MPC Concepts

7/28/2019 MPC Concepts

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EE392m - Spring 2005Gorinevsky

Control Engineering 14-1

Lecture 14 - Model Predictive ControlPart 1: The Concept

History and industrial application resource: Joe Qin, survey of industrial MPC algorithms

http://www.che.utexas.edu/~qin/cpcv/cpcv14.html

Emerging applications

State-based MPC Conceptual idea of MPC

Optimal control synthesis

Example Lateral control of a car

Stability

Lecture 15: Industrial MPC


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MPC concept MPC = Model Predictive Control

Also known as DMC = Dynamical Matrix Control

GPC = Generalized Predictive Control

RHC = Receding Horizon Control

Control algorithms based on

Numerically solving an optimization problem at each step

Constrained optimization typically QP or LP

Receding horizon control

More details need to be worked out for implementation


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Receding Horizon Control

At each time step, compute control by solving an open-loop optimization problem for the prediction horizon

Apply the first value of the computed control sequence

At the next time step, get the system state and re-compute

future input trajectory predicted future output

Plant

Model

prediction horizon

prediction horizon

Receding Horizon Control concept

current dynamic

system statesPlant

RHC


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Current MPC Use Used in a majority of existing multivariable control

applications Technology of choice for many new advanced multivariable

control application

Success rides on the computing power increase

Has many important practical advantages


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MPC Advantages Straightforward formulation, based on well understood

concepts Explicitly handles constraints

Explicit use of a model

Well understood tuning parameters

Prediction horizon

Optimization problem setup

Development time much shorter than for competing

advanced control methods Easier to maintain: changing model or specs does not require

complete redesign, sometimes can be done on the fly


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History First practical application:

DMC Dynamic Matrix Control, early 1970s at Shell Oil Cutler later started Dynamic Matrix Control Corp.

Many successful industrial applications

Theory (stability proofs etc) lagging behind 10-20 years.

See an excellent resource on industrial MPC

Joe Qin, Survey of industrial MPC algorithms

history and formulations

http://www.che.utexas.edu/~qin/cpcv/cpcv14.html


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Some Major Applications

From Joe Qin, http://www.che.utexas.edu/~qin/cpcv/cpcv14.html

1995 data, probably 1-2 order of magnitude growth by now


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Emerging MPC applications Nonlinear MPC

just need a computable model (simulation)

NLP optimization

Hybrid MPC discrete and parametric variables

combination of dynamics and discrete mode change

mixed-integer optimization (MILP, MIQP)

Engine control

Large scale operation control problems

Operations management (control of supply chain) Campaign control


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Emerging MPC applications Vehicle path planning and control

nonlinear vehicle models

world models

receding horizon preview


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Emerging MPC applications Spacecraft rendezvous with space station

visibility cone constraint

fuel optimality

Underwater vehicle guidance

Missile guidance

From Richards & How, MIT


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State-based control synthesis

Infinite horizon optimal control

( ) ( )

0

1

22

)(:subject to

min)1()()(

uu

uuryt

+

+=

)()()1( tButAxtx+=+

Consider single input system for better clarity

Solution = Optimal Control Synthesis

)()( tCxty =


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State-based MPC concept

Optimal control trajectories converge to (0,0) IfNis large, the part of the problem for t>Ncan be neglected

Infinite-horizon optimal control horizon-Noptimal control

x1

x2

t > N

Optimal

control

trajectories


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State-based MPC Receding horizon control;N-step optimal

( ) ( )

)()(

)()()1(

,)(:subject to

min)1()()(

0

1

22

tCxty

tButAxtx

uu

uuryJNt

t

=

+=+

+= +

+=

Solution Optimal Control Synthesis

[ ] )(SolverProblemMPC)( tutx


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Predictive Model Predictive system model

+

+

=

)(

)1(

Nty

ty

YM

Model matrices

+

+

=

)(

)1(

Ntu

tu

U M

FuHUGxY ++= initial condition response + control response

)(txx=

Predicted output Future control input Current state

(initial condition)

=

nCA

CAG M

=

=

)1()1()(

0)1()2(

00)1(

0

00

000

32 hNhNh

hh

h

BCABCA

CBH

NNK

MOMM

K

K

K

MOMM

K

K

)(tuu = computed atthe previous step

+

=

)1(

)3(

)2(

Nh

h

h

FM


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Computations Timeline

Assume that control u is applied and the state x is

sampled at the same instant t Entire sampling interval is available for computing u

x(t)

t

x(t+1)

t+1

Compute control based

onx(t). Apply it

u(t+1)

time

u(t)

Computed at the previous step


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MPC Optimization Problem Setup MPC optimization problem

FuHUGxY

uUDUDrUYYJ

TTT

++=

+=

,:subject tomin

0

Solution

[ ] )1()1(SolverQPProblem,MPC)( UtuUtx =+

This is a QP problem

=

000

010

011

K

MOMM

K

K

D

1st difference matrix


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QP solution QP Problem:

min2

1+=

UfQUUJ

bAU

TT

Standard QP codes can be used

)(tUU= Predicted

controlsequence

HHDrDQ TT +=

( )FuGxHf T +=

=I

IA0

1

1

ub

= M


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Linear MPC Nonlinearity is caused by the constraints

If constraints are inactive, the QP problem solution isfQU 1

=

( ) ( ))()()1(1

tFutGxHHHDrDltuTTTT

++=+

=

0

0

1

Ml

SuzKxzu 11 +=

KxSz

zu

1

1

1

=

( ) GHHHDrDlK TTTT 1+= ( ) FHHHDrDlS TTTT 1+=

This is linear state feedback

BuAxzx +=

Can be analyzed as a linear system, e.g., check eigenvalues

Ulu T=


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Nonlinear MPC Stability Theorem - from Bemporad et al (1994)

Consider a MPC algorithm for a linear plan with constraints. Assume

that there is a terminal constraint

x(t+N) = 0 for predicted statex and

u(t+N) = 0 for computed future control u

If the optimization problem is feasible at time t,

then the coordinate origin is stable.

Proof.

Use the performance index J as a Lyapunov function. It decreases

along the finite feasible trajectory computed at time t. Thistrajectory is suboptimal for the MPC algorithm, henceJdecreases

even faster.


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MPC Stability The analysis could be useful in practice

Theory says a terminal constraint is good MPC stability formulations

(Mayne et al,Automatica, 2000)

Terminal equality constraint Terminal cost function

Dual mode control infinite horizon

Terminal constraint set

Increase feasibility region

Terminal cost and constraint set


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Example: Lateral Control of a Car

Preview Control MacAdams driver model (1980)

Consider predictive control design

Simple kinematical model of a car driving at speed V

Lane direction

lateraldisplacement

y

x

V

u

Preview horizon

ua

aVyaVx

=

=

=

&

&&

sincos

lateral displacement

steering


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Lateral Control of a Car - Model

Assume a straight lane tracking a straight line

Linearized system: assume a


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Lateral Control of a Car - MPC

Formulate predictive model

MPC optimization problemFuHUGxY ++=

= )(

)()(tytatx

)()()1( tButAxtx +=+

=

= 2

5.0,

101

s

s

s VTTB

VTA

State-space system:

Observation: )()( tCxty = [ ]10=C

( ) ( )

,:subject to

min

0uU

DUDrUFuHUGxFuHUGxJTTT

+++++=

[ ] )1()1(QPMPC)( UtuUtx =+ Solution:


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Impulse Responses

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20IMPULSE RESPONSE FOR LATERAL ERROR

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5IMPULSE RESPONSE FOR HEADING ANGLE


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Lateral Control of a Car - SimulationSimulation

Results:

V= 50 mph

Sample time

of 200ms N= 20

All variables

in SI units

r= 1

0 1 2 3 4 5 6 7 80

0.5

1LATERAL ERROR

0 1 2 3 4 5 6 7 8-2

-1.5

-1

-0.5

0HEADING ANGLE (deg)

0 1 2 3 4 5 6 7 8-1

-0.5

0

0.5

1STEERING CONTROL (deg/s)

TIME (sec)


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Control Design Issues Several important issues remain

They are not visible in this simulation

Will be discussed in Lecture 15 (MPC, Part 2)

All states might not be available

Steady state error

Need integrator feedback

Large angle deviation

linearized model deficiency

introduce soft constraint

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