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Robotic manipulators that utilize the Cable-Pulley based Gravity Compensation Mechanism (CPGCM) have drawn increasing attention due to their exceptional features. In the current study, for the first time, the modeling and control of such systems will be addressed considering the flexibility of the cables. To this end, initially, the dominant dynamics of the cables are identified. Then, the Lagrangian of the whole robotic system including the flexibility of the cables is obtained and the system equations of motion are formed. To design the motion controller for the system, and suppress the vibrations stemmed from the inherent flexibility of the cables, the obtained dynamics is transformed into the standard form of the Singular Perturbation Theory (SPT). To achieve this goal, a combination of the inverse dynamics and Nonsingular Terminal Sliding Mode controller is designed, assuming the rigid model of the cable. A modifying term based on SPT is then added to the previous control signal to tackle the flexibility effects of the cables. The finite-time stability of the overall closed-loop controller is proven using Lyapunov's direct method. The proposed controller is robust to deal with both structured and unstructured uncertainties of the system. The obtained results confirm that the presented robust control algorithm guarantees the stability of the closed-loop system in the finite time. Moreover, it is able to suppress the arisen vibration because of the inherent axial flexibility of the cables in various operational conditions. [ABSTRACT FROM AUTHOR]

Spatially localized blooms of toxic plankton species have negative effects on other organisms owing to the production of toxins, mechanical damage, or other means. A two-prey (toxic and nontoxic phytoplankton) one-predator (zooplankton) Lotka–Volterra system with diffusion has been presented to understand the mechanism underlying the formation of spatial blooms of toxic plankton. In this study, we consider a one-dimensional (1D) system in which the ratio $ D $ of the diffusion rates of the predator and two prey, and the length of the spatial interval $ L $ are both sufficiently large while maintaining $ L^2/D $ as a constant. This system is reduced to a singularly perturbed system. The existence and stability properties of the 1D blooming stationary solutions are demonstrated by improving the previous results. [ABSTRACT FROM AUTHOR]

Marine-vehicle manipulators, which represent a kind of mechanical systems installed on marine surface or underwater vehicles, are mostly suffering from the problem of waves (or ocean currents)-caused base oscillations. The oscillations have a significant impact on system stability. Numerous control strategies have been investigated, but the majority of them are concentrated on the control's robust performance. This study focuses on an innovative marine-vehicle manipulator (ammunition transfer manipulator on warships) with novel compliant actuators (series elastic actuators), for which the control performance of convergence speed and flexible-vibration suppression should also be considered. To address these issues, this paper proposes a unique hybrid control based on the singular perturbation method, by which the control problem is decomposed into two time scales. In the slow time-scale, it is given a rapid trajectory tracking controller that integrates the computed torque method and the terminal sliding mode control law with a novel reaching law (variable power log reaching law). For the fast time-scale control, a derivative-type controller is used to achieve the suppression of the flexible vibrations. To demonstrate the effectiveness of the proposed control method, theoretical proofs and numerical simulations are both presented. According to our knowledge, this study presents the first control strategy for rapid robust control of marine-vehicle manipulators that are subject to base-oscillation-caused disturbance and compliant-actuator-induced flexible vibrations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the traveling wave solutions of a singular Keller-Segel system modeling chemotactic movement of biological species with logistic growth. We first show the existence of traveling wave solutions with zero chemical diffusion in $ \mathbb{R} $. We then show the existence of traveling wave solutions with small chemical diffusion by the geometric singular perturbation theory and establish the zero diffusion limit of traveling wave solutions. Furthermore, we show that the traveling wave solutions are linearly unstable in the Sobolev space $ H^1(\mathbb{R}) \times H^2(\mathbb{R}) $ by the spectral analysis. Finally we use numerical simulations to illustrate the stabilization of traveling wave profiles with fast decay initial data and numerically demonstrate the effect of system parameters on the wave propagation dynamics.

Background and Objectives: Robot manipulator with flexible-joints is very complex nonlinear system whose control is one of the most challenging issues in the control word. Therefore, the design of voltage-based controller in addition, using of order reduction methods can reduce the complexity of the control law in such systems. Methods: This paper proposes a voltage controller for flexible-joint robot manipulators based on singular perturbation method. The presented control approach has all three advantages of the singular perturbation method for model order reduction, the proper structure of Port-Hamiltonian systems, and voltage control strategy (VCS). In this approach, the robot manipulator model is divided into three sub-systems, slow, medium and fast sub-systems. Each of the sub-systems is controlled using separate controller. In addition, the stability of these sub-systems and ultimately the whole system are proved. Unlike other related works, in this work the tracking error system is considered from the beginning, and by singular perturbation method, a controller is designed to stabilize the tracking system. Moreover, in the suggested voltage-based controller unlike torque control strategy, the electrical model of actuators is used. Results: The main advantages of proposed approach are simple structure, using only velocity of motors, the position of the joints as a control signal and considering the electrical model of the actuators. So, practical implementation of this controller will be with much less effort, compared to the methods like feedback linearization or other controllers in related works. Moreover, using the Lyapunov-based method, the ultimate bounded stability of the closed loop system is proved. Then, some simulations are provided for tracking, regulation, robustness and response speed purposes. Conclusion: Since, the controllers for every sub-system are designed separately, also, the control signal parameters such as joints position, motors velocity, and motors current can be simply measured, therefore, the structure of designed controller is very simple and practically implementable. As the simulation results confirm, the performance of this controller is appropriate, even when external disturbances are present or the frequency of reference signal increases. At last, by comparison and analysis of the simulation results between the presented approach and a related work, the suitable tracking performance of the suggested controller is shown.

The microgrid with the high proportion of renewable sources has become the trend of the future. However, the negative features, such as renewable energy perturbation, nonlinear counterpart, and so on, are prone to causing the low-power quality of the ac microgrid. To deal with these problems, this article proposes an event-triggered consensus control approach. First, the nonlinear state-space function regarding the ac microgrid is built, which is further transformed into the standard linear multiagent model by using the singular perturbation method. It provides indispensable preprocessing for the direct application of advanced linear control approaches. Then, based on this standard linear multiagent model, the secondary consensus approach with the leader is designed to compensate for the output voltage deviation and achieve accurate power sharing. In order to decrease the communication among various distributed generators, the event-triggered communication method is further proposed. Meanwhile, the Zeno behavior is avoided through the theoretical proof. Finally, simulation results are presented to demonstrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

A virtual reaction scheme and the notion of quasi-independent is proposed in this paper to assist in model partitioning for the biochemical reactions with multiple time scales. Subsequently, the singular perturbation method is applied to the stochastic model, in particular the chemical master equation of the multiple time scales system to reduce the model dimension. As a result, a lower dimensional approximation for the chemical master equation is derived through this approach. Therefore, the high dimensional chemical master equation can be solved with a lower computational cost.

This paper is concerned with the stability of traveling wave fronts for a modified vector disease model. By applying geometric singular perturbation method and standard asymptotic theory, we first give the exact decay rates of the traveling wave fronts with small delay. Then by detailed spectral analysis, we can prove that the traveling wave fronts with noncritical speeds are nonlinearly exponentially stable in some exponentially weighted spaces. [ABSTRACT FROM AUTHOR]

We investigate a class of nonequilibrium media described by Langevin dynamics that satisfies the local detailed balance. For the effective dynamics of a probe immersed in the medium, we derive an inequality that bounds the violation of the second fluctuation-dissipation relation (FDR). We also discuss the validity of the effective dynamics. In particular, we show that the effective dynamics obtained from nonequilibrium linear response theory is consistent with that obtained from a singular perturbation method. As an example of these results, we propose a simple model for a nonequilibrium medium in which the particles are subjected to potentials that switch stochastically. For this model, we show that the second FDR is recovered in the fast switching limit, although the particles are out of equilibrium. [ABSTRACT FROM AUTHOR]

Virtual synchronous machines are inverters with a control algorithm that causes them to behave towards the power grid like synchronous generators. A popular way to realize such inverters are synchronverters. Their control algorithm has evolved over time, but all the different formulations in the literature share the same “basic control algorithm”. We investigate the equilibrium points and the stability of a synchronverter described by this basic algorithm, when connected to an infinite bus. We formulate a fifth order model for a grid-connected synchronverter and derive a necessary and sufficient condition for the existence of equilibrium points. We show that the set of equilibrium points with positive field current is a two-dimensional manifold that can be parametrized by the corresponding pair $(P,Q)$ , where $P$ is the active power and $Q$ is the reactive power. This parametrization has several surprizing geometric properties, for instance, the prime mover torque, the power angle and the field current can be seen directly as distances or angles in the $(P,Q)$ plane. In addition, the stable equilibrium points correspond to a subset of a certain angular sector in the $(P,Q)$ plane. Thus, we can predict the stable operating range of a synchronverter from its parameters and from the grid voltage and frequency. Our stability result is based on the intrinsic two time scales property of the system, using tools from singular perturbation theory. We illustrate our theoretical results with two numerical examples. [ABSTRACT FROM AUTHOR]

This article studies the trajectory tracking and active vibration control of flexible space manipulator without velocity measurements. Variable-speed control moment gyros (VSCMGs) are adopted as actuators for vibration suppression. Based on the singular perturbation method, the dynamics model of the manipulator system considering the uncertainties and disturbances is decoupled into two reduced-order subsystems: the slow subsystem and the fast subsystem. For the slow subsystem, neural network (NN) is used to approximate the unknown nonlinear functions. An NN-based adaptive integral sliding-mode observer is designed to estimate some unknown velocity variables. An observer-based slow controller is proposed. Considering the uncertainties and the unmeasured modal information of the fast system, a modified extended state observer (ESO) is designed for the estimation of the uncertainties and unknown fast variables. On the basis of the modified ESO, an adaptive feedback controller is proposed for the flexible motion. Lyapunov theory is used to prove the stability of the proposed controllers. The weighted robust pseudoinverse steering laws are designed for VSCMGs. The effectiveness of the proposed observer-based hybrid controller is verified via numerical simulations. [ABSTRACT FROM AUTHOR]

Background and Objectives: Robot manipulator with flexible-joints is very complex nonlinear system whose control is one of the most challenging issues in the control word. Therefore, the design of voltage-based controller in addition, using of order reduction methods can reduce the complexity of the control law in such systems. Methods: This paper proposes a voltage controller for flexible-joint robot manipulators based on singular perturbation method. The presented control approach has all three advantages of the singular perturbation method for model order reduction, the proper structure of Port-Hamiltonian systems, and voltage control strategy (VCS). In this approach, the robot manipulator model is divided into three sub-systems, slow, medium and fast sub-systems. Each of the sub-systems is controlled using separate controller. In addition, the stability of these sub-systems and ultimately the whole system are proved. Unlike other related works, in this work the tracking error system is considered from the beginning, and by singular perturbation method, a controller is designed to stabilize the tracking system. Moreover, in the suggested voltage-based controller unlike torque control strategy, the electrical model of actuators is used. Results: The main advantages of proposed approach are simple structure, using only velocity of motors, the position of the joints as a control signal and considering the electrical model of the actuators. So, practical implementation of this controller will be with much less effort, compared to the methods like feedback linearization or other controllers in related works. Moreover, using the Lyapunov-based method, the ultimate bounded stability of the closed loop system is proved. Then, some simulations are provided for tracking, regulation, robustness and response speed purposes. Conclusion: Since, the controllers for every sub-system are designed separately, also, the control signal parameters such as joints position, motors velocity, and motors current can be simply measured, therefore, the structure of the designed controller is very simple and practically implementable. As the simulation results confirm, the performance of this controller is appropriate, even when external disturbances are present or the frequency of reference signal increases. At last, by comparison and analysis of the simulation results between the presented approach and a related work, the suitable tracking performance of the suggested controller is shown. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with asymptotic solutions of a nonlinear boundary value problem which arises from laminar flow in a uniformly porous channel with expanding or contracting walls. For values of the wall suction Reynolds number, multiple solutions are observed. A method involving the inclusion of exponentially small terms in a perturbation series is mainly considered to obtain two of the solutions analytically. In addition, numerical solutions presented for each case agree well with asymptotic solutions, which illustrates that the asymptotic solutions constructed in this paper are more reliable.

In this paper we study the spruce-budworm interaction model with Holling's type II functional response. The existence, number and stability of equilibria are studied. Moreover, we prove the existence of relaxation oscillations by using singular perturbation method and give an asymptotic expression of the period of relaxation oscillations. Finally, the parameter ranges which allow the relaxation oscillations in several scenarios are explored and displayed by conducting numerical simulations. [ABSTRACT FROM AUTHOR]

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

Achieving a simplified model is a major issue in the field of fractional-order nonlinear systems, especially large-scale systems. So that in addition to simplifying the model, the outstanding features of the fractional-order modeling, such as memory feature, are preserved. This paper presented the homotopy singular perturbation method (HSPM) to reduce the complexity of the model and use the advantages of both models of the fractional order and the integer order. This method is a combination of the fractional-order singular perturbation method (FOSPM) and the homotopy perturbation method (HPM). Firstly, the FOSPM is developed for fractional-order nonlinear systems; then, a modification of the HPM is proposed. Finally, the HSPM is presented using a combination of these two methods. fractional-order nonlinear systems can be divided into two lower-order subsystems such as nonlinear or linear integer-order subsystem and linear fractional-order subsystem using this hybrid method. Convergence analysis of tracking error to zero is theoretically presented, and the effectiveness of the proposed method is also evaluated with two examples. Next, the number and location of equilibrium points are compared between the original system and the subsystems obtained from the proposed method. In the end, we show that the stability of fractional-order nonlinear system can be determined by investigating the stability of the subsystems using Theorem 3 and Lemma 2. [ABSTRACT FROM AUTHOR]

Handling a flexible object is more complicated than a rigid one since it involves the vibration of the object. Since vibration is known to lead to disturbance, discomfort, damage, and destruction; it needs to be suppressed. The system consists of two cooperative manipulators handling a flexible beam that is modelled in partial differential equation (PDE) form and employed a singular perturbation method to model the slow and fast subsystems. This paper presents a composite control comprising of the computed torque control (CTC) scheme for the slow subsystem and a velocity feedback control (VFC) for the fast subsystem that was developed based on the PDE model form so that two cooperative manipulators track the desired trajectories while suppressing the transverse vibrations of the beam. A stability analysis was carried out for each subsystem to satisfy Tikhonov's Theorem. The simulation results for slow subsystem showed that the tracking of positions and orientation have been achieved within 0.5 s with the root-mean-square error (RMSE) values of 0.002745 m, 0.02292 m, and 0.01563 rad for X-direction, Y-direction and the orientation, respectively. For the fast subsystem, the transverse vibration of the beam is completely suppressed within 0.8 s. The results proved that the proposed controller has worked well with the PDE model of cooperative manipulators to handle the flexible beam while suppressing its vibration. [ABSTRACT FROM AUTHOR]

The existence and multiplicity of similarity solutions for the steady, incompressible and fully developed laminar flows in a uniformly porous channel with two permeable walls are investigated. We shall focus on the so-called asymmetric case where the upper wall is with an amount of flow injection and the lower wall with a different amount of suction. The numerical results suggest that there exist three solutions designated as type |$I$|⁠ , type |$II$| and type |$III$| for the asymmetric case, type |$I$| solution exists for all non-negative Reynolds number and types |$II$| and |$III$| solutions appear simultaneously at a common Reynolds number that depends on the value of asymmetric parameter |$a$| and with the increase of |$a$| the common Reynolds numbers are decreasing. We also theoretically show that there exist three solutions. The corresponding asymptotic solution for each of the multiple solutions is constructed by the method of boundary layer correction or matched asymptotic expansion for the most difficult high Reynolds number case. These asymptotic solutions are all verified by their corresponding numerical solutions. [ABSTRACT FROM AUTHOR]

This paper presents the design of a singular-perturbation-based optimal guidance with constraints on terminal flight-path angle and angle of attack. By modeling the flight-control system dynamics as a first-order system, the angle of attack is introduced into the performance index as a state variable. To solve the resulting high-order optimal guidance problem analytically, the posed optimal guidance problem is divided into two sub-problems by utilizing the singular perturbation method according to two time scales: range, altitude, and flight-path angle are the slow time-scale variables while the angle of attack is the fast time-scale variable. The outer solutions are the optimal control of the slow-scale subsystem. Thereafter, by applying the stretching transformation, the fast-scale subsystem establishes the relationships between the outer solutions and acceleration command. Then, the optimal command can be obtained by solving the fast-scale subsystem also using the optimal control theory. The proposed guidance can achieve a near-zero terminal acceleration as well as a small miss distance. The superior performance of the guidance is demonstrated by adequate trajectory simulations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper presents the fault tolerant control (FTC) of a flexible joint robot using singular perturbation method in order to compensate for the lost performance due to the occurrence of actuator fault and the uncertainty. This FTC is based on Lyapunov redesign principle. The singular perturbation method is used to reduce the dynamic model of the flexible joint robot in a fast and slow subsystem. The time scale reduction of the flexible joint model is carried out when their joint stiffness is large enough and the singular perturbation parameter is set to zero. The fault-tolerant control structure in this paper is based on two parts. The first term described the composite control for the system without defect and without uncertainty which represents the sum between slow and fast controllers. While the second term of the fault tolerant command describes additive control designed to compensate for the fault effect of the actuator on the uncertain system. The additive approach is based on the Lyapunov theorem, which guarantees asymptotic stability despite the presence of actuator defects and the parametric uncertainty. The theoretical results are applied on a robot manipulator with a single flexible joint. [ABSTRACT FROM AUTHOR]

In this paper, active control of a 6-DOF space robot with flexible panels using singular perturbation method is investigated. Dynamic model of the system is established by using the Jourdain's velocity variation principle. Dynamic equation of the system is decomposed into a slow subsystem and a fast subsystem by using the singular perturbation method. A composite controller is used to make the space robot reach a specified position and suppress the elastic vibration of the panels. Simulation results indicate that the proposed model is effective to describe the dynamics of space robot; the designed composite controller can effectively make the robot reach a specified position and the elastic vibration of the panels may be suppressed simultaneously; the fast controller has a great influence on the control effect of the space robot system. [ABSTRACT FROM AUTHOR]

This paper proposes a practical method to realize multivariable full-state tracking control for industrial robots with elastic joints. Unlike existing methods, the proposed method does not require high-order derivatives of the link states such as acceleration and jerk. Therefore, the proposed method does not suffer from chatter related to inaccurate estimation of high-order derivatives. The method is derived by adopting a singular perturbation technique. A decoupled error dynamics is achieved by two decoupling control loops: a fast loop that controls the deflection error and a slow loop for tracking control on the link side. Our stability analysis based on a linear system shows that the proposed control system is stable as long as the fast system is at least twice as fast as the slow system. A practical method to select the gain is also presented such that the closed-loop poles are placed at the desired locations. In simulation, we compare the proposed method with feedback linearization. The results indicate that in an ideal scenario the proposed method can obtain a similar performance as feedback linearization. However, the proposed method obtains a superior performance in a realistic scenario. A real-world experiment with a six degree-of-freedom commercial industrial robot is carried out to further validate our approach. [ABSTRACT FROM AUTHOR]

We investigate an unsteady incompressible laminar micropolar flow in a semi-infinite porous pipe with large injection or suction through a deforming pipe wall. Using suitable similarity transformations, the governing partial differential are transformed into a coupled nonlinear singular boundary value problem. For large injection, the asymptotic solutions are constructed using the Lighthill method, which eliminates singularity of solution in the high order derivative. For large suction, a series expansion matching method is used. Analytical solutions are validated against the numerical solutions obtained by Bvp4c.

This work develops a new modular architecture that emulates a recentlydiscovered biological paradigm. It originates from the human brain where the information flows along two different pathways and is processed along two time scales: one is a fast neural network (NN) and the other is a slow network called the glial network (GN). It was found that the neural network is powered and controlled by the glial network. Based on our biological knowledge of glial cells and the powerful concept of modularity, a novel approach called artificial neuroglial Network (ANGN) was designed and an algorithm based on different concepts of modularity was also developed. The implementation is based on the notion of multi-time scale systems. Validation is performed through an asynchronous machine (ASM) modeled in the standard singularly perturbed form. We apply the geometrical approach, based on Gerschgorin’s circle theorem (GCT), to separate the fast and slow variables, as well as the singular perturbation method (SPM) to determine the reduced models. This new architecture makes it possible to obtain smaller networks with less complexity and better performance. [ABSTRACT FROM AUTHOR]

Abstract: In this work we propose a framework for incorporating occupant feedback towards temperature control of multi‐occupant spaces, and analyze it using singular perturbation theory. Such a system would typically have to accommodate occupants with different temperature preferences, and incorporate that with thermal correlation among multiple zones to obtain optimal control for minimization of occupant discomfort and energy cost. In current practice, an acceptable temperature set point for the occupancy level of the zone is estimated, and the control law is designed to maintain temperature at the corresponding set point irrespective of the changes in occupancy and the preferences of multiple occupants. Proposed algorithm incorporates active occupant feedback to minimize aggregate user discomfort and total energy cost. Occupant binary feedback in the form of hot/cold or thermal comfort preference input is used by the control algorithm. The control algorithm also takes the energy cost into account, trading it off optimally with the aggregate occupant discomfort. For convergence to the optimal, sufficient separation between the occupant feedback frequency and the temperature dynamics of system is necessary; in absence of which, the occupant feedback provided do not correctly reflect the effect of current control input value on occupant discomfort. Under sufficient time scale separation, using singular perturbation theory, we establish the stability condition of the system and show convergence of the proposed solution to the desired temperature that minimizes energy cost plus occupant discomfort. The occupants are only assumed to be rational in that they choose their own comfort range to minimize individual thermal discomfort. Optimization for a multi‐zone building also takes into account the thermal correlation among different zones. Simulation study with parameters based on our test facility, and experimental study conducted in the same building demonstrates performance of the algorithm under different occupancy and ambient conditions. [ABSTRACT FROM AUTHOR]

The diffusiophoretic migration of a circular cylindrical particle in a nonelectrolyte solution with a solute concentration gradient normal to its axis is analytically studied for a small but finite Péclet number P e . The interfacial layer of interaction between the solute molecules and the particle is taken to be thin, but the polarization of its mobile molecules is allowed. Using a method of matched asymptotic expansions, we solve the governing equations of conservation of the system and obtain an explicit formula for the diffusiophoretic velocity of the cylinder correct to the order P e 2 . It is found that the perturbed solute concentration and fluid velocity distributions have the order P e , but the leading correction to the particle velocity has the higher order P e 2 ln P e . The correction to the particle velocity to the order P e 2 can be either positive or negative depending on the polarization parameter of the thin interfacial layer, establishing that the solute convection effect is complicated and can enhance or retard the diffusiophoretic motion. The particle velocity at P e = 0.6 can be about 17% smaller or 0.2% greater than that at P e = 0 . Under practical conditions, the solute convection effect on the diffusiophoretic velocity is much greater for a cylindrical particle than for a spherical particle, whose leading correction has the order P e 2 .

This paper is concerned with multiple solutions of a singular nonlinear boundary value problem (BVP) on the interval [ 0 , 1 ] , which arises in a study of the laminar flow in a porous pipe with an expanding or contracting wall. For the singular nonlinear BVP, the correct boundary conditions are derived to guarantee that its linearization has a unique smooth solution. Then a numerical technique is proposed to find all possible multiple solutions. For the suction driven pipe flow with the expanding wall (e.g. α = 2 ), we find a new solution numerically and classify it as a type VI solution. The computed results agree well with what can be obtained by the bifurcation package AUTO. In addition, we also construct asymptotic solutions for a few cases of parameters, which agree well with numerical solutions. These serve as validations of our numerical results. Thus we believe that the numerical technique designed in the paper is reliable, and may be further applied to solve a variety of nonlinear equations that arise from other flow problems. [ABSTRACT FROM AUTHOR]

This paper is concerned with asymptotic solutions of a nonlinear boundary value problem (BVP), which arises in a study of laminar flow in a uniformly porous channel with retractable walls and an applied transverse magnetic field. For different ranges of the control parameters (i.e. α , Re and M ) arising in the BVP, four cases are considered using different singular perturbation methods. For the first case, unlike those in the existing literature, we make use of the Lighthill method and successfully construct an asymptotic solution with high-order derivatives at the center of the channel. For the second case, under large suction we consider M 2 = O (1) and M 2 = O ( Re ), respectively, which will further extend the applying range of asymptotic solutions. In other cases, asymptotic solutions with a boundary layer are successfully constructed. In addition, numerical solutions presented for each case agree well with asymptotic solutions, which illustrates that the asymptotic solutions constructed in this paper are more reliable. Finally, the influences of some parameters on flow field are discussed to develop a better understanding of the flow problem. [ABSTRACT FROM AUTHOR]

Optical tweezers have been widely utilized in biological sciences and biomedical engineering because of the attractive feature of manipulating microobjects without physical contacts. The integration of optical tweezers and robotic technologies has also led to the emergence of many robot-assisted optical manipulation systems, and a variety of motion control schemes have been reported to automated the process and improve the efficiency. However, the performance of existing control methods for optical tweezers is commonly limited by several issues: 1) presence of spatially varying trapping stiffness, which is difficult to model and identify; 2) requirements of high-order state variables, which are not measurable; and 3) effectiveness of trapping is only valid locally around the center of laser beam. These issues lead to the constructions of controllers, which are computationally involved. This paper presents a simple tracking control method for optical manipulation, which enables the laser beam to automatically trap and then manipulate the cell to track a time-varying trajectory, without high-order derivatives or construction of observers. The development of the proposed controller is based on the singular perturbation approach, by treating the fast manipulator or stage dynamics as a perturbation of the slow cell dynamics, such that the lowest control complexity is achieved. The exponential stability of the overall system that consists of the fast and slow subsystems is proved by using Tikhonov's theorem. Experimental results are presented to illustrate the performance of the proposed controller. [ABSTRACT FROM AUTHOR]

This paper presents the design of an additive fault tolerant control for nonlinear time-invariant singularly perturbed systems against actuator faults based on Lyapunov redesign principle. The overall system is reduced into subsystems with fast and slow dynamic behavior using singular perturbation method. The time scale reduction is carried out when the singular perturbation parameter is set to zero, thus avoiding the numerical stiffness due to the interaction of two different dynamics. The fault tolerant controller is computed in two steps. First, a nominal composite controller is designed using the reduced subsystems. Secondly, an additive part is combined with the basic controller to overcome the fault effect. The derived ε-independent controller guarantees asymptotic stability despite the presence of actuator faults. The Lyapunov stability theory is used to prove the stability provided the singular perturbation parameter is sufficiently small. The theoretical results are simulated using a numerical application. [ABSTRACT FROM AUTHOR]

Scalable and computationally efficient model of voltage source converter (VSC) that accurately captures the dynamics is key to the modelling, analysis and control of VSC–high‐voltage direct current system. In this study, a dynamic equivalent model of VSC including the dynamics of control system is obtained by using singular perturbation method for the first time. The reduced second‐order model with simple form and good generality captures the steady‐state and transient behaviours of original full model with a high degree of accuracy. This equivalent model retains the physical meaning of the variables, accommodates different control modes and significantly reduces the computational burden compared with the original full model. Singular perturbation method itself contains some systematic procedures to be able to improve the accuracy of the dynamic behaviour of the lower‐order model. The ability of the dynamic equivalent model to accurately describe the original dynamics of the system has been verified with numerical simulations in normal operation and on fault case associated with MTDC grids. [ABSTRACT FROM AUTHOR]

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 [ABSTRACT FROM AUTHOR]

This paper presents an application of robust control theory to an automotive Air Conditioning (A/C) system. A control-oriented model built using moving-boundary method is validated against experimental data collected on a vehicle chassis dynamometer, at constant engine speeds as well as on driving cycles. Next, an H ∞ controller is synthesized by formulating an optimization problem whose solution requires appropriate weighting functions selection. Singular perturbation method is utilized to remove states associated with fast dynamics in both model and controller. Both full-order and reduced-order H ∞ controllers are verified by simulation results obtained using the nonlinear A/C system model. It is demonstrated that the designed controller is capable of tracking the reference output trajectories while rejecting disturbances introduced on the boundary conditions of the heat exchangers. Furthermore, a preliminary study is performed to reveal the opportunity of designing a gain-scheduled H ∞ controller for global output tracking. [ABSTRACT FROM AUTHOR]

The standard approach to vibrational control applies first-order averaging methods to find an open-loop periodic input that stabilizes an unstable equilibrium point. While the capability of stabilization without feedback is appealing, this formulation has drawbacks from a design perspective. An alternative design framework based on stability maps for second-order linear periodic systems is not as general, but has significant potential advantages. The averaging approach only guarantees that a solution exists; the designer must then find that solution by other means. Furthermore, the frequencies required may be too high for practical implementation. Use of stability maps makes a broader class of stabilizing inputs accessible, allowing, for example, the use of lower frequency signals. Application to nonlinear and higher-order systems is demonstrated with two examples. The first is stabilization of the classical vertically forced inverted pendulum. The second is delay of a pitchfork bifurcation in a fourth-order nonlinear system. In the second example we show that the averaging-based approach necessarily fails to delay the bifurcation, while the alternative method achieves significant extension of the stable operating region. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability of traveling front solutions for a viscous Fisher-KPP equation. By applying geometric singular perturbation method, special Evans function estimates, detailed spectral analysis and C0 semigroup theories, each traveling front solution with wave speed c < -2√√′(0) is proved to be locally exponentially stable in some appropriate exponentially weighted spaces. [ABSTRACT FROM AUTHOR]

The cases of large Reynolds number and small expansion ratio for the asymmetric laminar flow through a two-dimensional porous expanding channel are considered. The Navier-Stokes equations are reduced to a nonlinear fourth-order ordinary differential equation by introducing a time and space similar transformation. A singular perturbation method is used for the large suction Reynolds case to obtain an asymptotic solution by matching outer and inner solutions. For the case of small expansion ratios, we are able to obtain asymptotic solutions by double parameter expansion in either a small Reynolds number or a small asymmetric parameter. The asymptotic solutions indicate that the Reynolds number and expansion ratio play an important role in the flow behavior. Numerical methods are also designed to confirm the correctness of the present asymptotic solutions. [ABSTRACT FROM AUTHOR]

In this article, a novel high gain observer (HGO)-based decentralised indirect adaptive fuzzy controller is developed for a class of uncertain affine large-scale nonlinear systems. By the combination of fuzzy logic systems and an HGO, the state variables are not required to be measurable. The proposed feedback and adaptation mechanisms guarantee that each subsystem is able to adaptively compensate for interconnections and disturbances with unknown bounds. It is ascertained using a singular perturbation method that all the signals of the closed-loop large-scale system stand uniformly ultimately bounded and the tracking errors converge to tunable neighbourhoods of the origin. Simulation results of correlated double inverted pendulums substantiate the effectiveness of the proposed controller. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, the asymptotic solution for the similarity equation of the laminar flow in a porous pipe with suction at expanding and contracting wall has been obtained using the singular perturbation method. However, this solution neglects exponentially small terms in the matching process. To take into account these exponentially small terms, a method involving the inclusion of exponentially small terms in a perturbation series was used to find the two solutions analytically. The series involving the exponentially small terms and expansion ratio predicts dual solutions. Furthermore, the result indicates that the expansion ratio has much important influence on the solutions. When the expansion ratio is zero, it is a special case that Terrill has discussed. [Copyright &y& Elsevier]

This brief presents an adaptive output regulator, an add-on type output regulator with a frequency adaptation algorithm. The frequency adaptation algorithm, based on an internal model principle, is used to identify the frequency of a sinusoidal disturbance. By adding the modified adaptive algorithm to the add-on output regulator, the adaptive output regulator can reject a biased sinusoidal disturbance with unknown bias, magnitude, phase, and frequency, while preserving the merits of the output regulator and the performance of the pre-existing controller. It is shown that the proposed method is stable for frequencies within a connected compact set. The stability of the closed-loop system is analyzed by the singular perturbation method and averaging theorem. The proposed controller is applied to an optical disk drive for verifying the performance of the controller. The performance of the proposed method is validated by both simulation and experimental results. [ABSTRACT FROM PUBLISHER]

The asymptotic behavior of solutions of a similarity equation for the laminar flow in a porous channel with suction at both expanding and contracting walls has been obtained by using a singular perturbation method. However, in the matching process, this solution neglects exponentially small terms. To take into account these exponentially small terms, a method involving the inclusion of exponentially small terms in a perturbation series was used to find two of the solutions analytically. The series involving the exponentially small terms and expansion ratio predicts dual solutions. Furthermore, the result indicates that the expansion ratio has much important influence on the solutions. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the stability of traveling front solutions for 2×2 quasi-linear relaxation systems with small diffusion rate. By applying geometric singular perturbation method, special Evans function estimates, detailed spectral analysis and semigroup theories, we prove that all the non-degenerate waves for semi-linear relaxation systems are locally exponentially stable in some exponentially weighted spaces. We also obtain the linear exponential stability of the non-degenerate waves for quasi-linear relaxation systems, where the wave strengths can be large. [Copyright &y& Elsevier]

Abstract: In this paper, a novel control strategy for a nonlinear boost inverter is proposed. The idea is based on generating an autonomous oscillator that does not need an external reference signal. This aim is achieved by using energy-shaping methodology with a suitable Hamiltonian function which defines the desired system behavior. A phase controller is added to the control law in order to achieve 180°-synchronization between both parts of the circuit as well as synchronize the voltage output with a pre-specified signal, e.g. synchronization with the electrical grid. An adaptive control is designed for dealing with the common problem of unknown load. In order to analyze the stability of the full system, singular perturbation approach is used. The resulting control is tested by means of simulations. [ABSTRACT FROM AUTHOR]

In this paper, we consider robust control of nonlinear systems, via inclusion nonlinear systems solution and H∞ controller using singular perturbation method. First, using technique for solving inclusion nonlinear systems, we transform the nonlinear system to ordinary nonlinear system. Then using normal form equations, we eliminate the nonlinear part of the system matrix of equations of system and transform it to a linear diagonal form. Separating new equations to slow and fast subsystems, due to the singular perturbation method and with the assumption of normboundedness of the fast dynamics, we can treat them as disturbance and design H∞ controller for a system with a lower order than the original one that stabilizes the overall closed loop system. The proposed method is applied to a nominal system. [ABSTRACT FROM AUTHOR]

Abstract: Primal-dual power control algorithms in optical networks must be designed properly to remain stable in the presence of time-delays. Control algorithms at the signal transmitters continuously update their channel powers in response to dynamic pricing information from the network links. The control algorithms optimize the optical signal-to-noise ratios (OSNRs) of the channels in a distributive and non-cooperative manner. We consider the case of a single link with multiple channels and a constant time-delay. These results also apply to single sink general network configurations if the time-delay represents the longest round-trip time in the network. We derive sufficient conditions for the stability of the closed loop system based on a tunable parameter in the control algorithm. The work combines singular perturbation theory with Lyapunov–Razumikhin time-delay techniques. We verify our results through simulations based on realistic network parameters. [Copyright &y& Elsevier]

Car following model is one of microscopic models for describing traffic flow. Through linear stability analysis, the neutral stability lines and the critical points are obtained for the different types of car following models and two modified models. The singular perturbation method has been used to derive various nonlinear wave equations, such as the Korteweg-de-Vries (KdV) equation and the modified Korteweg-de-Vries (mKdV) equation, which could describe different density waves occurring in traffic flows under certain conditions. These density waves are mainly employed to depict the formation of traffic jams in the congested traffic flow. The general soliton solutions are given for the different types of car following models, and the results have been used to the modified models efficiently. [ABSTRACT FROM AUTHOR]