Time-varying Dynamical Systems Research Articles

Furstenberg Family and Chaos for Time-Varying Discrete Dynamical Systems.

Assume that (Y,ρ) is a nontrivial complete metric space, and that (Y,g1,∞) is a time-varying discrete dynamical system (T-VDDS), which is given by sequences (gl)l=1∞ of continuous selfmaps gl:Y→Y. In this paper, for a given Furstenberg family G and a given T-VDDS (Y,g1,∞), G-scrambled pairs of points of the system (Y,g1,∞) (which contains the well-known scrambled pairs) are provided. Some properties of the set of G-scrambled pairs of a given T-VDDS (Y,g1,∞) are studied. Moreover, the generically G-chaotic T-VDDS and the generically strongly G-chaotic T-VDDS are defined. A sufficient condition for a given T-VDDS to be generically strongly G-chaotic is also presented.

Controllability of time-varying fractional dynamical systems with distributed delays in control

Abstract The objective of this study is to assess controllability results in the realm of Caputo fractional derivatives, specifically focusing on time-varying linear and nonlinear fractional dynamical systems with distributed control delays. In the setting of a linear system, using the Grammian matrix's positive definiteness makes it possible to ascertain both the necessary and sufficient conditions. It is possible to determine, for nonlinear systems, the sufficient conditions for the presence of a solution by making use of the Schauder fixed point theorem. In addition to that, the paper discusses the controllability of a nonlinear integrodifferential system for a particular case. A few examples and the graphs that correspond to them are shown here to make it easier to conceptualize the conclusions of the theoretical discussion.

Hybrid Nonlinear Passivity-Based Control Approach to Magnetic-Impulsive Spacecraft Attitude Regulation

This paper proposes two novel passivity-based control design frameworks for hybrid nonlinear time-varying dynamical systems involving an interacting amalgam of continuous-time and discrete-time dynamics whose dynamical properties evolve periodically over time. In this regard, the hybrid Kalman–Yakubovich–Popov conditions are employed in tandem with the passivity theorem to construct a three-step control design algorithm to render closed-loop dynamics stable. The proposed control architectures are subsequently exploited to regulate the attitude motion of spacecraft with magnetic and impulsive modes of operation. Practical considerations involved in implementing the proposed hybrid algorithms are then discussed in detail. Simulation results show significant improvement in the performance of the attitude control system in terms of system response, robustness, and the required magnetic and impulsive control usage as compared to a hybrid linear approach.

Controllability of the time-varying fractional dynamical systems with a single delay in control

In this article, we explored the controllability of fractional dynamical systems with a single delay in the control function with the Caputo fractional derivative. It is the first work in which the author studies the controllability of a time-varying fractional dynamical system with a delay in the control function. We develop the necessary and sufficient criteria for the solution representation of controllability of time-varying fractional linear dynamical systems by utilizing the Grammian matrix. We use Schauder’s fixed point theorem to establish sufficient conditions for the controllability of time-varying nonlinear fractional dynamical systems. With the help of successive approximation techniques, numerical examples validate the theoretical results.

Observability of Time-Varying Fractional Dynamical Systems with Caputo Fractional Derivative

Observability of Time-Varying Fractional Dynamical Systems with Caputo Fractional Derivative

Reachability of time-varying fractional dynamical systems with Riemann-Liouville fractional derivative

Reachability of time-varying fractional dynamical systems with Riemann-Liouville fractional derivative

A non-iterative partitioned computational method with the energy conservation property for time-variant dynamic systems

A non-iterative partitioned computational method with the energy conservation property is proposed in this study for calculating a large class of time-variant dynamic systems comprising multiple subsystems. The velocity continuity conditions are first assumed in all interfaces of the partitioned subsystems to resolve the interface link forces. The Newmark integration scheme is subsequently employed to independently calculate the responses of each system based on the obtained link forces. The proposed method is thus divided into two computational modules: multi-partitioned structural analyzers and an interface solver, providing a modular solution for time-variant systems. The proposed method resolves the long-standing problem of iterative computation required in partitioned time-variant systems. More specifically, the proposed method eliminates the need for time-variant matrix formation and the utilization of complex iterative procedures in partitioned computations, which significantly improves computational efficiency. The derivation process and theoretical demonstration of the proposed method are thoroughly presented through a representative example, i.e., a vehicle-rail-sleeper-ballast time-variant system. The proposed method’s accuracy, energy conservation property, and efficiency are systematically demonstrated in comparison with the results of the global model, highlighting its superior performance. A more general example provided in Appendix C demonstrates that the proposed method is not confined to the analysis of vehicle-rail-sleeper-ballast systems but applies to other structural dynamic systems.

LQG Graphon Mean Field Games: Analysis via Graphon-Invariant Subspaces

This paper studies approximate solutions to large-scale linear quadratic stochastic games with homogeneous nodal dynamics parameters and heterogeneous network couplings within the graphon mean field game framework in [2]-[4]. A graphon time-varying dynamical system model is first formulated to study the finite and then limit problems of linear quadratic Gaussian graphon mean field games (LQG-GMFG). The Nash equilibrium of the limit problem is then characterized by two coupled graphon time-varying dynamical systems. Sufficient conditions are established for the existence of a unique solution to the limit LQG-GMFG problem. For the computation of LQG-GMFG solutions, two methods are established and employed where one is based on fixed point iterations and the other on a decoupling operator Riccati equation; furthermore, two corresponding sets of solutions are established based on spectral decompositions. Finally, a set of numerical simulations on networks associated with different types of graphons are presented.

On use of train-track-subgrade dynamic model for investigating the train-induced cumulative deformation of subgrade and its dynamic effects

This work focuses on two research projects. The first project deals with the prediction of the train-induced cumulative deformation of the high-speed railway subgrade based on the train-track-subgrade dynamic model, and the second project deals with the long-term performance of the train-track-subgrade interaction system subject to the permanent deformation of the subgrade. The train-track-subgrade interaction is established as the model fundament, and the empirical model of soil cumulative deformation prediction is introduced to realize the joint analysis of the train-induced cumulative deformation of the subgrade and its effect on the performance of the time-varying dynamic system. From the numerical results, the train-induced cumulative deformation of subgrade shows a fast and then slow trend, and the deformation has periodic distribution in the longitudinal direction. Although the cumulative deformation of the subgrade for the ballastless track is small compared with that of ballasted track, its effect on the vertical response of the train-track-subgrade interaction system is obvious. With the development of cumulative deformation of subgrade, the vibrations of the car body are excited in the frequency band of 0∼100 Hz. Furthermore, the maximum car body vertical accelerations, the wheel-rail vertical forces, and the vertical accelerations of track and subgrade increase by about 34∼48.4%, 125 %, 400 %, and 66.7 %, respectively.

Fuzzy dynamic Bayesian network based on a discrete aggregation method for risk assessment of marine nuclear power platform hinge joints accidents

Hinge joints play a critical role in the positioning system of Marine Nuclear Power Platforms (MNPP) by facilitating the transmission of mooring forces and accommodating rotational motions of the hull. Failure of hinge joints during operation can result in severe consequences for both the environment and human safety. Therefore, this study aims to address the limitations associated with the dynamic characteristics and uncertain information pertaining to the risk systems of hinge joint structures in MNPP. To tackle the challenge of conflicting expert opinions in group decision-making, a novel method called the discrete aggregation method (DAM) is proposed. DAM utilizes fuzzy opinion aggregation, taking into account the degree of consensus among experts. This approach effectively overcomes the problem of conflicting opinions. Furthermore, building a hinge joint accidents model using Bow-tie (BT) for hazard identification, and then mapping the BT model to a dynamic Bayesian networks (DBN) model. Fuzzy set theory (FST) and DBN are employed to handle cognitive uncertainty caused by data limitations and incomplete knowledge. These techniques enable causal inference for time-varying dynamic risk systems. To validate the proposed method, a case study of the positioning system of the first MNPP in China is conducted. The dynamic fatigue failure probability of hinge joints is calculated using predictive technology based on Bayesian theory. Additionally, diagnostic analysis techniques are employed to accurately identify key events and their underlying causes. The analysis results provide valuable data support for on-site operations and management personnel, aiding in the development of effective risk mitigation measures.

Computing dispersion diagrams and forced responses of arbitrarily varying waveguides

Periodic structures can be used for vibration control thanks to their vibration attenuation stop bands. The dispersion diagram obtained for the structural unit cell can be used to characterize such wave attenuation bands. Several designs using spatially varying unit cells have recently been reported in the literature. Spatially varying properties can also be used to investigate the effect of variability on finite structures. Therefore, various methodologies to model spatially varying structures have been recently developed. This work presents new numerical methodologies to compute (i) dispersion diagrams and (ii) forced responses of one-dimensional spatially varying periodic waveguides. The first method extends previous work using a state-space formulation to compute dispersion relations using a linear time-varying system dynamics approach. The methodology is applied to elementary rods, Saint-Venant shafts, Euler–Bernoulli beams, and Timoshenko beams, and the results are validated with discretized spectral element models and the plane wave expansion method. The proposed approach is shown to be more efficient in cases where the plane wave expansion method presents a slow convergence. The second method is a recursive dynamic condensation to compute forced responses using the dynamic stiffness matrix of the unit cell. The results and computational times of the proposed recursive condensation are compared with the direct assembly of a global dynamic stiffness and its standard dynamic condensation. The two proposed methods are shown to be more efficient and accurate in the computation of dispersion diagrams and forced responses of one-dimensional structures for which the elastodynamic equations are known.

Time-varying stability and vibration of an embedded thin plate on a swept wing

The morphing wing aircraft, with good performances of flexibility and motility, has been paid more and more attention by researchers in recent years. During the unfolding process, the flow direction above the thin-walled plate structures embedded in the swept wing changes due to the relative movement of the plate with the outside airflow. The plate on the swept wing and the outside flow comprise a time-varying dynamic system. In this paper, based on the von Karman nonlinear plate theory and the linear potential flow theory, the dynamic equation of the plate interacting with the yawed subsonic flow is established and the time-varying dynamic properties of the system are analyzed. The natural frequency curve of the system during the unfolding process is obtained by solving the generalized eigenvalue problem. The vibration displacement time response of the thin plate is obtained by the Runge-Kutta method. The interaction between the flow velocity and the spread velocity of the swept wing will affect the stability of the plate, and the vibration amplitude of the plate will also be affected by the parameters of the plate.

Identification of linear time-varying dynamic systems based on the WKB method

This work proposes a new parameter identification method based on the Wentzel-Kramers-Brillouin (WKB) approximation for slow linear time-varying (LTV) dynamic systems. The considered time period is divided into a series of short time windows. In each time window, the assumption of “short time linearly varying” parameters is employed, and a nonlinear optimization problem is solved using the WKB results for the slow LTV dynamic system. A search algorithm is developed to find the optimal solution. In the identification process, only one type of response signal (displacement, velocity or acceleration) is required. Thus, numerical differentiation or integration of the measured signal, which leads to truncation or cumulative errors in noise environment, is avoided. The accuracy and robustness of the new identification method are validated by applying it to a particular LTV system with time-varying stiffness.

On the Absence of Spurious Local Trajectories in Time-Varying Nonconvex Optimization

In this article, we study the landscape of an online nonconvex optimization problem, for which the input data vary over time and the solution is a trajectory rather than a single point. To understand the complexity of finding a global solution of this problem, we introduce the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">spurious (i.e., nonglobal) local trajectory</i> as a generalization to the notion of spurious local solution in nonconvex (time-invariant) optimization. We develop an ordinary differential equation (ODE) associated with a time-varying nonlinear dynamical system which, at limit, characterizes the spurious local solutions of the time-varying optimization problem. We prove that the absence of spurious local trajectory is closely related to the transient behavior of the developed system. In particular, we show that if the problem is time-varying, the data variation may force all of the ODE trajectories initialized at arbitrary local minima at the initial time to gradually converge to the global solution trajectory. We study the Jacobian of the dynamical system along a local minimum trajectory and show how its eigenvalues are manipulated by the natural data variation in the problem, which may consequently trigger escaping poor local minima over time.

Minimal regret state estimation of time-varying systems

Kalman and H∞ filters, the most popular paradigms for linear state estimation, are designed for very specific specific noise and disturbance patterns, which may not appear in practice. State observers based on the minimization of regret measures are a promising alternative, as they aim to adapt to recognizable patterns in the estimation error. In this paper, we show that the regret minimization problem for finite horizon estimation can be cast into a simple convex optimization problem. For this purpose, we first rewrite linear time-varying system dynamics using a novel system level synthesis parametrization for state estimation, capable of handling both disturbance and measurement noise. We then provide a tractable formulation for the minimization of regret based on semi-definite programming. Both contributions make the minimal regret observer design easily implementable in practice. Finally, numerical experiments show that the computed observer can significantly outperform both H2 and H∞ filters.

Characterisation of driver longitudinal behaviour using an Unscented Kalman Filter

This paper presents a real-time driver characterisation algorithm that models the driver based on 10 parameters governing their control of vehicle longitudinal speed and acceleration. It utilises an open-loop longitudinal model in conjunction with an Unscented Kalman Filter (UKF). The algorithm can operate in real-time using velocity and acceleration measurements, in addition to a-priori knowledge of the path curvature and road features such as speed limit locations. The UKF also enables a system identification process to estimate average response of each driver. These identified parameters are subsequently fitted to independent measures of fuel economy and safety to demonstrate possible uses for the characterisations. In addition to the comprehensive characterisation of the drivers’ behaviour the work is also novel in its unique approach to identifying time-based parameters within a time-varying dynamic system (UKF). Possible applications include insurance black box assessments of driver aggressiveness and safety, and in autonomous driving algorithms by providing a model to closely replicate the specific driving styles of various users.

Security Control for Time-Varying Delay Systems Based on Random Switching Moving Defense Method in Cyber-Physical Environment

With the high digitization, informatization, and networking of practical complex systems, the security and reliability issue of systems control in the cyber-physical environment is becoming more and more critical, especially under the background of rampant network attacks nowadays. The security control for time-varying delay systems based on random switching moving defense method in a cyber-physical environment is studied in this paper. First, the security control model of one class of time-varying delay dynamic systems considering the actuator and controller attack problem is presented; second, controller design based on switching strategy for moving defense is studied; and then the security control scheme for time-varying delay systems based on random switching moving defense is presented for dealing with cyber attacks from malicious adversaries. Finally, multiple simulation results are presented to show the effectiveness of the proposed scheme.

Limiting Behavior of Hybrid Time-Varying Systems

Checking uniform attractivity of a time-varying dynamic system without a strict Lyapunov function is challenging as it requires the characterization of the limiting behavior of a set of trajectories. In the context of hybrid nonlinear time-varying systems, characterizing such limiting or convergent behaviors is even harder due to the complexity stemming from both continuous-time variations as well as discrete-time jumps. In this article, an extension of the standard hybrid time domain is introduced to define limiting behaviors, using set convergence, when time approaches either positive infinity or negative infinity. In particular, it is shown how to characterize limiting behaviors under the condition that an output signal approaches zero. Such limiting behaviors and their associated limiting systems can be used to verify uniform global attractivity. Particularly, a generalization of the classic Krasovskii–LaSalle theorem is obtained for hybrid time-varying systems. Two examples are used to demonstrate the effectiveness of the results.

Structure Learning-Based Task Decomposition for Reinforcement Learning in Non-stationary Environments

Reinforcement learning (RL) agents empowered by deep neural networks have been considered a feasible solution to automate control functions in a cyber-physical system. In this work, we consider an RL-based agent and address the issue of learning via continual interaction with a time-varying dynamic system modeled as a non-stationary Markov decision process (MDP). We view such a non-stationary MDP as a time series of conventional MDPs that can be parameterized by hidden variables. To infer the hidden parameters, we present a task decomposition method that exploits CycleGAN-based structure learning. This method enables the separation of time-variant tasks from a non-stationary MDP, establishing the task decomposition embedding specific to time-varying information. To mitigate the adverse effect due to inherent noises of task embedding, we also leverage continual learning on sequential tasks by adapting the orthogonal gradient descent scheme with a sliding window. Through various experiments, we demonstrate that our approach renders the RL agent adaptable to time-varying dynamic environment conditions, outperforming other methods including state-of-the-art non-stationary MDP algorithms.

Optimal Sensor Precision for Multirate Sensing for Bounded Estimation Error

We address the problem of determining optimal sensor precisions for estimating the states of linear time-varying discrete-time stochastic dynamical systems, with guaranteed bounds on the estimation errors. This is performed in the Kalman filtering framework, where the sensor precisions are treated as variables. They are determined by solving a constrained convex optimization problem, which guarantees the specified upper bound on the posterior error variance. Optimal sensor precisions are determined by minimizing the l1 norm, which promotes sparseness in the solution and indirectly addresses the sensor selection problem. The theory is applied to realistic flight mechanics and astrodynamics problems to highlight its engineering value. These examples demonstrate the application of the presented theory to a) determine redundant sensing architectures for linear time invariant systems, b) accurately estimate states with low-cost sensors, and c) optimally schedule sensors for linear time-varying systems.