Multi-timescale Structure Research Articles

Asymptotic optimality of a class of controlled non-Markov processes

Motivated by applications in power systems and problems arising in simulation of large scale complex system optimizations, this work is concerned with controlled stochastic switching systems. The system of interest displays a multi-time scale structure. In contrast to the so-called singularly perturbed diffusions and multi-scale Markov decision processes, controlled non-Markov processes (also known as non-Markov decision processes) are treated. The novelty of our work is the treatment of the non-Markov controlled processes and the time-scale used. The fast and slow processes are coupled through a stochastic differential equation. Using averaging, it is first shown that the non-Markov switching process has a weak limit that is a Markov decision process. Then asymptotic optimal control of the non-Markov process is obtained by using the limit process.

Idealized multiple-timescale model of cortical spreading depolarization initiation and pre-epileptic hyperexcitability caused by NaV1.1/SCN1A mutations.

NaV1.1 (SCN1A) is a voltage-gated sodium channel mainly expressed in GABAergic neurons. Loss of function mutations of NaV1.1 lead to epileptic disorders, while gain of function mutations cause a migraine in which cortical spreading depolarizations (CSDs) are involved. It is still debated how these opposite effects initiate two different manifestations of neuronal hyperactivity: epileptic seizures and CSD. To investigate this question, we previously built a conductance-based model of two neurons (GABAergic and pyramidal), with dynamic ion concentrations (Lemaire et al. in PLoS Comput Biol 17(7):e1009239, 2021. https://doi.org/10.1371/journal.pcbi.1009239 ). When implementing either NaV1.1 migraine or epileptogenic mutations, ion concentration modifications acted as slow processes driving the system to the corresponding pathological firing regime. However, the large dimensionality of the model complicated the exploitation of its implicit multi-timescale structure. Here, we substantially simplify our biophysical model to a minimal version more suitable for bifurcation analysis. The explicit timescale separation allows us to apply slow-fast theory, where slow variables are treated as parameters in the fast singular limit. In this setting, we reproduce both pathological transitions as dynamic bifurcations in the full system. In the epilepsy condition, we shift the spike-terminating bifurcation to lower inputs for the GABAergic neuron, to model an increased susceptibility to depolarization block. The resulting failure of synaptic inhibition triggers hyperactivity of the pyramidal neuron. In the migraine scenario, spiking-induced release of potassium leads to the abrupt increase of the extracellular potassium concentration. This causes a dynamic spike-terminating bifurcation of both neurons, which we interpret as CSD initiation.

Output feedback control design using Extended High-Gain Observers and dynamic inversion with projection for a small scaled helicopter

This paper presents the output feedback control design for a tracking problem of a small-scale helicopter in the presence of uncertainties. The dynamics of the helicopter is an underactuated mechanical system and the form of control inputs is nonaffine. A time-scale approach is suggested to cope with underactuated mechanical systems to overcome lack of the number of inputs. A newly developed dynamic inversion with projection is used to deal with nonaffine control inputs to increase the region of attraction as compared to linearized inputs, i.e., an affine control input form and to deal with actuator’s constraints and peaking in estimates from an Extended High-Gain Observer. The Extended High-Gain Observer is employed to quickly estimate model uncertainties and external disturbances. The singular perturbation method is used to analyze the stability of the closed-loop system in a multi-time scale structure. Based on the stability analysis, the design procedure for the proposed algorithm is presented for practical implementation. The effectiveness of the proposed control algorithm is shown via numerical simulations as well as experimental tests with a small-scale helicopter in an outdoor environment.

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

This paper is concerned with a multi-underactuated AUV three-dimensional (3-D) formation control method with a multi-time-scale structure in the presence of uncertain nonlinearities and environmental disturbances. First, a double-layer independent position-velocity fixed topology is adopted to overcome the drawback of narrow bandwidth of underwater communication. Then, an extended high-gain observer (EHGO) is utilized to provide compensation to uncertain nonlinearities and environmental disturbances. Moreover, the consensus theory is incorporated with an integral sliding mode controller to guarantee that the multi-underactuated AUV system can track the desired trajectory while maintaining the predefined formation. In addition, based on the analysis for the whole formation control system, the extended high-gain observer is on the ultra-fast time scale, the integral sliding mode controller is on the fast time scale, and the system state converges to the desired state on the slow time scale. Such multi-time scale structure allows independent analysis of the dynamics in each time scale, and the singular perturbation method is effectively utilized to establish the exponential stability of the equilibrium. Finally, simulation results are presented to illustrate the control performance.

Multi-time scale energy management framework for smart PV systems mixing fast and slow dynamics

We propose a multi-time scale energy management framework for a smart photovoltaic (PV) system that can calculate optimized schedules for battery operation, power purchases, and appliance usage. A smart PV system is a local energy community that includes several buildings and households equipped with PV panels and batteries. However, due to the unpredictability and fast variation of PV generation, maintaining energy balance and reducing electricity costs in the system is challenging. Our proposed framework employs a model predictive control approach with a physics-based PV forecasting model and an accurately parameterized battery model. We also introduce a multi-time scale structure composed of two-time scales: a longer coarse-grained time scale for daily horizon with 15-minutes resolution and a shorter fine-grained time scale for 15-minutes horizon with 1-second resolution. In contrast to the current single-time scale approaches, this alternative structure enables the management of a necessary mix of fast and slow system dynamics with reasonable computational times while maintaining high accuracy. Simulation results show that the proposed framework reduces electricity costs up 48.1% compared with baseline methods. The necessity of a multi-time scale and the impact on accurate system modeling in terms of PV forecasting and batteries are also demonstrated.

Modeling Long-Term Dependencies from Videos Using Deep Multiplicative Neural Networks

Understanding temporal dependencies of videos is fundamental for vision problems, but deep learning–based models are still insufficient in this field. In this article, we propose a novel deep multiplicative neural network (DMNN) for learning hierarchical long-term representations from video. The DMNN is built upon the multiplicative block that remembers the pairwise transformations between consecutive frames using multiplicative interactions rather than the regular weighted-sum ones. The block is slided over the timesteps to update the memory of the networks on the frame pairs. Deep architecture can be implemented by stacking multiple layers of the sliding blocks. The multiplicative interactions lead to exact, rather than approximate, modeling of temporal dependencies. The memory mechanism can remember the temporal dependencies for an arbitrary length of time. The multiple layers output multiple-level representations that reflect the multi-timescale structure of video. Moreover, to address the difficulty of training DMNNs, we derive a theoretically sound convergent method, which leads to a fast and stable convergence. We demonstrate a new state-of-the-art classification performance with proposed networks on the UCF101 dataset and the effectiveness of capturing complicate temporal dependencies on a variety of synthetic datasets.

Hierarchical Compression Reveals Sub-Second to Day-Long Structure in Larval Zebrafish Behavior.

Animal behavior is dynamic, evolving over multiple timescales from milliseconds to days and even across a lifetime. To understand the mechanisms governing these dynamics, it is necessary to capture multi-timescale structure from behavioral data. Here, we develop computational tools and study the behavior of hundreds of larval zebrafish tracked continuously across multiple 24-h day/night cycles. We extracted millions of movements and pauses, termed bouts, and used unsupervised learning to reduce each larva’s behavior to an alternating sequence of active and inactive bout types, termed modules. Through hierarchical compression, we identified recurrent behavioral patterns, termed motifs. Module and motif usage varied across the day/night cycle, revealing structure at sub-second to day-long timescales. We further demonstrate that module and motif analysis can uncover novel pharmacological and genetic mutant phenotypes. Overall, our work reveals the organization of larval zebrafish behavior at multiple timescales and provides tools to identify structure from large-scale behavioral datasets.

Fractal brain dynamics: from Mandelbrot to marmosets.

Fractal brain dynamics: from Mandelbrot to marmosets.

Big Ducks in the Heart: Canard Analysis Can Explain Large Early Afterdepolarizations in Cardiomyocytes

Early afterdepolarizations (EADs) are pathological voltage fluctuations that can occur in cardiac cells and are a potent source of potentially fatal arrhythmias. Recent works examining the mechanisms underlying EADs in minimal computational cardiac models have revealed that voltage-driven EADs are canard-induced mixed-mode oscillations whose properties are mediated by the rate at which these cells are paced. In this work, we analyze the mechanisms for the pacing-induced generation of different EAD behaviors in a reduced four-dimensional Luo--Rudy I model using slow-fast analysis. While previous explanations for EADs in this model have required manipulation of the underlying multitimescale structure, our approach does not and we find that the canard mechanism persists in generating EADs in this context. We also find that the canard mechanism gives a more complete explanation for the onset and properties of the EADs induced (e.g., EAD amplitude and number). In addition, we also find that the canards play an essential role in producing a richer set of behaviors than were seen in other minimal models, some of which have also been observed in experiments. These behaviors include pacing-induced termination of EADs, the periodic alternation of cardiac action potentials with and without EADs, as well as bistability between standard and EAD-containing action potentials at a fixed pacing rate. Finally, we show that this bistability can lead to hysteretic transitions between standard and arrhythmogenic action potentials under sufficiently slow oscillations in the pacing rate.

River discharge contribution to sea-level rise in the Yangtze River Estuary, China

Sea level changes in the Yangtze River Estuary (YRE) as a result of river discharge are investigated based on the monthly averaged river discharge from 1950 to 2011 at the Datong station. Quantification of the sea level contribution is made by model computed results and the sea level rates reported by the China Sea Level Bulletin (CSLB). The coastal modeling tool, MIKE21, is used to establish a depth-averaged hydrodynamic model covering the YRE and Hangzhou Bay. The model is validated with the measured data. Multi-year monthly river discharges are statistically calculated based on the monthly river discharges at Datong station from 1950 to 2011. The four characteristic discharges (frequency of 75%, 50% and 25%, and multi-year monthly) and month-averaged river discharge from 1950 to 2011 are used to study the seasonal and long-term changes of sea level. The computed sea level at the Dajishan and Lvsi stations are used to study the multi-time scale structure of periodic variation in different time scale of river discharge series. The results reveal that (1) the sea level rises as the river discharge increases, and its amplification decreases from upstream to the offshore. (2) The sea level amplification on the south coast is greater than that on the north coast. When river discharge increases by 20,000m3/s, the sea level will increase by 0.005–0.010m in most of Hangzhou Bay. (3) The sea level at the Dajishan station, influenced by river discharge, increased 0.178mm/y from 1980 to 2011. Correspondingly, the sea level rose at a rate of 2.6–3.0mm/y during the same period. These values were provided by the CSLB. The increase in sea level (1980–2011) at the Dajishan station caused by river discharge is 6.8–8.9% of the total increase in sea level. (4) The 19–20 year dominant nodal cycle of sea level at the Dajishan and Lvsi stations is in accord with 18.6 year nodal cycle of main tidal constituents on Chinese coasts. It implies that the sea-level change period on the coast of the Yangtze River is mainly controlled by tides.

Multi-Timescale Nonlinear Robust Control for a Miniature Helicopter

A new nonlinear control approach, which is applied to a miniature aerobatic helicopter through a multi-timescale structure, is proposed. Because of the highly nonlinear, unstable, and underactuated nature of a miniature helicopter, it is a challenge to design an autonomous flight control system that is capable of operating in the full flight envelope. To deal with unstable internal dynamics, the translational, rotational, and flapping dynamics of the helicopter (eleven degrees of freedom) are organized into a three-timescale, nonlinear model. The concepts of dynamic inversion and sliding manifold are combined together such that 1) the controller proposed is robust with respect to functional and parametric uncertainties, and 2) the settling time in faster modes is guaranteed to be less than the fixed step size of slower modes. A time-varying feedback gain, derived according to global stability and sliding manifold variations, is proved to be uniquely solvable based on the Perron-Frobenius Theorem. Partial uncertainties are explicitly taken into account in the nonlinear robust control design, and Monte Carlo simulations are used for validations under other sensor noises, model uncertainties, and a Federal Aviation Administration suggested gust condition.