OPTBASED CONTROL DESIGN

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OPT-BASED CONTROL DESIGN
When I complete this chapter, I want to be able
to do the following.

• Select appropriate optimization technology
• Apply methods for evaluating the possibility of
  achieving specified performance
• - Controllability
• - Process Performance
• Design a control strategy systematically

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OPT-BASED CONTROL DESIGN
We are here, and making progress all the time!

• Defining control objectives
• Controllability Observability
• Interaction Operating window
• The Relative Gain
• Multiloop Tuning
• Performance and the RDG
• SVD and Process directionality

• Robustness
• Integrity
• Control for profit
• Optimization-based design methods
• Process design
• - Series and self-regulation
• - Zeros (good/bad/ugly)
• - Recycle systems
• - Staged systems

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OPTIMIZATION-BASED CONTROL DESIGN
The following are some potential applications of
optimization in control design. 1. Controllability
2. Control performance bounds 3. Operating
window 4. Control for profit 5. Controller
tuning 6. Controller structure
Before we begin looking at applications, lets
review characteristics of easy difficult
optimization problems
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OPTIMIZATION-BASED CONTROL DESIGN
We desire to formulate the design problem (or
sub-problem) as a mathematical programming
problem. What are good/bad features of such a
problem?
Objective function Equality constraints Inequali
ty constraints Variable Bounds
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OPTIMIZATION-BASED CONTROL DESIGN
Convexity and the objective function. A function
of x (a vector) is convex if the following is
true.
For points x1 and x2 and 0 ? ? ? 1.
Is this function convex (over the region in the
figure)?
f(x)
x
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OPTIMIZATION-BASED CONTROL DESIGN
Convex region A region is convex if all points
on a straight line connecting any two points
within the region are also in the region.
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OPTIMIZATION-BASED CONTROL DESIGN
What can be said about the minimization of a
convex objective function over a convex region?
CONVEX PROGRAMMING The minimization of a convex
function over a convex region yields the
following condition A local minimum is a global
minimum!
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OPTIMIZATION-BASED CONTROL DESIGN
What classes of problems have this nice property?
Linear programming
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OPTIMIZATION-BASED CONTROL DESIGN
What classes of problems have this nice property?
Quadratic programming
The A matrix must have positive eigenvalues
Linear constraints
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OPTIMIZATION-BASED CONTROL DESIGN
What classes of problems have this nasty
properties?
Non-convex with multiple local optima
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OPTIMIZATION-BASED CONTROL DESIGN
What classes of problems have this nasty
properties?
Optimization involving many integer variables
Optimization with integer variables requires the
solution of many (perhaps very many) optimization
problems
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OPTIMIZATION-BASED CONTROL DESIGN

• Conclusions based on general properties of
  optimization. We seek problems with the
  following features.
• Convex objective function
• Convex region for the variables
• No integer variables

Difficult problem! Must chose formulations
carefully!

• Reality
• Important sub-problem - e.g., optimal tuning -
  has a non-convex objective function.
• Structure (e.g., loop pairing) involves integer
  decisions

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OPTIMIZATION-BASED CONTROL DESIGN
Lets survey some potential applications of
optimization in control design. 1. Controllability
2. Control performance bounds a. Linear b. Non-
linear c. Causal (feedback) control d. Linear
with fixed feedback 3. Operating
window 4. Control for profit 5. Controller
tuning 6. Controller structure a. Importance of
scenario definition b. MI - Linear programming
c. MI - NLP
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OPTIMIZATION-BASED CONTROL DESIGN
1. Controllability Can we determine the
controllability using optimization approaches?
Pointwise State Controllable - A system is
controllable if it is possible to adjust the
manipulated variables u(t) so that the system
will be forced from an arbitrary state x(t0) to
x(t1) in a finite time.
Functional Controllable - A system is
controllable if it is possible to adjust the
manipulated variables u(t) so that the system
will follow a (smooth) defined path from y(t0)
to y(t1) in a finite time.
Functional Controllable at Steady-state - A
system is controllable if it is possible to
adjust the manipulated variables uss so that the
system will achieve yss in a finite time.
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OPTIMIZATION-BASED CONTROL DESIGN
1. Controllability Can we determine the
controllability using optimization approaches?

• y is determined by the performance required
• disturbance d depends on the performance required
• System is controllable is optimal f 0
• deadtimes are easily included
• This is a convex quadratic programming problem

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OPTIMIZATION-BASED CONTROL DESIGN
2. Control Performance Bound Can we quickly
screen a process to determine if specified
control performance is achievable? (a) Linear
method - linear program

• The performance can be achieved if f 0
• No specific control structure is assumed
• The formulation allows non-causal control
• This is a lower bound on f (better than or equal
  to best performance)

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OPTIMIZATION-BASED CONTROL DESIGN
2. Control Performance Bound Can we quickly
screen a process to determine if specified
control performance is achievable? (b)
Non-linear method NLP

• The performance can be achieved if f 0
• No specific control structure is assumed
• The formulation allows non-causal control
• This is a lower bound on f (better than or equal
  to best performance)

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OPTIMIZATION-BASED CONTROL DESIGN
2. Control Performance Bound Can we quickly
screen a process to determine if specified
control performance is achievable? (c)
Non-linear method causal - NLP

• The performance can be achieved if f 0
• No specific control structure is assumed
• No guarantee that any fixed control law will
  achieve this performance
• This is a lower bound on f for feedback control

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OPTIMIZATION-BASED CONTROL DESIGN
2. Control Performance Bound Can we quickly
screen a process to determine if specified
control performance is achievable? (d) Linear
method - enforcing feedback

• The performance can be achieved if f 0
• A causal feedback controller (Linear Quadratic,
  K) is determined achieve this performance
• This is a classical optimal controller

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OPTIMIZATION-BASED CONTROL DESIGN
2. Control Performance Bound Can we quickly
screen a process to determine if specified
control performance is achievable? (d) Linear
method -Q-parameterization

• The performance can be achieved if f 0
• A causal feedback controller (Could be very high
  order) is determined achieve this performance
• This formulation allows more flexible performance
  specs on y and u as well as multiple disturbances

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OPTIMIZATION-BASED CONTROL DESIGN
3. Operating window We can determine the maximum
disturbance or set point change that can be
achieved in steady-state.

• The constraints could be violated during
  transient operation
• No specific control structure is assumed
• This is an upper bound on the window size for a
  specific control design

? the set point(s) or disturbance(s) changes
from nominal
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OPTIMIZATION-BASED CONTROL DESIGN
3. Operating window We can determine the maximum
disturbance or set point change that can be
achieved in steady-state.

• The constraints could be violated during
  transient operation
• The c( ) constraints represent the s-s effects
  of controllers
• Because of u-saturation, this can be a difficult
  problem

? the set point(s) or disturbance(s) changes
from nominal
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OPTIMIZATION-BASED CONTROL DESIGN
4. Control for profit Select design with
uncertain plant
Open-loop Backoff design, inputs set so that no
output violated
Manipulated variables are constant for all
scenarios
Objective function 13,825.48 /h
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OPTIMIZATION-BASED CONTROL DESIGN
4. Control for profit Select design with
uncertain plant
Feedback loop pairing, controllers have integral
control
Controlled variables are constant for all
scenarios
Objective function 16,996.73 /h ( 19)
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OPTIMIZATION-BASED CONTROL DESIGN
4. Control for profit Select design with
uncertain plant
Feedback loop pairing, controllers have integral
control variable structure
Controlled manipulated variables are depend on
scenario
Objective function 16,289.94 /h (15.2)
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OPTIMIZATION-BASED CONTROL DESIGN
5. Controller tuning Tune a control design with
the structure and algorithms previously selected.
Recall that this can be a complex problem!
Challenge most NL optimization will find one of
these, not necessarily the global optimum!
From Vlachos, Williams, and Gomm (IEE
Proc.-Control Theory Appl., 146, 58-64 (1999)
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OPTIMIZATION-BASED CONTROL DESIGN
6. Control structure Determine the best
interconnection of controlled and manipulated
variables.
a) Problem definition We have measures for many
facets of control performance. What is missing
is the transient behavior in the time domain.

• We cannot design in a vacuum we must have the
  scenarios, which are based on the control design
  form.
• CVs selected, range of set points
• MVs selected
• Disturbances known
• Limitations of equipment known
• Uncertainty estimate available
• Relative costs known

Requires an estimate of the dynamics needed for
transient analysis
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OPTIMIZATION-BASED CONTROL DESIGN
6. Control structure Determine the best
interconnection of controlled and manipulated
variables. (b) a linear programming approach

• Observations
• If the loop pairing has been decided, all
  variables are continuous
• If the tuning has been decided,
• If no constraints occur (especially saturation),
• The model is certain

The solution can be formulated as a linear
program - easily solved!
Many discrete tuning choices can be defined using
integer variables. The problem can be solved a
an MILP with each node shown on the next page.
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6. Control structure (b) a linear programming
approach
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OPTIMIZATION-BASED CONTROL DESIGN
6. Control structure Determine the best
interconnection of controlled and manipulated
variables. (b) a linear programming approach

• Advantages
• Easily solved convex sub-problems

• Disadvantages
• Many integer variables
• Loop pairing introduces more integer
• Cannot have saturation
• Applications limited to positive RGA

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OPTIMIZATION-BASED CONTROL DESIGN
6. Control structure Determine the best
interconnection of controlled and manipulated
variables. (c) sequential approach

• Observations
• Tuning is important and multiloop tuning can be
  very different from single-loop
• Noise is important (in some cases)
• Uncertainty prevents overly aggressive control
• Reasonable equipment capacities can lead to
  saturation
• In some cases, short-cut metrics substantially
  reduce the number of designs for detailed
  consideration.
• - RGA, RDG, etc.

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OPTIMIZATION-BASED CONTROL DESIGN
6. Control structure Determine the best
interconnection of controlled and manipulated
variables. (c) sequential approach

• General approach, integer one structure,
• solve optimal tuning at each node
• Enforce metrics as appropriate

Disadvantage Optimal tuning is non-linear and
non-convex current workaround, grid search first.
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6. Control structure Determine the best
interconnection of controlled and manipulated
variables. (c) sequential approach
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6. Control structure Determine the best
interconnection of controlled and manipulated
variables. (c) sequential approach
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OPTIMIZATION-BASED CONTROL DESIGN
6. Control structure Determine the best
interconnection of controlled and manipulated
variables. (c) sequential approach

• General approach, integer one structure,
• solve optimal tuning at each node
• Enforce metrics as appropriate

Lets apply this to a very small problem and seek
if can uncover some tricks.
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Realistic Scenarios
Directionality is critical
Saturation affects dynamics
D
MV
GP
GC
SP
CV
GP ?
-
Noise
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Full Transient Response with the Best Tuning
Damage equipment or Large capacity
Kc ? TI ?
D
MV
GP
GC
SP
CV
-
Noise
Overshot Oscillation Constraint
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Tractable Formulation
D
GP2
MV
GP
GC
CV
SP
-
Noise

• Linear dynamic plants
• Robustness
• One nominal plant
• One mismatch plant
• One controller tuned for both plants

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Tractable Formulation
D
MV
GP
GC
CV
SP
-
Noise

• Equation oriented formulation (not
  shooting)
• PI controllers

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Region Elimination Using Simulation
High or low mean aggressive or sluggish tuning
for Kc and TI
Do not know the controller sign for pairing with
negative RGA
Determine the optimal tuning within the selected
region using convex NLP solver
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Tractable Formulation
D
MV
GP
GC
CV
SP
-
Noise

• Different weightings represent different
  importance among controlled variables

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Tractable Formulation
D
MV
GP
GC
CV
SP
-
Noise

• Include manipulated variable saturation
• Need complementary constraints (nonconvex)
• Interior point method solver IPOPT

A. Watcher and L. T. Biegler (2002)
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Problem formulation
Min Q?E R??U s.t. Nominal plant
model Mismatch plant model PI controller
equation Saturation constraint Constraint
defining pairing (Integrity constraint e.g.
Positive RGA)
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Fluidized Catalytic Cracker(Partial Combustion
Mode)
Must be tightly controlled
Tris
Huge fluidized vessel
Tube reactor
Fcat
Trgn
Keep in safe range
Fair
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Fluidized Catalytic Cracker(Partial Combustion
Mode)
Plant model
Tris
Trgn
Key variable
Fcat
RGA
Fair
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Inverse Response of Tris-Fcat

• Initially, more hot catalyst is mixed with the
  feed oil at the riser inlet
• Ultimately, the regenerator temperature
  decreases, lowering the catalyst temperature

Partial combustion mode
Occurs when air deficient
Higher heat of reaction
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Positive RGA Pairing
Very good performance
Huge controller gain KCKP ? 106
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Positive RGA Pairing Mismatch model
?0.5F
Big controller gain KCKP ? 100
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Positive RGA Pairing Mismatch model Noise
600 extra capacity
Good performance
KCKP ? 1
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Cost for Capacity
Additional cost for maintain huge anti-surge
recycle at normal operation
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Positive RGA Pairing Mismatch model Noise
Constraint
Saturation
40 extra capacity
Slow dynamics
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Negative RGA Pairing Mismatch model Noise
Constraint
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Information for Trading off by Engineers
Integrity RGA
Dynamic performance
VS.
Industrial standard pairing (Shinnar, 1996)
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OPTIMIZATION-BASED CONTROL DESIGN
6. Control structure Determine the best
interconnection of controlled and manipulated
variables. (c) sequential approach

• Realistic scenario formulation for optimum
  controller tuning
• Trade-off between multiple criteria
• Able to select non-obvious designs
• Computationally intensive!!

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OPTIMIZATION-BASED CONTROL DESIGN
Lets survey some potential applications of
optimization in control design. 1. Controllability
2. Control performance bounds a. Linear b. Non-
linear c. Causal (feedback) control d. Linear
with fixed feedback 3. Operating
window 4. Control for profit 5. Controller
tuning 6. Controller structure a. Importance of
scenario definition b. MI - Linear programming
c. MI - NLP

• Summary
• Many useful applications of optimization in
  control design
• Some easily implemented
• Some require further research
• Dont be afraid of some calculations!

Proper problem definition critical