MPC of Nonlinear Systems
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Transcript MPC of Nonlinear Systems
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
Prediction
Horizon
current
step
max
u
min
M
past control
moves
Control Horizon
setpoint
y
model prediction
from k
new model prediction
actual outputs (past)
t k+1
current
step
P
Prediction
Horizon
max
u
min
past control
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 0.5iT,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
infusion
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