Second-order Nonlinear Dynamic Equations Research Articles

Extended set-membership filter for power system dynamic state estimation

A new method for power system dynamic state estimation is presented. It is based on the application of the extended set-membership filter (ESMF). The ESMF provides an on-line nonlinear guaranteed estimation that assumes that noise sources are unknown but bounded, rather than stochastic as in probabilistic estimation algorithms such as the extended Kalman filter (EKF) and the unscented Kalman filter (UKF). Thus, the ESMF provides 100% confidence to the estimated states for the safety and reliability of power systems. In this paper, the ESMF is derived and demonstrated by using two different single machine infinite bus (SMIB) systems. The performance of the ESMF is compared with the classical UKF by using one SMIB system modeled as a second-order nonlinear dynamic equation. The feasibility of the ESMF method with unknown inputs is also presented on the other SMIB system, in which the exciter output voltage may be unavailable but bounded. The ESMF method is also tested on a multi-machine system model to demonstrate its effectiveness.

Interval Oscillation Criteria for Second-Order Nonlinear Forced Dynamic Equations with Damping on Time Scales

By using the Riccati transformation technique and constructing a class of Philos-type functions on time scales, we establish some new interval oscillation criteria for the second-order damped nonlinear dynamic equations with forced term of the form(r(t)xΔ(t))Δ+p(t)xΔσ(t)+q(t)(xσ(t))α=F(t,xσ(t))on a time scale𝕋which is unbounded, whereαis a quotient of odd positive integer. Our results in this paper extend and improve some known results. Some examples are given here to illustrate our main results.

Kamenev-Type Oscillation Criteria of Second-Order Nonlinear Dynamic Equations on Time Scales

Using functions from some function classes and a generalized Riccati technique, we establish Kamenev-type oscillation criteria for second-order nonlinear dynamic equations on time scales of the form(p(t)ψ(x(t))k∘xΔ(t))Δ+f(t,x(σ(t)))=0. Two examples are included to show the significance of the results.

Oscillation criteria for second-order nonlinear dynamic equations

This paper concerns the oscillation of solutions to the second-order dynamic equation on a time scale which is unbounded above. No sign conditions are imposed on , , and . The function is assumed to satisfy and for . In addition, there is no need to assume certain restrictive conditions and also the both cases are considered. Our results will improve and extend results in (Baoguo et al. in Can. Math. Bull. 54:580-592, 2011; Bohner et al. in J. Math. Anal. Appl. 301:491-507, 2005; Hassan et al. in Comput. Math. Anal. 59:550-558, 2010; Hassan et al. in J. Differ. Equ. Appl. 17:505-523, 2011) and many known results on nonlinear oscillation. These results have significant importance to the study of oscillation criteria on discrete time scales such as , , , or and the space of harmonic numbers . Some examples illustrating the importance of our results are also included. MSC:34K11, 39A10, 39A99.

A train air brake force model: Locomotive automatic brake valve and brake pipe flow formulations

The goal of this study is to integrate an air brake model with efficient algorithms for train longitudinal force calculation that are based on trajectory coordinate formulations. The air brake model, developed in this investigation and presented in this paper and a companion paper, consists of the locomotive automatic brake valve, air brake pipe and car control unit (CCU). The proposed air brake force model accounts for the effect of the air flow in long train pipes as well as the effect of leakage and branch pipe flows. This model can be used to study the dynamic behavior of the air flow in the train pipe and its effect on the longitudinal train forces during brake application and release. The governing equations of the air pressure flow are developed using the general fluid continuity and momentum equations, simplified using the assumptions of one-dimensional isothermal flow. Using these assumptions, one obtains two coupled air velocity/pressure partial differential equations that depend on time and the longitudinal coordinate of the brake pipes. The partial differential equations are converted to a set of first-order ordinary differential equations using the finite element method. The resulting air brake ordinary differential equations are solved simultaneously with the train’s second-order non-linear dynamic differential equations of motion that are based on the trajectory coordinates. The train car non-linear dynamics is defined using a body track coordinate system that follows the car motion. The body track coordinate system translation and orientation are defined in terms of one parameter that describes the distance traveled by the car. The configuration of the car with respect to its track coordinate system is described using two translation coordinates and three Euler angles. The operation modes of the brake system considered in this investigation are the brake release mode and the brake application mode that includes service and emergency brakes. A detailed model of the locomotive automatic brake valve is presented in this investigation and used to define the inputs to the air brake pipe during the simulation. A simplified model of this valve is also proposed in order to reduce the computational time of the simulation. In a companion paper, the detailed CCU formulation is presented. The coupling between the air brake, locomotive automatic brake valve, CCUs and train equations is established and used in the companion paper in the simulation of the non-linear dynamics of long trains. The air brake formulations presented in the two companion papers are implemented in a computer code called Analysis of Train/Track Interaction Forces (ATTIF) which is developed for the analysis of longitudinal train forces.

Interval Oscillation Criteria of Second‐Order Nonlinear Dynamic Equations on Time Scales

Using functions in some function classes and a generalized Riccati technique, we establish interval oscillation criteria for second‐order nonlinear dynamic equations on time scales of the form . The obtained interval oscillation criteria can be applied to equations with a forcing term. An example is included to show the significance of the results.

Oscillatory Behavior of a Class of Second-order Nonlinear Dynamic Equations on Time Scales

The paper is devoted to the oscillation of a class of second-order nonlinear dynamic equations on time scales. By developing a generalized Riccati transformation technique, we establish some oscillation criteria for all solutions of the equations. Our results improve and extend some known results in the literature.

Oscillation of second-order dynamic equations

In this paper, we consider the second-order nonlinear dynamic equations on an isolated time scale . Our first goal is to establish a relationship between the oscillatory behaviour of these equations. Here we assume that τ : → . We also give two results about the behaviour of the linear form of the latter equation on a general time scale that is unbounded above. We use the Riccati transformation technique to obtain our results.

Necessary and Sufficient Conditions for Positive Solutions of Second‐Order Nonlinear Dynamic Equations on Time Scales

This paper is concerned with the existence of nonoscillatory solutions for the nonlinear dynamic equation on time scales. By making use of the generalized Riccati transformation technique, we establish some necessary and sufficient criteria to guarantee the existence. The last examples show that our results can be applied on the differential equations, the difference equations, and the q‐difference equations.

Asymptotic behavior of second-order nonlinear dynamic equations on time scales

In this paper, we consider the second-order nonlinear dynamic equation $(p(t)y^{\Delta}(t))^{\Delta}+f(t, y^{\sigma})g(p(t)y^{\Delta})=0,$ on a time scale $\mathbb{T}$. Our goal is to establish necessary and sufficient conditions for the existence of certain types of solutions of this dynamic equation. We apply results from the theory of lower and upper solutions for related dynamic equations and use several results from calculus.

Approximation of Solution of Some m-Point Boundary Value Problems on Time Scales

The method of upper and lower solutions and the generalized quasilinearization technique for second-order nonlinear m-point dynamic equations on time scales of the type , , , , , , are developed. A monotone sequence of solutions of linear problems converging uniformly and quadratically to a solution of the problem is obtained.

Oscillation criteria for nonlinear functional neutral dynamic equations on time scales

This paper is concerned with the oscillation of the second-order nonlinear functional dynamic equations on a time scale where γ is the quotient of odd positive integers, r(t), p(t), and are positive rd-continuous functions on , , and . We establish some new sufficient conditions for oscillation for the above equation. Several examples illustrating our results will be given.

Existence and uniqueness of solutions to a nonlinear elliptic boundary value problem on time scales

In this paper, we consider a boundary value problem (BVP) for second-order nonlinear partial dynamic equations on the time scale rectangles. Some explicit conditions are established that ensure existence and uniqueness of solution to the BVP under consideration.

Nonlinear oscillation of second-order dynamic equations on time scales

Interval oscillation criteria are established for a second-order nonlinear dynamic equation on time scales by utilizing a generalized Riccati technique and the Young inequality. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives.

Oscillation of second-order strongly superlinear and strongly sublinear dynamic equations

We establish some new criteria for the oscillation of second-order nonlinear dynamic equations on a time scale. We study the case of strongly superlinear and the case of strongly sublinear equations subject to various conditions.

Forced oscillation of second-order nonlinear dynamic equations on time scales

In this paper, by defining a class of functions, we establish some oscillation criteria for the second order nonlinear dynamic equations with forced term x �� (t) + a(t)f(x(q(t))) = e(t) on a time scale T. Our results unify the oscillation of the second order forced differential equation and the second order forced difference equation. An example is considered to illustrate the main results.

Oscillation criteria for a forced second-order nonlinear dynamic equation

In this paper, we will establish some new interval oscillation criteria for the forced second-order nonlinear dynamic equation on a time scale 𝕋 where γ ≥ 1. As a special case when 𝕋 = ℝ our results not only include the oscillation results for second-order differential equations established by Wong (Oscillation criteria for a forced second-order linear differential equation, J. Math. Anal. Appl. 231 (1999), pp. 233–240) and Nasr (Sufficient condition for the oscillation of forced super-linear second order differential equations with oscillatory potential, Proc. Amer. math. Soc. 126 (1998), pp. 123–125) but also improve these results. When 𝕋 = ℕ, 𝕋 = hℕ or 𝕋 = q N , i.e. for difference equations, generalized difference equations or q-difference equations our results are essentially new and also can be applied on different types of time scales, as illustrated in several examples.

On the Asymptotic Integration of Nonlinear Dynamic Equations

The purpose of this paper is to study the existence and asymptotic behavior of solutions to a class of second-order nonlinear dynamic equations on unbounded time scales. Four different results are obtained by using the Banach fixed point theorem, the Boyd and Wong fixed point theorem, the Leray-Schauder nonlinear alternative, and the Schauder fixed point theorem. For each theorem, an illustrative example is presented. The results provide unification and some extensions in the time scale setup of the theory of asymptotic integration of nonlinear equations both in the continuous and discrete cases.

Interval criteria for oscillation of nonlinear second-order dynamic equations on time scales

Interval oscillation criteria are established for a second-order nonlinear dynamic equation on time scales by utilizing a generalized Riccati technique and an integral averaging technique. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives.

Oscillation of second-order damped dynamic equations on time scales

The study of dynamic equations on time scales has been created in order to unify the study of differential and difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which may be an arbitrary closed subset of the reals. This way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained. In this paper, by employing the Riccati transformation technique we will establish some oscillation criteria for second-order linear and nonlinear dynamic equations with damping terms on a time scale T . Our results in the special case when T = R and T = N extend and improve some well-known oscillation results for second-order linear and nonlinear differential and difference equations and are essentially new on the time scales T = h N , h > 0 , T = q N for q > 1 , T = N 2 , etc. Some examples are considered to illustrate our main results.