7.1 Introduction

It should be evident from the discussion and various illustrations in Chapter 2 that even in its simplest form, the representation of dynamic and steady-state behavior of a bioprocess is multivariate in nature. Even a simple unstructured kinetic representation for biomass formation would require knowledge / monitoring / prediction of a minimum of two variables, namely concentrations / amounts of biomass and at least one specie (substrate) which leads to production of cell mass. Recognizing that biomass formation is the sum total of a large number of intracellular chemical reactions, each of which is catalyzed by an enzyme, and that activity of each enzyme is very sensitive to intracellular pH and temperature, one can appreciate that this simplest black box representation would be applicable only if the intracellular pH and temperature, and therefore indirectly the culture (composite of abiotic and biotic phases) pH and temperature were kept invariant. Considering that the pH and temperature in the biotic portion of the culture and culture as a whole, if left uncontrolled, would vary with time because of the large number of intracellular chemical reactions, it is obvious that maintaining the culture pH and temperature at desired values would require addition of an acid or a base and addition/removal of thermal energy (heating/cooling) as appropriate. Thus, even in the simplest scenario where the focus in the forefront is on formation of biomass and consumption of a single substrate, one must consider in the background manipulation of rates of acid/base addition and heating/cooling to keep culture pH and temperature at the desired values.

Having realized that one must always deal with multivariate problems when dealing with biological reactors, the dimension of the system representation will depend on the nature of that kinetic representation employed (complexity of the kinetic model if one is available or complexity of the bioprocess under consideration if a kinetic model is not available), mode of operation of bioreactor [whether batch, fed-batch, or continuous, with/without recycle (after selective removal of a portion of the bioreactor contents using a separation technique)], and other external influences, such as addition/removal of thermal energy, acid/base addition for pH control, mechanical agitation to promote mixing, and circulation of a gas phase to provide oxygen (in an aerobic bioprocess) or nitrogen (to maintain oxygen-free atmosphere in an anaerobic bioprocess) and remove carbon dioxide.

The three popular modes of operation of mechanically agitated reactors for cell cultivation are batch, fed-batch and continuous operations. Irrespective of the mode of operation with respect to culture, the bioreactors are always operated in continuous mode with respect to gas phase (gas phase continuously entering and leaving the reactor). A batch culture operation is characterized by no addition to and withdrawal from the culture of biomass, fresh nutrient medium and culture broth (with the exception of gas phase). A fed-batch culture operation is characterized by predetermined or controlled addition of nutrient medium in an otherwise batch operation (no withdrawal of culture). This operation allows for temporal variation in the supply of nutrients, thereby allowing tighter control of various cellular processes such as cell growth, nutrient uptake and production of target metabolites. The feed conditions (volumetric flow rate and composition) can be varied in a predetermined fashion (open-loop control) or by using feedback control. In a continuous culture operation, nutrients essential for growth are continuously fed and a portion of the culture is continuously withdrawn. The culture volume is controlled using a level controller. A continuous culture is usually preceded by a batch or fed-batch culture. If the mass flow rates of the bioreactor feed and bioreactor effluent are identical and time-invariant, a time-invariant (steady state) operation can be realized after sufficient time has elapsed from the start of continuous culture operation. As in fed-batch culture, the feed rate to a continuous bioreactor can be varied in a temporal sense in a predetermined fashion or using feedback control. Since the culture conditions (in a global sense) can be kept time-invariant, continuous cultures are easier to monitor and control. Unlike the operation of continuous processes employed for production of chemicals, long-term operation of continuous cultures is subject to many operating difficulties, including risks of contamination and loss in productivity due to cell washout in case of unanticipated disturbances and substantial changes in characteristics of the biotic phase. For this reason, batch and fed-batch culture operations are more common than continuous culture operations.

While the batch and fed-batch operations of a bioreactor are inherently transient, transients are also encountered in continuous bioreactor operations before attaining a steady state or during transition from one steady state (established under one set of feed and culture conditions) to another steady state (established under a different set of feed and culture conditions).

The effectiveness of the operation of a biological reactor (cell cultivation) depends on the outcome of the operation and what went into such operation. Some indicators of the outcome are characteristics associated with metabolites which are desired and/or are generated in significant amounts, biomass (cells), and significant material resources employed (substrates). These characteristics typically are amounts of these species in the biore-actor in batch and fed-batch operations and mass flow rates of these in continuous operation, both of which are related to concentrations of these species in the culture. The inventory of what went into the bioreactor operation will typically include cost of material resources, operating costs for the bioreactor, and the costs associated with separation and recovery of the desired product from the culture. The cost of raw material resources is proportional to the amount of nutrient medium (substrates) supplied to the culture. The operating costs will take into consideration costs associated with mechanical agitation, pumping of different fluids (feeding of nutrient medium and addition of acid/base and antifoam solutions), supply of a suitable gas phase (aeration in the case of an aerobic bioprocess) to the bioreactor, and control of certain culture parameters such as pH and temperature. The costs associated with downstream processing to recover the desired product at desirable concentration will depend on the culture composition and hence on the outcome of the bioreactor operation. The cost of separation and recovery of the desired product is always related inversely to its concentration in the culture.

The effectiveness of bioreactor operation is assessed via an objective function or a performance index which takes into account the price of the target product and the costs associated with generating that product (most or at least the prominent entries in what went into the bioreactor operation). For cost-effective operation of a bioprocess, one is interested in maximizing the objective function. The outcome of the bioreactor operation being decisively dependent on the trajectories of variables affecting the kinetics of the bioprocess (information contained in f in Eq. 7.1), these trajectories strongly influence the magnitude of the objective function. The trajectories that lead to maximization of the objective function can be attained using feedback control with appropriate controllers. Guiding the trajectories of influential culture variables at or very near their optimal values is accomplished by appropriate manipulation of some of the input variables. (All of these may not be the physical inputs to the process.) Identification of the optimal trajectories of the manipulated inputs can be accomplished using the optimal control theory.

The controlled variables are the output variables that influence the outcome of the process, which is assessed in terms of an objective function or performance index. Those inputs to the process which have the strongest influence on the controlled outputs are usually chosen as manipulated variables. The remaining inputs are either used for process optimization or cannot be influenced or even measured (disturbance variables). In this work, we will assume that there are m manipulated variables, rrid disturbances and no additional inputs used for optimization. In a multivariate system such as a biological reactor, one normally encounters multiple input variables (mt = m + m<j) and multiple output variables which decide the process outcome (p). To control the p outputs, it is essential to take into account how each of the mt inputs influences each of the p outputs in an uncontrolled process. One would anticipate that input (s) which have the greatest influence on a particular output should be manipulated by appropriate controllers to control the particular output. Control can be implemented by using multivariable or multi-loop controllers. In multivariable control, information from all controlled variables and disturbances is used together to compute all manipulated variable moves. In multi-loop control, many single-input, single-output (SISO) controllers are developed by pairing the appropriate manipulated and controlled variables. The collective operation of these SISO controllers controls the multivariable process. Multivariable controllers, such as linear quadratic Gaussian controllers (LQCs) and model predictive control (MPC) systems, are more effective than multiloop controllers and their use has increased in recent years.

The number of controllers involved in multi-loop control is min(mt, p). The issues to be resolved in multi-loop control are (1) how to pair input and output variables and (2) how to design the individual single-loop controllers. The decision on input-output pairings is based on the nature of process interactions (effect of an an input on multiple outputs). Even with the best possible input-output pairings, functioning of individual controller loops may be influenced by other control loops due to process interaction. Ideally, one would like all control loops to function independently. This requires use of decouplers (additional elements inserted between single-loop controllers and the process), so that the output from a controller used to control a particular output influences not only the manipulated input that is paired with that output, but also other inputs in order to eliminate the effects of interaction. The idea behind the use of decouplers is to make the controllers function independently in entirety.

This chapter starts with determination of optimal trajectories during bioprocess operation. Given a process model, this can be done by solving an appropriate open-loop optimization problem to maximize a particular objective function or performance index. The general procedure for identification of optimal open-loop control policies for nonlinear bioprocess models is provided in Section 7.2, which is followed by a detailed case study involving identification of optimal feeding policies for fed-batch cultures with varying complexity of kinetics. The general procedure discussed in Section 7.2 is applicable to both dynamic and steady-state operations of batch / fed-batch / continuous bioprocesses. A related problem, that of further potential improvement in performance of steady-state continuous bioprocesses via periodic variations in one or more inputs, is considered in Section 7.3. The discussion on criteria for superiority of periodic forcing is followed by a case study. Closed-loop feedback control based on state-space models involving multiple single-input, single-output (SISO) controllers is considered in Section 7.4. Methods for selection of multi-loop controller configuration and minimizing / eliminating effects of bioprocess interaction are considered in this section, the ultimate goal being independent functioning of various control loops. Multivariable control is discussed next in Sections 7.5 and 7.6. Identification of optimal feedback control strategies based on optimal open-loop trajectories (Section 7.2) is the focus of Section 7.5. The optimal feedback controllers have the traditional proportional, integral, and derivative modes of action, with controller parameters being functions of time. Finally, the more powerful and increasingly popular model-based multivariable control, the Model Predictive Control (MPC), is the subject of Section 7.6. The discussion of the general recipe for MPC is followed by a specific illustration of one of the MPC schemes, namely the Dynamic Matrix Control (DMC) and an introduction to nonlinear MPC.

Several review papers and books provide overviews of batch process control and its implementation in chemical process industries that complement this chapter. An early comparative assessment of the effectiveness of simple control techniques and dynamic optimization in the 1980s favors simple control tools for controlling fed-batch fermentation processes [261]. Industrial practice in control and diagnosis of batch processes has been reported [429]. The progress and challenges in batch process control have been discussed in various review papers [51, 265, 510] and assessed in the context of scheduling and optimization of batch process operations [501].

7.2 Open-Loop (Optimal) Control 7.2.1 Nonlinear Models of Bioreactor Dynamics

A popular form of operation of bioreactors employing living cells involves the use of a well-mixed reactor. The uniformity of composition and temperature in the reactor allows its representation as a lumped parameter system. The reactor dynamics can be described succinctly as dx.

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