Working with Uncertainty in Model Predictive Control Bob Bitmead University of California, San Diego Nonlinear MPC Workshop 4 April, 2005, Sheffield UK

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


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


MPC Applied to Jet Engine

Step in power demand

ConstraintsFuel flow

Exit nozzle area

Constraint-driven controller


MPC Applied to Jet Engine

Step in power demand

ConstraintsFuel flow

Exit nozzle area

Stage 3 pressureTwo inputs

One state


Message

MPC works in handling constraints on the model

With accurate state estimates

— this is fine for the real plant too


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


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


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


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)


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


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


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


Example

From Fukushima & Bitmead, Automatica, 2005, pp. 97-106


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


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 (,)


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 …


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


Collision avoidance with estimation


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


Back to the TrainsLow Performance plus String Instability


Relaxed Target SchedulesLow Performance but no string instability

Constraints not active


Improved CommunicationHigh performance, no string instability


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


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


Fair Congestion Control50 retransmissions per 1000 samples


Simulated Source Rates — Fair!

mean = 0.0012

variance = 0.0419

mean = 0.0013

variance = 0.0184

mean = 0.0013

variance = 0.0129


A Tougher Example

From Yan & Bitmead, Automatica, 2005 pp.595-604


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


Conclusions

MPC plus State EstimationTools for coordinated control performance

with managed communication complexity

Information architecture

Resource/bandwidth assignment

… as a function of system task


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


Constraints in design

The appeal of MPC is that it can handle constraintsConstraints provide a natural design paradigm

Lane keeping potential function


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



Documents

Working with Uncertainty in Model Predictive Control Bob Bitmead University of California, San Diego Nonlinear MPC Workshop 4 April, 2005, Sheffield UK