**1. Introduction**

Monitoring and control of biomass plays a vital role during fermentation process [1]. Low cost Production of ethanol/cellulase from lignocellulosic substances has been a topic of research for the past few years as it uses the carbon source from industrial or agricultural waste [2, 3]. These processes mostly involve fungi such as *Aspergillus,Trichoderma*. One of the main challenges during production of low cost cellulase is the estimation of biomass in the presence of insoluble solid substrates. The conventional methods such as dry weight filtration, optical density become unusable during fungal biomass estimation [4]. Methods such as monitoring DNA concentration, Image analysis were widely used. However, for the control of fermentation process, continuous monitoring of biomass is essential.

#### **2. Biomass estimation techniques**

The first principles model based on mass or energy balance equations is widely used in industries. Unstructured models such as monod model captures the process dynamics effectively only during log phase and on the other side, structured models have more parameters and are difficult to use [5]. The kinetic parameters in first

principles model such as specific growth rate, biomass yield are found using laborious experimentation methods [6].

New process monitoring approaches use several online sensors to determine the concentration of viable biomass that are found useful during control of fermentation process. A simple on-line method for fungal biomass estimation based on agitation rate has been evolved for DO stat cultures. The estimator is developed based on changes in dissolved oxygen concentration in the initial transient time and yield change. The estimation parameters are found using agitation rate at 20% of DO concentration [7].

**Dielectric spectroscopy** has been widely used for monitoring biomass during submerged fermentation. The working of capacitance probe is based on cell membrane polarization. When the microbial cells are placed in ionic solution and are subjected to alternating electric field, they will act as capacitor due to restriction of ion movement by cell membrane. The on-line capacitance value represents viable biomass as dead cells do not polarize [8]. The Biomass Monitor TM, model 214 M (Aber Instruments, Aberystwyth, UK), dual frequency version (0.2–1.0 MHz, and approximately 9.5 MHz) is the commonly used capacitance probe.

parameters. On comparison with other empirical models, neural networks are relatively less sensitive to noise and hence can be applied to process control systems with higher level of uncertainty [15]. During batch/fed-batch, ANN can be effectively used for estimation of biomass or product, optimization of fed-batch run and online control of bioprocess systems [16–18]. ANNs are suitable for many applications such as nonlinear filtering, prediction of output using input are widely used in the modeling of dynamic systems [19]. Several ANNs such as feed forward back-propagation neural network (FFBPNN) [20], Hopfield [21], radial basis function (RBF) networks [22], recurrent [23] and hybrid neural network (HNN) [24] found extensive application in bioprocess industries based on their functions. The BPNN with supervised learning has been reported in biofuel process modeling as shown in **Table 1**.

concentration

concentration

Lipid Productivity, biomass concentration

*, coefficient of determination.*

**Type**

**ANN structure**

— 1–2-1 — [3]

HNN 1–3-1 — [24]

FFBPNN 3–10-1 0.99 [25]

**R2 value**

**References**

**Input parameters Output parameters ANN**

*Soft Sensors for Biomass Monitoring during Low Cost Cellulase Production*

pH Biomass

*DOI: http://dx.doi.org/10.5772/intechopen.96027*

Sin (glucose) Biomass

Sin (glucose), sodium nitrate concentration, yeast extract

*Sin, initial substrate concentration; R2*

*Application of ANN in bioprocess industries.*

concentration

**Table 1.**

The Food and Drug Administration (FDA) of United States has initiated the online monitoring and closed loop control of a bioprocess via Process Analytical Technology (PAT) initiative. PAT highlights the concept of process understanding in order to deliver high quality products and this can be achieved by the design of accurate bioprocess models. The mathematical models are in general classified as

Black box models are input/output models that do not require *a priori* knowledge about the process and describe the system based on experimental data. An example of input/output model that has an output y which depends on past and

*B q*ð Þ

where *q* is the backward shift operator for the polynomials *A(q)* and *B(q)* as

ð Þ¼ *y t*ð Þ*<sup>i</sup> y ti*�*<sup>j</sup>*

*A q*ð Þ¼ <sup>1</sup> <sup>þ</sup> *<sup>a</sup>*1*q*�<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*2*q*�<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> *anq*�*<sup>n</sup>* (3)

*A q*ð Þ *u t*ð Þ*<sup>x</sup>* (1)

(2)

*y t*ð Þ¼ *<sup>x</sup>*

*q*�*<sup>j</sup>*

**3. Biomass modeling methods**

**3.1 Black box models**

**609**

black box, white box and gray box models.

present inputs is given in Eq. (1) as follows:

given in Eq. (2), Eq. (3) and Eq. (4)

Another method for estimation of viable biomass is the **radio-frequency (RF) impedance spectroscopy [9]**. This method provides useful information on the live cell concentration both in fixed as well as in dual frequency mode. To identify important changes in the process or to control the biomass at a constant level, the determination of on-line live cell concentration can be useful. **Electrochemical impedance spectroscopy (EIS)** is another method used to monitor biomass during fermentation [10]. The increase in biomass during cultivation is proportional to the increase in the double layer capacitance (Cdl), determined at frequencies below 1 kHz. A good correlation of Cdl with cell density is found and in order to get an appropriate verification of this method, different state-of-the-art biomass measurements are performed and compared. Since measurements in this frequency range are largely determined by the double layer region between the electrode and media, rather minor interferences with process parameters (aeration, stirring) are to be expected. It is shown that impedance spectroscopy at low frequencies is a powerful tool for cultivation monitoring. Though these dielectric spectroscopy, impedance spectroscopy techniques have been reported in literature, these methods require costlier instrumentation.

In recent years, soft sensors are widely used for the estimation of biomass. **Soft sensors** estimate the unknown state variable by using some other measured variables that influences the unknown state [11]. The data-driven methods widely used for the soft-sensor modeling are support vector machine, multiple least square support vector machine, neural network, deep learning, fuzzy logic and probabilistic latent variable models.

Artificial Neural Networks (ANN) based soft sensor have the capability to learn nonlinearity of the process using experimental plant data and thus can be used to estimate the state of bioprocess such as biomass concentration [12, 13]. The rapid development of algorithms and information technology is the major motivation behind the broad application of ANNs in research and development [14]. Currently, ANNs are employed in the prediction of various outcomes including process control, medicine, forensic science, biotechnology, weather forecasting, finance and investment and food science. However, it is noteworthy to state that the use of ANNs in biofuel production is currently in the early phases of its development. Generally, microbial fermentations exhibit non-linear relationships which could pose several problems during bioprocess modeling and optimization. The application of robust models such as ANN helps to capture this nonlinear behavior, and thus provides a model that links the process inputs to the corresponding output


#### *Soft Sensors for Biomass Monitoring during Low Cost Cellulase Production DOI: http://dx.doi.org/10.5772/intechopen.96027*

#### **Table 1.**

principles model such as specific growth rate, biomass yield are found using

New process monitoring approaches use several online sensors to determine the concentration of viable biomass that are found useful during control of fermentation process. A simple on-line method for fungal biomass estimation based on agitation rate has been evolved for DO stat cultures. The estimator is developed based on changes in dissolved oxygen concentration in the initial transient time and yield change. The estimation parameters are found using agitation rate at 20% of

**Dielectric spectroscopy** has been widely used for monitoring biomass during submerged fermentation. The working of capacitance probe is based on cell membrane polarization. When the microbial cells are placed in ionic solution and are subjected to alternating electric field, they will act as capacitor due to restriction of ion movement by cell membrane. The on-line capacitance value represents viable biomass as dead cells do not polarize [8]. The Biomass Monitor TM, model 214 M (Aber Instruments, Aberystwyth, UK), dual frequency version (0.2–1.0 MHz, and

Another method for estimation of viable biomass is the **radio-frequency (RF) impedance spectroscopy [9]**. This method provides useful information on the live cell concentration both in fixed as well as in dual frequency mode. To identify important changes in the process or to control the biomass at a constant level, the determination of on-line live cell concentration can be useful. **Electrochemical impedance spectroscopy (EIS)** is another method used to monitor biomass during fermentation [10]. The increase in biomass during cultivation is proportional to the increase in the double layer capacitance (Cdl), determined at frequencies below 1 kHz. A good correlation of Cdl with cell density is found and in order to get an appropriate verification of this method, different state-of-the-art biomass measurements are performed and compared. Since measurements in this frequency range are largely determined by the double layer region between the electrode and media, rather minor interferences with process parameters (aeration, stirring) are to be expected. It is shown that impedance spectroscopy at low frequencies is a powerful tool for cultivation monitoring. Though these dielectric spectroscopy, impedance spectroscopy techniques have been reported in literature, these methods

In recent years, soft sensors are widely used for the estimation of biomass. **Soft sensors** estimate the unknown state variable by using some other measured variables that influences the unknown state [11]. The data-driven methods widely used for the soft-sensor modeling are support vector machine, multiple least square support vector machine, neural network, deep learning, fuzzy logic and probabilis-

Artificial Neural Networks (ANN) based soft sensor have the capability to learn nonlinearity of the process using experimental plant data and thus can be used to estimate the state of bioprocess such as biomass concentration [12, 13]. The rapid development of algorithms and information technology is the major motivation behind the broad application of ANNs in research and development [14]. Currently, ANNs are employed in the prediction of various outcomes including process control, medicine, forensic science, biotechnology, weather forecasting, finance and investment and food science. However, it is noteworthy to state that the use of ANNs in biofuel production is currently in the early phases of its development. Generally, microbial fermentations exhibit non-linear relationships which could pose several problems during bioprocess modeling and optimization. The application of robust models such as ANN helps to capture this nonlinear behavior, and thus provides a model that links the process inputs to the corresponding output

approximately 9.5 MHz) is the commonly used capacitance probe.

laborious experimentation methods [6].

*Biotechnological Applications of Biomass*

DO concentration [7].

require costlier instrumentation.

tic latent variable models.

**608**

*Application of ANN in bioprocess industries.*

parameters. On comparison with other empirical models, neural networks are relatively less sensitive to noise and hence can be applied to process control systems with higher level of uncertainty [15]. During batch/fed-batch, ANN can be effectively used for estimation of biomass or product, optimization of fed-batch run and online control of bioprocess systems [16–18]. ANNs are suitable for many applications such as nonlinear filtering, prediction of output using input are widely used in the modeling of dynamic systems [19]. Several ANNs such as feed forward back-propagation neural network (FFBPNN) [20], Hopfield [21], radial basis function (RBF) networks [22], recurrent [23] and hybrid neural network (HNN) [24] found extensive application in bioprocess industries based on their functions. The BPNN with supervised learning has been reported in biofuel process modeling as shown in **Table 1**.

### **3. Biomass modeling methods**

The Food and Drug Administration (FDA) of United States has initiated the online monitoring and closed loop control of a bioprocess via Process Analytical Technology (PAT) initiative. PAT highlights the concept of process understanding in order to deliver high quality products and this can be achieved by the design of accurate bioprocess models. The mathematical models are in general classified as black box, white box and gray box models.

#### **3.1 Black box models**

Black box models are input/output models that do not require *a priori* knowledge about the process and describe the system based on experimental data. An example of input/output model that has an output y which depends on past and present inputs is given in Eq. (1) as follows:

$$y(t\_{\mathbf{x}}) = \frac{B(q)}{A(q)}u(t\_{\mathbf{x}}) \tag{1}$$

where *q* is the backward shift operator for the polynomials *A(q)* and *B(q)* as given in Eq. (2), Eq. (3) and Eq. (4)

$$q^{-j}(\mathcal{y}(t\_i)) = \mathcal{y}(t\_{i-j}) \tag{2}$$

$$A(q) = \mathbf{1} + a\_1 q^{-1} + a\_2 q^{-2} + \dots + a\_n q^{-n} \tag{3}$$

$$B(q) = b\_0 + b\_1 q^{-1} + b\_2 q^{-2} + \dots + b\_m q^{-m} \tag{4}$$

The input output experimental data is used to determine the values of variables a, b and the order of the polynomials *n* and *m*. Artificial Neural Network (ANN) models are another type of black box models that has wide application in bioprocess technology due to their ability to represent non-linear functions.

#### **3.2 White box models**

White box models are mechanistic models that are been used widely used incorporates the available process knowledge in the form of first principle model equations. As an example, the first principles model widely used for fungal fermentation during cellulase production are represented in Eq. (5), Eq. (6), Eq. (7) as follows [26].

$$\frac{dX}{dt} = \mu\_m \frac{\text{SX}}{K\_s + \text{S}} - k\_d X \tag{5}$$

When the First principles model is not accurate, Parallel configuration is a better choice as parallel hybrid model can compensate for the First principles model mismatch [29]. When the First principles model is accurate, serial configuration

A non-linear process such as bioreactor can be modeled using Fuzzy logic via Rule base analysis. Initially a model structure is chosen wherein the number of inputs and number of fuzzy sets per variable are found, then estimation of fuzzy membership function parameters with regard to its shape, position is carried out and finally the rule base mapping from fuzzy sets to functions are carried out. The fuzzy logic system draws decision with the help of Fuzzy Inference System which uses "If ..Then" rules along with OR, AND connectors. For example, if specific growth rate is high, biomass concentration is high; else if specific growth rate is low, biomass concentration is low. Fuzzy models render transparency and therefore have the advantage of interpretability when compared to other data- driven

methods. The popular fuzzy models in use are the Takagi-Sugeno and the Mamdani type. Takagi-Sugeno type fuzzy models are suitable for modeling the non-linearity of the process and can be represented as many linear models in parallel, wherein the sub-models are chosen based on some specified rule. These are generally well suited to perform mathematical analysis and can be applied in Multiple Input Single Output (MISO) systems. Also, in Takagi-Sugeno model, output membership function are either constant or linear. The Mamdani type fuzzy model has a fuzzy logic set as the output of each rule. These are well suited to perform with manual input and can be applied in Multiple Input Single Output (MISO) systems and Multiple Input Multiple Output (MIMO) systems. The output membership function is present and the output is not continuous. Moreover, the Mamdani type fuzzy model is more accurate than Takagi-Sugeno, but it requires estimation of huge number of

Hybrid Fuzzy models comprises both first principles model equations and Fuzzy models [31]. The parameter estimation of hybrid fuzzy models are commonly done by Kalman filter. Fuzzy models are identified from experimental process data commonly by Fuzzy Clustering Method (FCM). Improvements in accuracy and performance of fuzzy models can be achieved by implementation of FCM based on Artificial Bee Colony (FCMABC), FCM based on Chicken Swarm Oprimization

A case study to exhibit the Hybrid fuzzy modeling approach, the low cost cellulase production process is considered where the objective is to find the product concentration (cellulase) based on the interactions between the biomass and

seems to be a better choice as it offers better extrapolation [30].

*Soft Sensors for Biomass Monitoring during Low Cost Cellulase Production*

*Configurations of a hybrid model. (a) Parallel form (b) Serial form.*

*DOI: http://dx.doi.org/10.5772/intechopen.96027*

*3.3.1 Hybrid fuzzy models*

**Figure 1.**

parameters.

(FCMCSO), etc.

substrate.

**611**

$$\frac{dE\_t}{dt} = \frac{k\_1 X}{1 + \frac{S}{K\_i}} - k\_2 E\_t \tag{6}$$

$$\frac{d\mathbf{S}}{dt} = -\frac{1}{Y\_{X/\mathcal{S}}} \frac{d\mathbf{X}}{dt} - m\_s \mathbf{X} \tag{7}$$

where *X* represents the biomass concentration (g/l), *S* represents the substrate concentration (g/l), *μ<sup>m</sup>* is the maximum specific growth rate (1/h), *Ks* is the substrate saturation constant (g/l), *kd* is the cell death constant (1/h), *k*1, *k*<sup>2</sup> are rate constants for cellulase synthesis (IU/ml h) and cellulase decay (1/h), *Ki* is the substrate inhibition coefficient(g/l), *Et* is the total cellulase activity, *YX=<sup>S</sup>* is the stoichiometric biomass yield coefficient (g/g) and *ms* is the specific maintenance coefficient (1/h).

#### **3.3 Hybrid models**

Gray box models also known as hybrid models are considered as an effective tool for model identification. These models combine *a priori* knowledge of the process and black box representations. The black-box model can be ANN, Fuzzy, NARX, Neuro-Fuzzy etc. It provides the flexibility to develop model based on both process data and available knowledge about the process [27, 28]. Hybrid models provide higher estimation accuracy, interpretability and extrapolation. In particular, during fungal fermentation using lignocellulosic substrates, the kinetic parameters *μ<sup>m</sup>* and *YX=<sup>S</sup>* are generally assumed to be constant throughout the batch/fed-batch process. However, there might be a slight change in their values and accurate estimation of these kinetic parameters such as aids greatly in process control and product enhancement. The multivariate interactions during the process operation can be found using statistical design of experiments (DoE). The hybrid models are commonly represented in two configurations as parallel and cascade as shown in **Figure 1**.

The parallel configuration **Figure 1(a)** uses complete first principles model and the error between outputs of this model and real-time process are modeled. The serial configuration **Figure 1(b)** is used when there is less number of unknown parameters. This is the most frequently used hybrid model structure. The choice of hybrid configuration strongly depends on the First principles model structure.

*Soft Sensors for Biomass Monitoring during Low Cost Cellulase Production DOI: http://dx.doi.org/10.5772/intechopen.96027*

#### **Figure 1.**

*B q*ð Þ¼ *<sup>b</sup>*<sup>0</sup> <sup>þ</sup> *<sup>b</sup>*1*q*�<sup>1</sup> <sup>þ</sup> *<sup>b</sup>*2*q*�<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> *bmq*�*<sup>m</sup>* (4)

*Ks* <sup>þ</sup> *<sup>S</sup>* � *kdX* (5)

� *k*2*Et* (6)

*dt* � *msX* (7)

The input output experimental data is used to determine the values of variables a, b and the order of the polynomials *n* and *m*. Artificial Neural Network (ANN) models are another type of black box models that has wide application in bioprocess

White box models are mechanistic models that are been used widely used incorporates the available process knowledge in the form of first principle model equations. As an example, the first principles model widely used for fungal fermentation during cellulase production are represented in Eq. (5), Eq. (6), Eq. (7) as

*SX*

*dX*

*YX=<sup>S</sup>*

concentration (g/l), *μ<sup>m</sup>* is the maximum specific growth rate (1/h), *Ks* is the

where *X* represents the biomass concentration (g/l), *S* represents the substrate

substrate saturation constant (g/l), *kd* is the cell death constant (1/h), *k*1, *k*<sup>2</sup> are rate constants for cellulase synthesis (IU/ml h) and cellulase decay (1/h), *Ki* is the substrate inhibition coefficient(g/l), *Et* is the total cellulase activity, *YX=<sup>S</sup>* is the stoichiometric biomass yield coefficient (g/g) and *ms* is the specific maintenance

Gray box models also known as hybrid models are considered as an effective tool for model identification. These models combine *a priori* knowledge of the process and black box representations. The black-box model can be ANN, Fuzzy, NARX, Neuro-Fuzzy etc. It provides the flexibility to develop model based on both process data and available knowledge about the process [27, 28]. Hybrid models provide higher estimation accuracy, interpretability and extrapolation. In particular, during fungal fermentation using lignocellulosic substrates, the kinetic parameters *μ<sup>m</sup>* and *YX=<sup>S</sup>* are generally assumed to be constant throughout the batch/fed-batch process. However, there might be a slight change in their values and accurate estimation of these kinetic parameters such as aids greatly in process control and product enhancement. The multivariate interactions during the process operation can be found using statistical design of experiments (DoE). The hybrid models are commonly represented in two configurations as parallel and cascade as shown in

The parallel configuration **Figure 1(a)** uses complete first principles model and the error between outputs of this model and real-time process are modeled. The serial configuration **Figure 1(b)** is used when there is less number of unknown parameters. This is the most frequently used hybrid model structure. The choice of hybrid configuration strongly depends on the First principles model structure.

technology due to their ability to represent non-linear functions.

*dX dt* <sup>¼</sup> *<sup>μ</sup><sup>m</sup>*

*dS dt* ¼ � <sup>1</sup>

*dEt dt* <sup>¼</sup> *<sup>k</sup>*1*<sup>X</sup>* <sup>1</sup> <sup>þ</sup> *<sup>S</sup> Ki*

**3.2 White box models**

*Biotechnological Applications of Biomass*

follows [26].

coefficient (1/h).

**3.3 Hybrid models**

**Figure 1**.

**610**

*Configurations of a hybrid model. (a) Parallel form (b) Serial form.*

When the First principles model is not accurate, Parallel configuration is a better choice as parallel hybrid model can compensate for the First principles model mismatch [29]. When the First principles model is accurate, serial configuration seems to be a better choice as it offers better extrapolation [30].

#### *3.3.1 Hybrid fuzzy models*

A non-linear process such as bioreactor can be modeled using Fuzzy logic via Rule base analysis. Initially a model structure is chosen wherein the number of inputs and number of fuzzy sets per variable are found, then estimation of fuzzy membership function parameters with regard to its shape, position is carried out and finally the rule base mapping from fuzzy sets to functions are carried out. The fuzzy logic system draws decision with the help of Fuzzy Inference System which uses "If ..Then" rules along with OR, AND connectors. For example, if specific growth rate is high, biomass concentration is high; else if specific growth rate is low, biomass concentration is low. Fuzzy models render transparency and therefore have the advantage of interpretability when compared to other data- driven methods. The popular fuzzy models in use are the Takagi-Sugeno and the Mamdani type. Takagi-Sugeno type fuzzy models are suitable for modeling the non-linearity of the process and can be represented as many linear models in parallel, wherein the sub-models are chosen based on some specified rule. These are generally well suited to perform mathematical analysis and can be applied in Multiple Input Single Output (MISO) systems. Also, in Takagi-Sugeno model, output membership function are either constant or linear. The Mamdani type fuzzy model has a fuzzy logic set as the output of each rule. These are well suited to perform with manual input and can be applied in Multiple Input Single Output (MISO) systems and Multiple Input Multiple Output (MIMO) systems. The output membership function is present and the output is not continuous. Moreover, the Mamdani type fuzzy model is more accurate than Takagi-Sugeno, but it requires estimation of huge number of parameters.

Hybrid Fuzzy models comprises both first principles model equations and Fuzzy models [31]. The parameter estimation of hybrid fuzzy models are commonly done by Kalman filter. Fuzzy models are identified from experimental process data commonly by Fuzzy Clustering Method (FCM). Improvements in accuracy and performance of fuzzy models can be achieved by implementation of FCM based on Artificial Bee Colony (FCMABC), FCM based on Chicken Swarm Oprimization (FCMCSO), etc.

A case study to exhibit the Hybrid fuzzy modeling approach, the low cost cellulase production process is considered where the objective is to find the product concentration (cellulase) based on the interactions between the biomass and substrate.

The following assumptions are made during the batch/fed-batch operation of the bioreactor. The reactor is completely mixed and the feed flow rate (*F*) is known. Measurements for biomass concentration (*X*), Substrate concentration (*S*), product concentration (*P*) and volume (*V*) are available. The sampling interval is 30 minutes for these measurements.

The first principles model consists of 4 state equations for biomass concentration X (g/l), Substrate concentration S (g/l), Product concentration P (g/l) and volume of the bioreactor V (l) as defined in Eq. (8–11).

$$\frac{d\mathbf{x}}{dt} = \mu \mathbf{X} - \frac{F}{V} \mathbf{X} \tag{8}$$

base. The advantage of using fuzzy clustering method is that the experimental data is focused and from that, the fuzzy model with independent rules is developed. The fuzzy models for kinetic parameters are represented in **Figure 2** and the hybrid model output in comparison with experimental data is illustrated in **Figure 3**. The optimization of fuzzy model parameters will improve the performance of hybrid model.

Hybrid ANN models are combination of first principles and ANN wherein the ANNs are used to estimate the kinetic parameters (black box models) [32]. The hybrid model shown in **Figure 4** is a combination of neural network estimator with the Mass Balance equations (Mathematical model). The neural network estimator is capable of estimating the process parameters from the real time measurements and these kinetic parameters (μ and Ys/x) are updated in the mass balance equations to

In general, the kinetic parameters are determined offline from experimental data. Due to the ability of neural networks to learn and model non-linear relationships, the parameter values can be estimated after proper training. Neural network with varying number of hidden neurons has been trained and MSE between the actual data and estimated data are calculated. Network with less MSE has been selected to find optimal hidden neurons. In this case study, a neural network structure comprising of two layer feed-forward network with sigmoid hidden neuron and linear output neuron is used. The state variables Xt *X t*ð Þ and *S t*ð Þ S tð Þ are the inputs and the parameters *μ*^ and *Y*^ *<sup>S</sup>=<sup>X</sup>* are the outputs of the neural network. The parameters are found using the

S tð Þ

kS <sup>þ</sup> S tð Þ (15)

(16)

give the value of the state variables in the next time instant.

*Soft Sensors for Biomass Monitoring during Low Cost Cellulase Production*

*DOI: http://dx.doi.org/10.5772/intechopen.96027*

Eq. (15) and Eq. (16) given below for every time instant.

found experimentally to be 0.01 g/L.

*Fuzzy model for kinetic parameters μ and* Yp/s.

**Figure 2.**

**613**

μ ¼ μmax

where S(t) is the substrate concentration, *kS* – saturation constant which is

*<sup>Y</sup>*^ *<sup>S</sup>=<sup>X</sup>* <sup>¼</sup> *S t*ðÞ� *S t*ð Þ � <sup>1</sup>

YS*<sup>=</sup>*<sup>X</sup> <sup>¼</sup> St � St�<sup>1</sup> Xt � Xt�<sup>1</sup>

*X t*ðÞ� *X t*ð Þ � 1

*3.3.2 Hybrid ANN models*

$$\frac{d\mathbf{s}}{dt} = -q\_s X + \frac{F}{V}(\mathbf{S}\_{in} - \mathbf{S}) \tag{9}$$

$$\frac{dP}{dt} = q\_p X - P\left(\frac{F}{V} + K\right) \tag{10}$$

$$\frac{dV}{dt} = F$$

where *μ* is the specific growth rate (h�<sup>1</sup> ), *F* represents substrate feed rate, *Sin* is the substrate concentration in the feed, *qs* is the substrate consumption rate (h�<sup>1</sup> ), *qp* is the product formation rate (h�<sup>1</sup> ), *K* is the product decay constant (h�<sup>1</sup> ). No direct measurements were made for these kinetic parameter rates. Therefore, a fuzzy model structure is represented as in Eq. (12–14).

$$q\_s = \frac{\mu\_m S}{Y\_{\mathbf{x}/s}(K\_s + S)} + \frac{q\_p}{Y\_{p/s}} + m\_s X \tag{12}$$

$$\mu = f\_{f\_{\text{fuzzy}}}(\mathbb{S}, X) \tag{13}$$

$$q\_p = f\_{f\_{\text{fuzzy}}}(\mathbb{S}, X) \tag{14}$$

where *μm*, *Yx=<sup>s</sup>*, *Yp=<sup>s</sup>*, *Ks*, *ms* are constants.

The kinetic parameters *μ* and *qp* depends on *X* and *S* and their values are unknown. Hence, fuzzy models are developed to estimate their values. An extended Kalman filter is designed to obtain the estimated values of parameters. The filter is tuned by fixing the process noise covariance matrix Q as [0.001,0.001, 0.001, 0.054,0.003] and measurement error covariance matrix R as [0.1, 0.05, 0.03, 0.1]. The filter performance is evaluated by stability border criterion *λ* and significance level, *L<sup>α</sup>* (**Table 2**). Smaller values of these two criterions represent good tuning of the Kalman filter.

From the table, it is observed that the filter is tuned properly. In this work, the fuzzy sub-model identification for specific growth rate and product formation rate are done with fuzzy clustering. The basic idea is to form clusters (similar groups) with the available experimental data. Each cluster exhibit an independent rule in the rule


**Table 2.** *Results of Kalman filter.*

base. The advantage of using fuzzy clustering method is that the experimental data is focused and from that, the fuzzy model with independent rules is developed. The fuzzy models for kinetic parameters are represented in **Figure 2** and the hybrid model output in comparison with experimental data is illustrated in **Figure 3**. The optimization of fuzzy model parameters will improve the performance of hybrid model.
