Complexity

Volume 2017, Article ID 9391879, 16 pages

https://doi.org/10.1155/2017/9391879

## Neural Network-Based State Estimation for a Closed-Loop Control Strategy Applied to a Fed-Batch Bioreactor

^{1}Instituto de Ingeniería Química, Universidad Nacional de San Juan, CONICET, Av. Lib. San Martín Oeste, 1109 San Juan, Argentina^{2}Instituto de Automática, Universidad Nacional de San Juan, CONICET, Av. Lib. San Martín Oeste, 1109 San Juan, Argentina^{3}Facultad Regional Santa Fe, Universidad Tecnológica Nacional, CONICET, Lavaisse 610, Santa Fe, Argentina^{4}Instituto de Desarrollo Tecnológico para la Industria Química (INTEC (UNL-CONICET)), Güemes, 3450 Santa Fe, Argentina

Correspondence should be addressed to Mario Serrano; ra.ude.jsnu.if@onarrese

Received 15 June 2017; Accepted 9 July 2017; Published 5 September 2017

Academic Editor: Chenguang Yang

Copyright © 2017 Santiago Rómoli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The lack of online information on some bioprocess variables and the presence of model and parametric uncertainties pose significant challenges to the design of efficient closed-loop control strategies. To address this issue, this work proposes an online state estimator based on a Radial Basis Function (RBF) neural network that operates in closed loop together with a control law derived on a linear algebra-based design strategy. The proposed methodology is applied to a class of nonlinear systems with three types of uncertainties: (i) time-varying parameters, (ii) uncertain nonlinearities, and (iii) unmodeled dynamics. To reduce the effect of uncertainties on the bioreactor, some integrators of the tracking error are introduced, which in turn allow the derivation of the proper control actions. This new control scheme guarantees that all signals are uniformly and ultimately bounded, and the tracking error converges to small values. The effectiveness of the proposed approach is illustrated on the basis of simulated experiments on a fed-batch bioreactor, and its performance is compared with two controllers available in the literature.

#### 1. Introduction

In the last decade, several control strategies applied to different bioprocesses have been extensively investigated (e.g., [1–5]). In the particular case of fed-batch bioreactors, some nonlinear control strategies were developed to improve the process efficiency and the “batch-to-batch” reproducibility [6], as well as to deal with the unavoidable uncertainties typically present in bioprocesses [7].

Many articles have investigated the effects of model uncertainties on the performance of model-based nonlinear control strategies typically applied to bioprocesses. For example, the “yield coefficient of cell to glucose” in a bioprocess is a time-varying parameter often assumed as a constant. Also, the “specific growth rate” can be represented through the Monod-Haldane model [8], but it is possible that the model becomes inaccurate because of yeast growth inhibition due to an eventual presence of ethanol. In principle, three main types of uncertainties can be envisaged [9–13]: uncertain and/or time-varying model parameters, uncertain nonlinearities, and unmodeled dynamics. Additionally, nonmodeled external perturbations can disturb the process. In general, model uncertainties and external perturbations can cause severe risks in bioreactors, such as leading to undesirable operating conditions [14–16]. For instance, inadequate modeling of the inhibitory effect of a substrate can lead to a structural instability in the dynamical behavior of a bioprocess [15].

Adaptive control strategies are a valuable choice to control bioreactors due to their ability to compensate for parametric uncertainties. The Nussbaum gain method has been applied to control a bioprocess of unknown and time-varying control gain and parameters [17]. A fed-batch bioreactor control with simultaneous dynamic identification of unknown state and parameters has been investigated through an algorithm based on the auxiliary model principle with adaptive controllers [18]. A control strategy was proposed to track a desired profile for the biomass of a yeast culture in a high-density fed-batch bioreactor [19]. This last strategy was based on a model that included an uncertainty term estimated through an adaptive technique. Also, an adaptive nonlinear feedback controller has been designed to automatically estimate uncertain time-varying parameters in order to guarantee the closed-loop control of a waste water plant [20].

Fuzzy control systems have been successfully applied to several bioreactors. For instance, [12] considered an adaptive fuzzy-decentralized control for a system with unknown nonlinear uncertainties and dynamical uncertainties. The fuzzy system was used to estimate the nonlinear uncertainties. Also, a fuzzy controller for the alcoholic fermentation with* Saccharomyces cerevisiae *has been proposed [21]. It combined the capability of handling uncertainties with the ability of predictive control to improve the plant performance, making use of a neural network model of the nonlinear system. Optimal control strategies have also been applied to bioprocesses [22, 23]. In order to compensate for process uncertainties, this type of controllers incorporates an additional term to the bioprocess model (typically, in the specific growth rate), which is estimated through an optimization technique. Alternative strategies for controlling bioreactors under model uncertainties are trajectory-based control [24], hybrid control [25], and model predictive control [26]. Additionally, due to the inherent nature of a bioprocess, a control scheme based on a stochastic model seems to be adequate. The specific growth rate can be modeled as affected by a white noise representing environment uncertainties, and the mathematical model consists of a set of stochastic differential equations [27] (some applications can be consulted in [28, 29]). The application of this approach onto a real plant is quite difficult due to its complex design and implementation.

Based on a linear algebra methodology, a control strategy has been applied for trajectory tracking in a bioprocess [30]. The technique was used to calculate the control actions that ensure the tracking of independently determined optimal trajectories of some process variables. To this effect, a discrete approximation of the original process model must first be derived, and the control actions are then calculated through a least square problem. This methodology has been applied to a fed-batch bioprocess designed to produce a secreted protein, and it was originally modeled by Park and Ramirez [31]. Optimal profiles of such bioprocess have also been published [32].

Many nonlinear control techniques of bioprocesses assume that the feedback variables can be measured online [33, 34]. However, such hypothesis is often unrealistic and therefore some process variables must be estimated through virtual sensors (or soft sensors). There are several kinds of virtual sensors published in the literature which exhibit different abilities to deal with model mismatches and perturbations. In this work, an artificial intelligence-based observer has been chosen because this type of estimator is especially recommended for highly nonlinear systems in the presence of uncertainties [35].

This paper aims at designing a controller for the multivariable tracking of optimal profiles when the process is affected by uncertainties in the dynamic model and/or in the model parameters. Unmeasured process variables are estimated through a RBF neural network. The proposed approach is evaluated onto a bioprocess originally modeled by [31], with the optimal profiles determined by [32] in absence of model errors. Brief descriptions of the fed-batch bioreactor and the controller derived in [30] are presented in Appendix A.

In comparison with [30], the proposed approach prevents the polynomial uncertainties from affecting the tracking control. To achieve this aim, an integrative term is incorporated in the linear algebra methodology originally proposed. Then, the conditions in order for a system of linear equations to have exact solution are analyzed. The control action is obtained by solving the linear system, even though the original process model is nonlinear.

The main contribution is that the original method proposed herein ensures the convergence to zero of the tracking errors under additive uncertainties. Another important contribution is the inclusion of RBF neural network estimators: the control issue is addressed considering more realistic conditions with nonmeasurable variables. Further, the stability analysis is made using Lyapunov theory in discrete time.

The article is organized as follows. In Section 2, the controller design methodology and the convergence to zero of the tracking errors are presented. Section 3 shows briefly the description of the RBF estimator. A method to tune the controller is presented in Section 4, along with four testing cases designed to check the effectiveness of the new control law. Main conclusions of the work are summarized in Section 5.

#### 2. Controller Design

Derivation of the control strategy is based on the mathematical model presented in Appendix A. The effect of an additive uncertainty () is modeled by extending (A.4) as follows: where is the state vector, and are the input vectors, is the sample time, is the control action, and is an additive uncertainty. In principle, can be used to model either perturbed systems or a wide class of model mismatches. In what follows, a control law is designed to compensate for the tracking errors introduced by .

Taking into account the fact that the mismatch might depend on both system states and inputs, we here consider a real plant described by . The additive uncertainty can be expressed as , where is the discrete-time nonlinear system model. If** z** and** u** are assumed to be bounded and* f* is Lipschitz, then can be modeled as a bounded uncertainty [36–38].

With the additive uncertainty of (1), the fed-batch bioreactor model of (A.3) becomesThe control action for the secreted protein concentration, , given by of [30], isThe discrete value of , , is calculated by replacing (3) into the first row of (2):where and depend on and , respectively. Thus, the tracking error of isBy combining (4) and (5), one obtainsThe Taylor interpolation of at its desired value isBy replacing (7) into (6), the tracking error for* P*_{n} becomesIf a similar procedure is followed for , , and , then the error equation becomes [30]Alternatively, in vector form,where is a nonlinearity that tends to zero when tends to zero. Equations (9a) and (9b) suggest a direct effect of the additive uncertainty on the tracking error. Therefore, the presence of an uncertainty avoids a convergence of tracking errors to zero (see Theorem 1 of [30]).

##### 2.1. Single Integral Action

In order to reduce the effect of , some integrators for the tracking errors will be introduced, depending on the time variation hypothesis of . We will assume that is unknown and each of its components is an -order polynomial.

*Remark 1. *The first-order difference of is defined as , the second-order difference is defined as , and, as a rule, the -th order difference is defined as .

*Remark 2. *The -th difference of a -order polynomial is zero.

Consider first a constant uncertainty . Then, . In this case, an integrator for each state variable will force the error to converge to zero. Call the continuous time error in the state vector. Then, the integral of the tracking error is defined asHence, the control action (A.7) will be computed by assuming a new term in (A.5); that is,where** K** and are the proportional and integral design parameters, respectively.

By adding one integrator to (9b) and using (10), one obtainsRewriting (12) for the next sample time,Combining (10) and (12) and substituting them into (13),Alternatively,Therefore,** K** and are chosen in order to ensure the stability of the linear system represented in the left-hand side of (15). To achieve this condition, the roots of such polynomial must lie inside the unit circle. Then, when ; that is, the error tends to zero as long as uncertainties are constant.

Once the integral of the error has been added, the control action is calculated following the same design procedure based on linear algebra presented in [30]. Therefore, the new control law can be obtained through the least squares procedure as

##### 2.2. Multiple Integral Actions

Assume that the uncertainty can be modeled by a function with a second-order difference equal to zero; that is, . Similar to (10), a double integrator is now introduced, and new variables and are defined:In this case, the control action (A.7) will include an additional term in (A.5) as follows:where** K**,** k**_{1}, and** k**_{2} are the proportional, integral, and double-integral design parameters, respectively. With a similar procedure to that used for deriving (15) and after adding two integrators ( and ), the error dynamics can be expressed as As suggested by (19), under the assumption of a constant or linear varying uncertainty , the uncertainty has no influence on the error dynamics. As in the case of the single integrator, the controller parameters** K**,** k**_{1}, and** k**_{2} should be chosen in order to ensure the stability of the left-hand side of (19).

The previous derivation can be generalized for uncertainties that can be approximated with a -order polynomial; and, therefore, the influence of upon would be cancelled off by addition of integrators.

The linear algebra-based procedure enables the calculation of the following control law:

#### 3. Neural State Estimator

The proposed feedback control technique requires reliable estimates of several state variables. The state estimation error, , is defined aswhere is the vector of estimated state variables and is the vector of online measured states. The following neural estimator represents the nonlinear dynamic of the bioreactor described in (7):where is the input vector to the neural estimator, is the optimal weight vector, is the neural approximation error, is the RBF that represents each neuron in the hidden layers, subindex indicates the neuron number of the RBF, and is the maximum number of neurons. Every neuron of the hidden layer is modeled aswhere and are two parameter vectors that represent the centers and widths of the RBF, respectively.

In the bioreactor under study, biomass cannot be measured online and the protein concentrations cannot be measured at all; then a state estimator function based on (22) is determined as follows:From the difference between (22) and (24), the neural identification error can be described bywhere represents the neural weight vector estimation error and it is defined asIn order to train the neural network for identification, an offline data set of will be used. This training data will be obtained after following the same procedure presented in [39] and then applied in [40, 41]. A similar procedure was also used in [42]. The learning rule utilized to train the neural network is demonstrated in the next theorem.

Theorem 3. *The model of the fed-batch bioreactor (see (A.1) and (2)) can be approximated by the neural network (24) using a neuronal adjustment law defined by*

*Proof. *The demonstration of this theorem is presented in Appendix B.

As recommended by [35], the estimator was evaluated to test its performance and avoid biased estimates. Table 1 summarizes the implemented tests, and their results were evaluated through the average values of the mean squared errors (MSE) corresponding to , , and . The obtained results show an acceptable performance of the RBF neural network for estimating the nonmeasurable states. Note that none of the MSE values exceeds the thresholds proposed by [43], . Figure 1 shows the proposed closed-loop architecture. Note that the controller requires information on both measured and estimated variables.