Analysis And Control Of Networked Systems Using Structural And Measure-Theoretic Approaches

Network control theory provides a plethora of tools to analyze the behavior of dynamical processes taking place in complex networked systems. The pattern of interconnections among components affects the global behavior of the overall system. However, the analysis of the global behavior of large scale complex networked systems offers several major challenges. First of all, analyzing or characterizing the features of large-scale networked systems generally requires full knowledge of the parameters describing the system\u27s dynamics. However, in many applications, an exact quantitative description of the parameters of the system may not be available due to measurement errors and/or modeling uncertainties. Secondly, retrieving the whole structure of many real networks is very challenging due to both computation and security constraints. Therefore, an exact analysis of the global behavior of many real-world networks is practically unfeasible. Finally, the dynamics describing the interactions between components are often stochastic, which leads to difficulty in analyzing individual behaviors in the network. In this thesis, we provide solutions to tackle all the aforementioned challenges. In the first part of the thesis, we adopt graph-theoretic approaches to address the problem caused by inexact modeling and imprecise measurements. More specifically, we leverage the connection between algebra and graph theory to analyze various properties in linear structural systems. Using these results, we then design efficient graph-theoretic algorithms to tackle topology design problems in structural systems. In the second part of the thesis, we utilize measure-theoretic techniques to characterize global properties of a network using local structural information in the form of closed walks or subgraph counts. These methods are based on recent results in real algebraic geometry that relates semidefinite programming to the multidimensional moment problem. We leverage this connection to analyze stochastic networked spreading processes and characterize safety in nonlinear dynamical systems

A Unifying Framework for Strong Structural Controllability

This article deals with strong structural controllability of linear systems. In contrast to the existing work, the structured systems studied in this article have a so-called zero/nonzero/arbitrary structure, which means that some of the entries are equal to zero, some of the entries are arbitrary but nonzero, and the remaining entries are arbitrary (zero or nonzero). We formalize this in terms of pattern matrices, whose entries are either fixed zero, arbitrary nonzero, or arbitrary. We establish necessary and sufficient algebraic conditions for strong structural controllability in terms of full rank tests of certain pattern matrices. We also give a necessary and sufficient graph-theoretic condition for the full rank property of a given pattern matrix. This graph-theoretic condition makes use of a new color change rule that is introduced in this article. Based on these two results, we then establish a necessary and sufficient graph-theoretic condition for strong structural controllability. Moreover, we relate our results to those that exist in the literature and explain how our results generalize previous work.</p

From data and structure to models and controllers

Systems and control theory deals with analyzing dynamical systems and shaping their behavior by means of control. Dynamical systems are widespread, and control theory therefore has numerous applications ranging from the control of aircraft and spacecraft to chemical process control. During the last decades, a series of remarkable new control techniques have been developed. The majority of these techniques rely on mathematical models of the to-be-controlled system. However, the growing complexity of modern engineering systems complicates mathematical modeling. In this thesis, we therefore propose new methods to analyze and control dynamical systems without relying on a given system model. Models are thereby replaced by two other ingredients, namely measured data and system structure. In the first part of the thesis, we consider the problem of data-driven control. This problem involves the development of controllers for a dynamical system, purely on the basis of data. We consider both stabilizing controllers, and controllers that minimize a given cost function. Secondly, we focus on networked systems. A networked system is a collection of interconnected dynamical subsystems. For this type of systems, our aim is to reconstruct the interactions between subsystems on the basis of data. Finally, we consider the problem of assessing controllability of a dynamical system using its structure. We provide conditions under which this is possible for a general class of structured systems

Distributed infinite-horizon optimal control of continuous-time linear systems over network

This article deals with the distributed infinite-horizon linear-quadratic-Gaussian optimal control problem for continuous-time systems over networks. In particular, the feedback controller is composed of local control stations, which receive some measurement data from the plant process and regulates a portion of the input signal. We provide a solution when the nodes have information on the structural data of the whole network but takes local actions, and also when only local information on the network are available to the nodes. The proposed solution is arbitrarily close to the optimal centralized one (in terms of cost index) when a design parameter is set sufficiently large. Numerical simulation validate the theoretical results

Analysis of Structural Properties of Complex and Networked Systems

Over the past decades, science and society have been experiencing systems that tend to be increasingly sophisticated and interconnected. Although it would be challenging to understand and control complex systems fully, the analysis and control of such systems can be partially realized only after applying some reasonable simplifications. In particular, for the analysis of certain control properties, such as controllability, a complex system can be simplified to a linear structured system capturing an essential part of the structural information in that system, such as the existence or absence of relations between components of the system. This thesis has studied the effect of the interconnection structure of complex systems on their control properties following a structural analysis approach. More explicitly, we have analyzed strong structural properties of complex systems. The main contributions have been split into two parts:1. We have introduced a new framework for linear structured systems in which the relations between the components of the systems are allowed to be unknown. This kind of systems has been formalized in terms of pattern matrices whose entries are either fixed zero, arbitrary nonzero, or arbitrary. We have dealt with strong structural controllability and the solvability of the FDI problem of this kind of linear structured systems.2. We have introduced a novel framework for linear structured systems in which a priori given entries in the system matrices are restricted to take arbitrary but identical values. Several sufficient algebraic and graph theoretic conditions were established under which these systems are strongly structurally controllable.Finally, in the outlook subsection, we have suggested some future research problems concerning the analysis of strong structural properties of complex systems