- Email: [email protected]
Improvement of Transient Response by Means of Variable Setpoint Weighting
Cor yri ghl © IFAC 12th Triennial World Congress, Sydney, Auslralia, 1993
IMPROVEMENT OF TRANSIENT RESPONSE BY MEANS OF VARIABLE SETPOINT WEIGHTING C.C. Hang and L.S. Cao /)el'llr/ment of Electrical Engineerin j1, National University of Singapore, / 0 Kent Ridge Crescell{. Singapore 05//
Abstract. A new approach is introduced for speeding up the transient set-point response of a process controller by adapting the set-point weighting factor according to both the error signal and a measurable dynamic characteristics of thc process. For a PID controller, the initial set-point weighting factor, the times at which this factor should be varied and the adapted value of the set-point weighting factor, are all given by a set of correlation formulae which can be used in conjunction with the refined Z-N formulae. Analysis has also led to an equivalent block diagram more suitable for implementation as set-point filtering with a variable zero.
Key Words. Adaptive control; PID control; set-point weighting; step response; zeros. (Nove l Feature)
1. Introduction Ziegler-Nichols (Z-N) method is a well-known tuning method for PID controller. It has been found that the PID parameters are normally tuned to give good load disturbance response. When the dead time is small, however, the controller settings that give a good load disturbance response will give a set-point response that exhibits a large overshoot. The conventional solution is either to detune the controller gain (in which case the load di sturbance response will become sluggish), or to introduce set-point filtering (in which case the response will be somewhat delayed). A more recent solution is by means of setpoint weighting [1]. Applying set-point weighting to a Z-N tuned controller, we can reduce the overshoot drastically while the load disturbance response will not be affected. Both the methods of set-point filtering and set-point weighting, however, still suffer from one major disadvantage that the setpoint response speed is sacrificed .
Y.
Figure I. The New PID Controller with the Novel Self-tuning Variable Set-point Wcighting Factor
In thi s paper a controller is introduced which has a novel function of self-tuning variable set-point weighting to reduce the overshoot of the set-point response but without sacrificing the fast ri se time. In section 3, the variable set-point weighting function is introduced and automatic tuning of the weighting factors and their associated switching time is presented. The much improved set-point response is substantiated with si mulation. Concluding remarks and extensions to other co ntrollers are given in section 5.
.. ---- ----
I I I I I I
1-------y,
I I I
t---o+o{)-+-~
2. PID Control with Set-point Weighting (3)
The PID controller with set-point weighting is usually implemented as follows [1]:
,-_.~ or K
J
I (y, - y )dt-1d~1 dy lie = ke[(Py, - y ) +-
T,
/1
dt
Lr.
where Ue,y"y are the controller output, set-point and process output respectively, P is the set-point weighting factor and N is a filtering factor chosen between 3- /0; Without loss of generally, N is chosen to be 10 throughout this paper.
(b)
The block diagram of a typical PID controller with self-tuning variable set-point weighting is shown in Figure 1. It is evident
Figure 2. Equivalent Block Diagram of Figure I More Suitable for Implementati on
525
from Figure I that the load disturbance response is independent of the set-point weighting factor f3 as it does not act directly on the process output. Figure 2 shows the dynamic equivalent of Figure I . It is more suitable for practical implementation as f3 only acts on the change in set-point and hence the controller output will not change unnecessarily by the static value of the set-point when f3 is auto-tuned or adapted. It is also important to note from Figure 2 that the major function of f3 is to adjust the location of the zero in the transfer function of the filter for the set-point. f3=1 gives the conventional PlO control without set-point weighting. With f3=0, set-point filtering with a first order lag is resultant. When f3;JJ, a lead-Iag filter is resultant and when 0
Time
- 2 --~·'----~--~--~--~--~
012
10
Time
1.5~-~-~-~-~-~-~-~-~~
, ,
1.5,---~-~-~-~-~-~--~-~-~-
~
o
ee
0.5
a.
oL-_~_~_~_~_~_~_~
Time
_ _~_~~
o
10
6-~--~-~-~·--~-~-~-~--, ---,
Time
P=I
g
Figure 4. The Effect of Variable Set-point Weighting
.!!' U1
ec
1- as Process Ill: G(s) = - - - - -3
o u
(I +s)
o will be changed in process I by changing the dead time L, in
-2L--~_~_~_~_~_~_~_~_~~
o
10
process 1I by the order 11 and in process III by the numerator coefficient a . The Ziegler-Nichols closed loop tuning formulae are based on the empirical knowledge of the ultimate gain k, and ultimate period t, [2]. As the normalized dead time 0 and the normalized gain K are well correlated and hence can be interchanged [3], it suffices to state that similar formula can be derived in terms of K.
Time
Y, I 05
Process with normalized del!d ti"!..e 0-0.25
10
Time
Figure 3. The Effect of Different Set-point Weighting
in_t~
range.o.f
The Ziegler-Nichols formula for a process with small dead time produces good load disturbance response [3]. In this case only the set-point response needs to be improved and hence the self-tuning variable f3 formula will be based on a PlO controller tuned by following Z-N formula:
3. PID Control with Variable Set-point Weighting From the response of the process output y in Figure 3 it is found that while set-point weighting with 0
T, =!.....2
Process with 0.25-0.5
3.1 Process Characterization and Refined Z-N Formula
no~malized
dead time in the ra.nge o!
The load disturbance response of a process with large dead time produced by the Ziegler-Nichols formula is slow. In this case we revise the integral time to get a fast load disturbance response and use the self-tuning variable f3 formula based on following refined Z-N formula as shown in Figure 5 to improve the set-point response.
The normalized dead time 0 is defined as the ratio of the dead time L to the time constant T of the open-loop step response of the process [2]. In this paper it has been used to characterize the following processes. Process I: G(s) = _ _I_ e-'L (I + s)'
kc = 0.6k,
I
Process 11: G( s) = - (I + sf
7; = (-01888 + 0512)1,
526
point is changed. In this case the two switching instants are
T-~ d - 4
determined at e, = 0.7e(0) and em = (2 .0550 - 0.195)e(0) which divide the transient response into three periods and the values
where k. and t. are ultimate gain and period respectively, 0 is the normalized dead time of the process. Beyond 0 of 0.5, the refined Z-N formula [3] with fixed P is good enough and will not be further discussed in this paper. 0 .52
of P" p, and Pm can be used in the three periods respectively, as illustration in Figure 6.
Formulae for processes with normalized dead time in the range of 0.25-0.5
..---_---_---_--~--_____,
The corresponding values of switching instants are determined
0.5 ~---T.:.".:..: ' ~:>c..::. = O::::.5~(:::.":::N_='"::::nin~._=f"'.:..:P1_=D.:..:)~_ _ _ _ __1
at e, = 0.85e(0), and em The values of pare:
0.48
= (-43740 2 +4.4780 -
0.546)e(0).
p ,= 1
p, = 0.2
Pm = -14.2170] + 14.9520 2 -
0 . 44
4.2800 + 0.953
0.42
0.25
0.3
0 . 35 Normolized
0.4
0.45
In this section, the performance of the variable set-point weighting formula will be compared with the performance of the Ziegler-Nichols tuning formula. Two minimum phase processes(Proc.I, Il) and one non-minimum phase process (Proc III) will be used in the comparative studies.
0 .5
Oeodtime (8)
Figure 5. Refined Ziegler-Nichols Tuning ofPID Control for Proc. I, 11, III
3.2 Tuning of Variable Set-point Weighting Factor
The responses of the processes with dead time 0 smaller than 0.25 are shown in Figure 7. It can be seen that the ZieglerNichols tuning gives good load dis'turbance response but the set-point response has 50% or even higher overshoot. The superiority of variable set-point weighting is evident. It has the same speed as Ziegler-Nichols tuning with the overshoot reduced to less than 10%.
A large set-point weighting factor P, is proposed during the initial period of a set-point change, in order to maintain a fast rise time. Thereafter a smaller weighting factor p, is used to reduce the large overshoot and it is increased to a value Pm in the third period to eliminate the undershoot. A transient response with fast rise time, small overshoot and short settling time can thus be obtained as shown in Figure 6. It has been found from extensive simulation that two switching instants give the best results as a simple tuning formula for setting P and the switching instants in terms of 0 can be obtained. This cannot be achieved by using only one switching instant while the situation becomes too complicated with more than two switching instants.
Formulae for processes with normalized dead time the range of 0-0.25
(01
i~
Time 2 _ _ _~__~_______ ~(~ bl~_ _ _--~----~--_____,
The following values of P have been found in extensive simulation to work well for the all three processes:
P, = 1.1 p, =0.2
Pm = 0.6420 + 0.506 Set-point Response of the Process
'j
e(O
T ime
--
1.5 ..-_ _ _----_----_----'(~c)'----_--~_--~--_____,
-1
10
A
p,
20
25
30
35
Time
1- Ziegler-Nichols tuning formula ( ....) 2- Self-tuning variable set-point weighting (-)
/1 fJ=
15
p.
Figure 7. Comparison of Tuning Formulae of PID Control for Smaller Normalized Dead Time (a) Proc. I L=O.I (0=0.144 ) n=3 (0=0.221 ) (b) Proc. 11 (c) Proc. III a=0 .1 ( 0 =0.249 )
Time
Figure 6 . Definition of Switching Instants The switching instant is determined by the time at which the error e reaches a ratio of the initial error e(O) when the set-
527
40
e
The responses of the processes with dead time larger than 0.25 are shown in Figure 8. The new tuning formula with variable set-point weighting gives a much better set-point response as well as an improved load disturbance response compared to that tuned by the Ziegler-Nichols formula.
5. Conclusion The variable set-point weighting technique achieves a good reduction of the overshoot of the set-point response without sacrificing the rise time. The self-tuning formula of set-point weighting gives a systematic method of determining the switching instants and the suitable values of p. The formula is based on normalized dead time which is already available in most cases from the auto-tuning or adaptive part of the PID controller.
1.5r---~---~---'(';:'
31
;0., ....
1
Cl
v
J
e5 0.5
OL.L--~---~--~---"---~----' o 10
15
20
25
Reference
30
[I] Astram, K. J. and T Hagglund (1988): Automatic Tuning of PID Controllers, Instrument Society of America.
Time
1.5r---~-_~_ _--,-(b~)_ _~_-~_------,
[2] Astram, K. J., C C Hang, P. Persson and W . K. Ho (1992) : Towardl' lntelligent PID Control, Automatica, Vol. 138, No.2, 1-9.
v 10
30
20
40
50
[3] Hang, C C , K. J. Astram and W. K. Ho (1991): Refinement of the Ziegler-Nichols Tuning Formula, IEE Proceedings-D, Vol.138, No.2, 111-118.
60
Time
[4] Infelise, N. (1991) : A Clear Vision of Fuzzy Logic, Control Engineering, July, 28-30.
1.5,---_~--~--~--'(.;:.
[5] Shigemasa, T , Y . lino and M . Kanda (1987): Two degrees of Freedom PID Auto-tulling Controller, Proc. ISA Annual Conf. , USA, 703-711 . [6] Ziegler, J. G. and N. B . Nichols (1942) : Optimum Sellings for Automatic COlltrollers, Trans. ASME, Vol. 64, 759768 .
-0 . 5'---~--~--~-~--~----~---'
o
10
15
20
25
30
35
40
Time
1- Ziegler-Nichols tuning formula ( .... ) 2- Self-tuning variable set-point weighting (-)
Figure 8. Comparison of Tuning Formulae ofPID Control for Larger Normalized Dead Time (a) Proc. I (b) Proc. II (c) Proc. III
L=0.7 (0=0365) n=5 (0=0.412) a=0 .3 (0=0.312 )
4. Discussion It is observed from Figure 2 that the variable set-point weighting can be implemented as a lead-Iag set-point filter with a variable lead. The response of the filtered set-point Yif is shown in Figures 3 and 4. It is thus evident that the variable set-point weighting function aims to produce a filtered setpoint quite similar to that of fuzzy control [4] which implements the action of an experienced operator in reducing over-shoot while maximizing the response speed . The difference is that the proposed variable set-point weighting scheme is more systematic and is equipped with self-tuning hence eliminating the need of operator expertise which would vary from one installation to another. The variable set-point weighting function is also different from the simple two-degree-of-freedom approach [5] which only uses a constant filter which introduces a time lag to the setpoint response, or a complete decoupling of forward-loop servo controller from the feedback-loop regulator which is much more sophisticated and hence more difficult to adapt or auto-tune. The excellent responses for step changes in set-point are shown in Figures 7 and 8.
528
IMPROVEMENT OF TRANSIENT RESPONSE BY MEANS OF VARIABLE SETPOINT WEIGHTING C.C. Hang and L.S. Cao /)el'llr/ment of Electrical Engineerin j1, National University of Singapore, / 0 Kent Ridge Crescell{. Singapore 05//
Abstract. A new approach is introduced for speeding up the transient set-point response of a process controller by adapting the set-point weighting factor according to both the error signal and a measurable dynamic characteristics of thc process. For a PID controller, the initial set-point weighting factor, the times at which this factor should be varied and the adapted value of the set-point weighting factor, are all given by a set of correlation formulae which can be used in conjunction with the refined Z-N formulae. Analysis has also led to an equivalent block diagram more suitable for implementation as set-point filtering with a variable zero.
Key Words. Adaptive control; PID control; set-point weighting; step response; zeros. (Nove l Feature)
1. Introduction Ziegler-Nichols (Z-N) method is a well-known tuning method for PID controller. It has been found that the PID parameters are normally tuned to give good load disturbance response. When the dead time is small, however, the controller settings that give a good load disturbance response will give a set-point response that exhibits a large overshoot. The conventional solution is either to detune the controller gain (in which case the load di sturbance response will become sluggish), or to introduce set-point filtering (in which case the response will be somewhat delayed). A more recent solution is by means of setpoint weighting [1]. Applying set-point weighting to a Z-N tuned controller, we can reduce the overshoot drastically while the load disturbance response will not be affected. Both the methods of set-point filtering and set-point weighting, however, still suffer from one major disadvantage that the setpoint response speed is sacrificed .
Y.
Figure I. The New PID Controller with the Novel Self-tuning Variable Set-point Wcighting Factor
In thi s paper a controller is introduced which has a novel function of self-tuning variable set-point weighting to reduce the overshoot of the set-point response but without sacrificing the fast ri se time. In section 3, the variable set-point weighting function is introduced and automatic tuning of the weighting factors and their associated switching time is presented. The much improved set-point response is substantiated with si mulation. Concluding remarks and extensions to other co ntrollers are given in section 5.
.. ---- ----
I I I I I I
1-------y,
I I I
t---o+o{)-+-~
2. PID Control with Set-point Weighting (3)
The PID controller with set-point weighting is usually implemented as follows [1]:
,-_.~ or K
J
I (y, - y )dt-1d~1 dy lie = ke[(Py, - y ) +-
T,
/1
dt
Lr.
where Ue,y"y are the controller output, set-point and process output respectively, P is the set-point weighting factor and N is a filtering factor chosen between 3- /0; Without loss of generally, N is chosen to be 10 throughout this paper.
(b)
The block diagram of a typical PID controller with self-tuning variable set-point weighting is shown in Figure 1. It is evident
Figure 2. Equivalent Block Diagram of Figure I More Suitable for Implementati on
525
from Figure I that the load disturbance response is independent of the set-point weighting factor f3 as it does not act directly on the process output. Figure 2 shows the dynamic equivalent of Figure I . It is more suitable for practical implementation as f3 only acts on the change in set-point and hence the controller output will not change unnecessarily by the static value of the set-point when f3 is auto-tuned or adapted. It is also important to note from Figure 2 that the major function of f3 is to adjust the location of the zero in the transfer function of the filter for the set-point. f3=1 gives the conventional PlO control without set-point weighting. With f3=0, set-point filtering with a first order lag is resultant. When f3;JJ, a lead-Iag filter is resultant and when 0
Time
- 2 --~·'----~--~--~--~--~
012
10
Time
1.5~-~-~-~-~-~-~-~-~~
, ,
1.5,---~-~-~-~-~-~--~-~-~-
~
o
ee
0.5
a.
oL-_~_~_~_~_~_~_~
Time
_ _~_~~
o
10
6-~--~-~-~·--~-~-~-~--, ---,
Time
P=I
g
Figure 4. The Effect of Variable Set-point Weighting
.!!' U1
ec
1- as Process Ill: G(s) = - - - - -3
o u
(I +s)
o will be changed in process I by changing the dead time L, in
-2L--~_~_~_~_~_~_~_~_~~
o
10
process 1I by the order 11 and in process III by the numerator coefficient a . The Ziegler-Nichols closed loop tuning formulae are based on the empirical knowledge of the ultimate gain k, and ultimate period t, [2]. As the normalized dead time 0 and the normalized gain K are well correlated and hence can be interchanged [3], it suffices to state that similar formula can be derived in terms of K.
Time
Y, I 05
Process with normalized del!d ti"!..e 0-0.25
10
Time
Figure 3. The Effect of Different Set-point Weighting
in_t~
range.o.f
The Ziegler-Nichols formula for a process with small dead time produces good load disturbance response [3]. In this case only the set-point response needs to be improved and hence the self-tuning variable f3 formula will be based on a PlO controller tuned by following Z-N formula:
3. PID Control with Variable Set-point Weighting From the response of the process output y in Figure 3 it is found that while set-point weighting with 0
T, =!.....2
Process with 0.25-0.5
3.1 Process Characterization and Refined Z-N Formula
no~malized
dead time in the ra.nge o!
The load disturbance response of a process with large dead time produced by the Ziegler-Nichols formula is slow. In this case we revise the integral time to get a fast load disturbance response and use the self-tuning variable f3 formula based on following refined Z-N formula as shown in Figure 5 to improve the set-point response.
The normalized dead time 0 is defined as the ratio of the dead time L to the time constant T of the open-loop step response of the process [2]. In this paper it has been used to characterize the following processes. Process I: G(s) = _ _I_ e-'L (I + s)'
kc = 0.6k,
I
Process 11: G( s) = - (I + sf
7; = (-01888 + 0512)1,
526
point is changed. In this case the two switching instants are
T-~ d - 4
determined at e, = 0.7e(0) and em = (2 .0550 - 0.195)e(0) which divide the transient response into three periods and the values
where k. and t. are ultimate gain and period respectively, 0 is the normalized dead time of the process. Beyond 0 of 0.5, the refined Z-N formula [3] with fixed P is good enough and will not be further discussed in this paper. 0 .52
of P" p, and Pm can be used in the three periods respectively, as illustration in Figure 6.
Formulae for processes with normalized dead time in the range of 0.25-0.5
..---_---_---_--~--_____,
The corresponding values of switching instants are determined
0.5 ~---T.:.".:..: ' ~:>c..::. = O::::.5~(:::.":::N_='"::::nin~._=f"'.:..:P1_=D.:..:)~_ _ _ _ __1
at e, = 0.85e(0), and em The values of pare:
0.48
= (-43740 2 +4.4780 -
0.546)e(0).
p ,= 1
p, = 0.2
Pm = -14.2170] + 14.9520 2 -
0 . 44
4.2800 + 0.953
0.42
0.25
0.3
0 . 35 Normolized
0.4
0.45
In this section, the performance of the variable set-point weighting formula will be compared with the performance of the Ziegler-Nichols tuning formula. Two minimum phase processes(Proc.I, Il) and one non-minimum phase process (Proc III) will be used in the comparative studies.
0 .5
Oeodtime (8)
Figure 5. Refined Ziegler-Nichols Tuning ofPID Control for Proc. I, 11, III
3.2 Tuning of Variable Set-point Weighting Factor
The responses of the processes with dead time 0 smaller than 0.25 are shown in Figure 7. It can be seen that the ZieglerNichols tuning gives good load dis'turbance response but the set-point response has 50% or even higher overshoot. The superiority of variable set-point weighting is evident. It has the same speed as Ziegler-Nichols tuning with the overshoot reduced to less than 10%.
A large set-point weighting factor P, is proposed during the initial period of a set-point change, in order to maintain a fast rise time. Thereafter a smaller weighting factor p, is used to reduce the large overshoot and it is increased to a value Pm in the third period to eliminate the undershoot. A transient response with fast rise time, small overshoot and short settling time can thus be obtained as shown in Figure 6. It has been found from extensive simulation that two switching instants give the best results as a simple tuning formula for setting P and the switching instants in terms of 0 can be obtained. This cannot be achieved by using only one switching instant while the situation becomes too complicated with more than two switching instants.
Formulae for processes with normalized dead time the range of 0-0.25
(01
i~
Time 2 _ _ _~__~_______ ~(~ bl~_ _ _--~----~--_____,
The following values of P have been found in extensive simulation to work well for the all three processes:
P, = 1.1 p, =0.2
Pm = 0.6420 + 0.506 Set-point Response of the Process
'j
e(O
T ime
--
1.5 ..-_ _ _----_----_----'(~c)'----_--~_--~--_____,
-1
10
A
p,
20
25
30
35
Time
1- Ziegler-Nichols tuning formula ( ....) 2- Self-tuning variable set-point weighting (-)
/1 fJ=
15
p.
Figure 7. Comparison of Tuning Formulae of PID Control for Smaller Normalized Dead Time (a) Proc. I L=O.I (0=0.144 ) n=3 (0=0.221 ) (b) Proc. 11 (c) Proc. III a=0 .1 ( 0 =0.249 )
Time
Figure 6 . Definition of Switching Instants The switching instant is determined by the time at which the error e reaches a ratio of the initial error e(O) when the set-
527
40
e
The responses of the processes with dead time larger than 0.25 are shown in Figure 8. The new tuning formula with variable set-point weighting gives a much better set-point response as well as an improved load disturbance response compared to that tuned by the Ziegler-Nichols formula.
5. Conclusion The variable set-point weighting technique achieves a good reduction of the overshoot of the set-point response without sacrificing the rise time. The self-tuning formula of set-point weighting gives a systematic method of determining the switching instants and the suitable values of p. The formula is based on normalized dead time which is already available in most cases from the auto-tuning or adaptive part of the PID controller.
1.5r---~---~---'(';:'
31
;0., ....
1
Cl
v
J
e5 0.5
OL.L--~---~--~---"---~----' o 10
15
20
25
Reference
30
[I] Astram, K. J. and T Hagglund (1988): Automatic Tuning of PID Controllers, Instrument Society of America.
Time
1.5r---~-_~_ _--,-(b~)_ _~_-~_------,
[2] Astram, K. J., C C Hang, P. Persson and W . K. Ho (1992) : Towardl' lntelligent PID Control, Automatica, Vol. 138, No.2, 1-9.
v 10
30
20
40
50
[3] Hang, C C , K. J. Astram and W. K. Ho (1991): Refinement of the Ziegler-Nichols Tuning Formula, IEE Proceedings-D, Vol.138, No.2, 111-118.
60
Time
[4] Infelise, N. (1991) : A Clear Vision of Fuzzy Logic, Control Engineering, July, 28-30.
1.5,---_~--~--~--'(.;:.
[5] Shigemasa, T , Y . lino and M . Kanda (1987): Two degrees of Freedom PID Auto-tulling Controller, Proc. ISA Annual Conf. , USA, 703-711 . [6] Ziegler, J. G. and N. B . Nichols (1942) : Optimum Sellings for Automatic COlltrollers, Trans. ASME, Vol. 64, 759768 .
-0 . 5'---~--~--~-~--~----~---'
o
10
15
20
25
30
35
40
Time
1- Ziegler-Nichols tuning formula ( .... ) 2- Self-tuning variable set-point weighting (-)
Figure 8. Comparison of Tuning Formulae ofPID Control for Larger Normalized Dead Time (a) Proc. I (b) Proc. II (c) Proc. III
L=0.7 (0=0365) n=5 (0=0.412) a=0 .3 (0=0.312 )
4. Discussion It is observed from Figure 2 that the variable set-point weighting can be implemented as a lead-Iag set-point filter with a variable lead. The response of the filtered set-point Yif is shown in Figures 3 and 4. It is thus evident that the variable set-point weighting function aims to produce a filtered setpoint quite similar to that of fuzzy control [4] which implements the action of an experienced operator in reducing over-shoot while maximizing the response speed . The difference is that the proposed variable set-point weighting scheme is more systematic and is equipped with self-tuning hence eliminating the need of operator expertise which would vary from one installation to another. The variable set-point weighting function is also different from the simple two-degree-of-freedom approach [5] which only uses a constant filter which introduces a time lag to the setpoint response, or a complete decoupling of forward-loop servo controller from the feedback-loop regulator which is much more sophisticated and hence more difficult to adapt or auto-tune. The excellent responses for step changes in set-point are shown in Figures 7 and 8.
528