Fast Subsystem Research Articles

Turnover of hysteresis determines novel bursting in Duffing system with multiple-frequency external forcings

This paper investigates bursting dynamics of the Duffing system with multiple-frequency external forcings, in which novel bursting patterns (i.e., the so-called turnover-of-hysteresis-induced bursting patterns) can be observed. Typically, distinct oscillations are observed in the quasi-static processes of these bursting patterns. We show that the oscillations appear in the quasi-static processes because the equilibrium hysteresis curve of the fast subsystem becomes the one with twists and turns, and this forms a new route to bursting, which we call turnover of hysteresis. Besides, we investigate the effects of forcing frequencies and amplitudes on the turnover-of-hysteresis-induced bursting. Our study shows that the three frequency components observed in the bursting are decided by the forcing frequencies, while the transition of the bursting depends on the forcing amplitudes.

Two-Time Scale Virtual Sensor Design for Vibration Observation of a Translational Flexible-Link Manipulator Based on Singular Perturbation and Differential Games.

Effective feedback control requires all state variable information of the system. However, in the translational flexible-link manipulator (TFM) system, it is unrealistic to measure the vibration signals and their time derivative of any points of the TFM by infinite sensors. With the rigid-flexible coupling between the global motion of the rigid base and the elastic vibration of the flexible-link manipulator considered, a two-time scale virtual sensor, which includes the speed observer and the vibration observer, is designed to achieve the estimation for the vibration signals and their time derivative of the TFM, as well as the speed observer and the vibration observer are separately designed for the slow and fast subsystems, which are decomposed from the dynamic model of the TFM by the singular perturbation. Additionally, based on the linear-quadratic differential games, the observer gains of the two-time scale virtual sensor are optimized, which aims to minimize the estimation error while keeping the observer stable. Finally, the numerical calculation and experiment verify the efficiency of the designed two-time scale virtual sensor.

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.

Product-Form Stationary Distributions for Deficiency Zero Networks with Non-mass Action Kinetics.

In many applications, for example when computing statistics of fast subsystems in a multiscale setting, we wish to find the stationary distributions of systems of continuous-time Markov chains. Here we present a class of models that appears naturally in certain averaging approaches whose stationary distributions can be computed explicitly. In particular, we study continuous-time Markov chain models for biochemical interaction systems with non-mass action kinetics whose network satisfies a certain constraint. Analogous with previous related results, the distributions can be written in product form.

Set-membership methodology for model-based prognosis

This paper addresses model-based prognosis to predict Remaining Useful Life (RUL) of a class of dynamical systems. The methodology is based on singular perturbed techniques to take into account the slow behavior of degradations. The full-order system is firstly decoupled into slow and fast subsystems. An interval observer is designed for both subsystems under the assumption that the measurement noise and the disturbances are bounded. Then, the degradation is modeled as a polynomial whose parameters are estimated using ellipsoid algorithms. Finally, the RUL is predicted based on an interval evaluation of the degradation model over a time horizon. A numerical example illustrates the proposed technique.

Dynamic output feedback control for fast sampling discrete‐time singularly perturbed systems

This study is concerned with the dynamic output feedback control problem for fast sampling discrete-time singularly perturbed systems using the singular perturbation approach. Sufficient conditions in terms of linear matrix inequalities (LMIs) are presented to guarantee the existence of a dynamic output feedback controller for the corresponding slow and fast subsystems, respectively. The controller gains and the corresponding coefficient matrices can be obtained via solving the proposed LMIs. Thus, not only the high dimensionality and the ill condition are alleviated, but also the regularity restrictions attached to the Riccati-based solutions are avoided. The theoretical result demonstrates that the composite dynamic output feedback control designed through those of the slow and fast subsystems can stabilise the full-order discrete-time singularly perturbed systems. Finally, two real world practical examples are provided to show the effectiveness of the obtained results.

On the conditions of exponential stability in active disturbance rejection control based on singular perturbation analysis

ABSTRACTStability of active disturbance rejection control (ADRC) is analysed in the presence of unknown, nonlinear, and time-varying dynamics. In the framework of singular perturbations, the closed-loop error dynamics are semi-decoupled into a relatively slow subsystem (the feedback loop) and a relatively fast subsystem (the extended state observer), respectively. It is shown, analytically and geometrically, that there exists a unique exponential stable solution if the size of the initial observer error is sufficiently small, i.e. in the same order of the inverse of the observer bandwidth. The process of developing the uniformly asymptotic solution of the system reveals the condition on the stability of the ADRC and the relationship between the rate of change in the total disturbance and the size of the estimation error. The differentiability of the total disturbance is the only assumption made.

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.

Cross-Frequency rs-fMRI Network Connectivity Patterns Manifest Differently for Schizophrenia Patients and Healthy Controls.

Patterns of resting state fMRI functional network connectivity in schizophrenia patients have been shown to differ markedly from that of healthy controls. While some studies have explored connectivity within fixed frequency bands, the question of network phase synchrony across disparate frequency bands, or cross-frequency connectivity, has remained surprisingly underexplored. Computational modeling at the neuronal scale however has long acknowledged the existence of coupled fast and slow subsystems. Here we present preliminary evidence that cross-frequency coupling exists at the network level, that it patterns in meaningful ways over functional domains, and that this patterning differs between the healthy population and individuals with diagnosed schizophrenia.

Robust state estimation for singularly perturbed systems

ABSTRACTThis paper deals with the design of interval observers for singularly perturbed linear systems. The full-order system is first decoupled into slow and fast subsystems. Then, using the cooperativity theory, an interval observer is designed for the slow and fast subsystems assuming that the measurement noise and the disturbances are bounded and the singular perturbed parameter is uncertain. This decoupling leads to two observers that estimate the lower and upper bounds for the feasible state domain. A numerical example shows the efficiency of the proposed technique.

Efficient simulation of a slow-fast dynamical system using multirate finite difference schemes

We consider a system of ordinary differential equations describing a slow-fast dynamical system, in particular, a predator-prey system that is highly susceptible to local time variations. This model exhibits coexistence of predatorprey dynamics in the case when the prey population grows much faster than that of the predators with a quite diversified time response. For particular parametric values their interactions show a stable relaxation oscillation in the positive octant. Such characteristics are difficult to mimic using conventional time integrators that are used to solve systems of ordinary differential equations. To resolve this, we design and analyze multirate time integration methods to solve a mathematical model for a slow-fast dynamical system. Proposed methods are based on using extrapolation multirate discretisation algorithms. Through these methods, we reduce the integration time by integrating the slow sub-system with a larger step length than the fast sub-system. This allows us to efficiently solve multiscale ordinary differential equations. Besides theoretical results, we provide thorough numerical experiments which confirm that these multirate schemes outperform corresponding single-rate schemes substantially both in terms of computational work and CPU times.

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.

Canards in a minimal piecewise-linear square-wave burster.

We construct a piecewise-linear (PWL) approximation of the Hindmarsh-Rose (HR) neuron model that is minimal, in the sense that the vector field has the least number of linearity zones, in order to reproduce all the dynamics present in the original HR model with classical parameter values. This includes square-wave bursting and also special trajectories called canards, which possess long repelling segments and organise the transitions between stable bursting patterns with n and n + 1 spikes, also referred to as spike-adding canard explosions. We propose a first approximation of the smooth HR model, using a continuous PWL system, and show that its fast subsystem cannot possess a homoclinic bifurcation, which is necessary to obtain proper square-wave bursting. We then relax the assumption of continuity of the vector field across all zones, and we show that we can obtain a homoclinic bifurcation in the fast subsystem. We use the recently developed canard theory for PWL systems in order to reproduce the spike-adding canard explosion feature of the HR model as studied, e.g., in Desroches et al., Chaos 23(4), 046106 (2013).

Comparative evaluation of GHG emissions from the use of Miscanthus for bio-hydrocarbon production via fast pyrolysis and bio-oil upgrading

This study examines the GHG emissions associated with producing bio-hydrocarbons via fast pyrolysis of Miscanthus. The feedstock is then upgraded to bio-oil products via hydroprocessing and zeolite cracking. Inventory data for this study were obtained from current commercial cultivation practices of Miscanthus in the UK and state-of-the-art process models developed in Aspen Plus®. The system boundary considered spans from the cultivation of Miscanthus to conversion of the pyrolysis-derived bio-oil into bio-hydrocarbons up to the refinery gate. The Miscanthus cultivation subsystem considers three scenarios for soil organic carbon (SOC) sequestration rates. These were assumed as follows: (i) excluding (SOC), (ii) low SOC and (iii) high (SOC) for best and worst cases. Overall, Miscanthus cultivation contributed moderate to negative values to GHG emissions, from analysis of excluding SOC to high SOC scenarios. Furthermore, the rate of SOC in the Miscanthus cultivation subsystem has significant effects on total GHG emissions. Where SOC is excluded, the fast pyrolysis subsystem shows the highest positive contribution to GHG emissions, while the credit for exported electricity was the main ‘negative’ GHG emission contributor for both upgrading pathways. Comparison between the bio-hydrocarbons produced from the two upgrading routes and fossil fuels indicates GHG emission savings between 68% and 87%. Sensitivity analysis reveals that bio-hydrocarbon yield and nitrogen gas feed to the fast pyrolysis reactor are the main parameters that influence the total GHG emissions for both pathways.

Attitude tracking control for Mars entry vehicle via T-S model with time-varying input delay

This paper focuses on the attitude tracking control problem of Mars entry vehicle (MEV) with time-varying input delay. The original attitude dynamics of MEV is divided into slow subsystem and fast subsystem. For slow subsystem, the dynamic inversion method is used to generate the angular velocity command. For fast subsystem, a T-S fuzzy model is used to approximate it, and delays-dependent \(H_{\infty }\) attitude tracking control is applied to reduce the effects of delay on attitude dynamics. Specially, a decomposition coefficient of delay integral inequality is introduced in our proposed results, which may further reduce the design algorithm conservatism. Finally, numerical simulations are used to verify the effectiveness of the proposed method.

Early Afterdepolarizations with Growing Amplitudes via Delayed Subcritical Hopf Bifurcations and Unstable Manifolds of Saddle Foci in Cardiac Action Potential Dynamics.

Early afterdepolarizations (EADs) are pathological oscillations in cardiac action potentials during the repolarization phase and may be caused by drug side effects, ion channel disease or oxidative stress. The most widely observed EAD pattern is characterized by oscillations with growing amplitudes. So far, its occurence has been explained in terms of a supercritical Hopf bifurcation in the fast subsystem of the action potential dynamics from which stable limit cycles with growing amplitudes emerge. The novel contribution of this article is the introduction of two alternative explanations of EAD genesis with growing amplitudes that do not involve stable limit cycles in fast subsystems. In particular, we demonstrate that EAD patterns with growing amplitudes may alternatively arise due to a delayed subcritical Hopf bifurcation or an unstable manifold of a saddle focus fixed point in the full fast-slow system modelling the action potential. Our work extends the list of possible dynamical EAD mechanisms and may contribute to a classification of drug effects in preclinical cardiotoxicity testing.

A Hybrid Scheme of Synthesized Sliding Mode/Strain Rate Feedback Control Design for Flexible Spacecraft Attitude Maneuver Using Time Scale Decomposition

Presented herein is a new control approach for large angle attitude maneuver of flexible spacecraft. The singular perturbation theory (SPT) provides a useful tool for two time rate scale separation (mapping) of rigid and flexible body dynamics. The resulting slow and fast subsystems, enabling the use of two control approach for attitude (Modified Sliding Mode) and vibration Strain Rate Feedback (SRF) control of flexible spacecraft, respectively. An attractive feature of the present control approach is that the global stability of the entire system has been guaranteed while the controllers accomplished their tasks in coupled rigid/flexible dynamic domain without parasitic parameter interactions. Numerical simulations show the effectiveness of the present approach.

A Two-Time-Scale Approach for Discrete-Time Kalman Filter Design and Application to AHWR Flux Mapping

In large nuclear reactors such as the Advanced Heavy Water Reactor (AHWR), the core neutron flux distribution needs to be continuously monitored and displayed to the operator. This task is accomplished by an online Flux Mapping System, which employs a suitable algorithm to estimate the core flux distribution from the readings of a large number of in-core detectors. Most of the algorithms available today employ the Flux Synthesis method, Internal Boundary Condition method, and the method based on simultaneous least squares solutions of neutron diffusion and detector response equations. A common feature of these methods is the assumption that the neutron flux profile in the reactor is independent of time. Application of Kalman filtering-based approaches are also found though to a very limited extent. In this paper, we have formulated the task of flux-mapping problem in AHWR as a problem of optimally estimating the time-dependent neutron flux at a large number of mesh points in the core. The solution is obtained using the well-known Kalman filtering technique which works along with a space-time kinetics model of the reactor. However, the attempt to solve the Kalman filtering problem in a straightforward manner is not successful due to severe numerical ill-conditioning caused by the simultaneous presence of slow and fast phenomena typically present in a nuclear reactor. Hence, a grouping of state variables has been suggested whereby the original high-order model of the reactor is decoupled into a slow subsystem and a fast subsystem. Now according to the order of the slow and fast subsystems, the original time update and Kalman gain equations have also been decoupled into separate sets of equations for the slow and fast subsystems. The decoupled sets of equations could be solved easily. The proposed method has been validated in a number of typical transient situations. Overall accuracy in the estimation using the proposed methodology has been very good for mesh fluxes, channel fluxes, quadrant fluxes, and the core average flux.

Bursting phenomenon in a piecewise mechanical system with parameter perturbation in stiffness

In this paper the existence condition and generation mechanism of the possible bursting phenomenon in a piecewise mechanical system with different time scales are studied. As an example of mechanical systems, a piecewise linear oscillator with parameter perturbation in stiffness and subject to external excitation is examined. The order gaps between the time scales are considered in the model, which are related to the periodic excitation and the changing rates of the variables. The focus-type periodic bursting oscillation with two time scales is presented, and the corresponding generation mechanism is revealed by using slow–fast analysis method. Furthermore, the analytical solution of piecewise linear subsystem as well as the stability condition of the fast subsystem are explored to explain the transition of bursting behaviors coming from the variation of intrinsic parameter and external excitation. The results about bursting phenomenon and its generation mechanism would provide important theoretical basis on the mechanical manufacturing and engineering practice.

Evaluation of bifurcation phenomena in a modified Shen–Larter model for intracellular $$\hbox {Ca}^{\varvec{2+}}$$ Ca 2 + bursting oscillations

The present work describes an evaluation of the bifurcation phenomena in a modified Shen–Larter model based on calcium-induced calcium release and inositol triphosphate $$(\hbox {IP}_{3})$$ crosscoupling for calcium ion $$(\hbox {Ca}^{2+})$$ bursting oscillations. A time delay for negative $$\hbox {Ca}^{2+}$$ feedback on the $$(\hbox {IP}_{3})$$ receptor is added to the original Shen–Larter model, by introducing the proportion of receptors not inactivated by $$\hbox {Ca}^{2+}$$ as a new variable. Compared with the original model, the number of chaotic regions for a stimulation level r is significantly reduced, and regions of $$\hbox {Ca}^{2+}$$ oscillations (particularly bursting) appear to become slightly enlarged. Different topological types of bursting oscillations in this modified model are classified by fast/slow dynamical analysis and codimension-2 bifurcations of fast subsystem, when choosing better of two slow variables the free $$\hbox {Ca}^{2+}$$ concentration in the endoplasmic reticulum and the $$\hbox {IP}_{3}$$ concentration in the cytosol. Furthermore, classification and transition mechanisms of bursting $$\hbox {Ca}^{2+}$$ oscillations could help to understand or detect more distinctive oscillatory behaviors of real cells in response to different levels of stimulation.