Singular Perturbation Method Research Articles

Computationally efficient approach for simulation of multibody and hydraulic dynamics

A realistic real-time simulation of a complex system, such as an excavator, requires detailed description of the machinery and its components. To take into account the dynamics of entire systems, the model must encompass descriptions of non-mechanical systems, such as hydraulics. For the multibody systems, use of the semi-recursive methods has often been found to be the most efficient solution when the system size increases. For the hydraulic dynamics, in turn, the recently introduced application of the singular perturbation method is a potential candidate for the real-time applications. The main benefit of the application of the singular perturbation method over the conventionally used lumped fluid method is that it overcomes the challenges that the lumped fluid method encounters when numerical stiffness caused by small hydraulic volumes is present in the circuit. Objective of this paper is to improve a recently proposed monolithic formulation for the combined simulation of multibody and hydraulic dynamics via the introduction of the singular perturbation method. Results indicate that the proposed method improves efficiency and robustness when compared to the formulation proposed earlier.

A theory of reactant-stationary kinetics for a mechanism of zymogen activation

A theoretical analysis is performed on the nonlinear ordinary differential equations that govern the dynamics of a reaction mechanism of zymogen activation. The reaction consists of a primary non-observable zymogen activation reaction that it is coupled to an indicator (observable) reaction. The product of the first reaction is the enzyme of the indicator reaction, and both reactions are governed by the Michaelis–Menten reaction mechanism. Using singular perturbation methods, we derive asymptotic solutions that are valid under the quasi-steady-state and reactant-stationary assumptions. In particular, we obtain closed form solutions that are analogous to the Schnell–Mendoza equation for Michaelis–Menten type reactions. These closed-form solutions approximate the evolution of the observable reaction and provide the mathematical link necessary to measure the enzyme activity of the non-observable reaction. Conditions for the validity of the asymptotic solutions are also derived, and we demonstrate that these asymptotic expressions are applicable under reactant-stationary kinetics.

Order reduction in electrical power systems using singular perturbation in different coordinate systems

PurposeEfficient calculations of the transient behaviour after disturbances of large-scale power systems are complex because of, among other things, the non-linearity and the stiffness of the overall state equation system (SES). Because of the rising amount of flexible transmission system elements, there is an increasing need for reduced order models with a negligible loss of accuracy. With the Extended Nodal Approach and the application of the singular perturbation method, it is possible to reduce the order of the SES adapted to the respective setting of the desired tasks and accuracy requirements.Design/methodology/approachBased on a differential-algebraic equation for the electric power system which is formulated with the Extended Nodal Approach, the automatic decomposition into reduced order models is shown in this paper. The paper investigates the effects of different coordinate systems for an automatic order reduction with the singular perturbation method, as well as a comparison of results calculated with the full and reduced order models.FindingsThe eigenvalues of the full system are approximated sufficiently by the three subsystems. A simulation example demonstrates the good agreement between the reduced order models and the full model independent of the choice of the coordinate system. The decomposed subsystems in rotating coordinates have benefits as compared to those in static coordinates.Originality/valueThe paper presents a systematic decomposition based only on a differential-algebraic equation system of the electric power system into three subsystems.

The computation of biomarkers in pharmacokinetics with the aid of singular perturbation methods.

In pharmacokinetics, exact solutions to one-compartment models with nonlinear elimination kinetics cannot be found analytically, if dosages are assumed to be administered repetitively through extravascular routes (Tang and Xiao in J Pharmacokinet Pharmacodyn 34(6):807-827, 2007). Hence, for the corresponding impulsed dynamical system, alternative methods need to be developed to find approximate solutions. The primary purpose of this paper is to use the method of matched asymptotic expansions (Holmes Introduction to Perturbation Methods, vol 20. Springer Science & Business Media, Berlin, 2012), a singular perturbation method (Holmes, Introduction to Perturbation Methods, vol 20. Springer Science & Business Media, Berlin, 2012; Keener Principles of Applied Mathematics, Addison-Wesley, Boston, 1988), to obtain approximate solutions. With this method, we are able to rigorously determine conditions under which there is a stable periodic solution of the model equations. Furthermore, typical important biomarkers that enable the design of practical, efficient and safe drug delivery protocols, such as the time the drug concentration reaches the peak and the peak concentrations, are theoretically estimated by the perturbation method we employ.

A Class of Singular Perturbation Problem for the Nonlinear Differential-Integral Disturbed Evolution Equation

A class of nonlinear differential-integral singular perturbation problem for the disturbed evolution equations is studied. Using the singular perturbation method, the structure of solution to problem is discussed in the cases of two small parameters and under the suitable conditions. Firstly, the outer solution to boundary value problem is given. Secondly, constructing the non-singular coordinate system near the boundary, the variables of multiple scales is introduced to obtain the boundary layer corrective term for the solution. Then the stretched variable is applied to get the initial layer correction term. Finally, using the fix point theorem, the uniformly valid asymptotic expansion of the solution to problem is proved. The proposed method possesses the advantages of convenient use and high accuracy.

Editorial

The lead article for this issue of EJAM is Prof. Bernard J. Matkowsky's personalized survey-style article on the theory and application of singular perturbation methods to noisy dynamical systems in the limit of small noise. This article is based on his John von Neumann Prize lecture presented at the Society of Industrial and Applied Mathematics (SIAM) Annual Meeting in July 2017. The John von Neumann Lecture is awarded by SIAM for outstanding and distinguished contributions to the field of applied mathematical sciences and for the effective communication of these ideas to the community. From 1990–1996, Matkowsky was an inaugural editorial board member for EJAM.

Group controllability of two-time-scale multi-agent networks

As a basic concept in modern control theory, controllability reveals the fundamental structural characteristics of a dynamic system, and it also plays an important part in the analysis and control of a dynamic system. With the increasing complexity of multi-agent systems, the multi-agent networks can be divided into some subnetworks in terms of time scales. This paper concentrates on the group controllability of two-time-scale multi-agent networks, establishes the necessary and sufficient criterion of group controllability based on singular perturbation methods, and deduces easy-to-use group controllability criteria by using matrix theory and graph theory. Lastly, a simulation example is presented to illustrate the effectiveness of the proposed methods.

Postbuckling behavior of shear deformable anisotropic laminated cylindrical shell under combined external pressure and axial compression

Structural design of composite shells are more challenging than conventional metals due to the complex mechanical behavior and damage mechanisms which composite materials exhibit. Postbuckling analysis for a moderately thick anisotropic laminated cylindrical shell subjected to combined loadings of external pressure and axial compression is presented which extends the boundary layer theory of shell buckling. The governing equations are based on Reddy’s higher order shear deformation shell theory with von Kármán-Donnell-type of kinematic nonlinearity. Both nonlinear prebuckling deformations and initial geometric imperfections of the shell are taken into account. A two-step singular perturbation method is used to determine interactive buckling loads and postbuckling equilibrium paths. A verification study is conducted, and the validity of the formulation is established through comparison with results of nonlinear finite element software such as ABAQUS®. The internal physical mechanism of the shell geometric parameters on the buckling load and the postbuckling equilibrium path is obtained. The numerical illustrations concern the postbuckling response of perfect and imperfect, moderately thick, anisotropic laminated cylindrical shells with different load-proportional parameters. The analytical model can provide an effective tool to investigate postbuckling of composite shell structures.

Moment Lyapunov Exponent and Stochastic Stability of Binary Airfoil under Combined Harmonic and Non-Gaussian Colored Noise Excitations

Both periodic loading and random forces commonly co-exist in real engineering applications. However, the dynamic behavior, especially dynamic stability of systems under parametric periodic and random excitations has been reported little in the literature. In this study, the moment Lyapunov exponent and stochastic stability of binary airfoil under combined harmonic and non–Gaussian colored noise excitations are investigated. The noise is simplified to an Ornstein-Uhlenbeck process by applying the path-integral method. Via the singular perturbation method, the second-order expansions of the moment Lyapunov exponent are obtained, which agree well with the results obtained by the Monte Carlo simulation. Finally, the effects of the noise and parametric resonance (such as subharmonic resonance and combination additive resonance) on the stochastic stability of the binary airfoil system are discussed.

Optimal placement and active vibration control for piezoelectric smart flexible manipulators using modal H2 norm

The optimal placement and active vibration control for piezoelectric smart single flexible manipulator are investigated in this study. Based on the assumed mode method and Hamilton’s principle, the dynamic equation of the piezoelectric smart single flexible manipulator is established. Then, the singular perturbation method is adopted and the coupled dynamic equation is decomposed into slow (rigid) and fast (flexible) subsystems. After that, the couple optimal placement criterion of piezoelectric actuators is proposed on the base of modal H2 norm of the fast subsystem and the change rate of natural frequencies. Using an improved particle swarm optimization algorithm, the optimal placement of piezoelectric actuators is realized. Subsequently, in order to verify the validity and feasibility of the presented optimal placement criterion, the composite controller is designed for the active vibration control of the piezoelectric smart single flexible manipulator. Finally, numerical simulations and experiments are presented. The results demonstrate that the piezoelectric smart single flexible manipulator system has a better single modal controllability and observability and has a good result on the vibration suppression using the optimization results of actuators. The proposed optimal placement criterion and method are feasible and effective.

Command-Filtered Integral Backstepping Control of Longitudinal Flapping-Wing Flight

The current work is concerned with longitudinal trajectory tracking control and simulation of a flapping-wing micro air vehicle. For the purpose of accurate simulation studies, a nonlinear and time-periodic dynamic model that accounts for the wings inertial effects is developed using Lagrange’s equations for quasi coordinates. Moreover, the aerodynamic loads are formulated, based on quasi-steady assumptions, taking into account the rotational circulation as well as the leading-edge vortex. Using the averaging method, the dynamic model is transformed to a time-invariant one that is the basis for controller design. Employing two control inputs, the flapping frequency and bias, an integral backstepping control law is synthesized. To ensure smooth oscillation-free control signals, a low-pass command filter is introduced in the control law. The choice of the control signals is based on Lyapunov stability criterion, while stability of the error dynamics is proved by the singular perturbation method. Simulation studies indicate that the proposed backstepping controller successfully achieves trajectory tracking. In addition, simulation results highlight the role of integral backstepping in achieving robust performance against model uncertainties and external disturbances and the importance of the command filter in maintaining smooth feasible control signals.

Integrated guidance and control design of a flight vehicle with side-window detection

This paper considers the guidance and control problem of a flight vehicle with side-window detection. In order to guarantee the target remaining in the seeker's sight of view, the line of sight and the attitude of the flight vehicle should be under some constraints caused by the side-window, which leads to coupling between the guidance and the attitude dynamics model. To deal with the side-window constraints and the coupling, a novel Integrated Guidance and Control (IGC) design approach is proposed. Firstly, the relative motion equations are derived in the body-Line of Sight (LOS) coordinate system. And the guidance and control problem of the flight vehicle is formulated into an IGC problem with state constraints. Then, based on the singular perturbation method, the IGC problem is decomposed into the control design of the quasi-steady-state subsystem and the boundary-layer subsystem which can be designed separately. Finally, the receding horizon control is applied to the control design for the two subsystems. Simulation results show the effectiveness of the proposed approach.

Nonclassical Transonic Boundary Layers: Toward Overcoming Dead-End Situations in High-Speed Aerodynamics

Analytical models of unsteady free viscous-inviscid interaction of gas flows at transonic speeds, i.e., a transonic boundary layer with self-induced pressure (nonclassical boundary layer) are considered. It is shown that an adequate flow model can be constructed by applying methods of singular perturbations. The results of a comparative analysis of classical and regularized stability models for a boundary layer with self-induced pressure in the case of interaction at transonic speeds are overviewed.

Bifurcation structure of stationary solutions for a chemotaxis system with bistable growth

From the viewpoint of pattern formation, Keller–Segel systems with growth terms are studied. These models exhibit various stationary and spatio-temporal patterns which are caused by a combination of three effects: chemotaxis, diffusion and growth. In this paper, we consider Keller–Segel system with the cubic growth term known as the Allee effect in ecology and its shadow system in the limiting case that the mobility of biological population tends to infinity. We show the existence and stability of stationary solutions of the shadow system in one space dimension. Our proof is based on the bifurcation theory, a singular perturbation method and a level set analysis. We also show some numerical results on global structures of stationary solutions in the systems by using AUTO package. Moreover, we mention the difference in dynamics between Keller–Segel system with the cubic growth term and that with the logistic growth term with the aid of a computer.

Comparative Fault Response study of Synchronous Generator in the presence of Wind Generator using Singular Perturbation based Transient Stability Index

The presence of wind generator in interconnected power systems is an important aspect to be considered while studying transient faults. A deeper understanding of transient stability study is essential for the qualitative assessment of the system which includes wind generator. This understanding can be achieved by calculating transient stability index using singular perturbation method. For the calculation of this transient stability index, the state variables are modelled in a realistic approach of slow and fast time frames. In this study, the transient stability assessment results obtained using the method of singular perturbation are compared with the corresponding results calculated using catastrophe theory and linearized singular perturbation methods. In order to understand the implications of presence of a wind generator in the system during transient faults, the study also focuses on the effect of circuit breaker clearing time and generator transient reactance.

Waves, vortices and ligaments in fluid flows of different scales

An analysis of the complete solutions of the linearized system of fundamental equations for weakly dissipating media was performed by methods of singular perturbations. Regular functions of complete solutions describe waves of different types. Singularly perturbed solutions describe a wide class of ligaments–previously unknown high gradient components of flows supplementing well investigated propagating waves. All flow components that are internal waves, vortices and ligaments were observed in a stratified wake past a vertical plate. Ligaments are classified as linear precursors of shock waves.

Event-Based Time Varying Formation Control for Multiple Quadrotor UAVs with Markovian Switching Topologies

Time varying formation control problem for a group of quadrotor unmanned aerial vehicles (UAVs) under Markovian switching topologies is investigated through a modified dynamic event-triggered control protocol. The formation shape is specified by a time varying vector, which prescribes the relative positions and bearings among the whole agents. Instead of the general stochastic topology, the graph is governed by a set of Markov chains to the edges, which can recover the traditional Markovian switching topologies in line with the practical communication network. The stability proof for the state space origin of the overall closed-loop system is derived from the singular perturbation method and Lyapunov stability theory. An event-triggered formation control protocol in terms of a dynamically varying threshold parameter is delicately carried out, while acquiring satisfactory resource efficiency, and Zeno behavior of triggering time sequences is excluded. Finally, simulations on six quadrotor UAVs are given to verify the effectiveness of the theoretical results.

Almost-Global Formation Tracking Control for Multiple Vehicles With Disturbance Rejection

This paper studies the anti-disturbance control scheme for multiple vehicles formation tracking a specified trajectory. By communicating connections, these vehicles form a vehicle network. The communication topology of the network is a directed acyclic graph. Each node vehicle is modeled as a fully actuated rigid body on SE(3)(a Special Euclidean group) with an unknown constant disturbance, except the root node, which is used to specify the trajectory to be tracked. By means of Euclidean group operation and the convex combination theory on Euclidean group, the problem is first converted to the leader-follower tracking control for two vehicles. Then, without using local coordinates, an almost-globally exponential anti-disturbance tracking control law with the knowledge of the leader’s acceleration is designed by using Lyapunov stability theory which achieves formation tracking a given trajectory. To overcome the difficulties in practical obtaining acceleration information of the vehicle being tracked, a dynamic acceleration estimator is designed by using singular perturbation method, thereafter the formation tracking control law based on the acceleration estimator is constructed. For demonstration of the result, some numerical examples have been included.

Singularly Perturbed Networked Control Systems

We study networked control systems (NCSs) where the controller is given by a state-feedback law and the plant is modeled by a dynamical system evolving on two time-scales, representing a characterization by some slow and fast dynamics. When using the stability analysis frameworks for NCSs from the literature, this time-scale separation is ignored and, as a result, the slow dynamics are in general updated at the same rate as the fast dynamics, leading to many redundant transmissions of the slow dynamics. Therefore, we assume in this paper that the slow dynamics and fast dynamics can be transmitted separately over the network, allowing us to use techniques inspired by singular perturbation methods in the stability analysis. That is, we show by means of a Lyapunov-based proof how to obtain conditions on the transmission rates (expressed in maximal allowable transmission intervals (MATIs)) for the slow and fast dynamics separately such that stability of the NCS is guaranteed, based only on approximated models of the slow and the fast dynamics.

一类非线性奇异摄动自治微分系统的渐近解

A class of generalized Lienard singular perturbation systems were considered. Firstly, the reduced solution to the system was obtained. Next, the outer solution was constructed by means of the singular perturbation method. Then, a stretch variable was introduced and the initial layer corrective term was found. Finally, the arbitrary-order asymptotic analytic expansion of the system solution was given and the uniform validity of the solution was proved. The proposed method with the basic theory has wide application values.