1 selv-optimaliserende regulering anvendelser mot prosessindustrien, biologi og maratonløping...
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Selv-optimaliserende reguleringAnvendelser mot prosessindustrien, biologi og maratonløping
Sigurd Skogestad
Institutt for kjemisk prosessteknologi, NTNU, Trondheim
HC, 31. januar 2012
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Self-optimizing controlFrom key performance indicators to control of biological systems
Sigurd Skogestad
Department of Chemical Engineering
Norwegian University of Science and Technology (NTNU)
Trondheim
PSE 2003, Kunming, 05-10 Jan. 2004
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Outline• Optimal operation
• Implememtation of optimal operation: Self-optimizing control
• What should we control?
• Applications– Marathon runner
– KPI’s
– Biology
– ...
• Optimal measurement combination
• Optimal blending example
Focus: Not optimization (optimal decision making)But rather: How to implement decision in an uncertain world
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Optimal operation of systems
• Theory:– Model of overall system– Estimate present state– Optimize all degrees of freedom
• Problems: – Model not available and optimization complex– Not robust (difficult to handle uncertainty)
• Practice– Hierarchical system– Each level: Follow order (”setpoints”) given from level above– Goal: Self-optimizing
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Process operation: Hierarchical structure
PID
RTO
MPC
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Engineering systems
• Most (all?) large-scale engineering systems are controlled using hierarchies of quite simple single-loop controllers – Large-scale chemical plant (refinery)
– Commercial aircraft
• 1000’s of loops
• Simple components: on-off + P-control + PI-control + nonlinear fixes + some feedforward
Same in biological systems
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What should we control?
y1 = c ? (economics)
y2 = ? (stabilization)
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Self-optimizing Control
Self-optimizing control is when acceptable operation can be achieved using constant set points (c
s)
for the controlled variables c
(without re-optimizing when disturbances occur).
c=cs
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Optimal operation (economics)
• Define scalar cost function J(u0,d)
– u0: degrees of freedom
– d: disturbances
• Optimal operation for given d:
minu0 J(u0,d)subject to:
f(u0,d) = 0
g(u0,d) < 0
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Implementation of optimal operation
• Idea: Replace optimization by setpoint control
• Optimal solution is usually at constraints, that is, most of the degrees of freedom u0 are used to satisfy “active constraints”, g(u0,d) = 0
• CONTROL ACTIVE CONSTRAINTS!– Implementation of active constraints is usually simple.
• WHAT MORE SHOULD WE CONTROL?– Find variables c for remaining
unconstrained degrees of freedom u.
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Unconstrained variables
Cost J
Selected controlled variable (remaining unconstrained)
ccoptopt
JJoptopt
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Implementation of unconstrained variables is not trivial: How do we deal with uncertainty?
• 1. Disturbances d
• 2. Implementation error n
cs = copt(d*) – nominal optimization
nc = cs + n
d
Cost J Jopt(d)
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Problem no. 1: Disturbance d
Cost J
Controlled variable ccoptopt(d(d**))
JJoptopt
dd**
d d ≠≠ dd**
Loss with constant value for c
) Want copt independent of d
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Problem no. 2: Implementation error n
Cost J
ccss=c=coptopt(d(d**))
JJoptopt
dd** Loss due to implementation error for c
c = cs + n
) Want n small and ”flat” optimum
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Optimizer
Controller thatadjusts u to keep
cm = cs
Plant
cs
cm=c+n
u
c
n
d
u
c
J
cs=copt
uopt
n
) Want c sensitive to u (”large gain”)
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Which variable c to control?
• Define optimal operation: Minimize cost function J
• Each candidate variable c:
With constant setpoints cs compute loss L for expected disturbances d and implementation errors n
• Select variable c with smallest loss
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Constant setpoint policy:Loss for disturbances (“problem 1”)
Acceptable loss ) self-optimizing control
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Good candidate controlled variables c (for self-optimizing control)
Requirements:
• The optimal value of c should be insensitive to disturbances (avoid problem 1)
• c should be easy to measure and control (rest: avoid problem 2)
• The value of c should be sensitive to changes in the degrees of freedom
(Equivalently, J as a function of c should be flat)
• For cases with more than one unconstrained degrees of freedom, the selected controlled variables should be independent.
Singular value rule (Skogestad and Postlethwaite, 1996):Look for variables that maximize the minimum singular value of the appropriately scaled steady-state gain matrix G from u to c
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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)
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Self-optimizing Control – Marathon
• Optimal operation of Marathon runner, J=T– Any self-optimizing variable c (to control at constant
setpoint)?
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Self-optimizing Control – Marathon
• 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
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Further examples
• Central bank. J = welfare. c=inflation rate (2.5%)• Cake baking. J = nice taste, 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 spendingOptimal 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 identify what overall objective J the biological system has been
attempting to optimize
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Looking for “magic” variables to keep at constant setpoints.
How can we find them?
• Consider available measurements y, and evaluate loss when they are kept constant (“brute force”):
• More general: Find optimal linear combination (matrix H):
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Optimal measurement combination (Alstad)
• 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
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cs = y1s
MPC
PID
y2s
RTO
u (valves)
Follow path (+ look after other variables)
CV=y1 (+ u); MV=y2s
Stabilize + avoid drift CV=y2; MV=u
Min J (economics); MV=y1s
OBJECTIVE
Dealing with complexity
Main simplification: Hierarchical decomposition
Process control The controlled variables (CVs)interconnect the layers
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Outline
• Control structure design (plantwide control)
• A procedure for control structure designI Top Down
• Step 1: Define operational objective (cost) and constraints
• Step 2: Identify degrees of freedom and optimizate for disturbances
• Step 3: What to control ? (primary CV’s) (self-optimizing control)
• Step 4: Where set the production rate? (Inventory control)
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
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Main message
• 1. Control for economics (Top-down steady-state arguments)– Primary controlled variables c = y1 :
• Control active constraints
• For remaining unconstrained degrees of freedom: Look for “self-optimizing” variables
• 2. Control for stabilization (Bottom-up; regulatory PID control)– Secondary controlled variables y2 (“inner cascade loops”)
• Control variables which otherwise may “drift”
• Both cases: Control “sensitive” variables (with a large gain)!
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Process control:
“Plantwide control” = “Control structure design for complete chemical plant”
• 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
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. (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
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• 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!
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Main objectives control system
1. Stabilization2. Implementation of acceptable (near-optimal) operation
ARE THESE OBJECTIVES CONFLICTING?
• Usually NOT – Different time scales
• Stabilization fast time scale
– Stabilization doesn’t “use up” any degrees of freedom• Reference value (setpoint) available for layer above• But it “uses up” part of the time window (frequency range)
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cs = y1s
MPC
PID
y2s
RTO
u (valves)
Follow path (+ look after other variables)
CV=y1 (+ u); MV=y2s
Stabilize + avoid drift CV=y2; MV=u
Min J (economics); MV=y1s
OBJECTIVE
Dealing with complexity
Main simplification: Hierarchical decomposition
Process control The controlled variables (CVs)interconnect the layers
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Example: Bicycle ridingNote: design starts from the bottom
• Regulatory control: – First need to learn to stabilize the bicycle
• CV = y2 = tilt of bike• MV = body position
• Supervisory control: – Then need to follow the road.
• CV = y1 = distance from right hand side• MV=y2s
– Usually a constant setpoint policy is OK, e.g. y1s=0.5 m
• Optimization: – Which road should you follow? – Temporary (discrete) changes in y1s
Hierarchical decomposition
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Summary: The three layers
• Optimization layer (RTO; steady-state nonlinear model): • Identifies active constraints and computes optimal setpoints for primary controlled variables (y1).
• Supervisory control (MPC; linear model with constraints): • Follow setpoints for y1 (usually constant) by adjusting setpoints for secondary variables (MV=y2s) • Look after other variables (e.g., avoid saturation for MV’s used in regulatory layer)
• Regulatory control (PID): • Stabilizes the plant and avoids drift, in addition to following setpoints for y2. MV=valves (u).
Problem definition and overall control objectives (y1, y2) starts from the top.Design starts from the bottom.
A good example is bicycle riding:
• Regulatory control: • First you need to learn how to stabilize the bicycle (y2)
• Supervisory control: • Then you need to follow the road. Usually a constant setpoint policy is OK, for example, stay
y1s=0.5 m from the right hand side of the road (in this case the "magic" self-optimizing variable self-optimizing variable is y1=distance to right hand side of road)
• Optimization: • Which road (route) should you follow?
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Control structure 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
– Identify regions of active constraints
• 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?)
Understanding and using this procedure is the most important part of this course!!!!
y1
y2
Process
MVs
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Step 3. What should we control (c)? (primary controlled variables y1=c)
1. CONTROL ACTIVE CONSTRAINTS, c=cconstraint
2. REMAINING UNCONSTRAINED, c=?
What should we control?c = Hy
H
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Step 1. Define optimal operation (economics)
• What are we going to use our degrees of freedom u (MVs) for?• Define scalar cost function J(u,x,d)
– u: degrees of freedom (usually steady-state)– d: disturbances– x: states (internal variables)Typical cost function:
• Optimize operation with respect to u for given d (usually steady-state):
minu J(u,x,d)subject to:
Model equations: f(u,x,d) = 0Operational constraints: g(u,x,d) < 0
J = cost feed + cost energy – value products
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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= 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
value products
cost energy (heating+ cooling)
cost feed
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Optimal operation
1. Given feedAmount of products is then usually indirectly given and J = cost energy.
Optimal operation is then usually unconstrained:
2. Feed free Products usually much more valuable than feed + energy costs small.
Optimal operation is then usually constrained:
minimize J = cost feed + cost energy – value products
“maximize efficiency (energy)”
“maximize production”
Two main cases (modes) depending on marked conditions:
Control: Operate at bottleneck (“obvious what to control”)
Control: Operate at optimal trade-off (not obvious what to control to achieve this)
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Solution I (“obvious”): Optimal feedforward
Problem: UNREALISTIC!
1. Lack of measurements of d2. Sensitive to model error
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Solution II (”obvious”): Optimizing control
Estimate d from measurements y and recompute uopt(d)
Problem:COMPLICATED! Requires detailed model and description of uncertainty
y
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Solution III (in practice): FEEDBACK with hierarchical decomposition
When disturbance d:Degrees of freedom (u) are updated indirectly to keep CVs at setpoints
y
CVs: link optimization and control layers
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How does self-optimizing control (solution III) work?
• When disturbances d occur, controlled variable c deviates from setpoint cs
• Feedback controller changes degree of freedom u to uFB(d) to keep c at cs
• Near-optimal operation / acceptable loss (self-optimizing control) is achieved if– uFB(d) ≈ uopt(d)– or more generally, J(uFB(d)) ≈ J(uopt(d))
• Of course, variation of uFB(d) is different for different CVs c.
• We need to look for variables, for which J(uFB(d)) ≈ J(uopt(d)) or Loss = J(uFB(d)) - J(uopt(d)) is small
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Remarks “self-optimizing control”
1. Old idea (Morari et al., 1980):
“We want to find a function c of the process variables which when held constant, leads automatically to the optimal adjustments of the manipulated variables, and with it, the optimal operating conditions.”
2. “Self-optimizing control” = acceptable steady-state behavior (loss) with constant CVs.
is similar to
“Self-regulation” = acceptable dynamic behavior with constant MVs.
3. The ideal self-optimizing variable c is the gradient (c = J/ u = Ju)
– Keep gradient at zero for all disturbances (c = Ju=0)
– Problem: no measurement of gradient
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Step 3. What should we control (c)?Simple examples
What should we control?
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– Cost to be minimized, J=T
– One degree of freedom (u=power)
– What should we control?
Optimal operation - Runner
Optimal operation of runner
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Self-optimizing control: Sprinter (100m)
• 1. Optimal operation of Sprinter, J=T– Active constraint control:
• Maximum speed (”no thinking required”)
Optimal operation - Runner
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• 2. Optimal operation of Marathon runner, J=T
Optimal operation - Runner
Self-optimizing control: Marathon (40 km)
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Solution 1 Marathon: Optimizing control
• Even getting a reasonable model requires > 10 PhD’s … and the model has to be fitted to each individual….
• Clearly impractical!
Optimal operation - Runner
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Solution 2 Marathon – Feedback(Self-optimizing control)
– What should we control?
Optimal operation - Runner
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• 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
Optimal operation - Runner
Self-optimizing control: Marathon (40 km)
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Conclusion Marathon runner
c = heart rate
select one measurement
• Simple and robust implementation• Disturbances are indirectly handled by keeping a constant heart rate• May have infrequent adjustment of setpoint (heart rate)
Optimal operation - Runner
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Example: Cake Baking
• Objective: Nice tasting cake with good texture
u1 = Heat input
u2 = Final time
d1 = oven specifications
d2 = oven door opening
d3 = ambient temperature
d4 = initial temperature
y1 = oven temperature
y2 = cake temperature
y3 = cake color
Measurements Disturbances
Degrees of Freedom
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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)
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More on further examples
• 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 spendingOptimal 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