Get documents about "feeedback_xian_may09
Shared by: lanyuehua
-
Stats
- views:
- 2
- posted:
- 2/23/2012
- language:
- pages:
- 66
Document Sample
Feedback:
Still the simplest and best solution
Applications to self-optimizing control and stabilization of new operating regimes
Sigurd Skogestad
Department of Chemical Engineering
Norwegian University of Science and Technology (NTNU)
Trondheim
1 Xi’an May 2009
Trondheim, Norway
Xi’an
3
Arctic circle
North Sea
Trondheim
NORWAY SWEDEN
Oslo
DENMARK
GERMANY
UK
4
NTNU,
Trondheim
5
Outline
• I. Why feedback (and not feedforward) ?
• The feedback amplifier
• II. Self-optimizing control:
• How do we link optimization and feedback?
• What should we control?
• III. Stabilizing feedback control:
• Anti-slug control
• Conclusion
7
Example: AMPLIFIER
r G y
Amplifier
Want: y(t) = α r(t)
Solution 1 (feedforward):
G = α (adjust amplifier gain)
Very difficult to in practice
• Cannot get exact value of α
• Cannot easily adjust α online
• Do not get same amplification at all frequencies
8 • Problems with distortion and nonlinearity
Black’s feedback amplifier (1927)
r G y
Amplifier
K2
Measured y
Want: y(t) = α r(t)
Solution 2 (feedback):
G = k (any large amplifier gain, k > α)
K2 = 1/α (adjustable)
Closed-loop response
9 MAGIC! Independent of G, provided GK2 >> 1
Example: disturbance rejection
1
d
Gd k=10
u
G 25 time
y
Plant (uncontrolled system)
10
1. Feedforward control (measure d)
d
Gd
u
G
y
”Perfect” feedforward control: u = - G-1 Gd d
Our case: G=Gd → Use u = -d
11
1.Feedforward control: Nominal (perfect model)
d
Gd
u
G y
12
2. Feedback control
d
Gd
ys e
C u
G y
13
2. Feedback PI-control: Nominal case
d
Gd
ys e
C u
G y
Input u Output y
Feedback generates inverse!
Resulting output
14
Robustness comparison
• Gain error, k = 5, 10 (nominal), 20
• Time constant error, τ = 5, 10 (nominal), 20
• Time delay error, θ = 0 (nominal), 1, 2, 3
15
Robustness: Gain error,
k = 5, 10 (nominal), 20
1. FEEDFORWARD
2. FEEDBACK
16
Robustness: Time constant error,
τ= 5, 10 (nominal), 20
1. FEEDFORWARD
2. FEEDBACK
17
Robustness: Time delay error,
θ = 0 (nominal), 1, 2, 3
1. FEEDFORWARD
2. FEEDBACK
18
Conclusion: Why feedback?
(and not feedforward control)
• Simple: High gain feedback!
• Counteract unmeasured disturbances
• Reduce effect of changes / uncertainty (robustness)
• Change system dynamics (including stabilization)
• Linearize the behavior
• No explicit model required
• MAIN PROBLEM:
Potential instability (may occur “suddenly”) with time delay/RHP-zero
Unstable (RHP) zero: Fundamental problem with feedback!
Does not help with detailed model + state estimator (Kalman filter)…
20
Outline
• I. Why feedback (and not feedforward) ?
• II. Self-optimizing feedback control:
• How do we link optimization and feedback?
• What should we control?
• III. Stabilizing feedback control: Anti-slug control
• Conclusion
21
Optimal operation (economics)
• Define scalar cost function J(u0,x,d)
• u0: degrees of freedom
• d: disturbances
• x: states (internal variables)
• Optimal operation for given d.
Dynamic optimization problem:
minu0 J(u0,x,d)
subject to:
Model: f(u0,x,d) = 0
Constraints: g(u0,x,d) < 0
Here: How do we implement optimal operation?
22
1. ”Obvious” solution:
Optimizing control =
”Feedforward”
Estimate d and compute new uopt(d)
Probem: Complicated and
sensitive to uncertainty
23
2. In Practice: Feedback implementation
Issue:
What should we control?
24
Process control hierarchy
RTO
y1 = c ? (economics)
MPC
PID
25
What should we control?
• CONTROL ACTIVE CONSTRAINTS!
– Optimal solution is usually at constraints, that is, most of the degrees of
freedom are used to satisfy “active constraints”, g(u0,d) = 0
– Implementation of active constraints is usually simple.
• WHAT MORE SHOULD WE CONTROL?
– But what about the remaining unconstrained degrees of freedom?
– Look for “self-optimizing” controlled variables!
26
Self-optimizing Control
• Definition Self-optimizing Control
– Self-optimizing control is when acceptable
operation (=acceptable loss) can be achieved using
constant set points (cs) for the controlled variables c
c=cs
(without the need for re-optimizing when
disturbances occur).
27
Optimal operation – Runner
– Cost: J=T
– One degree of freedom (u=power)
– Optimal operation?
28
Optimal operation - Runner
Solution 1: Optimizing control
• Even getting a reasonable model
requires > 10 PhD’s … and
the model has to be fitted to each
individual….
• Clearly impractical!
29
Optimal operation - Runner
Solution 2 – Feedback
(Self-optimizing control)
– What should we control?
30
Optimal operation - Runner
Self-optimizing control: Sprinter (100m)
• 1. Optimal operation of Sprinter, J=T
– Active constraint control:
• Maximum speed (”no thinking required”)
31
Optimal operation - Runner
Self-optimizing control: Marathon (40 km)
• Optimal operation of Marathon runner, J=T
• Any self-optimizing variable c (to control at
constant setpoint)?
• c1 = distance to leader of race
• c2 = speed
• c3 = heart rate
• c4 = level of lactate in muscles
32
Optimal operation - Runner
Conclusion Marathon runner
select one measurement
c = heart rate
• Simple and robust implementation
• Disturbances are indirectly handled by keeping a constant heart rate
33 • May have infrequent adjustment of setpoint (heart rate)
Unconstrained optimum
Optimal operation
Cost J
Jopt
copt Controlled variable c
35
Unconstrained optimum
Optimal operation
Cost J d
Jopt
n
copt Controlled variable c
Two problems:
• 1. Optimum moves because of disturbances d: copt(d)
36
• 2. Implementation error, c = copt + n
Unconstrained optimum
Candidate controlled variables c
for self-optimizing control
Intuitive
1. The optimal value of c should be insensitive to disturbances (avoid
problem 1)
• Ideal self-optimizing variable is gradient, c = Jus
• Optimal value is always Ju=0 (gradient change sign at optimum)
2. Optimum should be flat (avoid problem 2 – implementation error).
Equivalently: Value of c should be sensitive to degrees of freedom u.
• “Want large gain”, |G|
• Or more generally: Maximize minimum singular value,
Good Good BAD
38
Unconstrained optimum
Quantitative steady-state: Maximum gain rule
Maximum gain rule (Skogestad and Postlethwaite, 1996):
Look for variables that maximize the scaled gain (Gs)
(minimum singular value of the appropriately scaled
steady-state gain matrix Gs from u to c)
39
Unconstrained optimum
Proof: Local analysis
cost J
c=Gu
uopt u
40
Unconstrained optimum
Optimal measurement combinations
Exact solutions for quadratic optimization problems
1. Nullspace method. No loss for disturbances (d)
2. Generalized (with noise n)
• c = Hy can be considered as linear invariants for the quadratic optimization
problem – which can be used for feedback implementation of optimal solution!
• Application: Explicit MPC
* V. Alstad, S. Skogestad and E.S. Hori, Optimal measurement combinations as controlled variables,
41 Journal of Process Control, 19, 138-148 (2009)
Example: CO2 refrigeration cycle
pH
J = Ws (work supplied)
DOF = u (valve opening, z)
Main disturbances:
d 1 = TH
d2 = TCs (setpoint)
d3 = UAloss
What should we control?
42
CO2 cycle: Maximum gain rule
43
Conclusion CO2 refrigeration cycle
44 Self-optimizing c= “temperature-corrected high pressure”
Outline
• I. Why feedback (and not feedforward) ?
• II. Self-optimizing feedback control: What should we control?
• III. Stabilizing feedback control: Anti-slug control
• IV. Conclusion
45
Application stabilizing feedback control:
Anti-slug control
Two-phase pipe flow
(liquid and vapor)
Slug (liquid) buildup
46
Slug cycle (stable limit cycle)
Experiments
performed by
the
Multiphase
Laboratory,
NTNU
47
Flow map with open valve
1
Steady flow
0.9 Steady/Pulsing
Pulsing flow
Pulsing flow Pulsing/Slugging
0.8
Riser slugging
0.7
0.6
Uso [m/s]
0.5
0.4
Riser slugging Steady flow
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
48 Usg [m/s]
Experimental mini-loop
49
z
p2
Experimental mini-loop
Valve opening (z) = 100%
p1
50
z
p2
Experimental mini-loop
Valve opening (z) = 25%
p1
51
z
p2
Experimental mini-loop
Valve opening (z) = 15%
p1
52
z
Experimental mini-loop: p2
Bifurcation diagram
p1
No slug
Valve opening z %
53 Slugging
Avoid slugging?
• Operate away from optimal point
• Design changes
• Feedforward control?
• Feedback control?
54
Design change z
p2
Avoid slugging:
1. Close valve (but increases pressure)
p1
No slugging when valve is closed
Valve opening z %
55
Design change
Avoid slugging:
2. Other design changes to avoid slugging
z
p2
p1
56
Design change
Minimize effect of slugging:
3. Build large slug-catcher
z
p2
p1
• Most common strategy in practice
57
Avoid slugging: 4. Feedback control?
Comparison with simple 3-state model:
Valve opening z %
Predicted smooth flow: Desirable but open-loop unstable
58
Avoid slugging:
4. ”Active” feedback control
ref
PC
z
PT
p
1
Simple PI-controller
59
Anti slug control: Mini-loop experiments
p1
[bar]
z
[%]
Controller ON Controller OFF
60
Anti slug control: Full-scale offshore
experiments at Hod-Vallhall field (Havre,1999)
61
Analysis: Poles and zeros Topside FT
Operation points: ρT
P1 DP Poles
z DP
-6.11 P1
0.175 70.05 1.94
0.0008±0.0067i
-6.21
0.25 69 0.96
0.0027±0.0092i
Zeros:
y
P1 [Bar] DP[Bar] ρT [kg/m3] FQ [m3/s] FW [kg/s]
z
-0.0034 3.2473 -0.0004 -4.5722 -7.6315
0.175 0.0142 0.0048 -0.0032 -0.0004
-0.0004 0
-0.0034 3.4828 -0.0004 -4.6276 -7.7528
0.25 0.0131 0.0048 -0.0032 -0.0004
-0.0004 0
62
Topside measurements: Ooops.... RHP-zeros or zeros close to origin
Stabilization with topside measurements:
Avoid RHP-zeros by using 2 measurements
• Model based control (LQG) with 2 top measurements: DP and
density ρT
63
Summary anti slug control
• Stabilization of smooth flow regime = $$$$!
• Stabilization using downhole pressure simple
• Stabilization using topside measurements possible
• Control can make a difference!
Thanks to: Espen Storkaas + Heidi Sivertsen + Håkon Dahl-Olsen + Ingvald Bårdsen
64
Conclusions
• Feedback is an extremely powerful tool
• simple
• robust
• Complex systems can be controlled by hierarchies (cascades) of single-
input-single-output (SISO) control loops
• Control the right variables to achieve ”self-optimizing control”
• Feedback can make new things possible
• Stabilization (anti-slug)
More details: See paper available at my home page
S. Skogestad. "Feedback: Still the simplest and best solution" Presented at ICIEA 2009, Xi’an, China, May 2009
65
Engineering systems
• Most (all?) large-scale engineering systems are controlled using
hierarchies of quite simple single-loop controllers
– Commercial aircraft
– Large-scale chemical plant (refinery)
• 1000’s of loops
• Simple components:
on-off + P-control + PI-control + nonlinear fixes + some feedforward
Same in biological systems
66
Self-optimizing control: Recycle process
J = V (minimize energy)
5
4
1
Given feedrate F0 and 2
column pressure:
Nm = 5 3
3 economic (steady- Constraints: Mr < Mrmax,
state) DOFs xB > xBmin = 0.98
67
DOF = degree of freedom
Recycle process: Control active constraints
Active constraint Remaining DOF:L
Mr = Mrmax
Active constraint
xB = xBmin
One unconstrained DOF left for optimization:
68 What more should we control?
Maximum gain rule: Steady-state gain
Conventional:
Looks good
Luyben snow-ball
rule: Not promising
economically
69
Recycle process: Loss with constant setpoint, cs
Large loss with c = F (Luyben rule)
Negligible loss with c =L/F
or c = temperature
70
Recycle process: Proposed control structure
for case with J = V (minimize energy)
Active constraint
Mr = Mrmax
Active constraint
xB = xBmin
Self-optimizing loop:
Adjust L such that L/F is constant
71
