raffaello d’andrea cornell university
DESCRIPTION
Design and Control of Interconnected Systems. Raffaello D’Andrea Cornell University. Examples. Power generation and distribution Vehicle platoons Satellite formation flight Paper processing Adaptive optics MEMS data storage Optical switching “Smart” structures - PowerPoint PPT PresentationTRANSCRIPT
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