Raffaello D’AndreaCornell University

Design and Control of Interconnected Systems

Examples•Power generation and distribution•Vehicle platoons•Satellite formation flight•Paper processing•Adaptive optics•MEMS data storage•Optical switching•“Smart” structures and so on...Common thread:

• Distributed sensing and actuation capabilities• Highly structured interconnection topology

General Problem Class

PLANT CONTROLLER

,1 , ,1 ,( , ), ( , )i i i L i i i Lw w w v v v

, ,i j j iw v

Stability, performance, robustnessRequirements:

Gi

vi

di

uiyi

zi

wi

Gi

uiyi

~wi~ vi

~

d z

ww

v

v

Basic building block, one spatial dimension

Simplest case: Homogeneous Systems

( , , )

( , )( , )

xw f x v dz

w w wv v v

PERIODIC CONFIGURATION

BOUNDARY CONDITIONS

INFINITE EXTENT SYSTEM

2D, 2D BOUNDARY CONDITIONS

2D, 1D BOUNDARY CONDITIONS

2D, NO BOUNDARY CONDITIONS

Results for linear and piece-wise linear systems

Theorem: If the following semidefinite program has a solution:

01

0N

ll

l Pq P

where N and the are fixed, and onlya function of the basic building block, then

lP

D’Andrea ’98, D’Andrea & Dullerud ‘03

|| || || ||dz all interconnected systems are well-posed, stable, and

d z

ww

v

v

d z

ww

v

v

y u

( , ),

( , )M

x xw w ww vv v vz d

y u

Basic building block: control design

Design controller that has the same structure as the plant:

y u

Kw

Kw

Kv

Kv

PERIODIC CONFIGURATION

2D, 2D BOUNDARY CONDITIONS

Properties of design

•Controller has the same structure as the plant

•Finite dimensional, convex optimization problem

•Optimization problem size is independent of the number of units

Arbitrary interconnections, heterogeneous components

Arbitrary interconnections, heterogeneous components

Theorem: the interconnected system is well-posed, stable, and if the following coupled semidefinite programs have a solution:|| || || ||dz

Langbort, Chandra, & D’Andrea ’03Chandra, Langbort, & D’Andrea ‘03

,

,0, ,, , , , , ,1 1

0, , 1i j

i j k i j k j iii j kkj k

NLP Pq q q i L

if the subsystems are not interconnected:0,i jN

Theorem: the interconnected system is well-posed, stable, and if the following coupled semidefinite programs have a solution:|| || || ||dz

Langbort, Chandra, & D’Andrea ’03Chandra, Langbort, & D’Andrea ‘03

,

,0, ,, , , , , ,1 1

0, , 1i j

i j k i j k j iii j kkj k

NLP Pq q q i L

if the subsystems are not interconnected:0,i jN

When working with linearized dynamics, results generalize tocontrol system design

Summary

• Semidefinite programming a powerful tool for controldesign and analysis of interconnected systems

• Generalization of powerful results for single systems:linear, piece-wise linear, nonlinear

• Leads to distributed semidefinite programs, whosestructure is captured by interconnection topology