ACC_Shams

Closed-loop PI/PID Controller Tuning for Stable and Unstable Processes

Shamsuzzoha, Moonyong Lee*, Hiroya Seki**

King Fahd University of Petroleum and Minerals, Saudi Arabia

*Yeungnam University, Kyongsan, Korea

**Tokyo Institute of Technology, Yokohama, Japan

2012 American Control Conference (ACC), 27th June 2012

MotivationDesborough and Miller (2001): More than 97% of controllers are PI/PID Vast majority of the PID controllers do not use D-action. PI controller: Only two adjustable parameters …

But still not easy to tuneMany industrial controllers poorly tuned

Ziegler-Nichols closed-loop method (1942) is popular, but Requires sustained oscillations Tunings relatively poor

Big need for a “fast and improved” closed-loop tuning procedure

2

Outline

1. Existing approaches to PID tuning

2. PID tuning rule for proposed study

3. Closed-loop setpoint experiment

4. Correlation between setpoint response and proposed PID-settings

5. Final choice of the controller settings (detuning)

6. Analysis and Simulation

7. Conclusion

3

1. Common approach:

PID-tuning based on open-loop model

Step 1: Open-loop experiment:Most tuning approaches are based on open-loop plant model

gain (k), time constant (τ) time delay (θ)

Problem: “Loose control” during identification experiment

Step 2: TuningMany approaches IMC-PID (Rivera et al., 1986): good for setpoint change SIMC-PI (Skogestad, 2003): Improved for integrating disturbances IMC-PID (Shamsuzzoha and Lee, 2007&2009) for disturbance

rejection for different type of processes Shamsuzzoha and Skogestad (2010): The setpoint overshoot method

(Closed-loop tuning method)

-

( )1

skeg s

s

4

Alternative approach:

PI-tuning based on closed-loop dataZiegler-Nichols (1942) closed-loop method Step 1. Closed-loop experiment Use P-controller with sustained oscillations. Record:1. Ultimate controller gain (Ku)2. Period of oscillations (Pu)

Step 2. Simple PI rules: Kc=0.45Ku and τI=0.83Pu.

Advantages ZN: Closed-loop experiment Very little information required Simple tuning rules

Disadvantages: System brought to limit of instability Many trials are required to obtain Ku Relay test (Åström) can avoid this problem but requires the feature of switching to on/off-

control Settings not very good: Aggressive for lag-dominant processes (Tyreus and Luyben) and

quite slow for delay-dominant process (Skogestad). Only for processes with phase lag > -180o (does not work on second-order)

5

Want to develop improved and simpler alternative to Z-N:

Closed-loop setpoint response with P-controller Use P-gain about 50% of

Z-N Identify “key parameters”

from setpoint response: Simplest to observe is first

peak!

py

pt

y sy

t0t

uy

sy

y

This work.

Improved closed-loop PID-tuning method

• Idea: Derive correlation between “key parameters” and proposed PID-settings for corresponding process

6

2. PID tuning rule based on IMC

ys

d

c gyu+

-

First-order process with time delay:

-

( )1

skeg s

s

PID controller:

PID controller based on IMC approach:

cτ =θ“Fast and robust” setting:

11c D

I

c s K ss

2

2cc

Kk

2D

I cτ =min , 4(τ +θ)2c

2

3cKk

min , 82I

2D

7

2I c

Skogestad (2003) recommended modifying the integral time

py

pt

y sy

t0t

uy

sy

Procedure:• Switch to P-only mode and make

setpoint change• Adjust controller gain to get

overshoot about 0.30 (30%)

Record “key parameters”:1. Controller gain Kc0

2. Overshoot = (Δyp-Δy∞)/Δy∞

3. Time to reach peak (overshoot), tp

4. Steady state change, b = Δy∞/Δys. Estimate of Δy∞ without waiting to settle:

Δy∞ = 0.45(Δyp + Δyu)

Advantages compared to ZN:* Not at limit to instability * Works on a simple second-order process.

3. Closed-loop setpoint experiment

Closed-loop step setpoint response with P-only control.

8

Various overshoots (10%-60%)

10 1

seg

s

0 5 10 15 20 25 300

0.25

0.5

0.75

1

1.25

time

OU

TP

UT

y

Setpoint

/=1 (Kc0

=0.855)

/=0.4 (Kc0

=0.404)

/=100 (Kc0

=79.9)

/=2 (Kc0

=1.636)

/=5 (Kc0

=4.012)

/=10 (Kc0

=8.0)

/=0.2 (Kc0

= 0.309)

/=0 (Kc0

= 0.3)

Closed-loop setpoint experiment

Overshoot of 0.3 (30%) with different τ’s

30%

τ=0

τ=100

τ=2

Small τ: Kc0 small and b small0 5 10 15 20 25

0

0.25

0.5

0.75

1

1.25

1.5

time

OU

TPU

T y

overshoot=0.10 (Kc0

=5.64)

overshoot=0.20 (Kc0

=6.87)

overshoot=0.30 (Kc0

=8.0)

overshoot=0.40 (Kc0

=9.1)

overshoot=0.50 (Kc0

=10.17)

overshoot=0.60 (Kc0

=11.26)

setpoint

9

Estimate of Δy∞ using undershoot Δyu

py

pt

y sy

t0t

uy

sy

y

Line: Δy∞ = 0.8947(Δyp+ Δyu)/2

0

0,2

0,4

0,6

0,8

1

1,2

0 0,2 0,4 0,6 0,8 1 1,2

(Δyp+ Δyu)/2

Δy ∞

overshoot=0.6

overshoot=0.5

overshoot=0.4

overshoot=0.3

overshoot=0.2

Data: 15 first-order with delay processes using 5 overshoots each (0.2, 0.3, 0.4, 0.5, 0.6). y s=1

Conclusion:Δy∞≈ 0.45(Δyp+Δyu)

10

4. Correlation between Setpoint response and proposed PID-settings

Goal: Find correlation between proposed PID-settings and “key parameters” from 90 setpoint experiments.

Consider 15 first-order plus delay processes:

τ/θ = 0.1, 0.2, 0.4, 0.8, 1, 1.5, 2, 2.5, 3, 5, 7.5, 10, 20, 50, 100

For each of the 15 processes: Obtain proposed PID-settings (Kc,τI)Generate setpoint responses with 6 different overshoots (0.10, 0.20,

0.30, 0.40, 0.50, 0.60) and record “key parameters”(Kc0, overshoot, tp, b)

-

( )1

seg s

s

11

Fixed overshoot: Slope Kc/Kc0 = A approx. constant, independent of the value of τ/θ

c

c0

K=A

K

Correlation Setpoint response and proposed PID-settings

Controller gain (Kc)

Kc0

Kc

Agrees with ZN (approx. 100% overshoot): Original: Kc/Kcu = 0.45 Tyreus-Luyben: Kc/Kcu = 0.33

90 cases: Plot Kc as a function of Kc0

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

kKc0

kKc

0.10 overshootkK

c=1.1621kK

c0

0.20 overshootkK

c=0.9701kK

c0

0.30 overshootkK

c=0.841kK

c0

0.40 overshootkK

c=0.7453kK

c0

0.50 overshootkK

c=0.6701kK

c0

0.60 overshootkK

c=0.6083kK

c0

12

2A= 1.55(overshoot) - 2.159(overshoot) + 1.35 Overshoots between 0.1 and 0.6 (should not be extended outside this range).

Conclusion: Kc = Kc0 A

A = slope

overshoot

0.1 0.2 0.3 0.4 0.5 0.6

0.7

0.8

0.9

1

1.1

overshoot (fractional)

A

y = 1.55*(overshoot)2 - 2.159*(overshoot) + 1.35

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I1

bτ =1.5A

1-bθ

Proposed PID-rules

Case 1 (large delay): τI1 = τ+θ/2 Case 2 (small delay): τI2 = 8θ

Case 1 (large delay):

c c0 c c0 c0kK =kK K K kK A

c0

bkK =

(1-b)

Correlation Setpoint response and proposed PID-settings

Integral time (τI)

(from steady-state offset)

Conclusion so far:

Still missing: Correlation for θ

I 1.5 ckK

14

c

2τ+θK =

3kθ

1.5 0.5ckK

I 1.5 ckK

(substitute τ I = τ+θ/2 into the proposed rule for Kc)

0.1 0.3 0.5 0.60

0.1

0.2

0.3

0.4

0.5

Overshoot

/t p

0.43 (I1

)

0.305 (I2

)

/=0.1

/=8

/=100

/=1

I I1 I p2 p

bτ =min(τ ,τ ) min 0.645A , 2.44t

1-bt

Correlation between θ and tp

py

pt

y sy

t0t

uy

sy

θ/tp

overshoot

Use:θ/tp = 0.43 for τI1 (large delay)θ/tp = 0.305 for τI2 (small delay)

Conclusion:

tp

θ

15

Derivative action (τD)

The derivative action can increase stability and improve the closed-loop performance.

Case I: For approximately integrating process (τ>> θ)

Case II: The processes with a relatively large delay (τ≈ θ)

Summary: The derivative action for both the cases i.e., τD1

and τD2 are approximately same

0.14 1

1-D p

bt if A

b

1

0.3050.15

2 2 2p

D p

tt

2 2

2

0.43 0.1433

2 3 3 3p

D p

tt

2D

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Selection of Controller Gain (Kc0)

An overshoot of around 0.3 is recommended,

Achieving the P-controller gain (Kc0) via trial and error can be sometime time consuming.

Let’s assume for the first closed-loop test P-controller gain of Kc01 is applied and resulting overshoot OS1 is achieved that is between 0.1 to 0.60 but not around 0.30.

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Selection of Controller Gain (Kc0)

18

First closed-loop test with P-controller gain of Kc01 is applied and resulting overshoot is OS1

Kc01=Kc0

Goal achieved!

Overshoot not around 0.30 but between 0.1 to 0.60

Overshoot around 0.30

2 21 1 01 01.55 OS 2.159 OS 1.35 1.55 OS 2.159 OS 1.35 c cK K

As it is mentioned earlier the proposed method is good agreement with the IMC-

PID for the overshoot around 0.3.

20 1 1 011.19 1.45 OS 2.02 OS 1.27 c cK K

Kc01=Kc0

Goal achieved!

Choice of detuning factor F: F=1. Good tradeoff between “fast and robust” (τc=θ)

F>1: Smoother control with more robustness F<1 to speed up the closed-loop response.

0c cK = K A F

p pIτ =min 0.645A ,b

t t1-b

2.44

F

2A= 1.55( ) - 2.15overshoot oversh9( ) + 1.3oot 5

From P-control setpoint experiment record “key parameters”:1. Controller gain Kc0



4. Steady state change, b = Δy∞/Δys

Proposed PID settings (including detuning factor F)

5. Summary setpoint overshoot method

D p

bτ = 0.14t if A 1

1-b

19

6. Analysis: Simulation PID-control

5 1

seg

s

First-order + delay process

t=0: Setpoint change t=40: Load disturbance

• ”in training set” • better response

0 20 40 60 800

0.25

0.5

0.75

1

1.25

time

OU

TP

UT

y

Shamsuzzoha and Skogestad method with F=1(overshoot=0.10)Shamsuzzoha and Skogestad method with F=1(overshoot=0.298)Shamsuzzoha and Skogestad method with F=1 (overshoot=0.599)Proposed method with F=1(overshoot=0.10)Proposed method with F=1(overshoot=0.298)Proposed method with F=1(overshoot=0.599)

20

sg e s

Integrating process

Analysis: Simulation PID-control

• ”in training set”

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

time

OU

TP

UT

y

Shamsuzzoha and Skogestad method with F=1 (overshoot=0.108)Shamsuzzoha and Skogestad method with F=1 (overshoot=0.302)Shamsuzzoha and Skogestad method with F=1 (overshoot=0.60)Proposed method with F=1 (overshoot=0.108)Proposed method with F=1 (overshoot=0.302)Proposed method with F=1 (overshoot=0.60)

21

1

1 0.2 1g

s s

Second-order process


• Not in ”training set”

0 2 4 6 8 100

0.25

0.5

0.75

1

1.25

1.5

time

OU

TP

UT

y

Shamsuzzoha and Skogestad method with F=1 (overshoot=0.127)Shamsuzzoha and Skogestad method with F=1 (overshoot=0.322)Shamsuzzoha and Skogestad method with F=1 (overshoot=0.508)Proposed method with F=1(overshoot=0.127)Proposed method with F=1(overshoot=0.322)Proposed method with F=1(overshoot=0.508)

22

1

1 0.2 1 0.04 1 0.008 1g

s s s s

High-order process



0 5 10 15 200

0.25

0.5

0.75

1

1.25

time

OU

TP

UT

y

Shamsuzzoha and Skogestad method with F=1 (overshoot=0.104)Shamsuzzoha and Skogestad method with F=1 (overshoot=0.292)Shamsuzzoha and Skogestad method with F=1(overshoot=0.598)Proposed method with F=1(overshoot=0.104)Proposed method with F=1(overshoot=0.292)Proposed method with F=1(overshoot=0.598)

23

21

1g

s s

Third-order integrating process



0 40 80 120 160 2000

1

2

3

4

5

time

OU

TP

UT

y

Shamsuzzoha and Skogestad method with F=1 (overshoot=0.106)Shamsuzzoha and Skogestad method with F=1 (overshoot=0.307)Shamsuzzoha and Skogestad method with F=1 (overshoot=0.610)Proposed method with F=1 (overshoot=0.106)Proposed method with F=1 (overshoot=0.307)Proposed method with F=1 (overshoot=0.610)

24

5 1

seg

s

First-order unstable process



0 20 40 60 800

0.5

1

1.5

2

time

OU

TP

UT

y

Shamsuzzoha and Skogestad method with F=1 (overshoot=0.10)Shamsuzzoha and Skogestad method with F=1 (overshoot=0.30)Shamsuzzoha and Skogestad method with F=1 (overshoot=0.607)Proposed method with F=1(overshoot=0.10)Proposed method with F=1(overshoot=0.30)Proposed method with F=1(overshoot=0.607)

25

6. Conclusion

From P-control setpoint experiment obtain: 1. Controller gain Kc0



4. Steady state change, b = Δy∞/Δys,

Estimate: Δy∞ = 0.45(Δyp + Δyu)

PID-tunings for “Revised Setpoint Overshoot Method”:

py

pt

y sy

t0t

uy

sy

y

c c0K = K A ,F2A= 1.55(overshoot) - 2.159(overshoot) + 1.35

I p p

bτ =min 0.645A t , 2.44t

1-b

F

F=1: Good trade-off between performance and robustnessF>1: Smoother F<1: Speed up

“Probably the fastest PID-tuning approach in the world”

D p

bτ = 0.14t if A 1

1-b

26

References Åström, K. J., Hägglund, T. (1984). Automatic tuning of simple regulators with specifications on phase and

amplitude margins, Automatica, (20), 645–651. Desborough, L. D., Miller, R. M. (2002). Increasing customer value of industrial control performance

monitoring—Honeywell’s experience. Chemical Process Control–VI (Tuscon, Arizona, Jan. 2001), AIChE Symposium Series No. 326. Volume 98, USA.

Kano, M., Ogawa, M. (2009). The state of art in advanced process control in Japan, IFAC symposium ADCHEM 2009, Istanbul, Turkey.

Rivera, D. E., Morari, M., Skogestad, S. (1986). Internal model control. 4. PID controller design, Ind. Eng. Chem. Res., 25 (1) 252–265.

Seborg, D. E., Edgar, T. F., Mellichamp, D. A., (2004). Process Dynamics and Control, 2nd ed., John Wiley & Sons, New York, U.S.A.

Shamsuzzoha, M., Skogestad. S. (2010). Report on the setpoint overshoot method (extended version) http://www.nt.ntnu.no/users/skoge/.

Skogestad, S., (2003). Simple analytic rules for model reduction and PID controller tuning, Journal of Process Control, 13, 291–309.

Tyreus, B.D., Luyben, W.L. (1992). Tuning PI controllers for integrator/dead time processes, Ind. Eng. Chem. Res. 2628–2631.

Yuwana, M., Seborg, D. E., (1982). A new method for on-line controller tuning, AIChE Journal 28 (3) 434-440.

Ziegler, J. G., Nichols, N. B. (1942). Optimum settings for automatic controllers. Trans. ASME, 64, 759-768.

Shamsuzzoha, M., Skogestad, S., (2010). The setpoint overshoot method: A simple and fast closed-loop approch for PI tuning”, Journal of Process Control 20 (2010) 1220–1234.

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Thank You

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