1. Unit 3-Tuning of Controllers

2. Introduction Tuning is a part of controller design of the loop.
In order to be able to use a controller, it must first be tuned to the system.
This tuning synchronizes the controller with the controlled variable, thus allowing the process to be kept at its desired operating condition.
A need for a loop to be tuned arises if it responds slowly, or if it oscillates too much, or if it has a steady-state error and most definitely if it’s unstable.

3. PID has to be tuned when,
Careful consideration was not given to the units of gains and other parameters.
The process dynamics were not well-understood when the gains were first set, or the dynamics have (for any reason) changed.
You (as designer or operator) think the controller can perform better.

4. Essentially all tuning methods are based on a model process for which optimal controller settings have been determined.
Tuning procedures are thus a matter of matching a real or simulated process to the tuning model.
Proper tuning of a controller is not only essential to its correct operation but will also greatly improve product quality, reduce scrap, shorten down-time and save money

5. Tuning a control loop is the adjustment of its control parameters (gain/proportional band, integral gain/reset, derivative gain/rate) to the optimum values for the desired control response.
The optimum behavior on a process change or set point change varies depending on the application.
Some processes must not allow an overshoot of the process variable from the set point. Other processes must minimize the energy expended in reaching a new set point.

6. Generally stability of response is required and the process must not oscillate for any combination of process conditions and set points.
Tuning of loops is made more complicated by the response time of the process; it may take minutes or hours for a set point change to produce a stable effect.

7. Gain
Gain, or as it is more commonly called, Proportional Band, simply amplifies the error between setpoint and measured value to establish a power level.
The term Proportional Band is one that expresses the gain of the controller as a percentage of the span of the instrument.
A 25% PB equates to a gain of 4; a 10% PB is a gain of 10.

9. Gain Contd…
The Proportional Band determines the magnitude of the response to an error.
If the Proportional Band is too small, meaning high gain, the system will oscillate through being over-responsive.
A wide Proportional Band, low gain, could lead to control "wander" due to a lack of responsiveness.
The ideal situation will be achieved when the Proportional Band is as narrow as possible without causing oscillation.

10. Reset
Integral action, or Automatic Reset, is probably the most important factor governing control at set point.
The integral term slowly shifts the output level as a result of an error between set point and measured value.
If the measured value is below set point the integral action will gradually increase the output power level in an attempt to correct this error.

13. Derivative
Derivative action, or Rate, provides a sudden shift in output power level as a result of a quick change in measured value.
If the measured value drops quickly the derivative term will provide a large change in output level in an attempt to correct the perturbation before it goes too far.
Derivative action is probably the most misunderstood of the three.
It is also the most beneficial in recovering from small perturbations.

15. TF of PID Controller
with C being a constant which depends on the bandwidth of the controlled system.
Traditionally, the output of the controller (i.e. the input to the process) is given by
where Pcontrib, Icontrib, and Dcontrib are the feedback contributions from the PID controller, defined below:

16. Where e (t) = Set point − Measurement (t) is the error signal, and Kp , Ki and Kd are contants that are used to tune the PID control loop:
Kp: Proportional Gain - Larger Kp typically means faster response since the larger the error, the larger the feedback to compensate.
Ki: Integral Gain - Larger Ki implies steady state errors are eliminated quicker.
Kd: Derivative Gain - Larger Kd decreases overshoot, but slows down transient response.

17. PID Control

18. In practice, most PID controllers employ 3 slightly different constants which correspond to these proportional, integral, and derivative gain.
Proportional Band - Often abbreviated Pb, this is the band where proportional gain acts upon. To get larger Kp Pb is decreased as
Integral Time - Often abbreviated It this is the time over which error is averaged. Because It has dimensions of time, one can conclude the following with dimensional analysis

19. Derivative Time - Often abbreviated Dt, this is the time over which the derivative of the error is evaluated. Because Dt has dimensions of time, one can conclude the following with dimensional analysis:

20. There are several methods for tuning a PID loop.
The choice of method will depend largely on whether or not the loop can be taken "offline" for tuning, and the response speed of the system.
If the system can be taken offline, the best tuning method often involves subjecting the system to a step change in input, measuring the output as a function of time, and using this response to determine the control parameters.

21. If the system must remain online, one tuning method is to first set the I and D values to zero.
Increase the P until the output of the loop oscillates.
Then increase I until oscillation stops.
Finally, increase D until the loop is acceptably quick to reach its reference.
A fast PID loop tuning usually overshoots slightly to reach the set point more quickly; however, some systems cannot accept overshoot.

22. Effects of increasing parameters
Rise Time
Overshoot
Settling Time
S.S. Error Kp
Decrease
Increase
Small Change Ki
Eliminate Kd

23. Tuning Criteria
Optimum decay ratio (1/4 wave decay).

24. Minimum Overshoot

25. Maximum Disturbance Rejection
The choice of methods depends upon the loop’s place in the process and its relationship with other loops.

26. Mathematical criteria
Mathematical methods—minimization of index
IAE - Integral of absolute value of error
ISE - Integral of error squared
ITAE - Integral of time times absolute value of error
ITSE - Integral of time times error squared

27. Ziegler and Nichols – open loop method
Ziegler and Nichols have developed PID tuning methods back in the early forties based on open loop tests.
The method allows to calculate PID parameters from the process parameters.

28. Procedure Step 1: Make an open loop plant test (e.g. a step test).
Step 2: Determine the process parameters: Process gain, dead time, time constant (see the diagram: draw a tangent through the inflection point and measure Td and Ts as shown.
Step 3: Calculate the parameters according to the formula.

29. Ideal Process

30. Steady-state gain or Process Gain
= (y2 - y1) / (u2 - u1)
Time delay or Dead Time, Td
Time constant, Ts

31. Formulas P: K = time constant / (process gain * deadtime)
PI: Proportional gain = 0.9 * K
Integral time = 3.3 * deadtime
PID: Proportional gain = 1.2 * K
Integral time = 2 * deadtime
Derivative time = 0.5 * deadtime

32. U U Y Y Td Ts

33. Open Loop - Controller Parameters
Gain
Reset
Derivative P
(Ts U )
(Td Y ) - PI
(0.9 Ts U ) (Td Y )
3.3 Td
PID
(1.2 Ts U ) (Td Y )
2.0 Td
0.5 Td

34. Advantages of the method
Only a single experimental test is needed.
It does not require trial and error
The controller settings are easily calculated.

35. Disadvantages of the method
Experiment is under open loop response and so disturbances may affect the results.
Results tend to be oscillatory.
Does not work well for complex responses - leads to inaccurate tuning model.

36. Controlling Real Process
In the real world, unfortunately, the response of a process to a change in one of its inputs seldom follows the first-order case required for the Z-N tuning.
In this case the tangent should be drawn at the point where the slope of the response is steepest.

37. Real Process

38. Controller parameter correlations
For a model system with delay time Td and first order lag T1, the fractional dead time Tf is defined as: Tf = Td/(Td + T1)
This is used as parameter to determine settings of dimensionless overall loop gain, the product of controller and process gains:
’ = controller gain * process gain
and dimensionless integral action time:
’I = controller reset time / (Td + T1)
Settings for PI controllers have been proposed by Lopez and Cianconne and Marlin.

39. Ciancone
Lopez
ZN (open) Tf
  ’
  ’I
1.1
0.23
5.8
0.4 -
0.1
0.5
8.1
0.33
0.2
1.8
3.1
0.6
3.6
0.66
0.3
0.72
2.1
0.7
1.0
1.7
0.8
1.35
1.32
0.70
0.91
0.9
1.65
0.59
0.67
1.98
0.42
0.60
0.43
2.31
0.32
0.53
0.25
2.64

40. Example Delay time = 5 Time Constant = 20 Gain = 3.5
Fractional Dead Time = 5 / (20 + 5) = 0.2

41. Ciacone Parameters: Lopez Parameters: ZN Open Loop: ’ = 1.8
’ = 1.8
’I = 0.23
Controller gain = 1.8 / 3.5 = 0.514
Controller reset = 0.23 * (20 + 5) = 5.75
Lopez Parameters:
’ = 3.1
’I = 0.6
Controller gain = 3.1 / 3.5 = 0.886
Controller reset = 0.6 * (20 + 5) = 15
ZN Open Loop:
’ = 3.6
’I = 0.66
Controller gain = 3.6 / 3.5 = 1.029
Controller reset = 0.66 * (20 + 5) = 16.5

42. Ziegler and Nichols – closed loop method
Place controller into low gain, no reset or derivative.
Gradually increase gain, making small changes in the set point, until oscillations start.
Adjust gain to make oscillations continue with a constant amplitude.
Note the gain (Ultimate gain Ku) and Period (Ultimate Period Pu)

43. Procedure
Step 1: Disable any D and I action of the controller (--> pure P-controller)
Step 2: Make a set point (SP) step test and observe the response
Step 3: Repeat the SP test with increased or decreased controller gain until a stable oscillation is achieved. This gain is called the "ultimate gain" Ku. (Ultimate gain is the gain at which the oscillations continue with a constant amplitude).

44. Step 4: Read the oscillation period Pu.
Step 5: Calculate the parameters according to the formulas.

45. Response at Ultimate Gain

46. Formulas PI: Proportional gain = 0.45 * Ku Integral time =Pu / 1.2
PID: Proportional gain = 0.6 * Ku
Integral time =Pu / 2
Derivative time = Pu / 8

47. Closed Loop - Controller Parameters
Gain
Reset
Derivative P
0.5 Ku - PI
0.45 Ku
1.2 / Pu
PID
0.6 Ku
2 / Pu
Pu / 8

48. For a variety of process, these settings were found to give a decay ratio of about ¼, a period of oscillation close to the ultimate period and a reasonable overshoot or peak error.
In PI control, the recommended gain is 10% lower than with only P control.
Integral action makes the system less stable because of phase lag in the controller.
The value of Ku in the equations are based on the tests with proportional control only and is not the true maximum gain setting for a system with other control actions.

49. When derivative control is added, the phase lead of the controller helps to stabilize the system and a higher gain and lower reset time are recommended.
For a process with several time constants, the maximum gain and the optimum gain may be doubled by the use of derivative action.
On the other hand derivative action has little effect on systems with a large dead time.

50. Practical Use
It is unwise to force the system into a situation where there are continuous oscillations as this represents the limit at which the feedback system is stable.
Generally, it is a good idea to stop at the point where some oscillation has been obtained.
It is then possible to approximate the period (Pu) and if the gain at this point is taken as the ultimate gain (ku) , then this will provide a more conservative tuning regime.

51. Damped oscillation method
This based on the ZN closed loop method.
The main disadvantage with the ZN closed loop method is that the plant conditions have to oscillate to obtain the parameters.
In many plants sustained oscillations for testing purposes are not allowable, and the ultimate frequency method cannot be used to secure the optimum settings.
However this method allows a damped oscillation which dies away as the process approaches a set point under control with non-optimal, but known controller settings.

52. The following modification is easy to follow and perhaps more accurate than the ultimate method.
By using only proportional control action and starting with a low gain, the gain is adjusted until the transient response of the closed loop shows a decay ratio of ¼.
The reset time and derivative time are based on the period of oscillation, P, which is always greater than the ultimate period Pu.

53. For PID control,
TR = P / 6
TD = P / 1.5
With the derivative and reset times at the above values, the gain for ¼ decay ratio is again established by transient response tests.

54. Procedure
Using proportional control only, find the controller settings which gives a response like the one shown in the figure - the response is kept within reasonable limits and reaches steady state.
The gain required to do this is called d , the damped gain.
Evaluate the period of the oscillations, Pd.

55. Damped Oscillation Response

56. To relate d to u it is necessary to estimate the increase in gain required to cause the system to oscillate continuously as in the ZN closed loop method. This is done by looking at the ratio of the peak-to-trough and following trough-to-peak distances.
u = d * first peak to trough distance / following trough to peak distance
Pu = Pd * 0.95

57. Damped Oscillation - Controller Parameters
Gain
Reset
Derivative P
u / 2 - PI
u / 2.2
Pu / 1.2
PID
u / 1.7
Pu / 2
Pu / 8

58. Reaction Curve Method
The Process Reaction Curve is obtained by removing the controller from the circuit and injecting a step input of power. (response of the open-loop system to step input)
The control loop can be opened at any point, but the usual place is between the controller and the valve.
The power level injected can be any convenient, safe amount, but should be introduced when the system is stable at room temperature.
The output recorded is usually an S-shaped response.

59. Process Reaction Curve

60. The time, L, is often referred to as the Lag Time and is considered to be the time necessary to overcome the thermal inertia of the load being heated.
A straight line drawn tangent to the Process Reaction Curve at the point of inflection will have a slope, R.
From these two terms, the PID values may be calculated by the equations.

61. Proportional Band is expressed as percent of instrument span whereas TI (integral) and TD (derivative) are time constants expressed in minutes.
P is the percent power level used as the step input divided by 100% (expressed as a fraction).

62. Procedure
Step 1: Gather data from open-loop plant response to unit step input
Step 2: Examine data set to find the maximum slope
Step 3: Then determine the parameters needed
Step 4: Finally, use tuning relations to generate PID constants

63. Flow chart depicting ZN - PRM

65. Measurements made from the reaction curve
X % Change of output
R %/min.     Rate of change at the point of inflection (POI)
D min.        Time until the intercept of tangent line and original process value

66. Process Reaction Curve - Controller Parameters
Gain
Reset
Derivative P
X/(R*D) - PI
0.9X/(R*D)
D/0.3
PID
1.2X/(R*D)
D/0.5
0.5*D

67. The effective lag L is always greater than any true time delay which may exist in the system.
The settings predicted by the reaction curve and the continuous cycling method generally agree to within 10 to 50%.

68. Cohen-Coon Method
The Cohen-Coon (C-C) method is used for first-order plus time delay models only.
Take the transfer function, G(s) = Ke-s/(s + 1)
For a PID controller, the following settings are chosen
Kc =  (16 + 3)/12K
I =  [32 + 6(/)]/[13 + 8(/)]
D = 4/[11 + 2(/)]

69. These settings are designed to give a decay ratio of 1/4.
That is, a C-C tuned controller will have a response with the second oscillation having one-fourth the amplitude of the first.

70. Practical Use
If the process delay is small (in the limit as it approaches zero) increasingly large controller gains will be predicted.
The method is therefore not suitable for systems where there is zero or virtually no time delay.

71. Controller Settings for Minimum Error Integral
ISE – used as criteria for convenience or to give more weight to large deviations
ITAE – weighs deviations more heavily as time increases
IAE – best for process control since penalty for poor control is generally a linear function of the error.
Difference in the controller settings obtained by using the three criteria are generally small.

72. Controller Synthesis - Time Domain
Time-domain techniques can be classified into two groups:
(a) Criteria based on a few points in the response
(b) Criteria based on the entire response, or integral criteria
Approach (a): settling time, % overshoot, rise time, decay ratio
Chapter 12
Process model
Several methods based on 1/4 decay ratio have been proposed: Cohen-Coon, Ziegler-Nichols

73. Performance Characteristics f or step response for an under damped process
Chapter 12

74. Comparison for ZN Vs CC
Chapter 12

75. Chapter 12

76. 1. Integral of square error (ISE)
2. Integral of absolute value of error (IAE)
3. Time-weighted IAE
Pick controller parameters to minimize integral.
IAE allows larger deviation than ISE (smaller overshoots)
ISE longer settling time
ITAE weights errors occurring later more heavily
Approximate optimum tuning parameters are correlated
with K, , 

77. IAE Tuning Relation

78. ISE Tuning Relation

79. ITAE Tuning Relation

80. Chapter 12

82. Internal Model Control Tuning Method
Internal model control (IMC) tuning is referred to a set of tuning procedures based on the internal model principle.
The underlying idea behind internal model methodologies is to compute a controller and/or to set its values relative to a prescribed response formulated as a prescribed (internal) model.
Internal model control (IMC) tuning rules have proven to yield acceptable performance and robustness properties when used in the control of typical processes (e.g., distillation columns, chemical reactors).

83. Lambda Method  = process dead time (seconds)
 = process lag time (seconds)
K = process gain (dimensionless)
= 2  used for aggressive but less robust tuning
 = 2( + ) used for more robust tuning

84. Controller Settings for Lambda Method
Controller Type
Controller Gain (no units)
Integral Time (seconds)
Derivative Time (seconds)
  PI control
                               t
not applicable
  PID control
                                     
q/2

85. The first step in using the IMC tuning is to compute, c , the closed loop time constant.
All time constants describe the speed or quickness of a response.
The closed loop time constant describes the desired speed or quickness of a controller in responding to a set point change.
Hence, a small (a short response time) implies an aggressive or quickly responding controller.

86. The closed loop time constants are computed as:
Aggressive Tuning: c is the larger of 0.1P or 0.8 θP
Moderate Tuning: c is the larger of 1.0 P or 8.0 θP
Conservative Tuning: c is the larger of 10 P or 80.0 θP

87. General conclusion for PID tuning
The controller gain should be inversely proportional to the products of the other gains in the feedback loop.
The controller gain should decrease as the ratio of time delay to dominant time constant increases.
The larger the ratio of time delay to dominant time constant is, the harder the system is to control.
The reset time and the derivative time should increase as the ratio of time delay to dominant time constant increases.

88. The ratio between derivative time and reset time is typically between 0.1 to 0.3.
The ¼ decay ratio is too oscillatory for process control.
If less oscillatory response is desired, the controller gain should decrease and reset time should increase.
Among IAE, ISE and ITAE, ITAE is the most conservative and ISE is the least conservative setting.