1. Effective Implementation of optimal operation using Self-optimizing control
Sigurd Skogestad
Institutt for kjemisk prosessteknologi
NTNU, Trondheim

2. Plantwide control = Control structure design
Not the tuning and behavior of each control loop,
But rather the control philosophy of the overall plant with emphasis on the structural decisions:
Selection of controlled variables (“outputs”)
Selection of manipulated variables (“inputs”)
Selection of (extra) measurements
Selection of control configuration (structure of overall controller that interconnects the controlled, manipulated and measured variables)
Selection of controller type (LQG, H-infinity, PID, decoupler, MPC etc.).
Previous work on plantwide control:
Page Buckley (1964) - Chapter on “Overall process control” (still industrial practice)
Greg Shinskey (1967) – process control systems
Alan Foss (1973) - control system structure
Bill Luyben et al. ( ) – case studies ; “snowball effect”
George Stephanopoulos and Manfred Morari (1980) – synthesis of control structures for chemical processes
Ruel Shinnar ( ) - “dominant variables”
Jim Downs (1991) - Tennessee Eastman challenge problem
Larsson and Skogestad (2000): Review of plantwide control

3. Plantwide control design procedure
I Top Down
Step 1: Define operational objectives (optimal operation)
Cost function J (to be minimized)
Operational constraints
Step 2: Identify degrees of freedom (MVs) and optimize for
expected disturbances
Step 3: Select primary controlled variables c=y1 (CVs)
Step 4: Where set the production rate? (Inventory control)
II Bottom Up
Step 5: Regulatory / stabilizing control (PID layer)
What more to control (y2; local CVs)?
Pairing of inputs and outputs
Step 6: Supervisory control (MPC layer)
Step 7: Real-time optimization (Do we need it?) y1 y2
MVs
Process

4. Process control: Implementation of optimal operation
y1s
MPC
PID
y2s
RTO
u (valves)

5. Outline Implementation of optimal operation
Paradigm 1: On-line optimizing control (including RTO)
Paradigm 2: "Self-optimizing" control schemes
Precomputed (off-line) solution
Examples
Control of optimal measurement combinations
Nullspace method
Exact local method
Summary/Conclusions

6. Optimal operation A typical dynamic optimization problem
Implementation: “Open-loop” solutions not robust to disturbances or model errors
Want to introduce feedback

7. Implementation of optimal operation
Paradigm 1: On-line optimizing control where measurements are used to update model and states
Process control: Steady-state “real-time optimization” (RTO)
Update model and states: “data reconciliation”
Paradigm 2: “Self-optimizing” control scheme found by exploiting properties of the solution
Most common in practice, but ad hoc

8. Implementation: Paradigm 1
Paradigm 1: Online optimizing control
Measurements are primarily used to update the model
The optimization problem is resolved online to compute new inputs.
Example: Conventional MPC, RTO (real-time optimization)
This is the “obvious” approach (for someone who does not know control)

9. Example paradigm 1: On-line optimizing control of Marathon runner
Even getting a reasonable model requires > 10 PhD’s  … and the model has to be fitted to each individual….
Clearly impractical!

10. Implementation: Paradigm 2
Paradigm 2: Precomputed solutions based on off-line optimization
Find properties of the solution suited for simple and robust on-line implementation
Most common in practice, but ad hoc
Most common approach: “Move optimization into controller layer”: Select controlled variables (CVs = c = Hy) H K
c = Hy
Setpoint cs
Input u
Optimizer (RTO)
Controller

11. Controlled variables “Turn optimization into feedback problem”
Need to select c = CV = Hy
Find regions of active constraints and in each region:
Control active constraints
Control “self-optimizing ” variables for the remaining unconstrained degrees of freedom
“inherent optimal operation”

12. Example: Runner One degree of freedom (u): Power Cost to be minimized
J = T
Constraints
u ≤ umax
Follow track
Fitness (body model)
Optimal operation: Minimize J with respect to u(t)
ISSUE: How implement optimal operation?

13. Solution 2 – Feedback (Self-optimizing control)
Optimal operation - Runner
Solution 2 – Feedback (Self-optimizing control)
What should we control?

14. Self-optimizing control: Sprinter (100m)
Optimal operation - Runner
Self-optimizing control: Sprinter (100m)
1. Optimal operation of Sprinter, J=T
Active constraint control:
Maximum speed (”no thinking required”)

15. Self-optimizing control: Marathon (40 km)
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

16. Implementation paradigm 2: Feedback control of Marathon runner
Simplest case:
select one measurement
c = heart rate
measurements
Simple and robust implementation
Disturbances are indirectly handled by keeping a constant heart rate
May have infrequent adjustment of setpoint (heart rate)

17. Implementation (in practice): Local feedback control!y
“Self-optimizing control:” Constant setpoints for c gives acceptable loss
Controlled variables c (z in book!)
Active constraints
“Self-optimizing” variables c
for remaining unconstrained degrees of freedom (u)
No or infrequent online optimization needed.
Controlled variables c are found based on off-line analysis. d
Local feedback:
Control c (CV)
Optimizing control
Feedforward

18. “Magic” self-optimizing variables: How do we find them?
Intuition: “Dominant variables” (Shinnar)
Is there any systematic procedure?

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

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

21. Candidate controlled variables c for self-optimizing control
Unconstrained optimum
Candidate controlled variables c for self-optimizing control
Intuitive
The optimal value of c should be insensitive to disturbances (avoid problem 1)
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,
BAD
Good

22. Quantitative steady-state: Maximum gain rule
Unconstrained optimum
Quantitative steady-state: Maximum gain rule u G c
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)

23. Why is Large Gain Good? J(u) J, c Loss Jopt copt Variation of u uopt u
Δc = G Δu
Jopt
copt
c-copt
Variation of u
uopt u
Controlled variable Δc = G Δu
Want large gain G:
Large implementation error n (in c) translates into small deviation of u from uopt(d)
- leading to lower loss

24. Ideal “Self-optimizing” variables
Unconstrained degrees of freedom:
Ideal “Self-optimizing” variables
Operational objective: Minimize cost function J(u,d)
The ideal “self-optimizing” variable is the gradient (first-order optimality condition) (Halvorsen and Skogestad, 1997; Bonvin et al., 2001; Cao, 2003):
Optimal setpoint = 0
BUT: Gradient can not be measured in practice
Possible approach: Estimate gradient Ju based on measurements y
Approach here: Look directly for c as a function of measurements y (c=Hy) without going via gradient
I.J. Halvorsen and S. Skogestad, ``Optimizing control of Petlyuk distillation: Understanding the steady-state behavior'', Comp. Chem. Engng., 21, S249-S254 (1997)
Ivar J. Halvorsen and Sigurd Skogestad, ``Indirect on-line optimization through setpoint control'', AIChE Annual Meeting, Nov. 1997, Los Angeles, Paper 194h. .

25. H

26. H

27. Optimal measurement combination
Candidate measurements (y): Include also inputs u H

28. Optimal H: Problem definition

29. No measurement noise (ny=0)
Nullspace method

30. Proof nullspace method
No measurement noise (ny=0)
Proof nullspace method
Basis: Want optimal value of c to be independent of disturbances
Find optimal solution as a function of d: uopt(d), yopt(d)
Linearize this relationship: yopt = F d
Want:
To achieve this for all values of  d:
To find a F that satisfies HF=0 we must require
Optimal when we disregard implementation error (n)
Amazingly simple!
Sigurd is told how easy it is to find H
V. Alstad and S. Skogestad, ``Null Space Method for Selecting Optimal Measurement Combinations as Controlled Variables'',
Ind.Eng.Chem.Res, 46 (3), (2007).

31. Loss evaluation (with measurement noise)
Controlled variables,
cs = constant + - K H ym c u u J
Loss d
Loss with c=Hym=0 due to
(i) Disturbances d
(ii) Measurement noise ny
Book: Eq. (10.12)
Proof: See handwritten notes 33

32. Optimal H (with measurement noise)
1. Kariwala: Can minimize 2-norm (Frobenius norm) instead of singular value
2. BUT still Non-convex optimization problem
(Halvorsen et al., 2003), see book p. 397
D : any non-singular matrix
Have extra degrees of freedom st
Improvement (Alstad et al. 2009)
Convex
optimization problem
Global solution
Do not need Juu
Analytical solution applies when YYT has full rank (w/ meas. noise):
Analytical solution

33. Special cases No noise (ny=0, Wny=0):
cs = constant + - K H y c u
Special cases
No noise (ny=0, Wny=0):
Optimal is HF=0 (Nullspace method)
But: If many measurement then solution is not unique
No disturbances (d=0; Wd=0) + same noise for all measurements (Wny=I):
Optimal is HT=Gy (“control sensitive measurements”)
Proof: Use analytic expression

34. Relationship to maximum gain rule
cs = constant + - K H y c u
“=0” in nullspace method (no noise)
“Minimize” in Maximum gain rule
( maximize S1 G Juu-1/2 , G=HGy )
“Scaling” S1-1 for max.gain rule

35. Special case: Indirect control = Estimation using “Soft sensor”
Indirect control: Control measurements c= Hy such that primary output y1 is constant
Optimal sensitivity F = dyopt/dd = (dy/dd)y1
Distillation: y=temperature measurements, y1= product composition
Estimation: Estimate primary variables, y1=c=Hy
y1 = G1 u, y = Gy u
Same as indirect control if we use extra degrees of freedom (H1=DH) such that (H1Gy)=G1

36. Summary 1: Systematic procedure for selecting controlled variables for optimal operation
Define economics (cost J) 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:
Identify “self-optimizing” controlled variables for remaining unconstrained degrees of freedom
A. “Brute force” loss evaluation
B. Sensitive variables: “Max. gain rule” (Gain= Minimum singular value)
C. Optimal linear combination of measurements, c = Hy
6. Identify switching policies between regions

37. Summary 2 Systematic approach for finding CVs, c=Hy
Can also be used for estimation
S. Skogestad, ``Control structure design for complete chemical plants'', Computers and Chemical Engineering, 28 (1-2), (2004).
V. Alstad and S. Skogestad, ``Null Space Method for Selecting Optimal Measurement Combinations as Controlled Variables'', Ind.Eng.Chem.Res, 46 (3), (2007).
V. Alstad, S. Skogestad and E.S. Hori, ``Optimal measurement combinations as controlled variables'', Journal of Process Control, 19, (2009)
Ramprasad Yelchuru, Sigurd Skogestad, Henrik Manum , MIQP formulation for Controlled Variable Selection in Self Optimizing Control IFAC symposium DYCOPS-9, Leuven, Belgium, 5-7 July 2010
Download papers: Google ”Skogestad”

38. Example: CO2 refrigeration cycle
Unconstrained DOF (u)
Control what?
c=? pH

39. 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!

40. Refrigeration cycle: Proposed control structure
Control c= “temperature-corrected high pressure”

41. 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