working with uncertainty in model predictive control

Working with Uncertainty in Model Predictive Control

Bob BitmeadUniversity of California, San Diego

Nonlinear MPC Workshop4 April, 2005, Sheffield UK

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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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