1. Self-optimizing control Theory

2. Outline Skogestad procedure for control structure design I Top Down
Step S1: Define operational objective (cost) and constraints
Step S2: Identify degrees of freedom and optimize operation for disturbances
Step S3: Implementation of optimal operation
What to control ? (primary CV’s)
Active constraints
Self-optimizing variables for unconstrained, c=Hy
Step S4: Where set the production rate? (Inventory control)
II Bottom Up
Step S5: Regulatory control: What more to control (secondary CV’s) ?
Step S6: Supervisory control
Step S7: Real-time optimization

3. Step S3: Implementation of optimal operation
Optimal operation for given d*:
minu J(u,x,d)
subject to:
Model equations: f(u,x,d) = 0
Operational constraints: g(u,x,d) < 0
→ uopt(d*)
Problem: Usally cannot keep uopt constant because disturbances d change
How should we adjust the degrees of freedom (u)?
What should we control?

4. “Optimizing Control” y

5. “Self-Optimizing Control”:Separate optimization and control
What should we control?
(What is c? What is H?)
Self-optimizing control: Constant setpoints give acceptable loss H y
C = Hy
H: Nonquare matrix
Usually selection matrix of 0’s and some 1’s (measurement selection)
Can also be full matrix (measurement combinations)

6. Definition of self-optimizing control
Acceptable loss )
self-optimizing control
“Self-optimizing control is when we achieve acceptable loss (in comparison with truly optimal operation) with constant setpoint values for the controlled variables (without the need to reoptimize when disturbances occur).”
Reference: S. Skogestad, “Plantwide control: The search for the self-optimizing control structure'', Journal of Process Control, 10, (2000).

7. Optimal operation - Runner
Marathon runner
c=heart rate
J=T
copt
select one measurement
c = heart rate
CV = heart rate is good “self-optimizing” variable
Simple and robust implementation
Disturbances are indirectly handled by keeping a constant heart rate
May have infrequent adjustment of setpoint (cs)

8. Unconstrained optimum
Optimal operation
Cost J
Jopt
copt
Controlled variable c

9. Unconstrained optimum
Optimal operation
Cost J d
Loss1
Jopt
Loss2 n
copt
Controlled variable c
Two problems:
Loss 1. Optimum moves because of disturbances d: copt(d)
Loss 2. Implementation error, c = copt + n

10. 2. UNCONSTRAINED VARIABLES: - WHAT MORE SHOULD WE CONTROL
2. UNCONSTRAINED VARIABLES: - WHAT MORE SHOULD WE CONTROL? - WHAT ARE GOOD “SELF-OPTIMIZING” VARIABLES?
Intuition: “Dominant variables” (Shinnar)
Is there any systematic procedure?
Sensitive variables: “Max. gain rule” (Gain= Minimum singular value)
“Brute force” loss evaluation
Gradient, Ju=0
Optimal linear combination of measurements, c = Hy

11. The ideal “self-optimizing” variable is the gradient, Ju
Unconstrained degrees of freedom
The ideal “self-optimizing” variable is the gradient, Ju
c =  J/ u = Ju
Keep gradient at zero for all disturbances (c = Ju=0)
Problem: Usually no measurement of gradient u
cost J
Ju=0
Ju<0
uopt Ju 14

12. u J
J>Jmin
Jmin
J<Jmin ?
Unconstrained optimum: NEVER try to control a variable that reaches max or min at the optimum
In particular, never try to control directly the cost J
Assume we want to minimize J (e.g., J = V = energy) - and we make the stupid choice os selecting CV = V = J
Then setting J < Jmin: Gives infeasible operation (cannot meet constraints)
and setting J > Jmin: Forces us to be nonoptimal (two steady states: may require strange operation)

13. Parallel heat exchangers What should we control?
Th1, UA1 T0
m [kg/s]
Th2, UA2 T2 T1 T α
1-α
1 Degree of freedom (u=split ®)
Objective: maximize J=T (= maximize total heat transfer Q)
What should we control?
If active constraint : Control it!
Here: No active constraint.
Control T?
NO!
Control T1=T2? Not quite
Optimal: Control Ju=0

14. Jäsche temperature
Can show that Ju=0 is equivalent to (see exercise for simplified case) TJ1 – TJ2 = 0 where TJ is «Jäschke temperature» for each branch: TJ1 = (T1-T0)2 / (Th1 – T0) TJ2 = (T2-T0)2 / (Th2 – T0) Th1 , Th2 = Inlet tempetaure on other side («hot») of heat exchanger T1 , T2 = outlet temperature T0 = feed temperature

16. c = JÄSCHKE TEMPERATURE

17. Unconstrained variablesJ
copt H
steady-state
control error
/ selection
controlled
variable
disturbance
measurement noise
Ideal: c = Ju
In practise: c = H y

18. Optimal measurement combination
CV=Measurement combination
Optimal measurement combination
Candidate measurements (y): Include also inputs u H
measurement noise
control error
disturbance
controlled
variable

19. Nullspace method No measurement noise (ny=0)
CV=Measurement combination
Nullspace method
Proof: Measurements: y = Gy u + Gyd d
Assume c=Hy,
dyopt = F dd
dcopt = HF dd
To make dcopt=0 for and dd, we must have HF=0

20. Nullspace method (HF=0) gives Ju=0
CV=Measurement combination
Nullspace method (HF=0) gives Ju=0
Proof:
Proof. Appendix B in: Jäschke and Skogestad, ”NCO tracking and self-optimizing control in the context of real-time optimization”, Journal of Process Control, (2011) .

21. Example. Nullspace Method for Marathon runner
CV=Measurement combination
Example. Nullspace Method for Marathon runner
u = power, d = slope [degrees]
y1 = hr [beat/min], y2 = v [m/s]
F = dyopt/dd = [ ]’
H = [h1 h2]]
HF = 0 -> h1 f1 + h2 f2 = 0.25 h1 – 0.2 h2 = 0
Choose h1 = 1 -> h2 = 0.25/0.2 = 1.25
Conclusion: c = hr v
Control c = constant -> hr increases when v decreases (OK uphill!)

22. Extension: ”Exact local method” (with measurement noise)
CV=Measurement combination
Extension: ”Exact local method” (with measurement noise)
General analytical solution (“full” H):
No disturbances (Wd=0) + same noise for all measurements (Wny=I):
Optimal is H=GyT (“control sensitive measurements”)
Proof: Use analytic expression
No noise (Wny=0): Cannot use analytic expression, but optimal is clearly
HF=0 (Nullspace method)
Assumes enough measurements: #y ¸ #u + #d
If “extra” measurements (>) then solution is not unique

23. Optimal measurement combination, c = Hy
With measurement noise
Optimal measurement combination, c = Hy
“=0” in nullspace method (no noise)
“Minimize” in Maximum gain rule
( maximize S1 G Juu-1/2 , G=HGy )
“Scaling” S1 27

24. Good candidate controlled variables c=Hy (for self-optimizing control)
CV=Single measurements
Good candidate controlled variables c=Hy (for self-optimizing control)
The optimal value of c should be insensitive to disturbances
Want small dcopt/dd = HF
The value of c should be sensitive to changes in the degrees of freedom
Want large gain G= dc/du = HGy
(same as wanting flat optimum)
c should be easy to measure and control
Proof: This keeps the gradient close to zero since
BAD
Good

25. Example: CO2 refrigeration cyclepH
J = Ws (work supplied)
DOF = u (valve opening, z)
Main disturbances:
d1 = TH
d2 = TCs (setpoint)
d3 = UAloss
What should we control?

26. CO2 refrigeration cycle
Step 1. One (remaining) degree of freedom (u=z)
Step 2. Objective function. J = Ws (compressor work)
Step 3. Optimize operation for disturbances (d1=TC, d2=TH, d3=UA)
Optimum always unconstrained
Step 4. Implementation of optimal operation
No good single measurements (all give large losses):
ph, Th, z, …
Nullspace method: Need to combine nu+nd=1+3=4 measurements to have zero disturbance loss
Simpler: Try combining two measurements. Exact local method:
c = h1 ph + h2 Th = ph + k Th; k = bar/K
Nonlinear evaluation of loss: OK!

27. CO2 cycle: Maximum gain rule

28. Refrigeration cycle: Proposed control structure
CV=Measurement combination
Refrigeration cycle: Proposed control structure
Control c= “temperature-corrected high pressure”

29. EXAMPLE: Recycle plant (Luyben, Yu, etc.)
Recycle of unreacted A (+ some B) 5
Feed of A 4 1
Given feedrate F0 and
column pressure: 2 3
Dynamic DOFs: Nm = 5
Column levels: N0y = 2
Steady-state DOFs: N0 = = 3
Product (98.5% B)

30. Recycle plant: Optimal operationmT
1 remaining unconstrained degree of freedom

31. Control of recycle plant: Conventional structure (“Two-point”: xD)LC LC xD XC XC xB LC
Control active constraints (Mr=max and xB=0.015) + xD

32. Luyben rule Luyben rule (to avoid snowballing):
“Fix a stream in the recycle loop” (F or D)

33. Luyben rule: D constantLC LC XC LC
Luyben rule (to avoid snowballing):
“Fix a stream in the recycle loop” (F or D)

34. A. Maximum gain rule: Steady-state gain
Conventional:
Looks good
Luyben rule:
Not promising
economically

35. How did we find the gains in the Table?
Find nominal optimum
Find (unscaled) gain G0 from input to candidate outputs:  c = G0  u.
In this case only a single unconstrained input (DOF). Choose at u=L
Obtain gain G0 numerically by making a small perturbation in u=L while adjusting the other inputs such that the active constraints are constant (bottom composition fixed in this case)
Find the span for each candidate variable
For each disturbance di make a typical change and reoptimize to obtain the optimal ranges copt(di)
For each candidate output obtain (estimate) the control error (noise) n
The expected variation for c is then: span(c) = i |copt(di)| + |n|
Obtain the scaled gain, G = |G0| / span(c)
Note: The absolute value (the vector 1-norm) is used here to "sum up" and get the overall span. Alternatively, the 2-norm could be used, which could be viewed as putting less emphasis on the worst case. As an example, assume that the only contribution to the span is the implementation/measurement error, and that the variable we are controlling (c) is the average of 5 measurements of the same y, i.e. c=sum yi/5, and that each yi has a measurement error of 1, i.e. nyi=1. Then with the absolute value (1-norm), the contribution to the span from the implementation (meas.) error is span=sum abs(nyi)/5 = 5*1/5=1, whereas with the two-norn, span = sqrt(5*(1/5^2) = The latter is more reasonable since we expect that the overall measurement error is reduced when taking the average of many measurements. In any case, the choice of norm is an engineering decision so there is not really one that is "right" and one that is "wrong". We often use the 2-norm for mathematical convenience, but there are also physical justifications (as just given!).
IMPORTANT!

36. B. “Brute force” loss evaluation: Disturbance in F0
Luyben rule:
Conventional
Loss with nominally optimal setpoints for Mr, xB and c

37. B. “Brute force” loss evaluation: Implementation error
Luyben rule:
Loss with nominally optimal setpoints for Mr, xB and c

38. Conclusion: Control of recycle plant
Active constraint
Mr = Mrmax
Self-optimizing
L/F constant: Easier than “two-point” control
Assumption: Minimize energy (V)
Active constraint
xB = xBmin

39. Recycle systems: Do not recommend Luyben’s rule of
fixing a flow in each recycle loop
(even to avoid “snowballing”)

40. Summary: Procedure selection controlled variables
Define economics and operational constraints
Identify degrees of freedom and important disturbances
Optimize for various disturbances
Identify active constraints regions (off-line calculations)
For each active constraint region do step 5-6:
5. Identify “self-optimizing” controlled variables for remaining degrees of freedom
6. Identify switching policies between regions

41. Importance of optimal operation for CO2 capturing process
Dependency of equivalent energy in CO2 capture plant verses recycle amine flowrate
An amine absorption/stripping CO2 capturing process*
*Figure from: Toshiba (2008). Toshiba to Build Pilot Plant to Test CO2 Capture Technology.

42. Case study
Control structure design using self-optimizing control for economically optimal CO2 recovery*
Step S1. Objective function= J = energy cost + cost (tax) of released CO2 to air
4 equality and 2 inequality constraints:
stripper top pressure
condenser temperature
pump pressure of recycle amine
cooler temperature
CO2 recovery ≥ 80%
Reboiler duty < 1393 kW (nominal +20%)
Step S2.
(a) 10 degrees of freedom: 8 valves + 2 pumps
4 levels without steady state effect: absorber 1,stripper 2,make up tank 1
Disturbances: flue gas flowrate, CO2 composition in flue gas + active constraints
(b) Optimization using Unisim steady-state simulator.
Mode I = Region I (nominal feedrate): No inequality constraints active
2 unconstrained degrees of freedom =10-4-4
Step S3 (Identify CVs). 1. Control the 4 equality constraints
2. Identify 2 self-optimizing CVs. Use Exact Local method and select CV set with minimum loss.
*M. Panahi and S. Skogestad, ``Economically efficient operation of CO2 capturing process, part I: Self-optimizing procedure for selecting the best controlled variables'', Chemical Engineering and Processing, 50, (2011).

43. Proposed control structure with given nominal flue gas flowrate (mode I)

44. Mode II: large feedrates of flue gas (+30%)
(kmol/hr)
Self-optimizing CVs in region I
Reboiler
duty
(kW)
Cost
(USD/ton)
CO2 recovery %
Temperature
tray no. 16 °C
Optimal nominal point
219.3
95.26
106.9
1161
2.49
+5% feedrate
230.3
1222
+10% feedrate
241.2
1279
+15% feedrate
252.2
1339
+19.38%, when reboiler duty saturates
261.8
1393
(+20%)
2.50
+30% feedrate (reoptimized)
285.1
91.60
103.3
2.65
region I
region II
max
Saturation of reboiler duty; one unconstrained degree of freedom left
Use Maximum gain rule to find the best CV among 37 candidates :
Temp. on tray no. 13 in the stripper: largest scaled gain, but tray 16 also OK

45. Proposed control structure with large flue gas flowrate (mode II = region II)
max

46. Conditions for switching between regions of active constraints (“supervisory control”)
Within each region of active constraints it is optimal to
Control active constraints at ca = c,a, constraint
Control self-optimizing variables at cso = c,so, optimal
Define in each region i:
Keep track of ci (active constraints and “self-optimizing” variables) in all regions i
Switch to region i when element in ci changes sign

47. Example – switching policies CO2 plant (”supervisory control”)
Assume operating in region I (unconstrained)
with CV=CO2-recovery=95.26%
When reach maximum Q: Switch to Q=Qmax (Region II) (obvious)
CO2-recovery will then drop below 95.26%
When CO2-recovery exceeds 95.26%: Switch back to region I !!!

48. Conclusion optimal operation
ALWAYS:
1. Control active constraints and control them tightly!!
Good times: Maximize throughput -> tight control of bottleneck
2. Identify “self-optimizing” CVs for remaining unconstrained degrees of freedom
Use offline analysis to find expected operating regions and prepare control system for this!
One control policy when prices are low (nominal, unconstrained optimum)
Another when prices are high (constrained optimum = bottleneck)
ONLY if necessary: consider RTO on top of this

49. Sigurd’s rules for CV selection
Always control active constraints! (almost always)
Purity constraint on expensive product always active (no overpurification):
(a) "Avoid product give away" (e.g., sell water as expensive product)
(b) Save energy (costs energy to overpurify)
Unconstrained optimum: NEVER try to control a variable that reaches max or min at the optimum
In particular, never try to control directly the cost J
- Assume we want to minimize J (e.g., J = V = energy) - and we make the stupid choice os selecting CV = V = J - Then setting J < Jmin: Gives infeasible operation (cannot meet constraints) - and setting J > Jmin: Forces us to be nonoptimal (which may require strange operation; see Exercise on recycle process)