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Design and Control of Interconnected Systems Raffaello D’Andrea Cornell University Design and Control of Interconnected Systems Raffaello D’Andrea Cornell University

Examples • Power generation and distribution • Vehicle platoons • Satellite formation flight • 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 CONTROLLER PLANT wi yi vi ~ wi Gi di zi yi General Problem Class CONTROLLER PLANT wi yi vi ~ wi Gi di zi yi ui ~ Gi ui Requirements: Stability, performance, robustness ~ vi

Simplest case: Homogeneous Systems Basic building block, one spatial dimension Simplest case: Homogeneous Systems Basic building block, one spatial dimension

PERIODIC CONFIGURATION PERIODIC CONFIGURATION

BOUNDARY CONDITIONS BOUNDARY CONDITIONS

INFINITE EXTENT SYSTEM INFINITE EXTENT SYSTEM

2 D, 2 D BOUNDARY CONDITIONS 2 D, 2 D BOUNDARY CONDITIONS

2 D, 1 D BOUNDARY CONDITIONS 2 D, 1 D BOUNDARY CONDITIONS

2 D, NO BOUNDARY CONDITIONS 2 D, NO BOUNDARY CONDITIONS

Results for linear and piece-wise linear systems Theorem: If the following semidefinite program has Results for linear and piece-wise linear systems Theorem: If the following semidefinite program has a solution: where N and the are fixed, and only a function of the basic building block, then all interconnected systems are well-posed, stable, and D’Andrea ’ 98, D’Andrea & Dullerud ‘ 03

Basic building block: control design Design controller that has the same structure as the Basic building block: control design Design controller that has the same structure as the plant:

PERIODIC CONFIGURATION PERIODIC CONFIGURATION

2 D, 2 D BOUNDARY CONDITIONS 2 D, 2 D BOUNDARY CONDITIONS

Properties of design • Controller has the same structure as the plant • Finite 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

Arbitrary interconnections, heterogeneous components Arbitrary interconnections, heterogeneous components

Theorem: the interconnected system is well-posed, stable, and if the following coupled semidefinite programs Theorem: the interconnected system is well-posed, stable, and if the following coupled semidefinite programs have a solution: if the subsystems are not interconnected: Langbort, Chandra, & D’Andrea ’ 03 Chandra, Langbort, & D’Andrea ‘ 03

Theorem: the interconnected system is well-posed, stable, and if the following coupled semidefinite programs Theorem: the interconnected system is well-posed, stable, and if the following coupled semidefinite programs have a solution: if the subsystems are not interconnected: Langbort, Chandra, & D’Andrea ’ 03 Chandra, Langbort, & D’Andrea ‘ 03 When working with linearized dynamics, results generalize to control system design

Summary • Semidefinite programming a powerful tool for control design and analysis of interconnected Summary • Semidefinite programming a powerful tool for control design and analysis of interconnected systems • Generalization of powerful results for single systems: linear, piece-wise linear, nonlinear • Leads to distributed semidefinite programs, whose structure is captured by interconnection topology