1 Plant Wide Brazil

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PLANTWIDE CONTROL

Sigurd Skogestad

Department of Chemical Engineering

Norwegian University of Science and Tecnology (NTNU)

Trondheim, Norway

01 April 2006


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Intro, DOFs, control objectives, self-op. control

What to control, Production rate,

stabilizing control, distillation example

Supervisory control. HDA example


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Contents

Overview of plantwide control

Selection of primary controlled variables based on economic : The llinkbetween 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 wecontrol")

- 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


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Trondheim, Norway


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Trondheim

Oslo

UK

NORWAY

DENMARK

GERMANY

North Sea

SWEDEN

Arctic circle


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NTNU,

Trondheim


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Main message

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


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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 ? (primary CVs) (self-optimizing control)

Step 4: Where set production rate?

II Bottom Up

Step 5: Regulatory control: What more to control (secondary CVs) ? Step 6: Supervisory control

Step 7: Real-time optimization

Case studies


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Idealized view of control

(Ph.D. control)


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Practice: Tennessee Eastman challenge

problem (Downs, 1991)

(PID control)


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How we design a control system for a

complete chemical plant?

Where do we start?

What should we control? and why?

etc.

etc.


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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 aprimitive, 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.


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Control structure design

Notthe tuning and behavior of each control loop,

But rather the control philosophy of the overall plant with emphasis on

thestructural 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


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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


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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 ECC03 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 simplification: Hierarchical structure

Need to define

objectives and identify

main issues for each

layer

PID

RTO

MPC


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Regulatory control (seconds)

Purpose: Stabilize the plant by controlling selected secondary

variables (y2) 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: Kc=1, XI=1 min (default)

Even wrong sign of gain Kc .


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Regulatory control...

Trend: Can do better! Carefully go through plant and retune important

loops using standardized tuning procedure

Exists many tuning rules, including Skogestad (SIMC) rules:

Kc = (1/k) (X1/ [Xc +U]) XI = min (X1, 4[Xc + U]), Typical: Xc=U

Probably the best simple PID tuning rules in the world Carlsberg

Outstanding structural issue: What loops to close, that is, whichvariables (y2) to control?


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Supervisory control (minutes)

Purpose: Keep primary controlled variables (c=y1) at desired values,

using as degrees of freedom the setpoints y2s 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


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Supervisory control...

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=y1 to control?


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Local optimization (hour)

Purpose: Minimize cost function J and:

Identify active constraints

Recompute optimal setpoints y1s 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


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Objectives of layers: MVs and CVs

cs = y1s

MPC

PID

y2s

RTO

u (valves)

CV=y1; MV=y2s

CV=y2; MV=u

Min J (economics);

MV=y1s


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Stepwise procedure plantwide control

I. TOP-DOWN

Step 1. DEGREES OF FREEDOM

Step 2. OPERATIONAL OBJECTIVES

Step 3. WHAT TO CONTROL? (primary CVs c=y1)

Step 4. PRODUCTION RATE

II. BOTTOM-UP (structure control system):Step 5. REGULATORY CONTROL LAYER (PID)

Stabilization

What more to control? (secondary CVs y2)

Step 6. SUPERVISORY CONTROL LAYER (MPC)

Decentralization

Step 7. OPTIMIZATION LAYER (RTO)

Can we do without it?


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Outline

About Trondheim and myself



I Top Down Step 1: Degrees of freedom


Step 3: What to control ? (self-optimzing control)


II Bottom Up

Step 5: Regulatory control: What more to control ? Step 6: Supervisory control


Case studies


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Step 1. Degrees of freedom (DOFs) for

operation (Nvalves):

To find all operational (dynamic) degrees of freedom

Count valves! (Nvalves)

Valves also includes adjustable compressor power, etc.

Anything we can manipulate!


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Steady-state degrees of freedom

Cost J depends normally only on steady-state DOFs

Three methods to obtain steady-state degrees of freedom (Nss):

1. Equation-counting

Nss = no. of variables no. of equations/specifications

Very difficult in practice (not covered here)

2. Valve-counting (easier!) Nss = Nvalves N0ss Nspecs N0ss = variables with no steady-state effect

3. Typical number for some units (useful for checking!)


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Steady-state degrees of freedom (Nss

):

2. Valve-counting

Nvalves = no. of dynamic (control) DOFs (valves)

Nss = Nvalves N0ss Nspecs : no. of steady-state DOFs

N0ss = N0y + N0,valves : no. of variables with no steady-state effect

N0,valves : no. purely dynamic control DOFs

N0y : no. controlled variables (liquid levels) with no steady-state effect

Nspecs: No of equality specifications (e.g., given pressure)


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Nvalves = 6 , N0y = 2 , Nspecs = 2, NSS = 6 -2 -2 = 2

Distillation column with given feed and pressure


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Heat-integrated distillation process

Nvalves = 11 w/feed , N0 = 4 levels , Nss = 11 4 =


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Nvalves = 11 (w/feed , N0 = 4 (levels , Nss = 11 4 = 7


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Heat exchanger with bypasses

CW

Nvalves = 3, N0valves = 2 (of 3), Nss = 3 2 = 1


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CW

Nvalves = 3, N0valves = 2 (of 3), Nss = 3 2 = 1


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Steady-state degrees of freedom (Nss):

3. Typical number for some process units

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 foradiabatic 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


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CW

Typical number heat exchanger Nss = 1


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Typical number,

Nss= 0 (distillation) + 2*1 (heat exchangers) = 2

Distillation column with given feed and pressure


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Typical number, Nss = 1 (feed) + 2*0 (columns) + 2*1

(column pressures) + 1 (sidestream) + 3 (hex) = 7


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HDA process

Mixer FEHE Furnace PFRQuench

Separator

Compressor

Cooler

StabilizerBenzeneColumn

TolueneColumn

H2 + CH4

Toluene

Toluene Benzene CH4

Diphenyl

Purge (H2 + CH4)


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HDA process: steady-state degrees of freedom

1

2

3

8

7

4

6

5

9

10

11

12

13

14

Conclusion: 14

steady-stateDOFsAssume given column pressures

feed:1.2

hex: 3, 4, 6

splitter 5, 7

compressor: 8

distillation: rest


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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


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Outline






Step 3: What to control ? (self-optimizing control)


II Bottom Up



Case studies


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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) < 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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Example: Paper machine drying section

~10m

water recycle

water

up to 30 m/s (100 km/h)

(~20 seconds)


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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 < dew point 10C (active not always controlled)

T along dryer (especially inlet) < bound (active?)

Remaining DOFs: minimize cost


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Optimal operation

1. Given feed

Amount 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)

Control: Operate at optimaltrade-off (not obvious how to do

and what to control)


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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


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Outline








II Bottom Up



Case studies


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Step 3. What should we control (c)?

(primary controlled variables y1=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


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Implementation of optimal operation

Optimal operation for given d*:

minu J(u,x,d)

subject to:Model equations: f(u,x,d) = 0

Operational constraints: g(u,x,d) < 0

uopt(d*

)

Problem: Usally cannot keep uopt constant because disturbances d change

How should be adjust the degrees of freedom (u)?


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Problem: Too complicated

(requires detailed model and

description of uncertainty)

Implementation of optimal operation (Cannot keep u0opt constant)

Obvious solution:

Optimizing control

Estimate d from measurements

and recompute uopt(d)


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In practice: Hierarchical

decomposition with separate layers

What should we control?


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Self-optimizing control:

When constant setpoints is OK

Constant setpoint


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Unconstrained variables:

Self-optimizing control

Self-optimizing control:

Constant setpoints cs givenear-optimal operation

(= acceptable loss L for expected

disturbances d and

implementation errors n)

Acceptable loss )self-optimizing control


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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.


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What should we control? Sprinter

Optimal operation of Sprinter (100 m), J=T

One input: power/speed

Active constraint control: Maximum speed ( no thinking required)


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What should we control? Marathon

Optimal operation of Marathon runner, J=T

No active constraints

Any self-optimizing variable c (to control at constantsetpoint)?


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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 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 spending

Optimal values obtained by benchmarking

Investment (portofolio management). J = profit. c = Fraction ofinvestment in shares (50%)

Biological systems: Self-optimizing controlled variables c have been found by naturalselection

Need to do reverse engineering :

Find the controlled variables used in nature

From this possibly identify what overall objective J the biological system hasbeen attempting to optimize

BREAK


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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?


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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)


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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Solution: Optimal operation distillation

Cost to be minimized


N=2 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:

1. Nactive

=2: Two active constraints (for example, xD, impurity

= max. xB,

impurity = max, TWO-POINT COMPOSITION CONTROL)

2. Nactive=1: One constraint active (1 remaining DOF)

3. Nactive=0: No constraints active (2 remaining DOFs)

Can happen if no purity specifications(e.g. byproducts or recycle)

Problem : WHAT SHOULD

WE CONTROL (TO SATISFY

THE UNCONSTRAINED DOFs)?

Solution:

Often compositions but

not always!


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Unconstrained variables:


Intuition: Dominant variables (Shinnar)

Is there any systematic procedure?


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Systematic procedure

Systematic: Minimize cost J(u,d*) w.r.t.

DOFs u.1. Control active constraints (constant setpoint

is optimal)

2. Remaining unconstrainedDOFs: 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

c = ? (economics)

y2 = ? (stabilization)


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The difficult unconstrainedvariables

Cost J

Selected controlled variable

(remaining unconstrained)

ccoptopt

JJoptopt

c


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Example: Tennessee Eastman plant

J

c = Purge rate

Nominal optimum setpoint is infeasible with disturbance 2

Oopss..

bends backwards

Conclusion: Do not use purge rate as controlled variable


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Optimal operation

Cost J

Controlled variable cccoptopt

JJoptopt

Two problems:

1. Optimum moves because ofdisturbances d: copt(d)

d

LOSS


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Optimal operation

Cost J

Controlled variable cccoptopt

JJoptopt

Two problems:

1. Optimum moves because of disturbances d: copt(d)

2. Implementation error, c = copt + n

d

n

LOSS


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Effect of implementation error on cost (problem 2)

BADGoodGood


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Example sharp optimum. High-purity distillation :

c = Temperature top of column

Temperature

Ttop

Water (L) - acetic acid (H)

Max 100 ppm acetic acid

100 C: 100% water

100.01C: 100 ppm99.99 C: Infeasible

U d d f f d


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Candidate controlled variables

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 temperaturelocations for distillation

Will show shortly: Can combine everything into the maximum gain rule:

Maximize scaled gain G = Go / span(c)

Unconstrained degrees of freedom:

span(c)

U t i d d f f d


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Optimizer

Controller that

adjusts u to keep

cm = cs

Plant

cs

cm=c+n

u

c

n

d

u

c

J

cs=copt

uopt

n


Justification for intuitively 2 and 3

Want the slope (= gain G0 from u to c) large

corresponds to flat optimum in c

Want small n


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Mathematic local analysis

(Proof of maximum gain rule)

u

cost

J

uopt


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Minimum singular value of scaled gain

Maximum gain rule (Skogestad andPostlethwaite, 1996):

Look for variables that maximize the scaled gain W(G)

(minimum singular value of the appropriately scaled

steady-state gain matrix Gfrom u to c)

W(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).


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Improved minimum singular value rule

for ill-conditioned plants

G: Scaled gain matrix (as before)

Juu

: Hessian for effect of us on cost

Problem: Juu can be difficult to obtain

Improved rule has been used successfully for distillation



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Maximum gain rule for scalar system



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Maximum gain rule in words

Select controlled variables c for which

theircontrollable range is large compared totheirsum of optimal variation and control error

controllable range = range c may reach by varying the inputs (=gain)

optimal variation: due to disturbance

control error= implementation error nspan


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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


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C. Optimal measurement combination (Alstad, 2002


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C. 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

Optimalwhen we disregard implementation error (n)


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Alstad-method continued

To handle implementation error: Use sensitive measurements, with

information about all independent variables (u and d)


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Summary unconstrained degrees of freedom:

Looking 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):


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Toy Example


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Toy Example


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Toy Example

EXAMPLE R l l t


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EXAMPLE: Recycle plant (Luyben, Yu, etc.)

1

2

3

4

5

Given feedrate F0 and

column pressure:

Dynamic DOFs: Nm = 5

Column levels: N0y = 2

Steady-state DOFs: N0 = 5 - 2 = 3


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Recycle plant: Optimal operation

mT

1 remaining unconstrained degree of freedom

C l f l l


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Control of recycle plant:

Conventional structure (Two-point: xD)

LCXC

LC

XC

LC

xB

xD

Control active constraints (Mr=max and xB=0.015) + xD


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Luyben rule

Luyben rule (to avoid snowballing):

Fix a stream in the recycle loop (F orD)


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Luyben rule: D constant

Luyben rule (to avoid snowballing):

Fix a stream in the recycle loop (F orD)

LCLC

LC

XC

A M i i l St d t t i


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A. Maximum gain rule: Steady-state gain

Luyben rule:

Not promising

economically

Conventional:

Looks good


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How did we find the gains in the Table?

1. Find nominal optimum

2. Find (unscaled) gain G0 from input to candidate outputs: ( c = G0 ( u.

In this case only a single unconstrained input (DOF). Choose at u=L

Obtain gain G0 numerically by making a small perturbation in u=L whileadjusting the other inputs such that the active constraints are constant(bottom composition fixed in this case)

3. Find the span for each candidate variable

For each disturbance di make a typical change and reoptimize to obtainthe optimal ranges (copt(di)

For each candidate output obtain (estimate) the control error (noise) n

span(c) = 7i |(copt(di)| + n

4. Obtain the scaled gain, G = G0 / span(c)

IMPORTANT!


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B. Brute force loss evaluation:

Disturbance in F0

Loss with nominally optimal setpoints for Mr, xB and c

Luyben rule:

Conventional


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B. Brute force loss evaluation:

Implementation error

Loss with nominally optimal setpoints for Mr, xB and c

Luyben rule:


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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)

Conclusion: Control of recycle plant


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Conclusion: Control of recycle plant

Active constraint

Mr = Mrmax

Active constraint

xB = xBmin

L/F constant: Easier than two-point control

Assumption: Minimize energy (V)

Self-optimizing


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Recycle systems:

Do not recommend Luybens rule offixing a flow in each recycle loop

(even to avoid snowballing)


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Summary: Self-optimizing Control

Self-optimizing control is whenacceptable operation can be achievedusingconstant set points (c

s) for the

controlled variables c (without the need tore-optimizing when disturbances occur).

c=cs


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Summary:

Procedure selection controlled variables

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 offreedom

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


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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 W(G)

MethodB: Evaluate loss L = J - Jopt

MethodC: Optimal linear measurement combination:(c = H(y where HF=0


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Outline



I Top Down

Step 1: Degrees of freedom Step 2: Operational objectives (optimal operation)

Step 3: What to control ? (self-optimzing control)


II Bottom Up

Step 5: Regulatory control: What more to control ?

Step 6: Supervisory control Step 7: Real-time optimization

Case studies


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Step 4. Where set production rate?

Very important!

Determines structure of remaining inventory (level) control system

Set production rate at (dynamic) bottleneck

Link between Top-down and Bottom-upparts


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Production rate set at inlet :

Inventory control in direction of flow


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Production rate set at outlet:

Inventory control opposite flow

d i i id


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Production rate set inside process


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Where set the production rate?

Very important decision that determines the structure of the rest of the

control system!

May also have important economic implications


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Often optimal: Set production rate at

bottleneck!

"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.

Reactor-recycle process:


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Given feedrate with production rate set at inlet



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Reactor recycle process:

Want to maximize feedrate: reachbottleneckin column

Bottleneck: max. vapor

rate in column

Reactor-recycle process with production rate set at inlet


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Reactor-recycle process with production rate set at inlet

Want to maximize feedrate: reach bottleneck in column

Bottleneck: max. vapor

rate in column

FC

Vmax

VVmax-Vs=Back-off

= Loss

Alt.1: Loss

Vs

Reactor-recycle process with increased feedrate:


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Alt.2 long loop

MAX

Reactor recycle process with increased feedrate:

Optimal: Set production rate at bottleneck

Reactor-recycle process with increased feedrate:


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Optimal: Set production rate at bottleneck

MAX

Alt.3: reconfigure



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Given feedrate with production rate set at bottleneck

F0s

Alt.3: reconfigure

(permanently)

Alt 4 M lti i bl t l (MPC)


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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!

Alt.4: Multivariable control (MPC)


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Conclusion production rate manipulator

Think carefully about where to place it!

Difficult to undo later

BREAK


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Outline



I Top Down





II Bottom Up


Step 6: Supervisory control Step 7: Real-time optimization

Case studies


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II. Bottom-up

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

Decentralized or multivariable control (MPC)?

Pairing?

Step 7. OPTIMIZATION LAYER (RTO)


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Step 5. Regulatory control layer

Purpose: Stabilize the plant using local SISO PID controllers

Enable manual operation (by operators)

Main structural issues:

What more should we control? (secondary cvs, y2)

Pairing with manipulated variables (mvs u2)

y1 = c

y2 = ?


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Regulatory loops

GKy2s u2

y2

y1

Key decision: Choice ofy2 (controlled variable)

Also important(since we almost always use single loops in the

regulatory control layer): Choice ofu2 (pairing)


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Example: Distillation

Primary controlled variable: y1 = c = xD, xB (compositions top, bottom)

BUT: Delay in measurement of x + unreliable

Regulatory control: For stabilization need control of (y2):

Liquid level condenser (MD) Liquid level reboiler (MB)

Pressure (p)

Holdup of light component in column

(temperature profile)

Unstable (Integrating) + No steady-state effect

Disturbs (destabilizes) other loops

Almost unstable (integrating)

TCTs

T-loop in bottom

Cascade control


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X

C

TC

FC

ys

y

Ls

Ts

L

T

z

X

C

distillation

With flow loop +

T-loop in top


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Degrees of freedom unchanged

No degrees of freedom lost by control of secondary (local) variables as

setpoints become y2s replace inputs u2 as new degrees of freedom

GKy2s u2

y2

y1

Original DOFNew DOF

Cascade control:


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Hierarchical control: Time scale separation

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 y2) is notmuch influenced by the presence of the upper (slow) layers (involving y1)

Reason: The frequency of the disturbance from the upper layer is well insidethe bandwidth of the lower layers

2. With the lower (faster) layer in place, the stability and performance of theupper (slower) layers do not depend much on the specific controller settingsused in the lower layers

Reason: The lower layers only effect frequencies outside the bandwidth of theupper layers

Objectives regulatory control layer


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Objectives regulatory control layer

1. Allow for manual operation2. 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 ofy2:

1. Control ofy2 stabilizes the plant

2. y2 is easy to control (favorable dynamics)

l f bili h l


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1. Control ofy2 stabilizes the plant

A. Mathematical stabilization (e.g. reactor):

Unstable mode is quickly detected (state observability) in themeasurement (y2) and is easily affected (state controllability) by theinput (u2).

Tool for selecting input/output: Pole vectors

y2: Want large elementin output pole vector: Instability easily detectedrelative to noise

u2: Want large elementin input pole vector: Small input usage requiredfor stabilization

B. Practical extended stabilization (avoid drift due to disturbance

sensitivity): Intuitive: y2 located close to important disturbance

Or rather: Controllable range fory2 is large compared to sum ofoptimal variation and control error

More exact tool: Partial control analysis


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Recall rule for selecting primary controlled variables c:

Controlled variables c for which theircontrollable range is large comparedto theirsum of optimal variation and control error

Control variables y2 for which theircontrollable range is large compared to

theirsum of optimal variation and control error

controllable range = range y2 may reach by varying the inputs

optimal variation: due to disturbances

control error= implementation error n

Restated for secondary controlled variables y2:

Want small

Want large

What should we control (y )?


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What should we control (y2)?

Rule: Maximize the scaled gain

General case: Maximize minimum singular value of scaled G

Scalar case: |Gs| = |G| / span

|G|: gain from independent variable (u2) to candidate controlled variable(y

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 (y1=c)

span (of y2) = optimal variation in y2 + control error for y2 Note optimal variation: This is often the same as the optimal variation used for selecting primarycontrolled variables (c).

Exception: If we at the fast regulatory time scale have some yet unused slower inputs (u1)which are constant then we may want find a more suitable optimal variation for the fast timescale.


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Minimize state drift by controlling y2

Problem in some cases: optimal variation for y2 depends on overallcontrol objectives which may change

Therefore: May want to decouple tasks of stabilization (y2) andoptimal operation (y1)

One way of achieving this: Choose y2 such that state drift dw/dd isminimized

w = Wx weighted average of all states

d disturbances

Some tools developed:

Optimal measurement combination y2=Hy that minimizes state drift(Hori) see Skogestad and Postlethwaite (Wiley, 2005) p. 418

Distillation column application: Control average temperature column

2 i l ( ll bili )


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2. y2 is easy to control (controllability)

1. Statics: Want large gain (from u2 to y2)

2. Main rule:y2 is easy to measure and located

close to available manipulated variable u2(pairing)

3. Dynamics: Want small effective delay (from u2 to y2) effective delay includes

inverse response (RHP-zeros)

+ high-order lags


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Rules for selecting u2 (to be paired with y2)

1. Avoid using variable u2 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 u2 to y2, should be good.


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Effective delay and tunings

= effective delay

PI-tunings from SIMC rule

Use half rule to obtain first-order model

Effective delay = True delay + inverse response time constant + halfof

second time constant + all smaller time constants

Time constant 1 = original time constant + halfof second time constant

NOTE: The first (largest) time constant is NOT important for controllability!


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Summary: Rules for selecting y2 (and u2)

1. y2 should be easy to measure

2. Control of y2 stabilizes the plant

3. y2 should have good controllability, that is, favorable dynamics for

control

4. y2 should be located close to a manipulated input (u2) (follows from

rule 3)

5. The (scaled) gain from u2 to y2 should be large

6. The effective delay from u2 to y2 should be small

7. Avoid using inputs u2 that may saturate (should generally avoidsaturation in lower layers)

Example regulatory control: Distillation

( t lid )


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(see separate slides)

5 dynamic DOFs (L,V,D,B,VT)

Overall objective:

Control compositions (xD and xB)

Obvious stabilizing loops:

1. Condenser level (M1)

2. Reboiler level (M2)

3. Pressure

E.A. Wolff and S. Skogestad, ``Temperature cascade control of distillation columns'', Ind.Eng.Chem.Res., 35, 475-484, 1996.

Selecting measurements and inputs for


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Selecting measurements and inputs for

stabilization: Pole vectors

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


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Example: Tennessee Eastman challenge


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Example: Tennessee Eastman challenge

problem


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2


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C t l fi ti l t


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Control configuration elements

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


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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 manipulatedinput computed by another decentralized controller.

Why simplified configurations?


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Why simplified configurations?

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

Cascade control


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(conventional; with extra measurement)

The reference r2 is an output from another controller

General case (parallel cascade)

Special common case (series cascade)

Series cascade


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Series cascade

1. Disturbances arising within the secondary loop (before y2) are corrected by thesecondary controller before they can influence the primary variable y1

2. Phase lag existing in the secondary part of the process (G2) is reduced by the secondaryloop. This improves the speed of response of the primary loop.

3. Gain variations in G2 are overcome within its own loop.

Thus, use cascade control (with an extra secondary measurement y2) when:

The disturbance d2 is significant and G1 has an effective delay

The plant G2 is uncertain (varies) or n onlinear

Design:

First design K2 (fast loop) to deal with d2 Then design K 1 to deal with d1

Tuning cascade


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Tuning cascade

Use SIMC tuning rules

K2 is designed based on G2 (which has effective delay U2)

then y 2 = T2 r2 + S2 d2 where S2 0 and T2 1 e-(U2+Xc2)s

T2: gain = 1 and effective delay = U2+Xc2

SIMC-rule: Xc2 U2

Time scale separation: Xc2 Xc1/5 (approximately)

K1 is designed based on G1T2 same as G1but with an additional delay U2+Xc2

y2 = T2 r2 + S2d2

Exercise: Tuning cascade


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Exercise: Tuning cascade

1. (without cascade, i.e. no feedback from y2).

Design a controller based on G1G

2. (with cascade)

Design K2

and then K1

Tuning cascade control


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Extra inputs


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Extra inputs

Exercise: Explain how valve position control fits into this

framework. As en example consider a heat exchanger with bypass

Exercise


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Exercise

Exercise:

(a) In what order would you tune the controllers?

(b) Give a practical example of a process that fits into this block diagram

Partial control


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Cascade control: y2 not important in itself, and setpoint (r2) is

available for control of y1

Decentralized control(using sequential design): y2 important in itself

Partial control analysis


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y1 = P1 u1 + Pr1 (y2s-n2) + Pd1 dP1 = G11 G12 G22-1 G21Pd1 = Gd1 G12 G22

-1 Gd2 - WANT SMALL

Pr1 = G12 G22-1

Primary controlled

variable y1 = c(supervisory control layer)

Local control of y2using u2(regulatory control layer)

Setpoint y2s : new DOF

for supervisory control

Partial control: Distillation


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y1 = P1 u1 + Pr1 (y2s-n2) + Pd1 d

P1 = G11 G12 G22-1 G21

Pd1 = Gd1 G12 G22-1

Gd2 - WANT SMALL

Pr1 = G12 G22-1

Supervisory control:

Primary controlledvariables y1 = c = (xD xB)

T

Regulatory control:

Control of y2=T

using u2 = L (original DOF)

Setpoint y2s = Ts : new DOF

for supervisory control

u1 = V

Limitations of partial control?


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Cascade control: Closing of secondary loops does not by itself impose

new problems

Theorem 10.2 (SP, 2005). The partially controlled system [P1 Pr1]

from [u1 r2] to y1

has no new RHP-zeros that are not present in the open-loop system [G11 G12]from [u1 u2] to y1

provided

r2 is available for control of y1

K2 has no RHP-zeros

Decentralized control (sequential design): Can introduce limitations.

Avoid pairing on negative RGA for u2/y2 otherwise Pu likely has a RHP-

zero

BREAK

Outline


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Outline



I Top Down



Step 3: What to control ? (primary CVs) (self-optimizing control)


II Bottom Up

Step 5: Regulatory control: What more to control (secondary CVs) ?

Step 6: Supervisory control


Case studies

Step 6 Supervisory control layer


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Step 6. Supervisory control layer

Purpose: Keep primary controlled outputs c=y1 at optimal setpoints cs

Degrees of freedom: Setpoints y2s in reg.control layer

Main structural issue: Decentralized or multivariable?

Decentralized control


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(single-loop controllers)

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

Multivariable control


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(with explicit constraint handling = MPC)

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

Outline


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Outline



I Top Down





II Bottom Up




Case studies

Step 7. Optimization layer (RTO)


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Step 7. Optimization layer (RTO)

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 self-

optimizing)

RTO not needed when

Can easily identify change in active constraints (operating region)

For each operating region there exists self-optimizing var

Outline


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Outline



I Top Down





II Bottom Up




Conclusion / References

Summary: Main steps


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Summary: Main steps

1. What should we control (y1=c=z)?

Must define optimal operation!

2. Where should we set the production rate?

At bottleneck

3. What more should we control (y2)?

Variables that stabilize the plant

4. Control of primary variables

Decentralized?

Multivariable (MPC)?

Conclusion


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Conclusion

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).

More examples and case studies


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p

HDA process

Cooling cycle

Distillation (C3-splitter)

Blending

References


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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 andControl, 21, 209-240.

Larsson, T., K. Hestetun, E. Hovland and S. Skogestad (2001), Self-optimizing control of alarge-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 controlstructure.J. Proc. Control10, 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, ComputersandChemical Engineering, 28, 219-234. (Special issue fromESCAPE12 Symposium, Haag,May 2002).

Documents

1 Plant Wide Brazil