Convergence Of Trajectories Research Articles

Complete Stability of Neural Networks With Extended Memristors.

The article considers a large class of delayed neural networks (NNs) with extended memristors obeying the Stanford model. This is a widely used and popular model that accurately describes the switching dynamics of real nonvolatile memristor devices implemented in nanotechnology. The article studies via the Lyapunov method complete stability (CS), i.e., convergence of trajectories in the presence of multiple equilibrium points (EPs), for delayed NNs with Stanford memristors. The obtained conditions for CS are robust with respect to variations of the interconnections and they hold for any value of the concentrated delay. Moreover, they can be checked either numerically, via a linear matrix inequality (LMI), or analytically, via the concept of Lyapunov diagonally stable (LDS) matrices. The conditions ensure that at the end of the transient capacitor voltages and NN power vanish. In turn, this leads to advantages in terms of power consumption. This notwithstanding, the nonvolatile memristors can retain the result of computation in accordance with the in-memory computing principle. The results are verified and illustrated via numerical simulations. From a methodological viewpoint, the article faces new challenges to prove CS since due to the presence of nonvolatile memristors the NNs possess a continuum of nonisolated EPs. Also, for physical reasons, the memristor state variables are constrained to lie in some given intervals so that the dynamics of the NNs need to be modeled via a class of differential inclusions named differential variational inequalities.

An improved whale optimization algorithm for UAV swarm trajectory planning

For the problem of trajectory planning in the small-scale unmanned aerial vehicle (UAV) swarms, classical intelligent algorithms and newly emerged bio-inspired algorithms often suffer from being trapped in local optima, leading to suboptimal solutions. Moreover, these algorithms fail to ensure optimal solutions and convergence speed in high-speed dynamic UAV networking scenarios. In response to these challenges, this paper proposes an Improved Whale Optimization Algorithm (IWOA) that considers the global perspective. To address the issue of initial solution diversity, an opposition-based learning method is introduced by constructing a reverse population, enhancing the diversity of the initial population. To overcome the problem of the algorithm getting trapped in local optima, random convergence factors, and end-point random neighborhood perturbations are incorporated to accelerate the global convergence speed of the IWOA and prevent it from getting stuck in local optima. Additionally, a grid digital elevation model is employed when constructing the experimental environment model, considering various physical constraints of the UAV swarm. This ensures that the simulation validation of the IWOA algorithm is closer to real-world scenarios. Simulation results indicate that, compared to commonly used algorithms, the IWOA can generate more optimal trajectories for drone swarm planning under constraints resembling real-world scenarios. It exhibits better performance in trajectory evaluation, convergence speed, and stability, thereby meeting the requirements of trajectory planning for small-scale UAV swarms. The proposed IWOA enhances UAV swarm coordination and efficiency, significantly impacting real-world applications in various fields.

Tikhonov regularized second-order plus first-order primal-dual dynamical systems with asymptotically vanishing damping for linear equality constrained convex optimization problems

In this paper, in the setting of Hilbert spaces, we consider a Tikhonov regularized second-order plus first-order primal-dual dynamical system with asymptotically vanishing damping for a linear equality constrained convex optimization problem. It is shown that convergence properties of the proposed inertial dynamical system depend upon the choice of the Tikhonov regularization parameter. When the Tikhonov regularization parameter decays rapidly to zero, we derive the fast convergence rates of the primal-dual gap, the objective residual, the feasibility violation and the gradient norm of the objective function along the generated trajectory. When the Tikhonov regularization parameter decreases slowly to zero, we prove the strong convergence of the primal trajectory of the Tikhonov regularized dynamical system to the minimal norm solution of the linear equality constrained convex optimization problem. Numerical experiments are performed to illustrate the efficiency of our approach.

Feedback mechanisms controlling Antarctic glacial-cycle dynamics simulated with a coupled ice sheet–solid Earth model

Abstract. The dynamics of the ice sheets on glacial timescales are highly controlled by interactions with the solid Earth, i.e., the glacial isostatic adjustment (GIA). Particularly at marine ice sheets, competing feedback mechanisms govern the migration of the ice sheet's grounding line (GL) and hence the ice sheet stability. For this study, we developed a coupling scheme and performed a suite of coupled ice sheet–solid Earth simulations over the last two glacial cycles. To represent ice sheet dynamics we apply the Parallel Ice Sheet Model (PISM), and to represent the solid Earth response we apply the 3D VIscoelastic Lithosphere and MAntle model (VILMA), which, in addition to load deformation and rotation changes, considers the gravitationally consistent redistribution of water (the sea-level equation). We decided on an offline coupling between the two model components. By convergence of trajectories of the Antarctic Ice Sheet deglaciation we determine optimal coupling time step and spatial resolution of the GIA model and compare patterns of inferred relative sea-level change since the Last Glacial Maximum with the results from previous studies. With our coupling setup we evaluate the relevance of feedback mechanisms for the glaciation and deglaciation phases in Antarctica considering different 3D Earth structures resulting in a range of load-response timescales. For rather long timescales, in a glacial climate associated with the far-field sea-level low stand, we find GL advance up to the edge of the continental shelf mainly in West Antarctica, dominated by a self-amplifying GIA feedback, which we call the “forebulge feedback”. For the much shorter timescale of deglaciation, dominated by the marine ice sheet instability, our simulations suggest that the stabilizing sea-level feedback can significantly slow down GL retreat in the Ross sector, which is dominated by a very weak Earth structure (i.e., low mantle viscosity and thin lithosphere). This delaying effect prevents a Holocene GL retreat beyond its present-day position, which is discussed in the scientific community and supported by observational evidence at the Siple Coast and by previous model simulations. The applied coupled framework, PISM–VILMA, allows for defining restart states to run multiple sensitivity simulations from. It can be easily implemented in Earth system models (ESMs) and provides the tools to gain a better understanding of ice sheet stability on glacial timescales as well as in a warmer future climate.

Design of adaptive barrier function-based backstepping finite time guidance control for interceptor-target systems

This study introduces an optimal finite-time backstepping sliding mode guidance law, leveraging an adaptive continuous barrier function to examine the interceptor-target system’s motion in cylindrical coordinates. The integration of sliding mode and backstepping controllers enhances the robustness and performance of the proposed guidance method amidst external disturbances and model uncertainties. This combination also improves transient response, reduces chattering, and lowers control efforts. The adaptive continuous barrier function employed in this method eliminates the need for prior knowledge of the upper bounds of uncertainties and disturbances. Additionally, it ensures that adaptation gains are not overestimated, leading to more accurate performance and complete elimination of chattering. The Lyapunov stability theory is used to prove the finite-time convergence of the state trajectories of the interceptor-target system to a predefined neighborhood around the origin, despite external disturbances and uncertainties. A genetic optimization algorithm is utilized to optimally select the parameters of the guidance method. The efficacy of the proposed technique is validated through simulation results and real-time experiments on the Baseline Speedgoat Real-Time Target Machine platform. The method’s effectiveness and performance are further demonstrated through analysis and simulation in the MATLAB/Simulink environment.

Evolution of Information Infrastructures in Healthcare as Convergence of Digital Trajectories

Evolution of Information Infrastructures in Healthcare as Convergence of Digital Trajectories

Accelerated optimization through time-scale analysis of inertial dynamics with asymptotic vanishing and Hessian-driven dampings

ABSTRACT Gradient optimization algorithms can be effectively studied from the perspective of ordinary differential equations (ODEs), where these algorithms are obtained through the temporal discretization of the continuous dynamics. This approach provides a powerful tool to comprehend acceleration phenomena in optimization, thanks to time-scale techniques and Lyapunov analysis. In the context of a Hilbert setting, our study focuses on the rapid optimization properties of inertial dynamics, which combine asymptotically vanishing viscous damping with Hessian-driven damping. These dynamics are derived as high-resolution ODEs from both the Nesterov accelerated gradient method and the Ravine method. For a general differentiable convex function f, choosing the viscous damping coefficient in the form of α / t with 3 $ ]]> α > 3 guarantees an inverse quadratic convergence rate of the values, i.e. o ( 1 / t 2 ) , the weak convergence of trajectories towards optimal solutions, and the fast convergence of gradients towards zero. By carefully scaling the dynamics temporally and based on the tuning of the damping coefficient in front of the Hessian term, we identify the limit dynamic as α becomes large. In particular, we examine the case where the limit dynamic corresponds to the Levenberg-Marquardt regularization of Newton's continuous method. This explanation accounts for the aforementioned fast convergence properties and provides fresh insights into the complexity of these methods, which play a central role in optimization for high-dimensional problems. The interplay between numerical algorithms and continuous-time ODEs plays a fundamental role in our analysis for the design and understanding of accelerated optimization algorithms. Numerical experiments are presented to illustrate and support the theoretical results.

Research of integrated spiral maneuvering and guidance based on virtual target

To address the issue of interception, maneuvering, and precise guidance during the final stage of gliding flight, a study was conducted on the design of integrated maneuvering and guidance using a spiral trajectory as the focus. Firstly, the spiral trajectory was decomposed into an optimal reference trajectory considering the glide angle constraint and a relative motion trajectory. Then, virtual circular moving targets were introduced to meet the requirements of spiral maneuvering and virtual trajectory moving targets were introduced to ensure trajectory convergence, based on the task division of different flight stages. Finally, the guidance rate of the reference trajectory was designed based on optimal control theory, and the guidance rate of the spiral maneuver was designed based on pure proportional guidance. Simulation results show that this guidance method can enable the vehicle to perform active spiral maneuvers, and control the landing deviation and impact angle deviation to within 2.5 m and 1 degree, respectively.

Guaranteed Trajectory Tracking under Learned Dynamics with Contraction Metrics and Disturbance Estimation

This paper presents a contraction-based learning control architecture that allows for using model learning tools to learn matched model uncertainties while guaranteeing trajectory tracking performance during the learning transients. The architecture relies on a disturbance estimator to estimate the pointwise value of the uncertainty, i.e., the discrepancy between a nominal model and the true dynamics, with pre-computable estimation error bounds, and a robust Riemannian energy condition for computing the control signal. Under certain conditions, the controller guarantees exponential trajectory convergence during the learning transients, while learning can improve robustness and facilitate better trajectory planning. Simulation results validate the efficacy of the proposed control architecture.

Diffusive lensing as a mechanism of intracellular transport and compartmentalization.

While inhomogeneous diffusivity has been identified as a ubiquitous feature of the cellular interior, its implications for particle mobility and concentration at different length scales remain largely unexplored. In this work, we use agent-based simulations of diffusion to investigate how heterogeneous diffusivity affects the movement and concentration of diffusing particles. We propose that a nonequilibrium mode of membrane-less compartmentalization arising from the convergence of diffusive trajectories into low-diffusive sinks, which we call 'diffusive lensing,' is relevant for living systems. Our work highlights the phenomenon of diffusive lensing as a potentially key driver of mesoscale dynamics in the cytoplasm, with possible far-reaching implications for biochemical processes.

Diffusive lensing as a mechanism of intracellular transport and compartmentalization

While inhomogeneous diffusivity has been identified as a ubiquitous feature of the cellular interior, its implications for particle mobility and concentration at different length scales remain largely unexplored. In this work, we use agent-based simulations of diffusion to investigate how heterogeneous diffusivity affects the movement and concentration of diffusing particles. We propose that a nonequilibrium mode of membrane-less compartmentalization arising from the convergence of diffusive trajectories into low-diffusive sinks, which we call ‘diffusive lensing,’ is relevant for living systems. Our work highlights the phenomenon of diffusive lensing as a potentially key driver of mesoscale dynamics in the cytoplasm, with possible far-reaching implications for biochemical processes.

Finite and fixed‐time stabilization of discrete‐time systems using passivity‐based control

AbstractIn this paper, we propose a new passivity‐based controller design technique for discrete‐time fully actuated systems. The controller establishes finite‐time and fixed‐time convergence of dynamical system trajectories to an equilibrium state. A new form of a dissipation function, selected a priori, is introduced to design a feedback control rule to achieve such convergence properties. An energy shaping and damping injection methodology is extended to achieve non‐asymptotic stabilization. A numerical example to validate the proposed methods is provided.

Deep reinforcement learning and decoupling proportional-integral-derivative control of a humanoid cable-driven hybrid robot

Aiming at the application of robots in service, medical treatment, rehabilitation, and other fields, a humanoid cable-driven hybrid robot by imitating the structure of human arm is designed in this article. The robot is composed of a cable–spool–pulley system, series mechanism, and coaxial spherical parallel mechanism, which can achieve six degrees of freedom movement in space. A challenge with cable driving is that the movement of the rear end joints (such as pitch and roll) can alter the length or tension of the cable driven by the frontend joints, resulting in joint coupling. This interference can lead to a decrease in the motion accuracy of the robotic arm. In addition, it also affects the durability of cables or mechanical components such as bearings and pulleys. Considering the joint motion coupling phenomenon caused by the cable-driven, the decoupling method is proposed, and the kinematic model of the robot is established. To solve the nonlinear coupling characteristics and the uncertainty of the dynamics parameters, a controller is proposed for the humanoid cable-driven hybrid robot, combining proportional-integral-derivative (PID) control based on decoupling method (DC-PID) and the double delay deep deterministic policy gradient (TD3) deep reinforcement learning algorithm. The trajectory tracking of the end-effector position and orientation are controlled by using DC-PID and TD3. And the simulation results show that the proposed control method has good trajectory tracking and convergence performance according to trajectory tracking error and training reward. Finally, the humanoid cable-driven hybrid robot prototype is developed. The experimental results of coupling compensation show that the average maximum error of the joint is reduced by 77.58% after considering the coupling compensation of the joint. The results of the controller validation experiments show that the DC-PID controller reduces the maximum error in each axis by 11.54%, 35.29%, and 40.16%, respectively, compared to the open-loop experiments.

Optimizing with Attractor: A Tutorial

This tutorial presents a novel search system—the Attractor-Based Search System (ABSS)—that can solve the Traveling Salesman Problem very efficiently with optimality guarantee. From the perspective of dynamical systems, a heuristic local search algorithm for an NP-complete combinatorial problem is a discrete dynamical system. In a local search system, an attractor drives the search trajectories into the vicinity of a globally optimal point in the solution space, and the convergence of local search trajectories makes the search system become a global and deterministic system. The attractor contains a small set of the most promising solutions to the problem. The attractor can reduce the problem size exponentially, and thus make the exhaustive search feasible. Therefore, this new search paradigm is called optimizing with attractor. The ABSS consists of two search phases: local search phase and exhaustive search phase. The local search process is used to quickly construct the attractor in the solution space, and the exhaustive search process is used to completely search the attractor to identify the optimal solution. Therefore, the exact optimal solution can be found quickly by combining local search and exhaustive search. This tutorial introduces the concept of an attractor in a local search system, and describes the process of optimizing with the attractor, using the Traveling Salesman Problem as the study platform.

Distributed control of autonomous watercraft dynamics using physicomimetics and robust synthesis for disturbance rejection

This study investigates the guidance and control of an autonomous swarm of surface watercraft with focusing on distributed control laws in scenarios with and without boat dynamics. Through comprehensive simulations, we explore the behavior of boats under varying conditions in terms of analyzing their trajectories, velocities, yaw rates, and angles. The inclusion of boat dynamics significantly impacts these parameters for affecting the efficiency and convergence of boat trajectories towards the homing point. We observe that boats operating under dynamic conditions show much smoother convergence and demonstrate improved collision avoidance behaviors although their movements are slower due to the increased complexities introduced. The introduction of noise in simulations enhances realism and robustness regarding mimicking real-world scenarios. Our findings emphasize the importance of considering boat dynamics in autonomous systems where offering insights into energy-efficient control strategies for surface watercraft. This research contributes to the understanding of swarm behavior and creates a path for practical applications in marine robotics.

Microscopic derivation of Vlasov–Dirac–Benney equation with short-range pair potentials

We present a probabilistic proof of the mean-field limit and propagation of chaos of a N-particle system in three dimensions with pair potentials of the form N3β−1ϕ(Nβx) for β∈[0,17] and ϕ∈ L∞(R3)∩L1(R3) . In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov–Dirac–Benney system with delta-like interactions. The proof is based on a Gronwall estimate for the maximal distance between the exact microscopic dynamics and the approximate mean-field dynamics. Thus our result leads to a derivation of the Vlasov–Dirac–Benney equation from the microscopic N-particle dynamics with a strong short range force.

Robust trajectory tracking of a 3-DOF robotic arm using a Super-Twisting Fast finite time Non-singular Terminal Sliding Mode Control in the presence of perturbations

Extensive research has focused on enhancing the efficiency and stability of robotic arms. Sliding mode control (SMC) is commonly used in industrial robots due to its robustness and simplicity. However, SMC approaches have challenges such as chattering and slow convergence rates which can compromise tracking accuracy. To address these issues, this paper proposes a novel Super-Twisting Fast Non-singular Terminal Sliding Mode Control (ST-FNTSMC) strategy for a 3-DOF arm robot. The proposed approach significantly improves trajectory tracking accuracy, robustness, and convergence time and eliminates chattering. The proposed controller was tested in the presence of model mismatches and external disturbances. The super-twisting methodology avoided chattering effects and increased robustness against perturbations. Two Lyapunov functions ensure closed system stability and finite-time convergence. The designed ST-FNTSMC controller is implemented in real-time using a Smart Man Robot manipulator. Its performance is compared to other sliding mode controllers, such as conventional PID Sliding Mode Control (PID-SMC), Non-singular Terminal Sliding Mode Control (NTSMC), and Fast Non-singular Terminal Sliding Mode Control (FNTSMC). Experimental results demonstrate the superior performance of the proposed controller, highlighting its effectiveness in improving the efficiency and stability of industrial robots.

Attack-tolerant control for Markovian jump systems with stochastic sampling: A sliding mode scheme

This work discusses the sampled-data sliding mode control problem for Markovian jump systems in the presence of Denial-of-Service (DoS) attacks, where the state signals are sampled stochastically and transmitted via vulnerable communication networks. The successful transmitted sequence with double randomness is established, and the probability distribution on both the attacked signals and stochastic delays are given. With the aid of the coordinate transformation, a linear sliding surface is constructed and the mode-dependent sliding mode controller is designed by means of only successfully transmitted state signals. Moreover, the resultant closed-loop systems composed of partial state and sliding mode variable are described as a continuous time-delay system with double randomness depending on the probability distribution characteristics of both the sampling periods and DoS attacks. Both the mean-square exponential ultimate boundedness of the closed-loop system and the reachability of the sliding surface are analyzed and the corresponding sufficient conditions are derived. Meanwhile, an optimization problem with genetic algorithm is formulated for achieving fast convergence of the state trajectory, small ultimate bound of the system state and small sliding bound. Finally, the simulation results via the numerical example are provided to illustrate the proposed sampled-data sliding mode control scheme.

Novel Stability Conditions for Nonlinear Monotone Systems and Consensus in Multiagent Networks

We introduce a novel definition of monotonicity, termed “type-K” in honor of Kamke, and study nonlinear type-K monotone dynamical systems possessing the plus-subhomogeneity property, which we call “K-subtopical” systems after Gunawardena and Keane. We show that type-K monotonicity, which is weaker than strong monotonicity, is also equivalent to monotonicity for smooth systems evolving in continuous-time, but not in discrete-time. K-subtopical systems are proved to converge toward equilibrium points, if any exists, generalizing the result of Angeli and Sontag about convergence of topical systems' trajectories toward the unique equilibrium point when strong monotonicity is considered. The theory provides an new methodology to study the consensus problem in nonlinear multi-agent systems (MASs). Necessary and sufficient conditions on the local interaction rule of the agents ensuring the K-subtopicality of MASs are provided, and consensus is proven to be achieved asymptotically by the agents under given connectivity assumptions on directed graphs. Examples in continuous-time and discrete-time corroborate the relevance of our results in different applications

Robust limit cycle control for finite-time generation of sustained oscillations in nonlinear systems with mixed dead-zone and saturation

This paper suggests a novel limit cycle control technique, based on the finite-time stability scheme, for producing sustained oscillations in a class of nonlinear complex systems. The considered systems are constrained to input dead-zone and saturation simultaneously and suffer from non-vanishing perturbations due to model uncertainties, and external disturbances. For this purpose, an innovative sliding manifold is proposed based on the topology of the wanted limit cycle which provides the conditions for developing the finite time stability concept for limit sets. With the aid of the finite-time command filter and the cooperation of the sliding mode control (SMC) method with the backstepping technique, a robust controller is constructed which guarantees the finite-time convergence of phase trajectories of the closed-loop system toward the wanted limit cycle. This leads to the creation of sustained oscillations, with the desirable amplitude and frequency, in the system's output. The stability analysis is proved based on the Lyapunov approach through the set stability concept. Finally, the effectiveness of the suggested methodology is confirmed by a variety of simulations.