Uncertain Initial Conditions Research Articles

Adaptive Neural Control With Guaranteed Performance for Mechanical Systems Under Uncertain Initial Conditions: A Time-Varying Neuron Approach

Adaptive Neural Control With Guaranteed Performance for Mechanical Systems Under Uncertain Initial Conditions: A Time-Varying Neuron Approach

Fault diagnosis of Discrete Event Systems under uncertain initial conditions

A new technique is presented for diagnosing faults in a Discrete Event System (DES) when the state of the system under diagnosis (SUD) is uncertain upon commencement of the diagnosis process. Specifically, a diagnoser is developed that detects, identifies, and isolates faults in a DES relative to an initial belief state: a set of states in which the SUD could be located at the diagnoser’s time of activation. The diagnoser does not need to be synchronously initialized with the SUD. Rather, it can be activated anytime before or after a fault in the SUD occurs, as it does not require knowledge of the SUD’s behavior pre-activation. This form of analysis can be understood as fault diagnosis under uncertain initial conditions. The construction procedure for the diagnoser is given and the new concept of semi-asynchronous diagnosability is introduced. The latter requires all failures in the SUD to be diagnosable by observing the behaviors of the SUD after the diagnoser is activated. Moreover, necessary and sufficient conditions for semi-asynchronous diagnosability of a DES are provided. The semi-asynchronous diagnosis technique presented herein is compared with classical synchronous diagnosis approaches. Illustrative examples are provided in order to explain the introduced concepts and the developed approach.

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.

Parametric study for model calibration of a friction-damped turbine blade with multiple test data

Model updating using multiple test data is usually a challenging task for frictional structures. The difficulty arises from the limitations of nonlinear models which often overlook the uncertainties inherent in contact interfaces and in actual test conditions. In this paper, we present a parametric study for the model calibration process of a friction-damped turbine blade, addressing the experimentally measured response variability in computational simulations. On the experimental side, a recently developed test setup imitating a turbomachinery application with mid-span dampers is used. This setup allows measuring multiple responses and contact forces under nominally identical macroscale conditions. On the computational side, the same system is modeled in a commercial finite element software, and nonlinear vibration analyses are performed with a specifically developed in-house code. In numerical simulations, the multivalued nature of Coulomb’s law, which stems from the inherent variability range of static friction forces in permanently sticking contacts, is considered to be the main uncertainty. As the system undergoes vibration, this uncertainty propagates into the dynamic behavior, particularly under conditions of partial slip in contacts, thus resulting in response variability. A deterministic approach based on an optimization algorithm is pursued to predict the limits of the variability range. The model is iteratively calibrated to investigate the sensitivity of response limits to contact parameters and assembly misalignment. Through several iterations, we demonstrate how uncertain initial contact conditions can be numerically incorporated into dynamic analyses of friction-damped turbine blades. The results show a satisfactory level of accuracy between experiments and computational simulations. This work offers valuable insights for understanding what influences test rig response and provides practical solutions for numerical simulations to improve agreement with experimental results.

Solution of generalized fractional Jaulent–Miodek model with uncertain initial conditions

This paper analyses a coupled system of generalized coupled system of fractional Jaulent–Miodek equations, including uncertain initial conditions with fuzzy extension. In this regard, an extension of the homotopy with a generalized integral algorithm is adopted for a class of time-fractional fuzzy Jaulent–Miodek models by mixing the fuzzy q-homotopy analysis algorithm with a generalized integral transform and Caputo fractional derivative. The triangular fuzzy numbers (TFNs)are expressed in double parametric form using κ-cut and r-cut and utilized to explain the uncertainties arising in the initial conditions of highly nonlinear differential equations with generalized Hukuhara differentiability (gH-differentiability). The TFNs are controlled by the κ-cut and r-cut, and the variability of uncertainty is examined using a “triangular membership function” (TMF). The results are analyzed by finding the solutions for different spatial coordinate values of time with κ-cut and r-cut for both lower and upper bounds and validated through numerical and graphical representations in crisp cases. Finally, it can be seen that the uncertain probability density function rapidly decreases at the left and right edges when the fractional order is increased, and it is observed that the obtained solutions are more accurate than the existing results through the Hermite wavelet method in the literature.

Prescribed Performance Tracking Control Under Uncertain Initial Conditions: A Neuroadaptive Output Feedback Approach.

This work is concerned with the prescribed performance tracking control for a family of nonlinear nontriangular structure systems under uncertain initial conditions and partial measurable states. By combining neural network and variable separation technique, a state observer with a simple structure is constructed for output-based finite-time tracking control, wherein the issue of algebraic loop arising from a nontriangular structure is circumvented. Meanwhile, by using an error transformation, the developed control scheme is able to ensure tracking with a prescribed accuracy within a pregiven time at a preassigned convergence rate under any bounded initial condition, eliminating the long-standing initial condition dependence issue inherited with conventional prescribed performance control methods, and guaranteeing the predeterminability of convergence time simultaneously. Two simulation examples also demonstrate the effectiveness of the presented control strategy.

A Numerical Study for Uncertainty in two Predators-One Prey Model

In this article, a mathematical model had been presented, which delved into the dynamics of a system featuring two predators and one prey, characterized by a growth rate of the predator dependent on a specific ratio. When facing uncertain initial conditions in the model, it had become imperative to adapt the mathematical descriptions by treating these initial conditions as random variables governed by specific distribution functions. The beta distribution had been employed to represent the stochastic initial state of the two predators-one prey model, allowing for an examination of how the random initial state of the prey had impacted the dynamic behavior of the system. Given the lack of an analytical solution for the model, computer simulations had been resorted to in order to delve into the dynamics of the system. Consequently, various properties of the resulting numerical solution had been thoroughly discussed, shedding light on the intricacies of the system's behavior. In addition, the research had offered invaluable insights into understanding the impact of uncertainty on predator-prey interactions and the subsequent stability and behavior of the ecological system in uncertain conditions, further bolstered by the numerical results obtained.

The main evolutionary pathways of massive hierarchical triple stars

Context. So far, stellar population studies have mainly focused on the evolution of single and binary stars. Recent observations show that triple and higher order multiple star systems are ubiquitous in the local population, especially among massive stars. Introducing three-body dynamical effects can influence the evolution of an individual stellar system and can therefore affect the predicted rates of astrophysical sources that are a product of stellar evolution. Therefore, predictions of triple star evolution are necessary for a more complete understanding of the evolutionary behaviour of stellar populations and their end products. Aims. We aim to constrain the main evolutionary pathways of massive hierarchical triple star systems and to quantify the effect of the third star on the evolution of the system. Methods. We model the massive triple star population by performing simulations of triple star evolution with the TRES code, which combines stellar evolution with secular evolution of triple systems, and explore how robust the predictions of these simulations are under variations of uncertain initial conditions. We focus on coeval, hierarchical stellar triples in pre-mass-transfer phases. Results. Interactions are common among massive triple stars. The majority of systems (65%–77%) experience a phase of mass transfer in the inner binary, often with a main sequence donor star. This differs significantly from isolated binary evolution, where mass transfer is less frequent (52.3% instead of 67% for our fiducial model) and the donors are typically post-main sequence stars. Initial constraints for dynamical stability as well as eccentricity oscillations driven by the third body facilitate the occurrence of interactions, such as mass transfer. The requirement of dynamical stability at formation places quite stringent constraints on allowed orbital properties, reducing uncertainties in triple evolution that resort from these initial conditions. Ignoring three-body dynamics during evolution of non-interacting triples leads to triple compact-object systems with stronger eccentricity oscillations and thereby likely over-predicts the merger rate of compact objects in such systems.

Mars orbit injection via aerocapture and low-thrust nonlinear orbit control

This research proposes a new strategy for Mars orbit injection, based on aerocapture and low-thrust nonlinear orbit control. The range of periapse altitudes that allow aerocapture is identified as a function of the hyperbolic excess velocity at Mars arrival, with reference to a large variety of atmospheric density profiles and different ballistic coefficients. This analysis proves that a safe periares altitudinal range leading to aerocapture in all atmospheric conditions does not exist. Three different correction maneuvers, aimed at avoiding both impact and escape, are identified. After the atmospheric arc, the spacecraft orbit exhibits large dispersions in terms of orbit elements. Therefore, the identification of an effective autonomous guidance strategy, capable of driving the spacecraft toward the desired operational orbit, is mandatory. To do this, low-thrust nonlinear orbit control is proposed as an effective option. A feedback law for the low-thrust direction and magnitude, with saturation of the thrust magnitude, is defined, and is proven to enjoy global stability properties. As a result, the spacecraft travels toward the operational orbit of interest, i.e. either (a) an areostationary orbit, (b) a quasi-synchronous inclined orbit, or (c) a low-altitude, sunsynchronous orbit. Monte Carlo simulations, with stochastic density profiles and uncertain initial conditions, point out that the strategy at hand is successful and allows reducing the overall propellant budget in comparison to direct orbit injection based on chemical propulsion. Moreover, the overall time of flight typically ranges from 45 to 140 days, and therefore it is much shorter than that required with the use of aerobraking. As a last advantage, low-thrust nonlinear orbit control allows achievement of a variety of operational orbits, with great accuracy.

Strategies for Controlling a Whole-Body Task With Uncertain Initial Conditions: Application to the Upstart on Bars.

The upstart is commonly used on bars in artistic gymnastics following a release and regrasp skill, where the gymnast will perform a flighted element before catching the bar. The variability of the flighted element leads to varying initial conditions prior to the upstart. The aim of the study was to understand how technique can be manipulated in order to ensure success at the task despite this variability. More specifically, the study aimed to quantify the ranges of initial angular velocity a gymnast could cope with in an upstart using (a)a fixed timing technique, (b)with one additional parameter to modify timings as a function of initial angular velocity, and (c)a further additional parameter to extend the range. Relationships were established, using computer simulation modeling, between the movement pattern parameters, which defined the technique, and the initial angular velocity of the upstart. A two-parameter relationship outperformed both the one-parameter relationship and the fixed timing solution in terms of the range of initial angular velocities the model could cope with. One of the two parameters governed the time by which the initiation of the shoulder extension should be reduced as a function of increased initial angular velocity, and the other parameter performed the same function for the remaining timing parameters at the hip and shoulder. The present study suggests that gymnasts, and, therefore, humans, may be able to modify movement patterns to cope with uncertain initial conditions using a relatively small number of parameters.

Nonlinear model predictive control leveraging quantum-inspired optimization in the three body problem with uncertainty

The optimization and control of spacecraft trajectories in presence of uncertainties and unmodeled disturbances represents a challenging problem. Future missions would benefit from robust control techniques to compute their trajectory in a complex, nonlinear environment and in presence of stochasticity. In this work the Stochastic Optimal Control problem in the Circular Restricted Three Body Problem is addressed and a method is proposed based on Nonlinear Model Predictive control (NMPC). This method presents the challenge of solving a nonlinear optimization problem on-board. A Quantum-inspired meta-heuristic optimization algorithm is tested in this context: the Diffusion Monte Carlo (DMC) method is based on a numerical techniques used in Quantum Mechanics, adapted to solve optimization problems. The NMPC with DMC is implemented, taking into account an estimate of the computational time required through the introduction of delayed actuation of the control. Two applications are presented featuring transfer trajectories between planar periodic orbits in the vicinity of the Moon; the proposed method proved able to successfully generate trajectories when disturbance accelerations and uncertain initial conditions are present.

A solving method for two-dimensional homogeneous system of fuzzy fractional differential equations

<abstract> <p>The purpose of this study is to extend and determine the analytical solution of a two-dimensional homogeneous system of fuzzy linear fractional differential equations with the Caputo derivative of two independent fractional orders. We extract two possible solutions to the coupled system under the definition of strongly generalized $ H $-differentiability, uncertain initial conditions and fuzzy constraint coefficients. These potential solutions are determined using the fuzzy Laplace transform. Furthermore, we extend the concept of fuzzy fractional calculus in terms of the Mittag-Leffler function involving triple series. In addition, several important concepts, facts, and relationships are derived and proved as property of boundedness. Finally, to grasp the considered approach, we solve a mathematical model of the diffusion process using proposed techniques to visualize and support theoretical results.</p></abstract>

A Symmetry-Based Unscented Particle Filter for State Estimation of a Ballistic Vehicle

The problem of state estimation for vehicles when an initial fix is highly uncertain and/or the number of sensors is not sufficient (and changes with time) is very relevant for both aircraft and spacecraft navigation. This work proposes a Locally Linearized Particle Filter based on a quaternion-adapted Unscented Kalman Filter to estimate the state of a vehicle with minimal sensors and uncertain initial conditions, exploiting geometrical symmetries. The algorithm is applied to a ballistic vehicle navigating towards a laser-illuminated target using on-board sensors, including a triad of accelerometers and gyroscopes, a barometric altimeter and a laser receiver. A symmetry around the vertical axis is identified; based on it, the algorithm becomes capable of solving the navigation problem, even with highly uncertain initial conditions and without enough sensor information; this second condition is particularly severe when the laser receiver is not yet obtaining data. The proposed navigation algorithm offers promising results in simulation, rapidly converging to an accurate estimate of the real trajectory when the laser receiver becomes active.

Solution of generalised fuzzy fractional Kaup–Kupershmidt equation using a robust multi parametric approach and a novel transform

This work considers a fuzzy fractional model of the Kaup–Kupershmidt equation with uncertain initial conditions. The uncertain behaviour of capillary gravity and dispersive waves is studied using a robust double parametric fuzzy approach termed the q-Homotopy analysis method with a generalised novel transform. The proposed method’s accuracy and efficiency have been explained by comparing the observation of fuzziness in the solution with several prior accessible results for the crisp case using the double parametric approach. This methodology provides a viable alternative to existing methods for various fuzzy fractional models.

Ensemble data assimilation using optimal control in the Wasserstein metric

An ensemble data assimilation method is proposed that is based on optimal control minimizing the cost of mismatch in the Wasserstein metric on the observation space. The new method achieved the optimal state without calculating the posterior distribution of the particle state and the particle states are evolved deterministically, which is easy to be implemented. The method is appropriate for systems in which multiple, noisy, partial observations are available (e.g. citizen weather stations or smart phones). The method is demonstrated for: (i) deterministic dynamics with uncertain initial conditions, (ii) multiple noisy observations of a randomly forced ordinary differential equation (ODE), (iii) observations from multiple sample paths from a stochastic differential equation (SDE). A bi-modal measure and a measure supported on a strange attractor are tested. The numerical results show that our method performs a relatively small Wasserstein distance which measures the approximation performance. But numerical implementation is a bit expensive due to the complexity of Wasserstein distance computation, especially with large set of particles.

A data-driven degradation prognostics approach for rolling element bearings

Degradation prognostic plays a crucial role in increasing the efficiency of health management for rolling element bearings (REBs). In this paper, a novel four-step data-driven degradation prognostics approach is proposed for REBs. In the first step, a series of degradation features are extracted by analyzing the vibration signals of REBs in time domain, frequency domain and time-frequency domain. In the second step, three indicators are utilized to select the sensitive features. In the third step, different health state labels are automatically assigned for health state estimation, where the influence of uncertain initial condition is eliminated. In the last step, a multivariate health state estimation model and a multivariate multistep degradation trend prediction model are combined to estimate the residence time in different health status and remaining useful life (RUL) of REBs. Verification results using the XJTU-SY datasets validate the effectiveness of the proposed method and show a more accurate prognostics results compared with the existing major approaches.

Set-Based B-Series

B-series were defined to unify the formalism of solutions for ordinary differential equations defined by series. Runge–Kutta schemes can be seen as truncated B-series, similar to Taylor series. In the prolific domain of reachability analysis, i.e., the process of computing the set of reachable states for a system, many techniques have been proposed without obvious links. In the particular case of uncertain initial conditions and/or parameters in the definition of differential equations, set-based approaches are a natural and elegant method to compute reachable sets. In this paper, an extension to B-series is proposed to merge these techniques in a common formalism—named set-based B-series. We show that the main properties of B-series are preserved. A validated technique, based on Runge–Kutta methods, able to compute such series, is presented. Experiments are provided in order to illustrate the proposed approach.

Initial value problems in Type-2 fuzzy environment

In real-world problems, uncertainty may not be ignored. Because in the actual scenario, the parameters involved may vary depending on a variety of factors. Accordingly, fuzzy numbers rather than crisp parameters may be used to deal with these uncertainties. It may be noted that the use of Type-2 fuzzy numbers (T2FN) over Type-1 fuzzy numbers (T1FN) is advantageous as the value of the membership function varies from observer to observer. There are several practical problems whose governing equations are initial value problems. The main goal of this investigation is to solve ordinary differential equations with type-2 fuzzy uncertain initial conditions in order to deal with such situations. This article presents an analytical method for solving type-2 fuzzy differential equations using Triangular Perfect Quasi Type-2 Fuzzy Numbers (TPQT2FN). A newly defined triple parametric form of the type-2 fuzzy number is used in this case. Finally, the proposed method is demonstrated using a few numerical examples and application problems.

Sawi Transform Based Homotopy Perturbation Method for Solving Shallow Water Wave Equations in Fuzzy Environment

In this manuscript, a new hybrid technique viz Sawi transform-based homotopy perturbation method is implemented to solve one-dimensional shallow water wave equations. In general, the quantities involved with such equations are commonly assumed to be crisp, but the parameters involved in the actual scenario may be imprecise/uncertain. Therefore, fuzzy uncertainty is introduced as an initial condition. The main focus of this study is to find the approximate solution of one-dimensional shallow water wave equations with crisp, as well as fuzzy, uncertain initial conditions. First, by taking the initial condition as crisp, the approximate series solutions are obtained. Then these solutions are compared graphically with existing solutions, showing the reliability of the present method. Further, by considering uncertain initial conditions in terms of Gaussian fuzzy number, the governing equation leads to fuzzy shallow water wave equations. Finally, the solutions obtained by the proposed method are presented in the form of Gaussian fuzzy number plots.

Lie symmetries applied to interval integration

In this paper, we propose a new approach for improving significantly existing guaranteed integration methods for state equations with uncertain initial conditions. We first find a tube that encloses the solution of the differential equation assuming that the initial state is known. Then, using Lie symmetries, we inflate the tube in order to contain the uncertainty associated with the initial state. The method is shown to be efficient on examples coming from reachability analysis and robotics.