Bulgarian Academy of Sciences Institute of Biophysics and

Bulgarian Academy of Sciences, Institute of Biophysics and Biomedical Engineering PREFERENCES AND DETERMINATION OF THE NOMINAL GROWTH RATE OF FED-BATCH PROCESS: CONTROL DESIGN OF COMPLEX PROCESSES SOFIA Yuri P. Pavlov, Peter Vassilev www. clbme. bas. bg yupavlov@clbme. bas. bg ilywrin@clbme. bas. bg 1

The objective of this investigation is development of comfortable tools and mathematical methodology that are useful for dealing with the uncertainty of human behavior and judgments in complex control problems. This investigation is based on 4 approaches: Equivalent Brunovsky form of the fed-batch model, Pontrjagin’s maximum principle, Sliding mode control, Utility theory and Stochastic programming. 2/20/2021 2

Main purpose ( ЦЕЛ НА ИЗСЛЕДВАНЕТО) Synchronized utilization of the 4 approaches to overcome difficulties arising from the biotechnological peculiarities, non linearity, singular optimal control and determination of good control solutions. 2/20/2021 3

MOTIVATION OF THE INVESTIGATION • Complex control problem -Assessment of human value for determination of the “best” technological conditions; • Non-linearity of the Monod kinetic model; • Singular optimal control; • Non observability of the Monod kinetic models. 4

CONTINUOUS PROCESS: Monod-Wang model I. PECULIARITIES 1. Description of the models; 2. The cultivation processes are different from the classical physical systems; 3. Appearance of non Gaussian noise in the system; 4. Measurements difficulties; 5. Complex systems need complex models. II. MODEL -X -biomass concentration, -S -substrate concentration, - -specific growth rate, -Ks - Mihaelis-Menten constant, -ν -white noise , -So substrate concentration in the feed, -m - coefficient, -D input –dilution coefficient, -μm (T, p. H) maximal value of the specific growth rate (as function of temperature T and the acidity p. H), 2/20/2021 -y -coefficient. 5

FED-BATCH FERMENTATION PROCESS: Monod-Wang data presentation (CLBME, Bulgarian Academy of Sciences, MOBPS) (от модел) 1. Substrate concentration S; 2. Specific growth rate ; MODEL (fed-batch-полупериодичен): • • • 2/20/2021 X -biomass concentration, S -substrate concentration, -specific growth rate, Ks - Mihaelis-Menten constant, ν -white noise , So substrate concentration in the feed, m - coefficient, F ”is the substrate-feed rate”, input , μm (T, p. H) maximal value of the specific growth rate (as function of temperature T and the acidity p. H), y -coefficient. 6

BEST GROWTH RATE: The inclusion of a value expert model as a part and criteria of a dynamical control system can be done with the expected Utility theory. 2/20/2021 • Data modelling 1. Substrate concentration S; 2. Specific growth rate ; • Objective function - U( ): 7

Human Judgment and Assessment of Human Value (Utility) in Complex Systems (Frontiers: Decision making, Subjective Preferences, Value, Expected Utility, Probability theory, Theory of Measurement ) The assessment bases on mathematically formulated axiomatic principles and stochastic procedures. The evaluation is a preferences-oriented machine learning procedure with restriction of the “certainty effect” and ”probability distortion” identified by Kahneman and Tversky (prospects theory). The uncertainty of the human preferences is eliminated as typically for the stochastic programming procedures. This evaluation is based on 3 approaches: Decision making theory, Utility theory of von Neuman and Potential function method of Aizerman, Braverman and Rozonoer. 8

EXPECTED UTILITY AND VALUE FUNCTION (mathematical definitions) (p q , (p, q) P 2 ) ( u(. )dp u(. )dq), p P, q P. Let X is the set of alternatives (X Rm). According to von Neumann & Morgenstern this formula means that the mathematical expectation of the expert utility function u(. ) is a measure for the expert preferences. These preferences are defined over the set of the probability distributions P ( P is defined over the set of the alternatives X). With is denote the preference relation over P (X P). The indifference relation ( ) is defined as: (x y) x y . A “value” function is a function u*(. ) for which is fulfilled: (x, y) X 2 , x y u*(x)>u*(y). Von Neumann and Morgenstern’s axioms: • I. The preference relation ( ) is negatively transitive and asymmetric one (weak order); • II. (Q P, 0< <1) (( P+(1 - )R) (( Q+(1 - )R)) (independence axiom); • III. (P Q, Q R) (( P+(1 - )R) Q), for , (0, 1) (Arhimed’s axiom); 9

UTILITY ASSESSMENT PROCEDURE • The expert compares the "lottery" x, y, with the alternative z X ("better- ", "worse - " or "can’t answer or equivalent - "): (1 ) 1 • x, y, • The expert compares the "lottery" <x, y, > with z (the “learning point” (x, y, z, )) and with the probability D 1(x, y, z, ) relates it to the set Au= (x, y, z, u(x)+( u(y))>u(z) , or with the probability D 2(x, y, z, ) - to the set Bu = (x, y, z, u(x)+( u(y))<u(z). At each “learning point” (x, y, z, ) a juxtaposition can be made: f(x, y, z, =1 for ( , f(x, y, z, =-1 for ( ) and f(x, y, z, =0 for ( (subjective characteristic of the expert which contain the uncertainty of expressing his/her preferences). • - ( or ) z. Let the "learning points" (the learning sequence) ((x, y, z, 1, (x, y, z, 2, …, (x, y, z, n, . . ) has the x, y, ( or ) z probability distribution F(x, y, z, . Then the probabilities D 1(x, y, z, ) and D 2(x, y, z, ) are the mathematical expectation of f(. ) over Au and Bu, respectively: D 1(x, y, z, )= M(f/x, y, z, , if M(f/x, y, z, >0, D 2(x, y, z, )=-M(f/x, y, z, , if M(f/x, y, z, <0. Let D'(x, y, z, ) is the random value: D'(x, y, z, )= D 1(x, y, z, ), M(f/x, y, z, >0; D'(x, y, z, )=-D 2(x, y, z, ), M(f/x, y, z, <0; D'(x, y, z, )=0, M(f/x, y, z, =0. • We approximate D'(x, y, z, ) by the function G(x, y, z, )=( g(x)+( - g(y)-g(z)), where. the function G(x, y, z, ) is positive over Au and negative over Bu depending on the degree of approximation of D'(x, y, z, ). In such case g(x) is an approximation of the empirical expert utility u*(. ). 10

APPROXIMATION OF THE EXPECTED UTILITY U*(. ) The expert answers has the presentation f(. ): f=D'+ , , M( x, y, z, )=0, M( 2/x, y, z, )<d, d R. Let the utility function u*(. ) is a “square-integrable function”: ( u*2(x)d. Fx< + ), where Fx is the probability distribution over X. The distribution Fx is defined by the probability distribution F(x, y, z, of the appearance of the learning points. The expected utility u(. ) fulfils: We note as ( i(x)) a family of polynomials and ri R. APPROXIMATION ALGORITHM: t=(x, y, z, ; 11

FIXATION OF THE SYSTEM IN A POSITION CONFORMING TO THE EQUIVALENT CONTROL The fixation is based on a “ time optimization” control. “Time optimization” control is solution of the next optimal problem (continuous process): max(U( (T))), where the variable is the third coordinate in the state vector of the model of the continuous process, ( [0, m], D [0, Dmax]. Here U( ) is an aggregation objective unimodal function. Possible choice is the expert utility function. Without control and fixation of the system on the equivalent control position. 2/20/2021 12

EQUIVALENT MODEL – CONTINUOUS PROCESS • With the use of the GS algorithm the non-linear Wang-Monod model is presented in Brunovsky normal form (нормална форма на Бруновски): • Diffeomorphic transformations: Transformation 2 2/20/2021 13

OPTIMAL CONTROL Equivalent BRUNOVSKY NORMAL FORM: • • Gardner, Robert B. ; Shadwick, William F. The GS algorithm for exact linearization to Brunovsky normalform. 1992, Text. Article, IEEE Trans. Autom. Control 37, No. 2, 224 -230 (1992). Elkin, V. Reduction of Non-linear Control Systems: A Differential Geometric Approach– Mathematics and its Applications, 472, Handbound, Kluwer, 1999. 2/20/2021 14

OPTIMIZATION PROBLEM – continuous process • The optimization problem is: • Hamilton based on the Brunovsky normal form : 2/20/2021 15

OPTIMAL CONTROL “Time minimization” Online journal: Bioautomation (2004, 2005 – Pavlov Yuri) 2/20/2021 16

OPTIMAL CONTROL - graphics 2/20/2021 17

Evaluation of the system-fed-batch profile 2/20/2021 18

SECOND ORDER SLIDING MODE CONTROL The scientists Emelyanov, Korovin and Levant evolve high-order sliding mode methods in control systems. Out to this approach the second order SM manifold becoms: Here is used the so cold “contraction” algorithm. After Emelyanov the SM control law is: 2/20/2021 19

Optimal profile: fed-batch process 2/20/2021 20

Optimal profile and Optimal Control Pavlov Y. (2008). Equivalent Forms of Wang-Yerusalimsky Kinetic Model and Optimal Growth Rate Control of Fed-batch Cultivation Processes, online journal Bioautomation, Vol. 11, Supplement, November, p. p. 1 – 12. 2/20/2021 21

CONCLUSIONS • This investigation permits utilization of Control theory and Utility theory to design a flexible methodology, useful in complex biotechnological processes and descriptions of the complex system “Technologist-Fed batch process”. • The possibilities of the second order SM are investigated. • Both controls SM and “time-minimization” synchronized utilization permits to overcome difficulties arising from the biotechnological peculiarities in order to obtain good control solutions. The inclusion of a value model as objective function as part of a dynamical system could be done with the expected utility theory. Such a utility objective function allows the user to • correct iteratively the control law in agreement with his value judgments. 2/20/2021 22

Applications • Pavlov Y. , K. Ljakova (2004). Equivalent Models and Exact Linearization by the Optimal Control of Monod Kinetics Models. Bioautomation, v. 1, Sofia, 42 – 56. • Pavlov Y. (2005). Equivalent Models, Maximum Principle and Optimal Control of Continuous Biotechnological Process: Peculiarities and Problems, Bioautomation, v. 2, Sofia, 24– 29. • Pavlov Y. (2007). Brunovsky Normal Form of Monod Kinetics Models and Growth Rate Control of a Fed-batch Cultivation Process, Bioautomation, v. 8, Sofia, 13 – 26. • Pavlov Y. (2008). Equivalent Forms of Wang-Yerusalimsky Kinetic Model and Optimal Growth Rate Control of Fed-batch Cultivation Processes, Bioautomation, Vol. 11, Supplement, November, p. p. 1 – 12. Thank you for your attention! 2/20/2021 23

МОДЕЛ НА ПРОЦЕСА НА ПОЛУЧАВАНЕ НА АЦЕТАТ - Микробиология Еквивалентен диференциален модел : • We discuss yeast’s C. blankii 35 continuos cultivation process. • The system parameters are as follows: μm=0. 776 [h-1], Ks= 14. 81 [g/l], Ko=1/1231 [ –], m=3. 51 [–], Se=0. 2625 [g/l], S 0=9 [g/l], y=0. 5584 [–], De=0. 01 [h-1]. 24

Bulgarian Academy of Sciences, CBME “Prof. Ivan Daskalov” • 2/20/2021 In chapter (3) is presented a control design for control and stabilization of the specific growth rate of fed-batch cultivation processes. The control design is based on Wang-Monod kinetic and on Wang. Yerusalimsky kinetic models and their equivalent Brunovsky normal form. The control is written based on information of the growth rate. The criterion for determination of the “best” growth rate is a utility function evaluated by the utilization of human preferences. By this way is obtained mathematical description of the complex system “technologist-biotechnological model”. The evaluation of the utility function is based on a scientific investigations and mathematical and programming results developed in Bulgaria, BAS by the author In the last two decades. 27

This work is partially supported by the Bulgarian National Science Fund under grant No. DID-02 -29 “Modelling Processes with Fixed Development Rules” 28

Thank You For Your Attention! 29