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Voltage instability is a major threat in power system operation. The growing presence of constant power loads significantly aggravates this issue, hence motivating the development of new analysis methods for both existence and stability of voltage equilibria. Formally, this problem can be cast as the analysis of solutions of a set of nonlinear algebraic equations of the form $f(x)=0$, where $f:\mathbb{R}^n \mapsto \mathbb{R}^{n}$, and the associated differential equation $\dot x=f(x)$. By invoking advanced concepts of dynamical systems theory and effectively exploiting its monotonicity, we exhibit all possible scenarios for existence, uniqueness and stability, of its equilibria. We prove that, if there are equilibria, there is a distinguished one that is locally stable and attractive, and we give some physically-interpretable conditions such that it is unique. Moreover, a simple on-line procedure to decide whether equilibria exist of not, and to compute the distinguished one is proposed. In addition, we show how the proposed framework can be applied to long-term voltage stability analysis in AC power systems, multi-terminal high-voltage DC systems and DC microgrids., Comment: Submitted to IEEE Transactions on Automatic Control. This version: September 21, 2018

A nonholonomic under-actuated robot with a bounded control range travels inside an unknown and unstructured 3D tunnel bounded by a generic 2D surface. The robot has access to the nearest point of the surface and measures the distance to the surface along any ray from a narrow beam emitted from the robot around the direction to the nearest point. The paper presents some mathematical facts underlying justification of the convergence and performance of a new navigation law that ensures constant advancement of the robot through the tunnel, along with respecting a given safety margin to its surface, driving the robot to the desired distance to it, and subsequently maintaining this distance., Comment: 15 pages

The paper presents some characteristics of unsteady scalar planar fields and studies their properties that are relevant to research on environmental extremum seeking and tracking environmental boundaries by mobile robots., Comment: arXiv admin note: substantial text overlap with arXiv:1502.02224

A single nonholonomic under-actuated mobile robot with a bounded control range travels in a 3D workspace. This workspace also hosts a compact unknown domain with a smooth boundary. The robot has access to the horizontal distance to this domain and to a certain direction (typically, vertical) in the space, along with its coordinate (typically, altitude) in this direction. We present a new navigation law that drives the robot to the desired distance to the domain and ensures subsequent full scan of its surface within a desired range of "altitudes". The proposed strategy is computationally inexpensive and achieves full scan at the lowest control level via generating the current control as a simple, reflex-like reaction to the current observation. The paper presents and proves key technical facts underlying mathematically rigorous analysis and justification of the proposed guidance approach.

We consider a single non-holonomic Dubins-like robot traveling with a constant longitudinal speed in an a priori unknown and unsteady planar environment. The robot should detect, locate, and track the boundary of a dynamic environmental scalar field. The field is measured by an on-board sensor in a point-wise fashion at the robot's location. The focus is on unsteady boundaries that evolve over time in an arbitrary fashion, including deformations, i.e., changes of shapes and sizes. We present a sliding mode control method for localizing and tracking such boundaries: the robot is steered to the boundary and circulates in its close proximity afterwards. The proposed control algorithm does not require estimation of the spatial gradient of the field and is non-demanding with respect to both computation and motion. The paper offers the proofs of technical facts required for rigorous justification of non-local convergence of the proposed control law, as well as theoretical illustrations of its performance in specific scenarios., Comment: 3 figures

We consider a single kinematically controlled robot with a bounded control range. The robot travels in a two-dimensional region supporting an unknown unsteady scalar field. A single sensor provides the field value at the current location of the robot. The paper is concerned with a reactive navigation strategy to drive the robot to the time-varying location where the field attains its spatial maximum, with subsequent keeping the robot close to this location. This strategy is based on a novel paradigm of kinematic control different from conventionally trying to align the velocity vector with the field gradient, does not employ estimation of the entire field gradient or derivative-dependent quantities, like the rate at which the available field reading evolves over time, and is non-demanding with respect to both computation and motion. The paper presents the proofs of the technical facts required to justify convergence of the proposed algorithm., Comment: 19 pages, 4 figures

We present a method for guidance of a Dubins-like vehicle with saturated control towards a target in a steady simply connected maze-like environment. The vehicle always has access to to the target relative bearing angle (even if the target is behind the obstacle or is far from the vehicle) and the distance to the nearest point of the maze if it is within the given sensor range. The proposed control law is composed by biologically inspired reflex-level rules. Mathematically rigorous analysis of this law is provided; its convergence and performance are confirmed by computer simulations and experiments with real robots.

This text presents the proofs of the technical facts underlying theoretical justification of the convergence and performance of the novel algorithm for reactive navigation of differential drive wheeled robots in dynamic uncertain environments. The algorithm restricts neither the natures nor the motions of the obstacles, they need not be rigid but conversely may deform. It does not consume data about the velocities, shapes, sizes, or orientations of the obstacles, and does not need a map of the environment or recognition of individual obstacles. The only information about the scene is the current distance to the nearest obstacle., Comment: 7 pages, 3 figures

We present technical results justifying a method for guidance of a Dubins-like vehicle with saturated control towards a target in a steady simply connected maze-like environment. The vehicle always has access to to the target relative bearing angle and the distance to the nearest point of the maze if it is within the given sensor range. The proposed control law is composed by biologically inspired reflex-level rules. Mathematically rigorous analysis of this law is provided; its convergence and performance are confirmed by computer simulations and experiments with real robots., Comment: 16 pages, 4 figures