1. EEN-E1040 Measurement and Control of Energy Systems Control I: Control, processes, PID controllers and PID tuning Nov 3rd 2016
If not marked otherwise, graphs and formulas in the presentation
are from Åström & Hägglund:
PID Controllers - Theory, Design, and Tuning (2nd Edition)

2. What is control?
Intentional modification of a system’s state via external means
Used for bringing a system (close) to a state the user desires
Applied to various quantities (temperature, flow, lighting etc.)
Can be manual or automatic, discrete (ON/OFF) or continuous (proportional)

3. Example: manual ON/OFF control
ACME Electric Heater

4. Example: automated ON/OFF control with a setpoint
ACME Electric Heater MK2
20.0
set
5.0
Indoor
temperature …
30.0 …
15.0
28.7 …

5. Example: automated PID (proportional integral-derivative) control with a setpoint
ACME Electric Heater MK3 PID
20.0
set
5.0
Indoor
temperature …
23.0 …
19.0
20.0 …

6. Process models
Let’s go back to the electric heater example and present it in the form of a diagram:
Measured
temperature
(process variable) y
Control
variable u
Setpoint
ysp Σ -1
Error e
Heating element
(process)
Controller

7. Process characterization
Static response: if control variable u is fed to a process, what will be the value of the process variable y?

8. Process characterization
Dynamic response: if control variable u is fed to a process, how will the value of the process variable y behave over time?

9. Process characterization
Dead time,
time constant,
Important aspects for control:
instability,
nonlinearity

10. Proportional control Let’s consider our model process from before: Σ
Heating element
(process)
Controller
Setpoint
ysp
Measured
temperature
(process variable) y
Control
variable u Σ -1
Error e

11. Proportional control
ON/OFF control  value of u same regardless of magnitude of e
Instability and oscillations
Proportional control: magnitudes of u and e linked
Overreaction to minor errors removed

12. Proportional control
Controllers with only proportional action have a steady-state error
For derivation, see Åström’s book, pp
To get rid of the error, we need something else…

13. PID controller - Background
(P)roportional-(I)ntegral-(D)erivative
By far the most common control algorithm
(over 90% of all industrial control loops)
When properly tuned, provides accurate control for a wide variety of processes

14. PID controller – Formulation
Let’s consider the controller output u as a function of e – the difference between the setpoint and the measured value: P I D

15. PID controller – Proportional action
With only proportional action, PID controller equation is reduced to
ub is controller bias/reset, often zero but can also be something else

16. PID controller – Static example of proportional actionΣ
Load disturbance l
Process output
Measurement noise n Σ x
Process
Controller
ysp
Measured
temperature
(process variable) y u Σ -1 e

17. PID controller – Static example of proportional action
From the relations obtained from the block diagram, we get
K = controller gain, Kp = process gain, KKp = loop gain
In a system with no measurement noise and zero controller bias, x = y (process variable)  high loop gain results in low steady-state error and resistance to load disturbances
Why not always set controller gain to a very high value?
In reality noise is always there and a high loop gain amplifies its effect
Deciding on an optimal loop gain is not easy, and it is always a tradeoff between different objectives for the control

18. PID controller – Dynamic processes
Almost every real process is dynamic
Time-dependent process gain and load disturbance
Setting loop gain too high leads to instability
How to solve steady-state error?

19. PID controller – Integral action
Always removes the steady-state error of the control
Steady state: constant error 
As long as error is nonzero, u will vary over time
Large Ti  slow convergence but less oscillation

20. PID controller – Derivative action
Dynamic closed loop systems often unstable due to dead time (process reacts to control too slowly)
To avoid this, prediction of future error needed
Consider a PD controller:
First order Taylor series expansion for the error:
Thus, control signal ~ linear approximation of error after time Td

21. PID controller – Derivative action

22. PID controller – When to use PI and PID
Derivative control quite often left out
PI sufficient for first order processes (ones that stabilize when control parameter is kept constant)
PID used for second order processes that involve oscillations and/or instability
Also useful for first order processes with dead time

23. PID controller – Caveats and limitations
Integrator windup
Occurs when control variable saturates due to actuator bounds
For example control valves can never be more than fully open
Typically happens with large sudden variations of setpoint
Can be avoided with modified control algorithm or limitations in setpoint variation
Higher order processes
Typically complicated processes with an order larger than 2 require more than PID for accurate control
Processes with large dead time
Error in the linear approximation via the D term gets large if the prediction needs to be made far into the future
Sophisticated predictive controllers better than PID in these cases

24. PID tuning – Process reaction curve
Open loop method
Controller set in manual mode
Requires process to be stable
How to do:
Start logging the process variable
Introduce a step change to the control variable
Wait until the system reaches a new steady state
Plot the logged data and analyse it to find the controller parameters

25. PID tuning – Process reaction curve
Graph and formulas taken from

26. PID tuning – Ziegler-Nichols closed loop method
Controller active during tuning
How to do:
Start logging the process variable
Remove integral and derivative action from the controller
Set the control gain to a constant value
Modify the setpoint slightly
Observe oscillations in the process variable
If dampening, increase gain
If amplifying, decrease gain
Repeat step 5 until stable oscillations are found
Record the gain value and the period of the stable oscillations

27. PID tuning – Ziegler-Nichols closed loop method
Graph and formulas taken from

28. PID tuning – Fine-tuning of a system
Effects of PID parameters when increasing them:
Rise Time
Overshoot
Settling time
Steady-state error
Stability K
Decrease
Increase
Very small effect
Degrade tI
Eliminate tD
No effect
Improve if tD small

29. Thank you!