An adjoint-based formalism for optimal design of time-delay systems and uncertainty quantification in stochastic systems

Ahsan, Zaid

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https://hdl.handle.net/2142/115483 Copy

Description

Title

An adjoint-based formalism for optimal design of time-delay systems and uncertainty quantification in stochastic systems

Author(s)

Ahsan, Zaid

Issue Date

2022-04-20

Director of Research (if dissertation) or Advisor (if thesis)

Dankowicz, Harry

Doctoral Committee Chair(s)

Dankowicz, Harry

Committee Member(s)

• Matlack, Kathryn
• Rapti, Zoi
• Salapaka, Srinivasa

Department of Study

Mechanical Sci & Engineering

Discipline

Mechanical Engineering

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Adjoints
• Delay
• Uncertainty

Abstract

The objectives of this dissertation are to develop computational techniques for i) tracking solutions to multi-segment, delay-coupled boundary-value problems, with particular emphasis on periodic orbits and quasiperiodic invariant tori for delay differential equations (DDEs), ii) solving single-objective optimal design problems along families of such solutions using a technique of successive continuation, and iii) characterizing the effects of noise on the asymptotic dynamics in the vicinity of transversally stable periodic and quasiperiodic solutions in deterministic limits of stochastic differential equations (SDEs). Considerable attention is placed on an algorithmic formalism and problem discretization that supports straightforward implementation in the Matlab-based software package coco. In the case of delay-coupled boundary-value problems, the analysis in this dissertation demonstrates how a Lagrangian formulation for constrained design optimization may be used to derive the necessary optimality conditions in terms of additional delay-coupled adjoint boundary-value problems that are linear in corresponding sets of Lagrange multipliers. Importantly, these adjoint boundary-value problems are shown to decompose into contributions associated with individual differential or algebraic constraints. As a consequence, their construction is found to conform with the staged construction paradigm of coco, and complex problems may be constructed by gluing together instances of simpler problems. In particular, this dissertation shows how a boundary-value problem for design optimization along families of quasiperiodic invariant tori may be constructed by gluing together multiple instances of single-segment trajectory problems using all-to-all boundary conditions. Several candidate implementations of this paradigm in a set of coco-compatible toolbox constructors are used to demonstrate such optimization according to a previously developed sequential methodology for finding stationary points of an objective function along implicitly defined manifolds. By their generality, these constructors are also shown to be compatible with problems from optimal control, phase response analysis, and continuation of homo- and heteroclinic trajectories. For transversally stable periodic orbits (limit cycles) and quasiperiodic invariant tori in the deterministic limit of SDEs, this dissertation proposes a novel covariance boundary-value problem to quantify the asymptotic dynamics near these objects in the presence of small amounts of Brownian noise. Here, the adjoint boundary-value problems associated with phase response analysis are used to construct a continuous family of projections onto transversal hyperplanes that are invariant under the linearized flow near the limit cycle or quasiperiodic invariant torus. The asymptotic dynamics in the presence of noise are shown to be represented by a stationary distribution whose restriction to individual hyperplanes is Gaussian with a covariance given by the solution to the corresponding covariance boundary-value problem. In the case of limit cycles, the analysis improves upon results in the literature through the explicit use of transversal projections and modifications that ensure uniqueness of the solution despite the lack of hyperbolicity along the limit cycle. These same innovations are then generalized to the case of a two-dimensional quasiperiodic invariant torus, as a model of the formalism required for a torus of arbitrary dimension. As in the case of DDEs, using a coco implementation of suitable toolbox constructors, the covariance analysis of a quasiperiodic invariant torus is shown to parallel the decomposition into multiple instances of single-segment trajectory problems using all-to-all boundary conditions derived for design optimization.

Graduation Semester

2022-05

Type of Resource

Thesis

Copyright and License Information

Copyright 2022 Zaid Ahsan

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