Time-varying Ordinary Differential Equations Research Articles

Time-varying uncertainty and disturbance estimator without velocity measurements: Design and application

The uncertainty and disturbance estimator (UDE) technique has been proven to be simple but effective in reconstructing unknown disturbances. However, classic UDE with a constant gain cannot deal with more complicated issues, such as the trade-off between steady-state and transient performance of the estimator, simultaneous attenuation of input disturbances and measurement noises, etc. Motivated by this observation, the design of UDE technique is extended to the time-varying situation and an implementable form of time-varying UDE (TV-UDE) is proposed. In particular, velocity measurements are assumed to be unavailable in the design. A time-varying ordinary differential equation is constructed to build the estimation relationship. A formula for the integration by parts is thereafter introduced to derive an implementable form of the TV-UDE. The classic UDE is shown to be a special case of the TV-UDE. The UDE is combined with a nominal dynamic output-feedback controller to yield a robust control solution to a class of uncertain second-order attitude control systems without velocity measurements. The design of a TV-UDE is discussed with a low initial gain and a higher final gain to avoid transient performance issues. Finally, numerical simulations and physical experiments are carried out on a 2-DOF AERO attitude helicopter platform to demonstrate the control performance of the solution. It is illustrated that the proposed design can effectively reduce the steady-state errors in both the trajectory tracking and target pointing tasks. Meanwhile, the input saturation and peaking phenomena associated with a high-gain estimation are avoided by applying the TV-UDE.

Reduction theorems for stability of compact sets in time-varying systems

Reduction theorems provide a framework for stability analysis that consists in breaking down a complex problem into a hierarchical list of subproblems that are simpler to address. This paper investigates the following reduction problem for time-varying ordinary differential equations on Rn. Let Γ1 be a compact set and Γ2 be a closed set, both positively invariant and such that Γ1⊂Γ2⊂Rn. Suppose that Γ1 is uniformly asymptotically stable relative to Γ2. Find conditions under which Γ1 is uniformly asymptotically stable. We present a reduction theorem for uniform asymptotic stability that completely addresses the local and global version of this problem, as well as two reduction theorems for uniform stability and either local or global uniform attractivity. These theorems generalize well-known equilibrium stability results for cascade-connected systems as well as previous reduction theorems for time-invariant systems. We also present Lyapunov characterizations of the stability properties required in the reduction theorems that to date have not been investigated in the stability theory literature.

Impact of ART-induced viral suppression on the HIV epidemic in Italy.

The present study aims to clarify the role of the fraction of patients under antiretroviral therapy (ART) achieving viral suppression (VS) (i.e. having plasma viral load below the detectability threshold) on the human immunodeficiency virus (HIV) epidemic in Italy. Based on the hypothesis that VS makes the virus untransmittable, we extend a previous model and we develop a time-varying ordinary differential equation model with immigration and treatment, where the naive and non-naive populations of infected are distinguished, and different compartments account for treated subjects virally suppressed and not suppressed. Moreover, naive and non-naive individuals with acquired immune deficiency syndrome (AIDS) are considered separately. Clinical data stored in the nationwide database Antiviral Response Cohort Analysis are used to reconstruct the history of the fraction of virally suppressed patients since highly active ART introduction, as well as to assess some model parameters. Other parameters are set according to the literature and the final model calibration is obtained by fitting epidemic data over the years 2003-2015. Predictions on the evolution of the HIV epidemic up to the end of 2035 are made assuming different future trends of the fraction of virally suppressed patients and different eligibility criteria for treatment. Increasing the VS fraction is found to reduce the incidence, the new cases of AIDS and the deaths from AIDS per year, especially in combination with early ART initiation. The asymptotic properties of a time-invariant formulation of the model are studied, and the existence and global asymptotic stability of a unique positive equilibrium are proved.

Admissible controls and controllable sets for a linear time-varying ordinary differential equation

In this paper, for a time optimal control problem governed by a linear time-varying ordinary differential equation, we give a description to check whether the set of admissible controls is nonempty or not by finite times.

Vibration behaviors of functionally graded rectangular plates with porosities and moving in thermal environment

A first known study is conducted on the vibrations of functionally graded material (FGM) rectangular plates with porosities and moving in thermal environment. The FGM plates contain porosities owing to the technical issues during the preparation of FGMs. Two types of porosity distribution, namely, even and uneven distribution, are considered. The geometric nonlinearity is taken into account by using von Kármán nonlinear plate theory. The out-of-plane equation of motion of the system is derived based on the D'Alembert's principle with the consideration of the thermal effect and longitudinal speed. Then the Galerkin method is employed to discretize the partial differential equation of motion to a set of ordinary differential equations. These time-varying ordinary differential equations are solved analytically by means of the method of harmonic balance. The accuracy of approximately analytical solutions is verified by the adaptive step-size fourth-order Runge–Kutta technique. Additionally, the stability of steady-state solutions is analyzed for the analytical solutions. Vibration characteristics such as natural frequency and nonlinear frequency response are shown. The present model is a hardening-spring system based on the nonlinear frequency response results. Effects of some key parameters are investigated on the vibration of rectangular FGM plates with porosities and moving in thermal environment.

External force field during diffusion - sedimentation in condensed matter

In this paper numerical description of the external force field during interdiffusion is discussed. Two methods of modeling the sedimentation process will be presented. Mainly, the steady state and evolutionary approximation. The proposed approaches will based on the generalized Darken method. The main equations will be the mass conservation and equation of Both presented approximations will based on the method of lines to solve numerically the ordinary differential equations (ODEs) system resulting from the space discretization (Schiesser, 1991). The uniform grid, which contained N mesh points along the x direction, respectively, was used and the concentrations and drift velocity were defined at points xk. The applied discretization led to the system of ODEs in time variable. The evolutionary method will based additionally on the adaptive step-size Runge–Kutta-Fehlberg modification. The six evaluations of the functions from the fifth-order Runge–Kutta algorithm were used to make other combinations implemented in the fourth-order Runge–Kutta method. The difference between these two estimates served as an estimate of the truncation error. Hence, the step size was adjusted (Press, 1992). The results of the proposed methods will be compared and discussed.motion. Thus the simultaneous action of backstress, external forces and of Kirkendall shift during interdiffusion in metals is described by self-consistent scheme. The diffusion flux is than determined by both, chemical and mechanical potentials - j=Di grad(ui).

Stochastic forced vibration analysis of a tapered beam with performance deterioration

This paper’s purpose is to investigate the stochastic dynamic behavior of a tapered cantilever beam, taking into account the deterioration of its performance during the working process. The equation of motion of the tapered beam is established by using Lagrange’s equation. By applying the assumed mode method, the equation of motion of the beam is discretized into a series of coupled linear time-varying ordinary differential equations. To describe the deterioration of the performance of the tapered cantilever beam, the mass and stiffness deteriorations of the beam are modeled by two independent Gamma processes. Based on the stochastic perturbation method, the response of the linear continuous system with stochastic parameters is investigated by superimposing the discretized modes. In the numerical simulation, the validity of the present study is confirmed, the effects of the cone angle (tapered ratio) of the tapered beam and the frequency of the external force on the response of the tapered cantilever beam are investigated, and the influences of the parameters of the two Gamma processes on the variance of the response are also discussed.

ANALYSIS ON GENERALIZED IMPACT ANGLE CONTROL GUIDANCE LAW

In this paper, a generalized guidance law with an arbitrary pair of guidance coefficients for impact angle control is proposed. Under the assumptions of a stationary target and a lag-free missile with constant speed, necessary conditions for the guidance coefficients to satisfy the required terminal constraints are obtained by deriving an explicit closed-form solution. Moreover, optimality of the generalized impact-angle control guidance law is discussed. By solving an inverse optimal control problem for the guidance law, it is found that the generalized guidance law can minimize a certain quadratic performance index. Finally, analytic solutions of the generalized guidance law for a first-order lag system are investigated. By solving a third-order linear time-varying ordinary differential equation, the blowing-up phenomenon of the guidance loop as the missile approaches the target is mathematically proved. Moreover, it is found that terminal misses due to the system lag are expressed in terms of the guidance coefficients, homing geometry, and the ratio of time-to-go to system time constant.

Nonlinear forced vibration analysis of postbuckled beams

A numerical solution methodology is proposed herein to investigate the nonlinear forced vibrations of Euler–Bernoulli beams with different boundary conditions around the buckled configurations. By introducing a set of differential and integral matrix operators, the nonlinear integro-differential equation that governs the buckling of beams is discretized and then solved using the pseudo-arc-length method. The discretized governing equation of free vibration around the buckled configurations is also solved as an eigenvalue problem after imposing the boundary conditions and some complicated matrix manipulations. To study forced and nonlinear vibrations that take place around a buckled configuration, a Galerkin-based numerical method is applied to reduce the partial integro-differential equation into a time-varying ordinary differential equation of Duffing type. The Duffing equation is then discretized using time differential matrix operators, which are defined based on the derivatives of a periodic base function. Finally, for any given magnitude of axial load, the pseudo -arc-length method is used to obtain the nonlinear frequencies of buckled beams. The effects of axial load on the free vibration, nonlinear, and forced vibrations of beams in both prebuckling and postbuckling domains for the lowest three vibration modes are analyzed. This study shows that the nonlinear response of beams subjected to periodic excitation is complex in the postbuckling domain. For example, the type of boundary conditions significantly affects the nonlinear response of the postbuckled beams.

Utilization of characteristic polynomials in vibration analysis of non-uniform beams under a moving mass excitation

Vibration of non-uniform beams with different boundary conditions subjected to a moving mass is investigated. The beam is modeled using Euler–Bernoulli beam theory. Applying the method of eigenfunction expansion, equation of motion has been transformed into a number of coupled linear time-varying ordinary differential equations. In non-uniform beams, the exact vibration functions do not exist and in order to solve these equations using eigenfunction expansion method, an adequate set of functions must be selected as the assumed vibration modes. A set of polynomial functions called as beam characteristic polynomials, which is constructed by considering beam boundary conditions, have been used along with the vibration functions of the equivalent uniform beam with similar boundary conditions, as the assumed vibration functions. Orthogonal polynomials which are generated by utilizing a Gram–Schmidt process are also used, and results of their application show no advantage over the set of simple non-orthogonal polynomials. In the numerical examples, both natural frequencies and forced vibration of three different non-uniform beams with different shapes and boundary conditions are scrutinized.

The Time Optimal Control with Constraints of the Rectangular Type for Linear Time-Varying ODEs

In this paper, we study a time optimal control problem of some linear time-varying ordinary differential equations, where the control constraint set is of the rectangular type. We derive a necessary and sufficient condition and provide an algorithm for the optimal time, as well as the optimal control for this problem. The basis consists of an equivalence theorem between the time optimal control problem and an associated norm optimal control problem, and the variational characterization of the latter problem.

Analytic Solutions of Generalized Impact-Angle-Control Guidance Law for First-Order Lag System

To examine the effects of system lag on performance of a generalized impact-angle-control guidance law, analytic solutions of the guidance law for a first-order lag system are investigated. Under the assumptions of a stationary target and a first-order missile system with constant speed and small flight-path angle, the analytic solutions are obtained by solving a third-order linear time-varying ordinary differential equation. The solutions are expressed by combinations of polynomial, logarithmic, and infinite power series functions. The analytic solutions provide an insight into the behavior of the missile near the target: the guidance command, the acceleration of the missile, and the velocity component perpendicular to the collision course tend to diverge as the missile approaches the target. Terminal misses due to the system lag are discussed using the analytic solutions, and effects of guidance coefficients on the terminal misses are examined. Linear and nonlinear simulations are performed to verify the proposed results.

Analytic solutions of optimal angularly constrained guidance for first-order lag system

Closed-form solutions of an optimal impact-angle-control guidance for a first-order lag system are derived. Under the assumptions of a stationary target and a first-order missile with constant speed and small flight path angle, they are obtained by solving a third-order linear time-varying ordinary differential equation. It is shown that the solutions are expressed by combinations of polynomial, logarithmic, and hypergeometric functions. Moreover, terminal misses due to the system lag and homing geometries for zero miss-distance have been studied. Linear and non-linear simulations are performed to verify the proposed results.

Coordinated Control of Space Robot Teams for the On-Orbit Construction of Large Flexible Space Structures

The construction of future space structures in Earth orbit such as solar power stations and space telescopes will require coordinated teams of autonomous space robots. These robot teams can excite undesirable vibrations in the structures while manipulating or assembling them. Controlling robot teams physically interacting with structural elements in space is challenging. The combined system dynamics are described by sets of nonlinear partial differential equations. Here, these dynamic equations are transformed into a set of linear time-varying ordinary differential equations. The control of the high-frequency robots is then decoupled from the control of the much lower frequency structures. This allows linear optimal control theory to be used to control the robots and minimize structural vibrations. Simulation and experimental studies shown here demonstrate the validity of the approach.

Comparison of Theory and Experiment in an Electron-Beam-Controlled High-Power TEA $\hbox{CO}_{2}$ Laser

In this paper, a theoretical model for the analysis of effective-discharge parameters in a high-power transversely excited atmospheric pressure CO2 laser with an electron beam preionization system is presented. The laser-discharge active volume and electron gun are modeled by nonlinear time-varying impedance. The lumped linear RLC circuits approximate the Marx generator and sustainer bank systems. By this method, the governing equations are reduced to a system of nonlinear time-varying ordinary differential equations that are solved numerically. Based on these equations, the electron beam and the main discharge parameters are evaluated. Our theoretical results are in good agreement with the experimental results.

Averaging of time-varying differential equations revisited

Employing comparison of integrals on a fast time scale, we offer a new criterion and simple proofs of the averaging principle for time-varying ordinary differential equations. The method allows straightforward extensions and generalizations. Comparisons with available criteria and estimates, along with examples and applications, are offered.

Stability Analysis for Systems with Impulse Effects

In the present paper the authors establish new conditions for the uniform stability and the uniform asymptotic stability of equilibria of systems with impulsive effects described by systems of nonlinear, time-varying ordinary differential equations. For the case when the corresponding systems without impulsive effects admit unstable properties, the above results are used to establish conditions under which the uniform stability even uniform asymptotic stability of equilibria of systems with impulsive effects can be caused by impulsive perturbations .

Stability radii for linear time-varying differential–algebraic equations with respect to dynamic perturbations

This paper is concerned with the robust stability for linear time-varying differential–algebraic equations. We consider the systems under the effect of uncertain dynamic perturbations. A formula of the structured stability radius is obtained. The result is an extension of a previous result for time-varying ordinary differential equations proven by Birgit Jacob [B. Jacob, A formula for the stability radius of time-varying systems, J. Differential Equations 142 (1998) 167–187].

Global optimization of linear hybrid systems with explicit transitions

The global optimization of hybrid systems described by linear time-varying ordinary differential equations is examined. A method to construct convex relaxations of general, nonlinear Bolza-type objective functions or constraints subject to an embedded hybrid system with explicit transitions is presented. The optimization problem can be solved using gradient-based algorithms in a branch and bound framework that is shown to be infinitely convergent when the implied state bounds are employed.

Input-to-State Stability for Nonlinear Time-Varying Systems via Averaging

We introduce two definitions of an averaged system for a time-varying ordinary differential equation with exogenous disturbances (“strong average” and “weak average”). The class of systems for which the strong average exists is shown to be strictly smaller than the class of systems for which the weak average exists. It is shown that input-to-state stability (ISS) of the strong average of a system implies uniform semi-global practical ISS of the actual system. This result generalizes the result of [TPA] which states that global asymptotic stability of the averaged system implies uniform semi-global practical stability of the actual system. On the other hand, we illustrate by an example that ISS of the weak average of a system does not necessarily imply uniform semi-global practical ISS of the actual system. However, ISS of the weak average of a system does imply a weaker semi-global practical “ISS-like” property for the actual system when the disturbances w are absolutely continuous and \(w, {\dot w}\in;{\scr L}_\infty\). ISS of the weak average of a system is shown to be useful in a stability analysis of time-varying cascaded systems.