trial-and-error refinement. In addition, the experiments required may be infeasible. For example, it may be impossible to achieve initial steady state for the Cohen–Coon method. If this is the case, then analytical or computer-simulation methods can be used.
Both analytical and computer-simulation methods require a fairly reliable mathematical model of the process. The analytical method involves sophisticated mathematical procedures, termed Laplace-domain synthesis and frequency-domain synthesis, to calculate the values of tuning parameters. If the model is too complex, the analytical method may be impractical. If this is the case, then a computer can use the model to simulate the process and the PID controller. Engineers can then find tuning parameters via the trial-and-error method using the simulation rather than the actual process.
Feedback control loops and PID controllers can control dissolved oxygen levels in activated sludge reactors (Corder and Lee, 1986). They can also control sludge age in an activated sludge process by manipulating the waste flowrate (Vaccari et al., 1988).
4.6 Cascade Control
An extension of PID control is cascade control. Cascade control uses two control loops to control a process: an inner “slave” control loop that has a physical controlled variable and an outer “master” control loop that does not. The controlled variable of the master controller is the setpoint of the slave controller. The slave controller must have a response time that is faster than the master controller so that the setpoint of the slave control loop does not change too fast and cause instability.
A common use of cascade control in wastewater treatment is when analytical instruments are used to control a process. In general, the analytical signal has a slower response time than a flow or pressure signal. In this instance, the flow or pressure can be used as the process variable of the slave loop for controlling those disturbances quickly. The analytical reading can be used as the process variable of the master to control the setpoint of the flow or pressure slave loop.
4.7 Rules-Based Control
Rules-based control can be considered a set of rules that are often implemented with a simple “if-then-else” structure. Rules-based control often can be applied with little quantitative knowledge of the actual processes to be controlled. Rules-based controllers are typically simple, transparent, intuitive, and easily understood. For instance, the header-pressure setpoint for an aeration control system might be periodically (e.g., every 5 minutes) adjusted by the following rule:
If (any valve position >95%) 0.05 psi)
Else if (any valve position <20%) then (decrease pressure setpoint by 0.05 psi)
Else (do not change pressure setpoint).
Other more complicated rule-based strategies have been implemented on biological nutrient removal (BNR) projects and real-time control of sewers. Vitasovic (2006) discussed the advantages and disadvantages of rules-based controllers compared to optimization controllers for real-time control of urban drainage systems. While optimization controllers can provide better control under some conditions, they require more work to implement and maintain and may not be justified in all conditions.
4.8 Model Predictive Control
Hill (2010) stated that the term, model predictive control (MPC), does not refer to a single specific control algorithm or strategy, but rather a class of control methods that compute a sequence of manipulated variable adjustments to optimize the future behavior of a process or facility. Model predictive control was originally developed to meet specialized control needs of petroleum refineries and power plants. Model predictive control technology can now be found in a wide range of applications, including chemical processing, food processing, aerospace, and paper and pulp industries. In the environmental utility field, model predictive control has potential for processes with poorly defined kinetics and long time delays. It can be used for processes with time constants that vary from seconds (blower control) to minutes (dissolved oxygen control) to days (sludge wasting).
Controller structures of MPC algorithms have a number of common elements including
• Explicit use of a model to predict process performance (outputs) at future time instants over an appropriate time period called the prediction horizon;
• Calculation of a control sequence to minimize an objective or cost function; and
• Use of a receding horizon where the MPC algorithm’s first output is used (i.e., sent to the control element or loop) and then another entire sequence is calculated at the next sampling interval.
The MPC algorithms differ among the models used to represent the process, the noise and cost functions to be minimized, how constraints are implemented, and what optimization algorithms are used. Methodology of all MPC controllers is characterized by the following strategy (Figure 7.5):
• Future outputs for the entire prediction horizon N are predicted at each sampling interval t using the process model. These predicted outputs y(t+k|t) for k = 1N depend on the known values up to the instant t (past inputs and outputs) and on future control signals u(t+k|t) for = 0…N-1. The notation y(t+k|t) means the value of the variable at the instant t+k calculated at instant t;
• The set of future control signals u(t+k|t) is calculated to minimize an objective or cost function. Most objective functions attempt to keep the process as close as possible to either a particular setpoint or a reference trajectory. The objective function is often a quadratic equation (least squares solution) of the errors between the predicted output signal and the reference trajectory;
FIGURE 7.5 Model predictive control strategy.
• The control signal u(t|t) is sent to the process while subsequent control signals are discarded; and
• At the next sampling interval t+1, steps 1 to 3 are repeated. In principle, the control signal calculated at time t+1, u(t+1|t+1), will not be identical to the control signal calculated at time t, u(t+1|t), because new information was available for the calculation performed at the later time. This is the receding horizon concept.
There are literally hundreds of references to industrial applications of MPC. Many of these are in nonmainstream journals and proceedings such as Automatica, Proceedings of the International Federation of Automatic Control, Institute of Electronics and Electronic Engineers Transactions on Automatic Control, and the European Journal of Control.
Model predictive control has not been widely adopted by environmental utilities. There are a number of barriers, both economic and technical, that could explain the lag between environmental utilities and their industrial counterparts in reference to adapting advanced control strategies. In general, environmental utilities do not experience the same economic pressures to innovate and face significant technical barriers in doing so. These barriers include the following:
• Unlike industrial processes, it is seldom possible to control inputs to a water resource recovery facility (WRRF). Future inputs must be estimated if they are to be used for MPC. This forecast adds additional error to the control strategy;
• Some inputs of importance are difficult to measure in real time. For instance, a common
