Singular Perturbation Method Research Articles

Load speed regulation in compliant mechanical transmission systems using feedback and feedforward control actions

The problem of controlling the load speed of a mechanical transmission system consisting of a belt-pulley and gear-pair is considered. The system is modeled as two inertia (motor and load) connected by a compliant transmission. If the transmission is assumed to be rigid, then using either the motor or load speed feedback provides the same result. However, with transmission compliance, due to belts or long shafts, the stability characteristics and performance of the closed-loop system are quite different when either motor or load speed feedback is employed. We investigate motor and load speed feedback schemes by utilizing the singular perturbation method. We propose and discuss a control scheme that utilizes both motor and load speed feedback, and design an adaptive feedforward action to reject load torque disturbances. The control algorithms are implemented on an experimental platform that is typically used in roll-to-roll manufacturing and results are shown and discussed.

Asymmetric blow-up for the SU(3) Toda system

We consider the so-called Toda system in a smooth planar domain under homogeneous Dirichlet boundary conditions. We prove the existence of a continuum of solutions for which both components blow up at the same point. This blow-up behavior is asymmetric, and moreover one component includes also a certain global mass. The proof uses singular perturbation methods.

Investigation of Spatial Control Strategies for AHWR: A Comparative Study

Large nuclear reactors such as the Advanced Heavy Water Reactor (AHWR), are susceptible to xenon-induced spatial oscillations in which, though the core average power remains constant, the power distribution may be nonuniform as well as it might experience unstable oscillations. Such oscillations influence the operation and control philosophy and could also drive safety issues. Therefore, large nuclear reactors are equipped with spatial controllers which maintain the core power distribution close to desired distribution during all the facets of operation and following disturbances. In this paper, the case of AHWR has been considered, for which a number of different types of spatial controllers have been designed during the last decade. Some of these designs are based on output feedback while the others are based on state feedback. Also, both the conventional and modern control concepts, such as linear quadratic regulator theory, sliding mode control, multirate output feedback control and fuzzy control have been investigated. The designs of these different controllers for the AHWR have been carried out using a 90th order model, which is highly stiff. Hence, direct application of design methods suffers with numerical ill-conditioning. Singular perturbation and time-scale methods have been applied whereby the design problem for the original higher order system is decoupled into two or three subproblems, each of which is solved separately. Nonlinear simulations have been carried out to obtain the transient responses of the system with different types of controllers and their performances have been compared.

Travelling wave analysis of a mathematical model of glioblastoma growth

In this paper we analyse a previously proposed cell-based model of glioblastoma (brain tumour) growth, which is based on the assumption that the cancer cells switch phenotypes between a proliferative and motile state (Gerlee and Nelander, 2012). The dynamics of this model can be described by a system of partial differential equations, which exhibits travelling wave solutions whose wave speed depends crucially on the rates of phenotypic switching. We show that under certain conditions on the model parameters, a closed form expression of the wave speed can be obtained, and using singular perturbation methods we also derive an approximate expression of the wave front shape. These new analytical results agree with simulations of the cell-based model, and importantly show that the inverse relationship between wave front steepness and speed observed for the Fisher equation no longer holds when phenotypic switching is considered.

Estimation and adaptive nonlinear model predictive control of selective catalytic reduction systems in automotive applications

The demands of high NOx conversion efficiency and low tailpipe ammonia slip for urea-based selective catalytic reduction (SCR) systems have been substantially increased in the past decade, as NOx emission legislations for Diesel engines are becoming more stringent than ever before. Since catalyst aging has a significant impact on SCR performance, robust and adaptive SCR control has been preferred for degraded SCR systems to realize emission control objectives. The purpose of this paper is twofold. Firstly, a robust ammonia coverage ratio observer was designed for estimating the ammonia coverage ratio reference for catalysts with different aging levels. An ammonia storage capacity observer was developed for estimating the actual ammonia storage capacity which can be reduced due to catalyst aging. An adaptive ammonia coverage ratio reference design was then developed to estimate the desired ammonia coverage ratio ranges at each instantaneous engine operating point for both single-cell and two-cell SCR systems at different aging levels based on a singular perturbation method. Secondly, to ensure the estimated ammonia coverage ratio falls in the desired ranges for most of engine operating conditions, robust nonlinear model predictive control (NMPC) algorithms were designed for both single-cell and two-cell SCR systems. Experimental data over US06 cycle were collected from a Diesel engine and aftertreatment system platform for controller verification. Simulation results under US06 test cycle demonstrate that the proposed NMPC algorithms were capable of consistently achieving high NOx conversion efficiency (>95.6%) and constrained tailpipe ammonia slip (<10ppm on average and <12ppm on the peak) for both fresh catalyst and aged catalyst with 30% loss of ammonia storage capacity.

On the sojourn time distribution in a finite population Markovian processor sharing queue

We consider a finite population processor sharing (PS) queue, with Markovian arrivals and an exponential server. Such a queue can model an interactive computer system consisting of a bank of terminals in series with a central processing unit. For systems with a large population N and a commensurately rapid service rate, or infrequent arrivals, we obtain various asymptotic results. We analyse the conditional sojourn time distribution of a tagged customer, conditioned on the number n of others in the system at the tagged customer's arrival instant, and also the unconditional distribution. The asymptotics are obtained by a combination of singular perturbation methods and spectral methods.We consider several space/time scales and parameter ranges, which lead to different asymptotic behaviours. We also identify precisely when the finite population model can be approximated by the standard infinite population M/M/1 - PS queue.

Output feedback performance recovery in the presence of uncertainties

We propose an output-feedback tracking controller for uncertain, nonaffine, nonlinear systems. The output feedback controller results in a closed-loop system with a three-time-scale structure; an extended high-gain observer estimates unmeasured states and uncertainties in the fastest time scale and dynamic inversion is used to deal with nonaffine inputs and input uncertainties in the intermediate time scale while the plant dynamics evolves in the slowest time scale. The dynamic inversion algorithm is based on sector conditions and results in exponential convergence of the inputs. Together with the extended high-gain observer, dynamic inversion results in performance recovery of a target system. The singular perturbation method is used to analyze the stability and performance of the system and numerical simulations are used to demonstrate the effectiveness of the control design.

Asymptotic computation of Greeks under a stochastic volatility model

We study asymptotic expansion formulae for numerical computation of Greeks (i.e. sensitivity) in finance. Our approach is based on the integration-by-parts formula of the Malliavin calculus. We propose asymptotic expansion of Greeks for a stochastic volatility model using the Greeks formula of the Black-Scholes model. A singular perturbation method is applied to derive asymptotic Greeks formulae. We also provide numerical simulation of our method and compare it to the Monte Carlo finite difference approach.

Optimal singular control for nonlinear semistabilisation

ABSTRACTThe singular optimal control problem for asymptotic stabilisation has been extensively studied in the literature. In this paper, the optimal singular control problem is extended to address a weaker version of closed-loop stability, namely, semistability, which is of paramount importance for consensus control of network dynamical systems. Three approaches are presented to address the nonlinear semistable singular control problem. Namely, a singular perturbation method is presented to construct a state-feedback singular controller that guarantees closed-loop semistability for nonlinear systems. In this approach, we show that for a non-negative cost-to-go function the minimum cost of a nonlinear semistabilising singular controller is lower than the minimum cost of a singular controller that guarantees asymptotic stability of the closed-loop system. In the second approach, we solve the nonlinear semistable singular control problem by using the cost-to-go function to cancel the singularities in the corresponding Hamilton–Jacobi–Bellman equation. For this case, we show that the minimum value of the singular performance measure is zero. Finally, we provide a framework based on the concepts of state-feedback linearisation and feedback equivalence to solve the singular control problem for semistabilisation of nonlinear dynamical systems. For this approach, we also show that the minimum value of the singular performance measure is zero. Three numerical examples are presented to demonstrate the efficacy of the proposed singular semistabilisation frameworks.

Identification and Control for Singularly Perturbed Systems Using Multitime-Scale Neural Networks

Many well-established singular perturbation theories for singularly perturbed systems require the full knowledge of system model parameters. In order to obtain an accurate and faithful model, a new identification scheme for singularly perturbed nonlinear system using multitime-scale recurrent high-order neural networks (NNs) is proposed in this paper. Inspired by the optimal bounded ellipsoid algorithm, which is originally designed for discrete-time systems, a novel weight updating law is developed for continuous-time NNs identification process. Compared with other widely used gradient-descent updating algorithms, this new method can achieve faster convergence, due to its adaptively adjusted learning rate. Based on the identification results, a control scheme using singular perturbation theories is developed. By using singular perturbation methods, the system order is reduced, and the controller structure is simplified. The closed-loop stability is analyzed and the convergence of system states is guaranteed. The effectiveness of the identification and the control scheme is demonstrated by simulation results.

Adaptive fault tolerant control of singularly perturbed systems with actuator faults

This paper deals with the adaptive fault tolerant control for linear time-invariant singularly perturbed systems against actuator failures and external disturbances. By time-scale decomposition using singular perturbation method, the full-order system is decomposed into slow and fast subsystems. An e-dependent fault tolerant controller for the global system is first designed. To avoid numerical stiffness, a simplification based on singular perturbation parameter decoupling is secondly carried out using the reduced subsystems and some manipulations of the Lyapunov equations. The resulting control system is fault tolerant in that, it provides guaranteed asymptotic stability in presence of external disturbances when all control components are operational as well as when actuator failures occur. The stability is guaranteed based on the Lyapunov stability theory provided the singular perturbation parameter is sufficiently small. A numerical example is given to illustrate the proposed method, where the efficiency of the developed approach will be compared with a reliable H∞ control technique.

Robust Control of Wind Turbines by Using Singular Perturbation Method and Linear Parameter Varying Model

The maximum power point tracking problem of variable-speed wind turbine systems is studied in this paper. The wind conversion systems contain both mechanical part and electromagnetic part, which means the systems have time scale property. The wind turbine systems are modeled using singular perturbation methodology. A linear parameter varying (LPV) model is developed to approximate the nonlinear singularly perturbed model. Then stability and robust properties of the open-loop linear singularly perturbed system are analyzed using linear matrix inequalities (LMIs). An algorithm of designing a stabilizing state-feedback controller is proposed which can guarantee the robust property of the closed-loop system. Two numerical examples are provided to demonstrate the effectiveness of the control scheme proposed.

Stability of the Horizontal Flight of an Aircraft

The stability of the horizontal flight of a light aircraft is studied using the singular-perturbation method. A numerical parameter is introduced into the equation of motion to correct for possible errors of modeling. A set of parameter values at which stability remains is obtained

Reconfigurable Control of Two-Time Scale Systems in Presence of Additive Faults

This work presents an adaptive approach for fault tolerant control of singularly perturbed systems, where both actuator and sensor faults are examined in presence of external disturbances. For sensor faults, an adaptive controller is designed based on an output-feedback control scheme. The feedback controller gain is determined in order to stabilize the closed-loop system in the fault free case and vanishing disturbance, while the additive gain is updated using an adaptive law to compensate for the sensor faults and the external disturbances. To correct the actuator faults, a state-feedback control method based on adaptive mechanism is considered. The both proposed controllers depend on the singular perturbation parameter ε leading to ill-conditioned problems. A well-posed problem is obtained by simplifying the Lyapunov equations and subsequently the controllers using the singular perturbation method and the reduced subsystems yielding to an ε-independent controller. The control scheme, designed based on the Lyapunov stability theory, guarantees asymptotic stability in presence of additive faults and external disturbances provided the singular perturbation parameter is sufficiently small. Finally, a numerical example is presented to demonstrate the effectiveness of the obtained results.DOI: http://dx.doi.org/10.5755/j01.itc.44.4.8532

Singular perturbations approach to localized surface-plasmon resonance: Nearly touching metal nanospheres

Metallic nano-structures characterised by multiple geometric length scales support low-frequency surface-plasmon modes, which enable strong light localization and field enhancement. We suggest studying such configurations using singular perturbation methods, and demonstrate the efficacy of this approach by considering, in the quasi-static limit, a pair of nearly touching metallic nano-spheres subjected to an incident electromagnetic wave polarized with the electric field along the line of sphere centers. Rather than attempting an exact analytical solution, we construct the pertinent (longitudinal) eigen-modes by matching relatively simple asymptotic expansions valid in overlapping spatial domains. We thereby arrive at an effective boundary eigenvalue problem in a half-space representing the metal region in the vicinity of the gap. Coupling with the gap field gives rise to a mixed-type boundary condition with varying coefficients, whereas coupling with the particle-scale field enters through an integral eigenvalue selection rule involving the electrostatic capacitance of the configuration. By solving the reduced problem we obtain accurate closed-form expressions for the resonance values of the metal dielectric function. Furthermore, together with an energy-like integral relation, the latter eigen-solutions yield also closed-form approximations for the induced-dipole moment and gap-field enhancement under resonance. We demonstrate agreement between the asymptotic formulas and a semi-numerical computation. The analysis, underpinned by asymptotic scaling arguments, elucidates how metal polarization together with geometrical confinement enables a strong plasmon-frequency redshift and amplified near-field at resonance.

Characterizing two-timescale nonlinear dynamics using finite-time Lyapunov exponents and subspaces

Finite-time Lyapunov exponents and subspaces are used to define and diagnose boundary-layer type, two-timescale behavior in the tangent linear dynamics and to determine the associated manifold structure in the flow of a finite-dimensional nonlinear autonomous dynamical system. Two-timescale behavior is characterized by a slow-fast splitting of the tangent bundle for a state space region. The slow-fast splitting is defined using finite-time Lyapunov exponents and vectors, guided by the asymptotic theory of partially hyperbolic sets, with important modifications for the finite-time case; for example, finite-time Lyapunov analysis relies more heavily on the Lyapunov vectors due to their relatively fast convergence compared to that of the corresponding exponents. The splitting is used to characterize and locate points approximately on normally hyperbolic center manifolds via tangency conditions for the vector field. Determining manifolds from tangent bundle structure is more generally applicable than approaches, such as the singular perturbation method, that require special normal forms or other a priori knowledge. The use, features, and accuracy of the approach are illustrated via several detailed examples.

Analytical study of acceleration waves on a nonlinear, externally damped string

This paper investigates the propagation of acceleration waves on a nonlinear, externally damped string. Using the singular perturbation method, the hyperbolic system of quasilinear partial differential equations of a string is analyzed. Employing this method provides an opportunity to acquire further physical insights into acceleration wave propagation by extracting the interactions between damping and nonlinearity and their effects on the system behavior as the wave propagates along the string. Considerable agreement is observed between the analytical model of this paper and numerical simulations.

Turbulent combustion modelling of a confined premixed jet flame including heat loss effects using tabulated chemistry

The present work addresses the coupling of a flamelet database, to a low-Mach approximation of the Navier–Stokes equations using scalar controlling variables. The model is characterized by the chemistry tabulation based on laminar premixed flamelets in combination with an optimal choice of the reaction progress variable, which is determined based on the computational singular perturbation (CSP) method. The formulation of the model focuses on turbulent premixed flames taking into account the effect of heat losses, but it is easily extended to partially premixed and non-premixed regimes. The model is designed for applications in both, Reynolds-averaged Navier–Stokes (RANS) as well as large-eddy simulations (LES) and results for the two methods are compared. A priori analysis of the database is presented to demonstrate the influence of the reaction progress definition and the chemistry tabulation is validated against a one-dimensional premixed laminar flame. The validation of the turbulent case is performed using a turbulent premixed confined jet flame subject to strong heat losses, in which the model shows a good overall performance.

A continuum of solutions for the SU(3) Toda System exhibiting partial blow-up: Figure 1.

In this paper we consider the so-called Toda System in planar domains under Dirichlet boundary condition. We show the existence of continua of solutions for which one component is blowing up at a certain number of points. The proofs use singular perturbation methods.

The asymptotic downstream flow of plane turbulent wall jets without external stream

The plane turbulent wall-jet flow without externally imposed stream is considered. It is assumed that the wall jet does not emerge from a second wall perpendicular to the velocity vector of the initial wall jet. The (kinematic) momentum flux$K(x)$of the wall jet decreases downstream owing to the shear stress at the wall. This investigation is based on the hypothesis that the total friction force on the wall is smaller than the total inflow momentum flux. In other words, the turbulent wall jet tends to a turbulent ‘half-free jet’ with a non-zero momentum flux$K_{\infty }\;(\text{m}^{3}~\text{s}^{-2})$far downstream. The fact that the turbulent half-free jet is the asymptotic form of a turbulent wall jet is the basis for a singular perturbation method by which the wall-jet flow is determined. It turns out that the ratio between the wall distance$y_{m}$of the maximum velocity and the wall distance$y_{0.5}$of half the maximum velocity decreases downstream to zero. Dimensional analysis leads immediately to a universal function of the dimensionless momentum flux$K(\mathit{Re}_{x})/K_{\infty }$that depends asymptotically only on the local Reynolds number$\mathit{Re}_{x}=\sqrt{(x-x_{0})K_{\infty }}/{\it\nu}$, where$x_{0}$denotes the coordinate of the virtual origin. When the values$K$and${\it\nu}$at the position$x-x_{0}$are known, the asymptotic momentum flux$K_{\infty }$can be determined. Experimental data on all turbulent plane wall jets (except those emerging from a second plane wall) collapse to a single universal curve. Comparisons between available experimental data and the analysis make the hypothesis$K_{\infty }\neq 0$plausible. A convincing verification, however, will be possible in the future, preferably by direct numerical simulations.