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Title:  Design principles for linear systems: Stability and optimality 
Author(s):  Kirkoryan, Artur 
Director of Research:  Belabbas, MohamedAli; Baryshnikov, Yuliy 
Doctoral Committee Chair(s):  DeVille, Lee 
Doctoral Committee Member(s):  Zharnitsky, Vadim 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Sparse Matrix Spaces
Stability Optimal Control Linear Dynamical Systems 
Abstract:  This thesis consists of two parts. Each of them deals with problems in the design of linear timeinvariant systems with certain prescribed properties, such as stability and cost optimality. The first part addresses theoretical questions arising in the design of autonomous decentralized systems. The network topology of such a system describes which agents are able to interact with each other. We study the following problem: For a specified network topology, can one find a set of interaction laws that yield stable dynamics for the ensemble of agents? We restrict our analysis to systems with strictly linear dynamics. This problem can also be referred to as the structural stability problem, seen as the counterpart to the structural controllability problem. In mathematical terms, we consider vector spaces of real square matrices for which every entry is either fixed at zero, or an arbitrary real number. We call them sparse matrix spaces, abbreviated SMS, and examine under what conditions they contain matrices for which all eigenvalues have strictly negative real parts. We call an SMS with this property stable. We estimate the proportion of stable SMS when their size approaches infinity and when the locations of the free variables are chosen independently at random. Using graph theory techniques, we also develop polynomialtime algorithms for extension of a given stable SMS to a stable SMS with up to two additional nodes. In the second part, we consider linear timeinvariant systems with control. The wellknown linear quadratic regulator (LQR) provides feedback controller that stabilizes the system while minimizing a quadratic cost function in the state of the system and the magnitude of the control. The optimal actuator design problem then consists of choosing an actuator that minimizes the cost incurred by an LQR. While this procedure guarantees a low overall cost incurred, it only takes into account the magnitude of the control signals the regulator sends to the actuator. Physical actuators are, however, also limited in their ability to follow rapid change in control signals. We show in this thesis how to design actuators so that the highfrequency content of the control signals is limited, while insuring stability and optimality of the resulting closedloop system. We also address optimal actuator design for linear systems with process noise. It is wellknown that the control that minimizes a quadratic cost in the state and control for a system with linear dynamics corrupted by additive Gaussian noise is of feedback type and its design depends on the solution of an associated Riccati equation. We consider here the case where the noise is multiplicative, by which we mean that its intensity is dependent on the state. We show how to derive the actuator that minimizes a linear quadratic cost. 
Issue Date:  20181206 
Type:  Thesis 
URI:  http://hdl.handle.net/2142/102941 
Rights Information:  Copyright 2018 Artur Kirkoryan 
Date Available in IDEALS:  20190208 
Date Deposited:  201812 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois