Alternative form with detuning factor F
SIMC-tunings QUIZ
Quiz: SIMC PI-tunings
y
y
Step response
t [s]Time t
(a) The Figure shows the response (y) from a test where we made a step change in the
input (Δu = 0.1) at t=0. Suggest PI-tunings for (1) τc=2,. (2) τc=10.
(b) Do the same, given that the actual plant is
QUIZ
Solution
Actual
plant:
QUIZ
Approximation of step response
Approximation ”bye eye”
SIMC-tunings
Kc=2.9, tauI=10
Kc=9.5, tauI=10
OUTPUT y
INPUT u
Tunings from Step response “by eye” model
Setpoint
change at t=0, input disturbance = 0.1 at t=50
Tunings from Half rule (Somewhat better)
Kc=2, tauI=5.5
Kc=6, tauI=5.5
QUIZ
Half-rule approach: Approximation of
zeros depends on tauc!
Some discussion points




Selection of τc: some other issues
Obtaining the model from step responses: How
long should we run the experiment?
Cascade control: Tuning
Controllability implications of tuning rules
Selection of c: Other issues

Input saturation.


Problem. Input may “overshoot” if we “speedup” the
response too much (here “speedup” = /c).
Solution: To avoid input saturation, we must obey max
“speedup”:
A little more on obtaining the model
from step response experiments

“Factor 5 rule”: Only dynamics
within a factor 5 from “control
time scale” (c) are important
1 ¼ 200
(may be neglected for c < 40)
0.9954
0.9953
0.9953

Integrating process (1 = 1)
Time constant 1 is not important if it is much larger
than the desired response time c. More
precisely, may use
1 =1 for 1 > 5 c
0.9952
0.9952
0.9951
0.9951
0.995
0.995
0.9949
0

Delay-free process (=0)
Delay  is not important if it is much smaller than
the desired response time c. More precisely,
may use
 ¼ 0 for  < c/5
10
20
30
40
50
¼1
(may be neglected for c > 5)
c = desired response time
60
time
Step response experiment: How
long do we need to wait?


RULE: May stop at about 10 times effective delay
FAST TUNING DESIRED (“tight control”, c = ):

NORMALLY NO NEED TO RUN THE STEP EXPERIMENT FOR LONGER THAN ABOUT 10 TIMES THE EFFECTIVE DELAY ()

EXCEPTION: LET IT RUN A LITTLE LONGER IF YOU SEE THAT IT IS ALMOST SETTLING (TO GET 1 RIGHT)


SIMC RULE: I = min (1, 4(c+)) with c =  for tight control
SLOW TUNING DESIRED (“smooth control”, c > ):

HERE YOU MAY WANT TO WAIT LONGER TO GET 1 RIGHT BECAUSE IT MAY AFFECT THE INTEGRAL TIME

BUT THEN ON THE OTHER HAND, GETTING THE RIGHT INTEGRAL TIME IS NOT ESSENTIAL FOR SLOW TUNING

SO ALSO HERE YOU MAY STOP AT 10 TIMES THE EFFECTIVE DELAY ()

“Integrating process” (c < 0.2 1):


Need only two parameters: k’ and 
From step response:
Response on stage 70 to step in L
Example.
Step change in u:
Initial value for y:
Observed delay:
At T=10 min:
Initial slope:
2.8
u = 0.1
y(0) = 2.19
 = 2.5 min
y(T)=2.62
2.7
2.6
2.5
y(t)
2.4
2.62-2.19
2.3
7.5 min
2.2
2.1
0
=2.5
t [min]
2
4
6
8
10
Example (from quiz)



Step response
Δu=0.1
Assume integrating process, theta=1.5; k’ = 0.03/(0.1*11.5)=0.026
SIMC-tunings tauc=2: Kc=11, tauI=14 (OK)
SIMC-tunings tauc=10: Kc=3.3, tauI = 46 (too long because process is not actually
integrating on this time scale!)
INPUT y
OUTPUT y
tauc=10
tauc=2
Cascade control
Cascade control
Tuning:
1. First tune TC
(based on response from V to T)
2. Close TC and tune CC
(based on response from Ts to xB)
Ts
Primary controller (CC)
sets setpoint to secondary
controller (TC).
TC
xB
CC
CC: Primary controller (“slow”):
TC: Secondary controller (“fast”):
y1 = xB (“original” CV),
y2 = T (CV),
u1 = y2s (MV)
u2 = V (“original” MV)
Cascade control
Tuning of cascade controllers
• Want to control y 1 (primary CV), but have “extra” measurement y2
• Idea: Secondary variable (y2) may be tightly controlled and this
helps control of y1.
• Implemented using cascade control: Input (MV) of “primary”
controller (1) is setpoint (SP) for “secondary” controller (2)
• Tuning simple: Start with inner secondary loops (fast) and move
upwards
• Must usually identify ”new” model ( G1’ = G1 G21 K2 (I+K2G22)-1 )
experimentally after closing each loop
•
One exception: Serial process, G21 = G22
2
– Inner (secondary-2) loop may be modelled with gain=1 and effective
delay=(c+)2
See next slide
Cascade control
Special case: Serial cascade
y2 = T2 r2 + S2d2, T2 = G2K2(I+G2K2)-1


K2 is designed based on G2 (which has effective delay 2)

then y2 = T2 r2 + S2 d2 where S2 ¼ 0 and T2 ¼1 · e-(2+c2)s
 T2: gain = 1 and effective delay = 2+c2
 NOTE: If delay is in meas. of y 2 (and not in G2) then T2 ¼ 1 ·e-c2s
 SIMC-rule: c2 ≥ 2
 Time scale separation: c2 ≤ c1/5 (approximately)
K1 is designed based on G1’ = G1T2
 same as G1 but with an additional delay 2+c2
Cascade control
Example: Cascade control serial process
d=6
ys
K1
y2s
K2
u
G2
y2
G1
y1
Use SIMC-rules!
Without cascade
With cascade
Cascade control
Tuning cascade control
Cascade control
Tuning cascade control: serial process

Inner fast (secondary) loop:




Outer slower primary loop:


Reduced effective delay (2 s instead of 6 s)
Time scale separation


P or PI-control
Local disturbance rejection
Much smaller effective delay (0.2 s)
Inner loop can be modelled as gain=1 + 2*effective delay (0.4s)
Very effective for control of large-scale systems
Setpoint overshoot method
Alternative closed-loop approach:
Setpoint overshoot method
Procedure:
•
Switch to P-only mode and make
setpoint change
•
Adjust controller gain to get overshoot
about 0.30 (30%)
ys
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 Ziegler-Nichols:
* Not at limit to instability
* Works on a simple second-order process.
y p
y u
y
y s

tp
t=0
Closed-loop step setpoint response with P-only control.
M. Shamsuzzoha and S. Skogestad, ``The setpoint overshoot method: A simple and fast method for closed-loop PID tuning'',
Journal of Process Control, 20, xxx-xxx (2010)
t
Setpoint overshoot method
Summary setpoint overshoot method
From P-control setpoint experiment 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
Proposed PI settings (including detuning factor F)
K c = K c0 A F
A= 1.152(overshoot) 2 - 1.607(overshoot) + 1.0 


b
τ I =min  0.86A
t p , 2.44t p F 


1-b

)


Choice of detuning factor F:

F=1. Good tradeoff between “fast and robust” (SIMC with τc=θ)

F>1: Smoother control with more robustness

F<1 to speed up the closed-loop response.
Setpoint overshoot method
Example: High-order process
P-setpoint experiments
1.25
OUTPUT y
1
0.75
g=
Closed-loop PI response
0.5
0.25
0
0
Proposed method with F=1 (overshoot=0.104)
Proposed method with F=1 (overshoot=0.292)
Proposed method with F=1 (overshoot=0.598)
SIMC (c=effective=0.148)
5
10
time
15
20
1
 s  1) 0.2s  1)0.04s  1)0.008s  1)
Setpoint overshoot method
Example: Unstable plant
2
Proposed method with F=1 (overshoot=0.10)
Proposed method with F=1 (overshoot=0.30)
Proposed method with F=1 (overshoot=0.607)
First-order unstable process
e s
g=
5s  1
OUTPUT y
1.5
1
• No SIMC settings available
Closed-loop PI response
0.5
0
0
20
40
time
60
80
CONTROLLABILITY
A comment on Controllability




(Input-Output) “Controllability” is the ability to
achieve acceptable control performance (with any
controller)
“Controllability” is a property of the process itself
Analyze controllability by looking at model G(s)
What limits controllability?
CONTROLLABILITY
Controllability
Recall SIMC tuning rules
1. Tight control: Select c= corresponding to
2. Smooth control. Select Kc ¸
Must require Kc,max > Kc.min for controllability
)
max. output deviation
initial effect of “input” disturbance
y reaches k’ ¢ |d0|¢ t after time t
y reaches ymax after t= |ymax|/ k’ ¢ |d0|
CONTROLLABILITY
Controllability
• More general disturbances. Requirement
becomes (for c=):
Following step
disturbance d0:
Time it takes for output y
to reach max. deviation
• Conclusion: The main factors limiting
controllability are
– large effective delay from u to y ( large)
– large disturbances (k’d |d0| / ymax large)
• Can generalize using “frequency domain”:
|Gd(j¢0.5/max)| ¢|d0| = |ymax|
CONTROLLABILITY
Example: Distillation column
Response to 20% increase in feed rate (disturbance) with no control
-3
16
x 10
Data for “column A”
Product purities:
xD = 0.99 § 0.002, xB =0.01 § 0.005
(mole fraction light component)
Small reboiler holdup, MB/F = 0.5 min
14
12
xB(t)
10
8
6
ymax=0.005
4
xD(t)
2
0
-2
0
1
2
3
4
5
6
7
8
9
10
time [min]
Max. delay in feedback
loop, max = 3/2 = 1.5 min
time to exceed bound = ymax/k’d |d| = 3 min
Controllability: Must close a loop with time constant (c) faster than 1.5 min to
avoid that bottom composition xB exceeds max. deviation
If this is not possible: May add tank (feed tank?, larger reboiler volume?)
to smooth disturbances
CONTROLLABILITY
Example: Distillation column
Increase reboiler holdup to MB/F = 10 min
Original holdup
-3
16
x 10
-3
16
14
Larger holdup
14
xB(t)
12
10
12
10
8
8
6
6
4
4
2
2
0
0
-2
x 10
0
1
2
3
3 min
4
5
6
7
8
9
10
-2
xB(t)
0
1
2
3
4
5
6
5.8 min
7
8
9
10
time [min]
With increased holdup: Max. delay in feedback loop: = 2.9 min
CONTROLLABILITY
Conclusion controllability
If the plant is not controllable then improved
tuning will not help
Alternatives


Change the process design to make it more controllable
1.

2.
Better “self-regulation” with respect to disturbances, e.g.
insulate your house to make y=Tin less sensitive to d=Tout.
Give up some of your performance requirements