Initial State Of System Research Articles

Robust adaptive prescribed performance control for nonlinear systems with arbitrary initial states

The initial state of nonlinear MIMO strict feedback systems is often unpredictable and random, and existing prescribed performance control (PPC) techniques can only guarantee system output constraints within a fixed initial value performance boundary. In this paper, an adaptive PPC method tailored for nonlinear systems was introduced, enabling the system stable tracking regardless of the arbitrary initial states. To guarantee that the tracking error consistently meets the prescribed performance boundary(PPB) envelope, we introduce a nonlinear mapping between the initial value of the prescribed performance functions(PPFs) and the system tracking errors, resulting in an asymmetric time-varying performance boundary. Building upon this foundation, an adaptive PPC scheme is proposed under the backstepping framework, integrated with a nonlinear disturbance observer (NDO). This method ensures that the tracking error remains within the desired PPB, regardless of external disturbances or system initial states. To prevent ‘complexity explosion’, a dynamic surface control (DSC) technology is employed to filter the virtual control signals of each subsystem. Furthermore, the ultimate uniform boundedness of the closed-loop signal has been demonstrated, and a practical flight vehicles roll control case was introduced to validate the efficacy of the proposed method.

Optimal Consensus Control for Continuous-Time Linear Multiagent Systems: A Dynamic Event-Triggered Approach.

This article investigates the optimal consensus problem for general linear multiagent systems (MASs) via a dynamic event-triggered approach. First, a modified interaction-related cost function is proposed. Second, a dynamic event-triggered approach is developed by constructing a new distributed dynamic triggering function and a new distributed event-triggered consensus protocol. Consequently, the modified interaction-related cost function can be minimized by applying the distributed control laws, which overcomes the difficulty in the optimal consensus problem that seeking the interaction-related cost function needs all agents' information. Then, some sufficient conditions are obtained to guarantee optimality. It is shown that the developed optimal consensus gain matrices are only related to the designed triggering parameters and the desirable modified interaction-related cost function, relaxing the constraint that the controller design requires the knowledge of system dynamics, initial states, and network scale. Meanwhile, the tradeoff between optimal consensus performance and event-triggered behavior is also considered. Finally, a simulation example is provided to verify the validity of the designed distributed event-triggered optimal controller.

Analytical insights into the interplay of momentum, multiplicity, and the speed of sound in heavy-ion collisions

We introduce a minimal model of ultracentral heavy-ion collisions to study the relation between the speed of sound of the produced plasma and the final particles' energy and multiplicity. We discuss how the particles' multiplicity Ntot and average energy Etot/Ntot is related to the speed of sound cs by cs2=dln(Etot/Ntot)/dlnNtot if the fluid is inviscid, its speed of sound is constant and all final particles can be measured. We show that finite rapidity cuts on the particles' multiplicity N and energy E introduce corrections between cs2 and dln(E/N)/dlnN that depend on the system's lifetime. We study analytically these deviations with the Gubser hydrodynamic solution, finding that, for ultrarelativistic bosons, they scale as the ratio of the freeze-out temperature TFO over the maximum initial temperature of the fluid T0; the nonthermodynamic aspect of these corrections is highlighted through their dependence on the system's initial conditions. Published by the American Physical Society 2024

Transient GI/MSP/1/N Queue.

A non-zero correlation between service times can be encountered in many real queueing systems. An attractive model for correlated service times is the Markovian service process, because it offers powerful fitting capabilities combined with analytical tractability. In this paper, a transient study of the queue length in a model with MSP services and a general distribution of interarrival times is performed. In particular, two theorems are proven: one on the queue length distribution at a particular time t, where t can be arbitrarily small or large, and another on the mean queue length at t. In addition to the theorems, multiple numerical examples are provided. They illustrate the development over time of the mean queue length and the standard deviation, along with the complete distribution, depending on the service correlation strength, initial system conditions, and the interarrival time variance.

Active particle motion in Poiseuille flow through rectangular channels.

We investigate the dynamics of a pointlike active particle suspended in fluid flow through a straight channel. For this particle-fluid system, we derive a constant of motion for a general unidirectional fluid flow and apply it to an approximation of Poiseuille flow through channels with rectangular cross- sections. We obtain a 4D nonlinear conservative dynamical system with one constant of motion and a dimensionless parameter describing the ratio of maximum flow speed to intrinsic active particle speed. Applied to square channels, we observe a diverse set of active particle trajectories with variations in system parameters and initial conditions which we classify into different types of swinging, trapping, tumbling, and wandering motion. Regular (periodic and quasiperiodic) motion as well as chaotic active particle motion are observed for these trajectories and quantified using largest Lyapunov exponents. We explore the transition to chaotic motion using Poincaré maps and show "sticky" chaotic tumbling trajectories that have long transients near a periodic state. We briefly illustrate how these results extend to rectangular cross-sectionswith a width-to-height ratio larger than one. Outcomes of this paper may have implications for dynamics of natural and artificial microswimmers in experimental microfluidic channels that typically have rectangular cross sections.

On the Dynamical Erasure of Initial Conditions in Multi-planetary Systems

Do sub-Neptunes assemble close to where we see them or do they form full-fledged farther away from their host star then migrate inwards? We explore this question using the distribution of measured orbital periods, one of the most fundamental observable parameters. Under disk-induced migration, planet occurrence rate is expected to decrease toward shorter orbital periods. Presently, the observed sub-Neptune period distribution is flat in log period, between 10 and 300 days. We show, using N-body integration, how post-disk dynamical instabilities and mergers in multi-planetary systems erase the initial conditions of migration emplaced in period distributions over 10 s to 100 Myr timescale, in rough agreement with an observational hint of the abundance of resonant pairs for systems younger than 100 Myr which drops dramatically for more evolved systems. We comment on caveats and future work.

Tipping points in coupled human–environment system models: a review

Abstract. Mathematical models that couple human behavior to environmental processes can offer valuable insights into how human behavior affects various types of ecological, climate, and epidemiological systems. This review focuses on human drivers of tipping events in coupled human–environment systems where changes to the human system can abruptly lead to desirable or undesirable new human–environment states. We use snowball sampling from relevant search terms to review the modeling of social processes – such as social norms and rates of social change – that are shown to drive tipping events, finding that many affect the coupled system depending on the system type and initial conditions. For example, tipping points can manifest very differently in human extraction versus human emission systems. Some potential interventions, such as reducing costs associated with sustainable behavior, have intuitive results. However, their beneficial outcomes via less obvious tipping events are highlighted. Of the models reviewed, we found that greater structural complexity can be associated with increased potential for tipping events. We review generic and state-of-the-art techniques in early warning signals of tipping events and identify significant opportunities to utilize digital social data to look for such signals. We conclude with an outline of challenges and promising future directions specific to furthering our understanding and informing policy that promotes sustainability within coupled human–environment systems. Non-technical summary. Mathematical models that include interactions between humans and the environment can provide valuable information to further our understanding of tipping points. Many social processes such as social norms and rates of social change can affect these tipping points in ways that are often specific to the system being modeled. Higher complexity of social structure can increase the likelihood of these transitions. We discuss how data are used to predict tipping events across many coupled systems.

Distributed event‐triggered fixed‐time formation tracking control for multi‐spacecraft systems based on adaptive immersion and invariance technique

AbstractIn this study, the problem of fixed‐time formation tracking for multi‐spacecraft systems without internal collisions was investigated. Aiming to ensure that the formation members can accurately realize and maintain the required configuration within the user‐given time, we designed a novel adaptive immersion and invariance (I&I)‐based control protocol. The novelty here lies in two aspects. First and foremost, this study was based on the adaptive I&I technique, combined with a new artificial potential function, to achieve the desired formation tracking without internal collision. Second, unlike the asymptotic convergence of the traditional I&I‐related works, to guarantee the fixed‐time stability, the proposed protocol introduced the prescribed performance control, which can also alleviate the possible system performance degradation caused by the non‐periodic control signal update of the event‐triggered mechanism. Further, the introduction of the event‐triggered mechanism can reduce the unnecessary information interaction. Lyapunov stability analysis shows that this proposed protocol can enable the defined implicit manifold to converge to the origin for the most initial conditions. Also, benefiting from the prescribed performance techniques, the convergence time can eliminate the dependence of the system on designed controller parameters or initial system conditions, relying only on the actual mission requirements. In addition, we adopted a linear extended state observer to deal with the parameter uncertainties and external disturbances, and used the I&I adaption to estimate the observer errors, thus further improving the system performance. Moreover, a new exponential‐type artificial potential function was designed to avoid close proximity between formation members and prevent internal collisions. Finally, numerical simulations were conducted to verify the theoretical results.

Prescribed finite-time stabilization of fuzzy neural networks with time-varying controller

This paper investigates the exponential and prescribed finite-time stabilization with time-varying controller. First, the constraints of boundedness and differentiability on time delays are simultaneously relaxed, the Lipschitz condition for activation function is also relaxed. Second, different from the traditional Lyapunov function, two different time-varying Lyapunov functions are respectively constructed to achieve the exponential and prescribed finite-time stabilization. Significantly, the exponential convergence rate and the settling time are constants that can be given in advance and are not affected by system parameters and initial states. In addition, the time-varying controllers have good tolerance for disturbance caused by discontinuous functions and the disturbance is perfectly resolved and does not affect the control performance. Especially, the form of controllers is relatively simple and there is not necessary to design the fractional-order controllers for prescribed finite-time stabilization. Furthermore, the exponential and prescribed finite-time stabilization for FNNs without delay are respectively established via continuous time-varying state feedback control. Finally, examples show the effectiveness of the proposed control methods.

Lorenz Attractors: Exploring its properties and the application value of chaos theories

Since Lorenz published Deterministic Nonperiodic Flow in 1963, Lorenz attractors and their equations have occupied an important position in the fields of mathematics, physics, meteorology and so on. Lorenz attractors reveal the aperiodic behavior and sensitivity to initial conditions in deterministic systems, which have attracted great attention in the scientific community. This paper deeply analyzes its characteristics, formation mechanism and performance in chaotic systems, and shows that it produces aperiodic behavior patterns in deterministic systems and is extremely sensitive to small changes in initial conditions, providing a new perspective for understanding the complexity and diversity of nature. Lorenz attractors are widely used in chaos theory, providing tools for chaos research and ideas for solving practical problems. It has shown its application potential in the field of meteorological prediction. It is expected to stimulate researchers interest in Lorenz attractors and chaos phenomena, promote the in-depth application and development of chaos theory in more fields, continue to reveal the mysteries of nature, and lead a new chapter in scientific exploration.

Effect of measurements on quantum speed limit

Given the initial and final states of a quantum system, the speed of transportation of state vector in the projective Hilbert space governs the quantum speed limit. Here, we ask the question: what happens to the quantum speed limit under continuous measurement process? We model the continuous measurement process by a non-Hermitian Hamiltonian which keeps the evolution of the system Schrödinger-like even under the process of measurement. Using this specific measurement model, we prove that under continuous measurement, the speed of transportation of a quantum system tends to zero. Interestingly, we also find that for small time scale, there is an enhancement of quantum speed even if the measurement strength is finite. Our findings can have applications in quantum computing and quantum control where dynamics is governed by both unitary and measurement processes.

Leader-following consensus of heterogeneous nonlinear multi-agent systems within arbitrary pre-specified time

Abstract The leader-following consensus problem of second-order heterogeneous nonlinear multi-agent systems (MASs) is studied in this paper. Based on the free-will arbitrary time (FWAT) theory, a piecewise distributed controller is developed to solve this problem within an arbitrary pre-specified convergence time. It should be noted that the pre-specified convergence time is independent of any system parameters and initial conditions. Furthermore, compared with the existing FWAT methods, consensus can still be maintained after the convergence time under the protocol designed in this work. Finally, numerical examples are given to illustrate the validity of the proposed method.

Experimental and numerical study of lateral vibration of a rotor–stator rubbing system

AbstractIn this paper, lateral vibration of a rotating system with rotor–stator rubbing is investigated experimentally and numerically. An experimental study is conducted using a simple test rig to model a horizontal rotating system with a fixed stator device to simulate rubbing. Furthermore, a lumped parameter model with two degrees of freedom is established with a viscoelastic contact model that considers the stiffness and damping properties of the stator. The fourth/fifth order Runge–Kutta technique is used to solve the nonlinear equations of motion of the rotor system. The effect of changing mass unbalance and radial clearance on the dynamic response is investigated experimentally and numerically. The observed results are presented in detail through bifurcation diagrams, frequency spectra, Poincaré’s points, orbit plots, and time waveforms. The experimental results show that changing the mass unbalance and radial clearance yield significant effects on the system response. The 2 × and 3 × harmonic components are found to serve as good indicators of increasing rub severity. The numerical results show the dynamic characteristics of rub responses such as periodic no-contact, periodic with contact, quasi-periodic, and chaotic responses. The response characteristics are very sensitive to the changes in system parameters and initial conditions. Both the experimental and numerical results show qualitatively that the lateral vibration response exhibits coexistence and alternation of periodic and chaotic responses. Also, quasiperiodic response shows up in some numerical case studies. The obtained results contribute toward a better diagnosis of rotor–stator rub fault in rotating machines.

Stochastic basins of attraction for uncertain initial conditions

Outcomes of nonlinear dynamical systems strongly depend on the initial conditions. However, it is common knowledge that it is not possible to fix the initial conditions in real experiments, with uncertainties being inherent to it. In this work, the problem of uncertain initial conditions in dynamical systems is investigated. These uncertainties are represented through four postulates, and a new definition of basin of attraction is deduced to quantify their effects on the global dynamics. It is shown that the new definition is a convolution between the uncertainty distribution and the deterministic basin of attraction, and a reduced-cost strategy to compute them is defined. A Helmholtz nonlinear oscillator is investigated by considering initial conditions with uniform and normal distributions, and a three-dimensional Hénon map is analyzed, too. Finally, the influence of the uncertainty level and the effectiveness of new basin definition are verified.

Compliant variable admittance adaptive fixed-time sliding mode control for trajectory tracking of robotic manipulators

Abstract This paper presents a compliant variable admittance adaptive fixed-time sliding mode control (SMC) algorithm for trajectory tracking of robotic manipulators. Specifically, a compliant variable admittance algorithm and an adaptive fixed-time SMC algorithm are combined to construct a double-loop control structure. In the outer loop, the variable admittance algorithm is developed to adjust admittance parameters during a collision to minimize the collision time, which gives the robot compliance property and reduce the rigid collision influence. Then, by employing the Lyapunov theory and the fixed-time stability theory, a new nonsingular sliding mode manifold is proposed and an adaptive fixed-time SMC algorithm is presented in the inner loop. More precisely, this approach enables rapid convergence, enhanced steady-state tracking precision, and a settling time that is independent of system initial states. As a result, the effectiveness and improved performance of the proposed algorithm are demonstrated through extensive simulations and experimental results.

Design of AR system tracking registration method using dynamic target light-field

In the process of tracking registration for an augmented reality (AR) system, it's essential first to obtain the system's initial state, as its accuracy significantly influences the precision of subsequent three-dimensional tracking registration. At this point, minor movements of the target can directly lead to calibration errors. Current methods fail to address the challenge of capturing the initial state of dynamic deformation in optically transparent AR systems effectively. To tackle this issue, the concept of a static light-field is expanded to a four-dimensional dynamic light-field, and a tracking registration method for an optical see-through AR system based on the four-dimensional dynamic light-field is introduced. This method begins by analyzing the relationship between the components of the optical see-through AR system and studying the impact of a dynamic target on the initial state model. Leveraging the fundamental principle of light-field correlation, the theory and model for four-dimensional dynamic light-field tracking registration are developed. A lot of experiments have confirmed the algorithm's accuracy, enhanced its stability, and demonstrated the superior performance of the three-dimensional tracking registration algorithm.

SYNTHESIS OF FRACTIONAL ORDER FUZZY OBSERVER FOR A FRACTIONAL ORDER TAKAGI-SUGENO MODEL WITH UNMEASURABLE PREMISE VARIABLES

In this article, we present a novel approach: a fractional-order fuzzy observer, an extension of the Thau-Luenberger observer, specifically designed for nonlinear systems characterized by commensurate non-integer order Takagi-Sugeno models. This work makes significant contributions in two key areas. Firstly, both the activation functions of the model and the Lipschitz fractional-order fuzzy observer are dependent on unmeasurable variables, particularly the system's state. Secondly, our proposed fuzzy observer explicitly incorporates the system's initial conditions. The stability conditions of the fractional-order fuzzy observer are expressed through Bilinear Matrix Inequalities, which are then converted into linear matrix inequalities (LMIs). Subsequently, numerical simulations are conducted to demonstrate the efficacy of our proposed estimator.

Some results regarding observability and initial state reconstruction for time-fractional systems

The aim of this study is to present the notion of observability for a specific class of linear time-fractional systems of Riemann-Liouville type with a differentiation order between 1 and 2. To accomplish this goal, we first define the concept of observability and its features, then we extend the Hilbert Uniqueness Method (HUM) to determine the system's initial state. This method converts the reconstruction problem into a solvability one, leading to an algorithm that calculates the initial state. The effectiveness of the proposed algorithm is demonstrated through numerical simulations, which are presented in the final section.

Exact periodic solution of the undamped Helmholtz oscillator subject to a constant force and arbitrary initial conditions

The Helmholtz oscillator is a nonlinear mixed‐parity oscillator that models the asymmetric vibrations of many engineering and scientific systems. This paper investigated a general and completely integrable form of the Helmholtz oscillator and derived its exact periodic solution from the first integral of the governing differential equation. The considered Helmholtz oscillator has general linear and nonlinear stiffness constants, is subject to a constant force, and has arbitrary initial conditions. Its exact time period was derived in terms of the complete elliptic integral of the first kind, while the exact displacement was derived in terms of the Jacobi elliptic sine function. The validity of the exact solutions was verified for various combinations of the system parameters and initial conditions by comparing with numerical solutions. The exact solutions and numerical solutions were found to match perfectly. Furthermore, the exact solution was applied to analyze the vibration response of real‐world systems that could be modeled as Helmholtz oscillators. The present exact solutions provide benchmark solutions that could be used to determine the accuracy of new and existing approximate solutions for the Helmholtz oscillator.

Fixed-time nonsingular terminal sliding mode control for trajectory tracking of uncertain robot manipulators

Fixed-time nonsingular terminal sliding mode control for trajectory tracking of uncertain robot manipulators