Controlling “Emergelent” Systems

Raffaello D’Andrea

Cornell University

INTERCONNECTED SYSTEMS Example: Formation Flight

Use upwash created by neighboring craft to provide extra lift

Formation Flight Test-bed

Interconnected Systems

•System consists of many units

•Sensing and actuation exists at every unit

•Units are coupled, either dynamically or through performance objectives

Some consideration for control design:

•Centralized control not desirable, nor feasible.

•Need tools for systems with very large number of actuators and sensors

•Robustness and reconfigurability

d z

ww

v

v ( , , )

( , , )

( , , )

( , ) ( , )

x f x v d

w g x v d

z h x v d

w w w v v v

BASIC BUILDING BLOCK: ONE SPATIAL DIMENSION

PERIODIC CONFIGURATION

BOUNDARY CONDITIONS

SPATIALLY CAUSAL SYSTEM

“INFINITE” EXTENT SYSTEMS

2D, 2D BOUNDARY CONDITIONS

2D, 1D BOUNDARY CONDITIONS

2D, NO BOUNDARY CONDITIONS

Performance theorem:

*S S TX X , X= 0

T T T T

T S

T

*

*

*

*

T

T S

*

*

0 0 0 0

0

0 0 0 00

0 00 0 0 0

0

0 0

X X X X

X X

X

X

X X

TT TT TS T

ST SS S TS ST SS S

T

TS

ST SS S TS ST SS S

T S T S

A A A B

A A B A A A B

B

A

A A B A A

I I

I II

I I

A B

C C D C C DI

0

z d if there exists

such that

Semi-definite Programming Approach

d z

ww

v

v

y u

( , , , )

( , , , )

( , , , )

( , , , )

x f x v d u

w g x v d u

z h x v d u

y l x v d u

BASIC BUILDING BLOCK: CONTROL DESIGN

Design controller that has the same structure as plant

PERIODIC CONFIGURATIONS

PERIODIC CONFIGURATION

SPATIALLY CAUSAL SYSTEMS

SPATIALLY CAUSAL SYSTEMS

INFINITE EXTENT SYSTEMS

INFINITE EXTENT SYSTEMS

BOUNDARY CONDITIONS

BOUNDARY CONDITIONS

2D, 2D BOUNDARY CONDITIONS

Theorem: There exists a controller which satisfies theperformance condition if and only if there exists T S T= = diag( ,X X X X ), X 0 T S T= = diag( ,Y Y Y Y ), Y 0

* *1 1*

1 11* *1 11

C0

A A BU C D U

B DI

I

Y Y Y+Y

* *1

* * *1 11

1 11

1

0A A B C

V B D VC D

II

X X X+X

T

T

X 0YI

I

Properties of design

•Implementation: distributed computation, limited connectivity

•Finite dimensional, convexoptimization problem

•Optimization problem size isindependent of the number of units

•Allows for real-time re-configuration

Decentralized Control

Distributed Control

Simulation results

•Distributed 0.24 60 seconds

•Decentralized 1.10 15 seconds

•Fully centralized 0.22 20 hours (4 wings)

Design time (P3, 1.2GHz)Worst Case L2

Intelligent Vehicle Systems

Example: RoboCup

• International competition: cooperation, adversaries, uncertainty – 1997: Nagoya Carnegie Mellon– 1998: Paris Carnegie Mellon– 1999: Stockholm Cornell– 2000: Melbourne Cornell– 2001: Seattle Singapore– 2002: Fukuoka Cornell

Develop hierarchy-based tools for designing high-performance controlled systems in uncertain environments

Approach:

•System level decomposition: temporal and spatial separation

•Embrace bottom up design

•Simplification of models via relaxations and reduction

•Propagation of uncertainty to higher levels

•Adoption of heuristics, coupled with verification

Objective:

Vehicle

System Level Decomposition

Low levelcontrol

Motion planning

High-levelreasoning

Vehicle

Low levelcontrol

Motion planning

High-levelreasoning

INFORMATION EXCHANGE

Example of bottom up designRelaxation and Simplified Dynamics:

X

Y

x x u

y y u

u

Restrict possible motions, design lower level systemsto behave like simplified dynamical model

Low levelcontrol

Motion planning

BACK-PASS PASS-PLAY

Highlights

Observations•Useful emergent behavior is the exception, not the norm

•Emergent behavior, when useful, is impressive and amazing

•Useful emergent behavior tends to be not very robust

•Reluctant to build upon emergent behavior without “understanding” it: no notion of reconfiguration and robustness

•Hierarchical decomposition, based on temporal and spatial separation, is a powerful paradigm

•Good tradeoff between reliability and performance seems to occur at the limits of our knowledge