Singular Perturbation Technique Research Articles

Output-feedback path-following control of underactuated AUVs via singular perturbation and interconnected-system technique

This paper focuses on the output-feedback control for path-following of underactuated autonomous underwater vehicles subject to multiple uncertainties and unmeasured velocities. First, a novel extended state observer is proposed to estimate the mismatched lumped disturbance and recover the unmeasured velocities. Based on this premise, to overcome the limitation of relying solely on the accurate kinematic model, a disturbance observer-based stabilizing controller is developed. The difference in bandwidths between the observer and the vehicle dynamics allows for a mathematical setup amenable to standard singular perturbation theory. In the fast mode, a kinematic observer is designed to reject system uncertainty caused by unknown attack angular velocity and prohibitive path-tangential angular velocity, using a novel physical perspective. In the slow mode, an interconnected-system control law is proposed by integrating the backstepping technique with the time scale decomposition method. Furthermore, the stability of the overall closed-loop system is established. Finally, simulation results are presented to demonstrate the effectiveness of the proposed method for path-following of underactuated autonomous underwater vehicles in the vertical plane.

Identification and Control of Flexible Joint Robots Based on a Composite-Learning Optimal Bounded Ellipsoid Algorithm and Prescribe Performance Control Technique

This paper presents an indirect adaptive neural network (NN) control algorithm tailored for flexible joint robots (FJRs), aimed at achieving desired transient and steady-state performance. To simplify the controller design process, the original higher-order system is decomposed into two lower-order subsystems using the singular perturbation technique (SPT). NNs are then employed to reconstruct the aggregated uncertainties. An adaptive prescribed performance control (PPC) strategy and a continuous terminal sliding mode control strategy are introduced for the reduced slow subsystem and fast subsystem, respectively, to guarantee a specified convergence speed and steady-state accuracy for the closed-loop system. Additionally, a composite-learning optimal bounded ellipsoid algorithm (OBE)-based identification scheme is proposed to update the NN weights, where the tracking errors of the reduced slow and fast subsystems are integrated into the learning algorithm to enhance the identification and tracking performance. The stability of the closed-loop system is rigorously established using the Lyapunov approach. Simulations demonstrate the effectiveness of the proposed identification and control schemes.

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.

Transient synchronization stability of photovoltaics integration by singular perturbation analysis

The integration of large-scale photovoltaics (PVs) into the power grid has significantly altered the transient synchronization dynamics of traditional power systems dominated by synchronous generators (SGs) and posed great challenges to modeling and analysis of PVs integration. In this paper, the transient synchronization stability of the PV-SG system is studied using the singular perturbation technique. Firstly, a nonlinear model of a PV-SG system is established to reveal the multiscale transient synchronization characteristics. Further, the full system is decomposed into a slow subsystem and a fast subsystem by the singular perturbation technique. The fast subsystem containing the dynamics of the DC voltage control, terminal voltage control, and phase-locked loop, and the slow subsystem containing the dynamics of rotor motion can perfectly reflect the dynamics of the full system within the electromagnetic and electromechanical timescales, respectively. The proposed model provides a clearer physical picture of dynamics in the PV-SG system within the electromagnetic and electromechanical timescales. Subsequently, the stability of the slow and fast subsystems is investigated using the energy function and eigenvalue analysis methods, respectively. Meanwhile, the impacts of various operating, control, and structural parameters on the transient synchronization stability are uncovered. Different from the most existing research endeavors on the wide simulations of the PVs integration, the impact of PVs on the synchronization dynamics of SGs without considering the dynamical characteristics of the PV system, and the transient synchronization stability analyses of the PLL-based voltage source converter systems, it is the key contribution to study the transient synchronization dynamical characteristics of the PV system and its interaction with the SG under different timescales. All these are helpful and easy to extend to more complicated PV-SG systems. Finally, the analysis results are validated by extensive simulations.

Localized pattern formation: semi-strong interaction asymptotic analysis for three components model

We investigate a three-component system involving the Belousov–Zhabotinsky reaction in water-in-oil microemulsions. Our goal is to investigate the connection between homoclinic snaking and semi-strength interaction in a three-variable reaction–diffusion system. A two-parameter bifurcation diagram of homogeneous, periodic and localized patterns is obtained numerically, and a natural asymptotic scaling for semi-strong interaction theory is found where an activator source term a = O ( δ 1 ) and b = O ( δ 1 ) , with δ 1 ≪ 1 being the diffusion ratio. Under this regime, singular perturbation techniques are used to construct localized steady-state patterns, and new types of non-local eigenvalue problems (NLEP) are derived to determine the stability of these patterns to O ( 1 ) time-scale instabilities. We extend the scope of the NLEP by considering a general scenario where both time-scaling parameters are non-zero. All analytical results are found to agree with numerics. Further numerical results are presented on the location of various types of breathing Hopf instability for localized patterns.

Modelling the Effect of the Exposure to Perfluorooctanoate (PFOA) and Perfluorooctane Sulfonate (PFOS) on Thyroid Homeostasis

Thyroid homeostasis is crucial for the human body. The imbalance of thyroid homeostasis might cause diseases such as hypothyroidism. Humans are exposed to PFOS/PFOA frequently since they have been used in various industrial products. As reported that PFOS/PFOA increase the metabolic clearance rate of thyroid hormones, we then develop a mathematical model in terms of a system of differential equations to investigate thyroid homeostasis based on the changes in the levels of thyrotropin-releasing hormone, thyroid-stimulating hormone and thyroid hormones when the effect of the exposure to PFOS and PFOA is also incorporated as well. The geometric singular perturbation technique is then employed to identify the possible dynamic behaviours obtained from the model. Numerical investigations are also presented to illustrate the results from theoretical analysis. Both theoretical and numerical results imply that a periodic behaviour that has been observed clinically in the pulsatile secretions of thyroid hormones, thyroid-stimulating hormone and thyrotropin-releasing hormone could be obtained from our model. In addition, the numerical experiment also shows that the levels of thyroid hormones and thyroid-stimulating hormone for the case when there is the effect of exposure to PFOS and PROA are lower than those of the case when there is no effect of the exposure to PFOS and PFOA which might lead to the imbalance of thyroid homeostasis.

Short note on a solution with large amplitude for the limiting system arising from the competition-diffusion system

This paper is concerned with positive solutions of the limiting system arising from the Shigesada–Kawasaki–Teramoto model with large interspecific competition rate. It has previously been suggested that the limiting system has positive solutions with large amplitude for a certain value of parameters. As a first step, the purpose of this paper is to discuss the precise spatial profile of such solutions by employing formal calculations by the singular perturbation technique.

Formation Control for Multiple Quadrotors Under DoS Attacks via Singular Perturbation

This paper looks into the topic of distributed formation control for a group of quadrotors that are vulnerable to denial-of-service (DoS) attacks. By preventing information interaction between adjacent quadrotors, the DoS attacks, which happen independently on individual communication channels, disrupt the formation mission. A decomposition-combination control framework is devised for the formation task. The corresponding high-order dynamic system is partitioned into slow and fast subsystems using the singular perturbation technique, which lessens the computing load and resolves the under-driven issue. A composition formation control scheme is then put forward, which includes a resilient slow-scale controller to tackle DoS attacks along with an optimal fast-scale controller to assure flying performance and stability. It should be noted that position and velocity information, which can be acquired and processed entirely by sub-controller in the proposed method, is the main content of communication between quadrotors, permitting more effective system execution. Finally, a slow-fast composition mechanism is provided, and simulation results of a system containing four quadrotors serve as validation for the suggested methods.

Existence results for a class of local and nonlocal nonlinear elliptic problems

In this paper, we study the $p$-Laplacian problems in the case where $p$ depends on the solution itself. We consider two situations, when $p$ is a local and nonlocal quantity. By using a singular perturbation technique, we prove the existence of weak solutions for the problem associated to the following equation $$ \begin{cases} -\mathrm{d}\mathrm{i}\mathrm{v}(|\nabla u|^{p(u)-2}\nabla u)+|u|^{p(u)-2}u=f & \mbox{in}\; \Omega\\ u=0& \mbox{on}\; \partial\Omega , \end{cases}$$ and also for its nonlocal version. The main goal of this paper is to extend the results established by M. Chipot and H. B. de Oliveira (Math. Ann., 2019, 375, 283-306).

Distributed extremum‐seeking based resource allocation algorithm with input dead‐zone

SummaryThis paper studies distributed resource allocation problem for agents with input dead‐zone, which is not considered in the existing work. At first, the primal problem is transformed to an auxiliary problem by using the exact penalty method to deal with local inequality constraints. It is assumed that the explicit expressions of cost functions and local inequality constraints are unknown to agents but the values of the cost and constraint functions can be obtained. Under such a setup, the extremum seeking control is used to estimate the gradient information. Thus, to obtain the optimal allocation, a novel distributed algorithm is designed by the virtue of the extremum seeking control and a dynamic compensating mechanism which is used to handle the effects of the input dead‐zone. Due to a two time‐scale structure of the designed distributed algorithm, the semi‐globally practically asymptotical convergence of all agents' decisions to the optimal allocation is obtained by the singular perturbation technique. Finally, numerical examples of economic dispatch in smart grids are given to verify the effectiveness of our proposed method.

Advanced sensorless nonlinear control strategy for grid connected photovoltaic systems via the two-time scale singular perturbation technique

This paper presents a new control strategy for grid connected photovoltaic (PV) system. The system consists of a solar panel, a boost converter, and a full-bridge inverter with L-filter connected to the grid. The control objectives are threefold: 1) tracking the maximum power point of the PV panel; 2) balancing the power exchange by regulating the DC link voltage; 3) ensuring a unity power factor in the grid. To achieve these objectives, a multi-loop controller is synthesised using a high-gain output feedback controller and the singular perturbation technique. The addressed control problem involves several difficulties including the existence of a number of state variables that are inaccessible to measurement. To address this problem, two-time-scale sliding mode observers are added which allow estimating, simultaneously, the solar panel current and the grid voltage. It is formally shown that all control objectives are asymptotically accomplished. This theoretical result is confirmed by numerical simulation tests.

Nash Equilibrium Seeking for Incomplete Cluster Game in the Cooperation–Competition Network

In this article, we investigate the problem of seeking Nash equilibrium (NE) in multiagent systems within cooperation–competition networks. Each agent aims to optimize a total cost function that accounts for its own interests as well as those of its cooperators, considering both cooperative and noncooperative interactions with other agents. It is worth noting that, unlike in existing N-coalition games, the agents in this study only have knowledge of whether they are cooperative or noncooperative with their neighboring agents; they do not have information about non-neighboring agents who might be cooperators. As a result, due to potential disconnections in the communication topology within the cluster, it is not possible to consider the entire cluster as a virtual player to optimize its objective functions. To address this issue, we developed an algorithm using the singular perturbation technique, which divides the system into two distinct timescales. We propose a novel estimation algorithm to estimate the total cost function of disconnected subnetworks within the fast system. In the slow system, the search for NE is based on a gradient algorithm, while the Lyapunov stability theory is utilized to analyze the convergence of the algorithms. Furthermore, we extend the problem to accommodate scenarios where multiple subnetworks exist within the network. Numerical simulations are conducted to demonstrate their capability for resolving the noncooperative game problem in cooperation–competition networks.

Advanced sensorless nonlinear control strategy for grid-connected photovoltaic systems via the two-time scale singular perturbation technique

Advanced sensorless nonlinear control strategy for grid-connected photovoltaic systems via the two-time scale singular perturbation technique

Robust Feedback Linearization Control Design for Five-Link Human Biped Robot with Multi-Performances

The study first proposes the difficult nonlinear convergent radius and convergent rate formulas and the complete derivations of a mathematical model for the nonlinear five-link human biped robot (FLHBR) system which has been a challenge for engineers in recent decades. The proposed theorem simultaneously has very distinctive superior advantages including the stringent almost disturbance decoupling feature that addresses the major deficiencies of the traditional singular perturbation approach without annoying “complete” conditions for the discriminant function and the global exponential stability feature without solving the impractical Hamilton–Jacobi equation for the traditional H-infinity technique. This article applies the feedback linearization technique to globally stabilize the FLHBR system that greatly improved those shortcomings of nonlinear function approximator and make the effective working range be global for whole state space, whereas the traditional Jacobian linearization technique is valid only for areas near the equilibrium point. In order to make some comparisons with traditional approaches, first example of the representative ones, that cannot be addressed well for the pioneer paper, is shown to demonstrate the fact that the effectiveness of the proposed main theorem is better than the traditional singular perturbation technique. Finally, we execute a second simulation example to compare the proposed approach with the traditional PID approach. The simulation results show that the transient behaviors of the proposed approach including the peak time, the rise time, the settling time and the maximum overshoot specifications are better than the traditional PID approach.

Distributed Nash Equilibrium Seeking Under Input Dead Zone

This article investigates the Nash equilibrium (NE) seeking problem for games in the presence of input dead-zone nonlinearity. The purpose of each player is to minimize its own cost function, which depends not only on its own decision variable, but also on decision variables of the other players. To obtain an NE, a novel two-time-scale distributed algorithm is designed. This new strategy consists of two parts: one is the fast compensating dynamics, which can rapidly compensate the effect of input dead zone; the other is the optimization dynamics that can distributively drive the players’ decision variables to achieve the NE. Since the proposed algorithm has a two-time-scale structure, singular perturbation techniques are used to prove that the players’ actions semiglobally practically asymptotically converge to the NE. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed strategy.

Criteria for the (in)stability of planar interfaces in singularly perturbed 2-component reaction–diffusion equations

We consider a class of singularly perturbed 2-component reaction–diffusion equations which admit bistable traveling front solutions, manifesting as sharp, slow–fast–slow, interfaces between stable homogeneous rest states. In many example systems, such as models of desertification fronts in dryland ecosystems, such fronts can exhibit an instability by which the interface destabilizes into fingering patterns. Motivated by the appearance of such patterns, we propose two versions of a 2D stability criterion for (transversal) long wavelength perturbations along the interface of these traveling slow–fast–slow fronts. The fronts are constructed using geometric singular perturbation techniques by connecting slow orbits on two distinct normally hyperbolic slow manifolds through a heteroclinic orbit in the fast problem. The associated stability criteria are expressed in terms of the nonlinearities of the system and the slow–fast–slow structure of the fronts. We illustrate and further elaborate on the general set-up by explicitly working out the existence and transversal (in)stability of traveling fronts in a number of example systems/models. We analytically establish the instability of invading bare soil/vegetation interfaces against transversal long wavelength perturbations in several dryland ecosystem models and numerically recover fingering vegetation patterns counter-invading an invading desertification front.

Geometric Singular Perturbation Analysis of a Multiple Time-scale Model for Diabetes and COVID-19 Comorbidity

More and more information on mortality and morbidity indicates that in order to fight the COVID-19 pandemic, it is important to focus our attention on comorbidities. Several reports evidence of how many elderly patients who become severely ill exhibit underlying illness such as cardiovascular disease, kidney disease, tumor, and more to our special attention here, type 2 diabetes. Better understanding of the mechanism underlying the comorbidity between different diseases requires merging models of systems across different time-scales. The model homogenization across multiple spatial and time scales poses an important challenge to researchers in the field of medical science. An approach that has been found relatively efficient in the analysis of such models is the use of singular perturbation technique. Here, we study a differential equation model system with multiple time scales which describes the diabetes and COVID-19 comorbidity. It tracks the changes in levels of plasma glucose, insulin, and functional-cells, incorporating insulin resistance and inflammation responses. The model is analyzed with the geometric singular perturbation technique, by which conditions on the system parameters may be derived to identify regions in which the system exhibits different dynamic behavior, whether the system would be stable, or eventually oscillate in a sustained fashion. Discussion of these conditions allows us to better understand how comorbidity mediates the development of life-threatening symptoms in a diabetic patient in order that proper care and treatment may be prescribed.

Bifurcation Analysis of Multiscale Malaria Model with Serratia AS1 Bacteria and Saturated Treatment

Recent experimental results have implied that Serratia AS1 bacteria efficiently suppress the development of P. falciparum, and rapidly disseminate via vertical transmission and sexual transmission throughout mosquito population. In this paper, we propose a new malaria model to investigate the roles of saturated treatment and AS1 bacteria in malaria transmission. One feature of the model is that human and mosquito populations dynamics run on disparate temporal scales. Singular perturbation techniques are used for the separation of fast and slow dynamics. It also enables us to overcome the difficulties of exploring bifurcations in higher dimensional system. We demonstrate that the reduced systems may undergo backward bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation of codimension 2, which indicates that the delayed effect for treatment has important implications for malaria control. Numerical simulations substantiate these analytical results and show epidemiological insights. Finally, the model without AS1 bacteria transmission is applied to simulate the accumulative malaria cases in Burundi. We observe that malaria may not be eliminated by a single control in Burundi. A variety of available control measures should be adopted to eradicate malaria, such as improving treatment rate, reducing the delayed effect for treatment, and possibly releasing mosquitoes that carry AS1 bacteria.

Indirect adaptive control of multi-input-multi-output nonlinear singularly perturbed systems with model uncertainties

In this paper, two indirect adaptive control schemes for a class of multi-input-multi-output nonlinear singularly perturbed systems with partially unknown models and parameters are presented. Firstly, the original system dynamic equation is reformulated into a new form with identity control gain matrices, and a new multi-time-scale singularity-free neural network is employed to represent the new dynamic equation. Subsequently, an online identification scheme is proposed to update neural network weights where a set of auxiliary weight error vectors are used such that better convergence property can be achieved. Based on identification results, a singularity-free singular perturbation controller is developed to control the unknown nonlinear system. By using the singularity-free neural network and singular perturbation technique, the complexity in controller design for a singularly perturbed system is reduced, and the potential singularity problem is avoided. Moreover, a singularity-free dynamic surface control scheme is also proposed, and the “explosion of complexity” issue is relieved. Compared to conventional direct adaptive dynamic surface control schemes which use gradient-like updating laws and tracking errors to train the neural networks, the singularity-free dynamic surface controller is designed indirectly and the neural networks are updated using the identification errors. Therefore, better identification and control performance is achieved, and the potential singularity problem is also circumvented. The stability of the closed-loop system is rigorously proved via the Lyapunov approach, and the effectiveness of proposed identification and control schemes is demonstrated by simulations.

Thermo-mechanical postbuckling analysis of sandwich cylindrical panels with functionally graded auxetic GRMMC core supported by elastic foundations

Modelling and analysis of compressive and thermal postbuckling of sandwich cylindrical panels supported by an elastic foundation are presented in this paper. The sandwich cylindrical panel is composed of auxetic graphene-reinforced metal matrix composite (GRMMC) core with negative Poisson's ratio (NPR) and two metal face sheets. Two cases of the compressive postbuckling of auxetic sandwich cylindrical panels under axial compression in thermal environments and the thermal postbuckling of auxetic sandwich cylindrical panels caused by uniform temperature rise are considered. In the GRMMC core, the volume fraction of graphene through the thickness domain is arranged as a piece-wise functionally graded (FG) pattern. The material properties of both metal face sheets and the GRMMC core are assumed to be temperature dependent. The governing equations of the auxetic sandwich cylindrical panels are formulated based on the framework of the Reddy's third order shear deformation theory. The von Kármán-type kinematic nonlinearity, the thermal effect, the panel-foundation interaction and the initial geometric imperfection of the panel are also taken into account. The postbuckling solutions are obtained by employing a singular perturbation technique along with a two-step perturbation approach. The effects of the FG pattern, the face sheet-to-core-to-face sheet thickness ratio, and foundation stiffness on the postbuckling behavior of sandwich cylindrical panels with FG auxetic GRMMC core are studied in detail.