Design principles for linear systems: Stability and optimality

This thesis consists of two parts. Each of them deals with problems in the design of linear time-invariant 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 polynomial-time 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 time-invariant systems with control. The well-known 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 high-frequency content of the control signals is limited, while insuring stability and optimality of the resulting closed-loop system. We also address optimal actuator design for linear systems with process noise. It is well-known 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.LimitedAuthor requested closed access (OA after 2yrs) in Vireo ETD syste