The process of anaerobic digestion in which waste biomass is transformed to methane by complex microbial communities has been modeled for more than 16 years by parametric gray box approaches that simplify process biology and do not resolve intracellular microbial activity. Information on such activity, however, has become available in unprecedented detail by recent experimental advances in metatranscriptomics and metaproteomics. The inclusion of such data could lead to more powerful process models of anaerobic digestion that more faithfully represent the activity of microbial communities. We augmented the Anaerobic Digestion Model No. 1 (ADM1) as the standard kinetic model of anaerobic digestion by coupling it to Flux-Balance-Analysis (FBA) models of methanogenic species. Steady-state results of coupled models are comparable to standard ADM1 simulations if the energy demand for non-growth associated maintenance (NGAM) is chosen adequately. When changing a constant feed of maize silage from continuous to pulsed feeding, the final average methane production remains very similar for both standard and coupled models, while both the initial response of the methanogenic population at the onset of pulsed feeding as well as its dynamics between pulses deviates considerably. In contrast to ADM1, the coupled models deliver predictions of up to 1,000s of intracellular metabolic fluxes per species, describing intracellular metabolic pathway activity in much higher detail. Furthermore, yield coefficients which need to be specified in ADM1 are no longer required as they are implicitly encoded in the topology of the species’ metabolic network. We show the feasibility of augmenting ADM1, an ordinary differential equation-based model for simulating biogas production, by FBA models implementing individual steps of anaerobic digestion. While cellular maintenance is introduced as a new parameter, the total number of parameters is reduced as yield coefficients no longer need to be specified. The coupled models provide detailed predictions on intracellular activity of microbial species which are compatible with experimental data on enzyme synthesis activity or abundance as obtained by metatranscriptomics or metaproteomics. By providing predictions of intracellular fluxes of individual community members, the presented approach advances the simulation of microbial community driven processes and provides a direct link to validation by state-of-the-art experimental techniques.

The sensitivity of optimally controlled systems to parameter variations is examined in this paper. Two general approaches toward compensating for these variations are employed. The first, for open-loop systems, involves augmenting the performance index with sensitivity terms and minimizing this combined index. The second approach. a n adaptive-type controller, involves estimating those parameter variations that have caused an observed state deviation, and adjusting the control policy in response to this measurement. Numerical exomples applied to continuous, discrete-time, and discrete-staged systems demonstrate the effectiveness of these approaches in adapting the control policy to parameter variations.In recent years there has been extensive literature dealing with different aspects of optimal control and its applications (1, 15, 2 5 ) . From a chemical engineering point of view however, the reliability of any mathematical model used in such control is probably only an approximation for a real process. Successful application of an optimal control policy calculated from such models requires some means of updating an initial model as time passes; alternatively, one might suggest the general form of a model and assume that deviations between the process and the model are due to variations in a number of characteristic parameters in the model. In the present work we shall investigate the influence of such parameter variations on control policies and the resulting trajectories.Let us consider that an optimal control policy for a given system has been calculated and is applied to that system. The states of that system will evolve along a trajectory in time, the nominal optimal trajectory. The effects of parameter variations on this trajectory can be manifested in two principal ways: ( 1 ) the state trajectory will deviate from the nominal trajectory (trajectory sensitivity) ; and (2) the performance index will change from that associ--ated with the nominal trajectory (performance index sensitivity).If the object of control is to bring the state to some desired end point, the first type of response can cause the target to be missed. Otherwise, this can merely alter the trajectory so that it passes through some undesirable region in state space (that is, high temperatures, pressures, etc.). The general aim of reducing trajectory sensitivity is to reduce this dispersion due to parameter variation. A common approach in this case is to define a timevarying sensitivity coefficient a ( t ) as the first-order change in the state vector x ( t ) due to variations of the parameter q, along some nominal trajectory ( 4 , 7, 14, 29) :These sensitivity coefficients, which can be generated for any number of parameters, can be treated as ordinary state vectors, and adjoined to x to form an augmented state X: (X = [x, a] T ) . Likewise, the original performance index can be augmented to include some positive-definite function of a, such as aTQa. Increasing the magnitude of Q a ( t ) = ex(t)/aq = q (1)Correspondence concerning this pape...

A discrete optimally sensitive controller is developed which yields steady-state feedback control of a three-stage biochemical reactor system in spite of system parameter variations. A feedback law is implemented that estimates those variations which cause the output states to deviate from nominal, and adjusts the control policy in light of these variations.