Singular Perturbation Research Articles

Adaptive Multi-Input Super Twisting Control for a Quadrotor: Singular Perturbation Approach

Singular perturbations easily result in the problem of high gain. Instead of high gain feedbacks, this work proposes an adaptive two-time-scale (TTS) super twisting control (STWC) algorithm for the attitude tracking of a singularly perturbed quadrotor system (SPQS). The bounds of uncertainties are assumed to be perturbed around their nominal values. First, a nominal STWC with slow and fast control gains is designed to achieve the finite-time attitude tracking for the nominal SPQS. The finite-time stability of the close-loop system is proven by utilizing the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -dependent Lyapunov method. Then, dynamically adapted slow and fast control gains initialized at their nominal values are used for the accurate finite-time convergence of tracking errors to the region around the origin, which avoids overestimating the values of control gains. Experimental results demonstrate the performance of the proposed controller in terms of attitude stabilization and tracking, and robustness to external disturbances.

Dynamic Modeling and Control Design Based on Singular Perturbation Theory for High-Order Wireless Power Transfer System

Dynamic Modeling and Control Design Based on Singular Perturbation Theory for High-Order Wireless Power Transfer System

Lowest-degree robust finite element schemes for inhomogeneous bi-Laplace problems

In this paper, we study the numerical method for the bi-Laplace problems with inhomogeneous coefficients; particularly, we propose finite element schemes on rectangular grids respectively for an inhomogeneous fourth-order elliptic singular perturbation problem and for the Helmholtz transmission eigenvalue problem. The new methods use the reduced rectangle Morley (RRM for short) element space with piecewise quadratic polynomials, which are of the lowest degree possible. For the finite element space, a discrete analogue of an equality by Grisvard is proved for the stability issue and a locally-averaged interpolation operator is constructed for the approximation issue. Optimal convergence rates of the schemes are proved, and numerical experiments are given to verify the theoretical analysis.

Existence of a periodic solution for a state-dependent delay differential equation with a singular perturbation

The idea of this article came from the works dones by Khung and Smith on the one hand in [2] for and Arino and Al in [1] for on the other hand, to proves the existence of a slowly oxillating preriodic solution by the fixed point theorem of Picard. I proposed the following question: Do the result remain valid if 1 ? by the same method with using the homotopy well defined link between the application of Khung and Smith and the application of Arino and and indeed, I came to prove existence of a slowly oxillating preriodic solution by the fixed point theorem of Picard, therefore the result obtained in this work is generalization of two special cases. A State-dependent Delay Differential equation play important role in recent years to modeling of several phenomenon, in dynamics of populations, in biology and even in the social life for example: When we take our driving test, we learn that the reaction time of our nervous system when driving is of the order of a few seconds, and that we must take care to maintain a sufficient distance between two cars following each other, from this example, we notice that the delay can be useful. Although this work has added a new addition to the scientific field which is a generalization and linking between two special cases, it remains somewhat limited because it takes special case of Arino and Al. Therefore in the futur, we are thinking about a new work that takes into account the general case of Arino and .

Dynamics of traveling waves for predator-prey systems with Allee effect and time delay

This article aims to establish the existence of traveling waves for a predator-prey system with Beddington-DeAngelis functional response, reproductive Allee effect, and time delay. We investigate the existence of solutions for a system with two special delay kernels by geometric singular perturbation theory, invariant manifold theory, and Fredholm orthogonality theory. In addition, we discuss the asymptotic behaviors of traveling waves with the aid of the asymptotic theory. For more information see https://ejde.math.txstate.edu/Volumes/2024/33/abstr.html

Adaptive Super-Twisting Sliding Mode Control for Robot Manipulators with Input Saturation.

The paper investigates a modified adaptive super-twisting sliding mode control (ASTSMC) for robotic manipulators with input saturation. To avoid singular perturbation while increasing the convergence rate, a modified sliding mode surface (SMS) is developed in this method. Using the proposed SMS, an ASTSMC is developed for robot manipulators, which not only achieves strong robustness but also ensures finite-time convergence. The boundary of lumped uncertainties cannot be easily obtained. A modified adaptive law is developed such that the boundaries of time-varying disturbance and its derivative are not required. Considering input saturation in practical cases, an ASTSMC with saturation compensation is proposed to reduce the effect of input saturation on tracking performances of robot manipulators. The finite-time convergence of the proposed scheme is analyzed. Through comparative simulations against two other sliding mode control schemes, the proposed method has been validated to possess strong adaptability, effectively adjusting control gains; simultaneously, it demonstrates robustness against disturbances and uncertainties.

Bifurcation of the travelling wave solutions in a perturbed (1 + 1)-dimensional dispersive long wave equation via a geometric approach

Choosing ${\kappa }$ (horizontal ordinate of the saddle point associated to the homoclinic orbit) as bifurcation parameter, bifurcations of the travelling wave solutions is studied in a perturbed $(1 + 1)$ -dimensional dispersive long wave equation. The solitary wave solution exists at a suitable wave speed $c$ for the bifurcation parameter ${\kappa }\in \left (0,1-\frac {\sqrt 3}{3}\right )\cup \left (1+\frac {\sqrt 3}{3},2\right )$ , while the kink and anti-kink wave solutions exist at a unique wave speed $c^*=\sqrt {15}/3$ for $\kappa =0$ or $\kappa =2$ . The methods are based on the geometric singular perturbation (GSP, for short) approach, Melnikov method and invariant manifolds theory. Interestingly, not only the explicit analytical expression of the complicated homoclinic Melnikov integral is directly obtained for the perturbed long wave equation, but also the explicit analytical expression of the limit wave speed is directly given. Numerical simulations are utilized to verify our mathematical results.

A Singular Tempered Sub-Diffusion Fractional Equation with Changing-Sign Perturbation

In this paper, we establish some new results on the existence of positive solutions for a singular tempered sub-diffusion fractional equation involving a changing-sign perturbation and a lower-order sub-diffusion term of the unknown function. By employing multiple transformations, we transform the changing-sign singular perturbation problem to a positive problem, then establish some sufficient conditions for the existence of positive solutions of the problem. The asymptotic properties of solutions are also derived. In deriving the results, we only require that the singular perturbation term satisfies the Carathéodory condition, which means that the disturbance influence is significant and may even achieve negative infinity near some time singular points.

Modeling of Chemical Tracers for Two-Phase Flow in Advective-Dominated Porous Media at Core Scale

Summary Chemical tracer modeling in porous media plays a key role in subsurface applications including oil recovery, aquifer remediation, and geothermal energy production. In oil reservoirs, chemical tracers are critical to quantifying the remaining oil saturation in porous media after displacing processes, enabling the correct evaluation of the sweep efficiency of recovery methods at the field scale. Even though the transport of solutes under single-phase flow has been modeled extensively with numerous solutions, there are no existing mathematical approaches to examine the displacement of solutes in two-phase flow conditions. Therefore, we present in this research work the first analytical solutions derived to model the transport of ideal and partitioning tracers in porous media with mobile water and oil phases. The models presented are derived from the classic study of fluid displacement by viscous forces and the analysis of dynamic phase distribution in porous media, where key transformation variables are introduced to simplify the nonlinear advection-dispersion equation (ADE) into a conventional partial differential expression. In our derivation process, it is recognized that the dispersion effect can be superimposed onto an ideal concentration front via a singular perturbation expansion, resulting in practical solutions that do not require complex numerical calculations or inversion methods. The solutions derived are verified with numerical simulations and validated with experimental data under different flow conditions for the transport of ideal and partitioning tracers, demonstrating that the complex mechanisms of hydrodynamic dispersion, partitioning, and adsorption are accurately modeled under two-phase flow. Thus, our solutions can be used to rapidly evaluate tracer transport under the existing flow conditions in porous media, significantly reducing the number of experiments and simulations to characterize and select the correct tracer to be used in field applications.

Universal slow dynamics of chemical reaction networks.

Understanding the emergent behavior of chemical reaction networks (CRNs) is a fundamental aspect of biology and its origin from inanimate matter. A closed CRN monotonically tends to thermal equilibrium, but when it is opened to external reservoirs, a range of behaviors is possible, including transition to a new equilibrium state, a nonequilibrium state, or indefinite growth. This study shows that slowly driven CRNs are governed by the conserved quantities of the closed system, which are generally far fewer in number than the species. Considering both deterministic and stochastic dynamics, a universal slow-dynamics equationis derived with singular perturbation methods and is shown to be thermodynamically consistent. The slow dynamics is highly robust against microscopic details of the network, which may be unknown in practical situations. In particular, nonequilibrium states of realistic large CRNs can be sought without knowledge of bulk reaction rates. The framework is successfully tested against a suite of networks of increasing complexity and argued to be relevant in the treatment of open CRNs as chemical machines.

A Novel Numerical Scheme for a Class of Singularly Perturbed Differential-Difference Equations with a Fixed Large Delay

A trigonometric spline based computational technique is suggested for the numerical solution of layer behavior differential-difference equations with a fixed large delay. The continuity of the first order derivative of the trigonometric spline at the interior mesh point is used to develop the system of difference equations. With the help of singular perturbation theory, a fitting parameter is inserted into the difference scheme to minimize the error in the solution. The method is examined for convergence. We have also discussed the impact of shift or delay on the boundary layer. The maximum absolute errors in comparison to other approaches in the literature are tallied, and layer behavior is displayed in graphs, to demonstrate the feasibility of the suggested numerical method.

Singular perturbation boundary and interior layers problems with multiple turning points

In the study of singularly perturbed boundary problems with turning points, the solution undergoes sharp changes near these points and exhibits various interior phenomena. We employ the matching asymptotic expansion method to analyze and solve a singularly perturbed boundary and interior layers problem with multiple turning points, resulting in a composite expansion that fits well with the numerical solution. The solution demonstrates a strong association with special functions, which is verified by the theory of differential inequalities.

Competition for resources in an exclusion model with biased lane-changing mechanism.

The motivation for the proposed work is drawn from the attachment-detachment observed in biological and physical transport processes that entail finite resources. We investigate the influence of limited particle availability on particle dynamics within two parallel totally asymmetric simple exclusion lanes, with one lane incorporating only particle detachment and the other considering particle attachment. We establish a theoretical framework by employing vertical mean-field theory in conjunction with singular perturbation technique. The analytical findings are supported by numerical and stochastic validation using a finite-difference scheme and the Gillespie algorithm. By utilizing these approaches, we scrutinize various stationary properties, including particle densities, phase boundaries, and particle currents for both lanes. Our analysis reveals that the complexity of the phase diagram exhibits a nonmonotonic trend in the number of stationary phases as the particle count increases. Each phase diagram is constructed with respect to the intrinsic boundary parameters, illustrating both bulk and surface transitions occurring within the lanes. The interplay between finite resources and coupling mechanisms gives rise to two phases involving upward shock in one of the lanes, while two phases exhibit synchronized downward shock in both lanes. Finally, we delve into shock dynamics to comprehend critical phase transitions occurring in the system.

Existence of Kink and Antikink Wave Solutions of Singularly Perturbed Modified Gardner Equation

In this paper, the singularly perturbed modified Gardner equation is considered. Firstly, for the unperturbed equation, under certain parameter conditions, we obtain the exact expressions of kink wave solution and antikink wave solution by using the bifurcation method of dynamical systems. Then, the persistence of the kink and antikink wave solutions of the perturbed modified Gardner equation is studied by exploiting the geometric singular perturbation theory and the Melnikov function method. When the perturbation parameter is sufficiently small, we obtain the sufficient conditions to guarantee the existence of kink and antikink wave solutions.

Periodic wave solutions for a KP-MEW equation under delay perturbation

In this paper we investigate the uniqueness of periodic wave solutions for a 4-parametric perturbed KP-MEW equation with local strong delay convolution kernel. Applying the geometric singular perturbation theory, a locally invariant manifold is established in a small neighborhood of the critical manifold to transform the singular perturbed system into a regular one. We prove the monotonicity of the ratio of two Abelian integrals and give all conditions for the uniqueness of periodic wave solutions. Moreover, we find that the amplitude and wavelength of this periodic wave change monotonously following monotonous variation of some parameters included in the perturbed KP-MEW equation as shown as in numerical simulations.

Modeling, Guidance, and Robust Cooperative Control of Two Quadrotors Carrying a “Y”-Shaped-Cable-Suspended Payload

This paper investigates the problem of cooperative payload delivery by two quadrotors with a novel “Y”-shaped cable that improves payload carrying and dropping efficiency. Compared with the existing “V”-shaped suspension, the proposed suspension method adds another payload swing degree of freedom to the quadrotor–payload system, making the modeling and control of such a system more challenging. In the modeling, the payload swing motion is decomposed into a forward–backward process and a lateral process, and the swing motion is then transmitted to the dynamics of the two quadrotors by converting it into disturbance cable pulling forces. A novel guidance and control framework is proposed, where a guidance law is designed to not only achieve formation transformation but also generate a local reference for the quadrotor, which does not have access to the global reference, based on which a cooperative controller is developed by incorporating an uncertainty and disturbance estimator to actively compensate for payload swing disturbance to achieve the desired formation trajectory tracking performance. A singular perturbation theory-based analysis shows that the proposed parameter mapping method, which unifies the parameter tuning of different control channels, allows us to tune a single parameter, ε, to quantitatively enhance both the formation control performance and system robustness. Simulation results verify the effectiveness of the proposed approach in different scenarios.

Weak convergence and stability of functional diffusion systems with singularly perturbed regime switching

This paper focuses on a class of functional diffusion systems with singularly perturbed regime switching, where the modulating Markov chain has a large state space and undergoes weak and strong interactions. By using the martingale method and weak convergence, this paper shows that the underlying system will weakly converge to a limit system, which is simpler than the original system. For a class of integro-differential diffusion system with singularly perturbed regime switching, as a class of special functional diffusion system, this paper demonstrates that if the limit system is moment exponentially stable, the original system with singular perturbation is also moment exponentially stable under suitable conditions. This result is interesting since the limit system is always simpler. Finally, an example is given to illustrate this result.

Singular Perturbations and Large Time Delays Through Accelerated Spline-Based Compression Technique

In the quest to solve the singularly perturbed delay differential equations (SPDDEs) involving large delay with integral boundary condition, the cubic spline in compression technique is explored for the study of dynamical systems to capture complex temporal phenomena in a wide range of scientific disciplines. The integral boundary condition is handled using Simpson's 1/3 rule and the scheme's applicability is validated by numerically experimenting with some problems at different values of mesh size and perturbation parameter. Numerical data are tabulated to show that the suggested approach is more accurate and is an improvement over the methods used in the literature. The insights gained from this research paper provide a foundation for further exploration and utilization of SPDDEs in understanding and predicting the behavior of complex systems across diverse scientific domains.

Improved Sliding Mode Variable Structure Control of Robot Manipulators with Flexible Joints based on Singular Perturbation

This paper discusses the kinematics analysis method of six link mechanism based on virtual prototype technology and Adams. Firstly, the kinematics mathematical model of the six-bar linkage is deduced. Then, based on the design and development process of virtual prototype technology, the virtual prototype model of the six-bar press is established in ADAMS software. Finally, the kinematics simulation analysis is carried out for the model. The proposed method provides a reference for the parametric and optimal design analysis of the multi bar linkage of the mechanical press.

Regularization of the Boundary Equilibrium Bifurcation in Filippov System with Rich Discontinuity Boundaries

This paper studies a particular type of planar Filippov system that consists of two discontinuity boundaries separating the phase plane into three disjoint regions with different dynamics. This type of system has wide applications in various subjects. As an illustration, a plant disease model and an avian-only model are presented, and their bifurcation scenarios are investigated. By means of the regularization approach, the blowing up method, and the singular perturbation theory, we provide a different way to analyze the dynamics of this type of Filippov system. In particular, the boundary equilibrium bifurcations of such systems are studied. As a consequence, the nonsmooth fold bifurcation becomes a saddle-node bifurcation, while the persistence bifurcation disappears after regularization.