Implementation of robust control theory to a continuous stirred tank bioreactor P. G. Georgieva, M. N. Ignatova Abstract An H-inf robust controller for a continuous stirred tank bioreactor (CSTB) is designed. The imple- mentation of H-inf framework in designing requires a linear plant model plus uncertainty description that has to be rearranged into a Linear Fractional Transformation structure. Therefore, the dynamical behavior of the non- linear plant is suitably approximated by linear state-space model with bounded parameter variations. The control problem considered is to guarantee robust stability and robust performance of the closed loop system. The per- formance speci®cations are de®ned through weighting functions. H-inf controller was successfully computed to certain process variations around the operating region. List of symbols Xt biomass concentration, g/l S sub t substrate concentration, g/l lS sub  speci®c growth rate, l/h Dt dilution rate, l/h S 0 external substrate concentration, g/l s Laplace operator WS 1 ; W 2 S frequency dependent performance weighting functions Ks controller Ss sensitivity function Ts complimentary sensitivity function A; B; C; D state-space model 1 Introduction The biotechnological processes have complex dynamical behavior, mathematically modeled by nonlinear differen- tial equations. Often the parameters or even the structure of the model are not perfectly known or change due to variations in the working conditions. Several control strategies are reported to have been successfully applied in bioprocess engineering [6]: these vary from the classical PI and PID controllers, through the advanced quadratic optimal control, to the more sophisticated self-tuning controllers, adaptive linearizing control and knowledge based (or expert) systems. An ef®cient way to cope simultaneously with different uncertainty reasons which could lead to instability and degradation of the process quality is to design an H-inf robustly performing controller. Though many signi®cant applications of advanced robust control are reported in the area of chemical processes, [7] very few results are avail- able for biological systems. In this paper we restrict our attention on the implemen- tation of the H-inf robust control theory to the class of continuous biotechnological processes. A detailed descrip- tion of all reactions in the bioreactor would lead to a high- order model. Instead of using such a rigorous model, we adopt a modeling strategy based on a reasonable simpli®- cation of the description, which preserves the essential structural properties of the process. The design is based on a linearized process model, which is augmented with a norm bounded uncertainty description to capture the discrepancy between the simpli®ed model of the process and the varia- tion in the process parameters. All performance speci®ca- tions are de®ned in the frequency domain. The design aim is to ®nd the best possible controller structure, which opti- mizes between the con¯icting objectives (tracking and ro- bustness), assuming the worst case of disturbance action. 2 Process model The process considered is a CSTB in which the growth of one population of microorganisms on a single limiting substrate is the reaction to be controlled, see Fig. 1. The reactor is continuously fed with the substrate in¯uent. The rate of out¯ow is equal to the rate of in¯ow and the volume of culture remains constant. It is considered that the feeding substrate is diluted in the water stream and the dilution rate is used as manipulated variable to control the process. The control signal is con®ned by physical constraints and can only be within a certain range. The output is assumed to be measured directly. The control objective is to maintain substrate concentration at a ®xed value despite variations in the feed substrate, which is regarded as unmeasured dis- turbance. Moreover the controller should track set-point step changes in the substrate concentration. Such reaction process describes a large class of real industrial cases. A typical representative of this class are activated sludge processes for biological puri®cation of contaminated water, where to regulate the substrate concentration is equivalent to control the pollution level in the ef¯uent of the plant. Received: 5 July 1999 P. G. Georgieva (&), M. N. Ignatova Institute of Control and System Research, Bulgarian Academy of Sciences, P.O. Box 79, 1113 So®a, Bulgaria This research was supported by the National Foundation ``Scienti®c Investigations'' by the Bulgarian Ministry of Education and Science, contract No. TN-715/97. Bioprocess Engineering 22 (2000) 563±568 Ó Springer-Verlag 2000 563