Singular Perturbation Model Research Articles

On the Scaling of the Cubic-to-Tetragonal Phase Transformation with Displacement Boundary Conditions

We provide (upper and lower) scaling bounds for a singular perturbation model for the cubic-to-tetragonal phase transformation with (partial) displacement boundary data. We illustrate that the order of lamination of the affine displacement data determines the complexity of the microstructure. As in (Rüland and Tribuzio in ESAIM Control Optim. Calc. Var. 29:68, 2023) we heavily exploit careful Fourier space localization methods in distinguishing between the different lamination orders in the data.

On the energy scaling behaviour of singular perturbation models with prescribed dirichlet data involving higher order laminates

Motivated by complex microstructures in the modelling of shape-memory alloys and by rigidity and flexibility considerations for the associated differential inclusions, in this article we study the energy scaling behaviour of a simplifiedm-well problem without gauge invariances. Considering wells for which the lamination convex hull consists of one-dimensional line segments of increasing order of lamination, we prove that for prescribed Dirichlet data the energy scaling is determined by theorder of lamination of the Dirichlet data. This follows by deducing matching upper and lower scaling bounds. For theupperbound we argue by providing iterated branching constructions, and complement this with ansatz-freelowerbounds. These are deduced by a careful analysis of the Fourier multipliers of the associated energies and iterated “bootstrap arguments” based on the ideas from [A. Rüland and A. Tribuzio,Arch. Rational Mech. Anal.243(2022) 401–431]. Relying on these observations, we study models involving laminates of arbitrary order.

Modeling, Fault Detection, and Fault-Tolerant Control for Nonlinear Singularly Perturbed Systems With Actuator Faults and External Disturbances

A fuzzy singular perturbation model with additive faults on control inputs and external disturbances, a fuzzy <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> fault detector (FHFD), and a fuzzy <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> fault-tolerant controller (FHFTC) are constructed for nonlinear singularly perturbed systems (NSPSs) with actuator faults and external disturbances, respectively. The sufficient conditions for FHFD and FHFTC existence are derived, respectively, which are described by a set of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> -independent linear matrix inequalities (LMIs), such that it can avoid the ill-conditioned problems caused by the tiny parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> . An application to a CE150 helicopter with abrupt and intermittent faults is provided to verify the above approaches. The innovation of the article lies in the following points: i) The fault detection and fault-tolerant control for NSPSs are handled under the united model, which can avoid the incomplete decoupling problem caused by decomposing NSPSs as the slow and fast subsystems; ii) the estimate values for actuator faults are regarded as a state vector of the FHFD, and the output feedback control law is introduced into FHFD, which make the FHFD not only can accurately detect faults but also can stabilize the closed-loop system; iii) a new method to transform the nonlinear matrix inequalities into the LMIs is proposed; and iv) the presented FHFTC can avoid the problems of missing or inaccurate status information caused by the sensor faults because it adopts the state feedback of the FHFD.

Asymptotic Modeling of a Viscous Laminar Flow Around Thin Airfoils: Resolution and Experimental Treatment in Case of Supersonic Flow

In this paper, starting from the conservation equations in a 2D orthogonal curvilinear coordinate system, an approached asymptotic model for perturbed potential flow equation is established. The study considers a high-speed steady laminar irrotational flow of a compressible and viscous fluid, over thin airfoils at low incidence in the standard atmosphere. This study is aimed to contribute to developing a fully two-dimensional general model that is efficiently applicable to industrial scale and scientific research, in a laminar flow, defined on thin geometries as it takes into consideration both compressibility and viscous effects. The singular perturbation model is solved in case of supersonic viscous flow, based on asymptotic and singular perturbation methods. Asymptotic matching of different layers is done through the triple deck theory. Experimental measurements have been conducted, with the AF300 supersonic wind tunnel, on a NACA 43013 airfoil model fitted with two pressure taps. The comparison of the local Mach numbers between the asymptotic model and the experimentation shows the accuracy of the model in case of laminar flow but diverge relatively as the flow becomes turbulent towards the trailing edge.

Existence and Stability of Equilibrium of DC Micro-Grid Under Master-Slave Control

In this paper, we analyze the existence and stability of equilibrium of dc micro-grids under the master-slave control (where some distributed generations (DGs) are under droop control with dual-loop (Droop-DGs) and some DGs are under MPPT control (MPPT-DGs)). Firstly, the power-flow equation of the dc micro-grids under master-slave control with CPLs is obtained. Then, we transform the solvability of the power-flow equation into the existence of a fixed point for a contraction mapping. Based on Banach's fixed point theorem, a sufficient condition to guarantee the existence of the power-flow solution in dc micro-grids is derived. The condition derived in this paper is not only useful for master-slave control but also for droop control. Besides, to calculate the power-flow solution, an iterative algorithm with exponential convergence rate is proposed. Secondly, we use a singular perturbation model to predict the qualitative behavior of the system near the equilibrium point. By analyzing eigenvalues of the boundary layer system and reduced-order system Jacobian matrix, the robust stable analytic conditions of the system are obtained. The effect of the sampling delay on the system stability is analyzed, the robust stability condition is obtained by using linear matrix inequality. The simulation results verify the correctness of the proposed conditions. The obtained conditions provide a reference for establishing a reliable dc micro-grid.

Robust fuzzy sliding mode control and vibration suppression of free-floating flexible-link and flexible-joints space manipulator with external interference and uncertain parameter

AbstractThe flexibility of the free-floating flexible space manipulator system’s link and joint may affect the control accuracy and cause vibrations. We studied the dynamics modeling, joint trajectory tracking control, and vibration suppressing problem of free-floating flexible-link and flexible-joints space manipulator system with external interference and uncertain parameter. The system’s dynamic equations are established using linear momentum conservation, angular momentum conservation, assumed mode method, and Lagrange equation. Then, the system’s singular perturbation model is established, and a hybrid control is presented. For the slow subsystem, a robust fuzzy sliding mode control is proposed to realize the joint desired trajectory tracking. For the fast subsystem, a speed difference feedback control and a linear-quadratic optimal control are designed to suppress the vibration caused by the flexible joints and the flexible link separately. The simulation comparison experiments under different conditions are taken. The simulate results demonstrate the proposed hybrid control’s validity.

Large-Signal Stability Improvement of DC-DC Converters in DC Microgrid

In DC microgrids, constant power loads (CPLs) reduce the effective damping of the DC-DC converter and may induce destabilizing effects into the DC-DC converter. To overcome such problems regarding CPL and ensure large-signal stability of DC-DC converters in DC microgrids, some feedforward terms are added to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$V$</tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$I$</tex-math></inline-formula> droop-based dual-loop controller for a DC-DC converter based on the large-signal model. It is proven that the feedforward terms can not only improve the transient response but also guarantee the exponential stability of the closed-loop system in the whole operating range in regards to a large-signal manner, which is verified by using a singular perturbation model. Moreover, a disturbance observer is designed to estimate the output current, thereby enabling the removal of the current measurement sensor. The proposed technique can be easily plugged into a pre-defined <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$V$</tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$I$</tex-math></inline-formula> droop-based dual-loop controller without an additional sensor being required. Ultimately, both simulation and experimental tests verify the effectiveness of the proposed method.

Fixed-Time Tracking Control for Flexible Joint Manipulator With Prescribed Performance Constraint

This paper studies the transient and steady performance constraints on flexible joint manipulator system with exogenous disturbance, and proposes a fixed-time prescribe performance tracking control method based on the singular perturbation model. For the fast subsystem, a subcontroller with superior transient performance is proposed that guarantees the suppression of flexible vibration in fixed-time. The remaining slow subsystem can be linked with rigid methods to achieve output constraints, a type of time-varying error constraint boundary is first constructed, which broadens the initial error constraint boundary of the time-varying tangent barrier Lyapunov function. Then, a slow subcontroller based on a non-singular fast-integration terminal sliding mode is constructed to make any initial tracking error converge in fixed-time. Finally, a nonlinear disturbance observer is used to counteract the exogenous disturbance. The simulation results show that the controller achieves the prescribe performance.

Data-based reduced-order modeling of nonlinear two-time-scale processes

This work focuses on data-based reduced-order modeling of nonlinear processes that exhibit time-scale multiplicity. Using time-series data from all the state variables of a nonlinear process, an approach that involves nonlinear principal component analysis and neural network function approximators is employed to identify the fast and slow process state variables as well as compute a nonlinear slow manifold function approximation in which the fast variables are “slaved” in terms of the slow variables. Subsequently, a nonlinear sparse identification approach is employed to calculate a dynamic model of nonlinear first-order ordinary differential equations that describe the temporal evolution of the slow process states. The method is applied to two chemical process examples, and the advantages of the proposed method over using only sparse identification for the original two-time-scale process are discussed. The approach can be thought of as a data-based analogue of the classical singular perturbation modeling of two-time-scale processes where explicit time-scale separation is assumed (and expressed in terms of a small positive parameter ε multiplying the time derivative of the fast states), and the reduced-order slow subsystem is calculated analytically.

Model-Based Control and Stability Analysis of Underactuated Autonomous Underwater Vehicles Via Singular Perturbations

Abstract This paper presents the control design and stability analysis for path-following of underactuated autonomous underwater vehicles (AUVs), with dynamics restricted to the horizontal plane. As illustration, the time-scale separation caused by different rates of numerous variables is exploited via a singular perturbation model formulation. On the basis of that, a time-scale decomposition method is used to decompose the full system into three-time scale subsystems. The three-time scale structure allows independent analysis of dynamics in each time scale. Therefore, control strategies are designed in each subsystem separately, leading to a reduction of control complexity and a relatively simple control law. This paper also demonstrates the asymptotic stability of the closed-loop system with a composite Lyapunov function candidate and provides alternative, simple but generic mathematical bounds on the singularly perturbed parameters. Finally, the simulation results are presented to illustrate the effective performance of proposed controller.

Singular Perturbation-Based Fault-Tolerant Control of the Air-Breathing Hypersonic Vehicle

This article studies the fault-tolerant control issue of the air-breathing hypersonic vehicle subject to the actuator fault. Through singular perturbation modeling, the longitudinal dynamics of the vehicle can be transformed into a three-time-scale singularly perturbed system. A new nonlinear diagnostic observer is designed to estimate the actuator fault first. Then a cyclic-small-gain principle-based distributed fault-tolerant control strategy is developed by regarding the singularly perturbed system as an interconnected system with three subsystems. The proposed distributed fault-tolerant controller can achieve the input-to-state stability of the longitudinal motion at the cruising condition with respect to the fault estimation error. The simulation results show the effectiveness of the proposed method.

Ensemble Kalman Filters for State Estimation and Prediction of Two-Time Scale Nonlinear Systems With Application to Gas Turbine Engines

In this brief, we propose and develop estimation, prediction, and health monitoring methodologies for nonlinear systems by modeling the damage and degradation mechanism dynamics as “slow” states that are augmented with the system “fast” states. This augmentation results in a two-time scale (TTS) nonlinear system that is utilized for the development of decoupled slow and fast health estimation and prediction modules within a health monitoring framework. Specifically, a TTS filtering approach based on ensemble Kalman filters is developed by taking advantage of the singular perturbation model reduction technique. Our proposed methodology is then applied to a gas turbine engine that is affected by degradation phenomenon due to the turbine erosion. Extensive comparative studies are conducted to validate and demonstrate the advantages and capabilities of our proposed methodology when compared to the well-known nonlinear particle filtering (PF) approach that is commonly utilized in the literature.

Perturbation Theory and Singular Perturbations for Input-to-State Multistable Systems on Manifolds

We consider the notion of input-to-state multistability, which generalizes input-to-state stability to nonlinear systems evolving on Riemannian manifolds and possessing a finite number of compact, globally attractive, invariant sets, and in addition satisfies a specific condition of acyclicity. We prove that a parameterized family of dynamical systems whose solutions converge to those of a limiting system inherits such input-to-state multistability property from the limiting system in a semiglobal practical fashion. A similar result is also established for singular perturbation models whose boundary-layer subsystem is uniformly asymptotically stable and whose reduced subsystem is input-to-state multistable. Known results in the theory of perturbations, singular perturbations, averaging, and highly oscillatory control systems are here generalized to the multistable setting by replacing the classical asymptotic stability requirement of a single invariant set with attractivity and acyclicity of a decomposable invariant one.

Tracking control design for a multi-degree underactuated flexible-cable overhead crane system with large swing angle based on singular perturbation method and an energy-shaping technique

A flexible-cable overhead crane system having large swing is studied as a multi-degree underactuated system. To resolve the system dynamics complexities, a second order singular perturbation (SP) formulation is developed to divide the crane dynamics into two one-degree underactuated fast and slow subsystems. Then, a control system is designed based on the two-time scale control (TTSC) method to: (a) transfer the payload to a desired location and decrease the payload swing, by a nonlinear controller for slow dynamics; and (b) suppress transverse vibrations of the cable, by a linear controller for fast dynamics. The nonlinear controller is designed based on an energy shaping technique according to the controlled Lagrangian method. To demonstrate the control system effectiveness, an example of the flexible cable crane systems with a lightweight payload is considered to perform simulations. In addition to the proposed control system, two other controllers; namely, a linear controller based on the linear–quadratic regulator method and a TTSC based on the approximate SP model and partial feedback linearization, are applied to the system for comparison. Also, by applying a disturbance force to the trolley and considering 10% uncertainty in crane parameters, the control performance against disturbances and parameter uncertainties is investigated.

Optimal control in teleoperation systems with time delay: A singular perturbation approach

The main goal of controller design in teleoperation systems is to achieve optimal performance, transparency and stability in presence of factors such as time delay in communication channel and modeling uncertainties. The teleoperation systems usually have complex dynamic. Consequently, differential equation solution of optimal control problem is difficult and complex for them. This paper presents a novel method for designing optimal controller based on singular perturbation framework for these systems. Firstly, we use the Taylor expansion to model the time delay, with considering time delay term; we derive a singular perturbation formulation for the teleoperation system. Using singular perturbation model and Chang decoupling transformation, singularly perturbed differential equations of optimal control problem is decomposed into the reduced order slow and fast differential equations. A formula is obtained that produces the solution of original differential equations of optimal control problem in terms of solutions of the slow and fast reduced order matrix differential equations. The reduced-order differential equations decrease the complexity of the optimal control problem for teleoperation systems. The simulations verify the effectiveness of the proposed control method and excellent performances tracking with high speed and small control signal.

Distributed quantized multi-modal H∞ fusion filtering for two-time-scale systems

In this paper, we deal with the distributed H∞ fusion filtering problem for a class of discrete-time system with significant modal difference in the time-scale. The nonstandard singularly perturbed model (SPM) is employed to describe the addressed two-time-scale system, where the small perturbation parameter ε reflects the discrepancies in the time-scale. Some sufficient conditions are provided to guarantee the existence of the parameters of the local filters for the SPM, where O(ε2) (the second-order correction terms in ε) is eliminated and an upper bound of ε is calculated. To enhance the filtering accuracy and scalability, the estimates provided by the local filter are transmitted to the fusion centre via the communication network. In the presence of the bandwidth constraint, a quantized fusion scheme is developed and the design of the fusion parameters is cast into a convex optimization problem. Finally, an example on the F-8 airplane is provided to verify the validity and applicability of the proposed methodology.

Observer-based two-time scale robust control of free-flying flexible-joint space manipulators with external disturbances

SUMMARYObserver-based two-time scale robust control is proposed for free-flying flexible-joint space manipulators with unknown payload parameters and bounded disturbances. The dynamic equations of a free-flying space manipulator with two flexible revolute joints were derived by the momentum conservation law and the Lagrange equations. A flexibility compensator was introduced to make the equivalent joint stiffness large enough, which made traditional singular perturbation approach applicable. Then, a singular perturbation model was formulated and a reduced-order controller is proposed. This controller consisted of a slow sub-controller and a fast flexible-joint sub-controller. To the slow subsystem, a sliding observer based robust slow sub-controller was proposed. By optimal linear quadratic regulator method, the fast sub-controller was designed with the estimated velocity by linear observer. This fast sub-controller could stabilize the fast subsystem around the equilibrium trajectory created by the slow subsystem under the effect of the slow control. Finally the numerical simulations were carried out, which showed that elastic joint vibrations had been stabilized effectively and good tracking performances had been achieved.

Modeling of an Epidemiological Model

The aim of this paper is to present mathematical modeling of the spread of infection in the context of the transmission of the human immunodeficiency virus HIV. The method is based on the models suggested by KHALLIL methods to reduce a complex model. Keywords: HIV/AIDS Epidemic, Singular Perturbation, Slow and Fast Variable Epidemic Models

Modelling and Composite Control of Single Flexible Manipulators with Piezoelectric Actuators

The piezoelectric actuators are used to investigate the active vibration control of flexible manipulators in this paper. Based on the assumed mode method, piezoelectric coupling model, and Hamilton’s principle, the dynamic equation of the single flexible manipulator (SFM) with surface bonded actuators is established. Then, a singular perturbation model consisted of a slow subsystem and a fast subsystem is formulated and used for designing the composite controller. The slow subsystem controller is designed by fuzzy sliding mode control method, and the linear quadratic regulator (LQR) optimal control method is used to design fast subsystem controller. Furthermore, the changing trends of natural frequencies along with the changes in the position of piezoelectric actuators are obtained through the ANSYS Workbench software, by which the optimal placement of actuators is determined. Finally, numerical simulations and experiments are presented. The results demonstrate that the method of optimal placement is feasible based on the maximal natural frequency, and the composite controller presented in this paper can not only realize the trajectory tracking of the SFM and has a good result on the vibration suppression.

Low‐frequency robust control for singularly perturbed system

This study is concerned with the low-frequency (LF) integral quadratic constraints (IQC) of the linear singularly perturbed model (SPM). The conditions satisfying LF IQC are derived in terms of linear matrix inequalities by using the generalised Kalman–Yakubovich–Popov lemma. As special cases of LF IQC, LF bounded realness and LF positive realness are exploited in detail. In addition, upper bound of the small perturbation parameter, which relates to the robustness of SPM, can be estimated. Numerical simulations demonstrate that the proposed method can reduce the conservatism and characterise the SPM more specifically by ignoring the unrelated frequency range.