MPC of Nonlinear Systems
• Motivation
• Challenging behavior
• Model Predictive Control
• Various Options
• EKF-based NMPC
• Multiple Model Predictive Control
• Summary
• Theory and Applications
B. Wayne Bequette
Challenging Behavior
Input Multiplicity
Output Multiplicity
Reactor Temperature
1.5
Output
a
1.0
b
c
0.5
Process Zero
to maximum flow
to minimum
flow
to minimum
flow
to maximum flow
0.0
-0.5
0
50
100
150
u
200
250
300
Connection with RHP zeros:
Sistu & Bequette, Chem. Eng. Sci. (1996)
Achievable performance is a strong
function of the operating condition
Jacket Flowrate
Region of instability
Russo & Bequette, AIChE J.
(1995)
Global Bifurcation Diagram
I
y
I
u
IV
II
II
y
y
u
IV
p2
III
III
y
u
V
V
y
u
Russo and Bequette, AIChE J. (1995)
p1
Design parameter Space III, IV, V:
Infeasible operating regions
u
past
future
Model Predictive
Control (MPC)
setpoint
y
model prediction
actual outputs (past)
P
tk
P redict ion
Horizon
current
step
max
u
min
past cont rol
moves
M
Control Horizon
setpoint
y
model prediction
from k
new model prediction
actual outputs (past)
t k+1
current
step
P
P redict ion
Horizon
max
u
min
past cont rol
moves
• Constraints
• Multivariable
• Time-delays
M
Control Horizon
• Objective function?
• Optimization technique?
• Model type?
• Disturbances?
• Initial cond./state est.?
Many Publications/Researchers




Will not attempt a reasonable overview
Every plenary speaker has worked on the topic!
Reviews

Bequette (1991)

Henson (1998)
Will focus on work of my graduate students
Our Approaches



Quadratic Objective Function
Models

Fundamental: numerical integration or collocation

Fundamental with linearization at each time step

Multiple model

Artificial neural network
State Estimates/Initial Conditions

Additive output disturbance (e.g. DMC)

Estimation horizon (optimization)

Extended Kalman Filter

Importance of stochastic states
Non-Convex
Problem
Sistu and Bequette, 1992 ACC
Input Multiplicity
Example
Sistu and Bequette, 1992 ACC
Additive
Disturbance
Assumption
Bequette, ADCHEM (1991)
Stability

Infinite Horizon


Terminal State Constraints


Michalska and Mayne (1993)
Quasi-Infinite Horizon


Mayne and Michalska (1990)
Dual Model (Region, State Feedback)


Meadows and Rawlings (1993)
Chen and Allgower (1998)
Numerical Lyapunov - Regions of Attraction

Sistu and Bequette (1995)
State Estimation

Output Disturbance (DMC, not a good idea)



Garcia (1984)
Extended Kalman Filter

Gattu and Zafiriou (1992)

Lee and Ricker (1994)
Estimation Horizon, Optimization

Ramamurthi et al. (1993)
EKF-based NMPC (Lee & Ricker, 1994)



Nonlinear Model
State Estimation: Extended Kalman Filter
Prediction



One integration of NL ODEs based on set of control moves
Perturbation (linear) model - effect of changes in control
moves
Optimization

SQP
Multi-rate EKF Implementation
Frequent temperature
Infrequent concentration
and/or MWD
Schley et al. NL-MPC, Ascona (1998), Prasad et al. J. Proc. Cont. (2002)
Multiple Model Predictive Control

Fundamental Model


ANN, other NL Empirical Model


Time consuming, often impractical (biomedical, etc.)
Much data required, large validation effort, “overfitting”
Multiple Model Predictive Control

Extension of multiple model adaptive control (MMAC)

MMAC developed for aircraft


Many flight conditions
Bank of possible linear models


Controller-model pairing
Switching vs. weighting
Multiple Model Predictive Control
Constrained MPC
r(k) Reference
Model
u(k)
y(k)
Optimization
Plant
Model
Bank
^y(k+1:P)
Prediction
+
1
^y (k)
+
-
i
2
+
^y(k)
y(k)
m
-
+
+
X
+
X
X
Rao et al. IEEE Eng. Med. Biol. Mag (2001)
wi(k)
Weight
Computation
i(k)
Multiple Models and Weighting
• Probabilities
Pi,k 
exp0.5iT,k K i ,k Pi ,k 1
Nm
 exp0.5
j 1
T
j, k
K j, k Pj ,k 1
, Pi,k  
• Weights
wi, k 
Pi ,k
Nm
P
j, k
j 1
for Pi, k   ,
wi, k  0 for Pi,k  
Example Comparison of MMPC with EKF-based NMPC
Cain
F
Constant V,T, 
A
A+A
B
C
D
Cb
F
Aufderheide et al., 2001 ACC
Aufderheide and Bequette, Comp.
Chem. Eng. (2003)
Feed Concentration Disturbance
Aufderheide et al., 2001 ACC
Feed Concentration Disturbance w/noise
Biomedical Control

blood pressure
cardiac output
drugs infused
Anesthesia

Adaptation

Multiple models
Constraints

Recovery time


Diabetes
sensors
controller
inf usion
pumps
glucose
setpoint

Blood glucose

s.c. measurement

Sensor recalibration

Meal disturbances
controller
pump
sensor
patient
Current Status of NMPC


Modeling: the biggest challenge

Fundamental: much effort, many parameters

Empirical: much data, range of conditions?
Estimation



Biased estimates
Adaptation

Parameter, operating condition changes

Failure detection and compensation
Cost-Benefit

Nonlinear vs. Better Performing Linear (e.g. not DMC)
Potential Techniques
Multiobjective Optimization-based MPC
Distributed: Multiple MPC



Individual optimization

Communicate solution
12
10
n
o
i
t
c
e
r
i
d
Birds
8
Bugs
6
y
4
2
0
2
4
6
8
10
x direction
12
14
16
18
Summary

Motivation: nonlinear behavior



Multiplicities
Nonlinear model predictive control

Various, including full NMPC

EKF-based NMPC

MMPC
Current and Future Work
El Dorado’s (Troy, NY, 1994)
Lou Russo
Ravi Gopinath
Kevin Schott
Wayne Bequette
Phani Sistu
Troy Pub and Brewery (1998)
Deepak
Nagrath
Wayne
Bequette
Matt
Schley
Manoel
Carvalho
Brian
Aufderheide
Vinay
Prasad
Venkatesh
Natarajan
Ramesh
Rao