The elements of the matrices T and A are best determined by choosing the elements of the scaling matrices W and V. The scaling matrices W and V will be partitioned matrices with the elements of each submatrix being identical and equal to the scaling factor for a particular output or manipulated input, as appropriate.
When constraints are involved, the objective function <p used for computing the control action sequence Au(fc) must be augmented with the constraint equations. This prevents the development of closed-form controller equations such as Eq. 7.159. The resulting quadratic program must be solved as a real-time optimization problem to identify the recommended control action sequence [176, 477, 507, 6811.
Batch and fed-batch bioprocesses are inherently transient operations. Although transients are also encountered in continuous bioprocesses, the focus usually is on operation at a desired steady-state. In a steady-state operation, the target trajectory reduces to a fixed set-point in the multidimensional space of output variables. For batch and fed-batch operations, which are more common compared to continuous operations, the target trajectories of the output variables will typically be time-variant, i.e., the set-point for a particular output will vary with time. One of the characteristics of industrial batch and fed-batch bioprocesses is that they are cyclic or repetitive. This characteristic allows the process operator and controllers the opportunity to make compensations based on errors from previous cycles. The target trajectories can therefore be updated from cycle to cycle as one gathers more knowledge of the bioprocess. Indeed, this idea, referred to by the generic name Iterative Learning Control (ILC) has been integrated into conventional model predictive control and applied to batch processes [327, 328, 682]. The integrated methodology is not only capable of eliminating persisting errors from previous cycles or runs, but also can respond to new disturbances as these occur during a particular cycle or run .
The DMC formulation uses a step response model which is limited to stable processes. To leverage the wealth of knowledge on state-space techniques, MPC algorithms based on state-space models were proposed by converting the step response models into state-space form [324, 339]. The state-space formulation for unstable linear processes has been addressed . A related formulation called generalized predictive control is based on transfer function models with the aim of handling unstable processes . Inequality constraints in an MPC lead to nonlinear feedback control laws. Considering the abundance of process nonlinearities in chemical and biochemical processes, the use of nonlinear process models in MPC formulation was a natural progress.
Many processes have significant nonlinearities that challenge successful implementation of linear MPC. This has motivated the development of nonlinear MPC (NMPC) which relies on the use of a nonlinear process model. NMPC has the potential of improving process operation, but it also provides challenging theoretical and practical problems mostly because of the nonlinear optimization problem that must be solved at each sampling instant in real time to compute the control moves.
Many nonlinear model representations were discussed in Section 4.3. Consider the general form expressed in Eqs. (4.44)-(4.45)
— = f (x(i), u(i)), y(t) = h (x(i), u (i)) (7.160)
that can be written in discrete time as x(fc + l) = f(x(fc),u(fc)), y(fc) = h(x(fc),u(fc)) fc = 0,1,2, • • ■
where x(fc) is a condensed form of the terminology x(ifc) used in Section 4.3.2. The optimization problem in NMPC formulation can be expressed as finding the values of u to optimize the objective function J subject to constraints [234, 381]
+ ^ L [y(fc + j\k), u(fc + j\k), Au(fc + j\k)}, j=o where Lq and L are nonlinear functions of their arguments, m is the control horizon, and p is the prediction horizon. y(k + j\k) denotes the value of the output y at time k + j computed from information available at time k and Au(k + j\k) = u(k + j\k) - u[k + j - l|fc). The functions L0 and L may represent a variety of objectives, including the minimization of the overall cost of process operation. For example, regulation to set points or tracking reference trajectories can be formulated as quadratic equations of the form
where yr(k) and ur(k) are the reference values for y and u, and Q, R, and S are positive definite weighting matrices. These weighting matrices, m, p and the sampling time are the tuning parameters of the NMPC. The prediction of output values y(fc + j\k) are based on state variables and the calculated input sequence. Hence, measurement or estimation of state variables is necessary. State estimation relies on the nonlinear model Eq. (7.161) and use of process information that reflects disturbance effects as discussed below. The solution of the NMPC problem yields the values for the input sequence (u(fc|fc), + l\k),- ■ ■ , u(k + m - ljfc)). Only the first input vector u(k\k) is implemented and the the real time optimization problem is solved again at the next sampling time.
Constraints Several constraints are imposed on inputs and outputs. Input (manipulated variable) constraints reflect actuator limitations such as saturation and rate-of-change restrictions such as rate of temperature increase:
Aumin < Au(fc + j\k) < Aumax j = 0, m - 1 . (7.165)
where umin and umax denote the minimum and maximum values of the inputs. Output constraints are associated with operational limitations such as equipment, materials, product properties, and safety considerations:
The nonlinear model in Eq. (7.161) is added as equality constraints:
x(A; + j + l|fc) = f(x(fc + j|fc),u(A + i|*;)) j=0,p-l y(k + j\k) = h(x(k + j\k)) j = l,p (7.167)
where x(/c|/c) = x(k) if the state variables are measured.
State and Disturbance Estimation The objective of the NMPC system is to drive process outputs (and inputs) to their reference (target) values. If the reference values used in Eqs. (7.163) are not chosen properly, unmeasured disturbances and modeling errors would cause offset. The offset problem can be handled by designing a disturbance estimator that provides an implicit integral control action [234, 381]. A simple method for incorporating integral action is to modify the reference values yr by shifting the set points with the disturbance estimates. The penalties on inputs are also eliminated in this method (R = 0). The output references are computed as
where ysp are the set points of outputs, y(k) are the measured values of outputs, y(k\k) are output estimates obtained from the nonlinear model Eq. (7.161), and d(fc) are the estimated disturbances. This disturbance model assumes that plant-model mismatch is attributable to a step disturbance in the output that remains constant over the prediction horizon . A method for incorporating integral action based on steady-state target optimization has been developed .
Simultaneous state and disturbance estimation can be performed by augmenting the state-space model:
where d(k) is a constant output disturbance. The augmented process model can be used for designing a nonlinear observer. A general theory for nonlinear observer design is not available, and input-output models are preferred over state-space models when full state feedback is not available. A list of NMPC applications with simulations and experimental studies is given in  along with a discussion of computational issues and future research directions.
Heuristic tuning guidelines are discussed in  and summarized in , For stable systems, the sampling interval should be selected to provide a compromise between on-line computation load and closed-loop performance. There is an inverse relationship between sampling interval and allowable modeling error. Smaller control horizons (m) yield more sluggish output responses and more conservative input moves. Large values of m increase the computation burden. Large prediction horizons (p) cause more aggressive control and heavier computation burden. The weighting matrices (Q, R, S) are dependent on the scaling of the problem. Usually they are diagonal matrices with positive elements. The parameter values can be tuned via simulation studies.
Computational constraints and stability of the controlled system are critical issues in NMPC. The need to solve the nonlinear programming problem in real time necessitate efficient and reliable nonlinear programming techniques and MPC formulations that have improved computational speed. Successive linearization of model equations, sequential model solution and optimization, simultaneous model solution and optimization are some of the approaches proposed in recent years [234, 381].
Batch and fed-batch bioprocesses typically exhibit large variations in the operating conditions during a cycle or a run. During different phases of batch and fed-batch operations, culture parameters such as pH, temperature, and substrate availability may change and these changes would substantially alter parameters in the bioprocess model. The performance of MPC depends critically on the predictive abilities of the process model employed for prediction. For the reasons mentioned above, building a nonlinear model for batch and fed-batch cultures is a cumbersome task . Empirical models are therefore appealing in model predictive control of batch and fed-batch bioprocesses. In fact, most of the practical applications of MPC have involved use of empirical models. Another way to circumvent the nonlinear first principles model development for application of MPC is to use artificial neural networks (ANNs). Models based on ANNs rely on data from previous runs or cultivations and are capable of extracting the relevant parameters and relationships from them . Model predictive control with ANNs has been used for on-line optimization of riboflavin production in a fed-batch bioprocess . A variant of this approach is the use of empirical reference trajectories and predictive models developed using the multivariate statistical methods discussed in Chapter 6. With landmark and trajectory alignment using dynamic time warping (DTW) and curve registration, the reference trajectories and predictive models developed show good promise for MPC.
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