Model-Based Adaptive Control of Bioreactors—A Brief Review

Abstract

This article summarizes the authors’ experiences in the development and application of the General Dynamical Model Approach related to adaptive linearizing control of biotechnological processes. Special attention has been given to some original, innovative solutions in model-based process control theory: new formalization of biotechnological process kinetics, derivation and tuning of the general software sensor of the full kinetics of biotechnological processes, and a general algorithm for fully adaptive linearizing control with software sensors. These theoretical solutions are the basis of three control strategies—fully adaptive control of the main substrate, partially adaptive control of intermediate metabolite, and recognition and stabilization of the desired physiological state based on the proposed theoretical solutions. Each strategy is illustrated in different case studies. The advantages and limitations of each of them are identified and discussed. The derived algorithms for monitoring and controlling the considered biotechnological processes are realized and included in a software platform named Interactive System for Education in Modelling and Control of Bioprocesses (InSEMCoBio). The InSEMCoBio modules and their main functions are discussed. The effectiveness of the proposed control strategies (achieving maximum productivity) has been proven through a series of simulation investigations of the considered case studies.

1. Introduction

Biotechnological processes (BTP) are carried out in bioreactors in which microorganisms grow by consuming different substrates—sources of carbon, oxygen, nitrogen, etc. Typically, bioreactors are connected to systems for controlling the physicochemical parameters of pH, temperature, and agitation, guaranteeing suitable environmental conditions for good microbial activity. In recent decades, BTPs have been the basis for the production of several primary and/or secondary metabolite products. Their applications are mainly in the pharmaceutical industry, the food industry, the production of biofuels [1,2], and the treatment of wastewater from organic waste [3].

It should be noted that bioreactors operate in three modes—batch, fed-batch, and continuous ones (Figure 1). In the batch mode, only physicochemical parameters are controlled. Fed-batch and continuous mode of cultivation allow for control of process variables—concentrations of biomass, substrates, and products.

One of the contemporary directions in the field of industrial biotechnology is the development of methods for monitoring and control, leading to real-time BTP optimization. The main problems related to the features of these processes are [4,5,6]:

• time-varying and nonlinear nature of BTPs;
• low level of reproducibility of the experiments;
• lack of reliable online sensors for most biochemical variables;
• significant model uncertainty (in cases where model-based control is used).

Different control strategies are applied to solve the problems listed above. In [7], an overview of the current progress and future perspectives of bioprocess control methods is presented. The authors present a comparison of the most frequently used control strategies and their structures, such as open and closed-loop control [8,9,10], fuzzy logic-based control [11,12], artificial neural network-based control [13,14], model predictive control [15,16,17,18,19], and model-based control [20,21]. Other control methods considered in the literature are sliding model control [22,23] and adaptive back-stepping control [24], etc. The limitations of neural-network-based methods are related to the huge amount of data required. Predictive model control requires an accurate real-time model. For many of the methods mentioned above, the computational cost is high.

To overcome the aforementioned features of these processes, the control strategies must achieve high accuracy, robustness, and adaptability concerning various types of disturbances (in the structure and parameters of the model, noise, etc.).

Special attention is paid to model-based adaptive control methods, where unknown process parameters and variables are estimated in real time based on software sensors (SS) [25,26]. The most widely used control approach among them is the General Dynamic Model (GDM) [5,6]. The GDM is a nonlinear operational model that describes the dynamics of processes based on a reaction scheme [5]. The deriving monitoring and control algorithms using GDM have simple structures, and the missing process information is obtained by SS. The GDM approach has a lot of successful applications [27,28,29,30,31,32,33,34,35,36].

The GDM structure consists of two parts—process kinetics ($\mathbf{K}\mathit{\phi }\left(\mathit{\xi }\right)$) and transport dynamics ($-D\mathit{\xi }-\mathit{Q}+\mathit{F}$), where K is the constant yield coefficient‘s matrix [g/g]; ξ is the vector of component concentrations dissolved in the culture broth [g/l]; D is the dilution rate [1/h]; Q is the rate of mass outflow from the reactor in the gaseous form [g/lh]; F is the mass feed rate of external substrates in the reactor [g/lh]; $\mathit{\phi }\left(\mathit{\xi }\right)$ is the vector of reaction kinetics (also called reaction rates [g/lh].

Usually, the process kinetics are unknown, and the transport dynamics are known. The major challenge is how to represent the kinetics of the process.

The GDM is presented as follows:

$$d\mathit{\xi }/\mathit{d}\mathit{t}=\mathbf{K}\mathit{\phi }\left(\mathit{\xi }\right)-D\mathit{\xi }-\mathit{Q}+\mathit{F}$$

(1)

In a control scheme, the GDM is precisely designed to achieve a linear closed loop (as seen in Figure 2) that is unconditionally stable regardless of the operating point or the trajectory of the transient feed batch. This type of control is called linearizing. In Figure 2, the nonlinear process is described by Equation (1) and the nonlinear controller by Equation (14).

Linearizing control laws derived from GDM assumes that all variables and parameters are known. In the absence of information about process parameters, SSs have to be designed. As a result, adaptive linearizing control is obtained.

The objective of this paper is to present some innovative solutions for the control of BTP that represent the application of the GDM approach. A comparison between the classic GDM approach and the proposed innovations is shown in Figure 3.

The authors focus their efforts on the GDM approach because compared to other nonlinear object control methods, it has the following advantages: (i) the well-known linear control theory can be applied because the resulting closed-loop control is linear; (ii) it simplifies the synthesis of control algorithms for all BTP carried out in stirred tank bioreactors.

A shortcoming of the GDM approach is that the operational model uses constant yield coefficients (Figure 3, GDM approach). This leads to inaccurate results due to the low reproducibility of bioprocesses and significant parameter time-varying parameter values. To deal with this problem, a new formalization of process kinetics is proposed in [4]. The idea is to consider the full kinetics of a biotechnological process in the structure of GDM as a completely unknown and time-varying parameter.

The authors propose innovative solutions related to the new formalization, which are presented in Section 2 as follows: derivation and tuning of the general SS of the complete kinetics of biotechnological processes, as well as a general algorithm for adaptive linearizing control with the SS. The theory is demonstrated through three case studies: fully adaptive, partially adaptive, and physiological state stabilization control strategies (Figure 3).

In Section 3, the idea of impulse adaptive control implemented based on an input marker is presented. The marker is defined as the difference between consumption and production rates of intermediate metabolites in a specific bioprocess, where maintaining the balance between these two rates is important. Two processes controlled by the impulse adaptive control are demonstrated.

A control design based on the monitoring of processes passing through different metabolic regimes (described by different sub-models) and its application is presented in Section 4. As a representative case, the batch fermentation process of E. coli is considered. A new key parameter for switching sub-models describing the different physiological states is proposed and successfully tested by simulations based on experimental data. The key parameter is the kinetics of acetate. This metabolite is measured in real time.

In Section 5, the advantages and limitations of the generalized structures and strategies of the control algorithms proposed in Section 2, Section 3 and Section 4 are discussed.

In Section 6, an interactive system for education in the modeling and control of biotechnological systems (InSEMCoBio) is presented. The system incorporates modern optimization algorithms for model identification of nonlinear systems, as well as algorithms for monitoring and controlling the BTP. A significant benefit of this user-friendly system is its application in the research or training of biotechnologists.

The main contributions to the development of the GDM approach are discussed in Section 7.

2. New Formalization of the Fermentation Kinetics and Its Application for Adaptive Linearizing Control Algorithm

The GDM of biotechnological processes carried out in a stirred tank reactor is presented by model (1). It is assumed that:

• A1. Each measured liquid phase component is related to one kinetic term only, considered an unknown time-varying parameter.
• A2. The transport dynamics are known.

Under assumptions A1 and A2, the GDM is presented as:

$$d{\mathit{\xi }}_{\mathit{m}}/dt={\mathit{\varphi }}_{\mathit{m}}-D{\mathit{\xi }}_{\mathit{m}}-{\mathit{Q}}_{\mathit{m}}+{\mathit{F}}_{\mathit{m}}$$

(2)

where ${\mathit{\xi }}_{\mathit{m}}$ is a vector of concentrations of measurable variables with dim(n_m) [g/l], ${\mathit{\varphi }}_{\mathit{m}}$—a vector of unknown time-varying parameters describing the kinetics of these variables with dim(n_m) [g/lh], ${\mathit{F}}_{\mathit{m}}$—mass feed rates in the bioreactor [g/lh], ${\mathit{Q}}_{\mathit{m}}$—a vector of rates of the mass outflow components [g/lh].

The vectors in Equation (2) are the subvectors of the corresponding vectors in Equation (1).

2.1. Structure of General Software Sensor of Measurable Variable’s Kinetics

The following structure of SS is proposed [37]:

$$d{\widehat{\mathit{\xi }}}_{\mathit{m}}/dt={\widehat{\mathit{\varphi }}}_m-D{\mathit{\xi }}_{\mathit{m}}-{\mathit{Q}}_{\mathit{m}}+{\mathit{F}}_{\mathit{m}}+\mathsf{\Omega }({\mathit{\xi }}_{\mathit{m}}-{\widehat{\mathit{\xi }}}_{\mathit{m}})$$

(3)

$$d{\widehat{\mathit{\varphi }}}_m/dt=\mathsf{\Gamma }({\mathit{\xi }}_{\mathit{m}}-{\widehat{\mathit{\xi }}}_{\mathit{m}})$$

(4)

where ${\widehat{\mathit{\varphi }}}_m$ are the estimates of unknown time-varying parameters [g/lh], ${\widehat{\mathit{\xi }}}_{\mathit{m}}$ are the estimates of the ${\mathit{\xi }}_{\mathit{m}}$ [g/l], Ω [1/h], and $\mathsf{\Gamma }$ [1/h²] are diagonal matrices whose elements are the tuning parameters of the estimator.

In [37], the analysis of the stability and convergence of the system (3) and (4) is part of this one related to the Fully Adaptive Linearizing Control Algorithm, shown below.

The result related to the single input-single output system (${\mathit{\xi }}_{\mathit{m}}={\xi }_1, {\mathit{F}}_{\mathit{m}}=F_1, {\mathit{\varphi }}_{\mathit{m}}={\varphi }_1, \mathsf{\Omega }={\mathsf{\omega }}_1, \mathsf{\Gamma }={\mathsf{\gamma }}_1)$ is shown here. The following asymptotical bounds of the estimation errors ${\tilde{\xi }}_1\left(t\right)$ and ${\tilde{\varphi }}_1$ are obtained:

$$\underset{t\to \infty }{\lim }\mathit{sup}\left|{\tilde{\xi }}_1\left(t\right)\right|\leqq \frac{2{\mathrm{m}}_{21}\mathsf{\delta }{\beta }_{11}}{\sqrt{{\mathsf{\omega }}_1^2-4{\mathsf{\gamma }}_1}}+\frac{{\beta }_{21}}{{\mathsf{\gamma }}_1}$$

(5)

$$\underset{t\to \infty }{\lim }\mathit{sup}\left|{\tilde{\varphi }}_1\left(t\right)\right|\leqq {\mathrm{m}}_{21}{\beta }_{11}+{\mathsf{\omega }}_1\frac{{\beta }_{21}}{{\gamma }_1}$$

(6)

with ${\beta }_{11}=D+{\mathsf{\omega }}_1$; ${\beta }_{21}={\mathrm{m}}_{21}\mathsf{\delta }+{\mathrm{m}}_{11}$; ${\beta }_{11}$, ${\beta }_{21}$ functional parameters [1/h]; $m_{11}$ and $m_{21}-$ upper bounds of the time derivative of ${\varphi }_1$ [g/lh²] and measurement noise of ${\xi }_1$ [g/l], respectively; $\mathsf{\delta }={\left(\frac{{\mathsf{\lambda }}_1}{{\mathsf{\lambda }}_2}\right)}^{{\mathsf{\lambda }}_1/{(\mathsf{\lambda }}_1-{\mathsf{\lambda }}_2)}-{\left(\frac{{\mathsf{\lambda }}_1}{{\mathsf{\lambda }}_2}\right)}^{{\mathsf{\lambda }}_2/{(\mathsf{\lambda }}_1-{\mathsf{\lambda }}_2)}$, ${\mathsf{\lambda }}_1$ [1/h], and ${\mathsf{\lambda }}_2$ [1/h] are the eigenvalues of the matrix of estimation error dynamics related to ${\mathsf{\omega }}_1$ and ${\mathsf{\gamma }}_1$ by:

$${\mathsf{\omega }}_1=-\left({\mathsf{\lambda }}_1+{\mathsf{\lambda }}_2\right),{\mathsf{\gamma }}_1={\mathsf{\lambda }}_1{\mathsf{\lambda }}_2 \mathrm{under} \mathrm{assumption} {\mathsf{\lambda }}_2<{\mathsf{\lambda }}_1<0 \mathrm{and} {\mathsf{\gamma }}_1\leqq {\mathsf{\omega }}_1^2/4$$

(7)

The formulation of the proving theorem for asymptotical bounds (5) and (6) is given in Appendix A.

2.2. Tuning Procedure

The optimal value of the estimator parameter ${\mathsf{\omega }}_1$ is considered as the one minimizing the asymptotic upper bound of ${\tilde{\varphi }}_1\left(t\right)$ from Equation (6). The following expression is obtained as a result [37]:

$${\mathsf{\omega }}_{1\mathrm{o}\mathrm{p}\mathrm{t}}=2\mathsf{\zeta }\sqrt{{\mathrm{m}}_{11}/{\mathrm{m}}_{21}}$$

(8)

The optimal value of the tuning parameter ${\mathsf{\gamma }}_1$ is chosen as follows:

$${\mathsf{\gamma }}_{1\mathrm{o}\mathrm{p}\mathrm{t}}={\mathsf{\omega }}_{1\mathrm{o}\mathrm{p}\mathrm{t}}^2/4{\mathsf{\zeta }}^2$$

(9)

where $\mathsf{\zeta }$ is the damping coefficient with a fixed value close to 1.

2.3. Fully Adaptive Linearizing Control Algorithm

It is well known that the control of bioprocesses can be accomplished by feeding limiting substrates. The following matrix of the differential equation describes the dynamics of the substrates in the bioreactor:

$$d{\mathit{\xi }}_{\mathit{c}\mathit{s}}/dt={\mathbf{K}}_{\mathbf{c}\mathbf{s}}{\mathit{\phi }}_{\mathit{c}\mathit{s}}\left(\mathit{\xi }\right)-D{\mathit{\xi }}_{\mathit{c}\mathit{s}}+{\mathit{F}}_{\mathit{c}\mathit{s}}$$

(10)

where ξ_cs ∈ ℝ^cs×1 [g/l] is a vector of the concentrations of the controlled feeding substrates (c_s indicates their number); K_cs ∈ ℝ^cs×n_cs [g/g] represents the yield coefficient matrix, related to the kinetics of ξ_cs (n_cs indicates the number of the reaction rates, describing the kinetics of ξ_cs); φ_cs ∈ ℝⁿ_cs^×1 [g/lh] stands for the reaction rates vector and F_cs ∈ ℝ^cs×1 [g/lh] is the mass feed rate of the substrate ξ_cs in the bioreactor.

The controlled output vector y is assumed to be the vector ξ_cs of the concentrations of the controlled substrates:

y = ξ_cs

It is assumed that:

• B1. The substrate concentrations (ξ_cs), the dilution rate (D), and the feed rates (F_cs) are measured online.
• B2. The term K_csφ_cs, describing the substrate’s ξ_cs consumption rates, is fully unknown.

Let us consider:

ϕ = −K_csφ_cs

where ϕ∈ℝ^cs×1 [g/lh] is considered as an unknown time-varying parameter vector.

In the case of assumptions A1 and A2, the well-known structure of the observer-based estimator can be represented in its general form as follows:

$$d\widehat{\mathit{y}}/dt=\widehat{\mathit{\varphi }}-D\mathit{y}+{\mathit{F}}_{\mathit{c}\mathit{s}}+\mathsf{\Omega }(\mathit{y}-\widehat{\mathit{y}})$$

(11a)

$$d\widehat{\mathit{\varphi }}/dt=\mathsf{\Gamma }(\mathit{y}-\widehat{\mathit{y}})$$

(11b)

where Ω, Γ ∈ ℝ^cs×cs are matrices whose components are the estimator design parameters.

Since the control objective is to set the substrate concentration ξ_cs to given values y* ∈ ℝ^cs×1 (constant or time-varying), Equation (10) is considered an input-output model. We choose the first-order reference model of the tracking error:

$$d\left({\mathit{y}}^{*}-\mathit{y}\right)/dt+\mathsf{\Lambda }\left({\mathit{y}}^{*}-\mathit{y}\right)=0$$

(12)

where Λ ∈ ℝ^cs×cs [1/h] is a diagonal matrix containing the control design parameters.

Now, combining Equations (10) and (12), we obtain:

$${\mathit{F}}_{\mathit{c}\mathit{s}}=\mathsf{\Lambda }\left({\mathit{y}}^{*}-\mathit{y}\right)-\mathit{\varphi }+D\mathit{y}+d{\mathit{y}}^{*}/\mathit{d}\mathit{t}$$

(13)

With the estimates $\widehat{\mathit{\varphi }}$ Equation (13) remains:

$${\mathit{F}}_{\mathit{c}\mathit{s}}=\mathsf{\Lambda }\left({\mathit{y}}^{*}-\mathit{y}\right)-\widehat{\mathit{\varphi }}+D\mathit{y}+d{\mathit{y}}^{*}/\mathit{d}\mathit{t}$$

(14)

The control algorithm (14) is valid for fed-batch cultivation processes. In continuous mode, it should be taken into account that D = F_cs/V.

The stability analysis of the system (11)–(14) is performed in [37]. The formulation of the proving theorem for asymptotical bounds is given in Appendix A.

This fully adaptive linearizing algorithm could be considered as an extension of the theory proposed by Bastin and Dochain [5] since the substrate consumption rates are presented as a fully unknown time-varying vector, ϕ, describing the whole process kinetics.

2.4. Applications

The two applications given below are case studies of fully adaptive control algorithms, as shown in Figure 3 in green.

2.4.1. Control of Gluconic Acid Production by Aspergillus niger

Following the procedure for control design proposed above, the following control law is derived for the process of gluconic acid production [37]:

$$d\widehat{G}/dt=\widehat{\varphi }-D{G}_m+D{G}_{in}+\mathsf{\omega }(G_m-\widehat{G})$$

(15a)

$$d\widehat{\varphi }/dt=\mathsf{\gamma }(G_m-\widehat{G})$$

(15b)

$$D=({\mathsf{\Lambda }}_1\left({\mathrm{G}}^{*}-G_m\right)-\widehat{\varphi })/( G_{in}-G_m)$$

(16)

where y = G, y_m = G_m and F_cs = DG_in, ϕ—glucose consumption rate, G—the glucose concentration [g/l], G_m—the glucose concentration measurements [g/l], G_in—glucose concentration in the feeding [g/l].

The purpose of the control is to set and maintain the glucose concentration (G) to a desired constant value (G*). The control algorithm is investigated by numerical simulations with a set point, G* = 3 g/l. In Figure 4, the results from the simulation are shown.

These simulations perfectly agree with the expected qualitative behavior of the closed-loop system and thus validate the proposed approach.

2.4.2. Alpha-Amylase Production by Bacillus subtilis

Following the procedure for control design proposed above, the following control law is derived for the considered case [38]:

$$d\widehat{G}/dt=\widehat{\varphi }-F(G_m-G_{in})+\mathsf{\omega }(G_m-\widehat{G})$$

(17)

$$d\widehat{\varphi }/dt=\mathsf{\gamma }(G_m-\widehat{G})$$

(18)

where ω and γ are the estimator design parameters, G_m = G + ε, ε is the additive measurement noise, and ϕ is the glucose consumption rate. The estimator (system (17) and (18)) design parameters are derived using the tuning procedure described by Equations (8) and (9).

The control law for the considered case is as follows:

$$F=V\left({\mathsf{\Lambda }}_1\right({\mathrm{G}}^{*}-G_m)-\widehat{\varphi })/(G_{in}-G_m)$$

(19)

In the paper [38], two discrete control algorithms are proposed, including software sensors of kinetics. Algorithm I has the advantage of being a fully adaptive one, estimating the value of the glucose consumption rate as a fully unknown time-varying parameter. Algorithm II shows better performance due to the application of two online measurements: biomass concentration with a sampling time of 0.01 h and glucose concentration of 0.5 h, while algorithm I uses only the second measurement.

Other applications of the proposed control design are related to PH control during continuous pre-fermentation of yogurt starter culture by strains S. thermophilus 13a and Lb. bulgaricus 2–11.

3. Marker-Based Pulse Control of Intermediate Metabolite Production and Consumption

In some bioprocesses, two main substrates limit the target production. One of them is an intermediate metabolite produced and consumed during the process. When it is necessary to maintain a balance between the consumption and production of an intermediate metabolite, a control marker can be defined as the difference between the consumption and production rates to switch on or switch off the control input, thus determining the impulse character of the control action.

The two applications given below are case studies of partially adaptive control algorithms, as shown in Figure 3 (in green).

3.1. Impulse Adaptive Control of Biopolymer Production by Mixed Culture

In [39], the mechanism of polyhydroxybutyrate (PHB) production by a mixed culture of Lactobacillus delbrulckii and Ralstonia eutropha is presented. The process control aims to receive more target product PHB using glucose as a feeding substrate. The intermediate metabolite, the lactate concentration, has to be maintained at an optimal level during the process. As a result, an increase in the concentration of R. eutropha, X₂, [g/l] and the production of PHB is achieved. Moreover, the growth of L. delbrulckii, X₁, [g/l] requires a low level of dissolved oxygen, O₂, [mg/l is also referred to as parts per million (ppm)] while the growth of X₂—a high O₂ level.

The optimal process control stabilizes the lactate concentration at an optimal level, ${\mathrm{L}}_{\mathrm{o}\mathrm{p}\mathrm{t}},$ [g/l] in the bioreactor. It is calculated theoretically by the expression ${\mathrm{L}}_{\mathrm{o}\mathrm{p}\mathrm{t}}=\sqrt{{\mathrm{K}}_{\mathrm{i}}{\mathrm{K}}_{\mathrm{P}}, }$ ${\mathrm{K}}_{\mathrm{i}}{,\mathrm{K}}_{\mathrm{P}}$—kinetic parameters [g/l].

Hence, SSs for lactate production rate, ${\Phi }_1$, [g/lh] and consumption rate, ${\Phi }_2$, [g/lh] are designed. Using this information, the growth of both microorganisms can be stimulated separately. The lactate production rate,${\Phi }_1$, is proportional to the X₁ growth rate and the lactate consumption rate, ${\Phi }_2$—to the X₂ growth rate. Each software sensor provides information on the growth rate of one microorganism. The difference between the estimates of ${\Phi }_1$ and ${\Phi }_2$, $\mathsf{\Delta }={\widehat{\Phi }}_1-{\widehat{\Phi }}_2$, is defined as a marker for stimulation of the L production or consumption rate leading the process to the target L_opt.

Switching over to the low O₂ level, as well as the inclusion of the glucose feed, takes place when Δ becomes negative, i.e., when it is necessary to stimulate the growth of L. delbrulckii. The levels of the glucose feed impulses, $F_{opt},$ [g/lh] are determined by the dynamical equation of lactate concentration, accepting zero dynamics, known kinetics, and the optimal value of the lactate concentration, L_opt:

$$F_{opt}=({\widehat{\Phi }}_1-{\widehat{\Phi }}_2)V/{\mathrm{L}}_{\mathrm{o}\mathrm{p}\mathrm{t}}$$

(20)

In the case of a positive Δ, the O₂ switches over to the high level. There is no substrate feed, i.e., the accumulated lactate is consumed by R. eutropha.

As a result, the lactate concentration tends to its optimal value, as shown in Figure 5c. The elapse of glucose is shown in Figure 5d. In Figure 5a, a good tracking elapse of estimated values of lactate production and consumption rates, ${\widehat{\Phi }}_1$ and ${\widehat{\Phi }}_2$, can be observed.

3.2. Impulse Control of Simultaneous Saccharification and Fermentation of Starch to Ethanol (SSFSE)

The main purpose of the control is to maintain as long as possible the glucose concentration in an equilibrium state, as observed during batch cultivation, where the production rate of the target product (ethanol) is the highest [40,41]. Glucose is an intermediate metabolite obtained through the enzymatic hydrolysis of starch. In this way, the starch is the control input variable.

Software sensors for the glucose production rate, ${\widehat{\Phi }}_1$, and consumption rate, ${\widehat{\Phi }}_2$, are used for recognizing this equilibrium state, and the difference between their measurements is defined as a marker $\Delta ={\widehat{\Phi }}_1-{\widehat{\Phi }}_2$. When the sign of the marker Δ is positive, the glucose production is higher than its consumption. A negative sign indicates the opposite situation. The main purpose is to observe the sign of the marker and to stimulate glucose production by starch feeding when the consumption is higher. Starch has to be added only when the marker is negative. The amplitude of the starch feed impulses can be calculated by the dynamical equation of glucose concentration, assuming zero dynamics of the glucose concentration:

$$F=({\widehat{\Phi }}_1-{\widehat{\Phi }}_2)V/G_m$$

(21)

Control scheme investigations are carried out through simulations. The process starts in the batch phase without using the marker and, therefore, without control input calculation until the glucose reaches an apparent equilibrium state (around 20 h). The calculation of the marker then starts, but the control is switched on only when the glucose production rate begins to decrease, which is around the 50th hour of fermentation, according to Figure 6a.

Figure 7 shows the simulation results for the ethanol concentration and the ethanol growth rate for the fed-batch SSFSE process using a starch input concentration of 50 g/l (containing 7.5 g/l of glucose available due to autoclaving), applying the adaptive control (21). In addition, the fed-batch results are compared to those for the batch process, which was open-loop simulated using the model given [41]. It can be seen that the ethanol concentration (and therefore the productivity) for the controlled fed-batch process is higher than the ethanol concentration reached under batch operation. Furthermore, it is important to remark that the ethanol production rate in the fed-batch process can be kept at higher values than for the batch, ensuring a more productive process.

The proposed control algorithm is derived based on glucose and starch sensors available in the industry. This allows control (21) to be applied in real-life experiments.

4. Monitoring and Control of Processes Passing through Different Metabolic Regimes

There exists a class of BTP in which the growth of microorganisms passes through two or more physiological states, which are described by different mathematical models. To recognize and identify the transition from one physiological state to another, it is necessary to determine a key parameter. In some cases, such a key parameter is the synthesis of an intermediate metabolite (acetate, ethanol, etc.) or some other parameter (main substrate consumption, etc.). Thus, monitoring of each of the physiological states is possible. Depending on the target product, the process is held in a state that leads to an increase in productivity.

In [42], the reaction scheme of the process could be presented as a set of the three main reactions (metabolic pathways) of fed-batch fermentation of E. coli that correspond to the following physiological states:

Oxidative growth on glucose, with a specific growth rate ${\mu }_1$ [1/h]:

$${\mathrm{k}}_{1}G+{\mathrm{k}}_5{O}_2 \stackrel{{\mu }_1}{\to }X+{\mathrm{k}}_{8}C$$

(22)

Fermentative growth on glucose, with a specific growth rate ${\mu }_2$ [1/h]:

$${\mathrm{k}}_{2}G+{\mathrm{k}}_6{O}_2 \stackrel{{\mu }_2}{\to }X+{\mathrm{k}}_{3}C+{\mathrm{k}}_{9}A$$

(23)

Oxidative growth on acetate, with a specific growth rate ${\mu }_3$ [1/h]:

$${\mathrm{k}}_{4}A+{\mathrm{k}}_7{O}_2 \stackrel{{\mu }_3}{\to }X+{\mathrm{k}}_{10}C,$$

(24)

where X, G, A, O₂ and C are, respectively, concentrations of the biomass, glucose, acetate, dissolved oxygen and carbon dioxide in the culture broth, [g/l], k₁–k₁₀—yield coefficients, [g/g], ${\mu }_1$,${\mu }_2$,${ \mu }_3$—specific growth rates related to the oxidative growth on glucose, fermentative growth on glucose, and oxidative growth on acetate, respectively.

A model of the process is presented by the reaction schemes (22)–(24) [42], including two sub-models that describe the oxidative-fermentative growth on glucose and oxidative growth on glucose and acetate. A new marker for switching the sub-models—kinetics of the acetate concentration, Rac, [g/lh], is proposed and tested successfully through simulations using experimental data. Rac values can be obtained online by calculations or software sensors using online information on the acetate concentration. The following boundary conditions are proposed for changing the regime of glucose from oxidative to oxidative-fermentative:

Rac = 0, G ≠ 0 oxidative growth on glucose

(25a)

Rac > 0, G ≠ 0 oxidative-fermentative growth on glucose

(25b)

i.e., if the acetate kinetics is zero and there is glucose in the culture medium, biomass growth depends only on the oxidative degradation of this substrate with rate ${\mu }_1$.

When the values of the acetate kinetics are positive, i.e., there is acetate production and the presence of glucose, the growth of microorganisms is determined by the aerobic and anaerobic consumption of glucose with rates ${\mu }_1$, ${\mu }_2$.

The negative values of Rac are markers for acetate consumption and biomass growth of this metabolite.

Two cases are considered, depending on the presence of glucose:

Rac < 0, G ≠ 0 oxidative growth on acetate and glucose

(26a)

Rac < 0, G = 0 oxidative growth on acetate

(26b)

When the values of acetate kinetics are negative, i.e., there is acetate consumption and the presence of glucose, the growth of microorganisms is determined by two rates, ${\mu }_1$, ${\mu }_3$. When the glucose concentration is zero, the biomass growth rate is ${\mu }_3,$ since only the oxidative growth on acetate is available.

Online monitoring of the three physiological states of the process is proposed in [42].

In Figure 8, the scheme of the software sensor design is shown.

Following the procedure for control design proposed in Section 2.3, a partially adaptive linearizing control low is derived for the considered process, characterized by two physiological states—oxidative and oxidative-fermentative growth on glucose [43].

The input/output model is presented as:

$$dG/dt=-{\mathrm{k}}_1{\widehat{R}}_{X1}-{\mathrm{k}}_2{\widehat{R}}_{X2}-D{G}_m+\left(F_{in}/V\right)G_{in}$$

(27)

where ${\widehat{R}}_{X1}$, ${\widehat{R}}_{X2}$, ${\widehat{R}}_{X3}$ [g/lh] are estimates of growth rates $R_{X1}$, $R_{X2}$, $R_{X3}$ [g/lh] related to oxidative growth in glucose, fermentative growth in glucose, and oxidative growth in acetate.

A first-order reference model is selected. In the case of stabilization, dG*/dt = 0 because G* is the constant set-point value of glucose concentration [g/l].

The reference model can be presented as follows:

$${\mathsf{\Lambda }}_1\left({\mathrm{G}}^{*}-G_m\right)=dG/dt$$

(28)

where ${\mathsf{\Lambda }}_1$ is the control tuning parameter [1/h].

The control algorithm is derived by substitution of the model (27) in (28):

$$F=\left(W\right(-{\mathsf{\Lambda }}_1\left({\mathrm{G}}^{*}-G_m\right)+{\mathrm{k}}_1{\widehat{R}}_{X1}+{\mathrm{k}}_2{\widehat{R}}_{X2})/(G_{in}-G_m)$$

(29)

where W is the weight of the culture medium [kg].

This type of control is partially adaptive since, in law (29), the evaluation of substrate kinetics $-{\mathrm{k}}_1{\widehat{R}}_{X1}-{\mathrm{k}}_2{\widehat{R}}_{X2}$ includes constant yield coefficients. The goal of control is to increase protein productivity, which is proportional to biomass. From an expert point of view, by maintaining a low constant value of G*, biomass concentration levels can be increased. The simulation results are shown in Figure 9.

A comparison of these results is realized in [43] with an open-loop control of the same process (Figure 10). Maintaining a constant value of G*, it can achieve almost the same concentration of biomass (target product) (Figure 9c) as in Figure 10, but it should be noted that the biomass concentration in the proposed control continues to grow until the end of the process, while in the other one, retention and decline in biomass concentration after 25 h was observed. The decrease may be due to the combination of factors that inhibit the growth of biomass—the increased biomass density during the fermentation process and the presence of acetate, which starts to be produced around the 20th hour. At the same time, the weight at the end of the process is smaller (Figure 9d) in comparison with Figure 10, which leads to a better efficiency of the proposed control.

5. Generalized Control Strategies and Structures

A significant contribution to the control synthesis theory is the new formalization of kinetics as a completely unknown nonstationary parameter, as presented in Section 2. Thus, errors assuming the constant yield coefficients are avoided.

The advantages and limitations of each structure are presented in

Table 1. General Dynamic Model Approach Developments.

No.	Control Strategy and Structure	Refs.	Description
1		[4,37,38]	Advantages: The process kinetics, ϕ (t), is presented as a fully unknown time-varying parameter. Optimal SS tuning is performed. The SS included in the control law makes it fully adaptive in terms of kinetics. Limitation: It is mainly applied for the stabilization of the limiting substrate.
2		[39,40,41]	Advantages: The control stabilizes an intermediate metabolite at an optimal value using a marker as the difference between consumption and production of that metabolite Limitation: The SS included in the control law makes it partially adaptive in terms of kinetics.
3		[42,43]	Advantages: Monitoring of physiological states in multi-rate processes by a marker of the kinetics of an intermediate metabolite. Recognition and stabilization of the desired physiological state. Limitation: The SS included in the control law makes it partially adaptive in terms of kinetics.

.

The role of software sensors in the process monitoring (recognition of different physiological states) and the synthesis of different control laws should be emphasized. For this purpose, the type of the generalized software sensor is derived, a procedure for its optimal setting is proposed, and its inclusion in the structure of a generalized fully adaptive linearizing control (Table 1, No. 1). As case studies for the first structure, the processes of gluconic acid and alpha-amylase production are considered.

In the processes of gluconic acid production, when the main carbon source (glucose) is exhausted, the resulting target product serves as a substrate for biomass growth. To avoid this effect, the phase is switched off by supplying the main carbon source.

The second structure is the partially adaptive control of an intermediate metabolite (Table 1, No. 2). The control stabilizes an intermediate metabolite at an optimal value using a marker—the difference between the consumption and production of that metabolite. Two case studies are discussed. In the first one, for ethanol production, the necessary substrate (glucose) is obtained from starch. To continue the phase of maximum production of the target product, a pulse control is employed, which is activated based on a proposed marker, as described in Section 3. Another type of pulse control is designed for the process of biopolymer (PHB) production by a mixed culture. Unlike the previous example, in this case, two control inputs—dissolved oxygen level and glucose feeding rate, are considered. The marker that activates the pulse control is calculated as the difference between the production and consumption of the intermediate metabolite (lactose). Thus, the maintained lactose concentration is at a theoretically calculated optimal value that guarantees the maximum production of the target product PHB.

The third structure is also a partially adaptive linearizing control (Table 1, No. 3). The role of an adaptive marker in recognizing the different physiological states of the E. coli protein production process is discussed in Section 4. A real-time estimation of the process kinetics is carried out. The derived control law, including the estimated kinetics, guarantees a higher productivity of the process compared with open-loop control.

The presented examples confirm the effectiveness of the proposed new GDM approach in reaching maximum productivity in various bioprocesses.

6. Interactive System for Education in Modelling and Control of Biotechnological Processes

Modern optimization algorithms, as well as the algorithms for monitoring and control discussed above, are incorporated into an interactive system called InSEMCoBio.

Figure 11 shows a diagram of the modules of InSEMCoBio and the connections between them. The modules with algorithms for identification, monitoring, and control are shown in yellow. Three types of optimization algorithms are developed and built into the system [43]. The model parameter identification can be performed by applying a genetic algorithm (GA), an evolutionary algorithm (EA), and a hybrid algorithm (GA-EA).

The adaptive control module includes two submodules. One contains software sensor algorithms, and the other control algorithms. The control can be carried out by activating one of the two modes of fermentation: fed-batch or continuous. The blocks in blue allow the user to choose the process and algorithms described above depending on the task at hand.

The first step is selecting one of the existing processes and relevant experimental data. The structure of the batch model for this process is then selected, as well as the optimization procedure, to perform the identification of the model parameters. The selection of the model structure and the optimization procedure can be repeated many times, each new result being compared with the previous ones. After completion of the identification procedures, the models with the most suitable structure and highest accuracy are stored.

The last group of modules (in pink) serves to store the obtained results. The optimal models for the batch fermentation mode are saved, and after analysis, the best one is selected. The model is used instead of the real process during the simulations. Moreover, this is the basis for deriving a model for the control. Finally, the monitoring and control results are displayed on the screen.

Using the system InSEMCoBio for research or training allows the user to become familiar with different approaches for model identification and control of a particular bioprocess—by activating a fed-batch or continuous mode of fermentation. After analyzing the simulation results, the user can then proceed to conduct experiments under laboratory conditions.

Some significant advantages of InSEMCoBio should be noted here: (i) it incorporates modern optimization algorithms for identifying models of nonlinear systems, as well as algorithms for monitoring and controlling biotechnological processes; (ii) the system has a user-friendly interface, and it is not necessary to understand programming to run the integrated algorithms and obtain the results one is interested in; and (iii) the system can be easily extended by implementing new processes, additional model structures, experimental data, etc.

7. Conclusions

In this paper, the newly proposed developments and applications of the GDM approach are discussed. Each of the mentioned innovative control designs is derived after analysis of the behavior of the respective culture during the batch phase. In this phase, the periods during which various metabolites are produced without external influence are clearly outlined. Depending on the target product, efforts are directed toward recognizing the phase of its production. The goal is to select a control action that lasts as long as possible in this phase, and in some cases, it is necessary to exclude parts of the natural phases.

The main contributions to the development of the GDM approach are the new formalization of the kinetics of biotechnological processes, as well as the development of a generalized software kinetic sensor and a generalized adaptive linearizing control algorithm. Based on this theory, three generalized structures and strategies for deriving control algorithms are proposed and summarized in Table 1. The advantages and limitations of the new structures and strategies described can be a guideline for choosing an appropriate control for a specific object.

In addition to the control laws discussed in the overview, they are included in the structure of the InSEMCoBio system. The advantages of using such a system are found in scientific research and student education.