Raffaello D’Andrea Cornell 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 Stability, performance, robustness Requirements: GiGi vivi didi uiui yiyi zizi wiwi GiGi uiui yiyi ~ wiwi ~ vivi ~

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

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: where N and the are fixed, and only a function of the basic building block, then D’Andrea ’98, D’Andrea & Dullerud ‘03 all interconnected systems are well-posed, stable, and

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

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

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

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

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