Geometric synthesis of a hybrid limit cycle for the stabilizing control of a class of nonlinear switched dynamical systems

International audienceThis paper proposes a new constructive method for synthesizing a hybrid limit cycle for the stabilizing control of a class of switched dynamical systems in IR 2 , switching between two discrete modes and without state discontinuity. For each mode, the system is continuous, linear or nonlinear. This method is based on a geometric approach. The first part of this paper demonstrates a necessary and sufficient condition of the existence and stability of a hybrid limit cycle consisting of a sequence of two operating modes in IR 2 which respects the technological constraints (minimum duration between two successive switchings, boundedness of the real valued state variables). It outlines the established method for reaching this hybrid limit cycle from an initial state, and then stablizing it, taking into account the constraints on the continuous variables. This is then illustrated on a Buck electrical energy converter and a nonlinear switched system in IR 2. The second part of the paper proposes and demonstrates an extension to IR n for a class of systems, which is then illustrated on a nonlinear switched system in IR 3

Uniform finite time stabilisation of non-smooth and variable structure systems with resets

This thesis studies uniform finite time stabilisation of uncertain variable structure and non-smooth systems with resets. Control of unilaterally constrained systems is a challenging area that requires an understanding of the underlying mechanics that give rise to reset or jumps while synthesizing stabilizing controllers. Discontinuous systems with resets are studied in various disciplines. Resets in states are hard nonlinearities. This thesis bridges non-smooth Lyapunov analysis, the quasi-homogeneity of differential inclusions and uniform finite time stability for a class of impact mechanical systems. Robust control synthesis based on second order sliding mode is undertaken in the presence of both impacts with finite accumulation time and persisting disturbances. Unlike existing work described in the literature, the Lyapunov analysis does not depend on the jumps in the state while also establishing proofs of uniform finite time stability. Orbital stabilization of fully actuated mechanical systems is established in the case of persisting impacts with an a priori guarantee of finite time convergence between t he periodic impacts. The distinguishing features of second order sliding mode controllers are their simplicity and robustness. Increasing research interest in the area has been complemented by recent advances in Lyapullov based frameworks which highlight the finite time Convergence property. This thesis computes the upper bound on the finite settling time of a second order sliding mode controller. Different to the latest advances in the area, a key contribution of this thesis is the theoretical proof of the fact that finite settling time of a second order sliding mode controller tends to zero when gains tend to infinity. This insight of the limiting behaviour forms the basis for solving the converse problem of finding an explicit a priori tuning formula for the gain parameters of the controller when and arbitrary finite settling time is given. These results play a central role ill the analysis of impact mechanical systems. Another key contribution of the thesis is that it extends the above results on variable structure systems with and without resets to non-smooth systems arising from continuous finite time controllers while proving uniform finite time stability. Finally, two applications are presented. The first application applies the above theoretical developments to the problem of orbital stabilization of a fully actuated seven link biped robot which is a nonlinear system with periodic impacts. The tuning of the controller gains leads to finite time convergence of the tracking errors between impacts while being robust to disturbances. The second application reports the outcome of an experiment with a continuous finite time controller

Coupling Rydberg atoms to superconducting microwave circuits

A hybrid circuit quantum electro-dynamics (circuit QED) setup consisting of helium atoms in high-n Rydberg states and superconducting co-planar waveguide (CPW) microwave resonators has been developed with the goal of performing hybrid quantum optics experiments with applications in quantum information processing. In this thesis an overview of the field of cavity QED is introduced, and numerical methods to calculate the atomic energy level structure and transition dipole moments in electric and magnetic fields are described. Using this background information, a new method for efficiently preparing high-n circular Rydberg states is presented. This was required to ac- cess circular-state–to–circular-state transition frequencies in experiments which are compatible with superconducting CPW microwave resonators. Experimental and numerical results demonstrating the implementation of this method for the preparation of the |n = 70, l = 69, ml = +69⟩ circular state in helium are reported. The design and fabrication of λ/4 superconducting CPW microwave resonators, compatible with these high-n circular Rydberg states of helium is then described. The effects of microwave driving power, temperature and magnetic field on the characteristics of these resonators are presented. Finally, experiments in which helium Rydberg atoms have been coherently coupled to the microwave field of a superconducting co-planar waveguide resonator are reported

Teixeira singularities in 3D switched feedback control systems

Abstract: This paper is concerned with the analysis of a singularity that can occur in threedimensional discontinuous feedback control systems. The singularity is the two-fold – a tangency of orbits to both sides of a switching manifold. Particular attention is placed on the Teixeira singularity, which previous literature suggests as a mechanism for dynamical transitions in this class of systems. We show that such a singularity cannot occur in classical single-input single-output systems in the Lur’e form. It is, however, a potentially dangerous phenomenon in multiple-input multiple-output switched control systems.The theoretical derivation is illustrated by means of a representative example

Generation of terahertz-modulated optical signals using AlGaAs/GaAs laser diodes

The Thesis reports on the research activities carried out under the Semiconductor-Laser Terahertz-Frequency Converters Project at the Department of Electronics and Electrical Engineering, University of Glasgow. The Thesis presents the work leading to the demonstration of reproducible harmonic modelocked operation from a novel design of monolithic semiconductor laser, comprising a compound cavity formed by a 1-D photonic-bandgap (PBG) mirror. Modelocking was achieved at a harmonic of the fundamental round-trip frequency with pulse repetition rates from 131 GHz up to a record-high frequency of 2.1 THz. The devices were fabricated from GaAs/AlGaAs material emitting at a wavelength of 860 nm and incorporated two gain sections with an etched PBG reflector between them, and a saturable absorber section. Autocorrelation studies are reported, which allow the device behaviour for different modelocking frequencies, compound cavity ratios, and type and number of intra-cavity reflectors to be analyzed. The highly reflective PBG microstructures are shown to be essential for subharmonic-free modelocking operation of the high-frequency devices. It was also demonstrated that the multi-slot PBG reflector can be replaced with two separate slots with smaller reflectivity. Some work was also done on the realisation of a dual-wavelength source using a broad-area laser diode in an external grating-loaded cavity. However, the source failed to deliver the spectrally-narrow lines required for optical heterodyning applications. Photomixer devices incorporating a terahertz antenna for optical-to microwave down-conversion were fabricated, however, no down-conversion experiments were attempted. Finally, novel device designs are proposed that exploit the remarkable spectral and modelocking properties of compound-cavity lasers. The ultrafast laser diodes demonstrated in this Project can be developed for applications in terahertz imaging, medicine, ultrafast optical links and atmospheric sensing