1. Electronic Control Systems Week 7 – PID Control
EET273
Electronic Control Systems
Week 7 – PID Control

2. PID Control
Readings: Ch. 29:6 – 29:9

3. Review - Controllers

4. Review – Controllers
Error is: the different between the system input (SP) and output (PV)
The purpose of feedback is to reduce system error
The purpose of a controller to process the system error in such a way that it reduces error quickly and efficiently
Proportional controllers work by multiplying the error by some scaling constant 𝐾 𝑃 , this is fairly effective but has limitations (creates offset and/or overshoot)

5. PID Control P – Proportional I – Integral D – Derivative
A controller can consist of 1 or more of these elements together, depending on the type of system/performance required
P controller
PI controller
PD controller
PID controller
We can use multiple types of controllers in parallel, and sum the outputs of each controller

6. Integrals An integral describes the area under a function
The area “S” describes the integral of the function f(x), integrated from a to b
Mathematically, this looks like this:
𝑎 𝑏 𝑓(𝑥) 𝑑𝑥
a and b are the “integration limits”

7. Integrals
Integrals can be computed directly using the methods of calculus, this will not be covered in this class
Integrals can also be approximated by dividing the area under a curve into boxes
Height of the box is the function value
Width of the box is some selected constant Δx
The smaller the Δx, the more accurate the approximation

8. Integral Controllers
For our purposes, the word “integral” can be used interchangeably with “accumulated sum”
An integral controller accumulates the error over time
If there is any steady state error in a system, the integral controller will continue to accumulate this error, and eventually drive it to zero
Because integral controllers only operate on accumulated changes in error, they are slower to respond to error than a proportional or derivative controller

9. Integral Controllers
Similar to a Proportional Controller, the amount of integral control is set by a constant 𝐾 𝑖
In some controllers, the constant 𝜏 𝑖 is used, rather than 𝐾 𝑖
𝐾 𝑖 = 1 𝜏 𝑖

10. Integral Control
Highly effective at controlling processes due to its ability to completely remove all system error
Slower response than proportional control (accumulating error takes time)
Too much integral control can cause oscillation
If 𝐾 𝑖 is set too high (or 𝜏 𝑖 too low, in other words), the integral controller’s output can saturate or “windup”
Integral “windup” can also occur if some external factor prevents the setpoint from being reached. In this case, the integral controller will continue accumulating, and eventually reach it’s maximum value

11. Derivatives
A derivative is mathematical function that describes the slope of another mathematical function
Ex #1:
𝑦=3𝑥
𝑦′=3
Ex #2:
𝑦=5𝑥 + 3
𝑦′=5

12. Derivatives
You can take multiple derivatives of the same function, called 2nd derivative, 3rd derivative, etc.
There are several types of notation for the derivative of a function
𝑦= 𝑥 2
𝑦 ′ = 𝑑y 𝑑𝑥 = 𝑦 =2𝑥
𝑦 ′′ = 𝑑 2 y 𝑑 𝑥 2 = 𝑦 =2

13. Derivatives
Several functions can have the same derivative, in fact every derivative has a “family” of anti-derivatives
Ex:
𝑦= 𝑥 2
𝑦′=2𝑥
𝑦= 𝑥

14. Derivatives – sines and cosines
Etc.

15. Derivatives of exponential functions
𝑦= 2 𝑥
𝑦= 3 𝑥

16. Derivatives of exponential functions
𝑒= …
𝑦= 𝑒 𝑥
𝑦′= 𝑒 𝑥
𝑦′′= 𝑒 𝑥

17. Derivatives
For our purposes, the word “derivative” can be used interchangeably with the word “slope”
Looking at the slope of the error allows our derivative controller to respond if there is a sudden change in error
This could be caused by a change in setpoint, a system switching on quickly, or a sudden disturbance in the system

18. Derivative Controller
While a proportional (P) controller is looking at the actual value of the system error, a derivative controller is looking at the slope of the error
Error is increasing quickly  lots of derivative control
Error is stable  very little derivative control
The amount of derivative control is set by the constant 𝐾 𝐷 , (or 𝜏 𝑑 )
Differential controllers are often used to respond to quick changes in error, and prevent the error from changing too quickly
This can help reduce overshoot coming from the P or I controller
Since the D controller is “keeping an eye out” for sudden changes, we can “get away with” more P and I than we could otherwise (and still avoid overshoot)

19. Derivative Controller
Differential controllers are often used to respond to quick changes in error, and prevent the error from changing too quickly
This can help reduce overshoot coming from the P or I controller
Since the D controller is “keeping an eye out” for sudden changes, we can “get away with” more P and I than we could otherwise (and still avoid overshoot)
Since D controllers respond to the error’s rate of change, they are more susceptible to high frequency noise – so be careful when implementing them!

20. PID Control
A PID controller is really just 3 controllers in parallel, a P, an I, and a D

21. PID Control Proportional: Integral: Derivatives:
Reacts to error in the present
Good for performing the bulk of error reduction
Results in “proportional-only offset”, which can reduced, but never eliminated with proportional only control
Integral:
Reacts to error in the past
Excellent at removing steady state errors
Slow to respond due to time needed to accumulate error
Derivatives:
Reacts to error in the future (anticipates error by looking at slope)
Starts working when error is changing quickly
Stops working when error is constant
Not good at eliminating steady state errors – if error is constant, D controller doesn’t care

22. PID Control

23. PID Control – some frightening math
Proportional Controller:
PI Controller:
PID Controller:

24. PID controllers – different configurations
P: Removes bulk of error, will always have offset (in a loaded system)
I: Completely removes offset (eventually), slower response
PI: Removes error quickly, and eliminates offset, may overshoot
PD: Removes bulk of error, D corrects for overshoot, will have offset
PID: Completely removes offset, responds quickly, D corrects for overshoot