Fedbatch Cultivation Control based on Genetic Algorithm PID

Fed-batch Cultivation Control based on Genetic Algorithm PID Controller Tuning Olympia Roeva 1, Tsonyo Slavov 2 1 Institute of Biophysics and Biomedical Engineering, BAS 2 Technical 20 -24. 08. 2010 University - Sofia Numerical Methods & Applications 2010

Presentation outline 1. Introduction 2. Theoretical background of the PID control algorithm 3. Description of the considered E. coli cultivation process 4. PID controller tuning using genetic algorithms 5. Results and discussion 20 -24. 08. 2010 Numerical Methods & Applications 2010

Introduction Commercially available controllers exist only for well established measurement systems as: v per p. H v temperature v stirrer speed v dissolved oxygen v etc. Bioprocesses characteristics: v highly changing dynamics, which is caused by Ø the non-linear growth of the cells Ø the metabolic changes Ø the overall metabolism changes GA are capable of handling problems with: v v v non-linear constraints multiple objectives dynamic components To achieve good closed-loop system performance GA based controller tuning is proposed. 20 -24. 08. 2010 Numerical Methods & Applications 2010

Theoretical background of the PID control algorithm Why PID control? § Simple, easy to use § Wide application: Ø petrochemicals, Ø pharmaceuticals, Ø food, Ø chemicals, Ø aerospace and semiconductors, Ø etc. Robust: insensitive to changes to plant parameters and disturbance § Over 90% of the control loops are PID. 20 -24. 08. 2010 Numerical Methods & Applications 2010

Theoretical background of the PID control algorithm Fig. 1. A typical structure of a PID control system Basic control actions: u(t) Proportional mode - (adjustable gain (amplifier)) Integral mode Derivative mode 20 -24. 08. 2010 - (eliminates steady-state error, can cause oscillations) - (effective in transient periods, provides faster response (higher sensitivity)) Numerical Methods & Applications 2010

Theoretical background of the PID control algorithm The mathematical description of discrete-time universal PID controller is: where k is the number of sample; u(k) - control signal; up(k), ui(k) and ud(k) - proportional, integral and derivative modes of control signal; r(k) – reference signal; y(k) - output signal; Kp proportional gain; Ti integral time; Td - derivative time; Td/N - time constant of first-order low pass filter; T 0 – sample time; b and c - weighting coefficients. Six tuning parameters - Kp, Ti, Td, b, c and N 20 -24. 08. 2010 Numerical Methods & Applications 2010

Description of the E. coli MC 4110 cultivation process The mathematical model can be represented by the following dynamic mass balance equations: Numerical values of the model parameters used in simulations are: µmax = 0. 55 h-1, k. S = 0. 01 g/l, YS/X = 0. 50. The controller is used to • control feed rate F and • maintain glucose concentration S at the desired set point of 0. 1 g/l. The optimal value of the PID controller parameters (Kp, Ti, Td, b, c and N) are found using GA. 20 -24. 08. 2010 Numerical Methods & Applications 2010

PID controller tuning using genetic algorithms Initialization of algorithm parameters Table 1. Genetic operators Table 2. Genetic parameters Operator Type Parameter Value encoding binary ggap 0. 90 fitness function linear ranking xovr 0. 80 selection function roulette wheel selection mutr 0. 01 crossover function double point nind 100 mutation function a bit inversion maxgen 200 reinsertion fitness-based 20 -24. 08. 2010 Numerical Methods & Applications 2010

PID controller tuning using genetic algorithms Representation of chromosomes Each chromosome is a sequence of k- parts each of them with n (encoding precision) genes. The range of the tuning parameters is considered as follows: Kp [0; 2], Ti [0; 1], Td [0; 0. 1], b [0; 1], c [0; 1] and N [0. 001; 1000] After several runs the range for the parameters is specified to: Kp [0. 4; 2], Ti [0. 005; 1] and Td [0. 003; 0. 1] 20 -24. 08. 2010 Numerical Methods & Applications 2010

PID controller tuning using genetic algorithms Objective function To evaluate the significance of the tuning procedure and controller performance four criteria are used: integrated squared error (IISE) integrated absolute error (IIAE) integrated time-weighted absolute error (IITAE) integrated squared time-weighted error (IISTE) The error e is the difference between the set-point and the estimated substrate concentration (Ssp - S). 20 -24. 08. 2010 Numerical Methods & Applications 2010

Results and discussion Table 3. Controller parameters, mean value (with noise) Case study Kp Ti Td b c N I value 1 0. 400 0. 985 0. 003 1. 000 1000. 000 IISE = 16. 1639 2 0. 404 0. 947 0. 003 0. 939 0. 871 778. 317 IISE = 16. 1510 1 0. 400 0. 985 0. 003 1. 000 1000. 000 IIAE = 38. 2181 2 0. 497 0. 936 0. 003 0. 838 0. 928 616. 359 IIAE = 38. 1324 1 0. 400 0. 987 0. 003 1. 000 1000. 000 IITAE = 110. 4505 2 0. 403 0. 935 0. 003 0. 946 0. 906 733. 913 IITAE = 110. 3550 1 0. 400 0. 978 0. 003 1. 000 1000. 000 IISTE = 755. 1833 2 0. 403 0. 939 0. 003 0. 865 0. 937 537. 913 IISTE = 754. 4549 20 -24. 08. 2010 Numerical Methods & Applications 2010

Results and discussion Comparison with the results obtained from the controller design of the same cultivation process reported in M. Arndt and B. Hitzmann, 2001 1 35 Substrate control - this report Substrate control - Arndt et all. 0. 8 30 Biomass - this report 0. 6 Biomass - Arndt et all. 25 0. 4 Substrate concentration, [g/l] Biomass concentration, [g/l] 20 15 10 5 0. 2 0 -0. 2 -0. 4 -0. 6 0 6 7 8 9 10 11 12 13 Time, [h] a) biomass concentration 14 15 6 7 8 9 10 11 12 13 Time, [h] b) substrate concentrations Fig. 1. Results of controller and process performance concerning Case 2 and IITAE 20 -24. 08. 2010 Numerical Methods & Applications 2010 14 15

Results and discussion 1 Substrate control - this report Substrate control - Arndt et all. 0. 8 0. 4 , Substrate concentration [g/l] 0. 6 0. 2 0 -0. 2 -0. 4 -0. 6 6 20 -24. 08. 2010 7 8 9 10 11 Time, [h] 12 13 Numerical Methods & Applications 2010 14 15

Results and discussion 0. 16 0. 8 Substrate control - this report Feed rate - this report Substrate control - Arndt et all. Feed rate - Arndt et all. 0. 7 0. 14 0. 6 0. 12 0. 1 Feed rate profile, [l/h] Substrate concentration, [g/l] 0. 5 0. 08 0. 4 0. 3 0. 2 0. 06 0. 1 0. 04 0 9 9. 1 9. 2 9. 3 9. 4 9. 5 Time, [h] 9. 6 9. 7 9. 8 c) substrate concentrations between 9 and 10 h 9. 9 10 6 7 8 9 10 11 12 13 14 Time, [h] d) resulting feed rate profiles Fig. 2. Results of controller and process performance concerning Case 2 and IITAE 20 -24. 08. 2010 Numerical Methods & Applications 2010 15

Results and discussion The maximum difference reported in M. Arndt and B. Hitzmann, 2001 is 0. 06 g/l. In parallel, the maximum difference achieved here is 0. 03 g/l. The resulting standard deviation and mean value concerning control variable are: in this report σs = 0. 0063 in M. Arndt and B. Hitzmann, 2001 σs = 0. 1513 ms = 0. 0967; ms = 0. 1306. The presented results indicate high quality and better performance of the designed control system. 20 -24. 08. 2010 Numerical Methods & Applications 2010

Thank You ! 20 -24. 08. 2010 Numerical Methods & Applications 2010