Singular Perturbation Method Research Articles

On the neutral stability curve for shallow conical shells subjected to lateral pressure

This work presents a detailed asymptotic description of the neutral stability envelope for the linear bifurcations of a shallow conical shell subjected to lateral pressure. The eighth-order boundary-eigenvalue problem investigated originates in the Donnell shallow-shell theory coupled with a linear membrane pre-bifurcation state, and leads to a neutral stability curve that exhibits two distinct growth rates. By using singular perturbation methods we propose accurate approximations for both regimes and explore a number of other novel features of this problem. Our theoretical results are compared with several direct numerical simulations that shed further light on the problem.

Finding Hankel singular values for singularly perturbed linear continuous‐time systems

This study presents an efficient algorithm for computing precisely the Hankel singular values of linear singularly perturbed systems. The algorithm is obtained in terms of reduced-order problems and avoids numerical ill-conditioning associated with singularly perturbed systems when the singular perturbation parameter is very small. The study compares the presented algorithm with the algorithm that exists in the control engineering literature and demonstrates its superiority. In addition, an analysis of system order reduction via balancing and singular perturbations is performed and it is concluded that the system order reduction via balancing is more general than the system order reduction obtained via the method of singular perturbations.

Impact of saturable distribution in compartmental PK models: dynamics and practical use

We explore the impact of saturable distribution over the central and the peripheral compartment in pharmacokinetic models, whilst assuming that back flow into the central compartiment is linear. Using simulations and analytical methods we demonstrate characteristic tell-tale differences in plasma concentration profiles of saturable versus linear distribution models, which can serve as a guide to their practical applicability. For two extreme cases, relating to (i) the size of the peripheral compartment with respect to the central compartment and (ii) the magnitude of the back flow as related to direct elimination from the central compartment, we derive explicit approximations which make it possible to give quantitative estimates of parameters. In three appendices we give detailed explanations of how these estimates are derived. They demonstrate how singular perturbation methods can be successfully employed to gain insight in the dynamics of multi-compartment pharmacokinetic models. These appendices are also intended to serve as an introductory tutorial to these ideas.

Additive fault tolerant control of nonlinear singularly perturbed systems against actuator fault

Abstract This paper presents the design of an additive fault tolerant control for nonlinear time-invariant singularly perturbed systems against actuator faults based on Lyapunov redesign principle. The overall system is reduced into subsystems with fast and slow dynamic behavior using singular perturbation method. The time scale reduction is carried out when the singular perturbation parameter is set to zero, thus avoiding the numerical stiffness due to the interaction of two different dynamics. The fault tolerant controller is computed in two steps. First, a nominal composite controller is designed using the reduced subsystems. Secondly, an additive part is combined with the basic controller to overcome the fault effect. The derived ε - independent controller guarantees asymptotic stability despite the presence of actuator faults. The Lyapunov stability theory is used to prove the stability provided the singular perturbation parameter is sufficiently small. The theoretical results are simulated using a numerical application.

Stochastic Optimal Controller Design of Decentralized Singularly Perturbed Unified Systems

이 연구에서는 연속시간, 이신시간과 단일 분산 특이변동 시스템에 대한 확률 최적 제어기 설계에 대한 비교작업을 수행하였다. 이 논문에서는 두 가지 목표가 고려되었는데 첫 번째는 특이변동 방법을 적용하여 완전 차수의 분산 특이변동 확률시스템으로부터 축소차수 시스템을 얻는 것이고 두 번째는 연속시간, 이산시간 그리고 단일 확률 시스템이 각각 적용되었을 때 상태귀환과 출력귀환 제어기를 이용하여 그 최적 제어기와 최적비용을 구하는 것이다. 축소차수 시스템의 획득과 최적 제어기-최적비용의 도출에 대한 타당성을 증명하기 위해, 상태귀환 확률제어기와 출력귀환 확률제어기에 대한 수치적인 데이터를 연속시간, 이산시간 그리고 단일시스템에 대한 예제들을 이용하여 각각 제시하였다.

Asymptotic solutions for laminar flow based on blood circulation through a uniformlyporous channel with retractable walls and an applied transverse magnetic field

This paper is concerned with asymptotic solutions of a nonlinear boundary value problem (BVP), which arises in a study of laminar flow in a uniformly porous channel with retractable walls and an applied transverse magnetic field. For different ranges of the control parameters (i.e. α,Re and M) arising in the BVP, four cases are considered using different singular perturbation methods. For the first case, unlike those in the existing literature, we make use of the Lighthill method and successfully construct an asymptotic solution with high-order derivatives at the center of the channel. For the second case, under large suction we consider M2=O(1) and M2=O(Re), respectively, which will further extend the applying range of asymptotic solutions. In other cases, asymptotic solutions with a boundary layer are successfully constructed. In addition, numerical solutions presented for each case agree well with asymptotic solutions, which illustrates that the asymptotic solutions constructed in this paper are more reliable. Finally, the influences of some parameters on flow field are discussed to develop a better understanding of the flow problem.

The uniqueness of limit cycles for classical Liénard equations of degree four under singular perturbations

The Lins-de Melo-Pugh conjecture for $n=4$ was proved in 2012, but the proof is complicated. In this article we give a simple proof for this problem under singular perturbations. We want to show the role of singular perturbation method in the study of qualitative theory of dynamical systems, and to show the application of a new formula of the slow divergence integral.

Development of a Reduced n-Decane/α-Methylnaphthalene/Polycyclic Aromatic Hydrocarbon Mechanism and Its Application for Combustion and Soot Prediction

In this work, a reduced n-decane/α-methylnaphthalene/polycyclic aromatic hydrocarbon (PAH) kinetic mechanism for calculation of combustion and soot behavior of diesel and its surrogate fuel was developed. This mechanism consists of the pyrolysis of n-decane (C10H22) and α-methylnaphthalene (C10H7CH3), C0–C3 core species reaction, and the formation of initial benzene and PAHs, including 77 species and 287 reactions. The pyrolysis reaction of fuel was obtained by pathway analysis from a detailed mechanism, while the reactions of core species and PAHs were reduced by a directed relation graph with error propagation, computational singular perturbation method, and direct sensitivity analysis, sequentially. The mechanism was validated by the mole fraction of main species and key PAH species in the ethylene premixed flame, ignition delay times of pure and mixed fuel in shock tubes, and the concentrations of major species in jet-stirred reactors. Finally, the new developed mechanism was coupled with a soot pheno...

Dynamic equivalent model of VSC based on singular perturbation

Scalable and computationally efficient model of voltage source converter (VSC) that accurately captures the dynamics is key to the modelling, analysis and control of VSC–high-voltage direct current system. In this study, a dynamic equivalent model of VSC including the dynamics of control system is obtained by using singular perturbation method for the first time. The reduced second-order model with simple form and good generality captures the steady-state and transient behaviours of original full model with a high degree of accuracy. This equivalent model retains the physical meaning of the variables, accommodates different control modes and significantly reduces the computational burden compared with the original full model. Singular perturbation method itself contains some systematic procedures to be able to improve the accuracy of the dynamic behaviour of the lower-order model. The ability of the dynamic equivalent model to accurately describe the original dynamics of the system has been verified with numerical simulations in normal operation and on fault case associated with MTDC grids.

Adaptive Fault Tolerant Control of Multi-time-scale Singularly Perturbed Systems

This paper studies the fault tolerant control, adaptive approach, for linear time-invariant two-time-scale and three-time-scale singularly perturbed systems in presence of actuator faults and external disturbances. First, the full order system will be controlled using e-dependent control law. The corresponding Lyapunov equation is ill-conditioned due to the presence of slow and fast phenomena. Secondly, a time-scale decomposition of the Lyapunov equation is carried out using singular perturbation method to avoid the numerical stiffness. A composite control law based on local controllers of the slow and fast subsystems is also used to make the control law e-independent. The designed fault tolerant control guarantees the robust stability of the global closed-loop singularly perturbed system despite loss of effectiveness of actuators. The stability is proved based on the Lyapunov stability theory in the case where the singular perturbation parameter is sufficiently small. A numerical example is provided to illustrate the proposed method.

Prediction of velocity distribution in straight open-channel flow with partial vegetation by singular perturbation method

A numerical analysis model based on two-dimensional shallow water differential equations is presented for straight open-channel flow with partial vegetation across the channel. Both the drag force acting on vegetation and the momentum exchange between the vegetation and non-vegetation zones are considered. The depth-averaged streamwise velocity is solved by the singular perturbation method, while the Reynolds stress is calculated based on the results of the streamwise velocity. Comparisons with the experimental data indicate that the accuracy of prediction is significantly improved by introducing a term for the secondary current in the model. A sensitivity analysis shows that a sound choice of the secondary current intensity coefficient is important for an accurate prediction of the depth-averaged streamwise velocity near the vegetation and non-vegetation interfaces, and the drag force coefficient is crucial for predictions in the vegetation zone.

Simulation of a helicopter’s main gearbox semiactive suspension with bond graphs

This paper presents a bond graph model of a helicopter’s semiactive suspension and the associated simulations. The structural and modular approach proposed with bond graph permits a systematic modeling of mechatronic multibody systems. This approach was carried out thanks to the use of the singular perturbation method, which is a variant of penalty formulation. The model is then built as an assembly of components or modules (rigid bodies and compliant kinematic joints) by following the structure of the actual system. The bond graph model of the passive suspension with fixed flapping masses has been verified with another multibody tool for three different excitations (pumping, roll, and yaw). Next, the passive model, augmented with electrical actuators and controllers, is called the semiactive suspension model. Simulations on the semiactive suspension model have been conducted.

Modeling and vibration control for a flexible pendulum inverted system based on a PDE observer

ABSTRACTThis study addresses the problem of trajectory control of a flexible pendulum inverted system on the basis of the partial differential equation (PDE) and ordinary differential equation (ODE) dynamic model. One of the key contributions of this study is that a new model is proposed to simplify the complex system. In addition, this study proposed a nonlinear PDE observer to estimate distributed positions and velocities along flexible pendulum. Singular perturbation method is proposed to solve the coupling system of nonlinear PDE observer. The nonlinear PDE observer is divided into a fast subsystem and a slow subsystem by the use of the singular perturbation method. To stabilise this fast subsystem, a boundary controller is proposed at the free end of the beam. The sliding-mode control method is proposed to design controller for slow subsystems. The asymptotic stability of both the proposed nonlinear PDE observer and controller is validated by theoretical analysis. The results are illustrated by simulation.

Two-time scale path following of underactuated marine surface vessels: Design and stability analysis using singular perturbation methods

The paper aims to develop a novel path following control law for nonlinear vessel models in two-time scales. For the 4-DOF path following model, we explore that the guidance dynamics are much slower than the ship motion dynamics. Based on such characteristic, a singular perturbation method is used to decompose the full system into two-time-scale subsystems. The two-time-scale structure allows independent analysis of dynamics in each time scale. Separate control strategies for the quasi-steady-state subsystem and the boundary layer subsystem are designed to stabilize the full system, yielding a rudder angle control law which is compact and uncomplicated to implement in practice. Singular perturbation methods are utilized to provide mathematical expressions for the upper bound of the singularly perturbed parameter and establish the exponential stability of the full system. The paper proves that the control law is robust in the presence of bounded perturbations and unmodeled dynamics, resulting states converge into an invariant set arbitrarily closed to the origin. Simulation results show the effectiveness and robustness of the proposed method for path following. The primary benefit of the proposed method is the simplicity of implementation.

Boundary Control for A Flexible Inverted Pendulum System Based on A Pde Model

AbstractA partial differential equation (PDE) model for a flexible inverted pendulum system (FIPS) is derived by the use of the Hamilton principle. To solve the coupling system model, a singular perturbation method was adopted. The PDE model was divided into a fast subsystem and a slow subsystem using the singular perturbation method. To stabilize the fast subsystem, a boundary control force was applied at the free end of the beam. It then was proven that the closed‐loop subsystem is appropriate and exponentially stable. For the slow subsystem, a sliding mode control method was employed to design a controller and the Linear Matrix Inequality (LMI) method was used to design the sliding surface. It then was shown that the slow subsystem is exponentially stable.

Singularities and wrinkling: The case of a concentrated force

The edge wrinkling of a uniformly stretched circular elastic plate subjected to a central concentrated load is considered within the framework of the Föppl–von Kármán nonlinear plate theory. Singular perturbation methods are employed to obtain a three-term asymptotic formula for the critical load in terms of a non-dimensional quantity that depends on the initial pre-stress. Comparisons between the analytical predictions and direct numerical simulations of the full bifurcation eigenproblem provide further insight into the accuracy and limitations of the derived results.

A dynamical model for bark beetle outbreaks

Tree-killing bark beetles are major disturbance agents affecting coniferous forest ecosystems. The role of environmental conditions on driving beetle outbreaks is becoming increasingly important as global climatic change alters environmental factors, such as drought stress, that, in turn, govern tree resistance. Furthermore, dynamics between beetles and trees are highly nonlinear, due to complex aggregation behaviors exhibited by beetles attacking trees. Models have a role to play in helping unravel the effects of variable tree resistance and beetle aggregation on bark beetle outbreaks. In this article we develop a new mathematical model for bark beetle outbreaks using an analogy with epidemiological models. Because the model operates on several distinct time scales, singular perturbation methods are used to simplify the model. The result is a dynamical system that tracks populations of uninfested and infested trees. A limiting case of the model is a discontinuous function of state variables, leading to solutions in the Filippov sense. The model assumes an extensive seed-bank so that tree recruitment is possible even if trees go extinct. Two scenarios are considered for immigration of new beetles. The first is a single tree stand with beetles immigrating from outside while the second considers two forest stands with beetle dispersal between them. For the seed-bank driven recruitment rate, when beetle immigration is low, the forest stand recovers to a beetle-free state. At high beetle immigration rates beetle populations approach an endemic equilibrium state. At intermediate immigration rates, the model predicts bistability as the forest can be in either of the two equilibrium states: a healthy forest, or a forest with an endemic beetle population. The model bistability leads to hysteresis. Interactions between two stands show how a less resistant stand of trees may provide an initial toe-hold for the invasion, which later leads to a regional beetle outbreak in the resistant stand.

A delayed stochastic volatility correction to the constant elasticity of variance model

The Black-Scholes model does not account non-Markovian property and volatility smile or skew although asset price might depend on the past movement of the asset price and real market data can find a non-flat structure of the implied volatility surface. So, in this paper, we formulate an underlying asset model by adding a delayed structure to the constant elasticity of variance (CEV) model that is one of renowned alternative models resolving the geometric issue. However, it is still one factor volatility model which usually does not capture full dynamics of the volatility showing discrepancy between its predicted price and market price for certain range of options. Based on this observation we combine a stochastic volatility factor with the delayed CEV structure and develop a delayed hybrid model of stochastic and local volatilities. Using both a martingale approach and a singular perturbation method, we demonstrate the delayed CEV correction effects on the European vanilla option price under this hybrid volatility model as a direct extension of our previous work [12].

The Term Structure of Interest Rates Under Heath–Jarrow–Morton Models with Fast Mean-Reverting Stochastic Volatility

This paper is a study of the term structure of interest rates based on the Heath–Jarrow–Morton (HJM) models with Hull–White volatility function. Under fast mean-reverting stochastic volatility, we obtain an analytic formula for an approximate bond price with estimated error using a Markovian transform method combined with a singular perturbation method. The stochastic volatility correction effect against time-to-maturity is revealed so that it can capture more of the complexities of the interest rate term structure.

Existence Results for Some Partial Integro-Differential Equations

In this note, we deal with semilinear integro-differential equations subject to homogeneous Dirichlet boundary conditions given on the boundaries of the sections. Even if the differentiation will be taken only in some directions, it is not possible to see the main problem parameterized by the other coordinates because of the non-local terms which also obliged the problem to be degenerate. We establish the existence of solutions by employing the singular perturbations method as a natural tool. The perturbed problems are classical, non-local, semilinear elliptic problems and the limits of the subsequences of their solutions, in weighted Sobolev type spaces, are solutions of the main problem. Some improvement, concerning the existence of the solutions and the convergence results depending on the weights, will be established. The paper also gives an idea about the study of the anisotropic singular perturbations in the framework of weighted spaces.