Multiple Input, Multiple Output II: Model Predictive Control By Peter Woolf ([email protected]) University of Michigan Michigan Chemical Process Dynamics.

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

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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