Multi-time-scale 3-D coordinated formation control for multi-underactuated AUV with uncertainties: Design and stability analysis using singular perturbation methods

Distributed formation control for multiple quadrotor UAVs under Markovian switching topologies with partially unknown transition rates

The "criticality hypothesis," based on observed scale-free correlations in neural activity, posits that the brain operates at a critical point of transition between two phases. However, the validity of this hypothesis is still debated. Here, employing a commonly used model of cortical dynamics, we find that a phase of long-range order (LRO) in neural activity may be induced by memory (time nonlocality) without invoking criticality. The cortical-dynamics model contains fast and slow time scales that govern the neural and resource (memory) dynamics, respectively. When the resource dynamics are sufficiently slow, we observe an LRO phase, which manifests in avalanche size and duration probability distributions that are fit well by power laws. When the slow and fast time scales are no longer sufficiently distinct, LRO is destroyed. Since this LRO phase spans a wide range of parameters, it is robust against perturbations, unlike critical systems.

The paper considers the eighth-order proton exchange membrane (PEM) fuel-cell mathematical model and shows that it has a multi-time scale property, indicating that the dynamics of three model state space variables operate in the slow time scale and the dynamics of five state variables operate in the fast time scale. This multi-scale nature allows independent controllers to be designed in slow and fast time scales using only corresponding reduced-order slow (of dimension three) and fast (of dimension five) sub-models. The presented design facilitates the design of hybrid controllers, for example, the linear-quadratic optimal controller for the slow subsystem and the eigenvalue assignment controller for the fast subsystem. The design efficiency and its high accuracy are demonstrated via simulation on the considered PEM fuel cell model.

Meta-equilibrium state of multi-species ambipolar diffusion and its relevance to polar mesopheric summer echoes

In recent years technology development for the design of electric and hybrid-electric vehicle systems has reached a peak, due to ever increasing restrictions on fuel economy and reduced vehicle emissions. An international race among car manufacturers to bring production hybrid-electric vehicles to market has generated a great deal of interest in the scientific community. The design of these systems requires development of new simulation and optimization tools. In this paper, a description of a real-time numerical environment for Virtual Proving Grounds studies for hybrid-electric vehicles is presented. Within this environment, vehicle models are developed using a recursive multibody dynamics formulation that results in a set of Differential-Algebraic Equations (DAE), and vehicle subsystem models are created using Ordinary Differential Equations (ODE). Based on engineering knowledge of vehicle systems, two time scales are identified. The first time scale, referred to as slow time scale, contains generalized coordinates describing the mechanical vehicle system that includs the chassis, steering rack, and suspension assemblies. The second time scale, referred to as fast time scale, contains the hybrid-electric powertrain components and vehicle tires. Multirate techniques to integrate the combined set of DAE and ODE in two time scales are used to obtain computational gains that will allow solution of the system’s governing equations for state derivatives, and efficient numerical integration in real time.

In this paper, we present a review about mathematical models of population and community dynamics incorporating individual behaviour. The considered models consist in coupling a game theoretic model, going on at a fast time scale and that describes the change of tactics among individuals in a population or in a community to a model going on at a slow time scale describing the effects in the long term of individual tactics on reproduction and mortality of the populations. As application, an extension of a stage-structured population with intra-individual behavioural variability is presented. Individuals can adopt three different tactics: hawk, dove and bully. We consider a stage-structured population with immature and mature individuals. A mature individual experiences the different tactics in his life and it is assumed that the game is played at a fast time scale in comparison to population dynamics. This fast part of the model is coupled to a slow part, which describes the growth of each sub-population. By use of aggregation of variables methods, we obtain a reduced system of two differential equations governing immature and total mature sub-populations at the slow time scale. The analysis of the model shows that when the cost of fightings is smaller, the proportion of hawks is larger. The total population size equilibrium shows a maximum with respect to the gain of the game.

We examined the role of the cerebellum in patients with tremor-dominant cervical dystonia by measuring the adaptive capacity of rapid reflexive eye movements (saccades). We chose the saccade adaptation paradigm because, unlike other motor learning paradigms, the real-time modification of saccades cannot "wait" for the sensory (visual) feedback. Instead, saccades rely primarily on the internal reafference modulated by the cerebellum. The saccade adaptation happens over fast and slow timescales. The fast timescale has poor retention of learned response, while the slow timescale has strong retention. Cerebellar defects resulting in loss of function affect the fast timescale but the slow timescale of saccade adaptation is retained. In contrast, maladaptive cerebellar disorders feature the absence of both fast and slow timescales. We were able to measure both timescales using noninvasive oculography in 6 normal individuals. In contrast, both timescales were absent in 12 patients with tremor-dominant cervical dystonia. These findings are consistent with maladaptive cerebellar outflow as a putative pathophysiological basis for tremor-dominant cervical dystonia.

Long-range effects, such as allostery, have evolved in proteins as a means of regulating function via communication between distal sites. An NMR-based perturbation mapping approach was used to more completely probe the dynamic response of the core mutation V54A in the protein eglin c by monitoring changes in picosecond to nanosecond aromatic side-chain dynamics and H/D exchange stabilities. Previous side-chain dynamics studies on this mutant were limited to methyl-bearing residues, most of which were found to rigidify on the picosecond to nanosecond time scale in the form of a contiguous "network". Here, high precision (13)C relaxation data from 13 aromatic side chains were acquired by applying canonical relaxation experiments to a newly developed carbon labeling scheme [Teilum et al. (2006) J. Am. Chem. Soc. 128, 2506-2507]. The fitting of model-free parameters yielded S (2) variability which is intermediate with respect to backbone and methyl-bearing side-chain variability and tau e values that are approximately 1 ns. Inclusion of the aromatic dynamic response results in an expanded network of dynamically coupled residues, with some aromatics showing increases in flexibility, which partially offsets the rigidification in methyl side chains. Using amide hydrogen exchange, dynamic propagation on a slower time scale was probed in response to the V54A perturbation. Surprisingly, regional stabilization (slowed exchange) 10-12 A from the site of mutation was observed despite a global destabilization of 1.5 kcal x mol (-1). Furthermore, this unlikely pocket of stabilized residues colocalizes with increases in aromatic flexibility on the faster time scale. Because the converse is also true (destabilized residues colocalize with rigidification on the fast time scale), a plausible entropy-driven mechanism is discussed for relating colocalization of opposing dynamic trends on vastly different time scales.

Aqueous hydroxyl group as the vibrational probe to access the hydrophobicity of amide derivatives

The motion control of a surface ship based on a four degrees of freedom (4-DoF) (surge, sway, roll, and yaw) maneuvering motion model is studied in this paper. A time-scale decomposition method is introduced to solve the path-following problem, implementing Rudder Roll Stabilization (RRS) at the same time. The control objectives are to let the ship to track a predefined curve path under environmental disturbances, and to reduce the roll motion at the same time. A singular perturbation method is used to decouple the whole system into two subsystems of different time scales: the slow path-following subsystem and the fast roll reduction subsystem. The coupling effect of the two subsystems is also considered in this framework of analysis. RRS control is only possible when there is the so-called bandwidth separation characteristic in the ship motion system, which requires a large bandwidth separation gap between the two subsystems. To avoid the slow subsystem being affected by the wave disturbances of high frequency and large system uncertainties, the L1 adaptive control is introduced in the slow subsystem, while a Proportion-Differentiation (PD) control law is adopted in the fast roll reduction subsystem. Simulation results show the effectiveness and robustness of the proposed control strategy.

In this study, acoustic pressure oscillations for single and coupled longitudinal acoustic modes in Solid Rocket Motor (SRM) are investigated using Multiple Time Scales (MTS) method. Two independent time scales are introduced. The oscillations occur on fast time scale whereas the amplitude and phase changes on slow time scale. Hopf bifurcation is employed to investigate the properties of the solution. The supercritical bifurcation phenomenon is observed for linearly unstable system. The amplitude of the oscillations result from equal energy gain and loss rates of longitudinal acoustic modes. The effect of linear instability and frequency of longitudinal modes on amplitude and phase of oscillations are determined for both single and coupled modes. For both cases, the maximum amplitude of oscillations decreases with the frequency of acoustic mode and linear instability of SRM. The comparison of analytical MTS results and numerical simulations demonstrate an excellent agreement.

This paper studies large deviations of a “fully coupled” finite state mean-field interacting particle system in a fast varying environment. The empirical measure of the particles evolves in the slow time scale and the random environment evolves in the fast time scale. Our main result is the path-space large deviation principle for the joint law of the empirical measure process of the particles and the occupation measure process of the fast environment. This extends previous results known for two time scale diffusions to two time scale mean-field models with jumps. Our proof is based on the method of stochastic exponentials. We characterise the rate function by studying a certain variational problem associated with an exponential martingale.

Red blood cells (RBCs) undergo remarkably large deformations when subjected to external forces but return to their biconcave discoid resting shape as the forces are withdrawn. In many experiments, such as when RBCs are subjected to a shear flow and undergo the tank-treading motion, the membrane elements are also displaced from their original (resting) locations along the cell surface with respect to the cell axis, in addition to the cell being deformed. A shape memory is said to exist if after the flow is stopped the RBC regains its biconcave shape and the membrane elements also return to their original locations. The shape memory of RBCs was demonstrated by Fischer [“Shape memory of human red blood cells,” Biophys. J. 86, 3304–3313 (2004)] using shear flow go-and-stop experiments. Optical tweezer and micropipette based stretch-relaxation experiments do not reveal the complete shape memory because while the RBC may be deformed, the membrane elements are not significantly displaced from their original locations with respect to the cell axis. Here we present the first three-dimensional computational study predicting the complete shape memory of RBCs using shear flow go-and-stop simulations. The influence of different parameters, namely, membrane shear elasticity and bending rigidity, membrane viscosity, cytoplasmic and suspending fluid viscosity, as well as different stress-free states of the RBC is studied. For all cases, the RBCs always exhibit shape memory. The complete recovery of the RBC in shear flow go-and-stop simulations occurs over a time that is orders of magnitude longer than that for optical tweezer and micropipette based relaxations. The response is also observed to be more complex and composed of widely disparate time scales as opposed to only one time scale that characterizes the optical tweezer and micropipette based relaxations. We observe that the recovery occurs in three phases: a rapid compression of the RBC immediately after the flow is stopped, followed by a slow recovery to the biconcave shape combined with membrane rotation, and a final rotational return of the membrane elements back to their original locations. A fast time scale on the order of a few hundred milliseconds characterizes the initial compression phase while a slow time scale on the order of tens of seconds is associated with the rotational phase. We observe that the response is strongly dependent on the stress-free state of the cells, that is, the relaxation time decreases significantly and the mode of recovery changes from rotation-driven to deformation-driven as the stress-free state becomes more non-spherical. We show that while membrane shear elasticity and non-spherical stress-free shape are necessary and sufficient for the membrane elements to return to their original locations, bending rigidity is needed for the “global” recovery of the biconcave shape. We also perform a novel relaxation simulation in which the cell axis of revolution is not aligned with the shear plane and show that the shape memory is exhibited even when the membrane elements are displaced normal to the imposed flow direction. The results presented here could motivate new experiments to determine the exact stress-free state of the RBC and also to clearly identify different tank-treading modes.

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.

The aim of this work is to present aggregation methods of hierarchically organized systems allowing one to replace the initial micro-system by a macro-system described by a few global variables. We also study the relations between the fast micro-dynamics and the slow macro-dynamics which can produce global properties. Emergence corresponds to a bottom-up coupling that is the result effected by a micro-level at a macro-level. As an example, we present prey-predator models with different time scales in an heterogeneous environment. A fast time scale is associated to the migration process on spatial patches and a slow time scale is associated to growth and interactions between the populations. Preys must go on spatial patches where resources are located and where predators can attack them. The efficiency of the predators to catch preys is patch dependent. Perturbation methods allow us to aggregate the initial system of differential equations for the patch sub-populations into a macro-system of two differential equations governing the total population densities. We study the case of density independent and density dependent migrations. In the latter case, we show that different functional responses can emerge in the macro prey-predator model as a result of the coupling between the slow and fast systems.