A new method to determine the periodic orbit of a nonlinear dynamic system and its period

Periodic motion is an important steady-state motion in the real world. In this paper, a new generalized shooting method for determining the periodic orbit of a nonlinear dynamic system and its period is presented by rebuilding the traditional shooting method. First, by changing the time scale, the period of the periodic orbit of a nonlinear system is drawn into the governing equation of this system explicitly. Then, the period is used as a parameter in the iteration procedure of the shooting method. The periodic orbit of the system and the period can be determined rapidly and precisely. The requirement of this method for the initial iteration conditions is not rigorous. This method can be used to analyze the forced nonlinear system and the parameter exciting system. As an example, the results of the Rössler equation for an eight-dimensional, nonlinear, flexible, rotor-bearing system are compared with those obtained by the Runge-Kutta integration algorithm. The validity of this method is verified by the numerical results obtained in the two examples.     

Existence of periodic orbits of stable saturated systems

The saturation of linear controllers produces the undesirable existence of equilibrium points or periodic orbits of the closed-loop system. This typical nonlinear behavior has been observed in real systems or by means of simulation of certain examples. However, there are only a few studies in which the properties of saturated systems have been examined rigorously and, a proof of the existence of periodic orbits created by the saturation of the controller is lacking. In this paper we choose an example of an open-loop stable linear control system with an stabilizing saturated linear feedback to prove rigorously the existence of a periodic orbit.     

Nonlinear dynamics by mode superposition

A mode superposition technique for approximately solving nonlinear initial-boundary-value problems of structural dynamics is discussed, and results for examples involving large deformation are compared to those obtained with implicit direct integration methods such as the Newmark generalized acceleration and Houbolt backward-difference operators. The initial natural frequencies and mode shapes are found by inverse power iteration with the trial vectors for successively higher modes being swept by Gram—Schmidt orthonormalization at each iteration. The subsequent modal spectrum for nonlinear states is based upon the tangent stiffness of the structure and is calculated by a subspace iteration procedure that involves matrix multiplication only, using the most recently computed spectrum as an initial estimate. Then, a precise time integration algorithm that has no artificial damping or phase velocity error for linear problems is applied to the uncoupled modal equations of motion. Squared-frequency extrapolation is examined for nonlinear problems as a means by which these qualities of accuracy and precision can be maintained when the state of the system (and, thus, the modal spectrum) is changing rapidly.The results indicate that a number of important advantages accrue to nonlinear mode superposition: (a) there is no significant difference in total solution time between mode superposition and implicit direct integration analyses for problems having narrow matrix half-bandwidth (in fact, as bandwidth increases, mode superposition becomes more economical), (b) solution accuracy is under better control since the analyst has ready access to modal participation factors and the ratios of time step size to modal period, and (c) physical understanding of nonlinear dynamic response is improved since the analyst is able to observe the changes in the modal spectrum as deformation proceeds.     

高维混沌系统的混合控制方法

将最短时间最优控制思想用于混沌轨道引导,通过可控域把混沌轨道引导和不稳定周期轨道的稳定化方法有机地结合起来,提出一种高维混沌系统的混合控制方法。利用该方法可实现高维混沌系统不稳定周期轨道之间的快速、有效切换。以双转子系统为例进行数字仿真,仿真结果表明了该算法的有效性。     

Periodic optimal control of nonlinear constrained systems using economic model predictive control

In this paper, we consider the problem of periodic optimal control of nonlinear systems subject to online changing and periodically time-varying economic performance measures using model predictive control (MPC). The proposed economic MPC scheme uses an online optimized artificial periodic orbit to ensure recursive feasibility and constraint satisfaction despite unpredictable changes in the economic performance index. We demonstrate that the direct extension of existing methods to periodic orbits does not necessarily yield the desirable closed-loop economic performance. Instead, we carefully revise the constraints on the artificial trajectory, which ensures that the closed-loop average performance is no worse than a locally optimal periodic orbit. In the special case that the prediction horizon is set to zero, the proposed scheme is a modified version of recent publications using periodicity constraints, with the important difference that the resulting closed loop has more degrees of freedom which are vital to ensure convergence to an optimal periodic orbit. In addition, we detail a tailored offline computation of suitable terminal ingredients, which are both theoretically and practically beneficial for closed-loop performance improvement. Finally, we demonstrate the practicality and performance improvements of the proposed approach on benchmark examples.     

含间隙弹簧振动系统的非线性模态特性

针对含间隙的两自由度弹簧-质量分段振动系统的非线性模态开展了研究.首先,解析确定了分段保守自治系统发生同相和反相模态运动的初始位移,并采用加权平均方法确定了分段振动系统的模态频率,及其在位形空间模态曲线.然后,采用数值方法求解了系统的非线性模态曲线和模态频率,与本文获得的解析模态频率比较,说明本文的结果较等效模态频率有更好的精度.研究结果表明:在一定的参数条件下,系统的非线性模态个数会高于系统的自由度数目,系统可能发生内共振,而产生多余模态.多余模态运动是两振子同向振动中含有异向振动,说明多余模态是在同相模态运动和反相模态运动之间转换的模态.     

A numerical technique to predict periodic and quasi-periodic response of nonlinear dynamic systems

A frequency domain based algorithm using Fourier approximation and Galerkin error minimization has been used to obtain the periodic orbits of large order nonlinear dynamic systems. The stability of these periodic response is determined through a bifurcation analysis using Floquet theory. This technique is applicable to dynamic systems having both analytic and nonanalytic nonlinearities. This technique is compared with numerical time integration and is found to be much faster in predicting the steady state periodic response.     

Hyperbolic automorphisms of tori and pseudo-random sequences

A method for generating pseudo-random sequences of d-dimensional vectors is considered; it is based on theergodic theory of periodic orbits in the sense of [2] for unstable dynamical systems such as the hyperbolic automorphisms of the d-dimensional Torus. Since these systems enjoy strong chaotic properties, their orbits are both dense andchaotic in some sense, however the ergodic property holds only for orbits having initial points with irrational coordinates, the remaining ones being periodic. Unfortunately, those orbits are the only ones that a computer is able to generate. Since a pseudo-random sequence in [0,1] ^d is a long periodic orbit which has chaotic behaviour similar in some sense to the one of aperiodic orbits, in this note, we shall prove lower and upper bounds for the length of the period of orbits of the hyperbolic automorphisms of the d-dimensional Torus, expressed in terms of the (rational) starting point. The algorithms proposed are free of computational error, since they work in integer arithmetic. Surprisingly the elimination of the round off errors turns out in anincrease of the length of the period. Statistical testing and the problem of estimating the discrepancy of the obtained sequences are also treated.     

Nonlinear H∞ control around periodic orbits

In this paper, we study the nonlinear H_∞ control of systems with periodic orbits. We develop the notion of an induced L₂ gain (so-called nonlinear H_∞ norm) for systems where the no-disturbance behavior of the system is a periodic orbit and provide conditions under which the induced L₂ gain of the system (around the orbit) can be made less than a specified value by state feedback. This work is a natural extension of results on nonlinear H_∞ control of nonlinear systems in a neighborhood of a stable equilibrium point to the periodic orbit case. Synthesis of a nonlinear H_∞ state feedback controller is facilitated by the use of transverse coordinates and, in particular, the transverse linearization of the system.     

Gait generation and control for biped robots with underactuation degree one

The paper develops a unified feedback control law for n degree-of-freedom biped robots with one degree of underactuation so as to generate periodic orbits on different slopes. The periodic orbits on different slopes are produced from an original periodic orbit, which is either a natural passive limit cycle on a specific slope or a stable periodic walking gait on level ground generated with active control. First, inspired by the controlled symmetries approach, a general result on gait generation on different slopes based on a periodic orbit on a specific slope is obtained. Second, the time-scaling control approach is integrated to reproduce geometrically same periodic orbits for biped robots with one degree of underactuation. The degree of underactuation is compensated by one degree-of-freedom in the temporal evolution that scales the original periodic orbit. Necessary and sufficient conditions are investigated for the existence and stability properties of periodic orbits on different slopes with the proposed control law. Finally, the proposed approach is illustrated by two kinds of underactuated biped robots: one has a passive gait on a specific ground slope and the other does not have a natural passive gait.     

基于变量反馈控制方法的混沌控制研究

将变量反馈控制(VFC)方法用于混沌杜芬方程的控制，通过调整反馈强度，可显著改善混沌现象，并可使系统由初始的混沌轨道被控变成非混沌的轨道，最终进入目标轨道。     

A novel method of periodic orbit computation in circular restricted three-body problem

Periodic orbits are fundamental keys to understand the dynamical system of circular restricted three-body problem, and they play important roles in practical deep-space exploration. Current methods of periodic orbit computation need a high-order analytical approximate solution to start the iteration process, thus making the computation complicated and limiting the types of periodic orbits that can be obtained. By utilizing the symmetry of the restricted three-body problem, a special kind of flow function is...     

Vibration predictions and verifications of disk drive spindle system with ball bearings

With areal recording density of hard disk drives (HDD) historically growing at an average of 60% per year, it is becoming increasingly more difficult to maintain the precise positioning required of the ever-smaller GMR heads to read and write data. Any unexpected vibration will cause the data written to a wrong data track, even the vibration amplitude is very small. Consequently, the dynamic behaviors of HDD spindle systems and their potential influence on track misregistration rate must be clearly understood. This paper is to apply an approach based on efficient component mode synthesis (CMS), incorporating multi-body system dynamics technology to predict dynamic characteristics of HDD ball-bearing spindle systems. First, the discrete governing equations of motion for HDD spindle systems, which consist of several flexible and rigid components, are derived through the use of Lagrangian equations. The elastic component modal frequencies and modal shape vectors are then obtained using a finite-element analysis. For ball bearing inherently defects, a mathematical model is used as a time-varying force, resulting in spindle vibrations. The time-varying force and component modal shape vectors are incorporated into the governing equations of the whole spindle systems. An implicit numerical integration method is used to obtain the forced vibration of the HDD spindle system. Finally, the dynamic responses of two typical HDD spindle systems are investigated numerically to predict the significant coupled vibration frequencies, mode shapes and resonance interactions. The results well agree with the solutions predicted by other analytical methods and the experimental results, respectively.     

Application of efficient composite methods for computing with certainty periodic orbits in molecular systems

Recently, we have proposed a technique for the computation of periodic orbits in molecular systems, based on the characteristic bisection method [Vrahatis et al., Comput. Phys. Commun. 138 (2001) 53]. The main advantage of the characteristic bisection method is that it converges with certainty within a given starting rectangular region. In this paper we further improve this technique by applying, on a surface of section of a Poincaré map, an iterative scheme based on the composition of the characteristic bisection method with other more rapid root-finding methods such as Newton's or Broyden's methods. Thus, the composite schemes compute rapidly with certainty periodic orbits of molecular systems. By applying these methods to the LiNC/LiCN molecular system we obtain promising results. We have reproduced previous results using considerable less CPU time. Also, we have located and computed new asymmetric families of periodic orbits.     

Analysis of Chattering in Systems With Second-Order Sliding Modes

A systematic approach to the chattering analysis in systems with second-order sliding modes is developed. The neglected actuator dynamics is considered to be the main cause of chattering in real systems. The magnitude of oscillations in nonlinear systems with unmodeled fast nonlinear actuators driven by second-order sliding-mode control generalized suboptimal (2-SMC G-SO) algorithms is evaluated. Sufficient conditions for the existence of orbitally stable periodic motions are found in terms of the properties of corresponding Poincare maps. For linear systems driven by 2-SMC G-SO algorithms, analysis tools based on the frequency-domain methods are developed. The first of these techniques is based on the describing function method and provides for a simple approximate approach to evaluate the frequency and the amplitude of possible periodic motions. The second technique represents a modified Tsypkin's method and provides for a relatively simple, theoretically exact, approach to evaluate the periodic motion parameters. Examples of analysis and simulation results are given throughout this paper.     

Vector field editing and periodic orbit extraction using Morse decomposition

Design and control of vector fields is critical for many visualization and graphics tasks such as vector field visualization, fluid simulation, and texture synthesis. The fundamental qualitative structures associated with vector fields are fixed points, periodic orbits, and separatrices. In this paper, we provide a new technique that allows for the systematic creation and cancellation of fixed points and periodic orbits. This technique enables vector field design and editing on the plane and surfaces with desired qualitative properties. The technique is based on Conley theory, which provides a unified framework that supports the cancellation of fixed points and periodic orbits. We also introduce a novel periodic orbit extraction and visualization algorithm that detects, for the first time, periodic orbits on surfaces. Furthermore, we describe the application of our periodic orbit detection and vector field simplification algorithms to engine simulation data demonstrating the utility of the approach. We apply our design system to vector field visualization by creating data sets containing periodic orbits. This helps us understand the effectiveness of existing visualization techniques. Finally, we propose a new streamline-based technique that allows vector field topology to be easily identified.     

LMI tools for eventually periodic systems

This paper is focused on the concept of an eventually periodic linear discrete-time system. We derive a necessary and sufficient analysis condition for checking open-loop stability and performance of such systems, and use this to derive exact controller synthesis conditions given eventually periodic plants. All the conditions derived are provided in terms of semi-definite programming problems. The motivation for this work is controlling nonlinear systems along prespecified trajectories, notably those which eventually settle down into periodic orbits and those with uncertain initial states.     

Periodic orbits in the Euler method for a class of delay differential equations

Recently, we have proved that Naimark-Sacker bifurcations occur in the Euler method applied to a delay differential equation [1]. By slightly modifying the proof, it is verified that the same result holds, e.g., for another equation obtained from the equation by a change of the dependent variable. However, in computer experiments, the Euler method presents different behavior for the two equations: an invariant circle is observed in the former case; a stable periodic orbit is observed in the latter case.

We show that it is reasonable to consider the behavior in the latter case as a weak resonance in the Naimark-Sacker bifurcation. More specifically, we study a class of delay differential equations which includes the second equation, paying attention to special periodic solutions found by Kaplan and Yorke [2], and prove constructively that the Euler method applied to each equation of the class has at least two periodic orbits. Numerical experiments are presented which indicate that one orbit is stable and the other orbit is unstable, whose unstable manifold forms an invariant circle.     

一个干摩擦Duffing振子的滑动分岔分析

分析了一个干摩擦Duffing振子在不同参数下混沌和周期轨及不同周期轨之间的共存与转换.干摩擦振子属于Filippov系统,会发生特有的粘滞现象.分析发现,从穿越轨线转换到粘滞--滑移轨线不仅可经穿越滑动分岔和切换滑动分岔实现,也可经两个相邻的穿越滑动分岔和多滑动分岔实现,而从粘滞-滑移轨线转换到非穿越轨线须经擦边滑动...     

永磁同步电动机中混沌运动的部分解耦控制

永磁同步电动机在参数处于特定区域时存在混沌现象,混沌的存在将使电机的性能变差,因此要设法消除其混沌运动.电机中混沌的现有控制方法的控制目标只能为周期点或平衡点,不能满足实际需要.为了解决这个问题,设计了基于非线性反馈的永磁同步电动机中混沌现象的部分解耦控制.该方法以直轴和交轴电压为控制变量,通过电机状态的非线性反馈将直轴和交轴电流方程中的耦合项解耦,同时使得直轴和交轴电流可以跟踪设定值.这种控制方式的结果是使系统具有唯一稳定平衡点,而这个平衡点的位置可以根据实际要求设置为任意点.在任意时刻对处于混沌状态的永磁同步电动机进行部分解耦控制,受控系统将稳定于设定平衡点,从而实现混沌的控制.文中分析了控制参数与系统响应快速性之间的关系,为参数选择提供了指导.仿真结果表明了理论分析的正确性和该控制方法的有效性.