Damping
The terms “overdamped,” “underdamped,” and “critically damped” are commonly used to characterize a response.
Overdamped.
A system is said to be overdamped when its responseapproaches the final value without ever crossing to the other side of the finalvalue.Figure 3.23 illustrates two responses that are overdamped. The upper oneis initially below the final value, and it approaches the final value from belowwithout ever going above the final value. The lower one is initially above thefinal value, and it approaches the final value from above without ever goingbelow the final value.
Underdamped.
A system is said to be underdamped when its responsecrosses the final value one or more times as it approaches the final value.Figure 3.23 also illustrates two responses that are underdamped.The presence of one or more sinusoidal components in the response leadsto underdamped behavior. When the sinusoidal component is not present, the response will be overdamped.
Behavior of Processes.
Underdamped behavior is rarely exhibited by the process itself or, more specifically, with the controller on manual. If the pipe is of sufficient length for the inertia of the flowing fluid to be significant, under-damped behavior will be observed in the pressure. How do you close a valveon a pipeline? Very slowly, otherwise the spike in pressure could have adverse consequences. But within process plants, the inertial effects are inconsequential, and most valves can be closed as quickly as possible with no adverse consequences. Underdamped behavior is so rarely exhibited by a process that the statement is sometimes made that processes are inherently overdamped. If youever observe underdamped behavior with the controller on manual, first understand the origin of the underdamped behavior.
Behavior of Control Loops.
For most control loops, underdamped is the desired behavior. As the process is overdamped, the control loop will exhibit overdamped behavior at low values of the controller gain. However, sufficiently increasing the controller gain will cause most loops to exhibit under-damped behavior.The responses in Figure 3.24 illustrate the effect of the controller gain on the performance of a typical control loop. The controller contains only the proportional mode. The responses are to a step change in the set point of one unit. Proportional only controllers do not line out with the controlled variableequal to the set point (we will leave this explanation for a subsequent chapter). But note that overdamped/underdamped behavior is based on the lineout value, not on the set point.For a controller gain of 0.5%, the behavior is overdamped. Increasingthe gain to 1.0% gives a small overshoot and underdamped behavior. For controller gains of 2.0% and 4.0%, the response clearly has a sinusoidal component. For gains less than about 6%, the amplitude of the sinusoidalcomponent decreases with each successive cycle (a damped sinusoid). Forgains greater than about 6%/%, the amplitude increases with each successivecycle (an undamped sinusoid).
Critically Damped.
Critically damped is where the transition from over-damped to underdamped occurs. For the responses in Figure 3.24, criticallydamped occurs at a controller gain of approximately 1.0%. The small overshoot exhibited by this response would have no adverse consequences on the process. A common definition of “good response” is one that responds as quicklyas possible but with “no overshoot.” Any overdamped response has no overshoot, but how do you know it is responding as quickly as possible? One approach is to permit the response to exhibit a very small overshoot, perhaps 5% or sometimes as much as 10%. We shall refer to such responses as “minimal overshoot,” but they usually pass for “no overshoot” or “critically damped.”