Invariance Principle Research Articles

Reynolds number effect on nonlinear aerodynamic characteristics of compressors and model predictions

The widespread use of unmanned aerial vehicles has increasingly drawn attention to the operating conditions of compressors at low Reynolds number conditions. The nonlinear aerodynamic characteristics of compressors vary significantly under different Reynolds numbers, making the prediction of these variations and the associated instability boundaries a critical issue. This paper starts with the bistable equilibrium surface equation of the compressor system and proposes a novel quantitative prediction method for the nonlinear aerodynamic characteristics of compressors at different Reynolds numbers. The method is based on the nonlinear characteristics of state and control parameters, combined with the principle of topological invariance and the radial basis function neural network. The establishment of the prediction method and its results indicate that the “bistable” nature observed in real compressor operating conditions shares the same topological properties as the equilibrium surface equation described by the cusp catastrophe model, which accounts for the multi-parameter influences on the state changes of the compressor system. The established equilibrium surface equation can conveniently characterize the different hysteretic behaviors of compressor rotational stall under multi-parameter influences and assess the size of hysteresis loops. By developing a high-precision topological mapping method, a topological mapping relationship between the compressor equilibrium surface equation and the nonlinear aerodynamic characteristics of real compressors has been constructed, allowing the theoretical compressor characteristics to predict unknown nonlinear aerodynamic features at various Reynolds numbers, with model prediction errors within 5%.

Deviation inequalities for the elephant random walk with random step sizes

Recently, the asymptotic behaviors of the elephant random walk have attracted a lot of attentions. Many interesting properties for the elephant random walk have been established in the past few years, such as strong invariance principles, central limit theorems, functional limit theorems and law of large numbers. In this paper, we give some deviation inequalities for the elephant random walk with random step sizes with various moment conditions. More precisely, we are interested in giving quantitative estimates for the convergence rates in the law of large numbers.

Self‐Triggered Formation Control of Heterogeneous Quadrotors on TSE(3)N in Presence of External Disturbances

ABSTRACTIn this paper, the formation control strategy is designed for a class of multi‐agent systems with heterogeneous quadrotor UAVs structure subjected to external disturbances and modeling uncertainties. Specifically, a self‐triggered scheme is developed to reduce network load, which computes the next triggering instant based on past and current transmitted states. Further, the unmeasurable states are estimated by applying an extended state observer. Based on the LaSalle's invariance principle, Lyapunov‐based approach is analyzed with some sufficient conditions proposed for the considered heterogeneous systems, while avoiding the Zeno phenomenon. Finally, we validate the effectiveness of the proposed algorithms through a simulation result.

Consensus-Based Formation Control for Heterogeneous Multi-Agent Systems in Complex Environments

The purpose of this paper is to develop formation control strategies for heterogeneous multi-intelligent-agent systems in complex environments, with the goal of enhancing their performance, reliability, and stability. Complex flight conditions, such as navigating narrow gaps in urban high-rise buildings, pose considerable challenges for agent control. To address these challenges, this paper proposes a consensus-based formation strategy that integrates graph theory and multi-consensus algorithms. This approach incorporates time-varying group consistency to strengthen fault tolerance and reduce interference while ensuring obstacle avoidance and formation maintenance in dynamic environments. Through a Lyapunov stability analysis, combined with minimum dwell time constraints and the LaSalle invariance principle, this work proves the convergence of the proposed control scheme under changing network topologies. Simulation results confirm that the proposed strategy significantly improves system performance, mission execution capability, autonomy, synergy, and robustness, thereby enabling agents to successfully maintain formation and avoid obstacles in both homogeneous and heterogeneous clusters in complex environments.

Load Energy Coupling-Based Underactuated Control Method on Trajectory Planning and Anti-Swing for Bridge Cranes

Abstract The closed-loop control methods of bridge crane systems adopt the real-time signal feedback to improve the performance of the systems. The energy-based closed-loop control method of bridge crane systems can be used to design the controller by constructing the energy function of the system, which avoids the direct analysis of the complex motion state of the systems and has the clear physical significance. However, the conventional energy-based control methods have some problems, including slow response, poor performance of eliminating pendulum and less parameters reflected by the control law. An novel load energy coupling-based underactuated control method for bridge cranes is proposed. The relationship between the energy of the bridge crane system and the energy of the load system is analyzed with the dynamic model of the two-dimensional bridge crane. The coupling function based on the load displacement and swing angle is constructed. The driving force of the system is obtained by Lyapunov method and the control law is designed. The boundedness and convergence of the closed-loop control system is explained by the principle of LaSalle invariance. With the comparison experiments and results analysis, it is shown that the proposed method can obtain the better performance of eliminating swing and realize the accurate positioning of the trolley. It also reflects this proposed method effectively reduces the dependence on model parameters and simplifies the control law.

Weak Quenched Invariance Principle for Random Walk with Random Environment in Time

Weak Quenched Invariance Principle for Random Walk with Random Environment in Time

Iterated invariance principle for random dynamical systems

Abstract We prove a weak iterated invariance principle for a large class of non-uniformly expanding random dynamical systems. In addition, we give a quenched homogenization result for fast-slow systems in the case when the fast component corresponds to a uniformly expanding random system. Our techniques rely on the appropriate martingale decomposition.

DYNAMICAL BEHAVIORS OF A FRACTIONAL-ORDER PREDATOR–PREY MODEL: INSIGHTS INTO MULTIPLE PREDATORS COMPETING FOR A SINGLE PREY

In this paper, we investigate the dynamical behaviors of a modified Bazykin-type two predator-one prey model involving the intra-specific and inter-specific competition among predators. A Caputo fractional-order derivative is utilized to include the influence of the memory on the constructed mathematical model. The mathematical validity is ensured by showing the model always has a unique, non-negative and bounded solution. Four kinds of equilibria are well identified which represent the condition when all populations are extinct, both two predators are extinct, only the first predator is extinct, only the second predator is extinct, and all populations are extinct. The Matignon condition is given to identify the dynamics around equilibrium points. The Lyapunov direct method, the Lyapunov function, and the generalized LaSalle invariant principle are also provided to show the global stability condition of the model. To explore the dynamics of the model more deeply, we utilize the predictor–corrector numerical scheme. Numerically, we find the forward bifurcation and the bistability conditions by showing the bifurcation diagram, phase portraits, and the time series. The biological interpretation of the analytical and numerical results is described explicitly when an interesting phenomenon occurs.

Collision‐free flocking control of uncertain multi‐agent systems: Combining current learning adaptive control and projection operator

AbstractAdaptive flocking control of multi‐agent systems faces challenges in handling uncertainties and ensuring safety. This paper aims to address these issues based on the hypothesis that the uncertain parameters are bounded. First, a concurrent learning adaptive control method relaxes the persistently excitation condition for parameter convergence, enabling adaptability with interval excitation only. Second, an element‐wise projection operator bounds parameter estimates within known intervals, precomputing collision avoidance conditions, and guaranteeing safety. Third, combining with the aforementioned methods, a distributed flocking algorithm incorporates limited sensing range in a moving region, achieving collision avoidance, connectivity, and cohesion via bounded potential functions. LaSalle's invariance principle shows that parameter estimates converge within bounds, collision avoidance conditions hold, and system stability is achieved. Simulations validate enhanced adaptability, guaranteed safety, and the expected cooperative flocking motion. The proposed approach addresses critical challenges for real‐world deployment of swarm technology.

An invariance principle for the 2d weakly self-repelling Brownian polymer

An invariance principle for the 2d weakly self-repelling Brownian polymer

Non-linear realizations and invariant action principles in higher gauge theory

Non-linear realizations and invariant action principles in higher gauge theory

A Novel Low-Winding-Loss Circulated Interleaving Structure for Planar Transformers Based on Current Invariance Principle

A Novel Low-Winding-Loss Circulated Interleaving Structure for Planar Transformers Based on Current Invariance Principle

On the Strongest Principles of Rational Belief Assignment

Abstract We show that in Polyadic Pure Inductive Logic the Invariance Principle, based on consideration of symmetry with respect to automorphisms, has only a trivial solution, namely the polyadic equivalent of Carnap’s $$c_0$$ c 0 . (This extends a result proved earlier in the unary case.) We then consider the Exchangeable Invariance Principle, a symmetry principle which is a weakening of the Invariance Principle and has been proven to be strictly stronger than the Permutation Invariance Principle. We show that the Exchangeable Invariance Principle follows from Spectrum Exchangeability, a principle not obviously based on symmetry but based on irrelevance, that is treating certain features as irrelevant for the purpose of belief assignment, and that the converse does not hold. We conclude that Spectrum Exchangeability is the strongest currently known rational principle of belief assignment in Pure Inductive Logic that does not lead to the aforementioned trivial solution.

Vibration Suppression and Adaptive Fault‐Tolerant Control for Three‐Dimensional Flexible Rotating Manipulator With Input Signal Constraints

ABSTRACTBased on a three‐dimensional flexible, rotating manipulator, this paper is dedicated to the issue of vibration suppression and angle planning subject to actuator failures and input signal constraints. Considering that a flexible link belongs to a distributed parameter system, the motion model is obtained by Hamilton's principle and described by partial differential equations (PDEs). A novel fault‐tolerant control law is proposed to eliminate the elastic deflection and vibration of the flexible link and to follow the desired angle of the rotating base and the manipulator with the appearance of actuator failures. The adaptive operator based on projection mapping is adopted to estimate the loss of the actuator. In addition, the hyperbolic tangent function is employed to constrain the input signal so that it is within an adjustable interval. The Lyapunov's method and LaSalle invariance principle are applied to prove the stability and convergence of the closed‐loop system. The simulation results are provided to demonstrate the effectiveness of the proposed method.

Analysis and optimal control of a fractional-order SEAIR epidemic model with two-strains

This study focuses on the analysis and optimal control of a fractional-order SEAIR epidemic model, which consists of two strains. The proposed model's well-posedness is evaluated by examining its existence, uniqueness, non-negativity, and boundedness. Furthermore, two basic regeneration numbers are computed, and the model's two equilibrium points are the endemic and disease-free equilibriums. Using suitable Lyapunov functions and LaSalle's invariance principle, we conduct a stability analysis to examine the global stability of these steady states. Ultimately, using Pontryagin's Maximum Principle, we created a time-dependent optimal control problem. We evaluated the impact of model parameters on the dynamics of disease transmission and determined the efficacy of control measures using numerical simulations.

Computational Study of a Fractional-Order HIV Epidemic Model with Latent Phase and Treatment

In this work, we propose and investigate a model of the dynamical behavior of HIV/AIDS transmission by considering a new compartment of the population with HIV: the latent asymptomatic class. The infection reproduction number that stabilizes the global dynamics of the model is evaluated. We analyze the model’s global asymptotic stability using the Lyapunov function and LaSalle’s invariance principle. To identify the primary factors affecting the dynamics of HIV/AIDS, a sensitivity analysis of the model parameters is conducted. We also examine a fractional-order HIV model using the Caputo fractional differential operator. Through qualitative analysis and applications, we determine the existence and uniqueness of the model’s solutions. We derive some results from the fixed-point theorem and Ulam–Hyers stability. Ultimately, the obtained numerical simulation results are in agreement with the analytical outcomes obtained from the model analysis. Our findings illustrate the efficacy of the fractional model in depicting the dynamics of the HIV/AIDS epidemic and offering critical insights for the formulation of effective control strategies. The results show that early intervention and treatment in the latent phase of infection can decrease the spread of the disease and its progression to AIDS, as well as increase the success of treatment strategies.

Advanced Control for Shipboard Cranes with Asymmetric Output Constraints

Considering the anti-swing control and output constraint problems of shipboard cranes, a nonlinear anti-swing controller based on asymmetric barrier Lyapunov functions (BLFs) is designed. First, model transformation mitigates the explicit effects of ship roll on the desired position and payload fluctuations. Then, a newly constructed BLF is introduced into the energy-based Lyapunov candidate function to generate nonlinear displacement and angle constraint terms to control the rope length and boom luffing angle. Among these, constraints with positive bounds are effectively handled by the proposed BLF. For the swing constraints of the unactuated payload, a carefully designed relevant constraint term is embedded in the controller by constructing an auxiliary signal, and strict theoretical analysis is provided by using a reductio ad absurdum argument. Additionally, the auxiliary signal effectively couples the boom and payload motions, thereby improving swing suppression performance. Finally, the asymptotic stability is proven using LaSalle’s invariance principle. The simulation comparison results indicate that the proposed method exhibits satisfactory performance in swing suppression control and output constraints. In all simulation cases, the payload swing angle complies with the 3° constraint and converges to the desired range within 6 s. This study provides an effective solution to the control challenges of shipboard crane systems operating in confined spaces, offering significant practical value and applicability.

Towards a blank design method for manufacturing big-tapered profiled ring disk by spinning-rolling process

Towards a blank design method for manufacturing big-tapered profiled ring disk by spinning-rolling process

Local and global stability of a fractional viral infection model with two routes of propagation, cure rate and non-lytic humoral immunity

A fractional viral model is proposed in this work, as fractional-order calculus is considered more suitable than integer-order calculus for modeling virological systems with inherent memory and long-range interactions. The model incorporates virus-to-cell infection, cell-to-cell transmission, cure rate, and humoral immunity. Additionally, the non-lytic immunological mechanism, which prevents viral reproduction and reduces cell infection, is included. Caputo fractional derivatives are utilized in each compartment to capture long-term memory effects and non-local behavior. It is demonstrated that the model has nonnegative and bounded solutions. Three equilibrium states are identified in the improved viral model: the virus-clear steady state $\mathcal{G}^{\circ }$, the immunity-free steady state $\mathcal{G}_{1}^{\star}$ and the infection steady state with humoral immunity $\mathcal{G}_{2}^{\star }$. The local stability of the equilibria is investigated using the Routh-Hurwitz criteria and the Matignon condition, while the global stability is shown through the Lyapunov approach and the fractional LaSalle invariance principle. Finally, the theoretical conclusions are validated by numerous numerical simulations.

Quantum chromodynamics Lagrangian density and SU(3) gauge symmetry: A fractional approach

Quantum ChromoDrynamics (QCD) is a quantum field theory, which describes the strong interaction. It explains how quarks and gluons combine to form hadrons. QCD is part of the standard model of particle physics, along with ElectroWeak (EW) theory. The aim of this study is to explore fractional orders in the QCD Lagrangian density using the Atangana–Baleanu fractional derivative. Starting from the fractional Euler–Lagrange equation, we were able to derive the equation of motion of the quark and gluon fields. Then, based on fractional QCD Lagrangian density, the fractional Hamilton equations were obtained. We demonstrate that the principle of local gauge invariance, a fundamental symmetry in QCD, is preserved under a fractional extension of the theory. Our findings indicate that the fractional equations of QCD encompass the classical equations as a specific case, offering a broader perspective on quark-gluon dynamics. The fractional QCD Lagrangian density provides new insights that are not accessible through classical derivatives, potentially enhancing our understanding of quark-gluon plasmas and contributing to advancements in collider phenomenology and precision measurements. This study opens new avenues for exploring the fundamental nature of the strong interaction and its implications for particle physics beyond the Standard Model.