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Transcript of Working with Uncertainty in Model Predictive Control Bob Bitmead University of California, San Diego...
Working with Uncertainty in Model Predictive Control
Bob Bitmead
University of California, San Diego
Nonlinear MPC Workshop
4 April, 2005, Sheffield UK
Sheffield April 4, 2005 2 of 31
Outline
Model Predictive ControlConstrained receding horizon optimal control
Based on full-state information or certainty equivalence
How do we include estimated states?Accommodate estimate error — tighten the constraints
Coordinated vehicles exampleVehicles solve local MPC problems
Interaction managed via constraints
Estimation error affects the constraints — back-offCommunication bandwidth affects state error
Control, Performance, Communication tied-in
Model Predictive Control Works
Full Authority Digital Engine Controller
(FADEC)
Commercial jet engine
Sheffield April 4, 2005 4 of 31
MPC with Constraints — Jet Engine
Full-Authority Digital Engine Controller (FADEC)
Multi-input/multi-output control 5x6
Constrained in Inputs - max fuel flow, rates of change
States - differential pressures, speeds
Outputs - turbine temperature
Control problem solved via Quadratic Programming (every 10 msec)
State estimator - Extended Kalman FilterState estimate used as if exact — Certainty Equivalence
SENSORS
ACTUATORSIGVs VSVs MFMV A8AFMV
N2T2 N25PS14
P25
PS3 T4B
0 1 2 14 16
25 3 4 49 5 56
6 8 9
STATIONS
Sheffield April 4, 2005 5 of 31
MPC Applied to Jet Engine
Step in power demand
ConstraintsFuel flow
Exit nozzle area
Constraint-driven controller
Sheffield April 4, 2005 6 of 31
MPC Applied to Jet Engine
Step in power demand
ConstraintsFuel flow
Exit nozzle area
Stage 3 pressureTwo inputs
One state
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Message
MPC works in handling constraints on the model
With accurate state estimates
— this is fine for the real plant too
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What if the estimates are not accurate?
Tighten the constraints imposed on the model
— to ensure their satisfaction on the plant
Remember. The MPC problem works on the estimate only
Sheffield April 4, 2005 9 of 31
Modifying constraints
Want and we have
Keep in MPC problem
x+ ^
x- ^
g
t
x̂
g-
t t+T
x ̂
x g
xt ˆ x t t
ˆ x t g t
Sheffield April 4, 2005 10 of 31
Handling uncertaintyTwo kinds of uncertainty
Modeling errorsState model is an inaccurate description of the real
system
State estimation errors
Remember the MPC constrained control calculation works with the model and not the real systemConstraints must be asserted on the real system
xk1 f (xk ,uk ) dk
dk dk1 dk
2 , dk1 xk , dk
2
ˆ x k|k ~ N xk ,k|k
Sheffield April 4, 2005 11 of 31
xk1Axk Buk Bddk
ABK xk B ˜ u k Bddk
ˆ Acxk B ˜ u k Bddk
dk dk
1
dk2
; dk
1 xk , dk2 1
Working with model error
Total Stability Theorem (Hahn, Yoshizawa)Uniform convergence rate of nominal system
+ bounds on model error bounds on state error
MPC formulation of Total StabilityRobust Control Lyapunov Function idea
degree of stability
model error bound
V (x) xT Px, P 0
QP AcT PAc , Q 0
a1 1 min (Q)max(P)
max P1/2Bd min (P)
a2 max P1/2Bd , bi P1/2B(:,i)
Sheffield April 4, 2005 12 of 31
Comparison model
a1degree of stability
a2 model error bound
bi P1/2B(:,i)
w1a1w a2 bi ˜ u i
i1
m
wt|t V (xt )xtT Pxt
Main lemma:
For any control
˜ u |t , [t, t T ]
V (x )w |t
The controlled behavior of dominates that of
w |t
x |t
Uses a control Lyapunov function for the unconstrained system
Sheffield April 4, 2005 13 of 31
Including constraints
xt1Acxt B ˜ u t Bddtˆ x 1Ac ˆ x B ˜ u , ˆ x |t xt
w1a1w a2 bi ˜ u i
i1
m , wt|t V (xt )
ˆ x i, |t i ˆ i ( ,w|t )
˜ u i, K ˆ x |t i i ˆ i ( ,w|t )
xi, i for [t, t T ]
˜ u i, Kx i i
three systems
real system
comparison system
model system
ˆ i( ,w|t ) Acs1Bdi (1,:)
T
s0
t
1
w( s | t)minP
Acs1Bdi (2,:)
T
1
ˆ i ( ,w|t ) KAcs1Bdi (1,:)
T
s0
t
1
w( s | t)minP
KAcs1Bdi (2,:)
T
1
Sheffield April 4, 2005 14 of 31
MPC with Comparison Model
min˜ u
J xt , ˜ u |t ˜ u |tT R ˜ u |t
t
tN 1
Subject to
ˆ x 1|t Ac ˆ x |t B ˜ u |t , ˆ x t|t xt
w1|t a1w |t a2 bi ˜ u i, |ti1
m , wt|t V (xt )xt
T Pxt
ˆ x i,1|t i ˆ i( 1,w|t )
˜ u i, |t Kˆ x i, |t i ˆ i( ,w|t )
w1|t , wtN |t 1
If feasible at t=0 thenFeasible for all t
Real system is stable, constrained and
xt a2
1 a1 1minP
for all t t f
Sheffield April 4, 2005 15 of 31
Example
From Fukushima & Bitmead, Automatica, 2005, pp. 97-106
Sheffield April 4, 2005 16 of 31
Working with state estimates
Kalman filtering frameworkGaussian state estimate errors
Probabilistic constraints are needed
State estimate error
Rework this as
The constrained controller will need to be cognizant of This is a non-(certainty-equivalence) controller
Information quality is of importance
Same concept of tightening constraints
P xi,t i
x |t ˆ x |t ~ N 0, |t
x |t ~ N( ˆ x |t , |t )
|t
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Approximately Normal State
Manage constraints by controlling the conditional mean stateUse the control independence
of
xni ~ N( ˆ x ni|n ,ni|n )
Pr(xni X ) ˆ x ni|n (,)
Sheffield April 4, 2005 18 of 31
Pause for breath
Our formulation so farModel errors
Tighten constraints on the nominal system
State estimate errorsTighten constraints to accommodate the estimate
covariance
Preserves the MPC structure and propertiesOriginal constraints inherited by real system
Perhaps with probabilistic measures
Feasibility and stability propertiesVia terminal constraint as usual
Some examples …
Sheffield April 4, 2005 19 of 31
The Shinkansen Example
One dimensional problemThree Shinkansen [Bullet Trains] on one track
Uncertainty in knowledge of other trains’ positionsUniformly distributed with known width
Follow the same reference with each train
Constraint — no crash with preceding trainLeader-follower strategy
Each solves an MPC problem with state estimation
Sheffield April 4, 2005 20 of 31
Collision avoidance with estimation
Sheffield April 4, 2005 21 of 31
Train coordination
All trains have the same scheduleOsaka to Tokyo in three hours
Depart at 09:00, arrive at 12:00
Each solves their own MPC problemMinimize departure from schedule
No-collision constraint
Estimates of other trains’ positions
Trains separate earlySeparation reflects quality of position knowledge
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Back to the TrainsLow Performance plus String Instability
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Relaxed Target SchedulesLow Performance but no string instability
Constraints not active
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Improved CommunicationHigh performance, no string instability
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Big Issues
Constraints
Quality of Information
Communication
Network and Control Architecture
Tools for systematic design of complex interacting dynamical systemsModel Predictive Control and State Estimation
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Single Node in Network
Queue length qt is the state variable
Constraint qt≤Q else retransmission required
Control signals are the source command data rates vi,t
Propagation delays di exist between sources and node
Available bit rate t is a random process
Model as an autoregressive process
qt1qt vi,t dii1
n t
P(qt≥Q)<0.05
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Fair Congestion Control50 retransmissions per 1000 samples
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Simulated Source Rates — Fair!
mean = 0.0012
variance = 0.0419
mean = 0.0013
variance = 0.0184
mean = 0.0013
variance = 0.0129
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A Tougher Example
From Yan & Bitmead, Automatica, 2005 pp.595-604
Sheffield April 4, 2005 30 of 31
Network Control
A variant of the train control problemMuch greater degree of connectivity — higher dimension
Improved performance is achievable by sending more frequent or more accurate state information upstream to control data flowsThis consumes network resources and must be managed
MPC and State Estimation (Kalman Filtering) tools prove of value
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Conclusions
MPC plus State EstimationTools for coordinated control performance
with managed communication complexity
Information architecture
Resource/bandwidth assignment
… as a function of system task
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Acknowledgements
Hiroaki Fukushima, Jun Yan, Tamer Basar, Soura Dasgupta, Jon Kuhl, Keunmo Kang
NSF, Cymer Inc
GE Global Research Labs, Pratt & Whitney,
United Technologies Research Center
My gracious UK and Irish hosts, IEEE
Sheffield April 4, 2005 33 of 31
Constraints in design
The appeal of MPC is that it can handle constraintsConstraints provide a natural design paradigm
Lane keeping potential function
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A Design Bonus
The MPC/KF design is much less sensitive to selection of design parameters than LQG
Constraints work well in design — simplicity
From Yan & Bitmead, Automatica, 2005 pp.595-604
Sheffield April 4, 2005 35 of 31
Sheffield April 4, 2005 36 of 31