working with uncertainty in model predictive control
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Working with Uncertainty in Model Predictive Control. Bob Bitmead University of California, San Diego. Nonlinear MPC Workshop. 4 April, 2005, Sheffield UK. Outline. Model Predictive Control Constrained receding horizon optimal control - PowerPoint PPT PresentationTRANSCRIPT
Working with Uncertainty in Model Predictive Control
Bob BitmeadUniversity of California, San Diego
Nonlinear MPC Workshop4 April, 2005, Sheffield UK
Sheffield April 4, 2005 2 of 31
OutlineModel Predictive Control
Constrained receding horizon optimal controlBased 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 problemsInteraction managed via constraintsEstimation error affects the constraints — back-off
Communication bandwidth affects state error
Control, Performance, Communication tied-in
Model Predictive Control Works
Full Authority Digital Engine Controller(FADEC)
Commercial jet engine
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MPC with Constraints — Jet EngineFull-Authority Digital Engine Controller (FADEC)Multi-input/multi-output control 5x6Constrained in
Inputs - max fuel flow, rates of changeStates - differential pressures, speedsOutputs - 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
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MPC Applied to Jet EngineStep in power demandConstraints
Fuel flowExit nozzle area
Constraint-driven controller
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MPC Applied to Jet EngineStep in power demand Constraints
Fuel flowExit nozzle areaStage 3 pressure
Two inputsOne 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
plantRemember. The MPC problem works on the estimate only
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Modifying constraintsWant 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
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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
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xk1Axk Buk Bddk ABK xk B ˜ u k Bddkˆ 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)
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Comparison model
a1degree of stabilitya2 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
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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, Kˆ x |t i i ˆ i ( ,w|t )
xi, i for [t, t T ]
˜ u i, Kx i i
three systemsreal 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
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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 K ˆ 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
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Example
From Fukushima & Bitmead, Automatica, 2005, pp. 97-106
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Working with state estimatesKalman filtering framework
Gaussian state estimate errorsProbabilistic 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 (,)
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Pause for breathOur formulation so far
Model errorsTighten 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 …
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The Shinkansen ExampleOne dimensional problem
Three Shinkansen [Bullet Trains] on one trackUncertainty in knowledge of other trains’ positions
Uniformly distributed with known width
Follow the same reference with each train
Constraint — no crash with preceding trainLeader-follower strategyEach solves an MPC problem with state estimation
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Collision avoidance with estimation
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Train coordinationAll trains have the same schedule
Osaka to Tokyo in three hoursDepart at 09:00, arrive at 12:00
Each solves their own MPC problemMinimize departure from scheduleNo-collision constraintEstimates 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 IssuesConstraints
Quality of InformationCommunication
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 variableConstraint 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 processModel 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
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Network ControlA variant of the train control problem
Much 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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AcknowledgementsHiroaki 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
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Constraints in designThe appeal of MPC is that it can handle constraints
Constraints provide a natural design paradigmLane keeping potential function
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A Design BonusThe MPC/KF design is much less sensitive to
selection of design parameters than LQGConstraints work well in design — simplicity
From Yan & Bitmead, Automatica, 2005 pp.595-604
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