Outline

1
Outline

• About Trondheim and myself
• Control structure design (plantwide control)
• A procedure for control structure design
• I Top Down
• Step 1 Degrees of freedom
• Step 2 Operational objectives (optimal
  operation)
• Step 3 What to control ? (self-optimizing
  control)
• Step 4 Where set production rate?
• II Bottom Up
• Step 5 Regulatory control What more to control
  ?
• Step 6 Supervisory control
• Step 7 Real-time optimization
• Case studies

2
Optimal operation (economics)

• What are we going to use our degrees of freedom
  for?
• Define scalar cost function J(u0,x,d)
• u0 degrees of freedom
• d disturbances
• x states (internal variables)
• Typical cost function
• Optimal operation for given d
• minuss J(uss,x,d)
• subject to
• Model equations f(uss,x,d) 0
• Operational constraints g(uss,x,d) lt 0

J cost feed cost energy value products
3
Optimal operation distillation column

• Distillation at steady state with given p and F
  N2 DOFs, e.g. L and V
• Cost to be minimized (economics)
• J - P where P pD D pB B pF F pV V
• Constraints
• Purity D For example xD, impurity max
• Purity B For example, xB, impurity max
• Flow constraints min D, B, L etc. max
• Column capacity (flooding) V Vmax, etc.
• Pressure 1) p given, 2) p free pmin p
  pmax
• Feed 1) F given 2) F free F Fmax
• Optimal operation Minimize J with respect to
  steady-state DOFs

cost energy (heating cooling)
value products
cost feed
4
Example Paper machine drying section
up to 30 m/s (100 km/h)
(20 seconds)
water
10m
water recycle
5
Paper machine Overall operational objectives

• Degrees of freedom (inputs) drying section
• Steam flow each drum (about 100)
• Air inflow and outflow (2)
• Objective Minimize cost (energy) subject to
  satisfying operational constraints
• Humidity paper 10 (active constraint
  controlled!)
• Air outflow T lt dew point 10C (active not
  always controlled)
• ?T along dryer (especially inlet) lt bound
  (active?)
• Remaining DOFs minimize cost

6
Optimal operation
minimize J cost feed cost energy value
products
Two main cases (modes) depending on marked
conditions

• Given feed
• Amount of products is then usually indirectly
  given and J cost energy. Optimal operation is
  then usually unconstrained
• Feed free
• Products usually much more valuable than feed
  energy costs small.
• Optimal operation is then usually constrained

maximize efficiency (energy)
Control Operate at optimal trade-off (not
obvious how to do and what to control)
maximize production
Control Operate at bottleneck (obvious)
7
Comments optimal operation

• Do not forget to include feedrate as a degree of
  freedom!!
• For paper machine it may be optimal to have max.
  drying and adjust the speed of the paper machine!
• Control at bottleneck
• see later Where to set the production rate

8
Outline

• About Trondheim and myself
• Control structure design (plantwide control)
• A procedure for control structure design
• I Top Down
• Step 1 Degrees of freedom
• Step 2 Operational objectives (optimal
  operation)
• Step 3 What to control ? (self-optimizing
  control)
• Step 4 Where set production rate?
• II Bottom Up
• Step 5 Regulatory control What more to control
  ?
• Step 6 Supervisory control
• Step 7 Real-time optimization
• Case studies

9
Step 3. What should we control (c)? (primary
controlled variables y1c)

• Outline
• Implementation of optimal operation
• Self-optimizing control
• Uncertainty (d and n)
• Example Marathon runner
• Methods for finding the magic self-optimizing
  variables
• A. Large gain Minimum singular value rule
• B. Brute force loss evaluation
• C. Optimal combination of measurements
• Example Recycle process
• Summary

10
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) lt 0

? uopt(d)
Problem Usally cannot keep uopt constant because
disturbances d change
How should be adjust the degrees of freedom (u)?
11
Implementation of optimal operation (Cannot keep
u0opt constant) Obvious solution Optimizing
control
Estimate d from measurements and recompute
uopt(d)
Problem Too complicated (requires detailed model
and description of uncertainty)
12
In practice Hierarchical decomposition with
separate layers
What should we control?
13
Self-optimizing control When constant setpoints
is OK
Constant setpoint
14
Unconstrained variables Self-optimizing control

• Self-optimizing control
• Constant setpoints cs give
• near-optimal operation
• ( acceptable loss L for expected disturbances d
  and implementation errors n)

Acceptable loss ) self-optimizing control
15
What cs should we control?

• Optimal solution is usually at constraints, that
  is, most of the degrees of freedom are used to
  satisfy active constraints, g(u,d) 0
• CONTROL ACTIVE CONSTRAINTS!
• cs value of active constraint
• Implementation of active constraints is usually
  simple.
• WHAT MORE SHOULD WE CONTROL?
• Find self-optimizing variables c for remaining
• unconstrained degrees of freedom u.

16
What should we control? Sprinter

• Optimal operation of Sprinter (100 m), JT
• One input power/speed
• Active constraint control
• Maximum speed (no thinking required)

17
What should we control? Marathon

• Optimal operation of Marathon runner, JT
• No active constraints
• Any self-optimizing variable c (to control at
  constant setpoint)?

18
Self-optimizing Control Marathon

• Optimal operation of Marathon runner, JT
• 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

19
Further examples self-optimizing control

• Marathon runner
• Central bank
• Cake baking
• Business systems (KPIs)
• Investment portifolio
• Biology
• Chemical process plants Optimal blending of
  gasoline

Define optimal operation (J) and look for magic
variable (c) which when kept constant gives
acceptable loss (self-optimizing control)
20
More on further examples

• Central bank. J welfare. u interest rate.
  cinflation rate (2.5)
• Cake baking. J nice taste, u heat input. c
  Temperature (200C)
• Business, J profit. c Key performance
  indicator (KPI), e.g.
• Response time to order
• Energy consumption pr. kg or unit
• Number of employees
• Research spending
• Optimal values obtained by benchmarking
• Investment (portofolio management). J profit. c
  Fraction of investment in shares (50)
• Biological systems
• Self-optimizing controlled variables c have
  been found by natural selection
• Need to do reverse engineering
• Find the controlled variables used in nature
• From this possibly identify what overall
  objective J the biological system has been
  attempting to optimize

BREAK
21
Summary so far Active constrains and
unconstrained variables

• Optimal operation Minimize J with respect to
  DOFs
• General Optimal solution with N DOFs
• Nactive DOFs used to satisfy active
  constraints ( is )
• Nu N Nactive. remaining unconstrained
  variables
• Often Nu is zero or small
• It is obvious how to control the active
  constraints
• Difficult issue What should we use the remaining
  Nu degrees of for, that is what should we control?

22
Recall Optimal operation distillation column

• Distillation at steady state with given p and F
  N2 DOFs, e.g. L and V
• Cost to be minimized (economics)
• J - P where P pD D pB B pF F pV V
• Constraints
• Purity D For example xD, impurity max
• Purity B For example, xB, impurity max
• Flow constraints min D, B, L etc. max
• Column capacity (flooding) V Vmax, etc.
• Pressure 1) p given, 2) p free pmin p
  pmax
• Feed 1) F given 2) F free F Fmax
• Optimal operation Minimize J with respect to
  steady-state DOFs

cost energy (heating cooling)
value products
cost feed
23
Solution Optimal operation distillation

• Cost to be minimized
• J - P where P pD D pB B pF F pV V
• N2 steady-state degrees of freedom
• Active constraints distillation
• Purity spec. valuable product is always active
  (avoid give-away of valuable product).
• Purity spec. cheap product may not be active
  (may want to overpurify to avoid loss of valuable
  product but costs energy)
• Three cases
• Nactive2 Two active constraints (for example,
  xD, impurity max. xB, impurity max,
  TWO-POINT COMPOSITION CONTROL)
• Nactive1 One constraint active (1 remaining
  DOF)
• Nactive0 No constraints active (2 remaining
  DOFs)

Problem WHAT SHOULD WE CONTROL (TO SATISFY
THE UNCONSTRAINED DOFs)? Solution Often
compositions but not always!
Can happen if no purity specifications (e.g.
byproducts or recycle)
24
Unconstrained variablesWhat should we control?

• Intuition Dominant variables (Shinnar)
• Is there any systematic procedure?

25
What should we control?Systematic procedure

• Systematic Minimize cost J(u,d) w.r.t. DOFs u.
• Control active constraints (constant setpoint is
  optimal)
• Remaining unconstrained DOFs Control
  self-optimizing variables c for which constant
  setpoints cs copt(d) give small (economic)
  loss
• Loss J - Jopt(d)
• when disturbances d ? d occur

26
The difficult unconstrained variables
Cost J
Jopt
c
copt
Selected controlled variable (remaining
unconstrained)
27
Example Tennessee Eastman plant
Conclusion Do not use purge rate as controlled
variable
28
Optimal operation
Cost J
d
LOSS
Jopt
copt
Controlled variable c

• Two problems
• 1. Optimum moves because of disturbances d
  copt(d)

29
Optimal operation
Cost J
d
LOSS
Jopt
n
copt
Controlled variable c

• Two problems
• 1. Optimum moves because of disturbances d
  copt(d)
• 2. Implementation error, c copt n

30
Effect of implementation error on cost (problem
2)
Good
BAD
Good
31
Example sharp optimum. High-purity distillation
c Temperature top of column
Water (L) - acetic acid (H) Max 100 ppm acetic
acid 100 C 100 water 100.01C 100
ppm 99.99 C Infeasible
32
Candidate controlled variables
Unconstrained degrees of freedom

• We are looking for some magic variables c to
  control.....What properties do they have?
• Intuitively 1 Should have small optimal range
  delta copt
• since we are going to keep them constant!
• Intuitively 2 Should have small implementation
  error n
• Intuitively 3 Should be sensitive to inputs u
  (remaining unconstrained degrees of freedom),
  that is, the gain G0 from u to c should be large
• G0 (unscaled) gain from u to c
• large gain gives flat optimum in c
• Charlie Moore (1980s) Maximize minimum singular
  value when selecting temperature locations for
  distillation
• Will show shortly Can combine everything into
  the maximum gain rule
• Maximize scaled gain G Go / span(c)

span(c)
33
Unconstrained degrees of freedomJustification
for intuitively 2 and 3
J
Optimizer
c
cs
Want small n
n
cmcn
n
Controller that adjusts u to keep cm cs
cscopt
u
c
Plant
d
uopt
u
Want the slope ( gain G0 from u to c) large
corresponds to flat optimum in c
34
Mathematic local analysis(Proof of maximum gain
rule)
35
Minimum singular value of scaled gain
Maximum gain rule (Skogestad and Postlethwaite,
1996) Look for variables that maximize the
scaled gain ?(G) (minimum singular value of
the appropriately scaled steady-state gain
matrix G from u to c)
?(G) is called the Morari Resiliency index (MRI)
by Luyben Detailed proof I.J. Halvorsen, S.
Skogestad, J.C. Morud and V. Alstad, Optimal
selection of controlled variables'', Ind. Eng.
Chem. Res., 42 (14), 3273-3284 (2003).
36
Maximum gain rule for scalar system
Unconstrained degrees of freedom
Juu Hessian for effect of us on cost
Problem Juu can be difficult to
obtain Fortunate for scalar system Juu does not
matter
37
Maximum gain rule in words
Select controlled variables c for which the gain
G0 (controllable range) is large compared
to its span (sum of optimal variation and
control error)

38
B. Brute-force procedure for selecting
(primary) controlled variables (Skogestad, 2000)

• Step 1 Determine DOFs for optimization
• Step 2 Definition of optimal operation J (cost
  and constraints)
• Step 3 Identification of important disturbances
• Step 4 Optimization (nominally and with
  disturbances)
• Step 5 Identification of candidate controlled
  variables (use active constraint control)
• Step 6 Evaluation of loss with constant setpoints
  for alternative controlled variables
• Step 7 Evaluation and selection (including
  controllability analysis)
• Case studies Tenneessee-Eastman,
  Propane-propylene splitter, recycle process,
  heat-integrated distillation

39
Unconstrained degrees of freedomC. Optimal
measurement combination (Alstad, 2002)
40
Unconstrained degrees of freedomC. Optimal
measurement combination (Alstad, 2002)

• Basis Want optimal value of c independent of
  disturbances )
• ? copt 0 ? d
• Find optimal solution as a function of d
  uopt(d), yopt(d)
• Linearize this relationship ?yopt F ?d
• F sensitivity matrix
• Want
• To achieve this for all values of ? d
• Always possible if
• Optimal when we disregard implementation error
  (n)

41
Alstad-method continued

• To handle implementation error Use sensitive
  measurements, with information about all
  independent variables (u and d)

42
Summary unconstrained degrees of freedomLooking
for magic variables to keep at constant
setpoints.How can we find them systematically?

• Candidates
• A. Start with Maximum gain (minimum singular
  value) rule
• B. Then Brute force evaluation of most
  promising alternatives.
• Evaluate loss when the candidate variables c are
  kept constant.
• In particular, may be problem with feasibility
• C. More general candidates Find optimal linear
  combination (matrix H)

43
Toy Example
44
Toy Example
45
Toy Example
46
EXAMPLE Recycle plant (Luyben, Yu, etc.)
5
4
1
Given feedrate F0 and column pressure
2
3
Dynamic DOFs Nm 5 Column levels N0y
2 Steady-state DOFs N0 5 - 2 3
47
Recycle plant Optimal operation
mT
1 remaining unconstrained degree of freedom
48
Control of recycle plantConventional structure
(Two-point xD)
LC
LC
xD
XC
XC
xB
LC
Control active constraints (Mrmax and xB0.015)
xD
49
Luyben rule
Luyben rule (to avoid snowballing) Fix a
stream in the recycle loop (F or D)
50
Luyben rule D constant
LC
LC
XC
LC
Luyben rule (to avoid snowballing) Fix a
stream in the recycle loop (F or D)
51
A. Maximum gain rule Steady-state gain
Conventional Looks good
Luyben rule Not promising economically
52
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 uL
• Obtain gain G0 numerically by making a small
  perturbation in uL 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
• span(c) ?i ?copt(di) n
• Obtain the scaled gain, G G0 / span(c)

IMPORTANT!
53
B. Brute force loss evaluation
Disturbance in F0
Luyben rule
Conventional
Loss with nominally optimal setpoints for Mr, xB
and c
54
B. Brute force loss evaluation
Implementation error
Luyben rule
Loss with nominally optimal setpoints for Mr, xB
and c
55
C. Optimal measurement combination

• 1 unconstrained variable (c 1)
• 1 (important) disturbance F0 (d 1)
• Optimal combination requires 2 measurements
  (y u d 2)
• For example, c h1 L h2 F
• BUT Not much to be gained compared to control of
  single variable (e.g. L/F or xD)

56
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
57
Recycle systems
Do not recommend Luybens rule of fixing a flow
in each recycle loop (even to avoid
snowballing)
58
Summary Self-optimizing Control

• Self-optimizing control is when acceptable
  operation can be achieved using constant set
  points (cs) for the controlled variables c
  (without the need to re-optimizing when
  disturbances occur).

ccs
59
Summary Procedure selection controlled variables

• Define economics and operational constraints
• Identify degrees of freedom and important
  disturbances
• Optimize for various disturbances
• Identify (and control) active constraints
  (off-line calculations)
• May vary depending on operating region. For each
  operating region do step 5
• Identify self-optimizing controlled variables
  for remaining degrees of freedom
• (A) Identify promising (single) measurements from
  maximize gain rule (gain minimum singular
  value)
• (C) Possibly consider measurement combinations if
  no promising
• (B) Brute force evaluation of loss for
  promising alternatives
• Necessary because maximum gain rule is local.
• In particular Look out for feasibility problems.
• Controllability evaluation for promising
  alternatives

60
Summary self-optimizing control

• Operation of most real system Constant setpoint
  policy (c cs)
• Central bank
• Business systems KPIs
• Biological systems
• Chemical processes
• Goal Find controlled variables c such that
  constant setpoint policy gives acceptable
  operation in spite of uncertainty
• ) Self-optimizing control
• Method A Maximize ?(G)
• Method B Evaluate loss L J - Jopt
• Method C Optimal linear measurement combination
• ?c H ?y where HF0

61
Outline

• Control structure design (plantwide control)
• A procedure for control structure design
• I Top Down
• Step 1 Degrees of freedom
• Step 2 Operational objectives (optimal
  operation)
• Step 3 What to control ? (self-optimzing
  control)
• Step 4 Where set production rate?
• II Bottom Up
• Step 5 Regulatory control What more to control
  ?
• Step 6 Supervisory control
• Step 7 Real-time optimization
• Case studies