Fxud x0xo71

with x denoting the state variables which represent the status of the cell culture in the bioreactor, and u and d representing the input variables which indirectly influence the status of the cell culture. The input variables are further classified into manipulated inputs (u) and disturbance variables (d). Let n, m and rn,i denote the number of state variables, manipulated inputs and disturbance variables. Not all state variables can be measured. Some of the state variables, which cannot be measured or can be measured less frequently, are estimated from measurements of other variables that are measured frequently by using estimators (Section 4.3). It must therefore be realized that only some of the state variables may be monitored or estimated. The set of variables which can be measured will be referred to as bioreactor outputs, y, with the number of outputs being p. The relations among the state variables and the output (measured) variables can then be succinctly stated as y = h(x). (7.2)

The functions f(-) and h(-) are in general nonlinear. But for mathematical convenience in developing the control equations, these functions are linearized. Linear state-space equations are discussed in Section 7.4.

7.2.2 Background on Optimal Control Theory

Whether the bioreactor is operating at a steady state or is exhibiting transients, one is interested in maximizing an appropriate objective function for cost-effectiveness of the operation. For the sake of generality, we consider here a transient bioreactor operation. During the time interval (0, tf), one may be interested in maximizing an objective function

The objective function in Eq. 7.3 is sufficiently general for a wide variety of practical problems. G(-) denotes the benefit generated at the end of the operation (tf) and g(-) the benefit materialized during the operation. The optimization may be cast as a minimization problem by defining G(-) and g(-) as costs. The objective function may also be written in terms of output variables y. Thus, for example, if

Jo then in view of relation 7.2, the objective function in Eq. 7.4 can be restated as in Eq. 7.3. Besides the constraints imposed bv the conservation equations (state equations) in Eqs. 7.1, the maximization of J may have to be achieved subject to the following integral constraints:

1 ftf 1 ftf

— 0(x, u)dt — 0 and — / ip{x, u)dt < 0. (7.5) if Jo if Jo where <j>(-) and tfti ) are appropriate linear or nonlinear functions. Integral constraints expressed in terms of outputs y, such as

/ V(y,u)dt = 0 and i / V(y,u)di<0 (7.6) if Jo if Jo can be readily expressed as in Eq. 7.5 in view of the relations in Eq. 7.2. As an example of the equality constraint in Eq. 7.5, consider a continuous culture operation. It is usually of interest to identify an optimal feed composition (for example, the substrate feed concentration, Sf) which will lead to maximization of an objective function, such as yield or productivity of the target metabolite. Economic considerations would dictate that this identification be done while keeping the throughput rate of the substrate (Gs) fixed. Different candidate continuous culture operations with variable Sf then would be subject to the integral constraint

[*' (FSf -Gs)dt = 0 ~ [ ' FSpdt = GS- (7.7) if Jo if Jo

The integral constraint in Eq. 7.7 is applicable for situations involving fixed F and Sf in an individual operation as well as time-varying F and Sf in an individual operation, as is the case of forced periodic operation of a continuous culture. As an example of the inequality constraint in Eq. 7.5, consider a batch, a fed-batch or a composite of batch and fed-batch culture operation. It may be desired to maximize an objective function such as the amount or yield of a target metabolite. This may have to be accomplished with a limited amount of substrate (Ms). The candidate operations would then be subject to the constraint

In a strictly batch operation, the nutrient medium containing substrate is added rapidly at the start of operation, i.e., at t = 0, the volumetric flow rate F being an impulse function in this operation.

Because of the considerable complexity that the constraints of the form in Eqs. 7.5 add to process optimization, we consider first situations involving constraints only on the manipulated inputs u. Each manipulated input then is considered to be bounded from above and below as

where umin and umax denote the lower and upper bounds, respectively. Maximization of the objective function is then accomplished via maximization of the Hamiltonian H with respect to u. The Hamiltonian is defined as [84]

with A being the vector of adjoint variables associated with Eqs. 7.1. The variation in A with time is described by d\T dH

dt dx

The influence of system equations Eq. 7.1 on H (and J) is transmitted by the adjoint variables. Eqs. 7.1 and 7.11 represent a set of ordinary differential equations. Their solution requires knowledge of each state and adjoint variable at some t (usually at /; — 0 or tf). Let x*, u*, tj, and J* denote the optimal values of x, u, i/ and J, respectively. Let the variations in x, u, tf and J for an arbitrary operation from their respective values for the optimal operation be expressed as 5tf = tf tj, 5J = J — J*, Sx = x — x*, and = u — u*. Identification of optimal u then involves expressing the variation in J (5 J) entirely in terms of variation in u (i5u). In general, 6J depends on ¿x(0), Sx(tf) and Stj as well. Trivializing the influences of these on 6 J leads to the following conditions [84]:

The conditions in Eqs. 7.12 and 7.13 are applicable for once-through operations, i.e., process operations where x(0) and x(i/) are independent (i.e., <5x(0) and Sx(tf) are not identical). In cyclic operation of a bioreactor, the operation modes under consideration here (batch, fed-batch and continuous) and certain sequences of these are repeated, with tf being the duration of a cycle. In this case, x(i) satisfy the periodic boundary conditions x(0) = x(tf) =>• <5x(0) = 5x(tf). (7.15)

The boundary condition for the adjoint variables are then obtained as dG

\i(tf) = ---1- Aj(0) if Xi(tf) is not specified. (7-16)

In view of the conditions above (Eqs. 7.12-7.14 and 7.16), 5J can be expressed as

If some components of u*(t) include segments (sections) where Ui — (uj)min or (uj)max, then <5uj(t) must be positive at (tit)min and Sui(t) must be negative at (uj)max. Since S J is expected to be non-positive, the following conditions must be satisfied on the optimal trajectory for u,, u*(t):

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