KC = the controller’s proportional gain (e.g., how far the accelerator is pressed or released per increment of error).
The response of the proportional control loop depends on the value of the gain. Suppose KC is 3.9 mm per km/h (0.25 in. per mph) above or below the setpoint speed. If this is too large, a drop in speed would cause the cruise control system to press the accelerator too far, increasing the speed too much, and then the system would release the accelerator too far, slowing the car too much. If it exceeds a critical value, a disturbance could cause the system to oscillate. So, KC affects the system’s stability, which is a crucial consideration in control system design.
If, on the other hand, KC is too small, a drop in speed would not be compensated for sufficiently, and the car’s speed would settle at a new value with a nonzero error. This steady-state error is called the controller offset (droop), which cannot be completely eliminated while maintaining a stable system, except by an integral control loop.
An integral control loop continuously adjusts the manipulated variable at a rate proportional to the amount of error until the offset is eliminated. As such, when the car starts moving up a hill, the proportional control loop presses the accelerator a preset distance, which partially compensates for the incline. The integral control loop continues pressing the accelerator until the setpoint speed is reached, which can completely eliminate the controller offset.
The combination of proportional and integral control is described as
Where
| tI | = | integral time constant (reset time) and |
| 1/tI | = | minutes per reset (i.e., how fast the controller increases its action in proportion to the amount of error). |
If 1/tI were large, the controller would press the accelerator rapidly whenever the speed dropped. As a result, the car would still be accelerating when the setpoint was reached and would overshoot the desired speed. During each overshoot, the controller would correct itself, causing the system to oscillate around the setpoint. Depending on KC and tI, the oscillations would either decrease until the system settles at the setpoint or they would increase, indicating that the system is unstable.
The third type of control in a PID controller is a derivative control loop, which adjusts the manipulated variable in proportion to the rate of change of the process variable. As such, when the car’s speed changes, the accelerator is moved in proportion to how fast the speed is changing. This helps dampen changes in response to large disturbances. (Derivative control is rarely needed in wastewater treatment applications.)
The equation for combined PID control is
Where tD is the derivative rate parameter (how much the controller responds based on the error’s rate of change). Therefore, in a PID controller, the proportional control loop responds to the control variable’s current value, the integral control loop responds to the control variable’s history, and the derivative control loop anticipates the control variable’s future values. Because PID is the most common control mechanism used, it is implied, if not explicitly stated, in many of the strategies in the remainder of this chapter.
Selecting appropriate proportional-gain, reset-time, and rate-of-change parameters for a PID controller can be difficult. One option is to calculate the parameters based on a simple analytical model of the process using the basic control theory found in textbooks. Another is to tune the control system experimentally by introducing a disturbance to the treatment process, observing the dynamic response, calibrating a simplified model based on that response, and then calculating the parameters based on the model, again using basic control theory. To compare the effectiveness of various controller settings, engineers typically use standardized controller-performance measures such as minimum offset, one-quarter decay ratio, or minimum integral square error (Stephanopolous, 1984).
Some of the most common methods for tuning a PID controller are experience-based principles, online trial and error, Cohen–Coon and Ziegler–Nichols analytical methods, and computer simulation. Experience is the tuning method of choice for common control loops (e.g., flow, level, pressure, or temperature). In flow control, for example, engineers typically set the proportional gain low to reduce the effects of noise, which are inherent in many flow meters (Luyben, 1973). They also set the integral reset time low to respond quickly to changes in setpoint error.
The online trial-and-error and Cohen–Coon and Ziegler–Nichols methods are experimental. During online trial and error, engineers repeatedly double KC until the process starts to oscillate (Luyben, 1973). The value of KC at this point is called the ultimate gain. They then set KC at half the ultimate gain. Then, they repeatedly double the integral control loop by repeatedly halving tI until the system begins oscillating again. They then set tI to twice that value. Finally, they increase tD until signal noise begins to affect the system and then set tD to half of that value. Engineers repeat this procedure using smaller changes in controller settings until the desired controller performance is achieved.
In the Cohen–Coon method, engineers first allow the process to achieve steady state without the controller (Stephanopolous, 1984). They then change the manipulated variable and plot the process variable’s response over time as the process returns to steady state. This plot is known as the process reaction curve, from which two measurements are made. This curve is used to estimate the process gain, process time constant, and dead time (a first-order relationship with dead time is assumed). Engineers then use these values to calculate the controller tuning parameters KC, tI, and tD.
The Ziegler–Nichols method involves using formulas to calculate tuning parameters based on KC measured in the trial-and-error method. Both the Cohen–Coon and Ziegler–Nichols methods have some practical shortcomings. They are not always accurate and must be followed by 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.