## Model Predictive Control

In the competitive global market, pharmaceutical and biotechnological companies are forced to increase the efficiency and productivity of well-established processes continuously. Changes amounting to a few percent in the final titers and product yields may lead to enormous benefits in large-scale cultivations. These goals can be achieved either by introducing more productive strains or by optimization of the cultivations. In the chemical and allied industries, the expectations for consistent attainment of high product quality, more efficient use of energy, and environmental impact of production activities have led to far stricter demands on control systems than can be met by traditional techniques alone. The response of these industries and academia to these challenges led to the development of a different control methodology, called Model Predictive Control (MPC). A built-in feature of MPC is the direct use of an explicit and separately identifiable process model. MPC is finding wide acceptance in applications in chemical and allied industries, because of its versatility, accommodation of nonlinear process models, and ability to handle variations in constraints in real time. MPC schemes use a process model for two key purposes, first to predict future process behavior explicitly, and the second to compute appropriate controller action required to drive the predicted outputs as close as possible to their respective desired values.

Industrial chemical, biotechnological and pharmaceutical processes are multivariable and nonlinear, and may exhibit difficult dynamic behavior due to time delays, inverse response, and open loop instability. Further, the process operations may be subject to constraints of all kinds to be satisfied by process inputs, process outputs, and certain state variables based on considerations for process economics and safety, environmental impact, and hardware (equipment) characteristics. An ideal controller required for optimal operation of these processes should be able to handle multivariable process interactions, time delays and other problematic dynamic behavior, input and output constraints, nonlinear process behavior, and influence of disturbance variables on the same, while optimizing the controller actions [438]. It should remain robust despite modeling errors and measurement noise and should be able to infer critical unmeasured information from whatever is available.

While no such ideal controller exists, the typical capabilities of MPCs come closest to the requirements for an ideal controller stated above. MPCs handle process interactions, time delays, inverse response and other difficult dynamics well. MPC utilizes a process model. A rigorous process model is not necessary since the MPC schemes can be based on non-parametric step-and impulse-response models. Being considered as an optimization, MPC is capable of meeting the control objectives by optimizing the control effort, while satisfying appropriate constraints. An attractive feature of MPC is compensation for the effect of measurable and unmeasurable disturbance variables. The compensation for measured disturbances is carried out in a feedforward mode, while that for unmeasured disturbances is carried out in a feedback fashion. A variety of references provide a good perspective of MPC [22, 174, 175, 176, 241, 376, 403, 420, 438, 477, 478]. MPC is best suited to processes with any of the following characteristics: (i) multiple input and output variables with significant interactions between single-input, single-output control loops, (ii) constraints in inputs and/or outputs, (iii) problematic dynamics such as long time delays, inverse response, and very large time constants. Although MPC is not inherently more or less robust than classical feedback control, it can be adjusted more easily for robustness.

Biotechnological processes have inherently slow dynamics. It therefore would take significant time for the full effect of each control action to be realized in the observable process outputs. It is therefore difficult to assess the full impact of the control actions taken in the past based only on the current output measurements. As a result, it is pertinent to consider how

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Figure 7.9. Example of elements in model predictive control: x — x: reference trajectory, y'\ o---o: predicted output, y\ A--A: measured output,

From BA Ogunnaike and WH Ray. Process Dynamics, Modeling, and Control. New York: Oxford University Press, Inc., 1994. Used by permission.

the process output will change in the future if no control action is taken (model-based prediction) and to target control action as a compensatory effect for what will need to be corrected after the full effects of the previously implemented control action have been completely realized. This is the motivation behind the MPC methodology.

The MPC design methodology consists of four elements: (i) specification of reference trajectories for the process outputs, (ii) model-based prediction of process outputs, (iii) model-based computation of control action, and (iv) update of error prediction for future control action. The variations in different MPC schemes are based primarily on how each element is implemented in the MPC scheme. The continuous-time process operation is comprised of successive time intervals. The four elements of MPC must be updated in each time interval. For this reason, it is convenient to work with discrete-time models for process and controllers. The discrete-time models are naturally well suited since most MPC schemes are implemented using digital computers. Techniques for transformation of continuous-time models into discrete-time models have been discussed earlier in Chapter 4.

The first element of MPC involves specification of the desired traiecto-

ries for the process outputs, y*(fc) (Figure 7.9). For an individual output, this can be a fixed set-point value or a trajectory. The second element involves prediction of trajectory of process outputs y in response to changes in the manipulated variables u in the absence of further control action. At the present time k (t — kT, T — sampling period), the behavior of the process is predicted over a horizon p. For discrete-time systems, this leads to prediction of y(k + 1), y(k + 2),... ,y(k + i) for i sample times into the future based on all actual past control actions u(fc), u(fc — 1),..., u(fc — j) (Figure 7.9). In the third element of MPC, the same model as that used in the second element is employed to calculate control trajectories that lead to optimization of a specified objective function, which typically may include minimization of the predicted deviation of the process outputs from the target trajectories over the prediction horizon and minimization of the expense for control effort in driving the process outputs to their respective target trajectories. This is equivalent to constructing and utilizing a suitable model inverse to predict trajectories of the manipulated inputs. This optimization must of course be accomplished while satisfying pre-specified operating constraints. This element therefore involves prediction of the control sequence u(k),u(k + 1),..., u(k + m — 1) required for achieving the desired output behavior p sampling times into the future [from t = kT to t = (k + p — 1)T] (Figure 7.9). Usually, the prediction horizon p is larger than the control horizon m. For computations, all control commands for times (k + m) to (k + p) are kept constant at their values at time (k + m — 1). This reduces the computational burden during real time optimization. The last element of MPC involves comparison of the output measurements ym(k) to model-predicted values of the same, y(fc). The prediction error e(k) = ym(k) — y(k) [not to be confused with the controller input, e(k) = ym(k) — y*(fc)] is then used to update future predictions y(fc + l),...,y (k + i).

The conventional MPC schemes have relied on linear process models. Incorporation of nonlinear process models within the MPC framework is relatively recent [54]. Here we briefly review the three more commonly used forms of discrete models, the finite convolution models based on impulse-and step-response function, state-space models, and the transfer function models. For easier understanding, these models are presented below for SISO systems and can be readily extended to MIMO systems.

There are two entirely equivalent forms of finite convolution models, namely, the impulse-response model and the step-response model. The former can be expressed as k l/(fe)=I>(t Mk-i), (7.141)

with g(i) being the impulse response functions of the process. The step-response model can be expressed as k y(fc) = ^/3(z)Au(fc-i), (7.142)

with (3{i) being the step-response functions for the process and Au(k) — u(k) —u(k — 1). For all real, causal systems, both g(0) and /3(0) are considered to be trivial, hence such systems will exhibit the mandatory one-step delay. For a process represented by the two model forms in Eqs. 7.141 and 7.142, the equivalency of the two models follows by equality of coefficients of u(k — i), i = 0,1,..., k, leading to the following relations.

Impulse or step response coefficients can be obtained directly from experimental data or from other parametric model forms, if available, such as those discussed in the following.

State-space models relate the variations in state variables and hence process outputs over time to the past history of state variables and to process inputs. The models may be in the form of differential equations (Eqs. 4.44, 4.45, 7.1 and 7.2) or difference equations (Eq. 4.51). The models in these forms are more advantageous for state estimation using Kalman filters and extended Kalman filters than the time series models. The reader should refer to Section 4.3 for discussion on continuous and discrete Kalman filters.

Time series models such as ARMA, provide relations among output values at the current and previous sampling times and input values at the current and prior sampling times k k y(k) = a(i)y(k - i) + b(i)u(k -%-d), (7.144)

The effect of time delay is included when d > 1. For real, causal processes, it follows that a(0) = b(0) = 0. The coefficients a(i) and b(i) and the time delay, d, in Eq. 7.144 must be identified by fitting the model to experimental process data. The linearized continuous-time versions of the nonlinear continuous-time state-space models, such as in Eqs. 7.104 and 7.106 obtained from linearization of Eqs. 7.1 and 7.2, can be transformed into time series models as in Eq. 7.144 with relative ease.

As the name suggests, in a transfer function model, the process outputs are related to the manipulated inputs by transfer functions. For a SISO process for example, the output y and the input u are related as y(z) = z~dAl%1)u{z)■ (7145)

A{z~v) and B(z~l) are appropriate polynomials in the ^-transform variable, z~l, and the term z~d incorporating the effect of process time delay of d sampling times (= dT). The parameters of the transfer function model must be determined from experimental data for the process. As stated earlier, a rigorous process model is not necessary since non-parametric step-and impulse-response models can be readily employed for MPC in the absence of rigorous process models.

The "model inverse" required for prediction of manipulated inputs at future sampling times is carried out numerically as the solution of an appropriate optimization problem. Only the first computed change in the manipulated inputs is implemented. At time k + 1, the computations for the four elements of MPC are repeated with the time horizon moved by one time interval (sampling time).

A variety of factors are responsible for the discrepancy observed between the actual output measurements and model-predicted values of the outputs. These include the effects of unmodeled and unmeasured disturbances, fundamental errors in model structure, and parameter uncertainties. Since it is difficult to independently assess these effects, it is a common strategy to attribute this discrepancy entirely to unmeasured disturbances, assume that this discrepancy will remain the same over the prediction horizon, until better information is available, and update model predictions by adding this discrepancy.

Two pioneering MPC schemes are the dynamic matrix control (DMC) and model algorithmic control (MAC) and derivatives of these, such as the quadratic dynamic matrix control (QDMC) [175] and IDCOM [383, 384, 503, 504, 505, 506]. In the following, we briefly discuss the basic formulation of DMC.

Dynamic Matrix Control (DMC). Let the current time instant be k. In the absence of further control action, let the process output take the following predicted values over the future horizon of p sampling times: y°(k), y°(k + 1),... ,y°(k + p— 1). Let the vector of the predicted p values be represented as y°(fc) = [y°(k) y°(k + 1) ... y°(k + P~ l)f. (7.146)

The argument in the vector above indicates the time origin of sequential predictions of the process outputs and the superscript ° indicates that the

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