Singular Perturbation Analysis Research Articles

Singular perturbation analysis of a two-site model for an epidemic in age-structured population

We formulate a model describing the dynamics for the spatial propagation of an SIS epidemic within a population, with age structure, living in an environment divided into two sites. An analysis of the model is given. We prove the existence of a unique disease free equilibrium (DFE) and its (local and global) stability. Further, we assume that fast infection processes and fast migration processes take place in the above-mentioned model; i.e. such processes last only a few days (less than a week). In opposition to such processes, demographic processes such as birth, death and maturation last quite a lot of years. Such a gap between the time scales gives rise to a multiple time scales model, in particular a singularly perturbed model. Through a singular perturbation analysis, based on Tikhonov theorem, we prove that for certain classes of initial conditions the nonlinear perturbed model can be approximated with very good accuracy by lower-dimensional linear models.

Phase-plane geometries in coupled enzyme assays

The determination of a substrate or enzyme activity by coupling one enzymatic reaction with another easily detectable (indicator) reaction is a common practice in the biochemical sciences. Usually, the kinetics of enzyme reactions is simplified with singular perturbation analysis to derive rate or time course expressions valid under the quasi-steady-state and reactant stationary state assumptions. In this paper, the dynamical behavior of coupled enzyme catalyzed reaction mechanisms is studied by analysis of the phase-plane. We analyze two types of time-dependent slow manifolds – Sisyphus and Laelaps manifolds – that occur in the asymptotically autonomous vector fields that arise from enzyme coupled reactions. Projection onto slow manifolds yields various reduced models, and we present a geometric interpretation of the slow/fast dynamics that occur in the phase–planes of these reactions.

Moving mesh simulation of contact sets in two dimensional models of elastic–electrostatic deflection problems

Numerical and analytical methods are developed for the investigation of contact sets in electrostatic–elastic deflections modeling micro-electro mechanical systems. The model for the membrane deflection is a fourth-order semi-linear partial differential equation and the contact events occur in this system as finite time singularities. Primary research interest is in the dependence of the contact set on model parameters and the geometry of the domain. An adaptive numerical strategy is developed based on a moving mesh partial differential equation to dynamically relocate a fixed number of mesh points to increase density where the solution has fine scale detail, particularly in the vicinity of forming singularities. To complement this computational tool, a singular perturbation analysis is used to develop a geometric theory for predicting the possible contact sets. The validity of these two approaches are demonstrated with a variety of test cases.

Distributed Generalized Nash Equilibrium Seeking Algorithm Design for Aggregative Games Over Weight-Balanced Digraphs.

In this paper, two aggregative games over weight-balanced digraphs are studied, where the cost functions of all players depend on not only their own decisions but also the aggregate of all decisions. In the first problem, the cost functions of players are differentiable with Lipschitz gradients, and the decisions of all players are coupled by linear coupling constraints. In the second problem, the cost functions are nonsmooth, and the decisions of all players are constrained by local feasibility constraints as well as linear coupling constraints. In order to seek the variational generalized Nash equilibrium (GNE) of the differentiable aggregative games, a continuous-time distributed algorithm is developed via gradient descent and dynamic average consensus, and its exponential convergence to the variational GNE is proven with the help of Lyapunov stability theory. Then, another continuous-time distributed projection-based algorithm is proposed for the nonsmooth aggregative games based on differential inclusions and differentiated projection operations. Moreover, the convergence of the algorithm to the variational GNE is analyzed by utilizing singular perturbation analysis. Finally, simulation examples are presented to illustrate the effectiveness of our methods.

Distributed sub-optimal resource allocation over weight-balanced graph via singular perturbation

In this paper, we consider distributed optimization design for resource allocation problems over weight-balanced graphs. With the help of singular perturbation analysis, we propose a simple sub-optimal continuous-time optimization algorithm. Moreover, we prove the existence and uniqueness of the algorithm equilibrium, and then show the convergence with an exponential rate. Finally, we verify the sub-optimality of the algorithm, which can approach the optimal solution as an adjustable parameter tends to zero.

Two queues with random time-limited polling

TWO QUEUES WITH RANDOM TIME-LIMITED POLLINGIn this paper, we analyse a single server polling model withtwo queues. Customers arrive at the two queues according to two independent Poisson processes. There is a single server that serves both queues withgenerally distributed service times. The server spends an exponentially distributed amount of time in each queue. After the completion of this residing time, the server instantaneously switches to the other queue, i.e., there is noswitch-over time. For this polling model we derive the steady-state marginal workload distribution, as well as heavy traffic and heavy tail asymptotic results. Furthermore, we also calculate the joint queue length distribution for the special case of exponentially distributed service times using singular perturbation analysis.

Computational singular perturbation analysis of super-knock in SI engines

Pre-ignition engine cycles leading to super-knock were simulated with a 48 species skeletal iso-octane mechanism to identify the dominant reaction pathways that are present in super-knock. To mimic pre-ignition, a deflagration front was generated via a hot spot that is placed over the piston at close proximity to the end-wall. Computational singular perturbation (CSP) was used to analyze the chemical dynamics at various in-cylinder locations: a point at the center of the cylinder where the deflagration front consumes the air/fuel mixture and two points located at 3 mm from the end-wall where super-knock and mild knock occur. The CSP analysis of the point at the center of the cylinder reveals weak two-stage ignition-like dynamics with a short second stage. At the other points, a pronounced two-stage ignition is displayed with a long second stage. A distinct contribution of formaldehyde (CH2O) at the second stage of ignition that adds to fast explosive modes in the super-knock points is not observed in the point at the center. A comparison between knock and super-knock analysis indicates that a similar set of reactions is responsible for the abnormal behavior but the fast explosive time scales are comparatively slower for knock, indicating lower reactivity, which results in the reduced intensity of knock. The analyzed results decoded important reactions responsible for the occurrence of super-knock.

Two-Time Scale Control of Flexible Joint Robots With an Improved Slow Model

This paper presents a novel multivariable two-time scale control method for flexible joint robots. The fundamental limitations of the conventional two-time scale control are clearly explained based on a singular perturbation analysis with a linearized dynamic model. Unlike the conventional methods, the bandwidth of the slow control can be made high and the derivatives of the rigid control are not required. The closed-form solution of a slow model that is more accurate than the rigid model is presented. It is shown that the modeling error of the proposed two-time scale control can be made arbitrarily small by the control gain only. A real-world experiment with a six degrees-of-freedom commercial industrial robot was performed to validate the presented approach.

A singular perturbation study of the Rolie-Poly model

A singular perturbation analysis for the Rolie-Poly model is presented for the limit of large relaxation time. The model describes entangled linear polymer melts and time scales for a Rouse time and reptation time. First, the case of the Rouse time approaching zero is named the nonstretching model, and the addition of a solvent model gives the small parameter to be the ratio of the retardation time to the slow reptation time. This analysis is instructive in formulating the perturbation analysis for the full Rolie-Poly model without the solvent, where the small parameter is the ratio of Rouse time to reptation time. The resulting formulae elucidate the characteristic features of thixotropic yield stress fluids, including yield stress hysteresis, delayed yielding and long term persistence of a decreased viscosity after cessation of flow. Two fundamental initial value problems are solved: startup of shear flow and cessation of shear flow. The scalings lead naturally to simplified dynamical systems that describe regimes of fast, slow and yielded dynamics, and show how the combination of these regimes can be used to describe the entire flow evolution.

Mechanisms Underlying Rate-Dependent Effects of State-Specific Binding of Sodium Channel Blockers in Cardiac Tissue: Insights from Idealized Models

Increased propensity for arrhythmias is a well-documented but poorly understood side-effect of drugs. Many arrhythmogenic drugs bind to and block ion-channels in a conformational state-specific manner, making it difficult to discern the fundamental mechanisms underlying pro-arrhythmic drug effects. We have developed mathematical models to examine the effects of state-specific Na+ channel blockers with various binding mechanisms on the rate-dependence of peak upstroke velocity, conduction velocity, and the vulnerable window of cardiac tissue. Specifically, we compared the effects of drugs that bind via the guarded receptor and gate immobilization models, which are idealized models consisting of motifs commonly found in modulated receptor models of drug binding. For both the guarded receptor and gate immobilization models, we considered drugs that bind to either inactivated or non-inactivated Na-channels. Our results suggest that drugs that bind to inactivated Na+ channels give rise to greater rate-dependent effects on peak upstroke velocity and conduction velocity than drugs that bind to non-inactivated Na+ channels. Additionally, we find drugs that bind to non-inactivated Na+ channels via the gate immobilization motif induce a reversal of the rate-dependence of peak upstroke velocity. By exploiting the idealized nature of guarded receptor and gate immobilization models and utilizing singular perturbation analysis, we provide an explanation of the fundamental mechanisms of rate-dependent drug effects. Finally, we show that a gate immobilization model with inactivated state binding captures the rate-dependent effects of lidocaine, demonstrating that idealized models can be useful models to study mechanisms of action of real drugs.

Spot Dynamics in a Reaction-Diffusion Model of Plant Root Hair Initiation

We study pattern formation in a 2-D reaction-diffusion (RD) sub-cellular model characterizing the effect of a spatial gradient of a plant hormone distribution on a family of G-proteins associated with root-hair (RH) initiation in the plant cell Arabidopsis thaliana. The activation of these G-proteins, known as the Rho of Plants (ROPs), by the plant hormone auxin, is known to promote certain protuberances on root hair cells, which are crucial for both anchorage and the uptake of nutrients from the soil. Our mathematical model for the activation of ROPs by the auxin gradient is an extension of the model of Payne and Grierson [PLoS ONE, 12(4), (2009)], and consists of a two-component Schnakenberg-type RD system with spatially heterogeneous coefficients on a 2-D domain. The nonlinear kinetics in this RD system model the nonlinear interactions between the active and inactive forms of ROPs. By using a singular perturbation analysis to study 2-D localized spatial patterns of active ROPs, it is shown that the spatial variations in the nonlinear reaction kinetics, due to the auxin gradient, lead to a slow spatial alignment of the localized regions of active ROPs along the longitudinal midline of the plant cell. Numerical bifurcation analysis, together with time-dependent numerical simulations of the RD system are used to illustrate both 2-D localized patterns in the model, and the spatial alignment of localized structures.

Automatic Tuning of Virtual Constraint-Based Control Algorithms for Powered Knee-Ankle Prostheses.

State-of-art powered prosthetic legs are often controlled using a collection of joint impedance controllers designed for different phases of a walking cycle. Consequently, finite state machines are used to control transitions between different phases. This approach requires a large number of impedance parameters and switching rules to be tuned. Since one set of control parameters cannot be used across different amputees, clinicians spend enormous time tuning these gains for each patient. This paper proposes a virtual constraint-based control scheme with a smaller set of control parameters, which are automatically tuned in real-time using an extremum seeking controller (ESC). ESC, being a model-free control method, assumes no prior knowledge of either the prosthesis or human. Using a singular perturbation analysis, we prove that the virtual constraint tracking errors are small and the PD gains remain bounded. Simulations demonstrate that our ESC-based method is capable of adapting the virtual-constraint based control parameters for amputees with different masses.

Multiloop Integral Controllability Analysis for Nonlinear Multiple-Input Single-Output Processes

Multi-loop integral control is still one of the most popular control strategies in industry due to its simplicity, efficiency, offset free tracking, and capability for fault tolerance. Skogestad and Morari introduced Decentralized Integral Controllability (DIC) to investigate the decentralized unconditional stability under multi-loop integral control for square systems. However, in engineering practice, some multivariable processes may not be square, which often utilize multiple redundant control inputs for the regulation of only one single output. This study extends the concept of Decentralized Integral Controllability to non-square systems, and presents sufficient conditions for Multiple-Input Single-Output nonlinear processes based on singular perturbation analysis. The proposed controllability analysis method is applied in the control of a real time temperature control system and achieves desired temperature tracking results.

Sliding Mode Based Extremum Seeking Control of Two-Phase Flow Micro-Thermal-Fluid Cooling Systems

The thermal-fluid control of a microchannel evaporator operating in the two-phase regime is considered. The control objective is to stabilize the fluid flow and maintain a minimal evaporator wall temperature by commanding the inlet flow rate using a pump. The first objective is achieved with a feedback controller using only mass flow rate measurement (via a lead filter). The second objective is more challenging as the achievable wall temperature depends on the heat transfer coefficient which is hard to characterize (heavily depends on the flow rate and heat load). An extremum seeking control based on sliding modes is proposed to minimize wall temperature and adjust the flow rate without considering heat transfer coefficient knowledge. Thus, considering a thermal-fluid model, the stability of the flow control loop is establish and the convergence to the extremum seeking control is analyzed via singular perturbation analysis and Tikhonov’s theorem. Simulation results demonstrate the efficacy of the proposed scheme.

A methodology for derivation of RCCE-reduced mechanisms via CSP

The development of reduced chemical mechanisms in a systematic way has emerged as a potential solution to the problem of incorporating the increasingly large chemical mechanisms into turbulent combustion CFD codes. In this work, a methodology is proposed for developing reduced mechanisms with Rate-Controlled Constrained Equilibrium (RCCE) via a Computational Singular Perturbation (CSP) analysis of counterflow non-premixed flamelets. An ordering of species for variable strain rates is derived by integrating over mixture fraction space a modified CSP pointer that depends on the timescale and mass fraction of each chemical species. Subsequently, a global set of kinetically controlled species is identified from weighting the local ordering for each strain rate. RCCE simulations with the derived reduced mechanisms for methane with 16 species and for propane with 27 species are compared with the integration of the detailed mechanisms GRI 1.2 and USC-Mech-II, respectively. The applicability of the methodology is demonstrated in non-premixed flames for several strain rates, in non-premixed flames ignited with a pilot in order to test the dynamics and ignition of the reduced schemes, in premixed flames for different equivalence ratios and subsequently in perfectly stirred reactors for ignition delay times for varying temperature, pressure and equivalence ratio. Overall very good agreement is obtained, indicating that the methodology can produce reliable mechanisms for different fuels and for a wide range of conditions, including dynamical behaviour and conditions different from those employed for the derivation of the mechanism.

A unified perturbation analysis framework for countable Markov chains

In this paper, we are devoted to singular perturbation analysis for discrete-time or continuous-time Markov chains. We modify and extend the drift condition method, well known for regular perturbation, to develop a new framework for singular perturbation analysis. Our results extend and improve the corresponding ones in [2] for singularly perturbed Markov chains by allowing a general perturbation form, less restrictive conditions, and more computable bounds. Our analysis covers the regular perturbation analysis, and hence unifies singular and regular perturbation analysis. Furthermore, our results are illustrated by two two-dimensional Markov chains, including a discrete-time queue and a continuous-time level dependent quasi-birth-death process.

Chemical model reduction under uncertainty

A general strategy for analysis and reduction of uncertain chemical kinetic models is presented, and its utility is illustrated in the context of ignition of hydrocarbon fuel–air mixtures. The strategy is based on a deterministic analysis and reduction method which employs computational singular perturbation analysis to generate simplified kinetic mechanisms, starting from a detailed reference mechanism. We model uncertain quantities in the reference mechanism, namely the Arrhenius rate parameters, as random variables with prescribed uncertainty factors. We propagate this uncertainty to obtain the probability of inclusion of each reaction in the simplified mechanism. We propose probabilistic error measures to compare predictions from the uncertain reference and simplified models, based on the comparison of the uncertain dynamics of the state variables, where the mixture entropy is chosen as progress variable. We employ the construction for the simplification of an uncertain mechanism in an n-butane–air mixture homogeneous ignition case, where a 176-species, 1111-reactions detailed kinetic model for the oxidation of n-butane is used with uncertainty factors assigned to each Arrhenius rate pre-exponential coefficient. This illustration is employed to highlight the utility of the construction, and the performance of a family of simplified models produced depending on chosen thresholds on importance and marginal probabilities of the reactions.

PyJac: Analytical Jacobian generator for chemical kinetics

Accurate simulations of combustion phenomena require the use of detailed chemical kinetics in order to capture limit phenomena such as ignition and extinction as well as predict pollutant formation. However, the chemical kinetic models for hydrocarbon fuels of practical interest typically have large numbers of species and reactions and exhibit high levels of mathematical stiffness in the governing differential equations, particularly for larger fuel molecules. In order to integrate the stiff equations governing chemical kinetics, generally reactive-flow simulations rely on implicit algorithms that require frequent Jacobian matrix evaluations. Some in situ and a posteriori computational diagnostics methods also require accurate Jacobian matrices, including computational singular perturbation and chemical explosive mode analysis. Typically, finite differences numerically approximate these, but for larger chemical kinetic models this poses significant computational demands since the number of chemical source term evaluations scales with the square of species count. Furthermore, existing analytical Jacobian tools do not optimize evaluations or support emerging SIMD processors such as GPUs. Here we introduce pyJac, a Python-based open-source program that generates analytical Jacobian matrices for use in chemical kinetics modeling and analysis. In addition to producing the necessary customized source code for evaluating reaction rates (including all modern reaction rate formulations), the chemical source terms, and the Jacobian matrix, pyJac uses an optimized evaluation order to minimize computational and memory operations. As a demonstration, we first establish the correctness of the Jacobian matrices for kinetic models of hydrogen, methane, ethylene, and isopentanol oxidation (number of species ranging 13–360) by showing agreement within 0.001% of matrices obtained via automatic differentiation. We then demonstrate the performance achievable on CPUs and GPUs using pyJac via matrix evaluation timing comparisons; the routines produced by pyJac outperformed first-order finite differences by 3–7.5 times and the existing analytical Jacobian software TChem by 1.1–2.2 times on a single-threaded basis. It is noted that TChem is not thread-safe, while pyJac is easily parallelized, and hence can greatly outperform TChem on multicore CPUs. The Jacobian matrix generator we describe here will be useful for reducing the cost of integrating chemical source terms with implicit algorithms in particular and algorithms that require an accurate Jacobian matrix in general. Furthermore, the open-source release of the program and Python-based implementation will enable wide adoption. Program summaryProgram Title: pyJacProgram Files doi:http://dx.doi.org/10.17632/mmr3z8j2m5.1Licensing provisions: MIT LicenseProgramming language: PythonExternal routines/libraries: required: NumPy; optional: Cython, Cantera, PyYAML, AdeptNature of problem: Automatic generation of source code to evaluate Jacobian matrices for chemical kinetic modelsSolution method: Chemical kinetic model interpreted from input file(s), and partial derivatives and necessary supporting functions are written to file based on a theoretical derivation.Additional comments: This paper describes pyJac v1.0.2, available at http://dx.doi.org/10.5281/zenodo.251144. Current version and support available at https://github.com/SLACKHA/pyjac/

Computational singular perturbation analysis of stochastic chemical systems with stiffness

Computational singular perturbation (CSP) is a useful method for analysis, reduction, and time integration of stiff ordinary differential equation systems. It has found dominant utility, in particular, in chemical reaction systems with a large range of time scales at continuum and deterministic level. On the other hand, CSP is not directly applicable to chemical reaction systems at micro or meso-scale, where stochasticity plays an non-negligible role and thus has to be taken into account. In this work we develop a novel stochastic computational singular perturbation (SCSP) analysis and time integration framework, and associated algorithm, that can be used to not only construct accurately and efficiently the numerical solutions to stiff stochastic chemical reaction systems, but also analyze the dynamics of the reduced stochastic reaction systems. The algorithm is illustrated by an application to a benchmark stochastic differential equation model, and numerical experiments are carried out to demonstrate the effectiveness of the construction.

Model reduction of AHWR space–time kinetics using balanced truncation

Study of transient behavior and design of control for large reactors, such as the Advanced Heavy Water Reactor (AHWR) requires accurate mathematical models. An 80th order space–time kinetics model which describes the core neutronics behavior of AHWR is available but it is nonlinear and stiff. The modern control design and analysis studies with this model are accompanied by serious numerical ill-conditioning problems. Hence, there is a strong motivation for obtaining a reduced order model which preserves the input–output behavior accurately.In this paper, we have explored the application of balanced truncation technique to obtain a reduced order model from the original high order model of the AHWR. The reduced order model thus obtained was examined from the view point of transient performance by comparing its response with that of the original model and it is found to yield a very good approximation. We also carried out the comparative study between different reduced order models of AHWR, namely, Davison’s technique, Marshall’s technique, singular perturbation analysis and balanced truncation by comparing their performances with respect to each other and with the original model. All of these methods are found to be effective, however the overall accuracy in the approximation using the balanced truncation approach is found to be far more superior.