Practical plantwide process control: PID tuning

Sigurd Skogestad, NTNU

Part 4: PID tuning

Part 2 (4h). PID controller tuning: It pays off to be systematic! 1. Obtaining first-order plus delay models

Open-loop step response From detailed model (half rule) From closed-loop setpoint response

2 . Derivation SIMC PID tuning rules Controller gain, Integral time, derivative time

 3. Special topics

Integrating processes (level control) Other special processes and examples When do we need derivative action? Near-optimality of SIMC PID tuning rules Non PID-control: Is there an advantage in using Smith Predictor? (No)

Examples 

Operation: Decision and control layers

cs = y1s

MPC

PID

y2s

RTO

u (valves)

CV=y1; MV=y2s

CV=y2; MV=u

Min J (economics); MV=y1s

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

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

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

Motivation for developing SIMC PID tuning rules1. 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.

SIMC PI tuning rule1. 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”

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; aggressive!)

Need a model for tuning

Model: Dynamic effect of change in input u (MV) on output y (CV)

First-order + delay model for PI-control

Second-order model for PID-control

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

MODEL

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

MODEL, Approach 1A

STEP IN INPUT u

RESULTING OUTPUT y

: Delay - Time where output does not change1: Time constant - Additional time to reach 63% of final changek = y(∞)/ u : Steady-state gain

Δy(∞)

Δu

MODEL, Approach 1A

Step response integrating process

Δy

Δt

MODEL, Approach 1A

Shams’ method: Closed-loop setpoint response with P-controller with about 20-40% overshoot

Kc0=1.5Δys=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/dysA = 1.152*Mo^2 - 1.607*Mo + 1.0r = 2*A*abs(b/(1-b))

%2. OBTAIN FIRST-ORDER MODEL:

k = (1/Kc0) * abs(b/(1-b))theta = tp*[0.309 + 0.209*exp(-0.61*r)]tau = theta*r

3. CAN THEN USE SIMC PI-rule

Example 2: Get k=0.99, theta =1.68, tau=3.03

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

Δy∞

MODEL, Approach 1B

2. 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

MODEL, Approach 2

MODEL, Approach 2

Example 1

Half rule

MODEL, Approach 2

original

1st-order+delay

MODEL, Approach 2

half rule

2

MODEL, Approach 2

original

1st-order+delay

2nd-order+delay

MODEL, Approach 2

Approximation of zeros

Alternative and improved method forf approximating zeros: Simple Analytic PID Controller Tuning Rules RevisitedJ Lee, W Cho, TF Edgar - Industrial & Engineering Chemistry Research 2014, 53 (13), pp 5038–5047

MODEL, Approach 2

To make these rules more general (and not only applicable to the choice c=): Replace (time delay) by c (desired closed-loop response time). (6 places)

c

c

c

c c

c

Derivation of SIMC-PID tuning rules PI-controller (based on first-order model)

For second-order model add D-action.For our purposes, simplest with the “series” (cascade) PID-form:

SIMC-tunings

Basis: Direct synthesis (IMC)

Closed-loop response to setpoint change

Idea: Specify desired response:

and from this get the controller. ……. Algebra:

SIMC-tunings

SIMC-tunings

NOTE: Setting the steady-state gain = 1 in T will result in integral action in the controller!

IMC Tuning = Direct Synthesis Algebra:

SIMC-tunings

Integral time Found: Integral time = dominant time constant (I = 1) (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 I = 30 gives poor disturbance responseTry reducing it!

gc

d

yu

SIMC-tunings

Integral Time

I = 1

Reduce I to this value:I = 4 (c+) = 8

SIMC-tunings

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

Integral time Want to reduce the integral time for “integrating” processes,

but to avoid “slow oscillations” we must require:

Derivation:

Setpoint response: Improve (get rid of overshoot) by “pre-filtering”, y’s = f(s) ys.

SIMC-tunings

Details: See www.nt.ntnu.no/users/skoge/publications/2003/tuningPID Remark 13 II

Conclusion: SIMC-PID Tuning Rules

One tuning parameter: c

SIMC-tunings

Some insights from tuning rules1. The effective delay θ (which limits the achievable closed-loop

time constant τc) is independent of the dominant process time constant τ1!

It depends on τ2/2 (PI) or τ3/2 (PID)

2. Use (close to) P-control for integrating process Beware of large I-action (small τI) for level control

3. Use (close to) I-control for fast process (with small time

constant τ1)4. Parameter variations: For robustness tune at operating point

with maximum value of k’ θ = (k/τ1)θ

SIMC-tunings

Cascade PID -> Ideal PID

SIMC-tunings

Selection of tuning parameter cTwo main cases1. TIGHT CONTROL: Want “fastest possible

control” subject to having good robustness• Want tight control of active constraints (“squeeze and shift”)

2. SMOOTH CONTROL: Want “slowest possible control” subject to acceptable disturbance rejection

• Want smooth control if fast setpoint tracking is not required, for example, levels and unconstrained (“self-optimizing”) variables

• THERE ARE ALSO OTHER ISSUES: Input saturation etc.

TIGHT CONTROL:

SMOOTH CONTROL:

SIMC-tunings

TIGHT CONTROL

Typical closed-loop SIMC responses with the choice c=

TIGHT CONTROL

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

IMC has I=1

Ziegler-Nichols is usually a bit aggressive

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

TIGHT CONTROL

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

2. Approximate as second-order model with k=1, 1 = 1, 2=0.2+0.02=0.22, =0.02+0.008 = 0.028Get SIMC PID-tunings (c=): Kc = 1 ¢ 1/(2¢ 0.028) = 17.9, I=min(1,8¢ 0.028) = 0.224, D=0.22

TIGHT CONTROL

Tuning for smooth control

Will derive Kc,min. From this we can get c,max using SIMC tuning rule

SMOOTH CONTROL

Tuning parameter: c = desired closed-loop response time

Selecting c= (“tight control”) is reasonable for cases with a relatively large effective delay

Other cases: Select c > for slower control smoother input usage

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

Question: Given that we require some disturbance rejection. What is the largest possible value for c ? Or equivalently: The smallest possible value for Kc?

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

Closed-loop disturbance rejection d0

ymax

-d0

-ymax

SMOOTH CONTROL

Kc

u

Minimum controller gain for PI-and PID-control:min |c(j)| = Kc

SMOOTH CONTROL

Rule: Min. controller gain for acceptable disturbance rejection:

Kc ¸ |ud|/|ymax| often ~1 (in span-scaled variables)

SMOOTH CONTROL

|ymax| = allowed deviation for output (CV)

|ud| = required change in input (MV) for disturbance rejection (steady state) = observed change (movement) in input from historical data

Rule: Kc ¸ |ud|/|ymax|

Exception to rule: Can have lower Kc if disturbances are handled by the integral action. Disturbances must occur at a frequency lower than 1/I

Applies to: Process with short time constant (1 is small) and no delay ( ¼ 0). For example, flow control

Then I = 1 is small so integral action is “large”

SMOOTH CONTROL

Summary: Tuning of easy loops Easy loops: Small effective delay ( ¼ 0), so closed-loop

response time c (>> ) is selected for “smooth control” ASSUME VARIABLES HAVE BEEN SCALED WITH

RESPECT TO THEIR SPAN SO THAT |u0/ymax| = 1 (approx.).

Flow control: Kc=0.2, I = 1 = time constant valve (typically, 2 to 10s; close to pure integrating!)

Level control: Kc=2 (and no integral action) Other easy loops (e.g. pressure): Kc = 2, I = min(4c, 1)

Note: Often want a tight pressure control loop (so may have Kc=10 or larger)

SMOOTH CONTROL

Conclusion PID tuningSIMC tuning rules

1. Tight control: Select c= corresponding to

2. Smooth control. Select Kc ¸

Note: Having selected Kc (or c), the integral time I should be selected as given above

3. Derivative time: Only for dominant second-order processes

PID: More (Special topics)

1. Integrating processes (level control)

2. Other special processes and examples

3. When do we need derivative action?

4. Near-optimality of SIMC PID tuning rules

5. Non PID-control: Is there an advantage in using Smith Predictor? (No)

April 4-8, 2004KFUPM-Distillation Control

Course 46

1. Application of smooth control Averaging level control

VqLC

Reason for having tank is to smoothen disturbances in concentration and flow. Tight level control is not desired: gives no “smoothening” of flow disturbances.

SMOOTH CONTROL LEVEL CONTROL

If you insist on integral actionthen this value avoids cycling

Proof: 1. Let |u0| = |q0| – expected flow change [m3/s] (input disturbance)

|ymax| = |Vmax| - largest allowed variation in level [m3]

Minimum controller gain for acceptable disturbance rejection: Kc ¸ Kc,min = |u0|/|ymax| = |q0| / |Vmax|

2. From the material balance (dV/dt = q – qout), the model is g(s)=k’/s with k’=1.Select Kc=Kc,min. SIMC-Integral time for integrating process:

I = 4 / (k’ Kc) = 4 |Vmax| / | q0| = 4 ¢ residence timeprovided tank is nominally half full and q0 is equal to the nominal flow.

More on level control

Level control often causes problems Typical story:

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

??? Explanation: Level is by itself unstable and

requires control.

LEVEL CONTROL

Level control: Can have both fast and slow oscillations Slow oscillations (Kc too low): P > 3¿I

Fast oscillations (Kc too high): P < 3¿I

Here: Consider the less common slow oscillations

LEVEL CONTROL

How avoid oscillating levels?

LEVEL CONTROL

• 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 ¢I by factor f=0.1¢(P0/I0)2

where P0 = period of oscillations [s]I0 = original integral time [s]0.1 ¼ 1/2

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

LEVEL CONTROL

LEVEL CONTROL

One tuning parameter: c

SIMC-tunings

2. Some special cases

Another special case: IPZ process

IPZ-process may represent response from steam flow to pressure

Rule T2: SIMC-tunings

These tunings turn out to be almost identical to the tunings given on page 104-106 in the Ph.D. thesis by O. Slatteke, Lund Univ., 2006 and K. Forsman, "Reglerteknik for processindustrien", Studentlitteratur, 2005.

SIMC-tunings

Note: Derivative action is commonly used for temperature control loops. Select D equal to 2 = time constant of temperature sensor

3. Derivative action?

Time delay process: Setpoint and disturbance responses same + input response same

θ=1

Optimal PI*

Pure time delay process: “Minor” improvement by adding D-action*

* D-action (ID-control) is equivalent to PI-control (with ¿I>0)

= I-control

Pure time delay process

) Two alternative “Improved SIMC”-rulesAlt. 1. Improved PI-rule (iSIMC-PI): Add θ/3 to 1

Alt. 2. Improved PID-rule (iSIMC-PID): Add θ/3 to 2

iSIMC-PI and iSIMC-PID are identical for pure delay process (¿1=0)iSIMC-PID is better for integrating process

Integrating process

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. Optimality of SIMC rules

How good are the SIMC-rules compared to optimal PI/PID?

Performance (J):

Robustness (Ms):

JIAE vs. Ms for optimal PI/PID (-) and SIMC (¢¢) for 4 processes

CONCLUSION: SIMC almost «Pareto-optimal»Note: “PID” weightings used for JIAE

5. Better with IMC, Smith Predictor or MPC?

Suprisingly, the answer is: NO, worse

The Smith Predictor

Where K is a “normal” controller

IMC controller

Special case of Smith Predictor where K is a PI controller with the parameters

Kc = ¿1/(k ¿c)¿I = ¿1

Kc =0Ki = Kc/¿I = 1/¿c

¿1 > 0 ¿1 = 0

(I-controller)

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

CONCLUSION: i-SIMC is generally better than IMC & SP!

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

Reason: SP & IMC can have multiple GM, PM, DM

CONCLUSION

Well-tuned PI or PID is better than Smith Predictor or IMC!!

Especially for integrating processes