Fig. 27.11. Proportional Integral controller performance using Z-N tuning rules

Derivative control action can improve the controller performance in slow processes. In this example, derivative action is not justified since the process responds fast to changes in the manipulated variable. Derivative action becomes beneficial if we include in the model the dynamics of a temperature sensor with a time constant of 0.3 s. Roughly speaking, the time constant tell us how fast the system responds, which is approximately 5 times the time constant. Therefore, the temperature sensor approximately takes 1.5 s to reach a new steady state after a disturbance. The ultimate gain of the modified model takes on the value 0.212 [GPM °C-1] and the ultimate period the value 1.2 s. Hence, the PI Z-N tuning results in Kc = 0.0954 and = 0.996. For proportional-integral-derivative tuning (PID tuning), the Z-N rules state that Kc = 0.6K„, rt = 0.5Pu, and rd = Pu/8; for our example this results in Kc = 0.126, r, = 0.6, and rd = 0.15. The response of both settings is shown in Fig. 27.12.

The figure shows that the derivative action reduces the oscillations, so that the measured variable settles at the set point value earlier than with PI control. The difference does not appear very marked in this case, but will be more noticeable in slower processes. However, care must be taken with noisy measurements, since the derivative action can cause undesirable oscillations and instability. In these cases, a low pass filter as described in Chap. 26, is essential.

PID is a good option for industrial control in standard, simple applications. However, the tuning of PID controllers can prove difficult when a process shows complex dynamics, interaction between different control loops, and large and variable time delays, such as occur in large scale SSF bioreactors. In these cases, even periodic attention and retuning do not assure good performance. Model Predictive Control (MPC), also known as Model Based Control, is a popular option nowadays in the process industries since it overcomes most of the limitations of PID control. We will present next the basics of this technique.

Fig. 27.12. Comparison of PI and PID controllers with Z-N tuning in the case where the temperature sensor has a time constant of 0.3 s

Fig. 27.12. Comparison of PI and PID controllers with Z-N tuning in the case where the temperature sensor has a time constant of 0.3 s

These control algorithms use a process model to predict future outputs, based on past input and output values. At each sampling time, future control movements are calculated that minimize a weighted function of predicted deviations from the set point and control movements. The general algorithm explicitly includes constraints on process inputs and outputs. In addition, MPC can be designed to include a different number of manipulated and controlled variables. While the implementation of these controllers in a real process is difficult and time consuming, they can operate without supervision over long periods. MPC would therefore be particularly useful to control large-scale SSF bioreactors. These algorithms have been widely presented and discussed in standard process control texts (Ogunnaike and Ray 1994) and specialized books (Maciejowski 2002).

The minimization problem can typically be formulated in the following form:

Kuk KuktC 1 1=1

Here, yk is the vector of predicted plant outputs at time interval k, rk is the vector of respective set points, and Auk the vector of control moves. The two matrices WD (one with superscript "y" and the other with superscript "Au") are diagonal matrices with weights that penalize the output deviations from the set points and the control movements respectively.

The process engineer can make some manipulated variables move more than others and get smaller deviations in some specified plant outputs, by adequately tuning the weights. In addition, the prediction horizon, P, defines the period over which the cost function will be minimized (see Fig. 27.13(a)); a large P assures a smooth and stable performance of the controller and should cover over 80% of the settling time (the time taken by the measured variable to settle into a new steady state value after an operating variable is moved). In turn, the control horizon, C, establishes the length of the sequence of future control moves (see Fig. 27.13(b)); heuristics suggests that C << P. The minimization problem also includes constraints on inputs and outputs, therefore, in the expression above, uL and yL represent the lower bounds, and uu and yu represent the upper bounds. This optimization problem does not have an analytical solution, except when a linear predictive model is used and no constraints are included, thus in the general case, the problem should be solved numerically.

new set point previous set point settling time prediction horizon M-►

measured outpi its predicted outputs past control calculated moves control moves

Auk.

Auk cont

&uk

AUk+2

Fig. 27.13. Model predictive control. (a) The deviations between the set point and the future outputs over the prediction horizon are minimized; this minimization is performed at each sample time k. (b) The algorithm calculates all control moves in the control horizon at each time k, although it applies only Auk. Next, at time k+1, all the calculations are repeated

In next chapter several simulation results are presented with applications of MPC to the control of SSF bioreactors. Some guidance on how to tune the algorithm is given and it is also illustrated how MPC can help to overcome the difficult dynamic behavior of SSF processes.

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