A ‘ closed loop ‘ consists of :
Simple On/Off control could be thought of as starting the engine (G(s)) of the car (Plant), pressing the accelerator (Output) to the floor until the desired speed (SP) of 80 kph is reached as indicated on the speedometer (PV), then taking your foot completely off the accelerator. When the speed drops below 80 kph the accelerator is again depressed to the floor until 80 kph is again reached.
Comparison to the Proportional Term
If we wish to drive from a standstill to 80 kph we can consider the procedure we adopt to achieve this to explain the Proportional term.
From a standstill we depress the accelerator pedal. The speed of the vehicle will increase and at a certain speed below our target speed of 80 kph we begin to ease off the accelerator pedal in order to prevent cruising past the desired speed. This easing off the accelerator pedal can be directly compared to entering the Proportional Band or the band relative to the required speed. If we do not look at the speedometer we will certainly drive at a speed that is not our desired speed and an error will result.
Comparison to the Integral Term
If we now look at the speedometer we see that we are low of our desired speed and using this visual feedback we correct for the error and begin to slowly depress the accelerator pedal. As a result our speed slowly increases to achieve the desired speed of 80 kph.
This procedure can be compared to the Integral Time of a PID controller.
Comparison to the Derivative Term
We are now cruising at our Setpoint of 80 kph and we continue to maintain this speed through visual feedback of the speedometer.
If we encounter a sudden change in slope of the road such as a hill we correct for the reduction in speed which would result by depressing the accelerator pedal more than would otherwise be the case for the slight reduction in speed we initially encounter. As the slope of the road levels off again we ease off the accelerator pedal more than would otherwise be the case for the slight increase in speed to avoid raising our speed too far beyond our target speed.
The amount of correction and time to reduce this correction to zero can be compared to the Derivative Time.
When we have driven a car for some time these procedures become second nature to us and we do not think about the technique we use to drive. Consider, however, the process of someone learning to drive and you will observe these descriptions in action.
Different process variables such as temperature, speed, pressure etc. all have different characteristics. For example a temperature on an extruder barrel responds very slowly whereas the speed responds much more quickly and the pressure can respond more quickly still.
Loop ResponseThe combination of the three terms can provide a stabilizing effect on a process only if the terms are correctly set. This is referred to as ‘ tuning the loop ‘. If we ignore the situation of loop oscillation, there are three categories of loop performance :
Under Damped
In this situation the terms are set to prevent oscillation but do lead to an overshoot of the Process Value followed by decaying oscillation to finally settle at the Setpoint. This type of response does give a minimum time to Setpoint but overshoot may cause problems in certain situations and the loop may be sensitive to sudden changes in Process Value. This will result in further decaying oscillations before settling once again.
Critically Damped
This represents and ideal situation where overshoot does not occur and the process responds to changes in a controlled, non oscillatory manner.
Over Damped
In this situation the loop responds in a controlled but sluggish manner which will result in a loop performance which is non ideal and unnecessarily slow. The balancing of the P, I and D terms depends totally upon the nature of the process to be controlled.
In a plastics extrusion example, a barrel zone will have a different response to a casting roll, drive loop, thickness control loop or pressure loop. In order to achieve the best performance from an extrusion line all loop tuning parameters must be set to their optimum values. Needless to say, many extrusion lines and industrial equipment are not set up to give their best performance.
Ziegler Nicholls: This method involves putting the loop into oscillation and measuring the oscillation period T. The oscillation is caused by setting the I and D terms to off and reducing the Proportional Band until the loop just oscillates. The setting of the P term at the onset of oscillation (Px) is used to determine the desired Proportional Band.
Cohen & Coon: In this method power is applied from a low Process Value and a note is made of the time before any response is observed ( dead time ) and the maximum rate of change of the Process Value. By combining this information with the final settling of the Process Value resulting from the power applied, the P, I and D values are calculated.
Popular Method (Rule of Thumb)
An easier method of manually determining the P, I and D values is to set the Integral and Derivative terms to off and to set the Proportional Band to a minimum. The resulting oscillation width Xosc and the oscillation period tosc can be used to determine the PID settings shown below.
P = 2.0 * Xosc
I = 1.5 * tosc
D = I / 5
These values result in a slightly over damped response and consequently all values can be reduced slightly to give a more responsive loop.
Avoiding Overshoot
Various techniques are used to avoid overshoot. The aim is to prevent the process value from exceeding the desired value or setpoint. In the example shown the proportional band is brought down to meet the approaching process value in order to reduce the output power and to smoothly bring the process value to the desired setting.
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