Search Results (101)

The Collatz conjecture is a famous unsolved problem in mathematics, known for its deceptively simple rules that generate complex, unpredictable behaviour. It can be efficiently modelled using a Petri net that represents its inverse graph, where each place corresponds to an integer and each transition encodes an inverse rule. The net, constructed up to a bound n, reveals the tree-like structure of predecessors and highlights properties such as recurrence, reachability, and liveness. Token flows simulate possible trajectories towards 1. This formal approach enables the investigation of the problem through discrete event systems theory and opens perspectives for parametric or inductive extensions beyond the bounded domain. The model proposed provides a structured framework for visualising and analysing the inverse dynamics of the conjecture. Some key numerical results highlight the challenges of working within a finite domain: for $n_{max}=1000$, the constructed Petri net comprises 1000 places and 667 transitions, including 417 source nodes (no predecessors), 333 sink nodes (no successors), and 218 isolated orphans, i.e., nodes only reachable via $\mathrm{Div}2$ transitions with no incoming $3n+1$ edge. Full article

Nowadays, internet connectivity suffers from instability and slowness due to optical fiber cable attacks across the seas and oceans. The optimal solution to this problem is using the Low Earth Orbit (LEO) satellite network, which can resolve the problem of internet connectivity and reachability, and it has the power to bring real-time, reliable, low-latency, high-bandwidth, cost-effective internet access to many urban and rural areas in any region of the Earth. However, satellite orbital placement (SOP) and navigation should be carefully designed to reduce signal impairments. The challenges of orbital satellite placement for LEO include constellation development, satellite parameter optimization, bandwidth optimization, consideration of signal impairment, and coverage optimization. This paper presents a comprehensive review of SOP and coverage optimization, examines prevalent issues affecting LEO internet connectivity, evaluates existing solutions, and proposes novel solutions to address these challenges. Furthermore, it recommends a machine learning solution for coverage optimization and SOP that can be used to efficiently enhance internet reliability and reachability for LEO satellite networks. This survey will open the gate for developing an optimal solution for global internet connectivity and reachability. Full article

This paper addresses the controller design problem for linear systems with time-varying delays. By constructing a novel Lyapunov–Krasovskii functional incorporating delay-partitioning techniques, we establish delay-dependent stability criteria for the solvability of the robust stabilization problem. The derived conditions are formulated as linear matrix inequalities (LMIs) that become affine upon fixing a single scalar parameter, thereby facilitating efficient numerical computation. Furthermore, these criteria guarantee that the reachable set of the closed-loop system remains bounded within a prescribed ellipsoid under zero initial conditions. The effectiveness and superiority of the proposed approach are demonstrated through two comparative numerical examples, including a benchmark problem with varying delay. Full article

Advanced air defense methods are essential to address the growing complexity of aerial threats. The increasing number of targets necessitates better defensive coordination, and a promising strategy involves the use of interceptors together to protect a specific area. This task fundamentally depends on accurately predicting their kinematic envelopes, or reachable sets. This paper presents a novel approach to determine the boundaries of reachable sets for aerodynamic interceptors, accounting for energy loss from drag, energy gain from thrust, variable acceleration limits, and autopilot dynamics. The devised numerical method approximates reachable sets for nonlinear problems using a constrained model predictive programming concept. Results demonstrate that explicitly accounting for input constraints, such as acceleration limits, significantly impacts the shape and area of the reachable boundaries. Furthermore, a sensitivity analysis was conducted to demonstrate the impact of parameter variations on the reachable set. Revealing the reachable set’s sensitivity to variations in thrust and drag coefficients, this analysis serves as a framework for considering parameter uncertainty and enables the evaluation of these effects prior to embedding the reachability boundaries into an offline database for guidance applications. The resulting boundaries, representing minimum and maximum ranges for various initial parameters, can be stored offline, allowing interceptors to estimate their own or allied platforms’ kinematic capabilities for cooperative strategies. Full article

This study develops an applied optimization method to address practical challenges in Laser Radar station planning for automotive Body-In-White (BIW) manufacturing inspection. Focusing on the spatially constrained industrial environments and complex measurement specifications, the work reformulates Laser Radar inspection planning as a multi-constrained optimization problem challenge. Firstly, a parametric geometric modeling approach is developed to define measurement spaces for individual features, accompanied by an innovative maximal complete subgraph mining algorithm to intelligently identify shared feasible measurement regions among multiple features. Secondly, kinematic equations are formulated using Denavit–Hartenberg (D-H) parameters, while a hierarchical bounding volume collision detection mechanism is integrated to establish a comprehensive constraint. Therefore, unified optimization method synergizing measurement coverage, robotic manipulator reachability, and operational safety requirements are proposed. Through experimental validations utilizing BIW (BIW) component inspection, the research has demonstrated its industrial applicability and has achieved a 92% measurement coverage with robot trajectories free of collisions. Compared with traditional manual planning methods, the proposed approach reduces the number of required inspection stations by 35% and improves the computational efficiency to meet industrial real-time deployment requirements. Experimental validation demonstrates the method’s effectiveness in measurement accuracy, operational safety, and equipment utilization for advanced manufacturing quality control systems. Full article

This work continues the study of a mathematical model for the motion of a mass–spring–damper system with friction. There, a two-mass model was constructed, its solvability established, the steady states investigated, and numerical simulations presented. The main interest here is in the modeling, analysis of the steady states, and simulation of a three-mass system—in particular, in the propagation of detachment or slip waves, which happen when the system transits from a stick state to a slip motion. The introduction of friction changes the problem into systems of three differential set-valued inclusions, which are mathematically and computationally very challenging. The analysis of the steady states shows the regions of stick, where there is enough frictional resistance that prevents motion. The proposed numerical methods are implemented, and the simulations show some representative types of system behavior, especially the cases of detachment waves. Some of the numerical simulations specifically support the theoretical analysis of slip initiation, reachability, and energy balance. Full article

In the post-COVID era, international business and tourism are quickly recovering from the global lockdown, with people and products traveling faster at higher frequency. This boosts the economy while facilitating the spread of pathogens, causing waves of COVID aftershock with new variants like Omicron XBB. Hence, continuous disinfection of our living environments becomes our first priority. Autonomous disinfection robots provide an efficient solution to this issue. Compared to human cleaners, disinfection robots are able to operate tirelessly in harsh environments without increasing the risk of cross-infection. In this paper, we propose the design of a new generation of the Smart Cleaner disinfection robot, which is equipped with both an Ultraviolet-C (UVC) light tower and a hydrogen peroxide (HP) aerosol dispenser. The safety of an autonomous disinfection robot has been a persistent problem, especially when they work in complex environments. To tackle this problem, Hamilton–Jacobi (HJ) reachability is adopted to construct a safety filter for motion control, which guarantees that the disinfection path taken by the robot is collision-free without severely compromising the optimality of control actions. The effectiveness of the developed robot has been demonstrated by conducting extensive experimental studies. Full article

The addition of telescopic joints will enhance the manipulator’s motion dexterity but increase the motion complexity, thereby potentially reducing the efficiency of motion planning. To address the issue, a manipulability priority principle is used to reduce the computational dimension, aiming to enhance computational efficiency. Firstly, an improved Monte Carlo algorithm combined with Weighted Ensemble Learning is proposed to rapidly obtain a well-distributed and information-complete reachable workspace under small-sample conditions. Then, all dexterous points are computed by the manipulability formula and the least squares algorithm is applied to obtain the boundary of the reachable workspace. Based on the boundary, a spatial quadratic surface-based segmentation algorithm is proposed to obtain the dexterous workspace corresponding to all dexterous points. Subsequently, the Telescopic Motion Computation (TMC) method is designed to transform the kinematics problem of a 9-DOF manipulator into a 7-DOF problem by precomputing the telescopic joint lengths, effectively reducing the computational time for inverse kinematics. Finally, simulations are performed to verify the effectiveness, dexterity, and efficiency of the TMC method. The results indicate that applying the TMC method to PInv, DLS, and PSO algorithms enhances the efficiency by 3.43, 3.53, and 3.64 times while maintaining the motion dexterity. Full article

This study addresses the completely distributed consensus control problem for the heterogeneous nonlinear multi-agent system (MAS) with disturbances under switching topology. First, a global sliding mode manifold (GSMM) is designed for the overall MAS dynamic, which maintains stability without oscillations during topology switching after achieving the sliding mode. Subsequently, a consensus sliding mode control protocol (SMCP) is proposed, adopting the common sliding mode control (SMC) format and ensuring the finite-time reachability of the GSMM under topology switching. Finally, the proposed GSMM and SMCP are applied to the formation control of multiple-wheeled mobile robots (WMRs), and simulation results confirm their feasibility and effectiveness. The proposed SMCP design demonstrates key advantages, including a simple control structure, complete robustness to matched disturbance, and reduced-order dynamics under the sliding mode. Full article

In this paper, we develop an exact schedulability test and sufficient infeasibility test for fixed-priority scheduling on multiprocessor platforms. We base our tests on presenting real-time systems as a Kripke model for dynamic real-time systems with sporadic non-preemptible tasks running on a multiprocessor platform and an online scheduler using global fixed priorities. This model includes states and transitions between these states, allows us to formally justify a polynomial-time algorithm for an exact schedulability test using the idea of backward reachability. Using this algorithm, we perform the exact schedulability test for the above real-time systems, in which there is one more task than the processors. The main advantage of this algorithm is its polynomial complexity, while, in general, the problem of the exact schedulability testing of real-time systems on multiprocessor platforms is NP-hard. The infeasibility test uses the same algorithm for an arbitrary task-to-processor ratio, providing a sufficient infeasibility condition: if the real-time system under test is not schedulable in some cases, the algorithm detects this. We conduct an experimental study of our algorithms on the datasets generated with different utilization values and compare them to several state-of-the-art schedulability tests. The experiments show that the performance of our algorithm exceeds the performance of its analogues while its accuracy is similar. Full article

This paper addresses the time-optimal control problem for a harmonic oscillator characterized by a time-dependent frequency. The primary objective is to determine the minimal time required to transition the system from an initial state, defined by a given position and velocity, to a specified final state, while ensuring that the frequency remains within prescribed bounds. The key challenge lies in identifying the optimal switching times between two available frequencies to meet all boundary conditions efficiently. By examining various boundary scenarios, constructing the reachable set of all admissible trajectories, and employing both analytical techniques and control theory, we develop a robust solution strategy. This work holds particular relevance for practical applications demanding rapid state transitions, such as mechanical vibration control and signal processing, where achieving time-optimal performance is critical. Furthermore, the methods presented are adaptable to a wide range of systems facing similar constraints, providing a versatile and effective framework for time-optimal control. Full article

This paper designs an online adjustment strategy for dynamic quantizer parameters and investigates the sliding mode control (SMC) problem for uncertain switched systems under a limited network communication bandwidth. Due to the limitation of the coding length in practical transmission, coding errors can significantly impact the system’s ideal performance. To address these issues, a dynamic quantizer is introduced to efficiently encode the system state while minimizing quantization error under the constraint of the finite code length. Additionally, a coding and decoding scheme based on dynamic quantization and a suitable sliding mode controller are designed to obtain a closed-loop switched system. Using Lyapunov functions and the average dwell time method, sufficient conditions are derived to guarantee the reachability of the sliding surface and the exponential ultimate bound (EUB) in the mean square for the closed-loop switched system, even in the presence of coding errors and data loss. The theoretical results are validated through numerical simulations, which demonstrate the effectiveness of the proposed approach. Full article

This work addresses the analysis and characterization of deadlocks in discrete-event systems modeled by labeled Petri nets (LPNs) with undistinguishable and unobservable transitions. To provide a solution for the notorious problem, it is essential to present an effective characterization in such a way that deadlock control and synthesis are technically and methodologically possible. To this end, we introduce the notion of dangerous implicit vectors (DIVs), which implicitly threaten the system deadlock-freedom. The set of dead markings is divided into two subsets: dead basis markings (DBMs) and dangerous implicit markings (DIMs). An algorithm is designed to compute the sets of DIVs and DIMs at a given basis state of a system. Moreover, by virtue of linear algebraic equations, we formulate sufficient conditions for identifying the existence of blocking markings in an LPN. Finally, an algorithm is developed to construct an observed graph that is a compendious presentation of the reachability graph of a net system, with respect to the existence of dead reaches. At the end of this paper, experiment results that illustrate the correctness and effectiveness of the reported solution are presented. Full article

This article introduces a new method for modelling and verifying the execution of timed security protocols (TSPs) and their time-dependent security properties. The method, which is novel and reliable, uses an extension of interpreted systems, accessible semantics in multi-agent systems, and timed interpreted systems (TISs) with dense time semantics to model TSP executions. We enhance the models of TSPs by incorporating delays and varying lifetimes to capture real-life aspects of protocol executions. To illustrate the method, we model a timed version of the Needham–Schroeder Public Key Authentication Protocol. We have also developed a new bounded model checking reachability algorithm for the proposed structures, based on Satisfiability Modulo Theories (SMTs), and implemented it within the tool. The method comprises a new procedure for modelling TSP executions, translating TSPs into TISs, and translating TISs’ reachability problem into the SMT problem. The paper also includes thorough experimental results for nine protocols modelled by TISs and discusses the findings in detail. Full article

As an extension of quantitative temporal logic, uncertain temporal logic essentially describes the temporal behavior of uncertain and incomplete systems, thus better solving search and decision-making problems in such systems. Fuzzy linear temporal logic (FLTL) is a focal point in uncertain temporal logic research. However, there are evident shortcomings in the current research outcomes. First, in previous FLTL studies, the practice of obtaining path reachability and formula satisfaction values independently and subsequently selecting the smaller of the two as the satisfaction value metric led to information loss. Furthermore, this simplistic information fusion approach fails to reflect the varying importance of these two types of information to the requirements. Second, computing path reachability and temporal logic formula satisfaction values separately may result in a mismatch between the two pieces of information with respect to the same path segment. Thus, the primary challenge lies in accurately integrating the satisfaction values of temporal logic formulas with the path reachability of the segments that yields these satisfaction values, utilizing various reasonable information synthesis methods, to ensure synchronization between path reachability and formula satisfaction values without incurring information loss. Additionally, it is crucial to reflect the different preference requirements for these two types of information. Moreover, the temporal logic formula characterizes system properties, with its sub-formulas delineating distinct sub-properties. Consequently, considering the varying importance preferences of sub-formulas is also significant. To address these deficiencies, we introduced quality constraint operators into FLTL, resulting in quality-constrained fuzzy linear temporal logic (QFLTL). This incorporation enables the synchronization and comprehensive fusion of path-reachability information and formula satisfaction values within the final semantic metric, thereby resolving the issues related to information synchronization and loss. Furthermore, it can accommodate the differing preference requirements between the two types of information and sub-properties during the information synthesis process. We defined the syntax and semantics of QFLTL and examined its expressive power and properties. Notably, we investigated the decidability of logical decision problems in QFLTL, encompassing validity, satisfiability, and model-checking issues. We proposed corresponding solution algorithms and analyzed their complexities. Full article