Fractional-order time-delay optimal control method for microbial intermittent fermentation process

Abstract

This invention discloses a fractional-order time-delay optimal control method for microbial batch fermentation processes, comprising the following steps: Step 1: Establishing a fractional-order time-delay system model for the batch fermentation process; Step 2: Establishing a fractional-order time-delay optimal control model; Step 3: Designing an optimal control method for the fractional-order time-delay optimal control model established in Step 2, thereby achieving optimal control of microbial batch fermentation production. This invention can provide a more accurate mathematical model for microbial fermentation processes and provide an optimal control strategy for improving the production efficiency of microbial batch fermentation processes. Therefore, it will significantly reduce the cost of actual production processes and improve their economic benefits.

Description

Technical Field

[0001] This invention relates to the field of microbial intermittent fermentation technology, specifically to a fractional-order time-delay optimal control method for microbial intermittent fermentation processes. Background Technology

[0002] Microbial fermentation refers to the process by which microorganisms, under suitable conditions, transform raw materials into products needed by humans through specific metabolic pathways. Microbial fermentation has many advantages, including mild reaction conditions, simple operation, and no pollution. Currently, it has been widely applied in the pharmaceutical, food, energy, and chemical industries. The microbial fermentation process is highly complex due to its nonlinearity, fractional order (microbial strains possess memory and genetic characteristics), and time lag (microbial strains require a certain growth time to react). Therefore, studying the optimal control of the fermentation process is crucial for efficient production. In summary, this invention not only enriches and promotes research on related theories and algorithms in operations research and cybernetics but also provides optimal control strategies for industrial microbial fermentation production, bringing considerable economic benefits.

[0003] Although significant progress has been made in both the theoretical and algorithmic aspects of integer-order time-delay optimal control and fractional-order optimal control problems, there is an urgent need to study a class of fractional-order time-delay optimal control problems in microbial fermentation production to improve the economic efficiency of the fermentation process. The simultaneous existence of fractional order and time delay makes the theoretical research and algorithm design of fractional-order time-delay optimal control problems extremely difficult. Currently, research on fractional-order time-delay optimal control problems is still in its early stages.

[0004] Secondly, the main drawback of current microbial batch fermentation production is the low production efficiency of the target product. Therefore, improving the production efficiency of the target product during batch fermentation is key to achieving industrial-scale production. While it's possible to improve the process through experimentation to increase efficiency, relying solely on experimental methods is extremely costly. Summary of the Invention

[0005] To address the nonlinearity, fractional order, and time delay characteristics of microbial fermentation, a fractional-order time-delay system model and an optimal control model for the microbial fermentation process are established. An optimal control method for microbial fermentation production is designed to improve economic efficiency.

[0006] The technical solution adopted in this invention is as follows:

[0007] A fractional-order time-delay optimal control method for microbial intermittent fermentation processes includes the following steps:

[0008] Step 1: Establish a fractional-order time-delay system model for the batch fermentation process;

[0009] Step 2: Establish a fractional-order time-delay optimal control model;

[0010] Step 3: Design the optimal control method for the fractional time delay optimal control model established in Step 2, thereby achieving optimal control of microbial intermittent fermentation production.

[0011] Preferably, in step 1, a fractional-order time-delay system model of the batch fermentation process is established, specifically:

[0012] 1.1 For microbial fermentation strains, the following form is established based on the fermentation process mechanism:

[0013]

[0014] Where t0 is the start time of the intermittent fermentation process; x1(t) is the concentration of the microbial strain; μ is the specific growth rate of the cells; and τ is the process time delay. It is the Caputo differential operator; It is the first derivative of y(t); The dilution rate of the process is denoted as u(t); the substrate injection rate is denoted as u(t); V0 is the initial volume of the fermentation broth; and r is a proportionality constant.

[0015] 1.2 For substrates of microbial fermentation, the following formal model is established:

[0016]

[0017] Where x2(t) is the substrate concentration; C s0 q1 represents the initial injection concentration of the substrate; q2 is the substrate consumption rate.

[0018] 1.3 For the target product of microbial fermentation, the following fractional time-delay differential equation model is established:

[0019]

[0020] Where x3(t) is the concentration of the target product; q3 is the product specific formation rate;

[0021] 1.4 During the intermittent fermentation process of microorganisms, the historical strain concentration, historical substrate concentration, and historical target product concentration are φ1(t), φ2(t), and φ3(t), respectively.

[0022] Preferably, in step 2, a fractional-order time-delay optimal control model is established, specifically:

[0023] 2.1 The substrate injection rate u(t) is selected as the control function, and the upper and lower bounds of the control function are given by u. min and u max ;

[0024] 2.2 Selecting the Control Target

[0025] J = Ψ(x(t) f Among them, t f It is the end point of the fermentation process; x(t) f ) is the state value at the terminal moment, that is, the concentration values ​​of the strain, substrate and target product at the terminal moment;

[0026] 2.3 Throughout the entire batch fermentation process [t0, t] f In this paper, the following fractional-order time-delay optimal control model is established:

[0027] min J=Ψ(x(t f ))

[0028]

[0029]

[0030]

[0031]

[0032] u(t)∈[u min u max ], t∈[t0, t f ]

[0033] Preferably, in step 3, an optimal control method is designed for the fractional-order time-delay optimal control problem, specifically:

[0034] 3.1 The fractional-order time-delay model corresponding to the strain, substrate, and target product is transformed into the following integral form:

[0035]

[0036] Among them, f1(x(s), x(s-τ), u(s))=μx1(s-τ)-Dx1(s);

[0037]

[0038] f3(x(s),x(s-τ),u(s))=q3x1(s-τ)-Dx3(s);

[0039] 3.2 The entire time process [t0, t] f Insert N grid points to satisfy

[0040] t k =kh, k=0,1,K,N

[0041] in

[0042] 3.3 At grid point t k Place

[0043]

[0044] 3.4 Calculate x i (t k Use the following formula

[0045]

[0046] in, d kl = (k-1);

[0047]

[0048] 3.5 Calculation and Use the following linear interpolation method:

[0049]

[0050]

[0051]

[0052] in,

[0053] 3.6 Using Taylor expansion, the calculation of x(t) is given. k Explicit iterative methods:

[0054] B k x(t k ) = C k

[0055] in,

[0056]

[0057]

[0058] 3.7 The optimal control problem of fractional time delay in microbial intermittent fermentation is transformed into the following parameter optimization problem:

[0059] min J=Ψ(x(t N ))

[0060] stB k x(t k ) = C k k = 1, 2, K, N

[0061] x i (t)=φ i (t), t≤t0,

[0062] u(t k )∈[u min u max ], k = 1, 2, K, N

[0063] 3.8 For explicit iterative methods with respect to the control variable u(t) k The gradient is calculated as follows:

[0064]

[0065] in,

[0066] 3.9 When q = 0, if k = 1, then

[0067]

[0068]

[0069] 3.10 If k > 1, then

[0070]

[0071]

[0072] 3.11 When q = 1, 2, KN, if q > k, then

[0073]

[0074] 3.12 When q = k, we have

[0075]

[0076]

[0077] 3.13 When q = k-1, we have

[0078]

[0079]

[0080] 3.14 When q < k-1, we have

[0081]

[0082]

[0083] 3.15 From the gradient derived above, the gradient of the objective function of the parameter optimization problem with respect to the control can be calculated as follows:

[0084]

[0085] 3.16 The gradient of the objective function with respect to control is embedded into a parameter optimization software package, such as FMINCON in MATLAB, to solve for the optimal control strategy of the batch fermentation process.

[0086] Compared with the prior art, the beneficial effects of the present invention are:

[0087] This invention fully explores the fractional-order and time-delay characteristics of biological systems, establishes an optimal control model for microbial intermittent fermentation processes based on fractional-order time-delay differential systems, and invents a numerical method for solving the optimal control of fractional-order time-delay processes. This invention can provide a more accurate mathematical model for microbial fermentation processes and offer optimal control strategies to improve the production efficiency of intermittent microbial fermentation processes. Therefore, it will significantly reduce the cost of actual production processes and improve their economic benefits. Detailed Implementation

[0088] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below. Obviously, the described embodiments are some embodiments of the present invention, but not all embodiments.

[0089] Therefore, the detailed description of the embodiments of the present invention provided below is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0090] This example uses intermittent microbial fermentation to produce 1,3-propanediol:

[0091] Step 1 establishes a fractional-order time-delay system model for the batch fermentation process of 1,3-propanediol, specifically:

[0092] 1.1 For microbial fermentation strains, the following form is established based on the fermentation process mechanism:

[0093]

[0094] Where t0 is the start time of the intermittent fermentation process; x1(t) is the concentration of the microbial strain; Δ1 is the specific growth rate of the cell; k1 is the maximum specific growth rate; τ is the Monod saturation constant; and τ is the process time delay. It is the Caputo differential operator; It is the first derivative of y(t); The dilution rate of the process is denoted as u(t); the substrate injection rate is denoted as u(t); V0 is the initial volume of the fermentation broth; and r is a proportionality constant.

[0095] 1.2 For glycerol, the substrate for microbial fermentation, the following formal model is established:

[0096]

[0097] Where x2(t) is the substrate concentration; C s0 The initial injection concentration of the substrate; Δ2 is the substrate consumption rate; m2 is the maintenance period of substrate consumption under substrate-limited conditions; Y2 is the maximum growth rate; Δ2 is the maximum increment of substrate consumption rate under substrate-sufficient conditions; k2 is the substrate saturation constant. The dilution rate of the process is denoted by ; u(t) is the injection rate of glycerol; V0 is the initial volume of the fermentation broth; and r is a proportionality constant.

[0098] 1.3 For the target product 1,3-propanediol from microbial fermentation, the following fractional-order time-delay differential equation model is established:

[0099]

[0100] Where x3(t) is the concentration of the target product 1,3-propanediol; Δ is the product specific generation rate; m3 is the maintenance period of product consumption under substrate-limited conditions; Y3 is the maximum product growth rate; Δ3 is the maximum increment of product consumption rate under substrate-sufficient conditions; k3 is the product saturation constant.

[0101] 1.4 During the intermittent fermentation process of microorganisms, the historical strain concentration, historical glycerol concentration, and historical 1,3-propanediol concentration are φ1(t), φ2(t), and φ3(t).

[0102] Step 2 establishes a fractional-order time-delay optimal control model, specifically:

[0103] 2.1 The injection rate of the substrate glycerol, u(t), is selected as the control function. The upper and lower bounds of the control function are given by u. min and u max ;

[0104] 2.2 Selecting the Control Target

[0105] J = Ψ(x(t) f ))=-x3(t f )

[0106] Among them, t f It is the end point of the fermentation process; x(t)f The concentration of 1,3-propanediol at the terminal time is the concentration of the strain, substrate glycerol, and 1,3-propanediol at the terminal time.

[0107] 2.3 Throughout the entire batch fermentation process [t0, t] f In this paper, the following fractional-order time-delay optimal control model is established:

[0108] min J=Ψ(x(t f ))

[0109]

[0110]

[0111]

[0112]

[0113] u(t)∈[u min u max ], t∈[t0, t f ]

[0114] Step 3 involves designing the optimal control method for the fractional-order time-delay optimal control problem, specifically:

[0115] 3.1 The fractional-order time-delay model corresponding to the bacterial strain, substrate glycerol, and 1,3-propanediol is transformed into the following integral form:

[0116]

[0117] Among them, f1(x(s), x(s-τ), u(s))=μx1(s-τ)-Dx1(s);

[0118]

[0119] f3(x(s),x(s-τ),u(s))=q3x1(s-τ)-Dx3(s);

[0120] 3.2 The entire time process [t0, t] f Insert N grid points to satisfy

[0121] t k =kh, k=0,1,K,N

[0122] in

[0123] 3.3 At grid point t k Place

[0124]

[0125] 3.4 Calculate x i (t k Use the following formula

[0126]

[0127] in, d kl =(kl);

[0128]

[0129] 3.5 Calculation and Use the following linear interpolation method:

[0130]

[0131]

[0132]

[0133] in,

[0134] 3.6 Using Taylor expansion, the calculation of x(t) is given. k Explicit iterative methods:

[0135] B k x(t k ) = C k

[0136] in,

[0137]

[0138]

[0139] 3.7 The fractional-order time-delay optimal control problem for the production of 1,3-propanediol by microbial batch fermentation is transformed into the following parameter optimization problem:

[0140] min J=Ψ(x(t N ))

[0141] stB k x(t k ) = C k k = 1, 2, K, N

[0142] x i (t)=φ i (t), t≤t0,

[0143] u(tk )∈[u min u max ], k = 1, 2, K, N

[0144] 3.8 For explicit iterative methods with respect to the control variable u(t) k The gradient is calculated as follows:

[0145]

[0146] in,

[0147]

[0148] 3.9 When q = 0, if k = 1, then

[0149]

[0150]

[0151] 3.10 If k > 1, then

[0152]

[0153]

[0154] 3.11 When q = 1, 2, KN, if q > k, then

[0155]

[0156] 3.12 When q = k, we have

[0157]

[0158]

[0159] 3.13 When q = k-1, we have

[0160]

[0161]

[0162] 3.14 When q < k-1, we have

[0163]

[0164]

[0165] 3.15 From the gradient derived above, the gradient of the objective function of the parameter optimization problem with respect to the control can be calculated as follows:

[0166]

[0167] 3.16 The gradient of the objective function with respect to control is embedded into a parameter optimization software package, such as FMINCON in MATLAB, to solve for the optimal control strategy of the batch fermentation process.

[0168] The above description is merely a preferred embodiment of the present invention and does not constitute any limitation on the present invention. Any simple modifications, alterations, and equivalent changes made to the above embodiments based on the technical essence of the invention shall still fall within the protection scope of the technical solution of the present invention.

Claims

1. A fractional-order time-delay optimal control method for microbial intermittent fermentation processes, characterized in that, Includes the following steps: Step 1: Establish a fractional-order time-delay system model for the batch fermentation process; 1.1 For microbial fermentation strains, the following form is established based on the fermentation process mechanism: Where t0 is the start time of the batch fermentation process; x1(t) is the concentration of the microbial strain; μ is the specific growth rate of the cells; and τ is the process time delay. Denotes the α-th order Caputo fractional derivative of x1(t); The dilution rate of the process; u(t) represents the substrate injection rate; V0 is the initial volume of the fermentation broth; r is a proportionality constant; 1.2 For substrates of microbial fermentation, the following formal model is established: Where x2(t) is the substrate concentration; C s0 q1 represents the initial injection concentration of the substrate; q2 is the substrate consumption rate. 1.3 For the target product of microbial fermentation, the following fractional time-delay differential equation model is established: Where x3(t) is the concentration of the target product; q3 is the product specific formation rate; 1.4 In the process of intermittent microbial fermentation, the historical strain concentration, historical substrate concentration, and historical target product concentration are respectively φ i (t), i = 1, 2, 3; Step 2: Establish a fractional-order time-delay optimal control model; 2.1 The substrate injection rate u(t) is selected as the control function, and the upper and lower bounds of the control function are given by u. min and u max ; 2.2 Selecting the Control Target J=Ψ(x(t f )) Among them, t f It is the end point of the fermentation process; x(t) f ) is the state value at the terminal moment, that is, the concentration values ​​of the strain, substrate and target product at the terminal moment; 2.3 Throughout the entire batch fermentation process [t0, t] f In this paper, the following fractional-order time-delay optimal control model is established: Step 3: Design the optimal control method for the fractional time delay optimal control model established in Step 2, thereby achieving optimal control of microbial intermittent fermentation production.

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