multiple input, multiple output ii: model predictive control by peter woolf ([email protected])...
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Multiple Input, Multiple Output II:Model Predictive Control
By Peter Woolf ([email protected])University of Michigan
Michigan Chemical Process Dynamics and Controls Open Textbook
version 1.0
Creative commons
P&ID
€
dV
dt= Fin − Fout
dT
dt=FinVTf −T( ) +
Q
ρc pV
Model
Goal: Control both LC1 and TC1 using Q and v1.
v1 strongly influences LC1, but also influences TC1Q strongly influences TC1, but depends on LC1 (volume)
Possible solutions:
(1) Decouple system and connect LC1 to v1 and TC1 to Q--> 2 PID controllers
Problem: controllers may fight as their objectives are not compatible.
(2) Develop a more sophisticated MIMO controller
MPC Philosophy:
Past: Use past data to create an accurate modelFuture: Use the model to predict the impact of future control eventsPresent: do the control action that is expected to yield the best long term outcome
Image from http://controls.engin.umich.edu/wiki/index.php/MPC
Chess Example for MPCWhite: operatorBlack: systemOperator’s move
Images and example from http://www.chessproblems.com/
Images and example from http://www.chessproblems.com/
Path 1:System takes operator’s queen w/ rook
Path 2:System takes operator’s queen w/ bishop
Images and example from http://www.chessproblems.com/
Path 1: System takes operator’s queen w/ rook
Operator moves knight
System moves rook to protect pawn
Operator moves knight and checkmate
Images and example from http://www.chessproblems.com/
Path 2:System takes operator’s queen w/ bishop
Operator moves bishop
System moves bishop to attack operator
Operator moves bishop and checkmate
Images and example from http://www.chessproblems.com/
Operator wins in both cases by sacrificing queen
Path 2
Path 1
Images and example from http://www.chessproblems.com/
Tree view: each column represents a move
Observations:• Sometimes a short term sacrifice yields a long term benefit (sacrifice queen to win the game)
• Avoid “win the battle, loose the war”
A more complex example
There are many paths, but some are shorter than others.
A controls exampleGoal: set yield to 2.5 g/L & minimize energy use
V1(open), v2(closed), v3(closed), v4(open)Yield=1.5 g, energy=250 W
Path 1:Increase yield and decrease energy
V1(open), v2(closed), v3(open), v4(open)Yield=1.5 g, energy=300 W
Path 2:maintain yield and increase energy
V1(open), v2(closed), v3(open), v4(closed)Yield=2.5 g, energy=50 W
Low energy, high productivity steady state
V1(open), v2(open), v3(open), v4(open)Yield=2.3 g, energy=250 W
V1(open), v2(open), v3(closed), v4(open)Yield=2.5 g, energy=230 W
higher energy, lower productivity steady state
Simple controllers will optimize based on “the next move” alone, thus will not go through less desirable states to get a larger return.
Can we learn from chess how to control our system better?
Need: (1)Rules and constraints of the game(2)Objective(3)Ability to “look ahead” to see the next best action
Model Predictive Control
MPC procedure
t=1 open open t=2 closed open t=3 open open t=4 closed closed
How do we search possible future actions?
Search by optimization of some objective function
Min[ Sum[ (predicted-desired)^2] ]
Can add constraints such as:(1)Heaters and valves with finite, positive
range(2)Actuators with finite states (open/closed
or high/medium/low)(3)Cost, energy, or expense limitsMuch of this can be done with Excel’s Solver function..
see class20.example.xls
Notes for Excel Solver and Integer Optimization
Key: Set up problem such that binary or integer values have a continuous interpretation
=IF($A$1=1,10,0) No -- solver will try values of 1.1 in an intermediate calculation and not find an appropriate value=IF($A$1>=1,10,0)
=$A$1*10Yes -- solver will try values of 1.1 to establish a gradient, and then constrain to binary or integer at the end
Alternate Models for MPCNeural Networks: Flexible
empirical model to fit time varying data to a model
Advantages: Model learned directly from data
Disadvantages: Only accurate in the domain in which the network was trained.
Figure from http://controls.engin.umich.edu/wiki/index.php/NN
Downsides of MPC• As implemented, the controller will
anticipate set point changes, which may not be desirable
• A grossly inaccurate model will yield poor control decisions (although the method is surprisingly robust)
• Predictions can be computationally demanding so requires fast computers and fast code to do in real time
Advantages of MPC• Incorporates in knowledge of the
system in making decisions• Realistic implementation of known
constraints• Anticipates longer term consequences
of controller actions• Simplifies or in some cases eliminates
controller design, instead replacing it with system modeling
Take Home Messages
• In some cases, simpler control architectures lead to short term gains and long term losses
• MPC is an increasingly popular and powerful method for control of complex chemical processes
• MPC models can be ODEs, neural networks, or other kinds of models