Theoretical Study and Optimization of the Biochemical Reaction Process byMeans of Feedback Control Strategy

e aim of this work is to present a theoretical analysis and optimization of a biochemical reaction process by means of feedback control strategy. To begin with, a mathematical model of the biochemical reaction process with feedback control is formulated. en, based on the formulated model, the analysis of system’s dynamics is presented. e optimization of the bioprocess is carried out, in order to achieve maximal biomass productivity. It is shown that during the optimization, the bioprocess with impulse effects loses the possibility of synchronization and strives for a simple continuous bioprocess. e analytical results presented in the work are validated by numerical simulations for the Tessier kinetics model.


Introduction
Chemical and bioprocess engineering play an important role in the production of many chemical products.In particular, bioreactor engineering as a branch of chemical engineering and biotechnology is an active area of research on bioprocesses, including among others development, control, and commercialization of new technology [1].e reaching of optimal results and the obtainment of maximal pro�ts require modern control strategies based on mathematical models or arti�cial intelligence methods [2].ere are many advantages with the use of mathematical models, and one of which is the possibility of testing process stability [3].According to different reactions and differential control technologies, many dynamic biochemical models in a chemostat have been established [4][5][6][7][8][9][10][11][12].However, there are many factors which affect the growth and reproduction of microorganisms in the bioprocess.e key variable is the biomass concentration, because, among other things, it provides information concerning biomass productivity.For that reason, almost all mathematical growth models contain the biomass as an important variable [13].Moreover, for aerobic microbes, the dissolved oxygen concentration (DOC) is an important growth factor.During the growth of microorganisms, the dissolved oxygen concentration depends on several factors, among them the biomass concentration, the concentration and type of substrate used, and the bioprocess conditions.Because a low level of dissolved oxygen concentration decreases the biomass yield and the speci�c growth rate, it is necessary to monitor and control the DOC level so that it stays within the appropriate range [14].
Since many biological phenomena such as bursting rhythm models in, for example, medicine, biology, pharmacokinetics, and frequency modulated systems exhibit impulsive effects [15], impulsive differential equations, which appear as a natural description of observed evolution phenomena of several real world problems, have been introduced in different kind of biological systems, for example, in population dynamics [16][17][18][19] and in biochemical process [20,21].It should be pointed out that in these works, the authors were concerned about the �xed time impulse effects, which has the rationality in describing the biological phenomena.While in some cases, using the state-dependent impulse effects to describe the biological phenomena is more appropriate.As far as the state-dependent impulse effect is concerned, Tang and Chen [22] introduced a Lotka-Voterra model, which is constructed according to the practices of IPM.By using analytical method, it is shown that there exists an orbitally asymptotically stable periodic solution with a maximum value no larger than the given economic threshold.Further, the complete expression of period of the periodic solution is given.More researches on the applications of the state-dependent impulse effects can be found in [23][24][25][26][27][28][29][30][31][32][33][34][35][36].In these researches, the authors analyzed the proposed system's dynamic behavior (e.g., the existence and stability of period-1 solution and the existence of period-2 solution) by applying the Poincaré principle and Poincaré-Bendixson theorem of the impulsive differential equation.Recently, in microbioprocess engineering, the dynamic properties of the kinetic models with impulse effects characterized by the universal microorganism growth rate and two different kinds of biomass yield are also analyzed theoretically by Sun et al. [37][38][39].
e objective of this work is to illuminate theoretical and practical aspects of the nonlinear analysis of a universal mathematical model of the biochemical reaction process.e paper is organized as follows.In Section 2, a universal mathematical model of the biochemical reaction process with any characteristics of growth kinetics is formulated under impulse effects.In Section 3, a qualitative analysis of the proposed model is presented.In Section 4, simulations with Tessier's kinetics are presented in order to verify the theoretical results and discuss the biological essence.Moreover, in Section 4, in order to optimize the biochemical reaction process, the conception of objective function is presented, and next the bioprocess optimization is put forward.Finally, in Section 5, we offer our conclusions.

Mathematical Model and Preliminaries
e general model of continuously culturing microorganism in a chemostat is given by the following form of differential equations [40]: where  = () denotes the biomass concentration (gL), and  = () the substrate concentration (gL) in the bioreactor medium at time ,  0 , and  0 denotes the initial biomass concentration and substrate concentration, respectively;  is the dilution rate (h −1 );  in is the in�uent substrate concentration (gL);   is the biomass yield (gg) de�ned as the ratio of the biomass  produced to the amount of substrate  consumed;   < 1 for biological constraints; the function () describes the biochemical kinetics, which are characterized by the cell concentration (), depending on one limiting substrate with concentration .Crooke et al. [6] pointed out that the biochemical kinetics expression plays an important role in the intrinsic oscillation mechanisms.So many different assumptions for the kinetics models are given in the literature, for example, Monod-type kinetics [3], that is, () =  max (  + ), where  max is the maximum speci�c growth rate and   is the substrate saturation constant; Tessier-type kinetics [41], that is, () =  max (1 −  −  ); Moser-type kinetics [41], that is, () =  max (1 +    − ) −1 , where  is a positive constant.It can be easily seen that these kinetic models satisfy the following properties: (i) (regularity) : ℝ + → ℝ + is continuously differential and (0) = 0, (ii) (monotonicity)  is monotonically increasing, that is,   0 for all   ℝ + ;  ′ () = 0 for some   ℝ + means that  ′ () = 0 for   , for all   0 1 and  1   2  ℝ + .
us, in the study that the kinetic models only satisfy the regularity, monotonicity, and convexity are considered.
According to the Herbert's and Pirt's models [41], if the substrate concentration is big enough (i.e.,     ), the biomass yield is constant.Because in continuous bioprocesses the optimal biomass productivity is obtained for bigger substrate concentration, in this work the constant biomass yield is assumed, that is,   =   max =  < 1.In addition, according to the design ideas of the bioreactor, the biomass concentration in the chemostat should be controlled to a set level  set , where 0 <  set ≤  CRI , and  CRI is the critical level of biomass concentration in the bioreactor medium.When the biomass concentration reaches the set level, part of the medium containing biomass and substrate is discharged from the bioreactor, followed by the input of the "clear" medium (i.e., the medium without substrate).erefore, system (1) can be modi�ed by introducing the impulsive state feedback control as follows: where   is the part of "clear" medium inputted into the bioreactor in each biomass oscillation cycle, which satis�es 0 ≤   ≤   max < 1, where   max is the maximum part of "clear" medium inputted into the bioreactor in each biomass oscillation cycle.Before presenting the main results, we recall the following de�nitions and lemmas �rst [31,42,43].
(() ()) is said to be period-1 solution if in a minimum cycle time there is one impulse effect.Similarly, (() ()) is said to be period-2 solution if in a minimum cycle time there are two impulse effects.
Γ is said to be orbitally stable, if for any   0, there exists   0, with the proviso that every solution (( ( of system (2) whose distance from Γ is less than  at    0 , will remain within a distance less than  from Γ for all    0 .Such a Γ is said to be orbitally asymptotically stable if, in addition, the distance of (( ( from Γ tends to zero as   .Moreover, if there exist positive constants  , and a real constant  such that ((( ( Γ   − for   , then Γ is said to be orbitally asymptotically stable and enjoys the property of asymptotic phase.direction of the orientation �eld determined by model ( 5) is pointing to the interior of .In addition, there is no singularity in the interior of , and the boundary.One of the boundary  is the impulse set of the model (5); the corresponding phase set satis�es (  ;  is also the no cut-arcs of model (5), and on , the direction of the orientation �eld determined by model (5) is pointing to the interior of .en, there must exist at least one period-1 solution of model (5)

Existence of
Proof.Firstly, it is obvious that all trajectories of system (2) starting from the region Ω  must interest with the segment  and then jump to the segment  if   )   in and  set   in    )).Next, we construct the closed region   Ω  such that all solutions of system (2) starting from Ω  enter into  and retain there.
It is obvious that the straight line  is described by    set   ), and  is described by    set  in , and both lines pass through the point , ).e derivative of the straight line , ) passing through the points   ),  set ) and   ) in ,   ) set ) along the trajectories of system ( 2) is in terms of  2)  ,  2) >  for the segment .
From the qualitative characteristic of system (2), we know that  2) >  for  and  2)   for .Besides,  2)   for    in .On the other hand, it follows from system (2) that )   and )   in   in    )  for  ∈ , +).Especially, when      ,   is the semitrivial solution of system (2).erefore, we have found a closed region , the boundary of which consists of , , , , , and ; see Figure 3(a).
Case (ii) (  >   ) and   ′    ′ ).Similar to the discussion of Case (i), the boundary of  can be constructed by ,   ′ ,  ′ , and ; see Figure 3(b).
In addition, there is no singularity in the region .erefore, it follows from the qualitative characteristics of the system that all the trajectories satisfying the conditions of theorem enter the closed region  and retain there.en it follows from Lemma 4 that system (2) has a period-1 solution.

Position of Period-1 Solution.
Suppose the period of the period-1 solution is .Let )  ), )  ) be such a -periodic solution.en )  ) + ) is also periodic.Denote     + )   + ),    ),     + )   + ), and    ).en from the -periodicity of the periodic solution and the third and fourth equations of system (2), we have        )  ,        )  , and     set .Without loss of generality, for  ∈ , , ) satis�es the relation en, we have  )   in +   +    in  exp ) .
Let   >   ) be the root of the following equation:  Next, we consider the following comparison system of system (2) without impulsive feedback control: It is easy to verify that system (13) has one positive equilibrium ( −1 (,  set , which is a center point.en all of its solutions are closed trajectories which satisfy where  is an arbitrary constant.e derivative of  1 (,  along the trajectories of system (2) is for  −1 (      .is implies that the trajectories of system (2) intersecting with the trajectories of system (13) pass through the trajectories of system (13) from the le to the right.In addition, the trajectory of system (13) passing through   ,  set ) (i.e.,    in ( 14)) intersects the line    −1 ) at two points  1  −1 ),   1 ) and , where      1, 2) satisfy that that is, Since Λ 1  ln   ) −     set < ln  set − 1, then we have . us, we have and   2 <  set <   1 , where  denotes Lambert  function de�ned in De�nition 3.
Next, we will prove that  1   * when the conditions of theorem are satis�ed.�et  be the intersection point of the line  =  * and  = 0,  1 , and let  2 be the intersection points of the line    −1 ( and the trajectory of system (23) passing through the point .If  set ≤ min{ * , (1 −   }, then we have (1 −    set ≤ .Since  2  (2 > 0,   1  > 0,   < 0,   > 0,   > 0, and   is semi-trivial solution of system (2).erefore, we get a closed region , the boundary of which consists of   1 ,  1 , , , , , and .In addition, there is no singularity in the region .erefore, it follows from the qualitative characteristics of the system that all the trajectories satisfying the conditions of theorem enter the closed region  and retain there.en, it follows from eorem 6 that system (2) has a period-1 solution in .Furthermore, we have  * ≤  1 ≤  in .If  1 ≥  −1 (, then  0 ≥  −1 (, in this case, we have 0 ≤  2 < 1. Else  * ≤  1 <  −1 ((1−  , in this case, we have −1 <  2 < 0. erefore, if the conditions of the theorem are satis�ed, then the period-1 solution ((, ( of system ( 2) is orbitally asymptotically stable and enjoying the property of asymptotic phase.

Discussion on the Period-𝑘𝑘 (𝑘𝑘 ≥ 2) Solution.
For the predator-prey model concerning IPM strategies, Tang and Cheke [23] proved that there is no periodic solution with order larger than or equal to three, except for one special case, by using the properties of the Lambert  function and Poincaré map.Moreover, ey showed that the existence of an order two periodic solution implies the existence of an order one periodic solution.Next, the existence of the period- ( ≥ ) solution is discussed.eorem 12. Suppose that  in >  −1 ((1−   and ( in −  −1 ((1 −    ≤  set < ( in −  −1 (.en, system (2) does not have period- ( ≥ ) solution.
Remark 13.From the proof of eorem 12, it can be seen that there is a possibility to exist a period-2 solution.Since it is difficult to obtain the exact expression of the solution to (2), then it will be challenging to give the sufficient conditions under which system (2) has a period-2 solution or not.

Applied Instance
Here, we will take the Tessier kinetics model [41], that is, (   max (1 − exp(−  , as the example of speci�c growth rate to verify the theoretical results.In addition,  max  0.5 [1h],    0.2 [gL], and     max  0.5 [gg] are used for the demonstration of system behavior.We assume in the following that  0  (0  0.2 [gL] and  0  (0  2 [g𝑥L].e vector graph of system (1) for   0.4 [1h] and  in 𝑆 20 [g𝑥L] are shown in Figure 4, from which it can be seen that the equilibrium (0.2, 9.84 is a stable node.
e time series and phase portrait for  set  10 [gL] is shown in Figure 5, it can be seen that no impulse occurs when  set > ( in +   ln(1 −  max   9.84 [g𝑥L].
Figure 6 gives the time series and phase portrait when  set 𝑆 6 [g𝑥L] and    0..It can be seen that the trajectory tends to be periodic.
Figure 7 shows the different positions of the period-1 solution for   0.4 [1h] and    0.1, 0., 0.5.e numerical simulations are consistent with the theoretical results obtained and presented in Section 3.

4.2.
Optimization of the Bioprocess.Next, we discuss aspects of the bioprocess optimization.where  OUT is the biomass productivity in the steady state.

Conclusions
In the paper the dynamic behavior of a universal biochemical reaction process with feedback control was analyzed, and it was shown that the stability of the bioprocess (i.e., the existence of the positive period-1 solution), depended on both the biomass yield and the microorganism growth rate.Furthermore, the conditions for the existence and stability of the system's period-1 solution were obtained (i.e., eorems 6 and 11).It also pointed out that the system (2) was not chaotic according to an analysis of period-2 solution existence.According to the theoretical results, the production of the microorganisms tended to be periodic.e microorganisms in the considered chemostat always kept the suitable growth rate.e biomass concentration could have been controlled to a given level between the critical and the optimal dissolved oxygen concentration.Moderation of the main bioprocess parameters in the selected biomass oscillation cycles for the Tessier kinetics model was presented.
Aer the analysis of the system stability, bioprocess optimization was covered in the work.In the �rst case, the optimization of a simple continuous bioprocess (i.e., bioprocess in which  > 0 and   = 0) was presented.In the second case of optimization, it was demonstrated that theoretically it was possible to obtain almost optimal biomass productivity without lost of synchronization of simultaneously performed bioprocesses (see (38) for small values of   set ). is was possible, when the substrate was dosed by both continuous (i.e.,  > 0) and pulse (i.e.,   > 0) method.In this case, the impulsive chemostat was obtained, where the biomass productivity mainly depended on , and the synchronization of simultaneously performed bioprocesses depended on   .e third case of optimization showed that during the optimization, the impulsive chemostat lost possibility of synchronization and strived for a simple continuous bioprocess (see (41)).Moreover, it was shown that improving the biomass productivity and synchronization of simultaneously performed bioprocesses were con�ict criteria.

− 1 F 1 :
e property of Lambert  function.

4. 1 .
Numerical Simulations.e simulations are carried out by changing one main parameter and �xing all other parameters.e following numerical simulations are given by the programs of Maple and Matlab sowares.e time interval is set two days, that is, 48 [h].

( 1 )
e Objective Function  OUT      in   set  =

F 5 :F 6 :F 7 :
e time series and phase portrait when    ,  in  2 gL and  set   gL.e time series and phase portrait when    h,    3,  in  2 gL, and  set  6 gL.Dependence of the biomass and substrate concentration on   .e phase portraits when    h,  in  2 gL,  set  6 gL, and    , 3, 5.
in region .and , , , , , , ,  are calculated at the point ((   (  . Period-1 Solution.e line    set intersects the isoclinal line    at the point , intersects the curve    at the point   ,  set ), and intersects the line    in at the point , where   satis�es that   +      set   in .(7) e line       ) set intersects the line    at the point  and intersects the line    in at the point .Denote     ,   ) set ) and     ,   ) set ), where        )  ) and        ) in .e impulsive set  imp ⊆   , )    )     in ,    set }, and the phase set  pha   imp ) ⊆   , )         ,       ) set }. Let the point  ′ be the primary image of , and denote the trajectories of the system starting from the points  and  that intersect the segment  at  ′ and ′ respectively.Denote the trajectory of the system between  and  ′ by   ′ , the trajectory of the system between  and  ′ by   ′ .eorem 6.If one of the conditions holds (i)      )   in and (11)ose that  in >   ) and  set   in    )).en the initial values of period-1 solution ), )) of system (2) satis�es     ) min  ,   }   + )      ) in   set ) and  + )      ) set , where   and   are determined by(7)and(11), respectively.