Modelling and simulation of a continuous process with feedback control and pulse feeding

Computers and Chemical Engineering 34 (2010) 976–984

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Modelling and simulation of a continuous process with feedback control and pulse feeding夽 Yuan Tian a,b,∗ , Lansun Chen a , Andrzej Kasperski c a b c

School of Mathematical Science, Dalian University of Technology, Dalian 116024, People’s Republic of China School of Information Engineering, Dalian University, Dalian 116622, People’s Republic of China Faculty of Mathematics, Computer Science and Econometrics, Bioinformatics Factory, University of Zielona Gora, Szafrana 4a, 65-516 Zielona Gora, Poland

a r t i c l e

i n f o

Article history: Received 8 May 2009 Received in revised form 31 August 2009 Accepted 8 September 2009 Available online 16 September 2009 Keywords: Continuous culture Feedback control State impulsive Variable biomass yield

a b s t r a c t This paper deals with feedback control of a microorganism continuous culture process with pulse dosage supply of substrate and removal of products. By the analysis of the dynamic properties and numerical simulation of the continuous process, the conditions are obtained for the existence and stability of positive period-1 solution of the system. It is also pointed out that there does not exist positive period-2 solution. The results simplify the choice of suitable operating conditions for continuous culture systems. It also gives the complete expression of the period of the positive period-1 solution, which provides the precise feeding period for a regularly continuous culture system to achieve the same stable output as a continuous culture system with feedback control in the same production environment. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Microorganisms play very important roles in nature and their activities have numerous industrial applications (see Rehm & Reed, 1981; Schugerl, 1987). In microbial as well as chemical processes, three different modes of operation, batch (Luedeking & Piret, 1959), fed-batch (Yammane & Shimizu, 1984) and continuous (Monod, 1950) have been applied. The batch process is most simple and easily carried out in industry production, while the continuous processes have been widely used to culture microbes for industrial and research purposes (see Hsu, Hubbell, & Waltman, 1977; Kubitschek, 1970). With respect to the microorganisms environment, the fed-batch process falls between batch and continuous operations (Yammane & Shimizu, 1984). In continuous culture systems, “turbidostat” and “chemostat” are two well-adopted laboratory apparatus (Buler, Hsu, & Waltman, 1985; Chen & Chen, 1993; DeLeenheer & Smith, 2003; De Leenheer, Levin, Sontag, & Klausmeier, 2006; Gouze & Robledo, 2005; Sun & Chen, 2007). In a turbidostat, fresh medium is delivered only when the microorganism concentration of the culture reaches some predetermined level, as measured by the biomass indicator. At this point, fresh medium

夽 This research is supported in part by National Natural Science Foundation of China (10771179). ∗ Corresponding author at: School of Mathematical Science, Dalian University of Technology, Linggong Road 2#, Dalian 116024, People’s Republic of China. E-mail address: [email protected] (Y. Tian). 0098-1354/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2009.09.002

is added to the culture and an equal volume of culture is removed. In a chemostat, the medium is delivered at a constant rate by a peristaltic pump or solenoid gate system, which ultimately determines the growth rate and microorganism concentration. The rate of media flow can be adjusted, and is often set at approximately 20% of culture volume per day (Kubitschek, 1970). A truly continuous culture will have the medium delivered at a constant volume per unit time. However, according to the researchers and chemical engineers’ experience pumps or solenoid gates used in industrial scale systems are inherently unreliable. For that reason it is difficult to adjust such production devices to deliver equal amounts of the medium to several cultures simultaneously. This way of the medium delivery is needed, if competition bioprocesses carried out in industrial scale are to be truly replicated. In order to solve the difficulty of delivering exactly the same amounts of medium to several cultures growing at once, a “practically” continuous approach can be taken. “Practically” means that, even if the flow is of pulse type, but if the frequency of impulses is high enough (as compared with the inertia of metabolic processes), the flow is still considered to be continuous, although in reality it is impulsive (Bailey & Ollis, 1986). To our best knowledge, there is no result on the microbial process with pulse dosage of substrate and pulse washout of part of bioreactor medium. On the other hand, systems with sudden perturbations are involving in impulsive differential equations. Authors, in (Liu & Chen, 2003; Sun & Chen, 2008; Tang & Chen, 2002; Tang & Chen, 2004a), introduced some impulsive differential equations in population dynamics and obtained some interest results. The research

Y. Tian et al. / Computers and Chemical Engineering 34 (2010) 976–984

on the chemostat model with impulsive perturbations was studied by Sun and Chen (2007). Tang and Chen (2004b) introduced a Lotka–Voterra model with state-dependent impulsion and analyzed the existence and stability of positive period-1 solution. Jiang, Lu, and Qian (2007a) and Smith (2001) have studied the statedependent models with impulsive state control, where the model has a first integral, and obtained the complete expression of the period of the periodic solution. Jiang, Lu, and Qian (2007b) and Zeng, Chen, and Sun (2006) have also discussed the models concerning integrated pest management (IPM), which have no explicit solution, by applying the Poincare principle and Poincare–Bendixson of the impulsive differential equation, respectively. However, the majority of the known results are just related to the systems with time-dependent impulsion. Few papers have discussed the microbial process using impulsive differential equation. According to carried out processes it is possible to receive high biomass yield by dosing the substrate in portions (Kasperski & Miskiewicz, 2008). Moreover, it is easy to prevent the process from the decrease of dissolved oxygen concentration (DOC) in the bioreactor medium below a low level by the monitoring of DOC oscillations. It is necessary because the low level of DOC decreases biomass yield and specific growth rate (Kasperski, 2008). On the other hand, with the growth of the microorganism and its concentration increasing in the bioreactor, the effect of inhibition between the production and other negative effect will occur when the biomass concentration reaches a critical value. For the purpose of continuously culturing the microorganism and decreasing the inhibition effect and the Crabtree effect, it is necessary to keep the biomass concentration lower than a certain level and dose the substrate into the bioreactor in portions. Thus, in this paper, we consider a microorganism continuous culture system, the sketch map of the apparatus can be seen in Fig. 1. The apparatus includes an optical sensing device which continuously monitors the biomass concentration in the bioreactor and two switches controlled by a computer. When the biomass concentration is lower a critical level, the switches are closed. In this case the biomass is increasing by consuming the substrate in the bioreactor. Once the biomass concentration reaches the critical level, two switches are opened, part of the medium containing biomass and substrate is discharged from the bioreactor, and the next portion of medium of a given substrate concentration is input. The rest of this paper is organized as follows. In Section 2 we introduce a continuous culture model with variable biomass yield and impulsive state feedback control. In Section 3, we obtain the conditions for the existence of positive period-1 solution by using the analytical method. We also give the complete expression of the period of period-1 solution. In addition, we show that no positive period-2 solution exists. Next in Section 4, we analyze the stability of the positive period-1 solutions by analogue of the Poincar criterion. In Section 5, we give the numerical simulations to verify the theoretical results, such as the existence of period-1 solutions, obtained in this paper and discuss the biological essence. Finally in Section 6 we present the conclusions.

model works for a single species growing in a continuously stirred homogeneous bioreactor

⎧ dX  S max ⎪ = X ⎪ ⎪ KS + S ⎨ dt dS

If the microorganisms’ growth proceeds in accordance with Monod’s kinetics model (see Alvarez-Ramirez, Alvarez, & Velasco, 2009; Lobry, Flandrois, Carret, & Pavemonod’s, 1992; Rehm & Reed, 1981; Schugerl & Bellgardt, 2000), i.e. according to dependence: (S) =

max S KS + S

and the linearly dependent YX/S = a + bS (a > 0, b ≥ 0) (see Crooke & Tanner, 1982; Crooke, Wel, & Tanner, 1980; Huang, 1990; Pilyugin & Waltman, 2003) is assumed, then the following mathematical

1



S

(1)

max =− X ⎪ ⎪ dt a + bS KS + S ⎪ ⎩ + +

X(0 ) = X0 , S(0 ) = S0

where X = X(t) denotes the biomass concentration and S = S(t) the substrate concentration in the bioreactor medium at time t, X0 and S0 denote the initial biomass concentration and substrate concentration in the bioreactor medium respectively, max is the maximum specific growth rate, KS , the saturation constant, a, b, the coefficients of the variable biomass yield. According to the design ideas of the bioreactor, the biomass concentration should be controlled to a certain level. When the biomass concentration X(t) in the bioreactor reaches the critical level XT , then part of the medium containing biomass and substrate is discharged from the bioreactor, and the next portion of medium of a given substrate concentration is input impulsively. Therefore, (1) can be modified as follows by introducing the impulsive state feedback control:

⎧ dX  SX max ⎪ = ⎪ ⎪ KS + S dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ dS = − 1 . max SX a + bS

dt

KS + S

⎪ X = −Wf X ⎪ ⎪ ⎪ ⎪ ⎪ S = Wf (SF − S) ⎪ ⎪ ⎩ + +

⎫ ⎪ ⎬ ⎪ ⎭

X < XT (2) X = XT

X(0 ) = X0 , S(0 ) = S0

where, SF is the concentration of the feed substrate which is input impulsively, and 0 < Wf < 1 is the part of biomass which is removed from the bioreactor in each biomass oscillation cycle. In the following, we mainly discuss the existence and stability of periodic solution of system (2). Before introducing the main results, we give the Analogue of Poincar criterion first. Theorem 1. (Bainov & Simeonov, 1993) (Analogue of Poincare’ Criterion) The T-periodic solution S = (t), X = (t) of system

dX

dS = P(S, X), = R(S, X), if (S, X) = / 0 dt dt X = ˛(S, X), S = ˇ(S, X), if (S, X) = 0

(3)

is orbitally asymptotically stable and enjoys the property of asymptotic phase if the multiplier 2 satisfies the condition |2 | < 1 and is unstable if |2 | > 1. Where 2 =

⎛

q 

k exp ⎝

k=1

k = 2. Model formulation and preliminaries

977

T  0

P+ ((∂ˇ/∂S)(∂/∂X) − (∂ˇ/∂X)(∂/∂S) + (∂/∂X)) P(∂/∂X) + R(∂/∂S)



+

⎞



∂P ∂R ((t), (t)) + ((t), (t)) dt ⎠ , ∂X ∂S



R+ (∂˛/∂X)(∂/∂S) − (∂˛∂S)(∂/∂X) + (∂/∂S) P(∂/∂X) + R(∂/∂S)

,

P+ = P((k+ ), (k+ )), R+ = R((k+ ), (k+ )) and P, R, ∂˛/∂X, ∂˛/ ∂S, ∂ˇ/∂X, ∂ˇ/∂S, ∂/∂X, ∂/∂S are calculated at the point ((k ), (k )) and q is the order of the periodic solution. 3. Existence of positive periodic solution of system (2) In this section, we mainly discuss the existence of periodic solution of system (2) by the analytic method. Before discussing the

978

Y. Tian et al. / Computers and Chemical Engineering 34 (2010) 976–984

Fig. 1. Schematic diagram of the analyzed process.

periodic solution of system (2), we should consider the qualitative characteristics of system (1). By (1) we have

T ), 1+ = ((t0 + T )+ ) and + = ((t0 + T )+ ). Then by the T 1 periodicity, we have

dX = −a − bS. dS

1+ = ((t0 + T )+ ) = (t0+ ) = 0 , Then

Therefore, X(X0 , S0 , S) = −aS −

+ = ((t0 + T )+ ) = (t0+ ) = 0 . 1

+

bS 2 2

+ X0 + aS0 +

bS02 2

0 − 1 = ((t0 + T ) ) − (t0 + T ) = −Wf (t0 + T ) = −Wf 1 = −Wf XT , +

.

(4)

The vector graph of system (1) can be seen in Fig. 2, from which we know that in the process the substrate concentration S is decreasing and the biomass concentration is increasing. If we do not adopt efficient control strategy, the microorganisms will finally consume the substrate and cause the whole process terminated. In order to not interrupt the culturing process and gain a stable output of the microorganism X, we need to discharge part of the bioreactor medium containing biomass and substrate, and add the medium of a given substrate concentration to the bioreactor when the biomass concentration reaches the critical level XT . Let S = (t), X = (t) be a T -period-1 solution of system (2). Denote 0 = (t0+ ), 0 = (t0+ ), 1 = (t0 + T ) = XT , 1 = (t0 +

0 − 1 = ((t0 + T ) ) − (t0 + T ) = Wf (SF − (t0 + T )) = Wf (SF − 1 ).

Therefore, 0 = (1 − Wf )XT ,

0 = Wf SF + (1 − Wf )1 .

(5)

¯ Similarly, let S = (t), ¯ X = (t) be a periodic-2 solution of sys¯ + ), ¯ 0 = (t ¯ 0+ ), t0 < t1 < t0 + T, ¯ 1 = tem (2). Denote ¯ 0 = (t 0 + + + ¯(t1 ) = XT , ¯ 1 = (t ¯ ¯ ¯ 0 + T) = ¯ 1 ),  = (t ), ¯ = (t ¯ + ), ¯ 2 = (t 1

1

1

1

¯ 0 + T )+ ) and ¯ + = ((t XT , ¯ 2 = (t ¯ 0 + T ), ¯ 2+ = ((t ¯ 0 + T )+ ). Then by 2 the T -periodicity, we have ¯ 0 + T )+ ) = (t ¯ + ) = ¯ 0 , ¯ 2+ = ((t 0

¯ + = ((t ¯ 0 + T )+ ) = (t ¯ 0+ ) = ¯ 0 . 2

Therefore, ¯ 0 = (1 − Wf )XT , ¯ 1+ = (1 − Wf )XT ,

¯ 0 = Wf SF + (1 − Wf )¯ 2 , ¯ + = Wf SF + (1 − Wf )¯ 1 . 1

(6)

3.1. Case 1. b = 0 In this case, system (2) is simplified into the following form:

⎧ dX  SX max ⎪ = ⎪ ⎪ KS + S dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ dS = − 1 . max SX dt

a

KS + S

⎫ ⎪ ⎬ ⎪ ⎭

⎪ S = Wf (SF − S) ⎪ ⎪ ⎪ ⎪ ⎪ X = −Wf X ⎪ ⎪ ⎩ + + X(0 ) = X0 ,

X < XT (7) X = XT

S(0 ) = S0

3.1.1. Period-1 solution

Fig. 2. Illustration of vector graph of system (1) when a = 0.5, b = 0.025, max = 0.3 and KS = 2.

Theorem 2. If the critical level XT and the feeding substrate concentration SF satisfy the condition XT < aSF , then system (7) has a unique periodic-1 solution with aS0 + X0 ≥ XT .

Y. Tian et al. / Computers and Chemical Engineering 34 (2010) 976–984

3.1.2. Expression of the period T of the period-1 solution In this subsection we will give a complete expression of period T of periodic solution ((t), (t)). It follows by the first equation of system (7) we have KS + S dt = dX, max SX

(8)

to the point P1k (k1 , XT ), with t = t|P k , in the counterclockwise

 Tk = t|P k − t|P k 1

T = tP1 − tP0

KS + SF − X/a dX max (SF − X/a)X

=

f )XT (1−W XT 

=

max SF



(1−Wf )XT

=

KS max SF



KS



ln

aSF − (1 − Wf )XT

 +

 

aSF − XT

−

1 max KS

max SF

1 X

dX 1 max

(a + bS)(KS + S) max SX(S)dS

0

k (XT )

(14)

(a + bS)(KS + S) . max SX(S)dS

Theorem 6. The period T of periodic solution ((t), (t)) of system (2) satisfies equation



Tk =

k (XT )

(a + bS)(KS + S) , max SX(S)dS

(15)

where ϕk (XT ) is determined by Eq. (33) and k (XT ) = Wf SF + (1 − Wf )ϕk (XT ), k = 1, 2.



+

k

The above result can be summarized in Theorem 6.

ϕk (XT )

1 1 + X aSF − X

1

ϕk (XT )

(9)

XT

0

k

=− =

Then travelling along S(X) from the point P0 (0 , (1 − Wf )XT ), with t = t|P0 , to the point P1 (1 , XT ), with t = t|P1 , in the counterclockwise direction yields



1

direction yields

and S(X) can be determined by the following equation X(t) . S(t) = SF − a

979



3.2.3. Period-2 solution ln(1 − Wf ).

Theorem 7.

There does not exist period-2 solution in system (2).

(10) 4. Asymptotic behavior of the period-1 solution The above result can be summarized in Theorem 3. Theorem 3. The period T of periodic solution ((t), (t)) of system (7) satisfies equation T=

KS max SF



ln

aSF − (1 − Wf )XT aSF − XT



−

SF + KS ln(1 − Wf ). max SF

(11)

3.1.3. Period-2 solution Theorem 4.

There does not exist period-2 solution in system (7).

3.2. Case 2. b > 0 3.2.1. Period-1 solution First, we give three variables which are commonly used in the following. Let D = a/b(1 − Wf ), u = aSF + bWf (SF )2 /2 and

According to the definitions of orbitally asymptotically stable and enjoys the property of asymptotic phase (Bainov & Simeonov, 1993), the following Theorems hold true. Theorem 8. If XT < aSF , then the T-periodic solution ((t), (t)) of system (7) is orbitally asymptotically stable and enjoys the property of asymptotic phase. Theorem 9. (1) If XT < u, then the T-periodic solution ((t), (t)) of system (2) is orbitally asymptotically stable and enjoys the property of asymptotic phase; (2) If SF > D and u ≤ XT < U, then one T-periodic solution ((t), (t)) of system (2) is orbitally asymptotically stable and enjoys the property of asymptotic phase, while the other is not stable; (3) If SF > D and XT = U, then the stability of the T-periodic solution ((t), (t)) of system (2) can not be determined by the theorem of Bainov and Simeonov.

2

U = u + (a − b(1 − Wf )SF ) /2b(2 − Wf ). Denote I(X, S) = X + aS + bS 2 /2. We have the following result. Theorem 5. (i) If one of the following two conditions holds: (1) XT < u; (2) SF > D and XT = U. Then system (2) has a unique period-1 solution with I(X0 , S0 ) ≥ XT ; (ii) If SF > D and u ≤ XT < U. Then system (2) has two period-1 solutions with I(X0 , S0 ) ≥ XT . 3.2.2. Expression of period T of the period-1 solution In this subsection we will give a complete expression of period T of periodic solution ((t), (t)). It follows from the second equation of (2) that dt = −

(a + bS)(KS + S) dS, max SX

(12)

and X(S) can be determined by the following equation X(S) = −aS −

2 b 2 b S + ak0 + (k0 ) + (1 − Wf )XT , 2 2

(13)

where k0 = Wf SF + (1 − D)k1 = Wf SF + (1 − Wf )ϕk (XT ) = k (XT ), k = 1, 2 and k1 is determined by Eq. (33). Then travelling along X(S) from the point P0k (k0 , (1 − Wf )XT ), with t = t|P k , 0

5. Discussions and numerical simulations We have analyzed theoretically the feedback control of microorganism continuous culture process for pule dosage supply of substrates and removal of products. The results are new and significant, which not only provide the possibility of a check of system dynamic property including the existence and stability of period1 solution for different microorganisms and several parameters, but also the possibility of a calculation of the period of the period1 solution. Moreover, the results provide a possibility of making simulation of real process according to the mathematical models determined in the article. In order to verify the received results, we will give the numerical simulations of systems (2). We will analyze the existence and stability of period-1 solution by changing one main parameter (i.e., XT , SF , X0 ) and fixing all other parameters. We assume that a = 0.5, Wf = 0.1, max = 0.3[1/h] and KS = 2[g/l]. We only verify the case of b > 0. Here we set b = 0.025. Then C = a/b(1 − Wf )  22.2[g/l]. We firstly check and show influence of XT changes on the existence and stability of period-1 solution. We set S0 = 2[g/l], SF = 6[g/l] and X0 = 1.6[g/l]. Then we have u = aSF + bWf (SF )2 /2  3.05[g/l], U = u + 2

(a − b(1 − Wf )SF ) /2b(2 − Wf )  4.45[g/l] and SF < C. The trajectory of the solution S(t), X(t) and the phase diagram (S(t), X(t))

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Y. Tian et al. / Computers and Chemical Engineering 34 (2010) 976–984

Fig. 3. The time series and portrait phase of system (2) for S0 = 2[g/l], SF = 6[g/l], X0 = 1.6[g/l] and XT = 2.6[g/l].

Fig. 4. The time series and portrait phase of system (2) for S0 = 2[g/l], SF = 6[g/l], X0 = 1.6[g/l] and XT = 3.6[g/l].

for XT = 2.6[g/l] and 3.6[g/l] are given in Figs. 3 and 4. According to Theorem 5, in Fig. 3 we can see that the solution tends to a stable periodic solution, i.e., we can achieve a stable output of the microorganisms. Fig. 4 indicates that the critical level is given so big that the biomass concentration can not reach it before the substrate is used up in the bioreactor. These simulations are consistent with the theoretical result of Theorem 5(i). Furthermore, the necessary condition for system (2) existing a period solution is XT < min{u, I(X0 , S0 )} = min{3.05, 2.65} = 2.65[g/l] by 5(i). Fig. 5 displays the period-1 solution of system (2) for SF = 6[g/l], XT = 2.6[g/l], S0 = 1.62[g/l] and X0 = 2.34[g/l]. The period T  0.864[h] calculated by Eq. (15). Secondly we check and show influence of SF changes on the existence and stability of period-1 solution. We set X0 = 1.6[g/l], XT = 2.6[g/l] and S0 = 2[g/l]. The trajectory of the solution S(t), X(t) and the phase diagram (S(t), X(t)) for SF = 6[g/l] and 4[g/l] are given in

Figs. 3 and 6, respectively. Fig. 3 indicates that the solution tends to a stable period solution. In Fig. 6, since the concentration of the feeding substrate is given so small that the substrate is used up before the biomass concentration reaching the critical level XT in some biomass oscillation cycle. So if we expect a stable output of microorganism, the feeding substrate concentration should be properly given. Thirdly we check and show influence of X0 changes on the existence and stability of period-1 solution. We set S0 = 2[g/l], SF = 6[g/l] and XT = 2.6[g/l]. The trajectory of the solution S(t), X(t) and the phase diagram (S(t), X(t)) for X0 = 1.2[g/l], 1.6[g/l] and 2[g/l] are given in Figs. 7, 3 and 8, respectively. In Fig. 7, the initial biomass concentration is given so small that the substrate is used up before the biomass concentration reaching the critical level, i.e., the impulsive effect does not happen. From Figs. 3 and 8 we can see that the solution tends to a stable periodic solution.

Fig. 5. The time series and portrait phase of system (2) for S0 = 1.62[g/l], SF = 6[g/l], X0 = 2.34[g/l] and XT = 2.6[g/l].

Fig. 6. The time series and portrait phase of system (2) for S0 = 2[g/l], SF = 4[g/l], X0 = 1.6[g/l] and XT = 2.6[g/l].

Y. Tian et al. / Computers and Chemical Engineering 34 (2010) 976–984

981

Fig. 7. The time series and portrait phase of system (2) for S0 = 2[g/l], SF = 6[g/l], X0 = 1.2[g/l] and XT = 2.6[g/l].

Fig. 8. The time series and portrait phase of system (2) for S0 = 2[g/l], SF = 6[g/l], X0 = 2[g/l] and XT = 2.6[g/l].

A potential application area of the culture bioreactor with the feedback control is the commercial and industrial production of the microorganism. The microorganism in the bioreactor always keeps the highest growth rate and the biomass concentration can be controlled to a given level. In this way, we can determine the rationality of the microorganism feedback concentration according to the conditions for the stability of the periodic solution, in other words, if we have the proper microorganism feedback concentration, we can achieve a stable output for a continuous culture system with feedback control in the circumstances of determined production environment. Furthermore, we obtain the feeding period for a regularly continuous culture system, which can help with carrying out the processes and can be used for example to check whether all measuring instruments (for example the photoelectricity system or the annunciator) are working well.

6. Conclusions In this study, the feedback control of microorganism continuous culture process for pulse dosage supply of substrates and removal of products was introduced and the dynamic property of this system was investigated. By analytical method, it was showed that if the parameters satisfy XT < aSF , then system (7) has a unique periodic solution ((t), (t)), which is orbitally asymptotically stable and enjoys the property of asymptotic phase. Furthermore, it was shown that (1) if XT < u, then system (2) has a unique periodic solution ((t), (t)), which is orbitally asymptotically stable and enjoys the property of asymptotic phase; (2) if SF > C and u ≤ XT < U, then system (2) has two periodic solutions, one of which is orbitally asymptotically stable and enjoys the property of asymptotic phase, while the other is not stable; (3) if SF > C and XT = U, then system (2) has a unique periodic solution ((t), (t)), the stability of which can not be determined by the theorem of Bainov and Simeonov. Meanwhile, numerical simulations to verify the theoretical results were presented. In addition, the complete expression of period of the periodic solution was given, by which the continuous culture model with the impulsive state feed-back control can be changed into a model with periodic control.

Appendix A. Definition 1. (Bainov & Simeonov, 1993) ((t), (t)) is said to be period-1 solution if in a minimum cycle time, there is one impulse effect. Similarly, ((t), (t)) is said to be period-2 solution if in a minimum cycle time, there are two impulse effects. Let be the orbit of the periodic solution ((t), (t)). Definition 2. (Bainov & Simeonov, 1993) is said to be orbitally stable, if for any ε > 0, there exists ı > 0, with the proviso that every solution (S(t), X(t)) of system (2) whose distance from is less than ı at t = t0 , will remain within a distance less than ε from

for all t ≥ t0 . Such an is said to be orbitally asymptotically stable if, in addition, the distance of (S(t), X(t)) from tends to zero as t → ∞. Moreover, if there exist positive constants ˛, ˇ and a real constant h such that (S(t), (X(t)), ) < ˛e−ˇt for t > t0 , then is said to be orbitally asymptotically stable and enjoys the property of asymptotic phase. The proof of Theorem 2 Proof.

By the first two equations of system (7), we have

dX = −adS.

(16)

Integrate two sides of Eq. (16) from t0 to t, we have X(t) = X0 + aS0 − aS(t).

(17)

It is easily to see from (17) that if the impulsive effect happens, the condition X0 + aS0 ≥ XT is necessary. For t ∈ (t0 , t0 + T ], the solution S = (t), X = (t) of system (7) satisfies the relation (5), i.e., (t) − 0 = a0 − a(t).

(18)

In particular, for t = t0 + T , we have (T ) − 0 = a0 − a(T ) or XT − 0 = a0 − a1 .

(19)

In view of Eq. (5), we have Wf XT = a(0 − 1 ) = aWf (SF − 1 ),

(20)

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Y. Tian et al. / Computers and Chemical Engineering 34 (2010) 976–984

which implies that 1 = SF − XT /a or 0 = Wf SF + (1 − Wf )(SF − XT /a). If XT < aSF , then we have 1 > 0. Therefore, if the critical level XT and the concentration of the feed substrate satisfy the condition XT < aSF , then system (7) has a unique periodic-1 solution with aS0 + X0 ≥ XT .  The proof of Theorem 4 Proof.

(21)

¯ − a(t). ¯ (t) − ¯ 1+ = a¯ + 1

¯ 1 − ¯ 0 = a¯ 0 − a¯ 1 .

A = bWf (2 − Wf )/2 > 0, B = aWf − bWf (1 − Wf )SF , C =

− a¯ 2 .

Let

D = a/b(1 − Wf ), u = aSF + bWf (SF )2 /2

and

U =u+

(a − b(1 − Wf )SF ) /2b(2 − Wf ). Denote I(S, X) = X + aS + bS 2 /2. If B2 − 4AC ≥ 0, i.e., XT < U, then solving Eq. (32), we have

(23)

B2 − 4AC = ϕ1 (XT ), 2A  2 −B + B − 4AC 21 = = ϕ2 (XT ). 2A

Substitute t = t0 + T into Eq. (22), we have =

where

(32)

(22)

Substitute t = t1 into Eq. (21), we have

a¯ + 1

It can be rewritten as a unitary quadratic equation about 1 as follows

Wf XT − aWf SF − bWf2 (SF )2 /2.

And for t ∈ (t1 , t0 + T ] we have

¯ 2 − ¯ 1+

1 b(20 − 21 ) 2 = aWf (SF − 1 ) + 12 b[(Wf SF + (1 − Wf )1 )2 − 21 ]. = a(0 − 1 ) +

Wf XT

A21 + B1 + C = 0,

For t ∈ (t0 , t1 ], by Eq. (16) we have

¯ ¯ (t) − ¯ 0 = a¯ 0 − a(t).

In view of Eq. (5), we have

(24)

By subtracting Eq. (23) from Eq. (24) we have

2

11

=

−B −



(33)

Case 1 B ≥ 0, i.e., SF ≤ D. In this case, 11 ≤ 0. If C < 0, i.e.,

− a¯ 0 + a¯ 1 − a¯ 2 . (¯ 2 − ¯ 1 ) + (¯ 0 − ¯ 1+ ) = a¯ + 1 By Eq. (6) we have

XT < u.

− a¯ 0 + a¯ 1 − a¯ 2 = 0 a¯ + 1

Then we have 21 > 0; Case 2 B < 0, i.e.,

(34)

or − ¯ 0 = ¯ 2 − ¯ 1 . ¯ + 1

(25)

− ¯ 0 )(¯ 2 − ¯ 1 ) ≥ 0. (¯ + 1

(26)

¯ − ¯ 0 = (1 − Wf )(¯ 1 − ¯ 2 ). Since X = (t), S= By Eq. (6) we have ¯ + 1 + = /  ¯ . Thus (  ¯ −  ¯ )(  ¯ −  ¯ )= (t) ¯ is period-2 solution, then ¯ + 0 0 2 1 1 1

−(1 − Wf )(¯ 2 − ¯ 1 )2 < 0, which contradicts to Eq. (26). Therefore, period-2 solution of system (7) does not exist.  The proof of Theorem 5

dX = −a − bS. dS

(27)

Integrate two sides of Eq. (27) from t0 to t, we have bS02 2

− aS(t) −

bS 2 (t) . 2

(28)

It is easily to see from (28) that if the impulsive effect happens, the condition X0 + aS0 +

bS02 2

≥ XT

(29)

is necessary. For t ∈ (t0 , t0 + T ], the solution S = (t), X = (t) of system (2) satisfies the relation (28), i.e., (t) − 0 = a0 +

b20 2

− a(t) −

b2 (t) . 2

(30)

b20 2

− a(T ) −

If 0 ≤ C < B2 /4A, i.e., u ≤ XT < U.

(36)

Then we have 11 ≥ 0 and 21 > 0; If (37)

Then we have 21 = 21 = −B/2A > 0. Thus, if conditions (29) and (34) hold, then system (2) has a unique periodic solution with one impulse effect per period; if conditions (29), (35) and (37) hold, then system (2) has a unique periodic solution with one impulse effect per period; if conditions (29), (35) and (36) hold, then system (2) has two periodic solutions with one impulse effect per period.  The proof of Theorem 7 Proof.

For t ∈ (t0 , t1 ], by Eq. (27) we have

b b ¯ (t) − ¯ 0 = a¯ 0 + ¯ 20 − a(t) ¯ − ¯ 2 (t). 2 2

(38)

And for t ∈ (t1 , t0 + T ] we have b b 2 ¯ (t) − ¯ 1+ = a¯ + + (¯ + ) − a(t) ¯ − ¯ 2 (t). 1 2 1 2

b b ¯ 1 − ¯ 0 = a¯ 0 + ¯ 20 − a¯ 1 − ¯ 21 . 2 2

b2 (T ) 2

(39)

(40)

Substitute t = t0 + T into Eq. (39), we have

or XT − 0 = a0 +

21 > 0.

Substitute t = t1 into Eq. (38), we have

In particular, for t = t0 + T , we have (T ) − 0 = a0 +

11 < 0,

XT = U.

By the first two equations of system (2) we have

X(t) = X0 + aS0 +

(35)

If C < 0, i.e., XT < u, then we have

Eq. (25) also implies that

Proof.

SF > D.

b20 2

− a1 −

b21 2

.

(31)

b b 2 ¯ 2 − ¯ 1+ = a¯ + + (¯ + ) − a¯ 2 − ¯ 22 . 1 2 1 2

(41)

Y. Tian et al. / Computers and Chemical Engineering 34 (2010) 976–984

By subtracting Eq. (40) from Eq. (41) we have + (¯ 2 − ¯ 1 ) + (¯ 0 − ¯ 1+ ) = a¯ + 1



b + 2 b (¯ ) − a¯ 0 + ¯ 20 2 1 2





orbitally asymptotically stable and enjoys the property of asymptotic phase.  + a¯ 1

The proof of Theorem 9



b b + ¯ 21 − a¯ 2 + ¯ 22 . 2 2 By Eq. (6) we have + a¯ + 1



b + 2 b (¯ ) − a¯ 0 + ¯ 20 2 1 2

or



(¯ + − ¯ 0 ) a + 1



+ a¯ 1 +





b 2 b ¯ − a¯ 2 + ¯ 22 2 1 2





=0



b + b (¯ + ¯ 0 ) = (¯ 2 − ¯ 1 ) a + (¯ 2 + ¯ 1 ) . 2 1 2

(42)

Eq. (42) also implies that (¯ + 1

− ¯ 0 )(¯ 2 − ¯ 1 ) ≥ 0.

(43)

Proof. According to Theorem 1 we calculate the multiplies 2 of the system (2) in variations corresponding to the T periodic solution ((t), (t)). Denote P0 (0 , 0 ), P1 (1 , 1 ), In where 1 = XT , 0 = (1 − Wf )XT , 0 = Wf SF + (1 − Wf )1 . system (2), since P(S, X) = max SX/(KS + S), R(S, X) = −1/(a + and bS).max SX/(KS + S), ˛(S, X) = −Wf X, ˇ(S, X) = Wf (SF − S) (S, X) = X − XT , then

¯ − ¯ 0 = (1 − Wf )(¯ 1 − ¯ 2 ). Since X = (t), S= By Eq. (6) we have ¯ + 1 + (t) ¯ is period-2 solution, then ¯ + = /  ¯ . Thus (  ¯ −  ¯ )(  ¯ ¯ 1) = 0 0 2− 1 1

∂˛ ∂ˇ ∂ˇ ∂ ∂˛ ∂ = −Wf , = 0, = 0, = −Wf , = 1, = 0. ∂X ∂S ∂X ∂S ∂X ∂S

=

1

+

The proof of Theorem 8

=

Proof. According to Theorem 1 we calculate the multiplies 2 of the system (7) corresponding to the T -periodic solution ((t), (t)). Denote P0 (0 , 0 ), P1 (1 , 1 ), where 1 = XT , 0 = (1 − Wf )XT , 0 = Wf SF + (1 − Wf )1 . In system (7), since P(S, X) = max SX/(KS + S), R(S, X) = −1/a.max SX/(KS + S), ˛(S, X) = −Wf X, ˇ(S, X) = Wf (SF − S) and (S, X) = X − XT , then

=

+ =

P(∂/∂X) + R(∂/∂S)

2

= 1 exp

= 1 exp

0 T 0 1 0



 

∂P ∂R ((t), (t)) + ((t), (t)) dt ∂X ∂S

max S dt+ KS +S 1 dX + X





0 1

0

T

b

max SX dt− 2 (a+bS) KS +S

−b dS + a + bS



1

0



Since 0 = Wf SF + (1 − Wf )1 , then we have 2 = (1 − Wf )

a + b(Wf SF + (1 − Wf )1 ) a + b1



1 =

0 (KS + 1 ) ; 1 (KS + 0 ) T

0 T



.

 



T

1 max XKS = 1 exp dt a (KS + S)2 0 1   1 0 1 KS = 1 exp dX + dS X S(K S + S) 0 0 1 1 KS + 0 = 1 · · · 0 0 KS + 1 = (1 − Wf ). max S dt − KS + S

b(1 − Wf )SF − a B2 − 4AC − B −B = , > 2A 2A b(2 − Wf )

i.e.,

∂P ∂R ((t), (t)) + ((t), (t)) dt ∂X ∂S



Since 0 < Wf < 1, then we have 0 < 2 < 1. Then by Theorem 1 we conclude that the T -periodic solution ((t), (t)) of system (7) is

bWf (2 − Wf )1 > Wf [(1 − Wf )bSF − a], which is equivalent to b(1 − (1 − Wf )2 )1 > (1 − Wf )(a + bWf SF ) − a, or 0 < (1 − Wf )

a + b(Wf SF + (1 − Wf )1 ) a + b1

< 1.

Thus we have |2 | < 1. For SF > D, if XT < u, then we have



1 =

0

T

 1 max XKS dt a+bS (KS +S)2

KS dS S(KS + S)

For SF ≤ D, if XT < u, then by Eq. (33), we have

max 0 0 /KS + 0 max 1 1 /KS + 1



0 (KS + 1 ) ; 1 (KS + 0 )

1 a + b0 1 KS + 0 · · · 0 a + b1 0 KS + 1 a + b0 = (1 − Wf ) . a + b1

P+ (1 − Wf )

= (1 − Wf )2

max 0 0 /KS + 0 max 1 1 /KS + 1

= 1 ·

P(∂/∂X) + R(∂/∂S)

= (1 − Wf )

P

= 1 exp

R+ (∂˛/∂X)(∂/∂S) − (∂˛/∂S)(∂/∂X) + (∂/∂S)

P

P+ (1 − Wf )

= 1 exp





P(∂/∂X) + R(∂/∂S)

 T 

P+ (∂ˇ/∂S)(∂/∂X) − (∂ˇ/∂X)(∂/∂S) + (∂/∂X)



R+ (∂˛/∂X)(∂/∂S) − (∂˛/∂S)(∂/∂X) + (∂/∂S)

= (1 − Wf )2

2



P(∂/∂X) + R(∂/∂S)

= (1 − Wf )

∂ˇ ∂ˇ ∂ ∂˛ ∂ ∂˛ = −Wf , = 0, = 0, = −Wf , = 1, = 0. ∂X ∂S ∂X ∂S ∂X ∂S

1





−(1 − Wf )(¯ 2 − ¯ 1 ) < 0, which contradicts to Eq. (43). Therefore, period-2 solution of system (2) does not exist. 

Therefore,



P+ (∂ˇ/∂S)(∂/∂X) − (∂ˇ/∂X)(∂/∂S) + (∂/∂X)

2

∂R 1 max XKS =− . , a (KS + S)2 ∂S

max XKS max SX ∂R 1 b . . , − = 2 K +S a + bS ∂S S (KS + S)2 (a + bS)

max S ∂P , = KS + S ∂X

Therefore,

max S ∂P , = KS + S ∂X

983

b(1 − Wf )SF − a B2 − 4AC − B −B , > = 2A 2A b(2 − Wf )



984

Y. Tian et al. / Computers and Chemical Engineering 34 (2010) 976–984

which implies that |2 | < 1; if u ≤ XT < U, then we have



1 =

b(1 − Wf )SF − a B2 − 4AC − B −B , > = 2A 2A b(2 − Wf )

which implies that |2 | < 1; while



1 =

−

b(1 − Wf )SF − a B2 − 4AC − B −B , < = 2A 2A b(2 − Wf )

which implies that 2 > 1; If XT = U, then we have 1 =

b(1 − Wf )SF − a −B = , 2A b(2 − Wf )

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