Existence and robustness of phase-locking in coupled Kuramoto oscillators under mean-field feedback

Motivated by the recent development of Deep Brain Stimulation (DBS) for neurological diseases, we study a network of interconnected oscillators under the influence of mean-field feedback and analyze the robustness of its phase-locking with respect to general inputs. Under standard assumptions, this system can be reduced to a modified version of the Kuramoto model of coupled nonlinear oscillators. In the first part of the paper we present an analytical study on the existence of phase-locked solutions under generic interconnection and feedback configurations. In particular we show that, in general, no oscillating phase-locked solutions can co-exist with any non-zero proportional mean-field feedback. In the second part we prove some robustness properties of phase-locked solutions (namely total stability). This general result allows in particular to justify the persistence of practically phase-locked states if sufficiently small feedback gains are applied, and to give explicit necessary conditions on the intensity of a desynchronizing mean-field feedback. Furthermore, the Lyapunov function used in the analysis provides a new characterization of the robust phase-locked configurations in the Kuramoto system with symmetric interconnections.     

Desynchronization of coupled limit-cycle oscillators through nonlinear output regulation

Synchronization is one of the most important phenomena in coupled oscillators. However, sometimes synchronization may be harmful and suppression of collective synchrony or desynchronization is desired. In this paper, we propose a control strategy for the desynchronization of an ensemble of all-to-all coupled Stuart–Landau oscillators. First, the desynchronization problem is redefined in the nonlinear output regulation framework. Then, we design an output regulator (stimulation) which forces Stuart–Landau oscillators (as a paradigm for limit-cycle oscillators) to track exogenous sinusoidal references with different phases. Finally, by considering some modifications, the initial version of the controller is improved to be more applicable in neuronal ensembles as an application of the desynchronization problem. The proposed stimulation is robust against variations of oscillators’ frequencies and can adapt itself with variations of the coupling strength. Mathematical analysis and simulation results reveal the efficiency of the proposed technique.     

Synchronisation of heterogeneous Kuramoto oscillators with sampled information and a constant leader

The Kuramoto oscillator has been widely studied because it can model a wide variety of engineering problems. Conditions for frequency synchronisation of a network of undirected coupled Kuramoto oscillators have been well established. However, the topology that describes the interaction between oscillators may not be connected due to environmental limitations or link failures. In this work we propose a consensus-based control strategy that forces the network to follow a virtual agent with a constant frequency, and where only few agents have access to the leader state. The controller is based on the exchange of information between the oscillators through a communication network. We introduce conditions for synchronisation and phase cohesiveness with the proposed control strategy that depends on the interaction between the physical topology and the communication topology. Finally, we study two applications to illustrate concepts presented in this work: (1) frequency synchronisation in smart grids; and (2) vehicle coordination.     

The solution set of the N-player scalar feedback Nash algebraicRiccati equations

We analyze the set of scalar algebraic Riccati equations (ARE) that play an important role in finding feedback Nash equilibria of the scalar N-player linear-quadratic differential game. We show that in general there exist at most 2^N-1 solutions of the ARE that give rise to a Nash equilibrium. In particular we analyze the number of equilibria as a function of the autonomous growth parameter and present both necessary and sufficient conditions for the existence of a unique solution of the ARE     

Stability analysis and synthesis for scalar linear systems with a quantized feedback

It is well known that a linear system controlled by a quantized feedback may exhibit the wild dynamic behavior which is typical of a nonlinear system. In the classical literature devoted to control with quantized feedback, the flow of information in the feedback was not considered as a critical parameter. Consequently, in that case, it was natural in the control synthesis to simply choose the quantized feedback approximating the one provided by the classical methods, and to model the quantization error as an additive white noise. On the other hand, if the flow of information has to be limited, for instance, because of the use of a transmission channel with limited capacity, some specific considerations are in order. The aim of this paper is to obtain a detailed analysis of linear scalar systems with a stabilizing quantized feedback control. First, a general framework based on a sort of Lyapunov approach encompassing known stabilization techniques is proposed. In this case, a rather complete analysis can be obtained through a nice geometric characterization of asymptotically stable closed-loop maps. In particular, a general tradeoff relation between the number of quantization intervals, quantifying the information flow, and the convergence time is established. Then, an alternative stabilization method, based on the chaotic behavior of piecewise affine maps is proposed. Finally, the performances of all these methods are compared.     

Finite‐time consensus tracking for harmonic oscillators using both state feedback control and output feedback control

This paper investigates the distributed finite‐time consensus‐tracking problem for coupled harmonic oscillators. The objective is to guarantee a team of followers modeled by harmonic oscillators to track a dynamic virtual leader in finite time. Only a subset of followers can access the information of the virtual leader, and the interactions between followers are assumed to be local. We consider two cases: (i) The followers can obtain the relative states between their neighbors and their own; and (ii) Only relative outputs between neighboring agents are available. In the former case, a distributed consensus protocol is adopted to achieve the finite‐time consensus tracking. In the latter case, we propose a novel observer‐based dynamic protocol to guarantee the consensus tracking in finite time. Simulation examples are finally presented to verify the theoretical analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

Output feedback control for systems with constraints and saturations: scalar control case

We consider the problem of regulating the state of a linear system within a prescribed region of the state-space in the presence of disturbances and control saturations. While prior work has focused on full state-feedback, we consider the case of noisy output feedback. For the scalar control case, we combine existing work on full state-feedback and set-valued observers to derive a computational procedure which determines a priori whether the desired constrained regulation is possible. Furthermore, we show that constrained regulation can always be achieved by a set-valued observer followed by static feedback.     

Synchronization in complex dynamical networks based on the feedback of scalar signals

This paper proposes a new approach of synchronization in complex dynamical networks. In this method, the scalar signals are used to instead the output variables of every node as the feedback variables and transmitted signals between every two coupling nodes. As a result, it not only simplifies the topological structure but also saves channel resources at the same time. Especially, some of the criteria are expressed in normal algebraic inequalities instead of matrix inequalities, which means that the original computational effort required is greatly decreased. Finally, several simulation examples are provided to show the effectiveness of the proposed results.     

A constrained scalar control for the motion of a system of oscillators with damping residual oscillations

The problem of scalar control of a finite or countable system of oscillators mounted on a common base is considered. The control is conducted by a scalar bounded force applied to the base. A wide circle of control problems for systems with distributed parameters and many-mass systems can be reduced to the considered statement. A method for solving the problem of constructing a control bounded in modulus that steers the base from a given state to the origin for a finite time and provides that there are no residual oscillations is developed.     

The effects of spike frequency adaptation and negative feedback on the synchronization of neural oscillators

There are several different biophysical mechanisms for spike frequency adaptation observed in recordings from cortical neurons. The two most commonly used in modeling studies are a calcium-dependent potassium current I(ahp) and a slow voltage-dependent potassium current, I(m). We show that both of these have strong effects on the synchronization properties of excitatorily coupled neurons. Furthermore, we show that the reasons for these effects are different. We show through an analysis of some standard models, that the M-current adaptation alters the mechanism for repetitive firing, while the afterhyperpolarization adaptation works via shunting the incoming synapses. This latter mechanism applies with a network that has recurrent inhibition. The shunting behavior is captured in a simple two-variable reduced model that arises near certain types of bifurcations. A one-dimensional map is derived from the simplified model.     

Detecting synchronisation of biological oscillators by model checking

We define a subclass of timed automata, called oscillator timed automata, suitable to model biological oscillators. Coupled biological oscillators may synchronise, as emerging behaviour, after a period of time in which they interact through physical or chemical means. We introduce a parametric semantics for their interaction that is general enough to capture the behaviour of different types of oscillators. We instantiate it both to the Kuramoto model, a model of synchronisation based on smooth interaction, and to the Peskin model of pacemaker cells in the heart, a model of synchronisation based on pulse interaction. We also introduce a logic, Biological Oscillators Synchronisation Logic (BOSL), that is able to describe collective synchronisation properties of a population of coupled oscillators. A model checking algorithm is proposed for the defined logic and it is implemented in a model checker. The model checker can be used to detect synchronisation properties of a given population of oscillators. This tool might be the basic step towards the generation of suitable techniques to control and regulate the behaviour of coupled oscillators in order to ensure the reachability of synchronisation.     

随机干扰下拓扑时变Kuramoto网络事件触发固定时间同步

本文针对一类网络拓扑时变且耦合通道存在随机干扰的Kuramoto型振子网络的同步问题, 设计了事件触 发机制下的固定时间控制算法. 针对耦合网络的拓扑在切换中保持连通的假设, 给出了一种具有多层结构的事件 触发分布式控制策略, 并能够抵消随机干扰对同步性能带来的负面影响. 利用随机Lyapunov稳定理论和固定时间稳 定性理论给出了达到实际固定时间相位同步的控制参数条件和同步调节时间的上界. 另外, 证明了所使用的事件 触发机制在引入双曲正切函数之后不会产生Zeno现象, 并给出了触发间隔的下界. 同时, 在推论中给出了本文提出 的控制方法也可以应对振子耦合拓扑图不连通情况的结论. 最后, 通过两组具有不同噪声强度的仿真实验验证了理 论分析结果.     

Synchronization analysis of coupled Lienard-type oscillators by averaging

Sufficient conditions for the synchronization of coupled Lienard-type oscillators are investigated via averaging technique. The coupling considered here is fixed, nonsymmetric, and nonlinear. Under the assumption that the interconnection topology defines a connected graph, it is shown that the solutions of oscillators converge arbitrarily close to each other, starting from initial conditions arbitrarily far apart, provided that the frequency of oscillations is large enough and the initial phases of oscillators all lie in an open semicircle. It is also shown that the nearly-synchronized oscillations always take place around some fixed magnitude independent of the initial conditions and the coupling functions.     

Iterative sound synthesis by means of cross-coupled digital oscillators

Abstract

Musicians have long been interested in using iterative processes to aid the composition of musical forms (macrostructure) and to synthesize sounds (microstructure). This paper introduces a new sound synthesis method exploring the non-linear behaviour of two iterative cross-coupled digital oscillators. It begins with a brief introduction to iterative systems followed by background information on previous attempts at using them for synthesizing sounds (e.g. feedback frequency and amplitude modulations). Next, it introduces our synthesis method and briefly explains how it has been implemented in a system for real-time composition and performance. The paper concludes with a discussion on how the system has been put into practice to compose and perform a number of works.     

An experimental study of improving performance by scalar replacement

We consider two computational fluid dynamics application codes known as HELIX and MAGI to study optimization by scalar replacement, i.e. loading array element values, those are used repeatedly, into scalars to enable them to be register allocated by the back-end compiler. This technique lowers the execution time by reducing the number of load/store instructions. HELIX computations are on structured grid and are regular, while MAGI computations are irregular. Our experiments compare three versions of HELIX—the original code, a version in which scalar replacement was performed by hand, and version in which scalar replacement was performed automatically by Memoria—a tool for performing scalar replacement developed at Rice University and Michigan Technological University. Our experiments show that scalar replacement improved performance of HELIX by 4–7% over and above performance obtained by using the highest level of optimization with vendor-supplied compilers on an SGI Origin, an SGI O2 workstation, an IBM SP2, and a Cray T3E. We also compare the original and a hand modified version of MAGI on a O2 workstation using three different data sets. Scalar replacement reduced execution time up to about 9%.     

Oscillatory synchronization requires precise and balanced feedback inhibition in a model of the insect antennal lobe

In the insect olfactory system, odor-evoked transient synchronization of antennal lobe (AL) projection neurons (PNs) is phase-locked to the oscillations of the local field potential. Sensory information is contained in the spatiotemporal synchronization pattern formed by the identities of the phase-locked PNs. This article investigates the role of feedback inhibition from the local neurons (LNs) in this coding. First, experimental biological results are reproduced with a reduced computational spiking neural network model of the AL. Second, the low complexity of the model leads to a mathematical analysis from which a lower bound on the phase-locking probability is derived. Parameters involved in the bound indicate that PN phase locking depends not only on the number of LN-evoked inhibitory postsynaptic potentials (IPSPs) previously received, but also on their temporal jitter. If the inhibition received by a PN at the current oscillatory cycle is both perfectly balanced (i.e., equal to the mean inhibitory drive) and precise (without any jitter), then the PN will be phase-locked at the next oscillatory cycle with probability one.     

Arbitrarily fast and robust tracking by feedback

The output of a singe-input-single-output linear feedback system with more than one pole in excess over the zeros in the loop transmission cannot track arbitrarily fast its input (by the root locus). In this work we extend the linear feedback so that some of the open loop poles may depend on the open loop gain; we call this new class quasi-linear feedback systems. We then derive time domain, pole-zero, and frequency domain conditions which ensure arbitrarily fast and robust tracking by quasi-linear feedback, for an arbitrary number of poles in excess over the zeros. We prove that in a particular case these conditions are equivalent, and that the boundedness in frequency of the closed loop transfer function is no longer necessary for achieving arbitrarily fast tracking. The robustness is to external disturbances and initial conditions, and the open loop has to be minimum phase. Some examples are presented which illustrate these results. They also show that this good performance can be obtained with a reduced control effort, and that quasi-linear feedback can alleviate the limitation on performance of non-minimum phase open loops.     

Decoupling of structured systems by parameter-independentprecompensation and state feedback

In this paper, structured systems described by state-space models are considered. For these systems, the entries of the state-space model matrices are assumed to be either fixed zeros or free independent parameters. We study the dynamic decoupling of structured linear systems. It turns out that a large class of systems which is not state feedback decouplable can be made decouplable by the addition of a simple structured precompensator. Such a precompensator then has the property to be independent of the system parameters. We give a necessary and sufficient condition for a structured system to be made feedback decouplable using a parameter independent precompensator. Furthermore, we prove that when such a solution exists, the precompensator can be chosen to be diagonal. In this case the precompensator is constructed via a simple combinatorial algorithm. Our approach is based on a graph representation of the structured system. All the results of this paper are related to some particular shortest sets of input-output paths in the graph associated with the structured system