Closed-Loop Tuning of Controllers ◾  89
amplitude decay ratio of about 1/49; that is, the first overshoot
was about 49 times higher than the second overshoot. It was
important to develop the optimum 7 to 1 pattern of response
without any significant influence from integral (reset) action
or derivative action in the controller. When there is no automatic reset, that is, no integral action in the controller, the first
peak in the process variable response may not get up to the
new setpoint, and the control loop can be left with offset from
the setpoint. So, a method was developed that looked for the
height between the first peak and valley in the process variable response to be 16% of the height of the step change. In
other words, for a step change in setpoint up 10°C, the height
between the first peak and valley is 1.6°C when the proportional gain is at the optimum value. Similarly, when the step
change in setpoint is down 10°C, the valley will be first, and
the height between the first valley and peak is 1.6°C. The size
of step change in setpoint in an actual controller needs to be
small enough so the manipulated variable (controller output)
does not saturate at 100% or 0%, because that can interfere
with the development of the height between the peak and valley and with the response time.
Figure 9.1 shows the pattern that can be recognized when
a control loop is tuned for minimum IAE (integral of absolute
error) after a setpoint change. The setpoint for the process
variable was increased by a step change from 50 to 60 at time
zero. The process variable crossed the new setpoint at 10 min
after the setpoint change and reached the first peak at 13.6
min. The time to reach the first peak is called the response
time in the descriptions that follow. The first peak height in
the process variable response was 61.6, and the first valley
height was 59.6; so the height between the first peak and valley was 20% of the size of the step change. The decay ratio
was 400; that is, the first peak was 61.600, and the second
peak was 60.004. The controller output, that is, the manipulated variable, stayed within the range of 0% and 100%. The
parameters used for the simulation were process gain (K p ) of