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Ph.D. Dissertation Defense: Alborz Alavian
Friday, August 11, 2017
1:30 p.m.
2460 A.V. Williams Bldg.
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ANNOUNCEMENT: Ph.D. Dissertation Defense

Name: Alborz Alavian


Committee members:

Professor Michael C. Rotkowitz, Chair

Professor Nuno Martins

Professor Perinkulam Krishnaprasad

Professor Andre Tits

Professor Nikhil Chopra, Dean's Representative

Title: Optimization-based Robustness and Stabilization in Decentralized Control

This dissertation pertains to the robustness and optimization of Finite Dimensional Linear Time Invariant (FDLTI) decentralized control systems. We mainly focus on decentralizations emerging from sparsity-induced information structures.

Two notions of stabilizability with respect to decentralized controllers are considered. First, the seminal result of Wang & Davison in 1973 regarding internal state stabilizability of decentralized system and its connection to the decentralized fixed-modes of the plant is revisited. This would be generalized to any arbitrary sparsity-induced information structure by providing comprehensive proofs to verify and show that those mode of the plant that are fixed with respect to the static controllers would remain fixed with respect to the dynamic ones. A complete proof is also provided to show that one can move any non-fixed mode of the plant to any arbitrary location within desired accuracy provided that they would remain symmetric in the complex plane. A synthesizing algorithm would then be distilled from the inductive proof. A second stronger notion of stability referred to as "non-overshooting stability" is then addressed. A key property called "feedthrough consistency" is derived, that when satisfied, makes extension of the centralized results to the decentralized case possible.

Synthesis of decentralized controllers to optimize an H-infinity norm for model-matching problems is considered next. A finite-dimensional parametrization of the infinite-dimensional controller is derived, and we show that once the poles are chosen for this parametrization, the remaining problem of coefficient optimization can be cast as a semidefinite program (SDP), and also demonstrate how to instead use first-order methods when the SDP is too large or when a first-order method is otherwise desired. This leaves the remaining choice of poles, for which we develop and discuss several methods for this dictionary selection phase.

Controllability of LTI systems with decentralized controllers is then studied. Whether an LTI system is controllable (by LTI controllers) with respect to a given information structure can be determined by testing for fixed modes, but this gives a binary answer with no information about robustness. Measures have already been developed to further determine how far a system is from having a fixed mode when one considers complex or real perturbations to the state-space matrices. These measures involve intractable minimizations of a non-convex singular value over a power-set, and hence cannot be computed except for the smallest of the plants. We derive another optimization problem involving a binary vector rather than the power-set minimization and prove its equality to the original metrics. An approximate form is also provided which enables us to utilize MIP techniques to derive scalable upper bounds, and polynomial-time lower bounds for these measures.

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