Question

Figure 2a shows the response of a process to a setpoint (SP) increase of 5°C for two sets of controller tuning parameters, Figure 2b show the response of the same process with the same two sets of controller tuning parameters in response to a disturbance. Curve A has Kc and t=T, values tuned for the performance objective of quarter amplitude decay ratio. Curve B is tuned for the performance objective of minimal overshoot. Answer the following question based on Figure 2. p p * min Hot Water Temperature, ' c Hot Water Temperature, ' c A 5.0 7.5 55.0+ + 0.0 55.0 30.0 40.0 10.0 20.0 Time, min 10.0 Time, min Figure 2 (a) (b) (c) (d) (e) Is this an example of P, PI, or PID control? Why? Tuning parameters not shown may be assumed to be zero. Which curve (A or B) is more aggressive? Why? Which curve (A or B) is better suited for setpoint tracking? Why? Which curve (A or B) is better suited for disturbance rejection? Why? Is curve A overdamped, underdamped, or critically damped? Why? Which curve (A, B, or neither) is guaranteed to have a characteristic equation with complex roots? Why? Which curve (A, B, or neither) has a characteristic equation with positive real roots? Why? The process is a utility providing hot water to a plant. The setpoint is the temperature of the hot water utility. The disturbances are system throughput and utility demand changes. Which is likely to be the more important control objective for this process: setpoint tracking or disturbance rejection? (h)

Answer 1

a. The curve A is an example of PI controller. From the diagram, it can be observed that there are oscillations, but there is no offset. The PI controller just does the same. It introduces oscillations into the system and removes the offset.

The curve B is based on PID controller as there are no oscillations. If the aim is to remove the offset as well as the oscillations, the derivative action is added to the PI controller, which results in PID controller.

b. Curve B is more aggressive. As the derivative action of PID controller is anticipatory. It helps the system preact for any upcoming disturbances.

c. Curve B is more suited for set point tracking as the curve B has the lesser value of $K_{c}$ . It can also be understood as lesser the vaue of $K_{c}$ , nearer is the destination. i.e. the curve reaches to its set point more quickly than the curve A.

d. Curve A is more suited for disturbance rejection as the value  of $K_{c}$ is more, so it will take more time for the curve A to respond to the disturbance. Curve B leads to the system being disturbed immediately as the value of $K_{c}$ is less.

e. The curve A is underdamped as there are oscillations.

f.Curve A is guaranteed to have a characteristic equation with complex roots as the value of $\xi <1$. The roots obtained as complex conjugates.

g. The curve B shows the behavior of an overdamped or critically damped system, that is either . For both the conditions, there will be real positive roots.

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h. The setpoint tracking is more important as in the industry the hot water utility for a particular process is fixed for a particular temperature. The system just has to make sure that the desired temperature is maintained.