1
Sigurd Skogestad
Department of Chemical Engineering
Norwegian University of Science and Tecnology (NTNU)
Trondheim, Norway
01 April 2006

2
Intro, DOFs, control objectives, self-op. control
What to control, Production rate
, stabilizing control, distillation example
Supervisory control. HDA example

3
• Overview of plantwide control
• Selection of primary controlled variables based on economic : The llink between the optimization (RTO) and the control (MPC; PID) layers
- Degrees of freedom
- Optimization
- Self-optimizing control
- Applications
- Many examples
• Where to set the production rate and bottleneck
• Design of the regulatory control layer ("what more should we control")
- stabilization
- secondary controlled variables (measurements)
- pairing with inputs
- controllability analysis
- cascade control and time scale separation.
• Design of supervisory control layer
- Decentralized versus centralized (MPC)
- Design of decentralized controllers: Sequential and independent design
- Pairing and RGA-analysis
• Summary and case studies

4

5
North Sea
Arctic circle
Trondheim
NORWAY
Oslo
SWEDEN
DENMARK
GERMANY
UK

6

7
• 1. Control for economics (Top-down steady-state arguments)
– Primary controlled variables c
• 2. Control for stabilization (Bottom-up; regulatory PID control)
– Secondary controlled variables (“inner cascade loops”)
• Both problems: “Maximum gain rule” useful for selecting controlled variables

8
• 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 ? (primary CV’s) (self-optimizing control)
• Step 4: Where set production rate?
II Bottom Up
• Step 5: Regulatory control: What more to control (secondary CV’s) ?
• Step 6: Supervisory control
• Step 7: Real-time optimization
• Case studies

9

10

11
• Where do we start?
• What should we control? and why?
• etc.
• etc.

12
• Alan Foss (“Critique of chemical process control theory”, AIChE
Journal,1973):
The central issue to be resolved ... is the determination of control system structure. Which variables should be measured, which inputs should be manipulated and which links should be made between the two sets?
There is more than a suspicion that the work of a genius is needed here, for without it the control configuration problem will likely remain in a primitive, hazily stated and wholly unmanageable form. The gap is present indeed, but contrary to the views of many, it is the theoretician who must close it.
• Carl Nett (1989):
Minimize control system complexity subject to the achievement of accuracy specifications in the face of uncertainty.

13
• 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.).
• That is: Control structure design includes all the decisions we need make to get from ``PID control’’ to “Ph.D” control

14
Process control:
• Large systems
• Each plant usually different – modeling expensive
• Slow processes – no problem with computation time
• Structural issues important
– What to control?
– Extra measurements
– Pairing of loops

15
• 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. (1975- ) – case studies ; “snowball effect”
• George Stephanopoulos and Manfred Morari (1980) – synthesis of control structures for chemical processes
• Ruel Shinnar (1981- ) “dominant variables”
• Jim Downs (1991) Tennessee Eastman challenge problem
• Larsson and Skogestad (2000): Review of plantwide control

16
• Control structure selection issues are identified as important also in other industries.
Professor Gary Balas (Minnesota) at ECC’03 about flight control at Boeing:
The most important control issue has always been to select the right controlled variables --- no systematic tools used!

RTO
17
MPC
PID
Need to define objectives and identify main issues for each layer

18
• Purpose : “Stabilize” the plant by controlling selected ‘’secondary’’ variables ( y
2
) such that the plant does not drift too far away from its desired operation
• Use simple single-loop PI(D) controllers
• Status: Many loops poorly tuned
– Most common setting: K c
=1, 
I
=1 min (default)
– Even wrong sign of gain K c
….

19
• Trend: Can do better! Carefully go through plant and retune important loops using standardized tuning procedure
• Exists many tuning rules, including Skogestad (SIMC) rules:
– K c
= (1/k) ( 
1
/ [  c
+  ]) 
I
= min ( 
1
, 4[  c
+  ]), Typical:  c
= 
– “Probably the best simple PID tuning rules in the world” © Carlsberg
• Outstanding structural issue : What loops to close, that is, which variables ( y
2
) to control?

20
• Purpose : Keep primary controlled variables (c=y
1 using as degrees of freedom the setpoints y
2s
) at desired values, for the regulatory layer.
• Status: Many different “advanced” controllers, including feedforward, decouplers, overrides, cascades, selectors, Smith Predictors, etc.
• Issues:
– Which variables to control may change due to change of “active constraints”
– Interactions and “pairing”

21
• Trend: Model predictive control ( MPC ) used as unifying tool.
– Linear multivariable models with input constraints
– Tuning (modelling) is time-consuming and expensive
• Issue: When use MPC and when use simpler single-loop decentralized controllers ?
– MPC is preferred if active constraints (“bottleneck”) change.
– Avoids logic for reconfiguration of loops
• Outstanding structural issue :
– What primary variables c=y
1 to control?

22
• Purpose : Minimize cost function J and:
– Identify active constraints
– Recompute optimal setpoints y
1s for the controlled variables
• Status: Done manually by clever operators and engineers
• Trend: Real-time optimization ( RTO ) based on detailed nonlinear steady-state model
• Issues:
– Optimization not reliable.
– Need nonlinear steady-state model
– Modelling is time-consuming and expensive

23
RTO
MPC
PID c s
= y
1s
Min J (economics);
MV=y
1s
CV=y
1
; MV=y
2s y
2s
CV=y
2
; MV=u u (valves)

24
I. TOP-DOWN
Step 1. DEGREES OF FREEDOM
Step 2. OPERATIONAL OBJECTIVES
Step 3. WHAT TO CONTROL? (primary CV’s c=y
1
Step 4. PRODUCTION RATE
)
II. BOTTOM-UP (structure control system):
Step 5. REGULATORY CONTROL LAYER (PID)
“Stabilization”
What more to control ? (secondary CV’s y
2
)
Step 6. SUPERVISORY CONTROL LAYER (MPC)
Decentralization
Step 7. OPTIMIZATION LAYER (RTO)
Can we do without it?

25
• 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-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

26
(N valves
)
To find all operational (dynamic) degrees of freedom
• Count valves! (N valves
)
• “Valves” also includes adjustable compressor power, etc.
Anything we can manipulate!

27
• Cost J depends normally only on steady-state DOFs
Three methods to obtain steady-state degrees of freedom (N ss
) :
1. Equation-counting
• N ss
= no. of variables – no. of equations/specifications
• Very difficult in practice (not covered here)
2. Valve-counting (easier!)
• N ss
• N
0ss
= N valves
– N
0ss
– N specs
= variables with no steady-state effect
3. Typical number for some units (useful for checking!)

28
(N ss
)
• N valves
= no. of dynamic (control) DOFs (valves)
• N ss
= N valves
• N
0ss
– N
= N
0y
0ss
– N specs
+ N
0,valves
: no. of steady-state DOFs
: no. of variables with no steady-state effect
– N
0,valves
– N
0y
: no. purely dynamic control DOFs
: no. controlled variables (liquid levels) with no steady-state effect
• N specs
: No of equality specifications (e.g., given pressure)

Distillation column with given feed and pressure
29
N valves
= 6 , N
0y
= 2 , N specs
= 2, N
SS
= 6 -2 -2 = 2

30
N valves
= 11 (w/feed), N
0y
= 4 (levels), N ss
= 11 – 4 =
7

31
N valves
= 11 (w/feed), N
0y
= 4 (levels), N ss
= 11 – 4 = 7

32
CW
N valves
= 3, N
0valves
= 2 (of 3), N ss
= 3 – 2 = 1

33
CW
N valves
= 3, N
0valves
= 2 (of 3), N ss
= 3 – 2 = 1

34
(N ss
)
• each external feedstream: 1 (feedrate)
• splitter: n-1 (split fractions) where n is the number of exit streams
• mixer: 0
• compressor, turbine, pump: 1 (work)
• adiabatic flash tank: 0 *
• liquid phase reactor: 1 (holdup-volume reactant)
• gas phase reactor: 0 *
• heat exchanger: 1 (duty or net area)
• distillation column excluding heat exchangers: 0 * + number of sidestreams
• pressure * : add 1DOF at each extra place you set pressure (using an extra valve, compressor or pump!). Could be for adiabatic flash tank, gas phase reactor, distillation column
* Pressure is normally assumed to be given by the surrounding process and is then not a degree of freedom

35
CW
“Typical number heat exchanger”
N ss
= 1

Distillation column with given feed and pressure
36
“Typical number”,
N ss
= 0 (distillation) + 2*1 (heat exchangers) = 2

37
Typical number, N ss
= 1 (feed) + 2*0 (columns) + 2*1
(column pressures) + 1 (sidestream) + 3 (hex) = 7

38
Compressor
Purge (H
2
+ CH
4
)
H
2
+ CH
4
Toluene
Mixer
Toluene
FEHE
Furnace PFR
Benzene
Cooler
CH
4
Toluene
Column
Benzene
Column
Stabilizer
Quench
Separator
Diphenyl

HDA process: steady-state degrees of freedom
8
7
39
3
1
2
13
14 12
Assume given column pressures
11
6
4
9
10
5
Conclusion: 14 steady-state
DOFs feed:1.2
hex: 3, 4, 6 splitter 5, 7 compressor: 8 distillation: rest

40
• Check that there are enough manipulated variables (DOFs) - both dynamically and at steady-state (step 2)
• Otherwise: Need to add equipment
– extra heat exchanger
– bypass
– surge tank

41
• 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

42
• What are we going to use our degrees of freedom for?
• Define scalar cost function J(u
0
– u
0
: degrees of freedom
,x,d)
– d: disturbances
– x: states (internal variables)
Typical cost function:
J = cost feed + cost energy – value products
• Optimal operation for given d: min uss subject to:
J(u
Model equations: ss
,x,d)
Operational constraints: f(u ss
,x,d) = 0 g(u ss
,x,d) < 0

43
Optimal operation distillation column
• Distillation at steady state with given p and F: N=2 DOFs, e.g. L and V
• Cost to be minimized (economics)
J = - P where P= p
D
D + p
B
B – p
F
F – p
V cost energy (heating+ cooling)
V value products cost feed
• Constraints
Purity D: For example
Purity B: For example, x
D, impurity x
B, impurity
· max
· max
Flow constraints:
Column capacity (flooding): V · V max
, etc.
Pressure:
Feed:
1) p given, min · D, B, L etc. · max
1) F given
2) p free: p min
· p · p max
2) F free: F · F max
• Optimal operation: Minimize J with respect to steady-state DOFs

44 water
up to 30 m/s (100 km/h)
(~20 seconds)
~10m water recycle

45
• 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 < dew point – 10C (active – not always controlled)
– ΔT along dryer (especially inlet) < bound (active?)
– Remaining DOFs: minimize cost

minimize J = cost feed + cost energy – value products
46
Two main cases (modes) depending on marked conditions:
1. Given feed
Amount of products is then usually indirectly given and J = cost energy.
Optimal operation is then usually unconstrained:
“maximize efficiency (energy)”
2. Feed free
Control: Operate at optimal trade-off (not obvious how to do and what to control)
Products usually much more valuable than feed + energy costs small.
Optimal operation is then usually constrained:
“maximize production”
Control: Operate at bottleneck (“obvious”)

47
• 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”

48
• 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

49
(primary controlled variables y
1
=c )
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

• Optimal operation for given d * :
u subject to:
Model equations:
Operational constraints: f(u,x,d) = 0 g(u,x,d) < 0
u opt
(d * )
50
Problem: Usally cannot keep u opt constant because disturbances d change
How should be adjust the degrees of freedom (u)?

51
Implementation of optimal operation (Cannot keep u
0opt
constant)
Estimate d from measurements and recompute u opt
(d)
Problem: Too complicated
(requires detailed model and description of uncertainty)

52
What should we control?

53
Constant setpoint

54
Unconstrained variables:
• Self-optimizing control:
Constant setpoints c s give
”near-optimal operation”
(= acceptable loss L for expected disturbances d and implementation errors n)
Acceptable loss ) self-optimizing control

55
c’s
• 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!
– c s
= 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.

56
• Optimal operation of Sprinter (100 m), J=T
– One input: ”power/speed”
– Active constraint control :
• Maximum speed (”no thinking required”)

57
• Optimal operation of Marathon runner, J=T
– No active constraints
– Any self-optimizing variable c (to control at constant setpoint)?

58
• Optimal operation of Marathon runner, J=T
– Any self-optimizing variable c (to control at constant setpoint)?
• c
1
= distance to leader of race
• c
2
= speed
• c
3
= heart rate
• c
4
= level of lactate in muscles

59
• 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 (selfoptimizing control)

60
• Central bank . J = welfare. u = interest rate. c=inflation 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

61
Summary so far: Active constrains and unconstrained variables
• Optimal operation: Minimize J with respect to DOFs
• General: Optimal solution with N DOFs:
– N active:
DOFs used to satisfy “active” constraints ( · is =)
– N u
= N – N active
. remaining unconstrained variables
Often: N u is zero or small
• It is “obvious” how to control the active constraints
• Difficult issue: What should we use the remaining N u for, that is what should we control?
degrees of

62
Recall: Optimal operation distillation column
• Distillation at steady state with given p and F: N=2 DOFs, e.g. L and V
• Cost to be minimized (economics)
J = - P where P= p
D
D + p
B
B – p
F
F – p
V cost energy (heating+ cooling)
V value products cost feed
• Constraints
Purity D: For example
Purity B: For example, x
D, impurity x
B, impurity
· max
· max
Flow constraints:
Column capacity (flooding): V · V max
, etc.
Pressure:
Feed:
1) p given, min · D, B, L etc. · max
1) F given
2) p free: p min
· p · p max
2) F free: F · F max
• Optimal operation: Minimize J with respect to steady-state DOFs

63
Solution: Optimal operation distillation
• Cost to be minimized
J = - P where P= p
D
D + p
B
B – p
F
F – p
V
• N=2 steady-state degrees of freedom
V
• Active constraints distillation:
– Purity spec. valuable product is always active (“avoid giveaway 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:
1. N active impurity
2. N active
3. N active
=2: Two active constraints (for example, x
D, impurity
= max,
“TWO-POINT” COMPOSITION CONTROL )
=1: One constraint active (1 remaining DOF)
=0: No constraints active (2 remaining DOFs)
= max. x
B,
Problem : WHAT SHOULD
WE CONTROL (TO SATISFY
THE UNCONSTRAINED DOFs
)?
Can happen if no purity specifications
(e.g. byproducts or recycle)
Solution:
Often compositions but not always!

64
• Intuition: “Dominant variables” (Shinnar)
• Is there any systematic procedure?

65
• Systemati c: Minimize cost J(u,d * ) w.r.t.
DOFs u
.
1. Control active constraints (constant setpoint is optimal)
2. Remaining unconstrained DOFs: Control
“self-optimizing” variables c for which constant setpoints c s
(economic) loss
= c opt
(d * ) give small
Loss = J - J opt
(d) when disturbances d ≠ d * occur c = ? (economics) y
2
= ? (stabilization)

66
Cost J
J opt c opt
Selected controlled variable
(remaining unconstrained)

67
J
Oopss..
bends backwards c = Purge rate
Nominal optimum setpoint is infeasible with disturbance 2
Conclusion: Do not use purge rate as controlled variable

Cost J d
LOSS
J opt
68 c opt Controlled variable c
Two problems:
• 1. Optimum moves because of disturbances d: c opt
(d)

Cost J d
69
LOSS
J opt n c opt Controlled variable c
Two problems:
•
• 1. Optimum moves because of disturbances d: c opt
(d)
2. Implementation error, c = c opt
+ n

70
Effect of implementation error on cost (“problem 2”)
Good
Good BAD

71
Example sharp optimum. High-purity distillation : c = Temperature top of column
T top
Temperature
Water (L) - acetic acid (H)
Max 100 ppm acetic acid
100 C: 100% water
100.01C: 100 ppm
99.99 C: Infeasible

72
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 c opt
– since we are going to keep them constant!
• Intuitively 2 : Should have small “implementation error” n span(c)
• Intuitively 3 : Should be sensitive to inputs u (remaining unconstrained degrees of freedom), that is, the gain G
– G
0
0
: (unscaled) gain from u to c from u to c should be large
– large gain gives flat optimum in c
– Charlie Moore (1980’s): 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 = G o
/ span(c)

73
Unconstrained degrees of freedom:
J
Optimizer c s c d
Controller that adjusts u to keep c m
= c s c m
=c + n u
Plant n n c s
=c opt
Want small n c u opt
Want the slope (= gain G
0 from u to c) large – corresponds to flat optimum in c u

74
cost
J u opt u

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)
75
 (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).

76
G: Scaled gain matrix (as before)
J uu
: Hessian for effect of u ’s on cost
Problem: J uu can be difficult to obtain
Improved rule has been used successfully for distillation

77
Unconstrained degrees of freedom:

78
Select controlled variables c for which their controllable range is large compared to their sum of optimal variation and control error controllable range = range c may reach by varying the inputs (=gain) optimal variation : due to disturbance span control error = implementation error n

79
B.
(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

80
Unconstrained degrees of freedom:
C.
(Alstad, 2002)

81
Unconstrained degrees of freedom:
C.
(Alstad, 2002)
Basis : Want optimal value of c independent of disturbances )
  c opt
= 0 ¢  d
• Find optimal solution as a function of d: u opt
(d), y opt
(d)
• Linearize this relationship:  y opt
= F  d
• F – sensitivity matrix
• Want:
• To achieve this for all values of  d:
• Always possible if
• Optimal when we disregard implementation error (n)

82
• To handle implementation error: Use “sensitive” measurements, with information about all independent variables (u and d)

83
Summary unconstrained degrees of freedom:
Looking for “ magic ” variables to keep at constant setpoints.
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):

84

85

86

(Luyben, Yu, etc.)
5
4
87
1
Given feedrate F
0 column pressure: and
Dynamic DOFs: N m
Column levels: N
0y
Steady-state DOFs: N
0
= 5
= 2
= 5 - 2 = 3
2
3

88
m T
1 remaining unconstrained degree of freedom

x
D
LC
LC
XC x
D
XC x
B
LC
89
Control active constraints
(M r
=max and x
B
=0.015)
+ x
D

90
Luyben rule (to avoid snowballing):
“Fix a stream in the recycle loop” ( F or D )

91
LC
LC
XC
LC
Luyben rule (to avoid snowballing):
“Fix a stream in the recycle loop” ( F or D )

92
A
Conventional:
Looks good
Luyben rule:
Not promising economically

93
1. Find nominal optimum
2. Find (unscaled) gain G
0 from input to candidate outputs:  c = G
• In this case only a single unconstrained input (DOF). Choose at u=L
0
 u.
• Obtain gain G
0 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)
IMPORTANT!
3. Find the span for each candidate variable
• For each disturbance d i the optimal ranges  c opt make a typical change and reoptimize to obtain
(d i
)
• For each candidate output obtain (estimate) the control error (noise) n
• span(c) =  i
|  c opt
(d i
)| + n
4. Obtain the scaled gain, G = G
0
/ span(c)

B.
0
Luyben rule:
Conventional
94
Loss with nominally optimal setpoints for M r
, x
B and c

B.
Luyben rule:
95
Loss with nominally optimal setpoints for M r
, x
B and c

96
C.
• 1 unconstrained variable (#c = 1)
• 1 (important) disturbance: F
0
(#d = 1)
• “Optimal” combination requires 2 “ measurements ” ( #y = #u + #d = 2)
– For example, c = h
1
L + h
2
F
• BUT: Not much to be gained compared to control of single variable
(e.g. L/F or x
D
)

Active constraint
M r
= M rmax
Self-optimizing
97
L/F constant: Easier than “two-point” control
Assumption: Minimize energy (V)
Active constraint x
B
= x
Bmin

98
(even to avoid “snowballing”)

99
Self-optimizing control is when acceptable operation can be achieved using constant set points (c s
) for the controlled variables c (without the need to re-optimizing when disturbances occur).
c=c s

10
0
1. Define economics and operational constraints
2. Identify degrees of freedom and important disturbances
3. Optimize for various disturbances
4. Identify (and control) active constraints (off-line calculations)
• May vary depending on operating region. For each operating region do step 5 :
5. Identify “self-optimizing” controlled variables for remaining degrees of freedom
1. (A) Identify promising (single) measurements from “maximize gain rule” (gain = minimum singular value)
• (C) Possibly consider measurement combinations if no promising
2. (B) “Brute force” evaluation of loss for promising alternatives
• Necessary because “maximum gain rule” is local.
• In particular: Look out for feasibility problems.
3. Controllability evaluation for promising alternatives

10
1
• Operation of most real system: Constant setpoint policy ( c = c s
)
– Central bank
– Business systems: KPI’s
– 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 - J opt
• Method C: Optimal linear measurement combination:
 c = H  y where HF=0

10
2
• 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

10
3
• Very important!
• Determines structure of remaining inventory (level) control system
• Set production rate at (dynamic) bottleneck
• Link between Top-down and Bottom-up parts

10
4

10
5

10
6

10
7
• Very important decision that determines the structure of the rest of the control system!
• May also have important economic implications

10
8
• "A bottleneck is an extensive variable that prevents an increase in the overall feed rate to the plant"
• If feed is cheap and available: Optimal to set production rate at bottleneck
• If the flow for some time is not at its maximum through the bottleneck, then this loss can never be recovered.

10
9
Reactor-recycle process:
Given feedrate with production rate set at inlet

11
0
Reactor-recycle process:
Want to maximize feedrate: reach bottleneck in column
Bottleneck: max. vapor rate in column

Reactor-recycle process with production rate set at inlet
Want to maximize feedrate: reach bottleneck in column
Bottleneck: max. vapor rate in column
11
1
V s FC
V max
V
V max
-V s
=Back-off
= Loss

11
2
Reactor-recycle process with increased feedrate:
Optimal: Set production rate at bottleneck
MAX

11
3
Reactor-recycle process with increased feedrate:
Optimal: Set production rate at bottleneck
MAX

Reactor-recycle process:
Given feedrate with production rate set at bottleneck
11
4
F
0s

11
5
• Can reduce loss
• BUT: Is generally placed on top of the regulatory control system
(including level loops), so it still important where the production rate is set!

• Think carefully about where to place it!
• Difficult to undo later
11
6
BREAK

11
7
• 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

11
8
• Determine secondary controlled variables and structure
(configuration) of control system (pairing)
• A good control configuration is insensitive to parameter changes
Step 5. REGULATORY CONTROL LAYER
5.1 Stabilization (including level control)
5.2 Local disturbance rejection (inner cascades)
What more to control? (secondary variables)
Step 6. SUPERVISORY CONTROL LAYER
D ecentralized or multivariable control (MPC)?
Pairing?
Step 7. OPTIMIZATION LAYER (RTO)

11
9
• Purpose : “Stabilize” the plant using local SISO PID controllers
• Enable manual operation (by operators)
• Main structural issues:
• What more should we control?
(secondary cv’s, y
2
)
• Pairing with manipulated variables (mv’s u
2
)
1
y
2
= ?

12
0
y
2s
u
2
y
1 y
2
Key decision : Choice of y
2
(controlled variable)
Also important (since we almost always use single loops in the regulatory control layer): Choice of u
2
( “pairing”)

12
1
• Primary controlled variable: y
1
= c = x
D
, x
B
(compositions top, bottom)
• BUT: Delay in measurement of x + unreliable
• Regulatory control: For “stabilization” need control of (y
2
):
– Liquid level condenser (M
D
)
Unstable (Integrating) + No steady-state effect
– Liquid level reboiler (M
B
)
– Pressure (p) Disturbs (“destabilizes”) other loops
– Holdup of light component in column
(temperature profile) Almost unstable (integrating)
T s
TC
T-loop in bottom

12
2
With flow loop +
T-loop in top y
T
L y s
X
C
T s
TC
L s
FC z
X
C

12
3
• No degrees of freedom lost by control of secondary (local) variables as setpoints become y
2s replace inputs u
2 as new degrees of freedom
Cascade control: y
2s
u
2
y
1 y
2
Original DOF

12
4
• With a “reasonable” time scale separation between the layers
(typically by a factor 5 or more in terms of closed-loop response time) we have the following advantages:
1. The stability and performance of the lower (faster) layer (involving y
2
) is not much influenced by the presence of the upper (slow) layers (involving y the bandwidth of the lower layers
1
)
Reason: The frequency of the “disturbance” from the upper layer is well inside
2. With the lower (faster) layer in place, the stability and performance of the upper (slower) layers do not depend much on the specific controller settings used in the lower layers
Reason: The lower layers only effect frequencies outside the bandwidth of the upper layers

12
5
1. Allow for manual operation
2. Simple decentralized (local) PID controllers that can be tuned on-line
3. Take care of “fast” control
4. Track setpoint changes from the layer above
5. Local disturbance rejection
6. Stabilization (mathematical sense)
7. Avoid “drift” (due to disturbances) so system stays in “linear region”
– “stabilization” (practical sense)
8. Allow for “slow” control in layer above (supervisory control)
9. Make control problem easy as seen from layer above
Implications for selection of y
2
:
1. Control of y
2
2. y
2
“stabilizes the plant” is easy to control (favorable dynamics)

12
6
2
A. “Mathematical stabilization” (e.g. reactor):
• Unstable mode is “quickly” detected ( state observability ) in the measurement ( y
2
) and is easily affected ( state controllability ) by the input (u
2
).
• Tool for selecting input/output: Pole vectors
– y
2
: Want large element in output pole vector: Instability easily detected relative to noise
– u
2
: Want large element in input pole vector: Small input usage required for stabilization
B. “Practical extended stabilization” (avoid “drift” due to disturbance sensitivity):
• Intuitive : y
2 located close to important disturbance
• Or rather: Controllable range for y
2 optimal variation and control error is large compared to sum of
• More exact tool: P artial control analysis

12
7
Recall rule for selecting primary controlled variables c:
Controlled variables c for which their controllable range is large compared to their sum of optimal variation and control error
Restated for secondary controlled variables y
2
:
Control variables y
2 for which their controllable range their sum of optimal variation and control error is large compared to controllable range = range y
2 may reach by varying the inputs optimal variation : due to disturbances
Want small control error = implementation error n
Want large

12
8
2
• General case: Maximize minimum singular value of scaled G
• Scalar case: |G s
| = |G| / span
• |G|: gain from independent variable (u
(y
2
) to candidate controlled variable
2
)
– IMPORTANT: The gain |G| should be evaluated at the (bandwidth) frequency of the layer above in the control hierarchy!
This can be very different from the steady-state gain used for selecting primary controlled variables (y
1
=c)
• span (of y
2
) = optimal variation in y controlled variables (c).
2
+ control error for y
2
– Note optimal variation: This is often the same as the optimal variation used for selecting primary
– Exception: If we at the “fast” regulatory time scale have some yet unused “slower” inputs (u
1
) which are constant then we may want find a more suitable optimal variation for the fast time scale.

12
9
2
• Problem in some cases : “optimal variation” for y
2 control objectives which may change depends on overall
• Therefore: May want to “decouple” tasks of stabilization (y
2 optimal operation (y
1
)
) and
• One way of achieving this : Choose y
2 minimized such that “state drift” dw/dd is
• w = Wx – weighted average of all states
• d – disturbances
• Some tools developed:
– Optimal measurement combination y
2
=Hy that minimizes state drift
(Hori) – see Skogestad and Postlethwaite (Wiley, 2005) p. 418
– Distillation column application: Control average temperature column

13
0
2
1. Statics: Want large gain
(from u
2 to y
2
)
2. Main rule: y
2 is easy to measure and located close to available manipulated variable u
2
(“pairing”)
3. Dynamics: Want small effective delay
(from u
2
• “effective delay” includes to y
2
)
• inverse response (RHP-zeros)
• + high-order lags

13
1
2
2
1. Avoid using variable u
2 that may saturate (especially in loops at the bottom of the control hieararchy)
• Alternatively: Need to use “input resetting” in higher layer
• Example: Stabilize reactor with bypass flow (e.g. if bypass may saturate, then reset in higher layer using cooling flow)
2. “Pair close”: The controllability, for example in terms a small effective delay from u
2 to y
2
, should be good.

13
2
• θ = effective delay
• PI-tunings from “SIMC rule”
• Use half rule to obtain first-order model
– Effective delay θ = “True” delay + inverse response time constant + half of second time constant + all smaller time constants
– Time constant τ
1
= original time constant + half of second time constant
– NOTE: The first (largest) time constant is NOT important for controllability!

13
3
2
2
1. y
2 should be easy to measure
2. Control of y
2 stabilizes the plant
3. y
2 should have good controllability, that is, favorable dynamics for control
4. y
2 should be located “close” to a manipulated input (u
2
) (follows from rule 3)
5. The (scaled) gain from u
2 to y
2
6. The effective delay from u
2 to y should be large
2 should be small
7. Avoid using inputs u
2 that may saturate (should generally avoid saturation in lower layers)

5 dynamic DOFs (L,V,D,B,VT)
Overall objective:
Control compositions (x
D and x
B
)
“Obvious” stabilizing loops:
1.
Condenser level (M
1
2.
Reboiler level (M
2
3.
Pressure
)
)
13
4
E.A. Wolff and S. Skogestad, ``Temperature cascade control of distillation columns'', Ind.Eng.Chem.Res.
, 35 , 475-484, 1996.

13
5
• Maximum gain rule is good for integrating (drifting) modes
• For “fast” unstable modes (e.g. reactor): Pole vectors useful for determining which input (valve) and output (measurement) to use for stabilizing unstable modes
• Assumes input usage (avoiding saturation) may be a problem

13
6

13
7

13
8

13
9

14
0

14
1

14
2

14
3

14
4

14
5
• Control configuration. The restrictions imposed on the overall controller by decomposing it into a set of local controllers
(subcontrollers, units, elements, blocks) with predetermined links and with a possibly predetermined design sequence where subcontrollers are designed locally.
Control configuration elements:
• Cascade controllers
• Decentralized controllers
• Feedforward elements
• Decoupling elements

14
6
• Cascade control arises when the output from one controller is the input to another. This is broader than the conventional definition of cascade control which is that the output from one controller is the reference command
(setpoint) to another. In addition, in cascade control, it is usually assumed that the inner loop K2 is much faster than the outer loop K1.
• Feedforward elements link measured disturbances to manipulated inputs.
• Decoupling elements link one set of manipulated inputs (“measurements”) with another set of manipulated inputs. They are used to improve the performance of decentralized control systems, and are often viewed as feedforward elements (although this is not correct when we view the control system as a whole) where the “measured disturbance” is the manipulated input computed by another decentralized controller .

14
7
• Fundamental: Save on modelling effort
• Other:
– easy to understand
– easy to tune and retune
– insensitive to model uncertainty
– possible to design for failure tolerance
– fewer links
– reduced computation load

14
8
(conventional; with extra measurement)
The reference r
2 is an output from another controller
General case (“parallel cascade”)
Special common case (“series cascade”)

14
9
1.
Disturbances arising within the secondary loop (before y
2
) are corrected by the secondary controller before they can influence the primary variable y
1
2.
Phase lag existing in the secondary part of the process (G
2
) is reduced by the secondary loop. This improves the speed of response of the primary loop.
3.
Gain variations in G
2 are overcome within its own loop.
•
Thus, use cascade control (with an extra secondary measurement y
The disturbance d
2 is significant and G
1 has an effective delay
2
) when:
• The plant G
2 is uncertain (varies) or n onlinear
Design:
• First design K
2
• Then design K
1
(“fast loop”) to deal with d
2 to deal with d
1

15
0
y
2
= T
2 r
2
+ S
2 d
2
• Use SIMC tuning rules
• K
2 is designed based on G
– then y
2
= T
2 r
2
+ S
2 d
2
2
(which has effective delay where S
2
¼ 0 and T
2
¼ 1 ¢ e -( 

2+
 c2)s
2
)
•
• T
2
: gain = 1 and effective delay = 
2
SIMC-rule:  c2
¸

2
• Time scale separation:  c2
·  c1
+  c2
/5 (approximately)
• K
1 is designed based on G
1
T
2
• same as G
1 but with an additional delay 
2
+  c2

15
1
1. (without cascade, i.e. no feedback from y
2
).
Design a controller based on G
1
G
2. (with cascade)
Design K
2 and then K
1

15
2

15
3
• Exercise: Explain how “valve position control” fits into this framework. As en example consider a heat exchanger with bypass

15
4
• Exercise:
(a) In what order would you tune the controllers?
(b) Give a practical example of a process that fits into this block diagram

15
5
• Cascade control : y
2 for control of y
1 not important in itself, and setpoint (r
• Decentralized control (using sequential design): y
2
2
) is available important in itself

15
6
Primary controlled variable y
1
= c
(supervisory control layer)
Local control of y
2 using u
2
(regulatory control layer )
Setpoint y
2s
: new DOF for supervisory control y
1
= P
1
u
1
+ P r1
(y
2s
-n
2
) + P d1
d
P
1
= G
11
– G
12
G
22
-1
G
21
P d1
= G d1
– G
12
G
22
-1
G d2
- WANT SMALL
P r1
= G
12
G
22
-1

15
7 u
1
= V
Supervisory control:
Primary controlled variables y
1
= c = (x
D x
B
) T
Regulatory control:
Control of y
2 using u
2
= T
= L (original DOF )
Setpoint y
2s
= T s
: new DOF for supervisory control y
1
= P
1
u
1
+ P r1
(y
2s
-n
2
) + P d1
d
P
1
= G
11
– G
12
G
22
-1
G
21
P d1
= G d1
– G
12
G
22
-1
G d2
- WANT SMALL
P r1
= G
12
G
22
-1

15
8
• Cascade control: Closing of secondary loops does not by itself impose new problems
– Theorem 10.2 (SP, 2005) . The partially controlled system [P
1
P r1
] from [u
1 from [u provided
1 r
2
] to y
1 has no new RHP-zeros that are not present in the open-loop system [G
11 u
2
] to y
1
• r
2 is available for control of y
1
• K
2 has no RHP-zeros
G
12
]
• Decentralized control (sequential design): Can introduce limitations.
– Avoid pairing on negative RGA for u
2
/y
2 zero
– otherwise P u likely has a RHP-
BREAK

15
9
• 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 ? (primary CV’s) (self-optimizing control)
• Step 4: Where set production rate?
II Bottom Up
• Step 5: Regulatory control: What more to control (secondary CV’s) ?
• Step 6: Supervisory control
• Step 7: Real-time optimization
• Case studies

16
0
• Purpose : Keep primary controlled outputs c=y
1
• Degrees of freedom: Setpoints y
2s at optimal setpoints c in reg.control layer s
• Main structural issue: Decentralized or multivariable?

16
1
Use for : Noninteracting process and no change in active constraints
+ Tuning may be done on-line
+ No or minimal model requirements
+ Easy to fix and change
- Need to determine pairing
- Performance loss compared to multivariable control
- Complicated logic required for reconfiguration when active constraints move

16
2
Use for : Interacting process and changes in active constraints
+ Easy handling of feedforward control
+ Easy handling of changing constraints
• no need for logic
• smooth transition
- Requires multivariable dynamic model
- Tuning may be difficult
- Less transparent
- “Everything goes down at the same time”

16
3
• 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

16
4
• Purpose: Identify active constraints and compute optimal setpoints (to be implemented by supervisory control layer)
• Main structural issue: Do we need RTO? (or is process selfoptimizing)
• RTO not needed when
– Can “easily” identify change in active constraints (operating region)
– For each operating region there exists self-optimizing var

16
5
• 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
• Conclusion / References

16
6
1. What should we control (y
1
=c=z)?
• Must define optimal operation!
2. Where should we set the production rate?
• At bottleneck
3. What more should we control (y
2
)?
• Variables that “stabilize” the plant
4. Control of primary variables
• Decentralized?
• Multivariable (MPC)?

16
7
Procedure plantwide control:
I. Top-down analysis to identify degrees of freedom and primary controlled variables (look for self-optimizing variables)
II. Bottom-up analysis to determine secondary controlled variables and structure of control system (pairing).

16
8
• HDA process
• Cooling cycle
• Distillation (C3-splitter)
• Blending

16
9
• Halvorsen, I.J, Skogestad, S., Morud, J.C., Alstad, V. (2003), “Optimal selection of controlled variables”, Ind.Eng.Chem.Res
., 42 , 3273-3284.
• Larsson, T. and S. Skogestad (2000), “Plantwide control: A review and a new design procedure”, Modeling, Identification and Control , 21 , 209-240.
• Larsson, T., K. Hestetun, E. Hovland and S. Skogestad (2001), “Self-optimizing control of a large-scale plant: The Tennessee Eastman process’’, Ind.Eng.Chem.Res
., 40, 4889-4901.
• Larsson, T., M.S. Govatsmark, S. Skogestad and C.C. Yu (2003), “Control of reactor, separator and recycle process’’, Ind.Eng.Chem.Res
., 42 , 1225-1234
• Skogestad, S. and Postlethwaite, I. (1996, 2005), Multivariable feedback control , Wiley
• Skogestad, S. (2000). “Plantwide control: The search for the self-optimizing control structure”. J. Proc. Control 10 , 487-507.
• Skogestad, S. (2003), ”Simple analytic rules for model reduction and PID controller tuning”,
J. Proc. Control , 13 , 291-309.
• Skogestad, S. (2004), “Control structure design for complete chemical plants”, Computers and Chemical Engineering , 28 , 219-234. (Special issue from ESCAPE’12 Symposium , Haag,
May 2002).
• … + more…..
See home page of S. Skogestad: http://www.nt.ntnu.no/users/skoge/