Dynamic Analysis and Control for a Bioreactor in Fractional Order

Abstract

In this paper, a mathematical model was developed to describe the dynamic behavior of a bioreactor in which a fermentation process takes place. The analysis took into account the bioreactor temperature controlled by the refrigerant fluid flow through the reactor jacket. An optimal LQR control acting in the water flow through a jacket was used in order to maintain the reactor temperature during the process. For the control design, a reduced-order model of the system was considered. Given the heat transfer asymmetry observed in reactors, a model considering the fractional order heat exchange between the reactor and the jacket using the Riemann–Liouville differential operators was proposed. The numerical simulation demonstrated that the proposed control was efficient in maintaining the temperature at the desired levels and was robust for disturbances in the inlet temperature reactor. Additionally, the proposed control proved to be easy to apply in real life, bypassing the singularity problem and the difficulty of initial conditions for real applications that can be observed when considering Riemann–Liouville differential operators.

1. Introduction

Bioreactors are commonly used in the industry for biochemical oxygen demand (BOD) and chemical oxygen demand (COD) reduction and ethanol production from glucose fermentation [1,2].

In alcoholic fermentation, glucose is degraded into ethanol and carbon dioxide through an anaerobic process. In the oxidative phase, some aromas are formed, while in the fermentation phase, the yeast metabolism converts sugars into ethyl alcohol and carbon dioxide and Saccharomyces cerevisiae, the yeast that is commonly used, as it is easy to use, and its biomass is generally reused [2,3]. In this process, temperature is an important variable to consider, since it acts directly in the fermentation efficiency and denaturation process of microorganism proteins in bioreactors [4].

Regarding the optimization of ethanol production in bioreactors, in addition to temperature control, it is necessary to monitor operating parameters such as the pH of the medium, the concentration of oxygen, the concentration of glucose, etc. [2]. Taking into account these variables, it is possible to obtain mathematical models for the fermentation process, and thus to design a control system that optimizes the production of ethanol [4,5,6,7].

A hybrid mathematical model for biological reactors was proposed by [8], to optimize and control microbial processes, by considering cellular intelligence. In [9], a fractional order model was proposed for a bioreactor system, considering a fractional differential equation for ‘biomass’ and an integer-order differential equation for the ‘substrate’ dynamics. For numerical analysis of gas and heat production by bioreactors, a mathematical model was proposed in [10].

Mathematical models that describe the behavior of bioreactors have helped in process control projects, enabling the optimization of ethanol production. In [7], a fractional order PID control was proposed for the reactor temperature control. In [11], non-linear control systems were considered seeking to preserve the influences of the system’s non-linearities. For control of chemical reactors in [12], a cascade control was proposed. Optimal control design was used by [13] in the controlling of the reactor jacket temperature. A PI-fuzzy controller was designed by [14] to regulate the temperature inside a fermentation tank. In [4], a control system for the fermentation process was proposed considering the use of an adaptive linear neural network (ADALINE) to control the process. A control system for a continuous bioreactor with an unknown reaction rate term subjected to input saturation is proposed in [15]. In [16], a PID controller for integrating systems with time delay was proposed using the direct synthesis method and multiple dominant pole-placement approach. The proposed control was applied in a nonlinear continuous stirred tank reactor model.

In [17], the non-linear control SDRE (state-dependent Riccati equation) was used to control the temperature of a bioreactor, by controlling the flow of liquid in the reactor jacket. A fractional order control system was proposed in [18] for a nonlinear mathematical model of a bioreactor modeled in fractional order.

Fractional order dynamic systems have shown a huge application perspective in several areas, such as control techniques, artificial intelligence, network science, mechanics, physical sciences, economics, biological medical treatment, physical systems, and so on [19,20,21,22,23,24,25,26]. In [27,28,29], the authors considered the application of an adaptive control to the chaotic behavior of neural network systems in fractional order. In [30], the sliding mode technique was applied to investigate the adaptive fuzzy finite-time backtracking control for a nonlinear fractional model. In [31], a delayed feedback controller was used to suppress the chaotic behavior of a fractional order hybrid optical model. Numerical simulations demonstrated the effectiveness of the proposed control. In [32], we proposed a structure of distributed interval observers for multi-agent systems of fractional order considering non-linearities in the presence of nonlinearity. Numerical simulations were presented to demonstrate the effectiveness of the proposed distributed range observer. In [33], a fractional fuzzy PID control was applied in the temperature maintenance of industrial systems. Numerical results showed that the proposed control system was robust for environmental changes caused by internal and external disturbances. In [34], a hybrid controller called controller (FO-F-PID) was proposed, composed of fractional order, fuzzy and proportional–integral–derivative components. The proposed control was applied in a power generation system, involving different independent power-producing units. In [35], the dynamic analysis and control of a generalized prey-predator delay stage structure model with a fear and prey refuge effect was performed. Numerical simulations demonstrated the influence of the order of the fractional derivative on the dynamics of the system and that the proposed control was efficient in controlling the emergence of Hopf bifurcation. In [36], a numerical investigation of a mathematical model of fractional order of diabetes and its complications was presented. In [37], the authors presented a proposal for the solutions of non-homogeneous fractional integral equations. In [38], it was possible to establish the existence and uniqueness of the boundary value problem solution for the non-integer variant of the classical Atkinson theorem in the oscillation of the Emden–Fowler equations, considering the Riemann–Liouville order derivative.

In this context, the objective of this work was to design a linear control system for the cooling fluid flow that passes through the fermenter jacket, in order to maintain the ideal temperature in the bioreactor, considering the asymmetric heat transfer effects observed in reactors [39]. Derivatives of fractional order were included in the model to represent the asymmetric heat transfer and hysteresis effect of temperature variation. The Riemann-Liouville differential operators were considered in this paper in the numerical simulations, as they are among the most popular and well-known operators, facilitating the reproduction of results by other researchers [40].

We considered the technique of linear control by the LQR (linear quadratic regulator) control and included fractional derivatives in the reactor and jacket temperature equations, thus including non-linearity in the heat exchange between the reactor and jacket.

The LQR control aims to design a practical control system that provides the desired operational performance and minimizes the operating cost. Moreover, according to [41,42], the LQR control has proven to be efficient in reactor control applications. The representation of heat transfer by fractional order models has received a lot of attention from researchers, due to its ability to more adequately represent the system’s non-linearity and memory effects [43,44,45]. Fractional order models have already been used in various fields of science and technology, such as heat conduction [46,47,48,49]. This paper contributes to the understanding of the reduced order control system, considering, in the LQR control design, only the equations that represent the temperature variation in the reactor and in the jacket. Such a strategy allows for easier control in real implementation and makes it possible to control the system efficiently, even with the inclusion of fractional derivatives in the reactor and jacket temperature equations, bypassing possible problems of singularities and determination of initial conditions observed in Riemann–Liouville differential operator applications. These problems are solved since the proposed control does not depend on fractional derivatives, making the system completely controllable.

This paper is organized as follows: Section 2 presents the mathematical model used and the system dynamics in the case in which reactor temperature is not controlled. In Section 3, the reactor temperature control is proposed, considering the application of the LQR control in the cooling liquid flow through the jacket, presenting the system dynamics with control, and the analysis of robustness for disturbances in the reactor inlet temperature. Section 4 presents the mathematical model considering both reactor and jacket temperature equations with fractional order derivatives. Numerical results of the system dynamics in fractional order, the efficiency of the proposed control for the system in fractional order, and its robustness for disturbances in the reactor inlet temperature are presented. Finally, the article is completed in Section 5.

2. Mathematical Model

Figure 1 shows the continuous fermentation reactor representation, with a stirring system and cooling jacket, and the variables considered in this paper.

Where C_O2, C_s, C_x and C_p are the concentration of oxygen, glucose, biomass and ethanol, respectively. T_ag, T_in and T_r, are the temperatures in the jacket, reactor inlet, and inside the reactor. F_ag is the coolant flow, F_i is the fermentation vessel inlet flow, and F_e is the fermenter outlet flow [4,17].

2.1. Mass Balance Equations

Considering Figure 1, it is possible to determine the mass and energy balance equations for the bioreactor system applied to ethanol production, assessing the influence of temperature, oxygen and the yeast concentrations [17].

For the mass balance equations, it was assumed that the variation in the microorganism growth in the bioreactor depends on the yeast growth rate and the microorganism concentration in the outlet.

$$\frac{d(C_x)}{dt}={\mu }_x{C}_x\frac{C_s}{K_s+C_s}{e}^{-K_p{C}_p}-\frac{F_e}{V}{C}_x$$

(1)

where ${\mu }_x$ is the specific growth rate (h⁻¹) defined as follows [17]:

$${\mu }_x=A_1{e}^{-\frac{E_{a1}}{R\left(T_r+273\right)}}-A_2{e}^{-\frac{E_{a2}}{R\left(T_r+273\right)}}$$

(2)

The variation of glucose concentration is given by:

$$\frac{d(C_S)}{dt}=-\frac{1}{R_{SX}}{\mu }_x{C}_x\frac{C_S}{K_s+C_S}{e}^{-K_p{C}_p}-\frac{1}{R_{SP}}{\mu }_P{C}_x\frac{C_S}{K_{S1}+C_S}{e}^{-K_{p1}{C}_p}+\frac{F_i}{V}{C}_{S,in}-\frac{F_e}{V}{C}_S$$

(3)

The variation in the concentration of ethanol inside the reactor can be related to the amount of ethanol produced and the fermenter output, and is defined as:

$$\frac{d(C_P)}{dt}={\mu }_P{C}_x\frac{C_S}{K_{S1}+C_S}{e}^{-K_{P1}{C}_P}-\frac{F_e}{V}{C}_P$$

(4)

According to [7], the maximum ethanol production depends on the ideal level of dissolved oxygen in the fermentation process; therefore, the rate of oxygen variation is given by the following equation:

$$\frac{d(C_{O_2})}{dt}=\left(k_{la}\right)\left(C_{O_2}^{*}-C_{O_2}\right)-r_{O_2}$$

(5)

where $k_{l\mathrm{a}}=k_{l{a}_0}{\left(1.024\right)}^{T_r-20}$.

The following relationship gives the oxygen concentration in the reactor:

$$C_{O_2}^{*}=\left(14.6-0.3943T_r+0.00714T_r^2-0.0000646T_r^3\right){10}^{-\sum H_i{I}_i}$$

(6)

The equilibrium oxygen level in the fermenter depends on the temperature and the coefficient of ionic forces, given by:

$$\begin{array}{l}{\displaystyle \sum }{H}_i{I}_i=0.5H_{Na}\frac{m_{NaCl}}{M_{NaCl}}\frac{M_{Na}}{V}+2H_{Ca}\frac{m_{CaC{O}_3}}{M_{CaC{O}_3}}\frac{M_{Ca}}{V}+2H_{Mg}\frac{m_{MgC{l}_2}}{M_{MgC{l}_2}}\frac{M_{Mg}}{V}\\ +0.5H_{Cl}\left(\frac{m_{NaCl}}{M_{NaCl}}+2\frac{m_{MgC{l}_2}}{M_{MgC{l}_2}}\right)\frac{M_{Cl}}{V}+2H_{C{O}_3}\frac{m_{CaC{O}_3}}{M_{CaC{O}_3}}\frac{M_{C{O}_3}}{V}+0.5H_H{10}^{-pH}\\ +0.5H_{OH}{10}^{-\left(14-pH\right)}\end{array}$$

(7)

However, during the fermentation process in the reactor, the yeast will consume oxygen, the consumed oxygen being defined as:

$$r_{O_2}=\frac{{\mu }_{O_2}{C}_x}{Y_{O_2}}\left(\frac{C_{O_2}}{K_{O_2}+C_{O_2}}\right)$$

(8)

The reactor temperature is given by equation:

$$\frac{d(T_r)}{dt}=\frac{F_i}{V}\left(T_{in}+273\right)-\frac{F_e}{V}\left(T_r+273\right)+\frac{r_{O_2}{\Delta \mathrm{H}}_r}{32{\mathsf{\rho }}_r{C}_{heat,r}}-\frac{K_T{A}_T\left(T_r-T_{ag}\right)}{V{\mathsf{\rho }}_r{C}_{heat,r}}$$

(9)

The temperature variation of the cooling jacket:

$$\frac{d(T_{ag})}{dt}=\frac{F_{ag}}{V_j}\left(T_{in,ag}-T_{ag}\right)+\frac{K_T{A}_T\left(T_r-T_{ag}\right)}{V_j{\mathsf{\rho }}_{ag}{C}_{heat,ag}}$$

(10)

where $V_j$ is the jacket volume (L).

Considering Equations (1), (3)–(5), (9) and (10), we obtain the following system of differential equations:

$$\left\{\begin{array}{l}\frac{d(C_x)}{dt}={\mu }_x{C}_x\frac{C_s}{K_s+C_s}{e}^{-K_p{C}_p}-\frac{F_e}{V}{C}_x\\ \frac{d(C_S)}{dt}=-\frac{1}{R_{SX}}{\mu }_x{C}_x\frac{C_S}{K_s+C_S}{e}^{-K_p{C}_p}-\frac{1}{R_{SP}}{\mu }_P{C}_x\frac{C_S}{K_{S1}+C_S}{e}^{-K_{p1}{C}_p}+\frac{F_i}{V}{C}_{S,in}-\frac{F_e}{V}{C}_S\\ \frac{d(C_P)}{dt}={\mu }_P{C}_x\frac{C_S}{K_{S1}+C_S}{e}^{-K_{P1}{C}_P}-\frac{F_e}{V}{C}_P\\ \frac{d(C_{O_2})}{dt}=\left(k_{la}\right)\left(C_{O_2}^{*}-C_{O_2}\right)-r_{O_2}\\ \frac{d(T_r)}{dt}=\frac{F_i}{V}\left(T_{in}+273\right)-\frac{F_e}{V}\left(T_r+273\right)+\frac{r_{O_2}{\Delta \mathrm{H}}_r}{32{\mathsf{\rho }}_r{C}_{heat,r}}-\frac{K_T{A}_T\left(T_r-T_{ag}\right)}{V{\mathsf{\rho }}_r{C}_{heat,r}}\\ \frac{d(T_{ag})}{dt}=\frac{F_{ag}}{V_j}\left(T_{in,ag}-T_{ag}\right)+\frac{K_T{A}_T\left(T_r-T_{ag}\right)}{V_j{\mathsf{\rho }}_{ag}{C}_{heat,ag}}\end{array}\right.$$

(11)

2.2. Numerical Simulations

In numerical simulations, we considered the system of differential equations previously described in Equation (11), fourth order Runge-Kutta method, with integration step (h = 0.01), and parameters: A₁ = 9.5 × 108, A₂ = 2.55 × 1033, A_T = 1, C_heat,ag = 4.18, C_heat,r = 4.18, E_a1 = 55,000, E_a2 = 220,000, k_la0 = 38, K_s = 1.03, K_s1 = 1.68, K_T = 3.6 × 10⁵, R = 8.31, R_SP = 0.435, R_sx = 0.607, H_Ca = −0.303, H_Cl = 0.844, H_CO3 = 0.485, H_H = −0.774, H_Mg = −0.314, H_Na = −0.550, H_OH = 0.941, m_CaCO3 = 100, m_MgCl2 = 100, m_NaCl = 500, M_Ca = 40, M_CaCO3 = 90, M_Cl = 35.5, M_CO3 = 60, Y_O2 = 0.97, ΔH_r = 518, µ_O2 = 0.5, µ_P = 1.79, ρ_ag = 1000, ρ_r = 1080 [4,15].

In Figure 2, the system (11) behavior without control can be seen, considering T_in and F_aq.

Figure 2 shows that the fermenter took approximately 100 h to stabilize and maintain its constant output. The results also show that in order to increase ethanol production, it is necessary to increase the yeast concentration, with more consumption of oxygen and glucose.

3. Temperature Control Strategy by LQR Control

Figure 3 shows the bioreactor representation considering the introduction of coolant flow control in the jacket.

Figure 3 shows that the control strategy of the reactor temperature acts in the control of the coolant flow control inside the jacket, thus allowing the cooling jacket to maintain the temperature inside the reactor by heat exchange.

Considering that, the main objective is to control the reactor temperature, which can simplify, for control project, the system from six differential equations to two differential equations, taking into account only the reactor and jacket temperature equations. The strategy used is similar to the one used successfully by [50].

The system can be considered as follows:

$$\left\{\begin{array}{l}\frac{d(T_r)}{dt}=-\left(\frac{F_e}{V}+\frac{K_T{A}_T}{V{\mathsf{\rho }}_r{C}_{heat,r}}\right)T_r+\frac{K_T{A}_T}{V{\mathsf{\rho }}_r{C}_{heat,r}}{T}_{ag}+\frac{F_i}{V}\left(T_{in}+273\right)-273\frac{F_e}{V}+\frac{r_{O_2}{\Delta \mathrm{H}}_r}{32{\mathsf{\rho }}_r{C}_{heat,r}}\\ \frac{d(T_{ag})}{dt}=\frac{K_T{A}_T}{V_j{\mathsf{\rho }}_{ag}{C}_{heat,ag}}{T}_r-\frac{K_T{A}_T}{V_j{\mathsf{\rho }}_{ag}{C}_{heat,ag}}{T}_{ag}-\frac{F_{ag}}{V_j}{T}_{ag}+\frac{F_{ag}}{V_j}{T}_{in,ag}\end{array}\right.$$

(12)

The system (12) can be presented by the following matrix form:

$$\frac{d(X)}{dt}=AX+G+BU$$

(13)

where $X=\left[\begin{array}{c}{T}_r\\ T_{ag}\end{array}\right]$, $A=\left[\begin{array}{cc}-\left(\frac{F_e}{V}+\frac{K_T{A}_T}{V{\mathsf{\rho }}_r{C}_{heat,r}}\right)& \frac{K_T{A}_T}{V{\mathsf{\rho }}_r{C}_{heat,r}}\\ \frac{K_T{A}_T}{V_j{\mathsf{\rho }}_{ag}{C}_{heat,ag}}& -\frac{K_T{A}_T}{V_j{\mathsf{\rho }}_{ag}{C}_{heat,ag}}\end{array}\right]$, $B=\left[\begin{array}{c}0\\ -\frac{T_{ag}}{V_j}+\frac{T_{in,ag}}{V_j}\end{array}\right]$, $U=F_{aq}$ and $G=\left[\begin{array}{c}\frac{F_i}{V}\left(T_{in}+273\right)-273\frac{F_e}{V}+\frac{r_{O_2}{\Delta \mathrm{H}}_r}{32{\mathsf{\rho }}_r{C}_{heat,r}}\\ 0\end{array}\right]$, where U is the feedback control, it is defined as follows:

$$U=-R^{-1}{B}^{\mathrm{T}}Pe=-Ke$$

(14)

where $e = [X-X^{*}]$, $X$ represents the system states and $X^{*}$ the desired states. The matrix $P$ is obtained by solving the Ricatti equation:

$$A^{T}P+PA-PB{R}^{-1}{B}^{T}P+Q=0$$

(15)

The functional cost for the control problem U is given by:

$$J=\frac{1}{2}{\displaystyle \underset{t_0}{\overset{\infty }{\int }}(e^{T}Qe+U^{T}RU)}dt$$

(16)

where Q and R are positive definite matrices.

The controllability is given by:

$$M=\begin{array}{cc}[B_{n\times m}& A_{n\times n}{B}_{n\times m}]\end{array}$$

(17)

If rank of the matrix M is 2, the system (12) is controllable. Considering the matrices A, B and M, it is possible to obtain: $A=\left[\begin{array}{cc}-0.1307& 0.0797\\ 1.7225& -1.7225\end{array}\right]$, $B=\left[\begin{array}{c}0\\ -0.1800\end{array}\right]$ and $M=\left[\begin{array}{cc}B& AB\end{array}\right]=\left[\begin{array}{cc}\begin{array}{c}0\\ -0.1800\end{array}& \begin{array}{c}-0.0144\\ 0.3100\end{array}\end{array}\right]$.

The rank of the matrix is 2, which ensures that the system (13) is controllable. Defining: $Q=35000\left[\begin{array}{cc}655000& 100\\ 100& 1\end{array}\right]$ and $R=\left[200\right]$. Considering $Fag = U$, and

$$U=-k_{1,1}\left(T_r-T_r^{*}\right)-k_{1,2}\left(T_{ag}-T_{ag}^{*}\right)$$

(18)

where $T_r^{*}$ and $T_{ag}^{*}$ are the desired temperatures for the reactor and jacket, and $k_{1,1}$ and $k_{1,2}$ are the gains of the control, obtained according to Equation (14).

According to [51,52], the optimum temperature for the alcoholic fermentation of Saccharomyces cerevisiae is 32 °C. In order to optimize alcoholic fermentation, the desired temperatures considered were $T_r^{*}=32 °\mathrm{C}$ and $T_{ag}^{*}=24 °\mathrm{C}$. Solving Equation (15) and replacing $T_r^{*}=32 °\mathrm{C}$ and $T_{ag}^{*}=24 °\mathrm{C}$ in Equation (18), the control signal is obtained:

$$U=10557.7595\left(T_r-32\right)+88.5189\left(T_{ag}-24\right)$$

(19)

Figure 4 shows the states’ variations with the proposed control: microorganism concentrations, glucose, alcohol and O₂ concentrations, as well as the reactor temperature and jacket temperature variation, taking into account the bioreactor mathematical model for the fermentation and ethanol production represented by Equation (11), and considering coolant flow rate variation (F_ag).

Figure 4 shows that the control maintained the ideal temperature at approximately 32 °C in the reactor, with less than ±5% error, which is considered to be an acceptable variation, according to [47]. As can be seen in Figure 4, in order to reduce the temperature of the reactor, it was necessary to rapidly increase the coolant flow in the jacket, reducing the temperature of the cooling agent and thus reducing the temperature of the reactor to 32 °C. The results also showed that when the reactor temperature was lower than 32 °C, the liquid flow was zero.

Table 1. Comparison of the variable values in equilibrium, with and without temperature control.

	${\mathit{C}}_{\mathit{x}}$	${\mathit{C}}_{\mathit{S}}$	${\mathit{C}}_{\mathit{p}}$	${\mathit{C}}_{{\mathit{O}}_2}$	${\mathit{T}}_{\mathit{r}}$	${\mathit{T}}_{\mathit{a}\mathit{g}}$
Without control	1.447	21.52	15.7	0.3484	36.01	30.01
With control	1.089	26.02	13.74	0.4371	32.01	24.18

presents a comparison of the process variables with and without the temperature control.

As can be seen in Table 1, with the proposed control, it was possible to maintain the reactor temperature at 32 °C, which provided an increase of 16.29% in the microorganisms’ productivity in generating ethanol (productivity coefficient ($\frac{C_p}{C_x}$): with 32 °C = 12.6 and 36 °C = 10), consuming 20.9% and 25.4% less glucose and O₂.

Proposed Control Sensitivity to Variations in the Reactor Inlet Temperature

To analyze the control robustness, a variation in the reactor inlet temperature was applied in the following form:

$$T_i\left(t\right)=25+\left(5sin\left(\frac{\pi }{12}t\right)\right)$$

(20)

In Figure 5, it is possible to observe the reactor temperature variation considering the perturbation according to Equation (20).

Figure 5 shows that the proposed control is robust for periodic variation in reactor inlet temperature (T_in), with less than 0.125% error. It can also be observed that the jacket temperature variation balances the reactor temperature close to the desired value, with variation between 21.04 and 27.33 °C. In addition, it is possible to observe that the coolant flow rate necessary to keep the reactor temperature close to the desired 32 °C varied between 29 and 157 L/h in a permanent regime case.

4. Dynamic Analysis and Control for the Fractional-Order Case

Given what was discussed in the present work until here, and taking into account that the heat exchange between the reactor and the jacket can be represented in fractional order, in the present section, both the reactor and jacket temperature equations were considered to have a fractional order derivative.

According to [53], differential equations may involve Riemann–Liouville differential operators of fractional order $q>0$, and they are generally considered in the following form [54,55,56]:

$$D^{q}x(t)=\frac{1}{\Gamma \left(m-q\right)}{\displaystyle \underset{t_0}{\overset{t}{\int }}\frac{x^{(m)}(x)}{{\left(t-x\right)}^{q-m+1}}}dx$$

(21)

where $m=\left[q\right]$, which is the first integer not smaller than q.

The system (11) in fractional order is described as follows:

$$\left\{\begin{array}{l}\frac{d^{q_1}{C}_x}{d{\tau }^{q_1}}={\mu }_x{C}_x\frac{Cs}{Ks+Cs}{e}^{-KpCp}-\frac{F_e}{V}{C}_x\\ \frac{d^{q_2}{C}_S}{d{\tau }^{q_2}}=-\frac{1}{R_{SX}}{\mu }_x{C}_x\frac{C_S}{Ks+Cs}{e}^{-KpCp}-\frac{1}{R_{SP}}{\mu }_P{C}_x\frac{C_S}{K_{S1}+Cs}{e}^{-K_{p1}{C}_p}+\frac{Fi}{V}{C}_{S,in}-\frac{F_e}{V}{C}_S\\ \frac{d^{q_3}{C}_p}{d{\tau }^{q_3}}={\mu }_P{C}_x\frac{C_S}{K_{S1}+Cs}{e}^{-K_{p1}{C}_p}-\frac{F_e}{V}{C}_P\\ \frac{d^{q_4}{C}_{O_2}}{d{\tau }^{q_4}}=(k_{la})(C_{O_2}^{*}-C_{O_2})-r_{O_2}\\ \frac{d^{q_5}{T}_r}{d{\tau }^{q_5}}=\frac{Fi}{V}(T_{in}+273)-\frac{Fe}{V}(T_r+273)+\frac{r_{O_2}\Delta {\mathsf{H}}_r}{32{\mathsf{\rho }}_r{\mathit{C}}_{heat,r}}-\frac{K_T{A}_T(T_r-T_{ag})}{\mathit{V}{\mathsf{\rho }}_r{\mathit{C}}_{heat,r}}\\ \frac{d^{q_6}{T}_{ag}}{d{\tau }^{q_6}}=\frac{Fag}{V_j}(T_{in,ag}-T_{ag})+\frac{K_T{A}_T(T_r-T_{ag})}{{\mathit{V}}_{\mathit{j}}{\mathsf{\rho }}_{ag}{\mathit{C}}_{heat,ag}}\end{array}\right.$$

(22)

where $q_1=q_2=q_3=q_4=1$, $q_5$ and $q_6$ represent the order of the derivatives, and $0.3<q_{5,6}<1$.

The numerical simulations for behavior analysis of the system (22) in fractional order uses the algorithm proposed by [33].

Figure 6 show the state variations considering the bioreactor mathematical model for fermentation and ethanol production with fractional order represented by Equation (22).

Considering the order variation of the fractional derivative in the reactor and in the jacket, it can be observed that for ($q_{5,6}<0.85$), the reactor temperature lower is than the ideal temperature of 32 °C, and for ($q_{5,6}>0.85$), the reactor temperature is higher than the desired value. We can also observe that the lower the value of ($q_{5,6}$), the lower the temperature inside the reactor, implying a reduction in alcohol production and an increase in glucose and oxygen concentrations.

The results are in accordance with what is expected for the system in fractional order, given that, according to [9], the dynamic responses of a bioreactor in fractional order are inherently slower than those obtained from the integer-order model, and their velocity increases with the order of the system.

4.1. Proposal of the Temperature Control Strategy by LQR Control in Fractional Order Case

In Figure 7, it is possible to observe the stabilization point of the system (22) in the case in which the system is under the action of the proposed control of the coolant flow in the jacket, and the reactor and jacket temperature derivatives are represented in fractional order.

As can be seen in Figure 7, the proposed control is efficient in bringing the temperature of the reactor to 32 °C for the cases in which the temperature of the reactor is above the considered setpoint. For cases in which the temperature was lower than 32 °C, the control was null, since the liquid flow control in the jacket was only able to cool the reactor. In this case, it can be seen that the control acted only in cases in which the order was higher than ($q_5=q_6=0.75$).

In Figure 8, the control applied to the system (22) is shown considering ($q_5=q_6=0.95$), which are the values that bring the reactor temperature close to 36 °C, the temperature in which the yeast proteins begin to denature [52].

Analyzing the results presented in Figure 8, it can be concluded that the proposed control is efficient in bringing the temperature of the reactor to 32 °C, keeping the temperature of the reactor at the ideal value for alcohol production even if the system is represented by a fractional order model.

4.2. Proposed Control Sensitivity to Variations in the Reactor Inlet Temperature in Fractional Order Case

In Figure 9, it is possible to observe the reactor temperature variation considering the disturbance of the inlet temperature according to Equation (20) and reactor and jacket temperature in fractional order considering $0.3\le q_{5,6}\le 1$.

As can be seen in the results presented in Figure 9, the control remained robust even when the system was represented in fractional order and suffered variations in the inlet temperature, as it was able to keep the reactor temperature at 32 °C, with and without disturbance. It could also be observed that the jacket temperature variation that balances the reactor temperature is similar to that already observed for the system with control in the integer-order model (Equation (11)).

5. Conclusions

Using numerical simulations, a dynamic analysis of a nonlinear mathematical model of an alcoholic fermentation reactor was presented. The use of the optimal LQR control made it possible to obtain a control system in a reduced order state space, allowing the maintenance of a constant reactor temperature by the variation of the water flow in the cooling jacket with only two fixed gains from the controller. In order to analyze the non-linearities of the heat exchange between the cooling jacket and the reactor, the use of a fractional order in the reactor temperature equation and in the coolant temperature equation was proposed.

The numerical results presented demonstrated that the proposed control strategy was efficient in keeping the interior temperature of the reactor at 32 °C and was robust for variations in the reactor inlet temperature, both in cases of fractional order or integer order. The numerical results are similar to those obtained experimentally as presented by [17], demonstrating the versatility of the proposed control to be used experimentally in future studies.

The results presented demonstrated that the proposed LQR control in reduced order contributes to the body of knowledge on temperature control of bioreactors, mainly considering the variation of the reactor temperature in fractional order. The presented control strategy can eliminate possible problems of singularities and problems of initial condition determination observed in the applications of the Riemann–Liouville differential operators for the cases in which control in fractional order is used in real systems. Such problems do not occur with the proposed control since it does not depend on fractional derivatives of the temperatures and given that systems in reduced order are completely controllable.