Singular Perturbation Approach Research Articles

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.

Synchronization Control of Networked Two-Timescale Dynamic Agents: A Singular Perturbation Approach

Synchronization Control of Networked Two-Timescale Dynamic Agents: A Singular Perturbation Approach

Adaptive Distributed Attitude Consensus of a Heterogeneous Multiagent Quadrotor System: Singular Perturbation Approach

Singular perturbations easily cause the problem of high gain control (HGC) of a multi-agent quadrotor system (MAQS), that easily excite hazardous high frequency oscillations in the system states. Without HGC, this work studies the distributed fixed-time attitude consensus control problem of multiple heterogeneous quadrotors in presence of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">unknown Lipschitz input disturbances and directed topology</i> . First, the second order sliding mode, composed of consensus errors of “slow” Euler angles and “fast” angular velocities, is designed to be two-time-scale (TTS). Then, an adaptive super twisting controller is designed to produce an integrated slow-fast control structure, and adjust controller gains in different time scales with respect to the disturbance gradients. On this basis, an <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 framework is built for the MAQS on multiple time scales, which provide a rigorous theoretical basis for the fast and robust attitude consensus, and estimate the reaching time subject to singular perturbations. Simulation results on a heterogeneous MAQS consisting of four agents are provided to verify the effectiveness and efficiency of the proposed controller.

A Batteryless Series Hybrid Electric Vehicle Control

Energy management/control in hybrid electric vehicles (HEVs) is of a prime for many electric vehicle manufacturers and researchers. In our previous work, a control based on the singular perturbation (SP) approach was designed and applied to a series HEV for managing/controlling its power. This power supply, an internal combustion engine (ICE) that was mechanically connected to an electric motor (EM), which is a permanent magnet synchronous motor (PMSM), through a flywheel. However, the energy losses, which were not taken into account in the system model and the control function design, caused a significant current control error. In this article, we propose improving the control system performance using an observer that estimates the energy losses in a closed loop system with the time delay. Stability of the closed loop system with the observer is analyzed using a SP method. The proposed control strategy that uses the observer of the energy losses is compared to the control approach of our previous work and its efficacy is demonstrated experimentally on a reduced power scale series HEV.

Three‐time scale multi‐objective optimal PID control with filter design

AbstractThis paper presents the design of a multi‐objective PID (proportional, integral, and derivative) controller in three time scales for a system with load disturbances and sensor noise. The key to this design method is to divide the problem into three time scales by following a singular perturbation approach, which allows for the optimization of fewer parameters in each time scale instead of all the three PID gains at once, hence less computation and design effort. The optimization objectives are the minimization of the overshoot and peak time and the maximization of the closed‐loop system's ability to reject noise and load disturbances. The impact of the integral action and filter on the optimal results is investigated by tuning the singular perturbation parameters. The obtained results show that the performance of the closed‐loop system recovers the optimal solution, which is based on the fast subsystem, as the singular perturbation parameters get sufficiently small.

Momentum-Based Distributed Continuous-Time Nonconvex Optimization of Nonlinear Multi-Agent Systems via Timescale Separation

This study addresses the distributed nonconvex optimization problem for nonlinear multi-agent systems over a weight-balanced and quasi-strongly connected graph. The purpose is to steer all agents to the optimum of a given global objective function with inputs and outputs on the basis of the actual partial information related to the input and output. Novel momentum-based distributed optimal coordinators are designed to achieve this objective, and the local objective functions should be analytic to replace its convexity. An interconnected system with different timescales is established by converting the overall closed-loop system involving the module of momentum-based distributed optimal coordinators and nonlinear multi-agent systems. The singular perturbation approach is applied to deal with the overall interconnected closed-loop system by timescale separation. A numerical example and an application example with four firefighting unmanned aerial vehicles (UAVs) verify the superiority and effectiveness of the proposed distributed method and extended algorithms. Three momentum-based gradient descent algorithms (basic momentum-based, momentum-based Newton, and projected momentum-based gradient flows) are compared and analyzed.

A singular perturbation approach for the control of electromagnetic actuators

The paper is devoted to the control of electromagnetic actuators. Compared to most results in the literature, the design explicitly takes into account the nonlinearities and the magnetic saturation. An approach based on singularly perturbed systems is proposed to take advantage of the different time scales of the model and provide a minimal complexity control law. Associated with an integral action, this control law ensures accuracy and robustness w.r.t. some disturbances and some unknown parameters of the actuator. The use of the singular perturbation method allows also to limit the implementation complexity compared to classical methods such as backstepping control. The synthesis of the control law is then tested through simulations that illustrate the efficiency of the method.

Coexistence of chaotic and non-chaotic attractors in a three-species slow–fast system

The dynamical complexity of conceptual few-species systems has long been attracting considerable attention. In particular, significant attention has been paid to the three trophic level systems to demonstrate that even a baseline prey/predator/top-predator system can exhibit complex dynamics. However, previous studies left many open questions. In this paper, we consider a generalization of the Hastings–Powell model that include intraspecific competition among the predators and top predators. To enhance the ecological relevance of the model, we incorporate different timescales for different species. Specifically, we aim to study how the dynamics of the system is affected in the presence of multiple timescales along the trophic levels. We employ the geometric singular perturbation approach to analyze the change in species abundance over three different timescales, slow, fast, and intermediate. We analytically determine the entry and exit points from the vicinity of the critical manifold; the corresponding values act as critical thresholds for a sudden, fast transition in population density. We show that the properties of this generalized form of the Hastings–Powell model are dynamically more rich than it was observed in previous studies; particularly, the system can exhibit bi-stability or tri-stability in certain parameter intervals. The existence of a homoclinic orbit in a limiting subsystem (prey–predator) indicates period-doubling cascades to chaos in the full system (prey–predator–top predator). We further investigate the impact of the intraspecific competition of predators and top predators on the system’s dynamics. Strong intraspecific competition among the predators and top predators shows periodic coexistence of all species. In contrast, weak competition can drive the system to exhibit tri-stability, which includes the coexistence of chaotic and periodic (of different periods) attractors.

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.

A Singular Perturbation Approach-Based Non-Cascade Sliding Mode Control for Surface-Mounted PMSMs

Motivated by the fact that electrical transients are rather fast compared with mechanical response, the traditional cascade control structure constituted by the inner current and outer speed loops is usually employed in the permanent magnet synchronous motors (PMSMs) servo control community. According to the above-mentioned time-scale characteristic of the PMSMs drive systems, this technique addresses the problems of the non-cascade sliding mode control (SMC) strategy for the surface-mounted PMSMs. Firstly, by appropriately introducing the singular perturbation theory, the corresponding mathematical equations are modeled as a singular perturbation system. Meanwhile, a composite sliding mode surface is constructed based on the Lyapunov equation, such that the system stability can be also guaranteed. Then, according to the exponential reaching law, a standard non-cascade SMC law is designed. Furthermore, an optimal nonlinear function-based tracking differentiator (TD) is presented to smooth the reference velocity value, while providing differential signals. As a result, a novel TD-based SMC strategy is synthesized by incorporating a nonlinear function, thus improving the inherent chattering phenomenon. Finally, a surface-mounted PMSM servo system is performed to illustrate the advantages and effectiveness of the proposed approaches. The main contribution of this paper is to present an alternative non-cascade SMC framework based on the singular perturbation approach, which provides a novel control structure for a PMSM speed regulation system.

Robust cooperative multiple flexible-joint arms control using the q-Bernstein-Schurer operators as the uncertainty approximator: A singular perturbation approach

The position-force tracking issue of an object being moved by collaborative Multiple Flexible-Joint arms, facing dynamic uncertainties, model nonlinearities, and unknown perturbations is studied in this paper. Toward this end, an adaptive control scheme applying the function approximation technique is suggested, enabling the object to track the reference trajectory. This capability arises from the universal approximation property of the FAT-based approaches. Herein, the q-Bernstein-Schurer operators are exploited to disturbances/uncertain dynamic approximations. Since the parameters of the system are not exactly recognized, adaptive rules are suggested for tuning the coefficients of the operator. The Lyapunov stability analysis guarantees uniformly ultimately bounded stability of all the error signals. Two Flexible-Joint manipulators transporting a rigid object are employed to validate the theoretical achievements. The state-of-the-art Chebyshev Neural Network approximator is also used to compare the suggested methodology. The outcomes exhibit the usefulness of the presented approach, handling the system even in the incidence of uncertainties and disturbances needless to the system’s state feedback for function approximation with a low computational burden.

Geometric Singular Perturbation Approach to Poisson-Nernst-Planck Systems with Local Hard-Sphere Potential: Studies on Zero-Current Ionic Flows with Boundary Layers

Geometric Singular Perturbation Approach to Poisson-Nernst-Planck Systems with Local Hard-Sphere Potential: Studies on Zero-Current Ionic Flows with Boundary Layers

Linear Active Disturbance Rejection Control for Two-Mass Systems Via Singular Perturbation Approach

This article presents a linear active disturbance rejection control (LADRC) scheme for two-mass systems (TMSs) based on a singular perturbation (SP) approach. In the proposed scheme, the dynamics of TMSs are separated into a quasi-steady-state system at a slow time scale and a boundary-layer system at a fast time scale via the SP method. The quasi-steady-state model is used as the nominal model to design the LADRC, where internal model control (IMC) rules are employed for prescribed tracking performance. This not only reduces the model order but also makes the gain tuning straightforward, which is crucial for practical use. Based on the boundary-layer model, the active damping method is designed for vibration suppression, which aims at achieving a higher stable bandwidth of the IMC to better attenuate the residual disturbance of linear extended state observer such that the tracking accuracy can be improved. Stability of the full system was analyzed via the extended Tikhonov's theorem. Comparative experiments were carried out to verify the effectiveness of the proposed scheme.

Disentangling how multiple traits drive 2 strain frequencies in SIS dynamics with coinfection

A general theory for competitive dynamics among many strains at the epidemiological level is required to understand polymorphisms in virulence, transmissibility, antibiotic resistance and other biological traits of infectious agents. Mathematical coinfection models have addressed specific systems, focusing on the criteria leading to stable coexistence or competitive exclusion, however, due to their complexity and nonlinearity, analytical solutions in coinfection models remain rare. Here we study a 2-strain Susceptible-Infected-Susceptible (SIS) compartmental model with co-infection/co-colonization, incorporating five strain fitness dimensions under the same framework: variation in transmissibility, duration of carriage, pairwise susceptibilities to coinfection, coinfection duration, and transmission priority effects from mixed coinfection. Taking advantage of a singular perturbation approach, under the assumption of strain similarity, we expose how strain dynamics on a slow timescale are explicitly governed by a replicator equation which encapsulates all traits and their interplay. This allows to predict explicitly not only the final epidemiological outcome of a given 2-player competition, but moreover, their entire frequency dynamics as a direct function of their relative variation and of strain-transcending global parameters. Based on mutual invasion fitnesses, we analyze and report rigorous results on transition phenomena in the 2-strain system, strongly mediated via endemic coinfection prevalence. We show that coinfection is not always a promoter of coexistence; instead, its effect to favour or prevent polymorphism is non-monotonic and depends on the type and level of phenotypic differentiation between strains. This framework offers a deeper analytical understanding of 2-strain competitive games in coinfection, with theoretical and practical applications in epidemiology, ecology and evolution.

Long-Term Regulation of Prolonged Epidemic Outbreaks in Large Populations via Adaptive Control: A Singular Perturbation Approach

In order to control highly-contagious and prolonged outbreaks, public health authorities intervene to institute social distancing, lock-down policies, and other Non-Pharmaceutical Interventions (NPIs). Given the high social, educational, psychological, and economic costs of NPIs, authorities tune them, alternatively tightening up or relaxing rules, with the result that, in effect, a relatively flat infection rate results. For example, during the summer of 2020 in parts of the United States, daily COVID-19 infection numbers dropped to a plateau. This letter approaches NPI tuning as a control-theoretic problem, starting from a simple dynamic model for social distancing based on the classical SIR epidemics model. Using a singular-perturbation approach, the plateau becomes a Quasi-Steady-State (QSS) of a reduced two-dimensional SIR model regulated by adaptive dynamic feedback. It is shown that the QSS can be assigned and it is globally asymptotically stable. Interestingly, the dynamic model for social distancing can be interpreted as a nonlinear integral controller. Problems of data fitting and parameter identifiability are also studied for this model. This letter also discusses how this simple model allows for a meaningful study of the effect of population size, vaccinations, and the emergence of second waves.

About switched affine system interconnected with fast LTI dynamics

We study an interconnection between a slow switched affine system and a fast LTI system. The lower dimensional slow and fast subsystems are computed via the singular perturbation approach. We consider the classical stabilization method based on the existence of a stable convex combination (while ignoring the fast dynamics) and we provide an LMI condition for checking ultimate boundedness when the fast dynamics is taken into account. Numerical example illustrates the main result.

Multivariable Bi-Objective Optimal Modification Adaptive Control for Turbofan Engine With Matched Uncertainty and Slow Actuator Dynamics

To address the multivariable control problem of turbofan engines with matched uncertainty and slow actuator dynamics (SAD), a bi-objective optimal modification adaptive control scheme is proposed. A singular perturbation approach is introduced to transform the plant and actuator dynamics into a reduced-order system with slow time coordinate. Based on this reduced-order system, the bi-objective optimal modification adaptive law (BOMAL) is deduced. Meanwhile, the stability of BOMAL is analyzed based on Lyapunov approach. The simulation results demonstrate that the proposed adaptive control method can effectively handle the control problem of turbofan engines with SAD. The superiority of the proposed bi-objective optimal modification adaptive controller is benchmarked with an LMI optimization gain scheduled controller and a <inline-formula> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula> synthesis controller.

Aggregative feedback optimization for distributed cooperative robotics

In this paper we propose Aggregative Tracking Feedback, i.e., a novel distributed feedback optimization law that steers network systems to a steady state, which is optimal according to an aggregative optimization problem. Aggregative optimization is a recently emerged distributed optimization framework in which the agents of a network minimize the sum of local objective functions. These functions depend both on local and aggregate decision variables (e.g., the barycenter). Motivated by this problem setup, we design a distributed feedback optimization law in which each agent reconstructs information not locally available while concurrently steering the network to an optimal steady state. We perform a system theoretical analysis based on a singular perturbation approach to show that Aggregative Tracking Feedback, in case of strongly convex objective functions, steers the network with a linear convergence rate to the problem minimum. Finally, we show some numerical simulations on a multi-robot surveillance scenario to validate the effectiveness of the proposed method.

Oriented manifold-based error tracking control for nonholonomic wheeled autonomous vehicles on Lie group for curvilinear approach

This paper considers the trajectory tracking control of wheeled autonomous vehicles (WAV) with slipping in the wheels, i.e., when the kinematic constraints are not satisfied. Usually, the coordinates system used to represent all control problems suggest invariant subspaces mutually orthogonal, but this approach can not be enough to treat curvatures significative large at different navigation speed. In order to get a slight im- provement on this topic, there are previous works showing that the kinematic problem (commonly associated with an outer loop) can be resynthesized by using other invariant subspaces, i.e., another representation of the configuration space. For this reason, the proposal reported here uses an oriented-manifold parametrized by a coordinate system on a curve viewpoint of the trajectory to describe the kinematic problem, however, the dynamic control law remains faithful to the singular perturbation approach with invariant subspaces mutually orthogonal, thus, it is possible to include the flexibility through a small factor in the dynamic model (well-known as ε), responsible to avoid the good-performance of the kinematic constraints. Only a common curvature-transformation between orthogonal and curve coordinates will be used to couple both approaches. Finally, it will be observed that when the controller is applied to the control scheme the behavior of the tracking is meaningfully improved.

Computational analysis of the effect of hydrogen peroxide addition on premixed laminar hydrogen/air flames

In the current work, the effect of H2O2 addition on the flame structure, laminar flame speed and NOx emissions is investigated in the context of 1D laminar premixed H2/air flames at Tu = 300 and 600 K, p = 1 and 30 atm, ϕ = 0.5. Mathematical tools from the computational singular perturbation approach are used in order to identify the key chemical and transport mechanisms. The H2O2 addition causes a significant increase to the laminar flame speed (sL), heat release (Q) and NOx emissions. Indicatively, 10% H2O2 addition (per fuel volume) at Tu = 300 K, p = 1 atm results in 72% increase of sL, 100% increase of Q, and 140% increase of the mass fraction of NO. Depending the conditions the flame structure is altered through the chain carrying reaction 10f (H2O2 + H → H2O + OH) or the chain branching 9f (H2O2 (+M) → 2OH (+M)); the first is favored at low temperatures/pressures while the latter is favored at sufficiently high temperatures/pressures. Both reactions boost the radical pool generation, therefore contributing to the broadening of the reaction zone. The reaction with the largest contribution to Q that is mostly affected (decreased) by the addition of H2O2 is reaction 21 (H + O2 (+M) ↔ HO2 (+M)). Moreover, the H2O2 addition enhances the stability of the flame. Finally, the increased production of NO is mainly associated with the increased temperature that is reached with the addition of H2O2.