# Adaptive Control of Biomass Specific Growth Rate in Fed-Batch Biotechnological Processes. A Comparative Study

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Biotechnological Process and Its Mathematical Model

_{x}(U/(g biomass)) is the specific protein activity; w (kg) is the culture broth weight; μ (1/h) is the biomass specific growth rate; q

_{s}(g/(gh)) is the glucose specific consumption rate; q

_{px}(U/(gh)) is the specific protein accumulation rate; T (°C) is the culture broth temperature; u (kg/h) is the substrate (glucose solution) feeding rate; F

_{smp}(kg/h) is the sampling rate; and s

_{f}(g/kg) is the glucose concentration in the feeding solution.

_{max}(1/h) is a maximal SGR parameter; K

_{s}(g/kg) is a Monod constant; K

_{i}(g/kg) is an inhibition constant; α (1/°C) is a model parameter that takes into account the influence of the temperature on the growth rate; and T

_{ref}(°C) is the reference temperature that is optimal for the growth phase.

_{s}assumes the proportional relationship between the cell growth and substrate consumption rates, and additionally takes into account the term that is related to maintenance of the vital functions of cells:

_{xs}(g/g) is a conversion yield coefficient; and m (g/(gh)) is a maintenance term.

_{px}takes into account the influence of the SGR and the actual protein activity:

_{px}(h) is a protein accumulation time constant; p

_{max}(μ) (U/(g biomass)) is the maximal specific protein activity as a function of SGR; K

_{m}(U/(g biomass)) is a proportionality coefficient; K

_{μ}(1/h) is a Monod constant of the protein accumulation; and K

_{iμ}(1/h) is an inhibition constant of the protein accumulation. The model identified from the process data has revealed that the maximal specific protein activity can be achieved while maintaining the SGR at ~0.1 (1/h).

_{ox}(g/g) is a conversion yield coefficient; and m

_{ox}(g/(gh)) is a maintenance term. This relationship was later used to develop an OUR-based state estimator of the specific growth rate.

#### 2.2. Adaptive Control Algorithms

#### 2.2.1. Adaptive PID Control Based on the Gain Scheduling Technique

_{0}and the time constant τ of the transfer function (14) are as follows:

_{C}is a proportional gain, τ

_{i}is an integral time constant, and ε is a tuning parameter (filter time constant). After taking into account Equations (15) and (16) and accepting the assumption that ${\mathrm{s}}_{\mathrm{f}}\gg s*$, the following tuning rules for the investigated process model were obtained:

_{s}, the inhibition term may be neglected. Despite that the µ(s) function (5) is nonlinear, one can assume that the relationship follows the linear law within this narrow range (i.e., the sensitivity Δµ/Δs may be set to some average value). In such case, the following adaptation rules of the PI controller for the entire control channel u-s-µ can be defined:

_{0}, κ

_{1}and κ

_{2}are the constants to be identified during the controller tuning procedure. For practical reasons, in the final expression (22), the variable term containing x* in the denominator is substituted by a constant κ

_{2}, as the term is significantly smaller than the one including OUR*/w*. This simplification holds even in the second phase of the process, where x* is at the largest. The tuning parameters K

_{C}and τ

_{i}are updated online every time the measured signals of OUR* and w* are received from the process.

_{i}is the ith sampling time interval.

_{i}) and w(t

_{i}). The initial value of x(t

_{0}) is obtained from the off-line measurements at the beginning of the process. During the process, the real-time estimator recursively calculates the online estimates of µ and x. Performance of the estimator in a fed-batch process is presented in Figure 5. In the plots shown, the induction takes place at the 8 h time point, when the SGR is sharply reduced by means of substrate limitation and the temperature drops. Starting from that moment, the process enters the protein production phase.

_{ox}and m

_{ox}were identified from historic process data collected from multiple runs. Within a series of cultivations, the parameter variation did not exceed 5%. Such variation does not significantly affect the model estimation quality and the µ estimates remain within the acceptable range. To further improve the estimation quality, the estimator may be enhanced by a real-time identification algorithm that makes use of the reference biomass measurements obtained during the process or, alternatively, by using more advanced-state estimation techniques [29,30]. Both the simulation and the real application results have demonstrated that the estimator (23) and (24) is suitable for practical application in the investigated processes. The state estimator was further applied in both investigated adaptive control systems.

#### 2.2.2. Model-Free Adaptive ANN-Based Control

_{C}.

_{set}(t) is the setpoint, y(t) is the controlled process variable, and e(t) is the tracking error.

_{set}(t) by manipulating the control action u(t); and (ii) the changing of the controller ANN weights w and h (see Figure 7) so that the controller can adapt to the changes in the process dynamics and efficiently reject the process disturbances.

_{j}(n) is the input of the jth activation function in the hidden layer; w

_{ij}(n) is the ijth weight in the hidden layer; E

_{i}(n) is the ith error signal; q

_{j}(n) is the output of the jth activation function in the hidden layer; o(n) is the output of the output layer; h

_{j}(n) is the jth weight of the output layer; o(t) is the continuous function of o(n); e(t) is the continuous error signal function; K

_{C}is the controller gain coefficient; and v(t) is the continuous function of the controller output (control variable).

_{ij}(n) is the change of the w

_{ij}(n) weight in the nth iteration; Δh

_{j}(n) is the change of the h

_{j}(n) weight in the nth iteration; η is a positive constant of the learning rate; and $\partial y(n)/\partial u(n)$ is the sensitivity of the process output y(t) with respect to the variations of the process input (control action) u(t).

_{C}, that need to be adjusted to achieve the best possible control performance.

#### 2.3. Numerical Simulation Techniques

## 3. Results

_{0}, κ

_{1}and κ

_{2}. For optimization, an evolutionary computation algorithm [33] was adapted and used. The MFA controller was initially tuned by selecting the parameters η and K

_{C}so that the process remained stable in a wide range of the operating conditions. The optimized values of the tuning coefficients are listed in Table 2.

_{C}and τ

_{i}, respectively. The variation span of the controller gain K

_{C}did not exceed 10%, as the parameter is proportional to the culture broth weight that changed moderately during the process. Integration time constant τ

_{i}follows the changes of the OUR profile and; therefore, reflects the significantly varying dynamics of the process.

_{ij}and h

_{j}of the MFA controller during the learning process. The step-wise changes of the weight values correspond to the SGR changes due to a setpoint change or disturbance, while the continuous drift was caused by slow changes of the related process state variables (i.e., biomass concentration and culture broth weight).

_{ij}and two weights h

_{j}. Further increasing the complexity of the MFA controller did not yield significantly better controller performance.

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Biomass (solid triangles) and glucose (solid circles) concentrations measured during two fed-batch cultivations (subplots (

**a**) and (

**b**), respectively), along with the corresponding model outputs (solid line and dashed line), determined from Equations (1)–(9).

**Figure 2.**Specific concentration of viral capsid protein VP1, p

_{x}(solid squares) obtained from two fed-batch cultivations (subplots (

**a**) and (

**b**), respectively), along with the corresponding model outputs (solid line) and the biomass specific growth rate, μ (dashed line), determined from Equations (1)–(9).

**Figure 3.**General structure of the adaptive proportional–integral–derivative (PID) control system for specific growth rate control.

**Figure 4.**µ(s) kinetics (Equation (5)): Possible range of s concentrations in the cultivation medium (

**a**), effective range of s concentrations (

**b**) during µ control by means of substrate limitation.

**Figure 5.**Performance of the specific growth rate estimator in a fed-batch process: Oxygen uptake rate, OUR (

**a**); culture broth weight, w (

**b**); specific growth rate, µ (

**c**); biomass concentration, x (

**d**) (dashed lines—measured online, solid lines—estimated online, squares—measured off-line).

**Figure 6.**General structure of the adaptive artificial neural network (ANN)-based control system for specific growth rate control.

**Figure 7.**Architecture of the ANN-based model-free adaptive (MFA) controller: e(t) is the tracking error; z

^{−1}is the discrete delay function; E

_{i}is the ith error signal; w

_{ij}is the ijth weight in the hidden layer; p

_{j}is the input of the jth activation function in the hidden layer; φ is a sigmoidal activation function; q

_{j}is the output of the jth activation function in the hidden layer; h

_{j}is the jth weight of the output layer; o(t) is the continuous function of the output; K

_{C}is the controller gain coefficient; and v(t) is the continuous function of the controller output.

**Figure 9.**Transient processes of the specific growth rate during a test run: Setpoint change (

**a**) and (

**c**); disturbance rejection (

**b**) and (

**d**); the setpoint (dotted line); adaptive PI-based control (dashed line); ANN-based control (solid line).

**Figure 10.**Evolution of the PI controller parameters K

_{C}(

**a**) and τ

_{i}(

**b**) during the fed-batch process.

**Figure 11.**Evolution of the ANN weights w

_{ij}(

**a**) and h

_{j}(

**b**) during the fed-batch process (one node in the hidden layer and one node in the output layer used).

**Figure 12.**Transient processes of the specific growth rate in a series of 10 test runs (randomly initialized ANN weights): Setpoint change (

**a**) and (

**c**); disturbance rejection (

**b**) and (

**d**); the setpoint (dotted line); ANN-based control (solid line).

Model Parameter | Value | Units |
---|---|---|

K_{i} | 93.8 ± 12.7 | g/kg |

K_{iµ} | 0.0174 ± 0.0012 | 1/h |

K_{m} | 751 ± 27 | U/g |

K_{s} | 0.01 ± 0.005 | g/kg |

K_{µ} | 0.61 ± 0.03 | 1/h |

m | 0.0242 ± 0.004 | g/(gh) |

m_{ox} | 0.05 ± 0.0025 | g/(gh) |

s_{f} | 151 | g/kg |

T_{px} | 1.5 ± 0.1 | h |

T_{ref} | 37 | °C |

Y_{ox} | 0.7 ± 0.01 | g/g |

Y_{xs} | 0.46 ± 0.01 | g/g |

α | 0.0495 ± 0.0025 | 1/°C |

µ_{max} | 0.737 ± 0.01 | 1/h |

Model Parameter | Value | Units |
---|---|---|

K_{C} | 0.4 | kg |

η | 1.5 | – |

κ_{0} | 0.14 | kg |

κ_{1} | 0.008 | g/kg |

κ_{2} | 0.8 | g/(kg h) |

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**MDPI and ACS Style**

Galvanauskas, V.; Simutis, R.; Vaitkus, V.
Adaptive Control of Biomass Specific Growth Rate in Fed-Batch Biotechnological Processes. A Comparative Study. *Processes* **2019**, *7*, 810.
https://doi.org/10.3390/pr7110810

**AMA Style**

Galvanauskas V, Simutis R, Vaitkus V.
Adaptive Control of Biomass Specific Growth Rate in Fed-Batch Biotechnological Processes. A Comparative Study. *Processes*. 2019; 7(11):810.
https://doi.org/10.3390/pr7110810

**Chicago/Turabian Style**

Galvanauskas, Vytautas, Rimvydas Simutis, and Vygandas Vaitkus.
2019. "Adaptive Control of Biomass Specific Growth Rate in Fed-Batch Biotechnological Processes. A Comparative Study" *Processes* 7, no. 11: 810.
https://doi.org/10.3390/pr7110810