Nonlinear feedforward control of bioreactors with input multiplicities

Abstract

A steady-state nonlinear feedforward controller (FFC) for measurable disturbances is designed for a continuous bioreactor, which is represented by Hammerstein type nonlinear model wherein the nonlinearity is a polynomial with input multiplicities. The manipulated variable is the feed substrate concentration (S f ) and the disturbance variable is the dilution rate (D). The productivity (Q=DP) is considered as the controlled variable. The desired value of Q=3.73 gives two values of feed substrate concentration. The nonlinearity in the gain is considered for relating output to the manipulated variable and separately for the relation between output to disturbance variable. The FFC is also designed for the overall linearized system. The performance of the FFC is evaluated on the nonlinear differential equation model. The FFC is also designed for the model based on a single nonlinear steady-state equation containing both D and S f . This nonlinear FFC gives the best performance. The nonlinear FFC is also designed by using only linear gain for the disturbance and nonlinear gain for the manipulated variable. Similarly, nonlinear FFC is also designed by using linear gain for the manipulated variable and the nonlinear gain for the disturbance variable. The performances of these FFC schemes are compared.List of symbols a 1 , a 2 de®ned by Eq. (20g) and Eq. (20h), respectively A, CThe denominator polynomials of the process and the disturbance model b 1 , b 2 de®ned by Eq. (20e) and Eq. (20f), respectively BThe numerator polynomial of the process c 1 , c 2 de®ned by Eq. (22a) and Eq. (22b), respectively D dilution rate (h ±1 ) D s steady-state value of D (g/L) G(z) output of the nonlinear gain system (block 4) g(z)G(z) ±Y s h 1 to h 4 de®ned by Eq. (20a) to Eq. (20d) K I substrate inhibition constant (g/L) K m substrate saturation constant (g/L)K p linear gain relating y to u K v linear gain relating y to v P product concentration (g/L) P m product saturation constant (g/L) Q product cells produced per unit time (g/L h) r 1 , r 2 de®ned by Eq. (22c) and Eq. (22d), respectively RThe numerator polynomial of the disturbance modelinput of the linear system (block 2) Y steady-state value of output y deviation value of the output y 1 output of the linear subsystem (block 4) X biomass concentration (g/L) a, b, d, coeffcients of linear discrete models [refer toEqs. (8a, 8b) and (9a, 9b)] r product yield parameter (g/g) w product yield parameter (h ±1 ) l speci®c growth rate (h ±1 ) l m maximum speci®c growth rate (h ±1 ) D deviation from the steady-state c yield factor for cell mass (g/g) IntroductionThe dynamics of bioreactors are represented by nonlinear differential-equation models. The design of linear feedforward controller (FFC) to compensate for the effect of measurable disturbances of linear systems is a well-developed ®eld [1,2]. The performance of the linear controller is adequate for linear or mildly nonlinear systems. However, for highly nonlinear systems, the performance of the linear controller will not be satisfactory....