Synchronization characteristics of controllable oscillators

This paper considers the problem of synchronizing a controllable oscillator with some external frequency and determines the conditions for synchronization. The requirements are determined analytically and the results are applied to several examples in various fields.     

The use of walsh functions in multivariable limit cycle prediction

The describing function method has played a leading role in the prediction of limit cycles in scalar systems; in recent years this approach has been extended to the multivariable case. Derived techniques assume the nonlinearity inputs to be sinusoidal. However, this condition is not always satisfied, and as an alternative, the present paper considers the use of a method which is based on a set of non-sinusoidal functions. Novel concepts related to the characteristic loci are defined and employed for the derivation of new graphical tests.     

Transverse contraction criteria for existence,stability, and robustness of a limit cycle

This paper derives a differential contraction condition for the existence of an orbitally-stable limit cycle in an autonomous system. This transverse contraction condition can be represented as a pointwise linear matrix inequality (LMI), thus allowing convex optimisation tools such as sum-of-squares programming to be used to search for certificates of the existence of a stable limit cycle. Many desirable properties of contracting dynamics are extended to this context, including the preservation of contraction under a broad class of interconnections. In addition, by introducing the concepts of differential dissipativity and transverse differential dissipativity, contraction and transverse contraction can be established for interconnected systems via LMI conditions on component subsystems.     

Synchronization of coupled harmonic oscillators with local interaction

This paper studies synchronization of coupled second-order linear harmonic oscillators with local interaction. We analyze convergence conditions over, respectively, directed fixed and switching network topologies by using tools from algebraic graph theory, matrix theory, and nonsmooth analysis. It is shown that the coupled harmonic oscillators can be synchronized under mild network connectivity conditions. Examples are given to validate the convergence conditions. The theoretical result is also applied to synchronized motion coordination of multi-agent systems as a proof of concept.     

Synchronization of coupled harmonic oscillators in a dynamic proximity network

In this paper, we revisit the synchronization problems for coupled harmonic oscillators in a dynamic proximity network. Unlike many existing algorithms for distributed control of complex dynamical networks that require explicit assumptions on the network connectivity, we show that the coupled harmonic oscillators can always be synchronized, without imposing any network connectivity assumption. Moreover, we also investigate the synchronization with a leader and show that all harmonic oscillators can asymptotically attain the position and velocity of the leader, again without any assumption on connectivity of the followers. Numerical simulation illustrates the theoretical results.     

Analysis of synchronization in a supermarket refrigeration system

In a supermarket refrigeration, the temperature in a display case, surprisingly, influences the temperature in other display cases. This leads to a synchronous operation of all display cases, in which the expansion valves in the display cases turn on and off at exactly the same time. This behavior increases both the energy consumption and the wear of components. Besides this practical importance, from the theoretical point of view, synchronization, likewise stability, Zeno phenomenon, and chaos, is an interesting dynamical phenomenon. The study of synchronization in the supermarket refrigeration systems is the subject matter of this work. For this purpose, we model it as a hybrid system, for which synchronization corresponds to a periodic trajectory.To examine whether it is stable, we transform the hybrid system to a single dynamical system defined on a torus. Consequently,we apply a Poincar ′e map to determine whether this periodic trajectory is asymptotically stable. To illustrate, this procedure is applied for a refrigeration system with two display-cases.     

Study of limit cycle in antagonistically coupled magneto-rheological actuators

In this paper, the presence of limit cycles in the behaviour of antagonistically coupled magneto-rheological (MR) actuators is investigated. The actuator considered in this paper offers high torque-to-mass and torque-to-inertia ratios, for inherent safe actuation. While the antagonistic arrangement is beneficial in improving the actuator performance and eliminating backlash, it may result in limit cycles when the actuator operates in a position control loop. The occurrence of a limit cycle depends on the parameters of the actuator as well as the controller. An in-depth analysis is carried out in this paper to establish a connection between the system parameters and the limit cycle occurrence. Moreover, sufficient conditions for avoiding limit cycles are derived specifically for a proportional-derivative (PD) controller. Simulations and experimental results validate the analysis and provide insights into conditions under which limit cycles are observed in the operation of antagonistic MR actuators.     

Generation of stable limit cycles in controllable linear systems

In this paper we prove that any controllable linear systems , admits a polynomial feedback u= u(x) such that the closed-loop system admits an orbitally asymptotically stable limit cycle.Moreover, we prove that for any positive integer n, there exists an nth-order polynomial, autonomous, ordinary differential equation with a unique limit cycle.     

Closed curve solutions and limit cycles in a class of second-order switched nonlinear systems

This paper studies the problem of finding the initial states for which the solution of a class of switched systems consisting of unstable second-order nonlinear subsystems is convergent. A method is described and applied to establish the regions in the plane where it is possible to define a switching law such that the solution of a class of switched nonlinear systems converges to the origin. We prove that, under certain conditions, these regions are delimited by closed curve solutions of the switched system. Furthermore, a sufficient condition for the closed curve solution to be a limit cycle is presented. Finally, a numerical example is included in order to illustrate the results.     

Synchronization in complex networks of phase oscillators: A survey

The emergence of synchronization in a network of coupled oscillators is a fascinating subject of multidisciplinary research. This survey reviews the vast literature on the theory and the applications of complex oscillator networks. We focus on phase oscillator models that are widespread in real-world synchronization phenomena, that generalize the celebrated Kuramoto model, and that feature a rich phenomenology. We review the history and the countless applications of this model throughout science and engineering. We justify the importance of the widespread coupled oscillator model as a locally canonical model and describe some selected applications relevant to control scientists, including vehicle coordination, electric power networks, and clock synchronization. We introduce the reader to several synchronization notions and performance estimates. We propose analysis approaches to phase and frequency synchronization, phase balancing, pattern formation, and partial synchronization. We present the sharpest known results about synchronization in networks of homogeneous and heterogeneous oscillators, with complete or sparse interconnection topologies, and in finite-dimensional and infinite-dimensional settings. We conclude by summarizing the limitations of existing analysis methods and by highlighting some directions for future research.     

Cluster formation in a time-varying multi-agent system

We introduce time-varying parameters in a multi-agent clustering model and we derive necessary and sufficient conditions for the occurrence of clustering behavior with respect to a given cluster structure. For periodically varying parameters the clustering conditions may be formulated in a similar way as for the time-invariant model. The results require the individual weights assigned to the agents to be constant. For time-varying weights we illustrate with an example that the obtained results can no longer be applied.     

混合系统改进的仿真时钟等步长同步方法及性能评价研究

通过对一类混合系统仿真中的连续系统、离散事件系统及推理决策系统各自仿真时钟的分析,得出了推理决策系统仿真依赖于其两个系统的结果,继而提出了用其中的离散事件 系统仿真时钟云同步具有等步长策略的连续系统仿真时钟改进的等步长方法,并给出了相应的软件实现方法与性能评价准则,解决了该类混合记真时钟的同步问题,减少了仿真误差,从而为该类混合系统的仿真提供理论依据。     

A detailed comparative analysis of all practical algorithms to compute lower bounds on the structured singular value

μ analysis is one of the most efficient techniques to evaluate the stability margins and the performance levels of linear time-invariant systems in the presence of structured time-invariant uncertainties. The exact computation of the structured singular value μ is known to be NP hard in the general case, but several methods have been developed in the last 30 years to compute accurate and reliable bounds. In this paper, all existing μ lower bound algorithms are reviewed and the most relevant ones are evaluated on a wide set of real-world benchmarks, corresponding to various fields of application, system dimensions and structures of the uncertainties. The results are thoroughly analyzed and simple improvements to the existing algorithms are proposed to approach the exact value of μ with a reasonable computation cost. Conclusions show that non-conservative values can be obtained in almost all cases. A brief extension to skew-μ analysis confirms the good results obtained in the classical μ case.     

非线性耦合统一混沌系统的同步

研究非线性耦合的两个统一混沌系统的同步问题．首先利用线性时变系统的稳定性理论,推出当两个统一混沌系统的误差系统渐近稳定时,耦合函数的参数选择范围,从而得出两个统一混沌系统全局渐近同步的充分条件．然后基于Routh-Hurwitz稳定性判别方法,同样得出了混沌系统同步的一个充分条件．通过数值仿真发现,根据第1种方法选择的参数能使混沌系统全局渐近同步;而依据第2种方法选择的参数,即使误差系统系数矩阵的瞬间特征值具有负实部,也会出现混沌同步失去的情况,从而表明了该分析方法的有效性．     

Prediction error methods for limit cycle data

Prediction error methods are considered for identification of the forward linear dynamics of nonlinear feedback closed-loop systems which operate in a perturbed stable limit cycle. A model of the signals measured in a neighborhood of the limit cycle is presented and shown to satisfy a quasistationarity property. Quasistationarity is then used to prove that prediction error methods are both convergent and consistent for our data model.     

大型超市制冷系统同步现象分析

在大型超市制冷系统耗能问题的研究中,由于系统通常采用分布式控制,控制方式往往忽视了各子系统之间的交互干扰而导致系统出现同步现象.将系统的非线性数学模型简化为一个二阶仿射切换系统模型,并在Matlab环境下进行了仿真,对系统的同步现象进行了分析.仿真结果表明,系统在运行过程中出现了同步现象,即各陈列柜温度趋于一致,各陈列柜蒸发器入口阀门的开闭状态趋于同步.通过分析,同步现象的出现使吸气总管的吸气压力出现了周期性较大的波动,容易造成压缩机频繁启停,增加了压缩机的损耗,不利于超市制冷系统的节能降耗.     

Characterization of limit cycles for self-regulating and integral processes with PI control and send-on-delta sampling

This work is focused on the study of limit cycles that appear in a control scheme which is based on the use of a PI controller with an event-based send-on-delta sampling (SOD). The processes investigated are integrator processes plus time delay (IPTD) and first and second order processes plus time delay, which are of interest because they are frequently used to model many industrial processes. The SOD sampling is characterized as a non-linearity of n levels with hysteresis. An algorithm to calculate the limit cycles properties is proposed, and then the results obtained in simulations are compared with experiments performed on a real plant, a distributed solar collector field at the Solar Platform of Almería (PSA, Spain).     

Analysis of robust performance with multiple objectives

Performance robustness with multiple objectives of linear control systems having structured norm-bounded uncertainty is considered. In order to deal with any two objective functions Ψ₁ and Ψ₂ associated with a single feedback control system, the combined criterion Ψ[Ψ₁ ψ₂]Ψ S 1 is often used. The paper, however, considers Ψ ψ₁ Ψ 1 and Ψ W, 11 1 directly. It is shown that it is possible to assess robust performance of two objectives by a single μ test if repeated nonscalar blocks are adequately introduced in the structure of μ. In other words, a performance robustness problem with multiple objectives is proved to be equivalent to a stability robustness problem with extra repeated uncertainty blocks. This equivalence theorem is applicable to various system and norm setups including sampled-data systems.     

Synchronization of complex switched networks with two types of delays

In this paper, a general switched model of complex network with two types of delays is presented. The complex switched network (CSN) contains switching behaviors on both its nodes and topology configuration. Different from those CSNs studied in the existing literatures, the two switching signals dominating the nodes and topology are desynchronized. The two types of delays are the time-varying system delay in nodes and the time-varying coupling delay between nodes and they have different values. The inherent synchronization properties of the CSN have been studied not subject to any controllers. By defining a piecewise Lyapunov-Krasovskii function and by using the improved free-weighting matrix method, global exponential synchronizations are obtained. Illustrated example is presented to show the effectiveness of the proposed result.