2. Optimization-based PI/PID control for SOPDT process

3. Summary on optimization-based PI/PID control

4. Best achievable IAE performance by PI/PID control
of FOPDT process

8. Optimal rise-time vs, IAE in PI/PID control of SOPDT process

9. Optimal rise-time vs, IAE in PI/PID control of SOPDT process

10. FOPDT

11. SOPDT

12. Loop transfer functions of IMC-PID Controllers
According to the IMC theory, nominal loop transfer function of a
control system that has an inverse-based controller will be of the
following:

13. IMC-PID for FOPDT process

14. Loop transfer functions of IMC-PID Controllers
FOPDT processes:
SOPDT processes:
the resulting loop transfer function becomes:

22. Inverse-based Controller Design
We should learn what happens to the Z-N tuned controllers?
How inverse-based controllers are synthesized?

23. Loop transfer functions of Inverse-based Controllers
Inverse-based synthesis approach is used
Target loop transfer function (LTF)
This LTF has satisfactory control performance as well as reasonable stability robustness
ko and a are selected to meet desired control specification
Defaulted value:
ko= a= GM = PM = 60 o

24. PI/PID Controllers Based on FOPDT Model
ko=0.65 , a=0.4
A direct synthesis approach is used
PI controller
ko=0.5
Controller parameters (actual PID)

27. PID Controller Based on SOPDT Model
Controller parameters (ideal PID)
ko=0.5

29. Gain margin vs. phase margin at a=0.4

31. Auto-tune Autotuning via relay feedback: Astrom and Hagglund (1984)
Referred as autotune variation (ATV): Luyben (1987)
Main advantage: under closed-loop

32. Apply Z-N or T-L tuning

33. MODEL-BASED CONTROLLERS DESIGN
Reduced order models
FOPDT
Monotonic step response
For zero offset, PI or PID controller is considered
Usage of PI or PID controller depend on:
The application occasions
The dynamic characteristics of given process
Processes are classified into two groups for controller tuning
- Underdamped SOPDT
Oscillatory step response

34. A: Ku > 1 Criterion for Classifying model order
In general, processes with overdamped or slightly underdamped SOPDT dynamics can be identified with FOPDT models for controller tuning
Q: When an SOPDT process could be reduced to an
FOPDT parameterization?
A: Ku > 1

36. PI/PID Controllers Based on FOPDT Model
ko=0.65 , a=0.4
A direct synthesis approach is used
PI controller
ko=0.5
Controller parameters (actual PID)
In terms of
ultimate data
(Ku = kp kcu)

37. Defaulted value: ko=0.55 a=0.4
PI controller
Defaulted value: ko= a=0.4
In terms of
ultimate data

38. PID Controller Based on SOPDT Model
Only PID controller is used for significant underdamped SOPDT dynamics, i.e.
Controller parameters (ideal PID)
The values of kp and need to be estimated in advance
ko=0.5
In terms of
ultimate data

39. Dynamic
Process
FOPDT Model
SOPDT Model
Group I
Group II
PID
Controller PI
Controller
PID
Controller

40. Estimation of process gain kp
Start the ATV test with a temporal disturbance to setpoint or process input
Define
and have
cycling responses

41. Estimation from is subject to error,
sometimes as high as 20%
From Fourier series expansion
Ultimate gain is computed exactly as:

42. Estimation of Apparent Deadtime
In an ATV test, two measured quantities are used to characterize the effect of the apparent deadtime
For SOPDT process, this two quantities are
functions of and

43. Underdamped SOPDT processes

44. Algorithm for estimation of apparent deadtime
Starting from a guessed value of
Calculate and , and feed them into networks to compute and
Check if the eq. holds
If not, increase the value
of until the above eq.
holds. At that time,
is the estimated apparent
deadtime

45. In ATV test, it provides and which are functions of and
Locate
on this figure
Zone I:
FOPDT parameterization
Zone II:
SOPDT parameterization

46. Initiate ATV test by a short period of manual disturbance and record y(t) and u(t) until constant cycling is attained
Compute kp and kcu
Estimate the apparent deadtime
Classify the process by the location of
If the process belongs to Group I, tune PI or PID controller based on FOPDT model parameterization
If the process belongs to Group II, tune PID controller based on SOPDT model parameterization

47. Examples

49. Ex. 3
Ex. 1
Ex. 2

50. Ideal PID controller with an extra filter
The value of kp and need to be known in advance

51. Optimal IAE Value for Set-point Tracking
PI control
PID control
These optimal systems have reasonable stability robustness
PI control gain margin = 2.6 For unit step set-point change (Huang and Jeng, 2002 )
phase margin = 55o
PID control gain margin = 2.1, phase margin = 60o

52. The optimal IAE value occurs at a phase margin about 30o to 50o
Optimal System for Disturbance Rejection
Control systems designed for optimal input disturbance response will give smaller gain margin and phase margin than those designed for optimal set-point response.
The optimal IAE value occurs at a phase margin about 30o to 50o
trade-off between disturbance performance and phase margin is not always needed
PI control
PID control

53. Optimal IAE Value for Disturbance Rejection
The smaller the gain margin is (i.e. less robust), the lower the optimal IAE value can achieve.
trade-off between disturbance performance and gain margin is needed
PI control
PID control
PI control
PID control
Gain margin