Methods and applications in networked control and Feedback control design for quantum systems

Two quite different topics are covered in the thesis. Methods and Applications in Networked Control The first topic is networked control, identification, and optimization, with particular interest in the design of individual control laws for multi-agent networked systems. These systems consist in a large, sometimes unknown and time-varying, number of agents. These agents can communicate, they interact with an underlying physical system by sensing and actuating it, and none of them has a complete knowledge of the system state and parameters. The main challenges in designing control, identification, and optimization algorithms for these systems are the need for scalability, the constraints in the communication capabilities of the agents, the need for robustness in case of agent failure, and the adaptivity to changes in the system (node appearance and disappearance, communication failures, change in the agents' placement, etc.). Some mathematical tools and methods from the literature are reviewed, and their application to the problem of networked control are explored. A series of original mathematical methods and algorithms have then be derived: - analysis results on the stability and scalability of consensus algorithms for unstable dynamics; - distributed parametric identification via least-square estimation; - distributed quasi-Newton methods for convex optimization; - randomized iterative methods for quadratic, linearly constrained optimization. These methods are presented via their application to some motivating examples of networked control systems. All of them are also of great interest per se, and can be considered challenging open problems in the field of networked control. They are the following: - leaderless, distributed, clock synchronization in networks of agents with extremely limited computational, communication, and energy resources; - distributed, online estimation of the wireless channel parameters for localization of mobile nodes via triangulation; - management architectures and distributed networked control laws for reactive power compensation (and other ancillary services) in the so called smart grids, with a particular focus on microgrids at the power distribution level. These open problems will be explored both as a motivating testbed for the proposed methods and as an outlook for future directions of investigation. Feedback Control Design for Quantum Systems The second topic is the problem of control of quantum dynamical systems, via the design of stabilizing feedback control law when quantum measurements are available. Discrete time models for quantum dynamics are reviewed, with a particular attention to those control problems that are typical key tasks in quantum information processing: - preparation of states of maximal information; - engineering of protected realization of quantum information, i.e. the realization of information encodings that preserve the fragile quantum states from the action of noise. These tasks can be casted into the more general problem of engineering stable quantum dynamical subspaces. In this part of the thesis the asymptotic behavior of discrete-time, Markovian quantum systems with respect to a subspace of interest, is analyzed. The results of this analysis, based on a Ljapunov approach, provide necessary and sufficient conditions on the dynamical model that ensure global asymptotic stability of a certain quantum subspace. It is then introduced a control scheme that allows modifying the underlying dynamics by indirectly measuring it and by applying unitary control actions conditioned on the outcome of the measurement. For this discrete-time feedback control scheme, an original design algorithm capable of stabilizing a target subspace is introduced. It is guaranteed that if the control problem is feasible, then the algorithm returns an effective control choice. In order to prove this result, a new technical tool is derived, namely a canonical QR matrix decomposition, which is also used to establish the control scheme potential for the simulation of open-system dynamics.