Multiple Input, Multiple Output II: Model Predictive Control By Peter Woolf University of Michigan Michigan Chemical Process Dynamics and Controls Open Textbook version 1.0 Creative commons

P&IDModel Goal: Control both LC1 and TC1 using Q and v1. v1 strongly influences LC1, but also influences TC1 Q 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 model Future: Use the model to predict the impact of future control events Present: do the control action that is expected to yield the best long term outcome Image from

Chess Example for MPC White: operator Black: system Operator’s move Images and example from

Path 1: System takes operator’s queen w/ rook Path 2: System takes operator’s queen w/ bishop

Images and example from 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 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 Operator wins in both cases by sacrificing queen Path 2 Path 1

Images and example from 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 example Goal: 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

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 limits Much 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*10 Yes -- solver will try values of 1.1 to establish a gradient, and then constrain to binary or integer at the end

Alternate Models for MPC Neural 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

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