Theory
Proportional term
The proportional term makes a change to the output that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant.
The proportional term is given by:
where
Pout: Proportional term of output
Kp: Proportional gain, a tuning parameter
e: Error = SP − PV
t: Time or instantaneous time (the present)
Integral term
The contribution from the integral term is proportional to both the magnitude of the error and the duration of the error. Integrating the error gives offset that should have been corrected previously. The accumulated error is then multiplied by the integral gain and added to the controller output.
The integral term is given by:
where
Iout: Integral term of output
Ki: Integral gain, a tuning parameter
e: Error = SP − PV
t: Time or instantaneous time (the present)
τ: a dummy integration variable
Derivative term
The rate of change of the process error is calculated by determining the slope of the error over time. The magnitude of the contribution of the derivative term to the overall control action is termed the derivative gain, Kd.
The derivative term is given by:
where
Dout: Derivative term of output
Kd: Derivative gain, a tuning parameter
e: Error = SP − PV
t: Time or instantaneous time (the present)
Experimental Rig and Instrumentation
Figure 1 represents a similar setup to the one used in the lab
Results & Data
Two different types of test were performed the first type of test one where the P, PI and PID controllers were used to control flow, data can be found below on graphs 1-8, and the second type of test one where water was heated to a set temperature using the P, PI and PID controllers data below on graphs 9-11
Graphs 1-8 have bottom x-axis in time in seconds and y-axis of flow
Graphs 9-12 have bottom x-axis in time in seconds and y-axis of Temperature in degrees Celsius
Graph 1
P controller: for the first test data graphed above, gain was set 1 after initial settling it continues to rise, has a steady state error between 0.2-0.3 therefore it cannot reach a zero errror.
Graph 2
P controller: for the second test gain is set to 1.5.Intial overshoot when over the line ,has a longer settling time thean test 1, has added distrubance shown by joints in graph again has slightly decreased steady state error to that of test 1.
Graph 3
PI controller: integration time of 5 much faster to settle than P controller, hits line therefore zero error and disturbances have a big impact but line is fixed faster. No steady state error.
Graph 4
PI controller: integration time from 5-0.7 starts with huge overshoot though settles quicker than test 3, it hits the line therefore no error, though disturbance have a greater impact with the reduce integration time.
Graph 5
PID controller: derivative time of 0.2, rise time is quicker, overshoot less, quicker settling time, graph is one of noise due to the electronic noise from humans or/and machines.
Graph 6
P controller: Gain of 1.4, drastic increase in intial overshoot, decrease in rise time, steady state error of roughly 0.1-0.2.
Graph 7
PI controller: Gain of 1.2, Time intergral of 1.5, massive overshoot though quickly settles, it is also effected greatly by disturbances but quickly returns to line, as on the line there is no steady state error
Graph 8
PI controller: Gain 1.2 intergral time 0.7 very quick settle, very close to zero error again no steady state error and No disturbances.
For the below graphs 9-12 the water temp was increase from about 20 °C to around 50 °C
Graph 9
P controller: Measuring the heating of water at 1.7kw to a temp of 58 °C the graph is linear and takes just under 19mins to reach a temp of 50°C though continues to hear water thereafter.
Graph 10
PI controller: Gain of 17.5 and integral time of 45mins looks like it hits 50 °C but doesn’t and has a log like graph and takes just over 13mins to reach near to 50°C though start at a temp of about 36°C.
Graph 11
PI controller: Gain of 15.8 and intergral time of 1.19 mins goes over 50°C when at roughly 59°C stops heating water and allows to cool. The graph looks like a log graph until towards the end when it begins to fall.
Test 12
PID: Gain of 21 integral time of 0.02 differential time of 0.18, no graph as would be similar to that of test 5, too much interference, spikes between 0,2.4,1.2 noise corrupting signal makes it look like a bad manual.
Discussion
Benefits and disadvantages of P controller: high proportional gain results in a large change in the output for a given change in the error. If the proportional gain is too high, the system can become unstable. If the proportional gain is too low, the control action may be too small to respond to system disturbances and when there are no disturbances, proportional control will not settle at its target value, but will retain a steady state error.
Benefits and disadvantages of PI controller: accelerates the movement of the process towards set point and eliminates the steady-state error, though since the integrator is responding to accumulated errors it can cause an overshoot of the set point value.
Benefits and disadvantages of PID controller: derivative control is used to reduce the magnitude of the overshoot produced by the integral component and improve the combined controller-process stability, though derivative control is prone to the effects of noise.
Steady state from the test performed affects all P controllers at differing levels dependent on gain. Steady state error is eliminated or near eliminated in all PI controller tests
Noise only affected PID in tests performed but the effect was extreme and lead to unstable pattern.
Conclusion
That the choice of controller P, PI, PID is dependent on the required application i.e. wouldn’t use PI or P for a lift as overshoot would be a disaster, that all PID controllers have their flaws and a combination of one or more would give the best value towards the set value without overshoot or high error.
A PID controller needs to be shielded as the effect of electrical noise from other machines and humans causes such an unstable effect.
Disturbances can have a massive effect on both P and PI and have to be taken into account in application as P is less effected takes longer to return to the line.
References
http://en.wikipedia.org/wiki/Ziegler%E2%80%93Nichols_method
http://en.wikipedia.org
David Sellers. "An Overview of Proportional plus Integral plus Derivative Control
