Least-squares integration of one-dimensional codistributions with application to approximate feedback linearization

We study the problem of approximating one-dimensional nonintegrable codistributions by integrable ones and apply the resulting approximations to approximate feedback linearization of single-input systems. The approach derived in this paper allows a linearizable nonlinear system to be found that is close to the given system in a least-squares (L ₂) sense. A linearly controllable single-input affine nonlinear system is feedback linearizable if and only if its characteristic distribution is involutive (hence integrable) or, equivalently, any characteristic one-form (a one-form that annihilates the characteristic distribution) is integrable. We study the problem of finding (least-squares approximate) integrating factors that make a fixed characteristic one-form close to being exact in anL ₂ sense. A given one-form can be decomposed into exact and inexact parts using the Hodge decomposition. We derive an upper bound on the size of the inexact part of a scaled characteristic one-form and show that a least-squares integrating factor provides the minimum value for this upper bound. We also consider higher-order approximate integrating factors that scale a nonintegrable one-form in a way that the scaled form is closer to being integrable inL ₂ together with some derivatives and derive similar bounds for the inexact part. This allows a linearizable nonlinear system that is close to the given system in a least-squares (L ₂) sense together with some derivatives to be found. The Sobolev embedding techniques allow us to obtain an upper bound on the uniform (L _∞) distance between the nonlinear system and its linearizable approximation. This research was supported in part by NSF under Grant PYI ECS-9396296, by AFOSR under Grant AFOSR F49620-94-1-0183, and by a grant from the Hughes Aircraft Company.     

Kalman decomposition for implicit linear systems

The Kalman decomposition for a general class of implicit systems is introduced and studied. The properties of strong reachability, observability, acceptance of all inputs and outer sequences, output-uniqueness, uniqueness, and trajectory observability of the original system are related to the properties of its subsystems obtained via the Kalman decomposition. The problems of transfer relation, regularizability by feedback, output feedback, and output injection are studied by means of this decomposition     

Scalable approach to uncertainty quantification and robust design of interconnected dynamical systems

Development of robust dynamical systems and networks such as autonomous aircraft systems capable of accomplishing complex missions faces challenges due to the dynamically evolving uncertainties coming from model uncertainties, necessity to operate in a hostile cluttered urban environment, and the distributed and dynamic nature of the communication and computation resources. Model-based robust design is difficult because of the complexity of the hybrid dynamic models including continuous vehicle dynamics, the discrete models of computations and communications, and the size of the problem. We will overview recent advances in methodology and tools to model, analyze, and design robust autonomous aerospace systems operating in uncertain environment, with stress on efficient uncertainty quantification and robust design using the case studies of the mission including model-based target tracking and search, and trajectory planning in uncertain urban environment. To show that the methodology is generally applicable to uncertain dynamical systems, we will also show examples of application of the new methods to efficient uncertainty quantification of energy usage in buildings, and stability assessment of interconnected power networks.     

On Hautus-type conditions for controllability of implicit linear discrete-time systems

Four notions of controllability for general (i.e., possibly nonregular) implicit linear discrete-time systems are considered. Relationships between them are studied. A Hautus-type characterization of all of these notions is also given.This work was performed under the auspices of RP.I.02: Teoria sterowania i optymalizacji ciagych ukadów dynamicznych i procesów dyskretnych.     

An adaptive algorithm for control of combustion instability

We propose an adaptive algorithm for control of combustion instability suitable for reduction of acoustic pressure oscillations in gas turbine engines, and main burners and augmentors of jet engines over a large range of operating conditions, and supply an experimental demonstration of oscillation attenuation, the first for a large industrial-scale gas turbine combustor. The algorithm consists of an Extended Kalman Filter based frequency tracking observer to determine the in-phase component, the quadrature component, and the magnitude of the acoustic mode of interest, and a phase shifting controller actuating fuel-flow, with the controller phase tuned using extremum-seeking. The paper also identifies a closed-loop model with phase-shifting control of combustion instability from experimental data; supplies stability analysis of the adaptive scheme based upon the identified model, and stable extremum-seeking designs used in experiments.     

Impact of exothermicity on steady and linearized response of a premixed ducted flame

Thermoacoustic instabilities arise in power generation devices such as gas turbines and aero-engines when acoustic modes couple with unsteady heat released due to combustion in a positive feedback loop. This work focuses on the development of a reduced order model for understanding flame dynamics in the case of flameholder-stabilized premixed combustion in a duct—a situation typical in many of these applications. Similar to earlier studies in reduced order modeling of this flow, we employ a G-equation formulation to obtain kinematical representation of the premixed flame and ignore the impact of the unsteady (vortical) fluid dynamics downstream of the flameholder. Unlike those studies, however, we retain the impact of combustion exothermicity in the form of a density jump and associated volume generation at the flame front as well as the steady portion of the baroclinic vortical effect. The reduced order model yields analytical solutions for the flame location and for linear transfer functions between imposed (acoustic) perturbation and combustion heat release. We validate these solutions against numerical simulations and other results in literature. The role of expansion (dilatation) and baroclinic aspects of exothermic effects are discussed in detail. Results show that for realistic density ratios across the flame, the flow is accelerated in the streamwise direction on account of combustion exothermicity and the effects of confinement. This not only alters the flame location but also changes the linearized dynamics of the flame and brings into question conclusions drawn from similar analyses in which exothermicity effects were neglected. This is discussed in the context of modeling and controlling thermoacoustic instabilities.     

Implicit linear discrete-time systems

This paper studies various properties of implicit linear discrete-time systems given by a linear difference equationEx _k+1 =Fx _k +Gu _k . The topics considered include a basic characterization of these subspaces which describe acceptance of all input sequences, the uniqueness property and regularity, and the notion of controllability. This work was performed under the auspices of Fund RP.I.02: Teoria sterowania i optymalizacji ciągłych układów dynamicznych i procesów dyskretnych.     

A homotopy algorithm for approximating geometric distributions by integrable systems

In the geometric theory of nonlinear control systems, the notion of a distribution and the dual notion of codistribution play a central role. Many results in nonlinear control theory require certain distributions to be integrable. Distributions (and codistributions) are not generically integrable and, moreover, the integrability property is not likely to persist under small perturbations of the system. Therefore, it is natural to consider the problem of approximating a given codistribution by an integrable codistribution, and to determine to what extent such an approximation may be used for obtaining approximate solutions to various problems in control theory. In this note, we concentrate on the mathematical problem of approximating a given codistribution by an integrable codistribution. We present an algorithm for approximating an m-dimensional non-integrable codistribution by an integrable one using a homotopy approach. The method yields an approximating codistribution that agrees with the original codistribution on an m-dimensional submanifold E₀ of ⁿ.     

On the existence and uniqueness of solutions for implicit linear systems on finite interval

In the present paper we study the problem of the existence and uniqueness of solutions of implicit systems considered on a finite interval of time. We consider two kinds of existence problems: input-acceptance and input-acceptance when the boundary conditions of a corresponding trajectory are set to zero, and two kinds of uniqueness problems: output uniqueness and output uniqueness when the boundary conditions of the corresponding trajectory are unknown. Geometric conditions for all of these notions are given, and the duality of these notions is studied.This research was supported by NSF Grant ECS-8805932.     

Hearing the clusters of a graph: A distributed algorithm

We propose a novel distributed algorithm to cluster graphs. The algorithm recovers the solution obtained from spectral clustering without the need for expensive eigenvalue/eigenvector computations. We prove that, by propagating waves through the graph, a local fast Fourier transform yields the local component of every eigenvector of the Laplacian matrix, thus providing clustering information. For large graphs, the proposed algorithm is orders of magnitude faster than random walk based approaches. We prove the equivalence of the proposed algorithm to spectral clustering and derive convergence rates. We demonstrate the benefit of using this decentralized clustering algorithm for community detection in social graphs, accelerating distributed estimation in sensor networks and efficient computation of distributed multi-agent search strategies.