Time-varying Differential Equations Research Articles

Lyapunov stability of suspension bridges in turbulent flow

In the era of sleek, super slender suspension bridges, facing the issue of stability against dynamic wind actions represents an increasingly complex challenge. Despite the significant progress over the last decades, the impact of atmospheric turbulence on bridge stability remains partially not understood, evoking the need for innovative research approaches. This study aims to address a gap in current research by investigating the random flutter stability associated with variations in the angle of attack due to turbulence, which has not formally been addressed yet. The present investigation employs the 2D rational function approximation model to express self-excited forces in a turbulent flow. The application of this type of models to bridge dynamics yields a viscoelastic coupled dynamic system characterized by memory effects and driven by broad-band long-time-scale noise, described here by a linear homogeneous time-variant differential equation, which shows apparent nonlinear features, and which has rarely been matter of research. Utilizing a Monte Carlo methodology, this work innovates in applying the largest Lyapunov exponent (LE) and the moment Lyapunov exponents (MLE) to study bridge random flutter stability. The calculation of LE and MLE under diverse turbulent wind conditions uncovers lower flutter stability than without turbulence effects. In most cases, sample and low-order p-th moment stability thresholds closely align with the bridge dynamic response pattern; therefore, the flutter critical wind speed is unequivocal. However, under certain turbulence scenarios, it is necessary to resort to MLE for a complete description of stability, evoking some additional consideration of which statistical moments should be considered for the engineering assessment of the flutter limit. Finally, this work provides a qualitative insight into the instability mechanisms by approximating the random parametric excitation with a sinusoidal gust and evaluating the time-periodic system stability via Floquet theory.

On the stability in variation of non-autonomous differential equations with perturbations

In this paper, we investigate the problem of stability in variation of solutions for nonautonomous differential equations. Some new sufficient conditions for the asymptotic or exponential stability for some classes of nonlinear time-varying differential equations are presented by using Lyapunov functions that are not necessarily smooth. The proposed approach for stability analysis is based on the determination of the bounds that characterize the asymptotic convergence of the solutions to a certain closed set containing the origin. Furthermore, some illustrative examples are given to prove the validity of the main results.

Trajectory tracking with integral action of switched periodic affine systems

This paper deals with integral control of a class of switched affine systems characterised by time-varying differential equations that, along with the discontinuities caused by the switching, also depend periodically on an exogenous parameter, which is an affine function of time. The goal is to design a state-dependent switching function with integral action that ensures global asymptotic tracking of a trajectory of interest. The integral action is responsible for providing robustness to the system against uncertainties and load variations, whenever the discrepancy between the model and the real system is sufficiently bounded. Due to the integrator, all the convex combinations of the dynamic matrices are non-Hurwitz. In this case, the main difficulty is to guarantee that the time derivative of the Lyapunov function is strictly negative definite, which is done by means of LMIs and conditions on the affine terms. To the best of our knowledge, this is the first Lyapunov-based switching rule with integral action able to ensure global asymptotic tracking of a pre-specified profile. When applied to AC circuits the advantage is to design the switching rule based on the original system, without resorting to averaged models. The theory is illustrated by the control of a three-phase AC-DC converter.

A class of time-varying differential equations for vibration research and application

<p>As a pivotal branch within the realm of differential equations, the theory of oscillation holds a crucial position in the exploration of natural sciences and the construction of modern control theory frameworks. Despite the extensive research conducted globally, focusing on individual or combined analyses of key elements such as explicit damping terms, positive and negative coefficients, time-varying delays, and nonlinear neutral terms, systematic investigations into the oscillatory behavior of even-order differential equations that concurrently embody these four complex characteristics remain scarce. This paper, by establishing reasonable assumptions, innovatively presents two crucial criteria, aiming to preliminary delve into the oscillation patterns of even-order differential equations under specific complex settings. In the course of the study, a variety of mathematical techniques, such as Riccati transformation, calculus scaling methods, and partial integration, have been utilized by the researchers to perform the necessary derivations and confirmations.</p>

On the Stability of Linear Time-Varying Differential Equations

On the Stability of Linear Time-Varying Differential Equations

A new adaptive co-site broadband interference cancellation method with auxiliary channel

With the boom of the communication systems on some independent platforms (such as satellites, space stations, airplanes, and vessels), co-site interference is becoming prominent. The adaptive interference cancellation method has been adopted to solve the co-site interference problem. But the broadband interference cancellation performance of traditional Adaptive Co-site Interference Cancellation System (ACICS) with large delay mismatching and antenna sway is relatively poor. This study put forward an Adaptive Co-site Broadband Interference Cancellation System With Two Auxiliary Channels (ACBICS-2A). The system model was established, and the steady state weights and Interference Cancellation Ratio (ICR) were deduced by solving a time-varying differential equation. The relationship of ICR, system gain, modulation factor, interference signal bandwidth and delay mismatching degree was acquired through an in-depth analysis. Compared with traditional adaptive interference cancellation system, the proposed ACBICS-2A can improve broadband interference cancellation ability remarkably with large delay mismatching and antenna sway for the effect of auxiliary channel. The maximum improved ICR is more than 25 ​dB. Finally, the theoretical and simulation results were verified by experiments.

Asymptotic behaviors of solutions of a class of time-varying differential equations

Abstract This paper deals with the problem of the global uniform stability of nonlinear time-varying systems in the presence of perturbations. The main novelty relies on the fact that the proposed approach for stability analysis allows for the computation of the bounds which characterize the asymptotic convergence of solutions to a small ball centered at the origin. Therefore, we generalize some results which have already be announced in the literature. Furthermore, we provide a numerical example to validate the effectiveness of our main result.

Model-free finite-horizon optimal tracking control of discrete-time linear systems

Conventionally, the finite-horizon linear quadratic tracking (FHLQT) problem relies on solving the time-varying Riccati equations and the time-varying non-causal difference equations as the system dynamics is known. In this paper, with unknown system dynamics being considered, a Q-function-based model-free method is developed to solve the FHLQT problem. First, an augmented system consisting of the controlled system and the desired trajectory system is formulated, and the FHLQT problem transforms to the finite-horizon linear quadratic regulator (FHLQR) problem with the augmented system. Then, a time-varying Q-function which depends explicitly on the control input is defined. With the defined time-varying Q-function, a model-free finite-horizon control method is developed to approximate the solutions of the time-varying Riccati equations of the transformed FHLQR problem. At last, simulation studies are carried out to verify the validity of the developed method.

A Note on Riccati Matrix Difference Equations

Discrete algebraic Riccati equations and their fixed points are well understood and arise in a variety of applications, however, the time-varying equations have not yet been fully explored in the literature. In this article we provide a self-contained study of discrete time Riccati matrix difference equations. In particular, we provide a novel Riccati semigroup duality formula and a new Floquet-type representation for these equations. Due to the aperiodicity of the underlying flow of the solution matrix, conventional Floquet theory does not apply in this setting and thus further analysis is required. We illustrate the impact of these formulae with an explicit description of the solution of time-varying Riccati difference equations and its fundamental-type solution in terms of the fixed point of the equation and an invertible linear matrix map, as well as uniform upper and lower bounds on the Riccati maps. These are the first results of this type for time varying Riccati matrix difference equations.

Stability of linear second-order time-varying differential equations via contractive polygons

A damped harmonic oscillator x ̈ ( t ) + a 1 x ̇ ( t ) + a 0 x ( t ) = 0 with a 0 , a 1 > 0 is known to be exponentially stable. We extend this result to time-varying positive coefficients a 0 ( t ) , a 1 ( t ) , t ≥ 0 , which are bounded from above and below and satisfy sup t ≥ 0 a 0 ( t ) < ( inf t ≥ 0 a 1 ( t ) ) 2 and we thus further extend the sufficient condition sup t ≥ 0 a 0 ( t ) ≤ 1 4 ( inf t ≥ 0 a 1 ( t ) ) 2 by Levin (1969). Under slightly weaker assumptions we show uniform stability.

Probability Density Evolution Algorithm for Stochastic Dynamical Systems Based on Fractional Calculus

With the increasing load and speed of trains, the problems caused by various random excitations (such as safety and passenger comfort) have become more prominent and thus arises the necessity to analyze stochastic dynamical systems, which is important in both academic and engineering circles. The existing analysis methods are inadequate in terms of computational accuracy, computational efficiency, and applicability in solving complex problems. For that, a new efficient and accurate method is used in this paper, suitable for linear and nonlinear random vibration analysis of large structures as well as static and dynamic reliability assessment. It is the direct probability integration method, which is extended and applied to the random vibration reliability analysis of dynamical systems. Dynamical models of the dynamic system and coupled system “three-car vehicle-rail-bridge” are established, the time-varying differential equations of motion are derived in detail, and the dynamic response of the system is calculated using the explicit Newmark algorithm. The simulation results show the influence of the number of representative points on the smoothness of the image of the probability density function and the accuracy of the calculation results.

On Symmetric Solutions to Linear Matrix Time-Varying Differential Equations

In this paper, we discuss when the solution to the initial value problem for a linear matrix time-varying differential equation is symmetric on a given interval. By symmetry, we mean that the solution does not change when transposed. Throughout the paper, we assume that the equation has coefficients of finite order of smoothness. We demonstrate that, in order to verify whether the solution to the initial value problem is symmetric on a given interval, it can be useful to construct two matrix sequences associated to the equation. Using these sequences, we prove a sufficient condition for the solution symmetry on a given interval. Assuming that the initial value problem for a linear matrix time-varying differential equation satisfies this condition, we derive a formula for a symmetric solution to this problem.

Solution of Time-Varying Index-2 Linear Differential Algebraic Control Systems Via A Variational Formulation Technique

This paper deals with finding an approximate solution to the index-2 time-varying linear differential algebraic control system based on the theory of variational formulation. The solution of index-2 time-varying differential algebraic equations (DAEs) is the critical point of the equivalent variational formulation. In addition, the variational problem is transformed from the indirect into direct method by using a generalized Ritz bases approach. The approximate solution is found by solving an explicit linear algebraic equation, which makes the proposed technique reliable and efficient for many physical problems. From the numerical results, it can be implied that very good efficiency, accuracy, and simplicity of the present approach are obtained.

Parameterization technique for solving variational formulation of index three linear time-varying singular control systems

This paper deals with finding an approximate solution to an index three time-varying linear differential algebraic control equations based on the theory of variational formulation. The critical points of the formulated variational functional subject to the differential- algebraic equation with the class of consistent initial conditions are shown to be the solution of the proposed index-three DAEs with control problem (singular control systems) and vise- versa. The given variational problem is also transformed from indirect method into direct method using generalized Ritz bases technique. From the numerical illustrations results, the efficiency and very good accuracy as well as the simplicity of the present approach are obtained.

Influence of common factors on vehicle bridge coupling vibration

As road and bridge construction industry becomes more prosperous, the vehicle’s speed and load capacity increase significantly. This paper aims at the problem of vehicle-bridge coupling vibration of highway bridges. In this paper, based on the basic theory of vehicle-bridge coupled vibration, the differential equation of vehicle-bridge coupled system is obtained. The mode decomposition method is adopted to transform the system differential equation into a matrix form. The second-order development function based on the built-in function was written by the principle of Runge-Kutta method, and the second-order time-varying differential equation group was solved. Taking the simply supported beam as an example, this paper studies the dynamic characteristics of bridge and vehicle under different speed and mass.

The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

A transform approach based on a variable initial time (VIT) formulation is developed for discrete-time signals and linear time-varying discrete-time systems or digital filters. The VIT transform is a formal power series in z−1, which converts functions given by linear time-varying difference equations into left polynomial fractions with variable coefficients, and with initial conditions incorporated into the framework. It is shown that the transform satisfies a number of properties that are analogous to those of the ordinary z-transform, and that it is possible to do scaling of z−i by time functions, which results in left-fraction forms for the transform of a large class of functions including sinusoids with general time-varying amplitudes and frequencies. Using the extended right Euclidean algorithm in a skew polynomial ring with time-varying coefficients, it is shown that a sum of left polynomial fractions can be written as a single fraction, which results in linear time-varying recursions for the inverse transform of the combined fraction. The extraction of a first-order term from a given polynomial fraction is carried out in terms of the evaluation of zi at time functions. In the application to linear time-varying systems, it is proved that the VIT transform of the system output is equal to the product of the VIT transform of the input and the VIT transform of the unit-pulse response function. For systems given by a time-varying moving average or an autoregressive model, the transform framework is used to determine the steady-state output response resulting from various signal inputs such as the step and cosine functions.

Characteristic model-based adaptive control for cryogenic wind tunnels

With the increasing requirement of Reynolds number simulation in wind tunnel tests, the cryogenic wind tunnel is considered as a feasible method to realize high Reynolds number. Characteristic model-based adaptive controller design method is introduced to flow field control problem of the cryogenic wind tunnel. A class of nonlinear multi-input multi-output (MIMO) system is given for theoretical research that is related to flow field control of the cryogenic wind tunnel. The characteristic model in the form of second-order time-varying difference equations is provided to represent the system. A characteristic model-based adaptive controller is also designed correspondingly. The stability analysis of the closed loop system composed of the characteristic model or the exact discrete-time model and the proposed controller is investigated respectively. Numerical simulation is presented to illustrate the effectiveness of this control method. The modeling and control problem based on characteristic model method for a class of MIMO system are studied and first applied to the cryogenic wind tunnel control field.

An Adaptive State Observer for Linear Time-varying Systems with Inaccurate Parameters

The design problem of an adaptive state observer for linear time-varying systems is considered. Some parameters of a time-varying plant are assumed to be unknown numbers multiplied by known functions of time. The approach proposed below is based on identification methods of adaptation. In other words, the main idea is to transform a mathematical model in the form of a linear time-varying differential equation to a static linear regression model containing unknown parameters. In this case, both the unknown parameters of the plant and its initial conditions are the unknown parameters of the regression model under consideration. Further, standard gradient methods or other parametric identification approaches are used to estimate the unknown parameters of the regression model and to construct the observer. The results of computer simulations illustrate the effectiveness of the new state observer design method.

Voltage Modulated DPC Strategy of DFIG Using Extended Power Theory under Unbalanced Grid Voltage Conditions

This paper develops a voltage modulated direct power control (VM-DPC) strategy of a doubly fed induction generator (DFIG) using extended power theory under unbalanced grid voltage conditions. By introducing the modulated voltage of the active and reactive powers and the non-linear VM controller, the proposed VM-DPC strategy enables the generator dynamics to change from the time-varying differential equations into the time-invariant ones. Then, in order to deal with the voltage unbalance, three available power feedbacks, a combination of real active power and extended reactive power, a combination of extended active power and real reactive power, a combination of extended active power and extended reactive power, are developed. Together with a conventional controller (PI + R and feedforwards) and a non-linear VM controller, the power feedbacks are well controlled to track their references with the achievement of the constant active power, the constant reactive power and the balanced stator current. The main advantage of the VM-DPC strategy is the introduction of the modulated voltage and the non-linear VM controller making the generator dynamics time-invariant, which is easy for applying various control methods. Furthermore, the application of extended power can avoid the sequence extractions and the power compensations under unbalanced grid voltage. Finally, the simulation results demonstrate the effectiveness of the developed VM-DPC strategy.

Pseudo-linear inertial navigation algorithm

A new method is illustrated for processing the output of a set of triad orthogonal rate gyros and accelerometers to reconstruct vehicle navigation parameters (attitude, velocity, and position). The paper introduces two vectors with dimensions 4 × 1 as velocity and position quaternions. The navigation equations for strapdown systems are nonlinear but after using these parameters, the navigation equations are converted into a pseudo-linear system. The new set of navigation equations has an analytical solution and the state transition matrix is used to solve the linear time-varying differential equations through time series. The navigation parameters are updated using the new formulation for strapdown navigation equations. Finally, the quaternions of velocity and position are converted into the original position and velocity vectors. The combination of the coning motion and a translational oscillatory trajectory is used to evaluate the accuracy of the proposed algorithm. The simulations show significant improvement in the accuracy of the inertial navigation system, which is achieved through the mentioned algorithm.