In this paper we propose a method for solving systems of nonlinear inequalities with predefined accuracy based on nonuniform covering concept formerly adopted for global optimization. The method generates inner and outer approximations of the solution set. We describe the general concept and three ways of numerical implementation of the method. The first one is applicable only in a few cases when a minimum and a maximum of the constraints convolution function can be found analytically. The second implementation uses a global optimization method to find extrema of the constraints convolution function numerically. The third one is based on extrema approximation with Lipschitz under- and overestimations. We obtain theoretical bounds on the complexity and the accuracy of the generated approximations as well as compare proposed approaches theoretically and experimentally.

Approximating a solution set of nonlinear inequalities

Event-triggered controllers are well known for guaranteeing the desired stability for a sampled-data system with minimum resource utilization. Over the past decade, the study has revealed that the overall performance improvement for a sampled-data system can be achieved by replacing the time-based sampling with an event-triggered one. The design of sliding mode control (SMC) in the event-triggering framework has also shown similar outcomes, especially for uncertain systems. There are different design strategies for event-triggered SMC available in the literature for networked dynamical systems that are potentially affected by uncertainties and transmission delays. The purpose of this survey article is to present the state of the art on event-triggered SMC and familiarize the readers with the design techniques with their pros and cons, since this will be very helpful to the researchers and engineers for implementing SMC using event-based feedback strategies.

A Survey on Event-Triggered Sliding Mode Control

The main focus of this study is to develop a finite-time super-twisting sliding mode control strategy for the quadrotor based on a higher-order sliding mode observer (HOSMO). 12 state variables are required to describe the motion of the quadrotor, of which six state variables, namely the position, altitude, and orientation, are assumed to be obtained from the sensors. The remaining state variables, i.e. the linear and angular velocities, are determined using the HOSMO. Besides, the HOSMO aids in determining the unknown bounded lumped disturbances acting on the quadrotor. The output of the HOSMO is utilised for implementing the finite-time super-twisting sliding mode controller (FTSTSMC). The proposed FTSTSMC ensures finite-time convergence of tracking error with chattering attenuation. The chattering analysis for a super-twisting algorithm is presented in this work. Moreover, the overall system stability is investigated using the Lyapunov theory, and an expression for the time of convergence of the tracking and estimation error is presented. The effectiveness of the proposed methodology is established using numerical simulations and its performance is compared to a finite-time sliding mode observer coupled with a combination of proportional–integral–derivative and continuous sliding-mode controller. This is then validated in real-time using the DJI Matrice 100.

https://ietresearch.onlinelibrary.wiley.com/doi/epdf/10.1049/iet-cta.2020.0348

Finite-time super twisting sliding mode controller based on higher-order sliding mode observer for real-time trajectory tracking of a quadrotor

This special issue of the Journal ofGlobalOptimization contains twelve high-quality research papers devoted to different aspects of global optimization such as theory, numerical methods and real-life applications. The papers included in this special issue are based on the presentations carefully selected by the guest editors among the talks delivered at the 2nd International Conference “Numerical Computations: Theory and Algorithms (NUMTA)” held in June 19–25, 2016 in Pizzo Calabro, Italy (the first NUMTA conference took place in Falerna, Italy in 2013). The NUMTA 2016 has been organized by the University of Calabria, Rende (CS), Italy, in cooperation with the Society for Industrial and Applied Mathematics, USA. The guest editors actively participated in the organization of the conference: the Program Committee of the NUMTA 2016 was chaired by Yaroslav D. Sergeyev, in their turn, Renato De Leone and Anatoly Zhigljavsky took part in the Program Committee. The goal of the NUMTA 2016 was creation of a multidisciplinary round table for an open discussion on numerical modeling nature by using traditional and emerging computational paradigms. Participants of this conference discussed several aspects of numerical computations and modeling from foundations of mathematics and computer science to advanced numerical techniques. A large part of presentations has been dedicated to optimization. Selected papers presented at the conference in the field of numerical analysis and respective applications have been published in the special issue of the international journal Applied Mathematics and Computation, Volume 318 (2018). In its turn, the present special issue contains articles dealing with global optimization. Let us give a brief description of the papers included in this special issue.

/pdf/guest-editors-preface-to-the-special-issue-devoted-to-the-bp47k7aovr.pdf

Guest editors’ preface to the special issue devoted to the 2nd International Conference “Numerical Computations: Theory and Algorithms”, June 19–25, 2016, Pizzo Calabro, Italy

Mathematical models for transmission dynamics of the novel COVID-2019 coronavirus, an outbreak of which began in December, 2019, in Wuhan are considered. To control the epidemiological situation, it is necessary to develop corresponding mathematical models. Mathematical models of COVID-2019 spread described by systems of nonlinear ordinary differential equations (ODEs) are overviewed. Some of the coefficients and initial data for the ODE systems are unknown or their averaged values are specified. The problem of identifying model parameters is reduced to the minimization of a quadratic objective functional. Since the ODEs are nonlinear, the solution of the inverse epidemiology problems can be nonunique, so approaches for analyzing the identifiability of inverse problems are described. These approaches make it possible to establish which of the unknown parameters (or their combinations) can be uniquely and stably recovered from available additional information. For the minimization problem, methods are presented based on a combination of global techniques (covering methods, nature-like algorithms, multilevel gradient methods) and local techniques (gradient methods and the Nelder–Mead method).

/pdf/mathematical-modeling-of-the-wuhan-covid-2019-epidemic-and-54v6frjoug.pdf

Mathematical Modeling of the Wuhan COVID-2019 Epidemic and Inverse Problems

This article addresses the problem of event-based consensus in a leader–follower multiagent system framework prone to external bounded disturbance. The proposed approach has three parts. The first part defines a novel measurement error based on sliding surface for super-twisting sliding-mode controller. The Lyapunov stability analysis is then used to derive a dynamic event-triggering condition for control updates. The event-based control updates guarantee stability along with the desired consensus amongst agents (robots). The second part derives a bound on reaching time to the sliding surface, thereby guaranteeing finite-time consensus control for each agent. The third part guarantees the admissibility of event-based control updates for each agent. The robustness of the proposed approach is validated through simulation and real-time experiments using three Pioneer P3-DX mobile robots in a multiagent framework. The real-time experimental results prove the reduction in computational burden of the entire system as control updates for two followers are found to be approximately $\text{28.33}\%$ and $\text{23.33}\%$ , respectively, in the presence of disturbances.

Finite-Time Robust Admissible Consensus Control of Multirobot System Under Dynamic Events

This paper presents a new numerical approach to manipulator workspace assessment. The method is based on the non-uniform covering concept in which the workspace is a solution for the system of non-linear inequalities with Lipschitz continuous functions. The illustrative example shows the performance of the approach and indicates the precision of the workspace assessment.

The non-uniform covering approach to manipulator workspace assessment

Issues of planning trajectory of parallel robots taking into account zones of singularity

The article presents and experimentally confirms two approaches to the problem of determining the working area of parallel robots using the example of a planar robot DexTAR with two degrees of freedom. The proposed approaches are based on the use of constraint equations of coordinates. In the first approach, the original kinematic equations of coordinates in the six-dimensional space (two coordinates describing the position of the output link and four coordinates – the rotation angles of the rods) followed by projecting the solution onto the two-dimensional plane is used. In the second approach, the system of constraint equations is reduced to a system of inequalities describing the coordinates of the output link of the robot, which are solved in a two-dimensional Euclidean space. The results of the computational experiments are given. As an algorithmic basis of the proposed approaches, the method of non-uniform coverings is used, which obtains the external and internal approximation of the solution set of equality/inequality systems with a given accuracy. The approximation is a set of boxes. It is shown that in the first approach, it is more efficient to apply interval estimates that coincide with the extremes of the function on the box, and in the second approach, grid approximation performs better due to multiple occurrences of variables in inequalities.

/pdf/approaches-to-the-determination-of-the-working-area-of-23p733ym5w.pdf

Larisa Rybak

Papers

Approximating a solution set of nonlinear inequalities

Finite-Time Robust Admissible Consensus Control of Multirobot System Under Dynamic Events

The non-uniform covering approach to manipulator workspace assessment

Issues of planning trajectory of parallel robots taking into account zones of singularity

Approaches to the Determination of the Working Area of Parallel Robots and the Analysis of Their Geometric Characteristics