On construction of a field of forces along given trajectories in the presence of random perturbations

In this paper, a force field is constructed along a given integral manifold in the presence of random perturbing forces. In this case, two types of integral manifolds are considered separately: 1) trajectories that depend on generalized coordinates and do not depend on generalized velocities, and 2) trajectories that depend on both generalized coordinates and generalized velocities. The construction of the force field is carried out in the class of second-order stochastic Ito differential equations. It is assumed that the functions in the right-hand sides of the equation must be continuous in time and satisfy the Lipschitz condition in generalized coordinates and generalized velocities. Also this functions satisfy the condition for linear growth in generalized coordinates and generalized velocities.These assumptions ensure the existence and uniqueness up to stochastic equivalence of the solution to the Cauchy problem of the constructed equations in the phase space, which is a strictly Markov process continuous with probability 1. To solve the two posed problems, stochastic differential equations of perturbed motion with respect to the integral manifold are constructed. Moreover, in the case when the trajectories depend on generalized coordinates and do not depend on generalized velocities, the second order equations of perturbed motion are constructed, and in the case when the trajectories depend on both generalized coordinates and generalized velocities, the first order equations of perturbed motion are constructed. And further, in both cases by Erugin’s method necessary and sufficient conditions for solving the posed problems are derived.

ON INVERSE STOCHASTIC RECONSTRUCTION PROBLEM

In this paper, general reconstruction problem in the class of second-order stochastic differential equations of the Ito type is considered for given properties of motion, when the control is included in the drift coefficient. And the form of control parameters is determined by the quasi-inversion method, which provides necessary and sufficient conditions for existence of a given integral manifold. Further, the solution of the Meshchersky’s stochastic problem is given, which is one of the inverse problems of dynamics and, according to the well-known Galiullin’s classification, refers to the restoration problem. It is assumed that random perturbations belong to the class of processes with independent increments. To solve the posed problem an equation of perturbed motion is drawn up by the Ito rule of stochastic differentiation. And, further, the Erugin method in combination with the quasi-inversion method is used to construct: 1) a set of control vector functions and 2) a set of diffusion matrices that provide necessary and sufficient conditions for a given second-order differential equation of Ito type to have a given integral manifold. The linear case of a stochastic problem with drift control is considered separately. In the linear setting, in contrast to the nonlinear formulation, the conditions of solvability in the presence of random perturbations from the class of processes with independent increments coincide with the conditions of solvability in a similar linear case in the presence of random perturbations from the class of independent Wiener processes. Also considered is the scalar case of the recovery problem with drift controls.

ON INVERSE STOCHASTIC RECONSTRUCTION PROBLEM

In this paper, general reconstruction problem in the class of second-order stochastic differential equations of the Ito type is considered for given properties of motion, when the control is included in the drift coefficient. And the form of control parameters is determined by the quasi-inversion method, which provides necessary and sufficient conditions for existence of a given integral manifold. Further, the solution of the Meshchersky’s stochastic problem is given, which is one of the inverse problems of dynamics and, according to the well-known Galiullin’s classification, refers to the restoration problem. It is assumed that random perturbations belong to the class of processes with independent increments. To solve the posed problem an equation of perturbed motion is drawn up by the Ito rule of stochastic differentiation. And, further, the Erugin method in combination with the quasi-inversion method is used to construct: 1) a set of control vector functions and 2) a set of diffusion matrices that provide necessary and sufficient conditions for a given second-order differential equation of Ito type to have a given integral manifold. The linear case of a stochastic problem with drift control is considered separately. In the linear setting, in contrast to the nonlinear formulation, the conditions of solvability in the presence of random perturbations from the class of processes with independent increments coincide with the conditions of solvability in a similar linear case in the presence of random perturbations from the class of independent Wiener processes. Also considered is the scalar case of the recovery problem with drift controls.

On stochastic inverse problem of construction of stable program motion

One of the inverse problems of dynamics in the presence of random perturbations is considered. This is the problem of the simultaneous construction of a set of first-order Ito stochastic differential equations with a given integral manifold, and a set of comparison functions. The given manifold is stable in probability with respect to these comparison functions.

A Robust Direct Parameter Identification of Exponentially Damped Low-Frequency Oscillation in Power Systems

Low-frequency oscillations in power systems can be modeled as an exponentially damped sinusoid (EDS) signal. Its frequency, damping factor, and amplitude are identified by the robust algorithm proposed in this paper. Under the condition of no noise, the exponentially convergent property of the proposed identification is proved by the use of time scale change, variable transformation, slow integral manifold, averaging method, and Lyapunov stability theorem in sequence. Under the condition of bounded additive noise, the antinoise performance of the identification of each parameter is investigated by the perturbed system theorem and error synthesis principle. The robustness of the proposed method is embodied in the following aspects: the exponential convergence for EDS signal with a wide range of frequency, especially with a rather low frequency; the boundary values of identification errors resulting from high-frequency sinusoidal noise of both frequency and damping factor can be adjusted by tuning the design parameters; and the different effects of the four design parameters on tracking performance and antinoise performance of each parameter identification. Simulation results demonstrate the performance of the algorithm and validate the conclusions.

Implementation of Dynamic Virtual Inertia Control of Supercapacitors for Multi-Area PV-Based Microgrid Clusters

In order to improve the dynamic stability of multi-area microgrid (MG) clusters in the autonomous mode, this study proposes a novel fuzzy-based dynamic inertia control strategy for supercapacitors in multi-area autonomous MG clusters. By virtue of the integral manifold theory, the interactive influence of inertia on dynamic stability for multi-area MG clusters is explored in detail. The energy function of multi-area MG clusters is constructed to further analyze the inertia constant. Based on the analysis of the mechanism, a control strategy for the fuzzy-based dynamic inertia control of supercapacitors for multi-area MG clusters is further proposed. For each sub-microgrid (sub-MG), the gain of the fuzzy-based dynamic inertia control is self-tuned dynamically, with system events being triggered, so as to flexibly and robustly enhance the dynamic performance of the multi-area MG clusters in the autonomous mode. To verify the effectiveness of the proposed control scheme, a three-area photovoltaic (PV)-based MG cluster is designed and simulated on the MATLAB/Simulink platform. Moreover, a comparison between the dynamic fuzzy-based inertial control method and an additional droop control method is finally presented to validate the advantages of the fuzzy-based dynamic inertial control approach.

Singular Perturbed Vector Field (SPVF) Applied to Complex ODE System with Hidden Hierarchy Application to Turbocharger Engine Model

In this paper, we present the concept of singularly perturbed vector field (SPVF) method and its application to spark ignition turbocharger engine. Given a mathematical/physical model, which consist of hidden multi-scale variables, the SPVF methods transfer (using the change of coordinates) and decompose such system to fast and slow subsystems. This decomposition enables one to apply different asymptotic methods such as the method of the integral manifold, homotopy analysis method, singular perturbation method, etc. The resulting subsystem enables us to understand better the complex system and to simplify the calculations. In addition, we investigated the stability of the equilibrium points of the model. This analysis has been done due to the SPVF method which reduces the complexity of the mathematical model.

A Robust Online Identification of Sustained Low Frequency Oscillation in Steady-State Power Systems

For sustained low frequency oscillations in steady-state power systems, an algorithm is proposed for precise online identification of oscillation frequency, oscillation amplitude, and fundamental amplitude. The algorithm consists of a robust low frequency estimator and a notch filter in parallel. The asymptotical convergence property is analyzed by slow integral manifold, averaging method, and Lyapunov stability theorem sequentially. The steady-state antinoise property is investigated by perturbed system theorem. The robust advantages of the proposed algorithm are embodied in the following aspects: the fundamental amplitude identification is little influenced by oscillation frequency and oscillation amplitude, both oscillation frequency identification and oscillation amplitude identification have small steady-state errors under high order harmonics or bounded noises, the ramp variations of both fundamental amplitude and oscillation amplitude are also significantly tracked, and three design parameters have different effects on identification performance of oscillation frequency, oscillation amplitude, and fundamental amplitude, respectively. Simulation results verify the validity.