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The emergence of feedforward periodicity for the fed-batch penicillin fermentation process
17th IFAC Workshop on Control Applications of Optimization 17th IFAC Workshop onOctober Control 15-19, Applications Yekaterinburg, Russia, 2018 of Optimization 17th IFAC IFAC Workshop Workshop onOctober Control 15-19, Applications ofonline Optimization Available at www.sciencedirect.com Yekaterinburg, Russia, 2018 of 17th on Control Applications Optimization 17th IFAC Workshop onOctober Control 15-19, Applications Yekaterinburg, Russia, 2018 Yekaterinburg, Russia, October 15-19, 2018 of Optimization Yekaterinburg, Russia, October 15-19, 2018
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IFAC PapersOnLine 51-32 (2018) 130–135
The emergence of feedforward periodicity for the fed-batch penicillin fermentation The emergence of feedforward periodicity for the fed-batch penicillin fermentation The emergence of feedforward periodicity for the fed-batch penicillin fermentation process The emergence of feedforward periodicity for the fed-batch penicillin fermentation process process process Chi Zhai*, Tong Qiu**, Ahmet Palazoglu***, Wei Sun****
Chi Zhai*, Tong Qiu**, Ahmet Palazoglu***, Wei Sun**** Chi Zhai*, Zhai*, Tong Tong Qiu**, Qiu**, Ahmet Ahmet Palazoglu***, Wei Wei Sun**** Sun**** Chi Palazoglu***, Chi Zhai*, Tong Qiu**, Ahmet Palazoglu***, Wei Sun**** * Beijing University of Chemical Technology, Beijing, 100029China (e-mail: [email protected]). * Beijing University of Chemical Technology, Beijing, 100029China (e-mail: [email protected]). ** Tsinghua University, Beijing, 100084 China (e-mail:[email protected]) * Beijing Beijing University University of Chemical Chemical Technology, Beijing, 100029China (e-mail: [email protected]). [email protected]). * of Technology, Beijing, 100029China (e-mail: ** Tsinghua University, Beijing, 100084 China (e-mail:[email protected]) * Beijing University of Chemical Technology, Beijing, 100029China (e-mail: [email protected]). ***University of California, Davis, CA 95616 USA (e-mail:[email protected]) ** Tsinghua University, Beijing, 100084 China (e-mail:[email protected]) ** Tsinghua University, Beijing, 100084 China (e-mail:[email protected]) ***University California, Davis, CABeijing, 95616 (e-mail:[email protected]) ** Tsinghua University, Beijing, 100084 ChinaUSA (e-mail:[email protected]) *** Beijing University ofof Technology, 100029China (e-mail: [email protected]) ***University ofChemical California, Davis, CABeijing, 95616 USA (e-mail:[email protected]) ***University of California, Davis, CA 95616 USA (e-mail:[email protected]) *** Beijing University of Chemical Technology, 100029China (e-mail: [email protected]) ***University of California, Davis, CA 95616 USA (e-mail:[email protected]) *** Beijing Beijing University University of of Chemical Chemical Technology, Technology, Beijing, Beijing, 100029China 100029China (e-mail: (e-mail: [email protected]) [email protected]) *** *** Beijing University of Chemical Technology, Beijing, 100029China (e-mail: [email protected]) Abstract: In this paper, focus is on identifying the feedforward structure for the emergence of input Abstract: In this optimal paper, focus is onand identifying the feedforward structure for the emergence periodicity at the situation, the fed-batch penicillin fermentation process is applied of forinput case Abstract: In In this optimal paper, focus focus is on onand identifying the feedforward feedforward structure for for the emergence emergence of input Abstract: this paper, is identifying the structure the of periodicity at the situation, the fed-batch penicillin fermentation process is applied forinput case Abstract: In this paper, focus is onand identifying feedforward structuremethods for the are emergence of input study. Through information of optimal control, thethe reversed system analysis constructed, and periodicity at the optimal situation, the fed-batch penicillin fermentation process is applied for periodicity at the optimal situation, andcontrol, the fed-batch penicillin fermentation processare is constructed, applied for case case study. Through information of optimal the reversed system analysis methods and periodicity at the optimal situation, and the fed-batch penicillin fermentation process is applied for case the criterion for the emergency of optimal periodic control (OPC) is built, and possibly, analytical study. Through information of optimal control, the reversed system analysis methods are constructed, and study. Through information of optimal control,periodic the reversed system analysis methods are constructed, and the criterion for the emergency of optimal control (OPC) is built, and possibly, analytical study. Through information of optimal control, the system analysis methods are constructed, and method to compute outemergency the OPC problem would be reversed proposed. the criterion for the of optimal periodic control (OPC) is built, and possibly, analytical the criterion for theoutemergency of optimal periodic control (OPC) is built, and possibly, analytical method to compute the OPC problem would be proposed. the criterion for the emergency of optimal periodic control (OPC) is built, and possibly, analytical method compute out the problem would proposed. method to compute out the OPC OPC problem would be be proposed. © 2018, to IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: optimal control, bang-singular-bang, OPC. method to compute the OPC problem would be proposed. Keywords: optimal out control, bang-singular-bang, OPC. Keywords: Keywords: optimal optimal control, control, bang-singular-bang, bang-singular-bang, OPC. OPC. Keywords: optimal control, bang-singular-bang, OPC. are held constant (Jobses et al., 1986). The ethanol produced 1. INTRODUCTION are held al., 1986). The ethanol produced in the constant fermentor(Jobses is a et secondary metabolite, and the 1. INTRODUCTION are held constant (Jobses et al., 1986). The ethanol are held constant (Jobses et al., 1986). The ethanol produced produced in the fermentor is a secondary metabolite, andfactor the 1. INTRODUCTION INTRODUCTION are held constant (Jobses etsecondary al., 1986). The ethanol produced of which becomes a negative feedback 1. Ever since M. Turing’s (1952) conjecture of morphogenesis accumulation in the fermentor is a metabolite, and the in the fermentor is abecomes secondary metabolite, andfactor the 1. INTRODUCTION Ever since M. Turing’s (1952) conjecture of morphogenesis accumulation of which a negative feedback in the fermentor is abecomes secondary metabolite, andfactor the and affects the reaction adversely, causing oscillations to in thesince reaction-diffusion systems, efforts have been made to accumulation of which a negative feedback Ever M. Turing’s (1952) conjecture of morphogenesis accumulation of reaction which becomes a negative feedback factor Ever since M. Turing’s (1952) conjecture of morphogenesis in the reaction-diffusion systems, efforts have been made to and affects the adversely, causing oscillations to accumulation of which becomes a negative feedback factor emerge. As is shown in Fig.1, the product may become an Ever since M. Turing’s (1952) conjecture of morphogenesis explore the formation of self-organized patterns in nature and and the reaction adversely, causing oscillations to in the systems, have made to and affects affects theshown reaction adversely, causingmay oscillations to in the reaction-diffusion reaction-diffusion systems, efforts efforts have been been madeand to emerge. explore the formation (1978) of self-organized patterns in nature Asfactor is Fig.1, the product become an affects theshown reaction adversely, causing oscillations to inhibitive and in generate feedback structure for the in the reaction-diffusion systems, efforts have made to and industry. I. Prigogine claimed that when been open systems emerge. As is in Fig.1, the product may become an explore the formation of self-organized patterns in nature and emerge. As is shown in Fig.1, the product may become an explore the formation (1978) of self-organized patterns in nature and inhibitive factor and generate feedback structure for the industry. I. Prigogine claimed that when open systems emerge. As is shown in Fig.1, the product may become an bioprocess. On the other hand, some bioprocess, like explore the formation of self-organized patterns in nature and are far-from-equilibrium and nonlinear, structurally inhibitive factor factor and other generate feedback structure for like the industry. I. Prigogine Prigogine (1978) (1978) claimed claimed that when when open open systems bioprocess. inhibitive and generate feedback for the industry. I. that systems are far-from-equilibrium and by nonlinear, structurally On the some structure bioprocess, inhibitive fermentation, factor and other generate structure for like may hand, be feedback substrate-inhibitive and the industry. I. Prigogine (1978) claimed that symmetric-breaking when open systems penicillin organized patterns would form bioprocess. On the hand, some bioprocess, are far-from-equilibrium and nonlinear, structurally bioprocess. On the other hand, some bioprocess, like are far-from-equilibrium and nonlinear, structurally organized patterns would them form by symmetric-breaking may be substrate-inhibitive and like the bioprocess. On thesource other hand, bioprocess, supply of afermentation, carbon could formsome feedforward loops for are far-from-equilibrium and the nonlinear, structurally bifurcations; and named dissipative structure penicillin penicillin fermentation, may be substrate-inhibitive substrate-inhibitive and the the organized patterns would them form by symmetric-breaking penicillin fermentation, may be and organized patterns would form by symmetric-breaking supply of a carbon source could form feedforward loops for bifurcations; and named the dissipative structure penicillin fermentation, may be substrate-inhibitive and the the process. organized patterns would form by symmetric-breaking systems. While, different bifurcation sources (Angeli et al., supply of a carbon source could form feedforward loops bifurcations; anddifferent named bifurcation them the the sources dissipative structure supply of a carbon source could form feedforward loops for for bifurcations; and named them dissipative structure the process. systems. While, (Angeli et al., of a carbon source could form feedforward loops for bifurcations; anddifferent named them thesystems dissipative structure 2004) may cause the biological to evolve to supply the process. systems. While, bifurcation sources et the process. Knowing the mechanism of the fermentation process assists systems. While, different bifurcationsystems sources (Angeli (Angeli et al., al., 2004) may cause the biological to evolve to the process. systems. While, different bifurcation sources (Angeli et and al., sophisticated structures, making it hard to understand thedesign mechanism of the strategies. fermentation assists 2004) the biological systems to to 2004) may may cause cause the making biological systems to evolve evolve and to Knowing formulating /operation Forprocess the productsophisticated structures, it hard to understand Knowing the thedesign mechanism of the the strategies. fermentation process assists Knowing mechanism of fermentation process assists 2004) may cause the biological systems to evolve to model complex bioprocesses in reality. In this paper, the idea formulating /operation For the productsophisticated structures, making it hard to understand and Knowing the mechanism of the fermentation process assists sophisticated structures, making it hard to understand and inhibitive processes, product selective membrane fermentor model complex bioprocesses in reality. In this paper, the idea formulating design /operation strategies. For the productformulating design /operation strategies. For the productsophisticated structures, making it hard to understand and of “optimal control” is used for reference to study the inhibitive processes, product selective membrane fermentor model complex bioprocesses in In the formulating design /operation strategies. For theto productmodel complex bioprocesses in reality. reality. In this this paper, paper, the idea idea or extractive fermentor are options to be adopted remove of “optimal control” is used for reference to study the inhibitive processes, product selective membrane fermentor inhibitive processes, product selective membrane fermentor model complex bioprocesses inthe reality. In this paper, thefrom idea fermentation processes, is originated fermentor are options to be adopted to remove of “optimal control” iswhere used foranalogy reference to study study the or inhibitive processes, membrane fermentor of “optimal control” is used for reference to the theextractive inhibitive factor inproduct time, byselective so doing, the feedback loop is fermentation processes, where the analogy is originated from or extractive fermentor are options to be adopted to remove or extractive fermentor are options to be adopted to remove of “optimal control” is used for reference to study the two aspects of concern. Firstly, the mechanism for many the inhibitive factor in time, by so doing, the feedback loop is fermentation processes, where the analogy is originated from or extractive fermentor are options to be adopted to remove fermentation processes, where the analogy is originated from opened and the oscillation is eliminated. For the substratetwo aspects processes, of concern.where Firstly, thecontrol mechanism for many the factor in by so feedback loop the inhibitive inhibitive factor in time, time, is byeliminated. so doing, doing, the the feedback loop is is fermentation the analogy is structures, originated from bioprocesses is analogous to typical as is opened and the oscillation For thebesubstratetwo aspects of of concern. Firstly, Firstly, thecontrol mechanism for many many the inhibitive factor in by eliminated. so doing, the feedback loop is two aspects concern. the mechanism for inhibitive processes, thetime, fed-batch operation could adopted bioprocesses is analogous to typical structures, as is opened and the oscillation is For the substrateopened and the oscillation is eliminated. For the substratetwo aspects of concern. Firstly, the mechanism for many shown in Fig. 1; secondly, by natural adaption and evolution, inhibitive processes, the fed-batch operation could be adopted bioprocesses is1;analogous analogous to typical control structures, as is is opened and the oscillation is eliminated. thebesubstratewhen the purpose isthe to fed-batch alleviate substrateFor inhibition of the bioprocesses is to typical structures, as shown in Fig.always secondly, byoptimally, natural control adaption and evolution, inhibitive processes, operation could adopted inhibitive processes, fed-batch operation could be adopted bioprocesses is1;analogous typical control as of is when behaveto hence,structures, knowledge theOversupply purpose isthe to alleviate substrate inhibition ofmore the shown in by natural and evolution, inhibitive processes, the fed-batch operation could be adopted process. of a carbon source results in shown in Fig. Fig.always 1; secondly, secondly, byoptimally, natural adaption adaption and evolution, bioprocesses behave hence, knowledge of when the theOversupply purpose is is to to alleviate substrate inhibition ofmore the when purpose alleviate substrate inhibition of the shown in Fig.always 1; secondly, byoptimally, natural adaption and evolution, optimization may provide valuable information for the process. of a carbon source results in bioprocesses behave hence, knowledge of when theOversupply purpose and is to alleviate substrate inhibition ofwhile the bioprocesses always behave optimally, hence, knowledge of biomass growth lower penicillin formation; optimization may provide valuable information for the process. of a carbon source results in more process. Oversupply of a carbon source results in more bioprocesses always behave optimally, hence, knowledge of processes to be studied. growth lower penicillin formation; optimization may provide process. Oversupply a carbon source results in while more optimization may provide valuable valuable information information for for the the biomass undersupply resultsand in of slower biomass growth, and eventually, processes to be studied. biomass growth and lower penicillin formation; while growth lowerbiomass penicillin formation; while optimization may provide valuable information for the biomass undersupply resultsand in slower growth, and eventually, processes biomass growth and lower Therefore, penicillin formation; while processes to to be be studied. studied. slower penicillin formation. an optimal input undersupply results in growth, and eventually, undersupply resultsformation. in slower slower biomass biomass growth, and eventually, processes to be studied. slower penicillin Therefore, an optimal input undersupply results in slower biomass growth, and eventually, profile is existed for the fed-batch operation. slower penicillin formation. Therefore, an optimal slower ispenicillin Therefore, optimal input input profile existed forformation. the fed-batch operation.an slower penicillin formation. Therefore, an optimal input profile is existed for the fed-batch operation. profile is existed for the is fed-batch operation. A fed-batch operation a semi-batch operation, in which profile is existed for the is fed-batch operation. A fed-batch operation a semi-batch operation, in during which substrate is added eitherisintermittently continuously A fed-batch fed-batch operation semi-batchor operation, in during which in which A operation isintermittently aa semi-batch operation, substrate is added either orharvested continuously A fed-batch operation is a semi-batch operation, in which the operation course and the product is only at the substrate is added added either intermittently orharvested continuously during substrate is either intermittently or continuously during the operation course and the product is only at the substrate is added either intermittently or continuously during end of the run. Analytically, the optimal feeding profile is the operation operation course and the the product product is harvested harvested only at the the the course and is only at end of thebyrun. Analytically, the optimal feeding profile is the operation course and the product is harvested only at the obtained formulating the Hamiltonian, and then end of of the the run. run. Analytically, the optimal feeding feeding profile is end Analytically, the optimal profile is formulating thethe Hamiltonian, and then But the end of theby run. Analytically, the optimal feeding profile is Fig. 1. Feedforward/feedback mechanism of the inhibitive obtained maximum principle provides optimization criterion. obtained by formulating the Hamiltonian, and then the obtained by formulating the Hamiltonian, and then the Fig. 1. inFeedforward/feedback mechanism of the inhibitive maximum principle provides theHamiltonian, optimization criterion. But by formulating the factors the fermentation processes. the presence of lower andthe upper bounds forand the then substrate Fig. 1. inFeedforward/feedback Feedforward/feedback mechanism of of the the inhibitive inhibitive obtained maximum principle provides the optimization optimization criterion. But Fig. 1. mechanism maximum principle provides the criterion. But factors the fermentation processes. the presence of lower and upper bounds for the substrate Fig. 1. Feedforward/feedback mechanism of the inhibitive maximum principle provides the optimization criterion. But stream causes the control to be bang-singular-bang structure factors in in the the fermentation processes. the presence presence of lower and upper bounds for for the the substrate substrate factors fermentation processes. the of lower and upper bounds Oscillatory patterns are typical in nature but the evolutionary stream causestoof the control to upper be bang-singular-bang structure factors in the fermentation processes. the presence lower bounds for the substrate and difficult deal withand mathematically (Modak et al., 1986). Oscillatory patterns are typical in remain nature but the evolutionary stream causes the control to be bang-singular-bang structure stream causes the control to be bang-singular-bang structure origins of oscillatory phenomena a mystery (Tu et al., and difficult to deal with mathematically (Modak et al., 1986). Oscillatory patterns are typical in nature but the evolutionary stream causes the control to be bang-singular-bang structure Oscillatory patterns are typical in remain nature abut the evolutionary While numerically, by mathematically orthogonal collocation on finite origins of oscillatory phenomena mystery (Tu et al., and difficult to deal with (Modak et al., 1986). Oscillatory are typical nature but the evolutionary and difficult to deal with mathematically (Modak et on al., 1986). 2006). It oscillatory ispatterns well known that in Hopf bifurcation causes the While numerically, by orthogonal collocation finite origins of phenomena remain mystery (Tu et et the al., and origins of oscillatory phenomena remain aa mystery (Tu al., difficult to deal with (Modak al., 1986). elements (Cuthrell andmathematically Biegler, 1989), theet on dynamic 2006). It is well known that Hopf bifurcation causes While numerically, by orthogonal collocation finite origins of oscillatory phenomena remain a mystery (Tu et al., While numerically, by orthogonal collocation on finite system to generate self-oscillatory patterns. Some elements (Cuthrell and Biegler, 1989), the dynamic 2006). It It to is well well known self-oscillatory that Hopf Hopf bifurcation bifurcation causes the While 2006). is known that causes the numerically, orthogonal collocation finite optimization problemby can be converted to the a on nonlinear system generate patterns. Some elements (Cuthrell and Biegler, 1989), dynamic 2006). is processes, well known that Hopfproduction bifurcation the optimization elements (Cuthrell and Biegler, 1989), the dynamic fermentation e.g., ethanol bycauses Z. mobilis, problemand can Biegler, be converted a nonlinear system It to to generate self-oscillatory patterns. Some elements (Cuthrell 1989), to dynamic system generate self-oscillatory patterns. Some fermentation processes, e.g., ethanol production by Z. mobilis, optimization problem can can be be converted converted to the a nonlinear nonlinear system to generate self-oscillatory patterns. Some optimization problem to a may behave oscillatory state trajectories even if all the inputs fermentation processes, state e.g., ethanol ethanol production by Z. mobilis, fermentation processes, e.g., production may behave oscillatory trajectories even if by all Z. themobilis, inputs optimization problem can be converted to a nonlinear fermentation processes, e.g., ethanol production by Z. mobilis, may may behave behave oscillatory oscillatory state state trajectories trajectories even even if if all all the the inputs inputs may behave oscillatory state trajectories evenofifAutomatic all the inputs 2405-8963 © IFAC (International Federation Control) Copyright © 2018, 2018 IFAC 130Hosting by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic 2018 responsibility IFAC 130Control. Copyright Copyright © 130 10.1016/j.ifacol.2018.11.367 Copyright © 2018 2018 IFAC IFAC 130 Copyright © 2018 IFAC 130
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programming (NLP) problem, which is especially prominent for dealing with singular control problem (San, 1988).
are [0, 40], [0, +∞], [0, 100] and [0, 10], respectively; and the initial values are [1.5, 0, 0, 7].
This work is focused on identifying the feedforward structure for the emergence of input periodicity at the optimal situation, just the opposite to output periodicity as the Z. mobilis fermentation process has indicated. The optimal periodic control (OPC) problem has been studied by many researchers (Speyer et al., 1984), but the structural origins that causes the optimal input profile periodic has not been discussed before. Obviously, the presence of lower and upper bounds for the input is one of the key reasons that lead the drug delivery system (Varigonda et al., 2008) to be OPC, but the penicillin fermentation model is affine, which directly leads the optimization to be bang-bang nature. While, input periodicity may emerge if some optimal conditions are loosen for the penicillin fermentation case. Through this case study, the criterion for the emergency of OPC is built, and possibly, analytical method to compute out the OPC problem is proposed.
2.2 Analysis on the optimal fed-batch operation
2. PROCESS MODEL AND BANG-SINGULAR-BANG CONTROL PRELIMINARY
H λT a(x) bU
(3)
where the adjoint vector, λT, is defined by the follows, λ ' H / x axT λ
(4)
(5)
where g(tf) = -P(tf)V(tf) is the objective to be minimized. Minimization of the Hamiltonian is considerably simplified when U(t) appears linearly in it. If the coefficient of U, λTb, is positive, one can pick the input control to the lower boundary; while, if λTb is negative, on can choose the upper boundary. Up to now, it is the bang-bang control problem. However, the situation of singular control may appear when λTb is identically zero over a finite time interval, then the minimum principle fails to give U(t) during the interval. Therefore, the optimal fed-batch penicillin fermentation process is a bangsingular-bang control problem,
P (t f )V (t f ),
s.t : X )U , S FV P )U , S FV
(2)
The optimization problem is to determine the optimal substrate profile, U(t), which is subject to the upper and lower constraints given by Eq. (1). The Pontryagin minimum principle yields an equivalent problem, which is to minimize the Hamiltonian defined as follows,
λi (t f ) g / xi (t f ); i 1, 2,3
u ( t ), t f
P ' ( S ) X K deg P (
x(0) x0
And λ must satisfy the terminal conditions given as follows,
The fermentation model (Modak et al., 1986)) is described by mass balance of the volume (V), concentrations of biomass (X), product (P) and substrate (S). The dynamic optimization problem can be formulated as provided in Eq. (1), where the penicillin production is maximized, and the system subject to constraints of the process model and boundary conditions with respect to the state and manipulate variables.
X ' ( X , S) X (
x ' a(x) bU
T
2.1 The penicillin fermentation model
min
The state vector x = ∑(X, P, S, V) is affine to the control U, one can give the general dynamics of the penicillin process as follows,
(1)
X X ms S S U S ' ( X , S ) X 1 (S ) , YX S YP S K m S S V F U , V' SF
U (t )opt
where S S , , ( S ) max ( S ) 1 K S S K P in KX S
( X , S ) max ( X , S )
X L X (t ) X U , S L S (t ) S U ,V L V (t ) V U ,U L U (t ) U U , t f L t f t f U
where tf is the end-time of the fed-batch process; μ(X, S) is the specific biomass growth rate and ρ(S) is the specific penicillin production rate. μmax = 0.11h-1 is maximum specific biomass growth rate; ρmax = 0.0055h-1 is maximum specific production rate; KX = 0.006g/g is saturation parameter for biomass growth; KP = 0.0001g/l is saturation parameter for production; Kin = 0.1g/l is inhibition parameter for production; Kdeg = 0.01h-1 is the product degradation rate; Km = 0.0001g/l is saturation parameter for maintenance consumption; mS = 0.029h-1 is maintenance consumption rate; Yx/s = 0.47g/g is yield factor for substrate to biomass; YP/S =1.2g/g is the yield factor for substrate to product and SF = 500g/l substrate feeding concentration. Boundary conditions for X, P, S and V
131
U U U (t )singular U L
λT b 0 λT b 0 λT b 0
(6)
The singular substrate feed rate, U(t)singular, can be determined by recognizing that during the singular interval λTb is identically zero, hence its time derivatives must also vanish, which lead to relation as follows, U (t )singular [(a / x)c (c / x)a] / λ T (c / x)b where : c (a / x)b
(7)
In addition, the Hamiltonian on the optimal path is also an unknown constant H*, hence three equations linear to λ(t) are satisfied, λ T a H * T λ b 0 T λ c 0
(8)
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In general, Eq. (8) forms a two-point boundary value problem in which the initial conditions for the state variables and the final conditions for the adjoint variables are known. The desired optimal substrate feeding profile consists of maximum, minimum, and singular values with the singular flow rate being a function of the state and adjoint variables.
conditions of x(t) for each element. Also, the Lagrange polynomial has the desirable property that for xKx(t): xKx (t ) xi j
(12)
which is due to the Lagrange condition ϕk(tik) = δkj,, where δkj is the Kronecker delta. This polynomial form allows the direct bounding of the states and controls, e.g., path constraints can be imposed on the problem formulation. Using K=Kx=Ku point orthogonal collocation on finite elements as shown in Fig. 2, and by defining the basic functions, so that they are normalised over each element ∆ζi, one can write the residual equation as follows,
To compute out the optimal bang-singular-bang problem, one must know the exact sequence of optimal feed rate (maximum, minimum or singular) and solve the two-point boundary value problem posed by Eqs. (2) and (4)-(7). The computational algorithms are provided (Lim et al., 1986), while, a more universally applicable method is to convert the optimal control problem by discretization and solve the NLP problem numerically (Zhai et al., 2017).
Kx
i r (ti k ) xi j j ( k ) i f (xi k , ui k , ti k )
3. NUMERICAL OPTIMAL INPUT PROFILE
(13)
j 1
3.1 Problem formulation
where i = 1, 2…, NE, j = 1, 2…, ϕj(τk) = dtϕj/dτk, and together with ϕj(τ), θj(τ), terms (basis functions), they are calculated beforehand, since they depend only on the Legendre root locations. Note that tik = ζi+ζiτk. This form is convenient to work with when the element lengths are included as decision variables. The element lengths are also used to find possible points of discontinuity for the control profiles and to ensure that the integration accuracy is within a numerical tolerance. Additionally, the continuity of the states is enforced at element (ζi, i = 2, ..., NE), but it is allowed that the control profiles to have discontinuities at these endpoints as follows,
In order to derive the NLP problem, the differential (integral, algebraic) equations are converted into algebraic equations using orthogonal collocation on finite elements, and the residual equations are then formed as a set of algebraic equations. For optimal residuals, the roots of the shifted Legendre polynomial are set as the collocation points, and the Lagrange polynomial is used to approximate the differentiation and integration terms. The mathematical development is as follows: Consider the initial-value problem over a finite element i with time t∈(ζi, ζi+1), and the dynamic system is as follows,
xiK x ( i ) xiKx1 ( i ) i 2,..., NE Kx
x ' f (x, u, t )
xi 0 xi 1, j j ( 1) i 2,..., NE; j 2,..., K x
(9)
(14)
j 0
2
xL
1
i
xi ,1 , ui ,1 xi ,2 , ui ,2
x
i xi ,0
x
These equations extrapolate the polynomial xiK1 (t ) to the endpoints of its element and provide accurate initial conditions for the next element and polynomial xiK (t ) . The bounds can be done in similar way.
i
i 1
NE
xU
x
Ku
uKi u ( i ) ui , j j ( 0); i 2,..., NE; j 2,..., K u
i 1
j 1
xi1,0
Ku
uKi u ( i 1 ) ui , j j ( 1); i 2,..., NE; j 2,..., K u (15) j 1
Fig. 2. Collocation on finite elements for state profiles, control profiles and element lengths (Kx, Ku = 2).
Note that the constraints on continuity of the states are out of practical concerns, since the states x = ∑(X, P, S, V) being discontinuous will be unexplainable in the real world; while designing the endpoint of each control element to be discontinuous is practical, and those points with abrupt change of control action is actually the singular control.Then a set of nonlinear equations are generated, which can be solved by NLP solvers, such as sequential quadratic programming (Zhai et al., 2015) or SNOPT.
The solution is approximated by Lagrange polynomials over elementi (ζi ≤ t ≤ ζi+1) as follows, Kx
xK x (t ) xi j j (t ); j (t ) j 0
Ku
uKu (t ) ui j j (t ); j (t ) j 1
Kx
(t ti k ) i j ti k )
(10)
(t ti k ) i j ti k )
(11)
(t
k 0, j
Ku
(t
k 1, j
3.2 Results
where i =1, 2…, NE. k = 0 to j meaning that k starts form 0 and k ≠ j, NE is the number of elements. Also, xKx(t) is a (Kx+1)th order degree piecewise polynomial and uKu(t) is piecewise polynomial of order Ku. The polynomial approximating the state x takes into account the initial
The fed-batch penicillin process is of Mayer form (no integral terms in the objective function), Lagrange multipliers are applied to deal with the equality constraints, but before which, the inequality constraints are relaxed by adding extra dimensions. The Kuhn-Tucher conditions are used to derive 132
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the expression of the multipliers. Now, the system becomes a standard NLP problem. The search for the optimal of a nonlinear system is difficult, but one can seek the less attractive alternative by searching the optimal of the locally quadratic approximation, where Hessian matrix leads directly to the optimal of the approximated system, and the optimal of the original system can be obtained by this sequential quadratic searching procedure.
a
Biomass concerntration (g/l)
30
An analytically-based optimal control profile was obtained by the method stated in section 2.2. This solution derives from the variational conditions, but also requires repeated numerical solution to solve the two-point boundary value problem. The values for the points of control profile discontinuity, final time and the optimal value of the objective function are 11.21, 28.79 and 124.9 and -87.05, respectively, and the control profile is presented in Fig. 3.
20
15
10
0
0
20
40
60
80
100
120 125
80
100
120 125
80
100
120 125
80
100
120 125
Time (h) 9
b 8
Product concentration (g/l)
7
45
Substrate feeding profile (g/h)
25
5
50
Obj: -87.05
40
133
35 30 25
6 5 4 3 2
20
1
15 10
0
0
20
40
0
60
Time (h)
5 0
11.21
20
28.79
40
60
80
100
30
120 125
c
Time (h) 25
Substrate concentration (g/l)
Fig. 3. Analytical-based optimal substrate input profile. Set the analytically-based case above as a typical case for the subsequent numerical investigation, where tf = 124.9h-1. Choose Kx = Ku = 3, NE = 10, both the tolerance of the objective function and the constraints as e-5, and the total tolerance as 5e-5. The control profile of Fig. 4 is different from that of Fig. 3 because this system is not feedback controllable. From section 2.2 we know that, to compute out the optimal control profile, the sequence of maximum, singular and minimum need to decide beforehand, which means the dynamic reverse is underdetermined, and the mapping of optimal output to the control input has multiple routes. Numerically, the choice of initial control profile, collocation points and number of elements affects the profiler of the control.
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Fig. 5. Numerical-based state profiles.
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Fig. 4. Numerical-based optimal substrate input profile. 133
The corresponding state profile for x = ∑(X, P, S, V) are presented in Fig.5 (a-d). it is obvious that the inflexion point at t* = 29.8h separates the fed-batch operation into two folds: before t*, the fermentor adjust itself with varying substrate input rates, where biomass is accumulated in an optimal
IFAC CAO 2018 134 Yekaterinburg, Russia, October 15-19, 2018 Chi Zhai et al. / IFAC PapersOnLine 51-32 (2018) 130–135
5.4
fashion and no product is harvested; after t*, the substrate is consumed mainly to produce penicillin product and the substrate input are maintained at a (almost) steady level.
a
Substrate feeding profile (g/h)
Assume the product penicillin do not affect the metabolism, when the fermentor volume is considerable large, the volume gain by substrate input is insignificant during the time interval [t*, tf], then, one can speculate that the state change of the key factors, (X, S), are held constant for the well-mixed fermentor. To this end, the assumed fed-batch fermentator is almost identical to the continuous CSTR, except that V and P are accumulated with time.
5.2 5.1 5 4.9 4.8 4.7
One can find that the above assumption is not exactly fit to the numerical results as is shown in Fig. 5 (a), where the biomass is not constant during interval [t*, tf]. While the slight decrease of X is due to the dilution effect caused by constant input of substrate stream, meaning that the assumption of V approaches infinite is not proper here. Still, this assumption and speculation procedure is important because the following conclusion can be drawn,
4.6
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b Substrate feeding profile (g/h)
dP C; t [t*, t f ] dt
Obj: -36.5 Xmax: 16.08
5.3
(16)
Where C is the constant, meaning that constant production of penicillin is maintained during the time interval.
Obj: -82.2 Xmax: 31.63
18
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4. THE EMERGENCE OF FEEDFORWARD PERIODICITY
8
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It is obvious that the derivative of the objective function is critical for the NLP solving strategies, because it decides the optimal searching direction. While deliberately omitting this condition leads to locally suboptimal input profiles as shown in Fig. 6. In general, the substrate feeding profiles in Fig.6 also can be separated into tow folds by t*, and after dynamic transient during the interval [0, t*), the “normal” working state is built, but unlike Fig. 3 and 4, periodic inputs profiles emerge around the initial input profile U0(t).
35
Substrate feeding profile (g/h)
c
As to the emergence of input periodicity, one may speculate that it is caused by Hopf bifurcation: when the input-output relation of fed-bath fermentation process is reversed, for example, if the output is known, which in fact is the optimal penicillin productivity, and then viewing Fig. 1 from the right hand side to the left hand side, feedforward periodicity is similar to the mechanism of feedback periodicity. Since Hopf bifurcation is detected for the self-oscillatory Z. mobilis fermentation model, one may expect Hopf points are also detected of the reversed fed-batch fermentation process.
Obj: -80.21 Xmax: 33.33
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35
If the optimal condition is loosening to the suboptimal situation as our current case indicates, one will find that it is in fact the OPC problem (the local, suboptimal situation can be converted to exact optimal problem by adding constraints on the optimization problem, e.g., forcing the input boundary to (4.6, 5.4) for the case shown in Fig. 6-a). Under dynamic reverse perspectives, the optimal problem may provide a new way to solve the OPC problem: the critical condition for the emergence of the OPC is actually the Hopf point, and compute out the period of the limit cycle (Zhai et al., 2017b) is relatively easy compared to the current numerical OPC strategies (Varigonda et al., 2008).
30 25 20 15 10 5 0
0
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40
60
80
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120
Time (h)
Fig. 6. Numerical suboptimal substrate input profile, where periodic inputs emerge. a: U0(t) = 5g/h; b: U0(t) = 12g/h; c: U0(t) = 25g/h; d: U0(t) = 40g/h. 134
IFAC CAO 2018 Yekaterinburg, Russia, October 15-19, 2018 Chi Zhai et al. / IFAC PapersOnLine 51-32 (2018) 130–135
The time-invariant output for the fed-batch process is the constant production rate exhibit in Eq. (16), when the assumption in section 3.2 is satisfied. The difference of Fig. 4 and Fig. 6 may be caused by the state variable (X, S), by substituting the condition of Eq. (16) into the dynamic system in Eq. (1), one can obtain the ordinary differential equations as follows, ( S ) X K deg P * C X ' ( X , S ) X P*/X S ' ( X , S ) X ( S ) X ms S X ( S ) X K deg P * C YX S YP S K m S P * / S F S
135
of biological positive-feedback systems. PNAS., 101(7), 1822-1827. Cuthrell, J.E., Biegler, L.T., (1989). Simultaneous optimization and solution methods for batch reactor control profiles. Comput. Chem. Eng., 13, 49-52. Gao, N., Zhai, C.,，Sun W., et al., (2015). Equation Oriented Method for Rectisol Wash Modeling and Analysis. Chinese Journal of Chemical Engineering, 23(9), 15301535.
(17)
Jobses, M.L., Egberts, G.T.C., Luyben, K.C.A.M., Roels, J.A., (1986). Fermentation Kinetics of Zymomonas mobilis at High Ethanol Concentrations: Oscillations in Continuous Cultures. Biotechnology and Bioengineering, 868-877, 1986.
where P* is the nominal product concentration, since Kdeg is very small while V is sufficiently big, changes of P are insignificant. In the following analysis, P at tf is chosen for P*. Numerical bifurcation analysis (Zhai et al., 2017b) is implemented on the system represented in Eq. (17), and the range of C being tested can be determined by the information from Fig. 5-b. however, no Hopf points are detected, which means that the periodicity is not caused by the dynamic system itself.
Modak, J.M., Lim H.C., Tayeb Y.J., (1986). General characteristics of optimal feed rate profiles for various fed-batch fermentation processes, Biotechnology and bioengineering, 28(9): 1396-1407. Prigogine, I., (1978). Time, structure, and fluctuations. Science, 201(4358), 777-785.
Further study on the NLP algorithm and it can be found that the periodic profiles in Fig. 6 are caused by the cut of the whole time interval as shown in Fig. 2, roughly 10 orbits with identical period is obtained because the elements, NE, is set as 10. Since singular control may generate at the endpoint of each element and similar trajectories are obtained for each elements. However, the analysis in this section is not useless, when the substrate input profile is pre-determined as shown in Fig. 6, the production may not be the optimal, but in some situations, e.g., Fig.6-c, the penicillin concentration is higher than the optimal situation. Moreover, the affine system represented in Eq. (2) may cause the optimal control to be bang-singular-bang structure, but no optimal periodic control can be obtained analytically.
San, K.Y., Stephanopoulos, G., (1989). Optimization of fedbatch penicillin fermentation: A case of singular optimal control with state constraints, Biotechnology and bioengineering, 34(1), 72-78. Speyer, J., Evans, R., (1984). A second variational theory for optimal periodic processes. IEEE Transactions on Automatic Control, 29(2), 138-148. Tu, B.P., McKnight, S.L., (2006). Metabolic cycles as an underlying basis of biological oscillations. Nature Reviews Molecular Cell Biology, 7, 696–701. Turing, A.M., (1952). The Chemical Basis of Morphogenesis. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237(641), 37-72.
5. CONCLUSIONS
Varigonda, S., Georgiou, T.T., Siegel, R.A., Daoutidis, P., (2008). Optimal periodic control of a drug delivery system. Computers & Chemical engineering, 32(10), 2256-2262.
In this paper, the fed-batch penicillin process is investigated for its bang-singular-bang control structure and numerical solution by infinite elements method is detailed. It turned out that the feedforward periodicity is caused by the numerical computing strategies, but the methods to study OPC by dynamic inverse are constructed, which may be of a reference for other systems.
Zhai, C., Sun W., et al., (2017a). Dynamic optimization of penicillin fermentation with constraint on wastewater discharge. Chemical Engineering Transactions, 61, 499504. Zhai, C., Palazoglu, A., Wang, S., Sun, W., (2017b). Strategies for the Analysis of Continuous Bioethanol Fermentation under Periodical Forcing. Ind. Eng. Chem. Res., 56(14), 3958-3968.
ACKNOWLEDGEMENTS The authors gratefully acknowledge the following institutions for support: the National Natural Science Foundation of China (Grant No. 21576015); the National Basic Research Program of China, 973 program (Grant No. 2013CB733600).
REFERENCES Angeli, D., Ferrell, J.E., Sontag, E.D., (2004). Detection of multistability, bifurcations, and hysteresis in a large class 135
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The emergence of feedforward periodicity for the fed-batch penicillin fermentation The emergence of feedforward periodicity for the fed-batch penicillin fermentation The emergence of feedforward periodicity for the fed-batch penicillin fermentation process The emergence of feedforward periodicity for the fed-batch penicillin fermentation process process process Chi Zhai*, Tong Qiu**, Ahmet Palazoglu***, Wei Sun****
Chi Zhai*, Tong Qiu**, Ahmet Palazoglu***, Wei Sun**** Chi Zhai*, Zhai*, Tong Tong Qiu**, Qiu**, Ahmet Ahmet Palazoglu***, Wei Wei Sun**** Sun**** Chi Palazoglu***, Chi Zhai*, Tong Qiu**, Ahmet Palazoglu***, Wei Sun**** * Beijing University of Chemical Technology, Beijing, 100029China (e-mail: [email protected]). * Beijing University of Chemical Technology, Beijing, 100029China (e-mail: [email protected]). ** Tsinghua University, Beijing, 100084 China (e-mail:[email protected]) * Beijing Beijing University University of Chemical Chemical Technology, Beijing, 100029China (e-mail: [email protected]). [email protected]). * of Technology, Beijing, 100029China (e-mail: ** Tsinghua University, Beijing, 100084 China (e-mail:[email protected]) * Beijing University of Chemical Technology, Beijing, 100029China (e-mail: [email protected]). ***University of California, Davis, CA 95616 USA (e-mail:[email protected]) ** Tsinghua University, Beijing, 100084 China (e-mail:[email protected]) ** Tsinghua University, Beijing, 100084 China (e-mail:[email protected]) ***University California, Davis, CABeijing, 95616 (e-mail:[email protected]) ** Tsinghua University, Beijing, 100084 ChinaUSA (e-mail:[email protected]) *** Beijing University ofof Technology, 100029China (e-mail: [email protected]) ***University ofChemical California, Davis, CABeijing, 95616 USA (e-mail:[email protected]) ***University of California, Davis, CA 95616 USA (e-mail:[email protected]) *** Beijing University of Chemical Technology, 100029China (e-mail: [email protected]) ***University of California, Davis, CA 95616 USA (e-mail:[email protected]) *** Beijing Beijing University University of of Chemical Chemical Technology, Technology, Beijing, Beijing, 100029China 100029China (e-mail: (e-mail: [email protected]) [email protected]) *** *** Beijing University of Chemical Technology, Beijing, 100029China (e-mail: [email protected]) Abstract: In this paper, focus is on identifying the feedforward structure for the emergence of input Abstract: In this optimal paper, focus is onand identifying the feedforward structure for the emergence periodicity at the situation, the fed-batch penicillin fermentation process is applied of forinput case Abstract: In In this optimal paper, focus focus is on onand identifying the feedforward feedforward structure for for the emergence emergence of input Abstract: this paper, is identifying the structure the of periodicity at the situation, the fed-batch penicillin fermentation process is applied forinput case Abstract: In this paper, focus is onand identifying feedforward structuremethods for the are emergence of input study. Through information of optimal control, thethe reversed system analysis constructed, and periodicity at the optimal situation, the fed-batch penicillin fermentation process is applied for periodicity at the optimal situation, andcontrol, the fed-batch penicillin fermentation processare is constructed, applied for case case study. Through information of optimal the reversed system analysis methods and periodicity at the optimal situation, and the fed-batch penicillin fermentation process is applied for case the criterion for the emergency of optimal periodic control (OPC) is built, and possibly, analytical study. Through information of optimal control, the reversed system analysis methods are constructed, and study. Through information of optimal control,periodic the reversed system analysis methods are constructed, and the criterion for the emergency of optimal control (OPC) is built, and possibly, analytical study. Through information of optimal control, the system analysis methods are constructed, and method to compute outemergency the OPC problem would be reversed proposed. the criterion for the of optimal periodic control (OPC) is built, and possibly, analytical the criterion for theoutemergency of optimal periodic control (OPC) is built, and possibly, analytical method to compute the OPC problem would be proposed. the criterion for the emergency of optimal periodic control (OPC) is built, and possibly, analytical method compute out the problem would proposed. method to compute out the OPC OPC problem would be be proposed. © 2018, to IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: optimal control, bang-singular-bang, OPC. method to compute the OPC problem would be proposed. Keywords: optimal out control, bang-singular-bang, OPC. Keywords: Keywords: optimal optimal control, control, bang-singular-bang, bang-singular-bang, OPC. OPC. Keywords: optimal control, bang-singular-bang, OPC. are held constant (Jobses et al., 1986). The ethanol produced 1. INTRODUCTION are held al., 1986). The ethanol produced in the constant fermentor(Jobses is a et secondary metabolite, and the 1. INTRODUCTION are held constant (Jobses et al., 1986). The ethanol are held constant (Jobses et al., 1986). The ethanol produced produced in the fermentor is a secondary metabolite, andfactor the 1. INTRODUCTION INTRODUCTION are held constant (Jobses etsecondary al., 1986). The ethanol produced of which becomes a negative feedback 1. Ever since M. Turing’s (1952) conjecture of morphogenesis accumulation in the fermentor is a metabolite, and the in the fermentor is abecomes secondary metabolite, andfactor the 1. INTRODUCTION Ever since M. Turing’s (1952) conjecture of morphogenesis accumulation of which a negative feedback in the fermentor is abecomes secondary metabolite, andfactor the and affects the reaction adversely, causing oscillations to in thesince reaction-diffusion systems, efforts have been made to accumulation of which a negative feedback Ever M. Turing’s (1952) conjecture of morphogenesis accumulation of reaction which becomes a negative feedback factor Ever since M. Turing’s (1952) conjecture of morphogenesis in the reaction-diffusion systems, efforts have been made to and affects the adversely, causing oscillations to accumulation of which becomes a negative feedback factor emerge. As is shown in Fig.1, the product may become an Ever since M. Turing’s (1952) conjecture of morphogenesis explore the formation of self-organized patterns in nature and and the reaction adversely, causing oscillations to in the systems, have made to and affects affects theshown reaction adversely, causingmay oscillations to in the reaction-diffusion reaction-diffusion systems, efforts efforts have been been madeand to emerge. explore the formation (1978) of self-organized patterns in nature Asfactor is Fig.1, the product become an affects theshown reaction adversely, causing oscillations to inhibitive and in generate feedback structure for the in the reaction-diffusion systems, efforts have made to and industry. I. Prigogine claimed that when been open systems emerge. As is in Fig.1, the product may become an explore the formation of self-organized patterns in nature and emerge. As is shown in Fig.1, the product may become an explore the formation (1978) of self-organized patterns in nature and inhibitive factor and generate feedback structure for the industry. I. Prigogine claimed that when open systems emerge. As is shown in Fig.1, the product may become an bioprocess. On the other hand, some bioprocess, like explore the formation of self-organized patterns in nature and are far-from-equilibrium and nonlinear, structurally inhibitive factor factor and other generate feedback structure for like the industry. I. Prigogine Prigogine (1978) (1978) claimed claimed that when when open open systems bioprocess. inhibitive and generate feedback for the industry. I. that systems are far-from-equilibrium and by nonlinear, structurally On the some structure bioprocess, inhibitive fermentation, factor and other generate structure for like may hand, be feedback substrate-inhibitive and the industry. I. Prigogine (1978) claimed that symmetric-breaking when open systems penicillin organized patterns would form bioprocess. On the hand, some bioprocess, are far-from-equilibrium and nonlinear, structurally bioprocess. On the other hand, some bioprocess, like are far-from-equilibrium and nonlinear, structurally organized patterns would them form by symmetric-breaking may be substrate-inhibitive and like the bioprocess. On thesource other hand, bioprocess, supply of afermentation, carbon could formsome feedforward loops for are far-from-equilibrium and the nonlinear, structurally bifurcations; and named dissipative structure penicillin penicillin fermentation, may be substrate-inhibitive substrate-inhibitive and the the organized patterns would them form by symmetric-breaking penicillin fermentation, may be and organized patterns would form by symmetric-breaking supply of a carbon source could form feedforward loops for bifurcations; and named the dissipative structure penicillin fermentation, may be substrate-inhibitive and the the process. organized patterns would form by symmetric-breaking systems. While, different bifurcation sources (Angeli et al., supply of a carbon source could form feedforward loops bifurcations; anddifferent named bifurcation them the the sources dissipative structure supply of a carbon source could form feedforward loops for for bifurcations; and named them dissipative structure the process. systems. While, (Angeli et al., of a carbon source could form feedforward loops for bifurcations; anddifferent named them thesystems dissipative structure 2004) may cause the biological to evolve to supply the process. systems. While, bifurcation sources et the process. Knowing the mechanism of the fermentation process assists systems. While, different bifurcationsystems sources (Angeli (Angeli et al., al., 2004) may cause the biological to evolve to the process. systems. While, different bifurcation sources (Angeli et and al., sophisticated structures, making it hard to understand thedesign mechanism of the strategies. fermentation assists 2004) the biological systems to to 2004) may may cause cause the making biological systems to evolve evolve and to Knowing formulating /operation Forprocess the productsophisticated structures, it hard to understand Knowing the thedesign mechanism of the the strategies. fermentation process assists Knowing mechanism of fermentation process assists 2004) may cause the biological systems to evolve to model complex bioprocesses in reality. In this paper, the idea formulating /operation For the productsophisticated structures, making it hard to understand and Knowing the mechanism of the fermentation process assists sophisticated structures, making it hard to understand and inhibitive processes, product selective membrane fermentor model complex bioprocesses in reality. In this paper, the idea formulating design /operation strategies. For the productformulating design /operation strategies. For the productsophisticated structures, making it hard to understand and of “optimal control” is used for reference to study the inhibitive processes, product selective membrane fermentor model complex bioprocesses in In the formulating design /operation strategies. For theto productmodel complex bioprocesses in reality. reality. In this this paper, paper, the idea idea or extractive fermentor are options to be adopted remove of “optimal control” is used for reference to study the inhibitive processes, product selective membrane fermentor inhibitive processes, product selective membrane fermentor model complex bioprocesses inthe reality. In this paper, thefrom idea fermentation processes, is originated fermentor are options to be adopted to remove of “optimal control” iswhere used foranalogy reference to study study the or inhibitive processes, membrane fermentor of “optimal control” is used for reference to the theextractive inhibitive factor inproduct time, byselective so doing, the feedback loop is fermentation processes, where the analogy is originated from or extractive fermentor are options to be adopted to remove or extractive fermentor are options to be adopted to remove of “optimal control” is used for reference to study the two aspects of concern. Firstly, the mechanism for many the inhibitive factor in time, by so doing, the feedback loop is fermentation processes, where the analogy is originated from or extractive fermentor are options to be adopted to remove fermentation processes, where the analogy is originated from opened and the oscillation is eliminated. For the substratetwo aspects processes, of concern.where Firstly, thecontrol mechanism for many the factor in by so feedback loop the inhibitive inhibitive factor in time, time, is byeliminated. so doing, doing, the the feedback loop is is fermentation the analogy is structures, originated from bioprocesses is analogous to typical as is opened and the oscillation For thebesubstratetwo aspects of of concern. Firstly, Firstly, thecontrol mechanism for many many the inhibitive factor in by eliminated. so doing, the feedback loop is two aspects concern. the mechanism for inhibitive processes, thetime, fed-batch operation could adopted bioprocesses is analogous to typical structures, as is opened and the oscillation is For the substrateopened and the oscillation is eliminated. For the substratetwo aspects of concern. Firstly, the mechanism for many shown in Fig. 1; secondly, by natural adaption and evolution, inhibitive processes, the fed-batch operation could be adopted bioprocesses is1;analogous analogous to typical control structures, as is is opened and the oscillation is eliminated. thebesubstratewhen the purpose isthe to fed-batch alleviate substrateFor inhibition of the bioprocesses is to typical structures, as shown in Fig.always secondly, byoptimally, natural control adaption and evolution, inhibitive processes, operation could adopted inhibitive processes, fed-batch operation could be adopted bioprocesses is1;analogous typical control as of is when behaveto hence,structures, knowledge theOversupply purpose isthe to alleviate substrate inhibition ofmore the shown in by natural and evolution, inhibitive processes, the fed-batch operation could be adopted process. of a carbon source results in shown in Fig. Fig.always 1; secondly, secondly, byoptimally, natural adaption adaption and evolution, bioprocesses behave hence, knowledge of when the theOversupply purpose is is to to alleviate substrate inhibition ofmore the when purpose alleviate substrate inhibition of the shown in Fig.always 1; secondly, byoptimally, natural adaption and evolution, optimization may provide valuable information for the process. of a carbon source results in bioprocesses behave hence, knowledge of when theOversupply purpose and is to alleviate substrate inhibition ofwhile the bioprocesses always behave optimally, hence, knowledge of biomass growth lower penicillin formation; optimization may provide valuable information for the process. of a carbon source results in more process. Oversupply of a carbon source results in more bioprocesses always behave optimally, hence, knowledge of processes to be studied. growth lower penicillin formation; optimization may provide process. Oversupply a carbon source results in while more optimization may provide valuable valuable information information for for the the biomass undersupply resultsand in of slower biomass growth, and eventually, processes to be studied. biomass growth and lower penicillin formation; while growth lowerbiomass penicillin formation; while optimization may provide valuable information for the biomass undersupply resultsand in slower growth, and eventually, processes biomass growth and lower Therefore, penicillin formation; while processes to to be be studied. studied. slower penicillin formation. an optimal input undersupply results in growth, and eventually, undersupply resultsformation. in slower slower biomass biomass growth, and eventually, processes to be studied. slower penicillin Therefore, an optimal input undersupply results in slower biomass growth, and eventually, profile is existed for the fed-batch operation. slower penicillin formation. Therefore, an optimal slower ispenicillin Therefore, optimal input input profile existed forformation. the fed-batch operation.an slower penicillin formation. Therefore, an optimal input profile is existed for the fed-batch operation. profile is existed for the is fed-batch operation. A fed-batch operation a semi-batch operation, in which profile is existed for the is fed-batch operation. A fed-batch operation a semi-batch operation, in during which substrate is added eitherisintermittently continuously A fed-batch fed-batch operation semi-batchor operation, in during which in which A operation isintermittently aa semi-batch operation, substrate is added either orharvested continuously A fed-batch operation is a semi-batch operation, in which the operation course and the product is only at the substrate is added added either intermittently orharvested continuously during substrate is either intermittently or continuously during the operation course and the product is only at the substrate is added either intermittently or continuously during end of the run. Analytically, the optimal feeding profile is the operation operation course and the the product product is harvested harvested only at the the the course and is only at end of thebyrun. Analytically, the optimal feeding profile is the operation course and the product is harvested only at the obtained formulating the Hamiltonian, and then end of of the the run. run. Analytically, the optimal feeding feeding profile is end Analytically, the optimal profile is formulating thethe Hamiltonian, and then But the end of theby run. Analytically, the optimal feeding profile is Fig. 1. Feedforward/feedback mechanism of the inhibitive obtained maximum principle provides optimization criterion. obtained by formulating the Hamiltonian, and then the obtained by formulating the Hamiltonian, and then the Fig. 1. inFeedforward/feedback mechanism of the inhibitive maximum principle provides theHamiltonian, optimization criterion. But by formulating the factors the fermentation processes. the presence of lower andthe upper bounds forand the then substrate Fig. 1. inFeedforward/feedback Feedforward/feedback mechanism of of the the inhibitive inhibitive obtained maximum principle provides the optimization optimization criterion. But Fig. 1. mechanism maximum principle provides the criterion. But factors the fermentation processes. the presence of lower and upper bounds for the substrate Fig. 1. Feedforward/feedback mechanism of the inhibitive maximum principle provides the optimization criterion. But stream causes the control to be bang-singular-bang structure factors in in the the fermentation processes. the presence presence of lower and upper bounds for for the the substrate substrate factors fermentation processes. the of lower and upper bounds Oscillatory patterns are typical in nature but the evolutionary stream causestoof the control to upper be bang-singular-bang structure factors in the fermentation processes. the presence lower bounds for the substrate and difficult deal withand mathematically (Modak et al., 1986). Oscillatory patterns are typical in remain nature but the evolutionary stream causes the control to be bang-singular-bang structure stream causes the control to be bang-singular-bang structure origins of oscillatory phenomena a mystery (Tu et al., and difficult to deal with mathematically (Modak et al., 1986). Oscillatory patterns are typical in nature but the evolutionary stream causes the control to be bang-singular-bang structure Oscillatory patterns are typical in remain nature abut the evolutionary While numerically, by mathematically orthogonal collocation on finite origins of oscillatory phenomena mystery (Tu et al., and difficult to deal with (Modak et al., 1986). Oscillatory are typical nature but the evolutionary and difficult to deal with mathematically (Modak et on al., 1986). 2006). It oscillatory ispatterns well known that in Hopf bifurcation causes the While numerically, by orthogonal collocation finite origins of phenomena remain mystery (Tu et et the al., and origins of oscillatory phenomena remain aa mystery (Tu al., difficult to deal with (Modak al., 1986). elements (Cuthrell andmathematically Biegler, 1989), theet on dynamic 2006). It is well known that Hopf bifurcation causes While numerically, by orthogonal collocation finite origins of oscillatory phenomena remain a mystery (Tu et al., While numerically, by orthogonal collocation on finite system to generate self-oscillatory patterns. Some elements (Cuthrell and Biegler, 1989), the dynamic 2006). It It to is well well known self-oscillatory that Hopf Hopf bifurcation bifurcation causes the While 2006). is known that causes the numerically, orthogonal collocation finite optimization problemby can be converted to the a on nonlinear system generate patterns. Some elements (Cuthrell and Biegler, 1989), dynamic 2006). is processes, well known that Hopfproduction bifurcation the optimization elements (Cuthrell and Biegler, 1989), the dynamic fermentation e.g., ethanol bycauses Z. mobilis, problemand can Biegler, be converted a nonlinear system It to to generate self-oscillatory patterns. Some elements (Cuthrell 1989), to dynamic system generate self-oscillatory patterns. Some fermentation processes, e.g., ethanol production by Z. mobilis, optimization problem can can be be converted converted to the a nonlinear nonlinear system to generate self-oscillatory patterns. Some optimization problem to a may behave oscillatory state trajectories even if all the inputs fermentation processes, state e.g., ethanol ethanol production by Z. mobilis, fermentation processes, e.g., production may behave oscillatory trajectories even if by all Z. themobilis, inputs optimization problem can be converted to a nonlinear fermentation processes, e.g., ethanol production by Z. mobilis, may may behave behave oscillatory oscillatory state state trajectories trajectories even even if if all all the the inputs inputs may behave oscillatory state trajectories evenofifAutomatic all the inputs 2405-8963 © IFAC (International Federation Control) Copyright © 2018, 2018 IFAC 130Hosting by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic 2018 responsibility IFAC 130Control. Copyright Copyright © 130 10.1016/j.ifacol.2018.11.367 Copyright © 2018 2018 IFAC IFAC 130 Copyright © 2018 IFAC 130
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131
programming (NLP) problem, which is especially prominent for dealing with singular control problem (San, 1988).
are [0, 40], [0, +∞], [0, 100] and [0, 10], respectively; and the initial values are [1.5, 0, 0, 7].
This work is focused on identifying the feedforward structure for the emergence of input periodicity at the optimal situation, just the opposite to output periodicity as the Z. mobilis fermentation process has indicated. The optimal periodic control (OPC) problem has been studied by many researchers (Speyer et al., 1984), but the structural origins that causes the optimal input profile periodic has not been discussed before. Obviously, the presence of lower and upper bounds for the input is one of the key reasons that lead the drug delivery system (Varigonda et al., 2008) to be OPC, but the penicillin fermentation model is affine, which directly leads the optimization to be bang-bang nature. While, input periodicity may emerge if some optimal conditions are loosen for the penicillin fermentation case. Through this case study, the criterion for the emergency of OPC is built, and possibly, analytical method to compute out the OPC problem is proposed.
2.2 Analysis on the optimal fed-batch operation
2. PROCESS MODEL AND BANG-SINGULAR-BANG CONTROL PRELIMINARY
H λT a(x) bU
(3)
where the adjoint vector, λT, is defined by the follows, λ ' H / x axT λ
(4)
(5)
where g(tf) = -P(tf)V(tf) is the objective to be minimized. Minimization of the Hamiltonian is considerably simplified when U(t) appears linearly in it. If the coefficient of U, λTb, is positive, one can pick the input control to the lower boundary; while, if λTb is negative, on can choose the upper boundary. Up to now, it is the bang-bang control problem. However, the situation of singular control may appear when λTb is identically zero over a finite time interval, then the minimum principle fails to give U(t) during the interval. Therefore, the optimal fed-batch penicillin fermentation process is a bangsingular-bang control problem,
P (t f )V (t f ),
s.t : X )U , S FV P )U , S FV
(2)
The optimization problem is to determine the optimal substrate profile, U(t), which is subject to the upper and lower constraints given by Eq. (1). The Pontryagin minimum principle yields an equivalent problem, which is to minimize the Hamiltonian defined as follows,
λi (t f ) g / xi (t f ); i 1, 2,3
u ( t ), t f
P ' ( S ) X K deg P (
x(0) x0
And λ must satisfy the terminal conditions given as follows,
The fermentation model (Modak et al., 1986)) is described by mass balance of the volume (V), concentrations of biomass (X), product (P) and substrate (S). The dynamic optimization problem can be formulated as provided in Eq. (1), where the penicillin production is maximized, and the system subject to constraints of the process model and boundary conditions with respect to the state and manipulate variables.
X ' ( X , S) X (
x ' a(x) bU
T
2.1 The penicillin fermentation model
min
The state vector x = ∑(X, P, S, V) is affine to the control U, one can give the general dynamics of the penicillin process as follows,
(1)
X X ms S S U S ' ( X , S ) X 1 (S ) , YX S YP S K m S S V F U , V' SF
U (t )opt
where S S , , ( S ) max ( S ) 1 K S S K P in KX S
( X , S ) max ( X , S )
X L X (t ) X U , S L S (t ) S U ,V L V (t ) V U ,U L U (t ) U U , t f L t f t f U
where tf is the end-time of the fed-batch process; μ(X, S) is the specific biomass growth rate and ρ(S) is the specific penicillin production rate. μmax = 0.11h-1 is maximum specific biomass growth rate; ρmax = 0.0055h-1 is maximum specific production rate; KX = 0.006g/g is saturation parameter for biomass growth; KP = 0.0001g/l is saturation parameter for production; Kin = 0.1g/l is inhibition parameter for production; Kdeg = 0.01h-1 is the product degradation rate; Km = 0.0001g/l is saturation parameter for maintenance consumption; mS = 0.029h-1 is maintenance consumption rate; Yx/s = 0.47g/g is yield factor for substrate to biomass; YP/S =1.2g/g is the yield factor for substrate to product and SF = 500g/l substrate feeding concentration. Boundary conditions for X, P, S and V
131
U U U (t )singular U L
λT b 0 λT b 0 λT b 0
(6)
The singular substrate feed rate, U(t)singular, can be determined by recognizing that during the singular interval λTb is identically zero, hence its time derivatives must also vanish, which lead to relation as follows, U (t )singular [(a / x)c (c / x)a] / λ T (c / x)b where : c (a / x)b
(7)
In addition, the Hamiltonian on the optimal path is also an unknown constant H*, hence three equations linear to λ(t) are satisfied, λ T a H * T λ b 0 T λ c 0
(8)
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In general, Eq. (8) forms a two-point boundary value problem in which the initial conditions for the state variables and the final conditions for the adjoint variables are known. The desired optimal substrate feeding profile consists of maximum, minimum, and singular values with the singular flow rate being a function of the state and adjoint variables.
conditions of x(t) for each element. Also, the Lagrange polynomial has the desirable property that for xKx(t): xKx (t ) xi j
(12)
which is due to the Lagrange condition ϕk(tik) = δkj,, where δkj is the Kronecker delta. This polynomial form allows the direct bounding of the states and controls, e.g., path constraints can be imposed on the problem formulation. Using K=Kx=Ku point orthogonal collocation on finite elements as shown in Fig. 2, and by defining the basic functions, so that they are normalised over each element ∆ζi, one can write the residual equation as follows,
To compute out the optimal bang-singular-bang problem, one must know the exact sequence of optimal feed rate (maximum, minimum or singular) and solve the two-point boundary value problem posed by Eqs. (2) and (4)-(7). The computational algorithms are provided (Lim et al., 1986), while, a more universally applicable method is to convert the optimal control problem by discretization and solve the NLP problem numerically (Zhai et al., 2017).
Kx
i r (ti k ) xi j j ( k ) i f (xi k , ui k , ti k )
3. NUMERICAL OPTIMAL INPUT PROFILE
(13)
j 1
3.1 Problem formulation
where i = 1, 2…, NE, j = 1, 2…, ϕj(τk) = dtϕj/dτk, and together with ϕj(τ), θj(τ), terms (basis functions), they are calculated beforehand, since they depend only on the Legendre root locations. Note that tik = ζi+ζiτk. This form is convenient to work with when the element lengths are included as decision variables. The element lengths are also used to find possible points of discontinuity for the control profiles and to ensure that the integration accuracy is within a numerical tolerance. Additionally, the continuity of the states is enforced at element (ζi, i = 2, ..., NE), but it is allowed that the control profiles to have discontinuities at these endpoints as follows,
In order to derive the NLP problem, the differential (integral, algebraic) equations are converted into algebraic equations using orthogonal collocation on finite elements, and the residual equations are then formed as a set of algebraic equations. For optimal residuals, the roots of the shifted Legendre polynomial are set as the collocation points, and the Lagrange polynomial is used to approximate the differentiation and integration terms. The mathematical development is as follows: Consider the initial-value problem over a finite element i with time t∈(ζi, ζi+1), and the dynamic system is as follows,
xiK x ( i ) xiKx1 ( i ) i 2,..., NE Kx
x ' f (x, u, t )
xi 0 xi 1, j j ( 1) i 2,..., NE; j 2,..., K x
(9)
(14)
j 0
2
xL
1
i
xi ,1 , ui ,1 xi ,2 , ui ,2
x
i xi ,0
x
These equations extrapolate the polynomial xiK1 (t ) to the endpoints of its element and provide accurate initial conditions for the next element and polynomial xiK (t ) . The bounds can be done in similar way.
i
i 1
NE
xU
x
Ku
uKi u ( i ) ui , j j ( 0); i 2,..., NE; j 2,..., K u
i 1
j 1
xi1,0
Ku
uKi u ( i 1 ) ui , j j ( 1); i 2,..., NE; j 2,..., K u (15) j 1
Fig. 2. Collocation on finite elements for state profiles, control profiles and element lengths (Kx, Ku = 2).
Note that the constraints on continuity of the states are out of practical concerns, since the states x = ∑(X, P, S, V) being discontinuous will be unexplainable in the real world; while designing the endpoint of each control element to be discontinuous is practical, and those points with abrupt change of control action is actually the singular control.Then a set of nonlinear equations are generated, which can be solved by NLP solvers, such as sequential quadratic programming (Zhai et al., 2015) or SNOPT.
The solution is approximated by Lagrange polynomials over elementi (ζi ≤ t ≤ ζi+1) as follows, Kx
xK x (t ) xi j j (t ); j (t ) j 0
Ku
uKu (t ) ui j j (t ); j (t ) j 1
Kx
(t ti k ) i j ti k )
(10)
(t ti k ) i j ti k )
(11)
(t
k 0, j
Ku
(t
k 1, j
3.2 Results
where i =1, 2…, NE. k = 0 to j meaning that k starts form 0 and k ≠ j, NE is the number of elements. Also, xKx(t) is a (Kx+1)th order degree piecewise polynomial and uKu(t) is piecewise polynomial of order Ku. The polynomial approximating the state x takes into account the initial
The fed-batch penicillin process is of Mayer form (no integral terms in the objective function), Lagrange multipliers are applied to deal with the equality constraints, but before which, the inequality constraints are relaxed by adding extra dimensions. The Kuhn-Tucher conditions are used to derive 132
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35
the expression of the multipliers. Now, the system becomes a standard NLP problem. The search for the optimal of a nonlinear system is difficult, but one can seek the less attractive alternative by searching the optimal of the locally quadratic approximation, where Hessian matrix leads directly to the optimal of the approximated system, and the optimal of the original system can be obtained by this sequential quadratic searching procedure.
a
Biomass concerntration (g/l)
30
An analytically-based optimal control profile was obtained by the method stated in section 2.2. This solution derives from the variational conditions, but also requires repeated numerical solution to solve the two-point boundary value problem. The values for the points of control profile discontinuity, final time and the optimal value of the objective function are 11.21, 28.79 and 124.9 and -87.05, respectively, and the control profile is presented in Fig. 3.
20
15
10
0
0
20
40
60
80
100
120 125
80
100
120 125
80
100
120 125
80
100
120 125
Time (h) 9
b 8
Product concentration (g/l)
7
45
Substrate feeding profile (g/h)
25
5
50
Obj: -87.05
40
133
35 30 25
6 5 4 3 2
20
1
15 10
0
0
20
40
0
60
Time (h)
5 0
11.21
20
28.79
40
60
80
100
30
120 125
c
Time (h) 25
Substrate concentration (g/l)
Fig. 3. Analytical-based optimal substrate input profile. Set the analytically-based case above as a typical case for the subsequent numerical investigation, where tf = 124.9h-1. Choose Kx = Ku = 3, NE = 10, both the tolerance of the objective function and the constraints as e-5, and the total tolerance as 5e-5. The control profile of Fig. 4 is different from that of Fig. 3 because this system is not feedback controllable. From section 2.2 we know that, to compute out the optimal control profile, the sequence of maximum, singular and minimum need to decide beforehand, which means the dynamic reverse is underdetermined, and the mapping of optimal output to the control input has multiple routes. Numerically, the choice of initial control profile, collocation points and number of elements affects the profiler of the control.
15
10
5
0
-5
0
20
40
60
Time (h) 10
d 9.5
Volume (l)
9
55 50 45
Substrate feeding profile (g/h)
20
8.5
8
40 7.5
35 30
7
25
0
20
40
60
Time (h)
20
Fig. 5. Numerical-based state profiles.
15 10 5
0
20
40
60
80
100
120 125
Time (h)
Fig. 4. Numerical-based optimal substrate input profile. 133
The corresponding state profile for x = ∑(X, P, S, V) are presented in Fig.5 (a-d). it is obvious that the inflexion point at t* = 29.8h separates the fed-batch operation into two folds: before t*, the fermentor adjust itself with varying substrate input rates, where biomass is accumulated in an optimal
IFAC CAO 2018 134 Yekaterinburg, Russia, October 15-19, 2018 Chi Zhai et al. / IFAC PapersOnLine 51-32 (2018) 130–135
5.4
fashion and no product is harvested; after t*, the substrate is consumed mainly to produce penicillin product and the substrate input are maintained at a (almost) steady level.
a
Substrate feeding profile (g/h)
Assume the product penicillin do not affect the metabolism, when the fermentor volume is considerable large, the volume gain by substrate input is insignificant during the time interval [t*, tf], then, one can speculate that the state change of the key factors, (X, S), are held constant for the well-mixed fermentor. To this end, the assumed fed-batch fermentator is almost identical to the continuous CSTR, except that V and P are accumulated with time.
5.2 5.1 5 4.9 4.8 4.7
One can find that the above assumption is not exactly fit to the numerical results as is shown in Fig. 5 (a), where the biomass is not constant during interval [t*, tf]. While the slight decrease of X is due to the dilution effect caused by constant input of substrate stream, meaning that the assumption of V approaches infinite is not proper here. Still, this assumption and speculation procedure is important because the following conclusion can be drawn,
4.6
0
20
40
60
80
100
120
Time (h) 20
b Substrate feeding profile (g/h)
dP C; t [t*, t f ] dt
Obj: -36.5 Xmax: 16.08
5.3
(16)
Where C is the constant, meaning that constant production of penicillin is maintained during the time interval.
Obj: -82.2 Xmax: 31.63
18
16
14
12
10
4. THE EMERGENCE OF FEEDFORWARD PERIODICITY
8
0
20
40
60
80
100
120
Time (h)
It is obvious that the derivative of the objective function is critical for the NLP solving strategies, because it decides the optimal searching direction. While deliberately omitting this condition leads to locally suboptimal input profiles as shown in Fig. 6. In general, the substrate feeding profiles in Fig.6 also can be separated into tow folds by t*, and after dynamic transient during the interval [0, t*), the “normal” working state is built, but unlike Fig. 3 and 4, periodic inputs profiles emerge around the initial input profile U0(t).
35
Substrate feeding profile (g/h)
c
As to the emergence of input periodicity, one may speculate that it is caused by Hopf bifurcation: when the input-output relation of fed-bath fermentation process is reversed, for example, if the output is known, which in fact is the optimal penicillin productivity, and then viewing Fig. 1 from the right hand side to the left hand side, feedforward periodicity is similar to the mechanism of feedback periodicity. Since Hopf bifurcation is detected for the self-oscillatory Z. mobilis fermentation model, one may expect Hopf points are also detected of the reversed fed-batch fermentation process.
Obj: -80.21 Xmax: 33.33
30
25
20
15
10
5
0
20
40
60
80
100
120
Time (h) 40
d
Obj: -73.8 Xmax: 37.47
Substrate feeding profile (g/l)
35
If the optimal condition is loosening to the suboptimal situation as our current case indicates, one will find that it is in fact the OPC problem (the local, suboptimal situation can be converted to exact optimal problem by adding constraints on the optimization problem, e.g., forcing the input boundary to (4.6, 5.4) for the case shown in Fig. 6-a). Under dynamic reverse perspectives, the optimal problem may provide a new way to solve the OPC problem: the critical condition for the emergence of the OPC is actually the Hopf point, and compute out the period of the limit cycle (Zhai et al., 2017b) is relatively easy compared to the current numerical OPC strategies (Varigonda et al., 2008).
30 25 20 15 10 5 0
0
20
40
60
80
100
120
Time (h)
Fig. 6. Numerical suboptimal substrate input profile, where periodic inputs emerge. a: U0(t) = 5g/h; b: U0(t) = 12g/h; c: U0(t) = 25g/h; d: U0(t) = 40g/h. 134
IFAC CAO 2018 Yekaterinburg, Russia, October 15-19, 2018 Chi Zhai et al. / IFAC PapersOnLine 51-32 (2018) 130–135
The time-invariant output for the fed-batch process is the constant production rate exhibit in Eq. (16), when the assumption in section 3.2 is satisfied. The difference of Fig. 4 and Fig. 6 may be caused by the state variable (X, S), by substituting the condition of Eq. (16) into the dynamic system in Eq. (1), one can obtain the ordinary differential equations as follows, ( S ) X K deg P * C X ' ( X , S ) X P*/X S ' ( X , S ) X ( S ) X ms S X ( S ) X K deg P * C YX S YP S K m S P * / S F S
135
of biological positive-feedback systems. PNAS., 101(7), 1822-1827. Cuthrell, J.E., Biegler, L.T., (1989). Simultaneous optimization and solution methods for batch reactor control profiles. Comput. Chem. Eng., 13, 49-52. Gao, N., Zhai, C.,，Sun W., et al., (2015). Equation Oriented Method for Rectisol Wash Modeling and Analysis. Chinese Journal of Chemical Engineering, 23(9), 15301535.
(17)
Jobses, M.L., Egberts, G.T.C., Luyben, K.C.A.M., Roels, J.A., (1986). Fermentation Kinetics of Zymomonas mobilis at High Ethanol Concentrations: Oscillations in Continuous Cultures. Biotechnology and Bioengineering, 868-877, 1986.
where P* is the nominal product concentration, since Kdeg is very small while V is sufficiently big, changes of P are insignificant. In the following analysis, P at tf is chosen for P*. Numerical bifurcation analysis (Zhai et al., 2017b) is implemented on the system represented in Eq. (17), and the range of C being tested can be determined by the information from Fig. 5-b. however, no Hopf points are detected, which means that the periodicity is not caused by the dynamic system itself.
Modak, J.M., Lim H.C., Tayeb Y.J., (1986). General characteristics of optimal feed rate profiles for various fed-batch fermentation processes, Biotechnology and bioengineering, 28(9): 1396-1407. Prigogine, I., (1978). Time, structure, and fluctuations. Science, 201(4358), 777-785.
Further study on the NLP algorithm and it can be found that the periodic profiles in Fig. 6 are caused by the cut of the whole time interval as shown in Fig. 2, roughly 10 orbits with identical period is obtained because the elements, NE, is set as 10. Since singular control may generate at the endpoint of each element and similar trajectories are obtained for each elements. However, the analysis in this section is not useless, when the substrate input profile is pre-determined as shown in Fig. 6, the production may not be the optimal, but in some situations, e.g., Fig.6-c, the penicillin concentration is higher than the optimal situation. Moreover, the affine system represented in Eq. (2) may cause the optimal control to be bang-singular-bang structure, but no optimal periodic control can be obtained analytically.
San, K.Y., Stephanopoulos, G., (1989). Optimization of fedbatch penicillin fermentation: A case of singular optimal control with state constraints, Biotechnology and bioengineering, 34(1), 72-78. Speyer, J., Evans, R., (1984). A second variational theory for optimal periodic processes. IEEE Transactions on Automatic Control, 29(2), 138-148. Tu, B.P., McKnight, S.L., (2006). Metabolic cycles as an underlying basis of biological oscillations. Nature Reviews Molecular Cell Biology, 7, 696–701. Turing, A.M., (1952). The Chemical Basis of Morphogenesis. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237(641), 37-72.
5. CONCLUSIONS
Varigonda, S., Georgiou, T.T., Siegel, R.A., Daoutidis, P., (2008). Optimal periodic control of a drug delivery system. Computers & Chemical engineering, 32(10), 2256-2262.
In this paper, the fed-batch penicillin process is investigated for its bang-singular-bang control structure and numerical solution by infinite elements method is detailed. It turned out that the feedforward periodicity is caused by the numerical computing strategies, but the methods to study OPC by dynamic inverse are constructed, which may be of a reference for other systems.
Zhai, C., Sun W., et al., (2017a). Dynamic optimization of penicillin fermentation with constraint on wastewater discharge. Chemical Engineering Transactions, 61, 499504. Zhai, C., Palazoglu, A., Wang, S., Sun, W., (2017b). Strategies for the Analysis of Continuous Bioethanol Fermentation under Periodical Forcing. Ind. Eng. Chem. Res., 56(14), 3958-3968.
ACKNOWLEDGEMENTS The authors gratefully acknowledge the following institutions for support: the National Natural Science Foundation of China (Grant No. 21576015); the National Basic Research Program of China, 973 program (Grant No. 2013CB733600).
REFERENCES Angeli, D., Ferrell, J.E., Sontag, E.D., (2004). Detection of multistability, bifurcations, and hysteresis in a large class 135