predictive capabilities coupled with a modified Kalman Filter is used. The system’s dynamical evolution is used as feedback to the model assisting in the prediction. In this manner, model inaccuracies are compensated, accounting also for slow time-dependent behavior (drift) of the system. This “corrected” model is used to estimate the bifurcation location coupled with the criticality conditions (second and third modules) that define the bifurcation sought. The location of the bifurcation point is estimated using a Newton−Raphson solver. The final module involves a control policy that seeks a parameter sequence that, after a few iterations, will drive the system to the bifurcation. An Artificial Neural Network (ANN) is used to build the static component of the reference model. ANN’s have proven, for over two decades, to be a good alternative for describing complex dynamical transitions from experimental data.18,19 The ANN is trained off-line by using time-series collected from the experiment, and confirming a prediction of the bifurcation scenario consistent with the dynamical transitions implicit in the data. In this case, prediction of a pair of period doubling bifurcations of periodic orbits is achieved, as described below. The ANN is built in the form of a discrete mapping based on a next-maximum map representation which is appropriate for pinpointing local bifurcations of period orbits. For discrete mappings, a period-doubling bifurcation is located at the solution of the following conditions:

II. EXPERIMENTAL SETUP The experimental system used consisted of an EG&G Princeton Applied Research Model 616 electrochemical cell with three electrodes: anode, cathode, and the reference electrode. The electrodissolution of copper in phosphoric acid under potentiostatic conditions16,17 is used for the illustration of the protocol application. The anode (working electrode), is a rotating copper disk (5 mm diameter) shrouded by epoxy resin. This electrode was operated at 6500 rpm for all experiments. The electrolyte solution consists of 125 mL of an orthophosphoric acid (85%, Merck), maintained at a constant temperature of −12 ± 0.5 °C. The cathode is a platinum foil 10 cm2 area. The distances between anode and reference electrode and anode and cathode were fixed at 25 mm. The anodic potential is measured relative to a standard saturated calomel reference electrode (SCE). Figure 1 shows a schematic of this experimental setup. An EG&G Princeton Applied Research potentiostat, Model 263A, was used to manipulate the anodic potential applied to the system (V), which serves as both the control and the bifurcation parameter. The dynamical system response is described by observing the anodic current C. Time series were collected and stored in a computer by sampling the anodic

Ci − fi (C ,V ) = 0 det(I + J ) = 0

(i = 1, ..., n)

(1) (2)

where f represents the mapping contained in the reference model, C is the system state (observed current for the experiment), V is the parameter (applied anodic voltage), I is the identity matrix, and J is the Jacobian of the model mapping. The solutions of these equations, Ccr and Vcr, constitute the bifurcation point coordinates. 536



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manipulating this desired finite horizon, one can prevent the emergence of abrupt changes in the control parameter, smoothing out the approach to the bifurcation thus improving the performance of the bifurcation detection protocol. The receding horizon control policy is formulated as an optimum control problem for a finite horizon, generating an open loop control trajectory of the state. Further details can be found in references.22,23 Overall convergence is declared when the experiment has been driven to the bifurcation vicinity within a preset tolerance, for three consecutive recalculations of the correction terms described above. The control policy is designed to retain the system dynamics in this vicinity of the bifurcation point for as long as desired.

The successful application of this protocol depends directly on the reference model robustness because the estimated bifurcation coordinates are used as set point of the control problem posed in the fourth module. In this sense, by adding adaptation capabilities, based upon the modified Kalman Filter, one provides a mechanism for modeling errors compensation, thus improving the robustness of the overall modeling strategy. The Kalman filter20 was proposed in the 60s as a mechanism to filter out noise in linear dynamical systems.21 It is a predictivecorrective numerical tool, and an optimum filter in the sense that it minimizes the estimated covariance of the error in experimental measurements for linear systems. The modified version takes into account the nonlinearity of the system by using the static model as predictor of the curvature of the vector field. Thus, the feedback correction is modified as follows:12 − Ĉ k

=

− f (Ĉk − 1 ,V )

(3)

zk = f (zk̂ − 1 ,V ) Ĉ k =

− Ĉ k

+ K k (z k −

IV. EXPERIMENTAL RESULTS Potentiostatic scans were carried out to locate the window in the parameter space where the period doubling bifurcations take place. In this window of potential, between 463 and 490 mV, current time series at different fixed anodic potentials were collected. These data were recorded from a single experimental run. The time series were discretized by constructing the next minimum maps. These next minimum data were used as training set for the ANN static reference model. The training set was formed by thousand triads of potential V, minima current Cn, and next minima current Cn+1 data. This model predicts the next minimum in the current oscillations using the current minimum and the applied voltage as inputs. The ANN architecture is a feed-forward four layers regular network with two neurons in the input layer, eight in the hidden layers and one in the output layer. In Figure 2, orbit diagrams for the training set and the corresponding ANN prediction are shown. The ANN training is carried out by minimizing the squared errors between the predicted and the experimental data on the training set. A hundred iterations in the algorithm of conjugate gradient were used in the minimization. A second data set, different from training set, is required to test the prediction and declare convergence. For the model used here, the prediction error was below 3% for all points in both the training and test sets. After a period doubling bifurcation, period-2 dynamics are observed. Figure 3 shows the experimentally observed period-2 autonomous oscillations of the copper electro-oxidation recorded at 470 mV. The figure shows the current time series (a), attractor reconstruction using time-delays (b), and next minimum map representation (c) of the data. Using these tools, and the data from the full scan, one can pinpoint that the transition from the single periodic orbit to the double-loop period orbit to occurs around 468 mV for the experimental set used to train the ANN static reference model. In the absence of modeling errors, one may be tempted to use this model as reference for automatic bifurcation location directly on the experiment. However, most electrochemical systems exhibit a time depending condition known as drift, resulting from surface changes during the oxidation of the metals (e.g., pitting24). Recently,25 an empirical procedure to attenuate the drift effect in an electrochemical system has been presented based on applying a compensation signal for the “poisoning” process that prevents the reestablishment of the coverage of the adsorbed species. However, the signal enhancement achieved, although increases significantly the time window of occurrence for some experimentally observed types of behavior, may not be sufficient for elucidating

(4) − Ĉ k )

(5)

where f is the reference model, zk is the observed state of the experimental system, K is the Kalman gain, Ĉ −k is the a priori prediction of the state, Ĉ k is the corrected prediction of the state. The updating of the covariance matrix remains the same as for the original Kalman scheme. A complete description of the scheme can be found in.15 In this manner, the correction term (zk − Ĉ −k ) is now an estimator of modeling errors that is the subsequent feedback for the static model prediction, thus achieving an adaptive corrected prediction. The correction term is continuously updated as the experiment progresses. Each experimental-state recording, minimum peak in the current time-series, is used to make a corrected prediction of the next minimum with the reference model coupled with the modified Kalman filter. For the second module, bifurcation estimation, the correction term is kept constant, by running the experiment at a fix parameter value and waiting until the covariance update converges (usually after four or five experimental points). Once the overall protocol converges to the bifurcation, this constant correction term is recalculated to verify this convergence. The above-described scheme compensates for the inaccuracies of the model in predicting the state evolution; however, model inaccuracies may also be present in the estimation of the bifurcation location due to the poor prediction of the dependence on the parameter. Motivated by the theory of normal forms of local bifurcations, all postulating linear parameter dependence in the bifurcation vicinity, a linear compensation of this dependence is used. It is also used as a constant for the bifurcation estimation module (second module), and it is calculated as the difference between the static model prediction of the derivative of the state with respect to the parameter, and its numerically calculated counterpart using the experimental data. Such compensation is, once again, recalculated after the protocol convergence is declared, to test the accuracy of the bifurcation location estimation. For the fourth module of the bifurcation protocol, a receding horizon control law is used as control strategy. The receding horizon is based on seeking an optimum trajectory to reach the bifurcation22,23 in a predetermined number of iterations. By 537



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Figure 2. Orbit diagrams for (a) training set taken from the experimental data and (b) reference model (ANN) predictions.

bifurcations occurrence and location. The drift, as a uncharacterized time dependent condition, remains an important challenge for any bifurcation detection algorithm. For our experiment, the drift translates in the “shift” of the observed period-2 state. As observed in Figure 2, the bifurcation point, originally recorded at 470 mV, shifts by a few millivolts in the parameter window after running the experiment continuously for a few minutes. Moreover, the location of the bifurcation can be different from one experimental run to another because of the electrochemical drift. In this case, the difference is the result of minute changes in the electrode surface as prepared for the experiment. As a result, the parameter window at which a particular bifurcation is observed may be shifted a few millivolts, and this window may be slightly expanded or compressed. Thus, the prediction of the reference ANN model bifurcation scenario will be similar but at a neighboring location, in parameter-state space, with respect to the observed scenario in any other experimental run except the one used to collect the training data. These unavoidable differences are compensated by the coupling of the model prediction with the modified Kalman Filter, as illustrated below. Owing to the presence of two different period-doubling bifurcation points in close vicinity to each other in the parameter space, under the experimental conditions described in section III, the starting point will define to which bifurcation the system is driven by the protocol. Figure 4 illustrates the protocol application with the potential set at 465 mV, a point

Figure 3. Autonomous period-2 dynamics of the anodic copper electro-oxidation in phosphoric acid under potentiostatic conditions: (a) current time series; (b) attractor reconstruction using a time-delay τ = 0.05 s; (c) next minimum map.

closer to the leftmost bifurcation point, as starting point for the bifurcation location protocol. Figure 4 shows the trajectory experimentally observed. The protocol finds the presence of the bifurcation at 468.5 mV, at a current value of 0.644 mA and maintains the system in the close vicinity of the bifurcation limited by the precision of the data-acquisition system. The figure also shows the time series of the current collected and the reconstructed single loop attractor. Numerical calculation of the multiplier of the trajectory confirms the bifurcation location. Figure 5 shows the trajectory for a starting condition closer to the right-most bifurcation point. The potential was set at 475 538



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Figure 4. Application of the bifurcation detection protocol. The starting point was the single loop periodic orbit observed at 465 mV. (a) Time-series of the final state in the vicinity of the bifurcation, (b) phase-space trajectory, and (c) route to the bifurcation driven by the protocol assisted by the hybrid reference model.

Figure 5. Application of the bifurcation detection protocol. The starting point was the double-loop periodic orbit observed at 475 mV. (a) Time-series of the final state in the vicinity of the bifurcation, (b) phase-space trajectory, and (c) route to the bifurcation driven by the protocol assisted by the hybrid reference model.

mV. Once again the protocol smoothly drives the system to the bifurcation finding it for a potential value of 478 mV, and a current value of 0.68 mA. Figure 5 shows also the time series of the current collected once convergence was declared, along with the attractor reconstruction. Numerical calculation of the multipliers of the map representation, once again, confirms the presence of the underlying bifurcation. Note that the values at which the bifurcations are found for the runs presented in Figures 4 and 5 differ slightly from those predicted by the ANN model (Figure 2), as expected by virtue of the drift discussed above. Even though this drift effect is not reproduced by the empirical ANN model, the generic reference

model coupled with the modified Kalman filter is robust in the sense that it can locate the bifurcation and maintain the system near this unstable point in experiments with different drifts. The experimental runs were repeated at different days in conditions where drift induced shifts as large as 10 mV were observed without a significant loss in performance of the protocol. 539



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Electrochemistry; Conway, B. E., Bockris, J. O., White, R. E., White, R. E., Vayenas, C. G., Eds.; Modern Aspects of Electrochemistry; Springer: New York, 2002; Vol. 32, pp 1−142. (14) Kevrekidis, I. G.; Rico-Martínez, R.; Ecke, R.; Farber, R. M.; Lapedes, A. S. Physica D 1994, 71, 342−362. (15) Ramrez-Á lvarez, E.; Rico-Martínez, R.; Parmananda, P. J. Chem. Phys. A 2010, 114, 12819−12824. (16) Kiss, I. Z.; Gáspár, V.; Nyikos, L.; Parmananda, P. J. Phys. Chem. 1997, 101, 8668−8674. (17) Schell, M.; Albahadily, F. N. J. Chem. Phys. 1989, 90, 822−828. (18) Chu, S. R.; Shoureshi, R.; Tenorio, M. IEEE Control Systems Magazine 1990, 10, 31−35. (19) Chen, S.; Billings, S. A. Int. J. Control 1992, 56, 319−346. (20) Kalman, R. E. Trans. ASME, J. Basic Eng. 1960, 35−45. (21) Xiong, K.; Liang, T.; Yongjun, L. Acta Astronaut. 2011, 68, 843− 852. (22) Acar, L. Some examples of the decentralized receding horizon control. Proc. 31st IEEE Conf. Decision Control 1992, 2, 1356−1359. (23) Alessandri, A.; Baglietto, M.; Battistelli, G. Automatica 2008, 44, 1753−1765. (24) Frankel, G. S. J. Electrochem. Soc. 1998, 145, 2186−2198. (25) Nagao, R.; Sitta, E.; Varela, H. J. Phys. Chem. C 2010, 114, 22262−22268.

V. CONCLUSIONS A successful automatic experimental location of perioddoubling bifurcations during the copper electro-dissolution has been demonstrated. A protocol relying on the predictive qualities of a hybrid modeling strategy, allowing compensation of unmodeled time dependencies like the drift effect exhibited by the electrochemical system was employed for the location of the bifurcation points. Previously, the same hybrid modeling strategy, consisting in the coupling of a reference model with semiquantitative capabilities with a modified Kalman filter, was also successfully applied to a nontrivial chaos promotion problem.15 These illustrations suggest that the strategy may serve as a generic model reference platform for manipulation, control and optimization of nonlinear systems. Currently, we are exploring a hybrid modeling strategy for possible application to the manipulation of systems with spatiotemporal dynamics, as well as applications that involve noise provoked resonances (coherence resonance). In parallel, we are extending the efforts described here for devising robust control strategies for the automatic experimental detection of local catastrophic bifurcations (saddle-node type) in the presence of similar time-dependent unmodeled components as the ones described in this work.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge the Mexican Council of Science and Technology (CONACYT) for continuous support for this work. We are grateful to Prof. Katharina Krischer for her valuable insight on electrochemical systems.

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REFERENCES

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Detecting Bifurcations in an Electrochemical Cell Employing an

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