Optimality of PID control for process control applications

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Optimality of PID control for process control applications

Sigurd SkogestadChriss Grimholt

NTNU, Trondheim, Norway

AdCONIP, Japan, May 2014

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Trondheim, Norway

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Operation hierarchy

CV1

MPC

PID

CV2

RTO

u (valves)

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Outline

1.Motivation: Ziegler-Nichols open-loop tuning + IMC

2.SIMC PI(D)-rule

3.Definition of optimality (performance & robustness)

4.Optimal PI control of first-order plus delay process

5.Comparison of SIMC with optimal PI

6.Improved SIMC-PI for time-delay process

7.Non-PID control: Better with IMC / Smith Predictor? (no)

8.Conclusion

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PID controller

•Time domain (“ideal” PID)

•Laplace domain (“ideal”/”parallel” form)

•For our purposes. Simpler with cascade form

•Usually τD=0. Then the two forms are identical.

•Only two parameters left (Kc and τI)

•How difficult can it be to tune???– Surprisingly difficult without systematic approach!

e

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Trans. ASME, 64, 759-768 (Nov. 1942).

Disadvantages Ziegler-Nichols:1.Aggressive settings2.No tuning parameter3.Poor for processes with large time delay (µ)

Comment:Similar to SIMC for integrating process with ¿c=0:Kc = 1/k’ 1/µ¿I = 4 µ

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Disadvantage IMC-PID:1.Many rules2.Poor disturbance response for «slow» processes (with large ¿1/µ)

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Motivation for developing SIMC PID tuning rules

1.The tuning rules should be well motivated, and preferably be model-based and analytically derived.

2.They should be simple and easy to memorize.

3.They should work well on a wide range of processes.

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2. SIMC PI tuning rule1.A

pproximate process as first-order with delay• k = process gain• ¿1 = process time constant• µ = process delay

2.Derive SIMC tuning rule:

Reference: S. Skogestad, “Simple analytic rules for model reduction and PID controller design”, J.Proc.Control, Vol. 13, 291-309, 2003

c ¸ - : Desired closed-loop response time (tuning parameter)

Open-loop step response

Integral time rule combines well-known rules:IMC (Lamda-tuning): Same as SIMC for small ¿1 (¿I = ¿1)Ziegler-Nichols: Similar to SIMC for large ¿1 (if we choose ¿c= 0)

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Derivation SIMC tuning rule (setpoints)

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Effect of integral time on closed-loop response

I = 1=30

Setpoint change (ys=1) at t=0 Input disturbance (d=1) at t=20

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SIMC: Integral time correction•S

etpoints: ¿I=¿1(“IMC-rule”). Want smaller integral time for disturbance rejection for “slow” processes (with large ¿1), but to avoid “slow oscillations” must require:

•Derivation:

•Conclusion SIMC:

15Typical closed-loop SIMC responses with the choice c=

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SIMC PI tuning rule

c ¸ - : Desired closed-loop response time (tuning parameter)•For robustness select: c ¸

Two questions:• How good is really the SIMC rule?• Can it be improved?

Reference: S. Skogestad, “Simple analytic rules for model reduction and PID controller design”, J.Proc.Control, Vol. 13, 291-309, 2003

“Probably the best simple PID tuning rule in the world”

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How good is really the SIMC rule?

Want to compare with:

•Optimal PI-controller

for class of first-order with delay processes

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•Multiobjective. Tradeoff between

– Output performance – Robustness– Input usage– Noise sensitivity

High controller gain (“tight control”)

Low controller gain (“smooth control”)

• Quantification– Output performance:

• Rise time, overshoot, settling time

• IAE or ISE for setpoint/disturbance

– Robustness: Ms, Mt, GM, PM, Delay margin, …

– Input usage: ||KSGd||, TV(u) for step response

– Noise sensitivity: ||KS||, etc.

Ms = peak sensitivity

J = avg. IAE for setpoint/disturbance

Our choice:

3. Optimal controller

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IAE = Integrated absolute error = ∫|y-ys|dt, for step change in ys or d

Cost J(c) is independent of:1. process gain (k)2. setpoint (ys or dys) and disturbance (d) magnitude3. unit for time

Output performance (J)

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4. Optimal PI-controller: Minimize J for given Ms

Optimal PI-controller

Chriss Grimholt and Sigurd Skogestad. "Optimal PI-Control and Verification of the SIMC Tuning Rule". Proceedings IFAc conference on Advances in PID control (PID'12), Brescia, Italy, 28-30 March 2012.

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Optimal PI-settings vs. process time constant (1 /θ)


Ziegler-Nichols

Ziegler-Nichols

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Ms=2

Ms=1.2

Ms=1.59|S|

frequency


Optimal sensitivity function, S = 1/(gc+1)

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Ms=2


4 processes, g(s)=k e-θs/(1s+1), Time delay θ=1.Setpoint change at t=0, Input disturbance at t=20,

Optimal closed-loop response

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Ms=1.59


Setpoint change at t=0, Input disturbance at t=20,g(s)=k e-θs/(1s+1), Time delay θ=1


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Ms=1.2


Setpoint change at t=0, Input disturbance at t=20,g(s)=k e-θs/(1s+1), Time delay θ=1


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Infeasible

UninterestingPareto-optimal PI

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Optimal performance (J) vs. Ms


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5. What about SIMC-PI?

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SIMC: Tuning parameter (¿c) correlates nicely with robustness measures

Ms

GM

PM

¿c=µ ¿c=µ

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What about SIMC-PI performance?

34Comparison of J vs. Ms for optimal and SIMC for 4 processes

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Conclusion (so far): How good is really the SIMC rule?

•Varying C gives (almost) Pareto-optimal tradeoff between performance (J) and robustness (Ms)

C = θ is a good ”default” choice

•Not possible to do much better with any other PI-controller!

•Exception: Time delay process

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6. Can the SIMC-rule be improved?

Yes, for time delay process

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Optimal PI-settings vs. process time constant (1 /θ)


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Optimal PI-settings (small 1)

Time-delay processSIMC: I=1=0

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Step response for time delay process

θ=1

Optimal PI

NOTE for time delay process: Setpoint response = disturbance responses = input response

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Pure time delay process

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Two “Improved SIMC”-rules that give optimal for pure time delay process

1. Improved PI-rule: Add θ/3 to 1

1. Improved PID-rule: Add θ/3 to 2

42Comparison of J vs. Ms for optimal-PI and SIMC for 4 processes

CONCLUSION PI: SIMC-improved almost «Pareto-optimal»

7. Better with IMC or Smith Predictor?

Surprisingly, the answer is:

NO, worse

Smith Predictor

K: Typically a PI controller

Internal model control (IMC): Special case with ¿I=¿1

Fundamental problem Smith Predictor: No integral action in c for integrating process

c

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Optimal SP compared with optimal PI

SP = Smith Predictor with PI (K)

¿1=20 since J=1 for SPfor integrating process

¿1=20

¿1=0

¿1=8

¿1=1

Small performance gain with Smith Predictor

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Additional drawbacks with Smith Predictor•N

o integral action for integrating process•S

ensitive to both positive and negative delay error•W

ith tight tuning (Ms approaching 2): Multiple gain and delay margins

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Step response, SP and PI

g(s) = k e¡ s

s+1

µ= 1

y

time time time

Smith Predictor: Sensitive to both positive and negative delay error

SP = Smith Predictor

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Delay margin, SP and PI

SP = Smith Predictor

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8. Conclusion Questions for 1st and 2nd order processes with delay:

1.How good is really PI/PID-control?– Answer: Very good, but it must be tuned properly

2.How good is the SIMC PI/PID-rule?– Answer: Pretty close to the optimal PI/PID, – To improve PI for time delay process: Replace 1 by 1+µ/3

3.Can we do better with Smith Predictor or IMC?– No. Slightly better performance in some cases, but much worse delay margin

4.Can we do better with other non-PI/PID controllers (MPC)?– Not much (further work needed)

•SIMC: “Probably the best simple PID tuning

rule in the world”

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11th International IFAC Symposium on Dynamics and Control of Process and

Bioprocess Systems (DYCOPS+CAB). 06-08 June 2016

Location: Trondheim (NTNU)

Organizer: NFA (Norwegian NMO) + NTNU (Sigurd Skogestad, Bjarne Foss, Morten Hovd, Lars Imsland, Heinz Preisig, Magne Hillestad, Nadi Bar),

Norwegian University of Science and Technology (NTNU), Trondheim

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