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PID-tuning using the SIMC rules

Sigurd Skogestad

Department of Chemical Engineering, NTNU, Trondheim, Norway

Salamanca, Spain, Feb. 2017

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PID Tuning using the SIMC rules Sigurd Skogestad NTNU, Trondheim, Norway Abstract: Although the proportional-integral-derivative (PID) controller has only three parameters, it is not easy, without a systematic procedure, to find good values (settings) for them. In fact, a visit to a process plant will usually show that a large number of the PID controllers are poorly tuned. In general, much better results are obtained if one is willing to take a systematic approach and invest some time. The following two-step procedure works well: Step 1. Obtain a first- or second-order plus delay model. Step 2. Derive model-based SIMC controller settings. With the SIMC method, PI-settings result if we start from a first-order model, whereas PID-settings result from a second-order model. The SIMC method is based on classical ideas presented earlier by Ziegler and Nichols (1942), the IMC PID-tuning paper by Rivera et al. (1986), and the closely related direct synthesis tuning rules in the book by Smith and Corripio (1985). The Ziegler-Nichols settings result in a very good disturbance response for integrating processes, but are otherwise known to result in rather aggressive settings (Tyreus and Luyben 1992) (Astrom and Hagglund 1995), and also give poor performance for processes with a dominant delay. On the other hand, the analytically derived IMC-settings of Rivera et al. (1986) are known to result in poor disturbance response for integrating processes (Chien and Fruehauf 1990), (Horn et al. 1996), but are robust and generally give very good responses for setpoint changes. The SIMC tuning rule works well for both integrating and pure time delay processes, and for both setpoints and load disturbances. It is actually close to the optimum as can be seen by evaluating the Pareto-optimality of the SIMC method with respect to he conflicting objectives of performance and robustness. The results with PID control are generally also better than with the model-based Smith Predictor, even with processes with large time delays. This is surprising, and it shows that if one puts enough effort into the PID tuning then there is little benefit in considering more complex controllers, including MPC.

•  •  •  •  • 

Trondheim

Oslo

UK

NORWAY

DENMARK

GERMANY

North Sea

SWEDEN

Arctic circle

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Operation hierarchy

CV1

MPC

PID

CV2

RTO

u (valves)

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Outline 1.  First-order plus delay model. 2.  PID-control. 3.  SIMC PI(D)-rule 4.  Optimal PI controller 5.  Comparison of SIMC with optimal PI 6.  Improved SIMC-PI for time-delay process 7.  Non-PID control of first-order plus delay process: Better

with IMC / Smith Predictor / MPC? (no) 8.  Example: Level control 9.  Conclusion

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1. Need a model for tuning •  First-order + delay model for PI-control

•  Second-order model for PID-control

–  Recommend: Use second-order model only if ¿2>µ

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Step response experiment n  Make step change in one u (MV) at a time n  Record the output (s) y (CV)

MODEL, Approach 1

MV = manipulated variable (input) CV = controlled variable (output)

u y

Identify k, τ1 and θ from step response

k’=k/τ1

STEP IN INPUT u (MV)

RESULTING OUTPUT y (CV)

θ: Delay - Time where output does not change τ1: Time constant - Additional time to reach 63% of final change k : steady-state gain = Δ y(1)/Δ u k’ : slope after response “takes off” = k/τ1

If doesn’t settle: Don’t need k and τ1.Stop experiment at about 8 x θ and assume integrating process

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Step response integrating process

Δy

Δt

Closed-loop setpoint response with P-controller with about 20-40% overshoot

Kc0=1.5 Δys=1

Δyu=0.54 Δyp=0.79

tp=4.4

1.  OBTAIN DATA IN RED (first overshoot and undershoot), and then:

tp=4.4, dyp=0.79; dyu=0.54, Kc0=1.5, dys=1 dyinf = 0.45*(dyp + dyu) Mo =(dyp -dyinf)/dyinf % Mo=overshoot (about 0.3) b=dyinf/dys A = 1.152*Mo^2 - 1.607*Mo + 1.0 r = 2*A*abs(b/(1-b)) %2. OBTAIN FIRST-ORDER MODEL: k = (1/Kc0) * abs(b/(1-b)) = 0.99 theta = tp*[0.309 + 0.209*exp(-0.61*r)] = 1.68 tau = theta*r = 3.03

Ref: Shamssuzzoha and Skogestad (JPC, 2010) + modification by C. Grimholt (PID-book 2012)

Δy∞

MODEL, Approach 2

Similar to Ziegler-Nichols experiment but get more information

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Model reduction of more complicated model

•  Start with complicated stable model on the form

•  Want to get a simplified model on the form

•  Most important parameter is the “effective” delay θ

MODEL, Approach 3

13 MODEL, Approach 3. HALF RULE

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Example 1

Half rule

MODEL, Approach 3

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original 1st-order+delay

MODEL, Approach 3

half rule

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MODEL, Approach 3

original 1st-order+delay 2nd-order+delay

MODEL, Approach 3. HALF RULE

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2. 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 settings 2. No tuning parameter 3. 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 rules 2. 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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3. SIMC PI tuning rule 1.  Approximate process as first-order with delay (e.g., use “half rule”)

•  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 (*) “Probably the best simple PID tuning rules in the world”

τc ¸ -θ : 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; aggressive!)

Derivation SIMC tuning rule. Basis: Direct synthesis (IMC) for setpoints

Closed-loop response to setpoint change:

Idea: Specify desired response: and from this get the controller. ……. Algebra:

NOTE: Setting T(0)=1 (steady-state gain = 1) gives integral action.

Derivation SIMC tuning rule (setpoints)

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Integral time •  Found: Integral time = dominant time constant (τI = τ1) (IMC-rule) •  Works well for setpoint changes •  Needs to be modified (reduced) for integrating disturbances

Example. “Almost-integrating process” with disturbance at input:

G(s) = e-s/(30s+1) Original integral time τI = 30 gives poor disturbance response Try reducing it!

gc

dyu

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Effect of integral time on closed-loop response

IMC-rule: τI = τ1=30

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

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SIMC: Integral time correction •  Setpoints: ¿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:

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Conclusion: SIMC-PID Tuning Rules

One tuning parameter: τc Note: • Recommend derivative action (PID) only for «dominant» second-order proceses» with τ2>θ. • Otherwise, add τ2 to θ and use PI.

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Selection of tuning parameter τc n  Tuning parameter: τc = desired closed-loop response time

n  Choice τc=θ (“tight control”) gives good balance between performance (IAE) and robustness

n  Other cases: Select τc > θ (“smooth control”) for q  slower control q  smoother input usage

n  less disturbing effect on rest of the plant q  less sensitivity to measurement noise q  better robustness

S. Skogestad, ``Tuning for smooth PID control with acceptable disturbance rejection'', Ind.Eng.Chem.Res, 45 (23), 7817-7822 (2006).

31 TIGHT CONTROL

32 Typical closed-loop SIMC responses with the choice τc=θ

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Example. Integrating process with delay=1. G(s) = e-s/s. Model: k’=1, θ=1, τ1=1 SIMC-tunings with τc with =θ=1:

IMC has τI=1

Ziegler-Nichols is usually a bit aggressive

Setpoint change at t=0c Input disturbance at t=20

1.  Approximate as first-order model: k=1, τ1 = 1+0.1=1.1, θ=0.1+0.04+0.008 = 0.148 Get SIMC PI-tunings (τc=θ): Kc = 1 ¢ 1.1/(2¢ 0.148) = 3.71, τI=min(1.1,8¢ 0.148) = 1.1

2.  Approximate as second-order model: k=1, τ1 = 1, τ2=0.2+0.02=0.22, θ=0.02+0.008 = 0.028

Get SIMC PID-tunings (τc=θ): Kc = 1 ¢ 1/(2¢ 0.028) = 17.9, τI=min(1,8¢ 0.028) = 0.224, τD=0.22

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

Ms

GM

PM

τc /θ τc /θ

DM= Δθ /θ

3

1 1

2

1.6 1 1

60o

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But: 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:

4. Optimal PID controller

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Ms=2

Ms=1.2

Ms=1.59 |S|

frequency

Optimal PI-controller

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

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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) magnitude 3.  unit for time

Output performance (J)

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Ms=1.59

Optimal PI-controller

Setpoint change at t=0, Input disturbance at t=20, g(s)=k e-θs/(τ1s+1), Time delay θ=1

Optimal closed-loop response

IAE

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Ms=1.2

Optimal PI-controller

Setpoint change at t=0, Input disturbance at t=20, g(s)=k e-θs/(τ1s+1), Time delay θ=1

Optimal closed-loop response IAE

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Optimal PI-controller: Minimize J=IAE 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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Ms=2

Optimal PI-controller

4 processes, g(s)=k e-θs/(τ1s+1), Time delay θ=1. Setpoint change at t=0, Input disturbance at t=20,

Optimal closed-loop response

Infeasible

Uninteresting

Optimal performance (J) vs. Ms

Optimal PI-controller

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Input usage (TV) increases with Ms

TVys TVd

Optimal PI-controller

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Setpoint / disturbance tradeoff

Pure time delay process: J=1, No tradeoff (since setpoint and disturbance the same)

Optimal controller: Emphasis on disturbance d

Optimal PI-controller

Ms=1.59

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Setpoint / disturbance tradeoff

Optimal for setpoint: ¿I=¿1 (except time delay process) Integrating process (¿1=1): No integral action

Optimal PI-controller

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Optimal PI-settings vs. process time constant (τ1 /θ)

Optimal PI-controller

Ziegler-Nichols

Ziegler-Nichols

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5. What about SIMC-PI?

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

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Optimal PI-settings vs. process time constant (τ1 /θ)

Optimal PI-controller

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

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

•  τC = 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 (small τ1)

Time-delay process SIMC: τI=τ1=0

0.33

Optimal PI-controller

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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 τ1

1. Improved PID-rule: Add θ/3 to τ2

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

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

7. Better with IMC, Smith Predictor or MPC?

n  Surprisingly, the answer is: n  NO, often worse

Smith Predictor

di

do

ysp K G�e�✓s y

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⇣1� e�✓̂s

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�

e�

u + +

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 SP for 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 •  No integral action for integrating process •  Sensitive to both positive and negative delay error •  With tight tuning (Ms approaching 2): Multiple gain and delay margins

10�1 100 101 10210�1

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101

100

|L|

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L(s) = e�(1+✓✏)s

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Frequency, !

Magnitude

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Step response, SP and PI

y

time time time

Smith Predictor (IMC): Sensitive to both positive and negative delay error

Nominal. τ1 = θ = 1

θ=1(nominal)

θ=1.43 θ=0.57

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Delay margin, SP and PI

SP = Smith Predictor

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8. Example: Level control

n  Level control often causes problems n  Typical story:

q  Level loop starts oscillating q Operator detunes by decreasing controller gain q  Level loop oscillates even more q  ......

n  ??? n  Explanation: Level is by itself unstable and requires

control.

VqLC

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Level control: Can have both fast and slow oscillations

•  Slow oscillations (Kc too low): Period > 3¿I

•  Fast oscillations (Kc too high due to delay): Period < 3¿I

Here: Consider the less known slow oscillations

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How avoid slowly oscillating levels? • Simplest: Use P-control only (no integral action)• If you insist on integral action, then make sure

the controller gain is sufficiently large• If you have a level loop that is oscillating then

use Sigurds rule (can be derived):

To avoid oscillations, increase Kc ¢τI by factor f=0.1¢(P0/τI0)2

where P0 = period of oscillations [s]τI0 = original integral time [s]0.1 ¼ = 1/π2

.

.

Use SIMC with large τc

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Case study oscillating level

•  We were called upon to solve a problem with oscillations in a distillation column

•  Closer analysis: Problem was oscillating reboiler level in upstream column

•  Use of Sigurd’s rule solved the problem

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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 τ1 by τ1+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”