Practical plantwide process
control: PID tuning
Sigurd Skogestad, NTNU
Thailand, April 2014
Part 2: PID tuning
Part 2 (4h). PID controller tuning: It pays off to be systematic!
1.
Obtaining first-order plus delay models



2
Open-loop step response
From detailed model (half rule)
From closed-loop setpoint response
. Derivation SIMC PID tuning rules

3.
Controller gain, Integral time, derivative time
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
RTO
cs = y1s
Min J (economics);
MV=y1s
CV=y1; MV=y2s
MPC
y2s
PID
CV=y2; MV=u
u (valves)
PID controller
e

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!
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
Disadvantage IMC-PID (=Lambda tuning):
1.Many rules
2.Poor disturbance response for «slow» processes (with large ¿1/µ)
Motivation for developing SIMC
PID tuning rules
1.
2.
3.
The tuning rules should be well motivated, and
preferably be model-based and analytically
derived.
They should be simple and easy to memorize.
They should work well on a wide range of
processes.
SIMC PI tuning rule
1.
Approximate process as first-order with delay (e.g., use “half rule”)



2.
k = process gain
¿1 = process time constant
µ = process delay
Derive SIMC tuning rule*:
Open-loop step response
c ¸ - : Desired closed-loop response time (tuning parameter)
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!)
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”
MODEL
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, Approach 1A
1. Step response experiment


Make step change in one u (MV) at a time
Record the output (s) y (CV)
MODEL, Approach 1A
Δy(∞)
RESULTING OUTPUT y
STEP IN INPUT u
Δu
: Delay - Time where output does not change
1: Time constant - Additional time to reach
63% of final change
k =  y(∞)/ u : Steady-state gain
MODEL, Approach 1A
Step response integrating process
Δy
Δt
MODEL, Approach 1B
Shams’ method: Closed-loop setpoint response
with P-controller with about 20-40% overshoot
Kc0=1.5
Δys=1
Δy∞
1. OBTAIN DATA IN RED (first overshoot
and undershoot), and then:
tp=2, dyp=1.23; dyu=0.91, Kc0=60, dys=1
Δyp=0.79
Δyu=0.54
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))
theta = tp*[0.309 + 0.209*exp(-0.61*r)]
tau = theta*r
3. CAN THEN USE SIMC PI-rule
tp=4.4
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)
MODEL, Approach 2
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
2
half rule
MODEL, Approach 2
original
1st-order+delay
2nd-order+delay
MODEL, Approach 2
Approximation of zeros
c
c
c
c
c
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
Alternative and improved method forf approximating zeros:
Simple Analytic PID Controller Tuning Rules Revisited
J Lee, W Cho, TF Edgar - Industrial & Engineering Chemistry Research 2014, 53 (13), pp 5038–5047
SIMC-tunings
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
NOTE: Setting the steady-state gain = 1 in T will result in integral action in the controller!
SIMC-tunings
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
d
c
u
g
y
Example. “Almost-integrating process” with disturbance at input:
G(s) = e-s/(30s+1)
Original integral time I = 30 gives poor disturbance response
Try reducing it!
SIMC-tunings
Integral Time
I = 1
Reduce I to this value:
I = 4 (c+) = 8 
Setpoint change at t=0
Input disturbance at t=20
SIMC-tunings
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 “prefiltering”, y’s = f(s) ys.
Details: See www.nt.ntnu.no/users/skoge/publications/2003/tuningPID Remark 13 II
SIMC-tunings
Conclusion: SIMC-PID Tuning Rules
One tuning parameter: c
SIMC-tunings
Some insights from tuning rules
1.
2.
3.
4.
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)
Use (close to) P-control for integrating process
 Beware of large I-action (small τI) for level control
Use (close to) I-control for fast process (with small time
constant τ1)
Parameter variations: For robustness tune at operating point
with maximum value of k’ θ = (k/τ1)θ
Cascade PID -> Ideal PID
SIMC-tunings
SIMC-tunings
Selection of tuning parameter c
Two main cases
1. TIGHT
CONTROL:
Want “fastest possible
TIGHT CONTROL:
control” subject to having good robustness
•
2.
SMOOTH CONTROL:
CONTROL: Want “slowest possible
control” subject to acceptable disturbance rejection
•
•
Want tight control of active constraints (“squeeze and shift”)
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
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 = 0.148
Get 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 = 0.028
Get SIMC PID-tunings (c=): Kc = 1 ¢ 1/(2¢ 0.028) = 17.9, I=min(1,8¢ 0.028) = 0.224, D=0.22
SMOOTH CONTROL
Tuning for 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?
Will derive Kc,min. From this we can get c,max using SIMC tuning rule
S. Skogestad, ``Tuning for smooth PID control with acceptable disturbance rejection'', Ind.Eng.Chem.Res, 45 (23), 7817-7822 (2006).
SMOOTH CONTROL
Closed-loop disturbance rejection
d0
-d0
ymax
-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 ¸ |u0|/|ymax|
often ~1 (in span-scaled variables)
|ymax|
= allowed deviation for output (CV)
|u0|
= required change in input (MV) for disturbance rejection (steady state)
= observed change (movement) in input from historical data
SMOOTH CONTROL
Rule: Kc ¸ |u0|/|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 closedloop 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(4c, 1)

Note: Often want a tight pressure control loop (so may have
Kc=10 or larger)
Conclusion PID tuning
SIMC 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.
2.
3.
4.
5.
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)
April 4-8, 2004
KFUPM-Distillation Control Course
46
SMOOTH CONTROL LEVEL CONTROL
1. Application of smooth control

Averaging level control
q
V
LC
If you insist on integral action
then this value avoids cycling
Reason for having tank is to smoothen disturbances in concentration and flow.
Tight level control is not desired: gives no “smoothening” of flow disturbances.
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 time
provided tank is nominally half full and q0 is equal to the nominal flow.
LEVEL CONTROL
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
How avoid 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 ¢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
LEVEL CONTROL
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
SIMC-tunings
2. Some special cases
One tuning parameter: c
SIMC-tunings
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.
3. Derivative action?
Note: Derivative action is commonly used for temperature control loops.
Select D equal to 2 = time constant of temperature sensor
BUT: Improvement possible for pure
time delay process
Optimal PI
θ=1
Time delay process: Setpoint and disturbance responses same + input response same
Pure time delay process
Two “Improved SIMC”-rules that give
optimal for pure time delay process
1. Improved PI-rule (iSIMC PI): Add θ/3 to 1
2. Improved PID-rule (iSIMC PID): Add θ/3 to 2
iSIMC PID is better for integrating process
Integrating process
Out put s, y
3
G(s) =
i SI M C P I D
2
do
1
1 −s
se
di
SI M
C
op t P P I
I
op t P I D
0
0
10
20
T ime, t
30
40
0
− 0.5
SI M
−1
− 1.5
i SI M C P I D
op t P I D
I
C P
Input s, u
0.5
op
0
10
20
T ime, t
tP
I
30
40
4. Optimality of SIMC rules
How good are the SIMC-rules compared to optimal PI/PID?

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.
Our choice:
J = avg. IAE for
setpoint/disturbance
Ms = peak sensitivity
Performance (J):
di
ysp
e
K (s)
−
u
do
+
+
G(s)
1.5
y
1.5
do
di
1
I AE do
0.5
T ime, t
0
I AE di
0.5
T ime, t
0
2
− 0.5
Error, e(t )
Error, e(t )
1
4
6
8
10
2
− 0.5
4
6
8
10
Robustness (Ms):
Comparison of J vs. Ms for optimal and SIMC for 4 processes
CONCLUSION: i-SIMC almost «Pareto-optimal»
5. Better with IMC, Smith
Predictor or MPC?


Suprisingly, the answer is:
NO, worse
The Smith Predictor
Where K is a “normal” PI controller
IMC controller
Special case of Smith Predictor where K is a PI controller with the parameters
tau1 > 0
Kc = tau1/(k tau_c)
tau_I = tau1
tau1 = 0
Kc =0
Ki = Kc/tau_I = 1/tau_c
Comparison of J vs. Ms for optimal and SIMC for 4 processes
CONCLUSION: i-SIMC is generally better than IMC & SP!

In addition: SP & IMC usually have much lower (worse) delay margin
than PI/PID

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
