Kepler’s equation and limit cycles in a class of PWM feedback control systems

The aim of this paper is to point out some new results concerning the ripple instability in the closed-loop control system using pulse width modulators (PWM), with natural sampling, as power amplifier. The presented analysis, based on the dual-input describing function method and the theoretical framework of Kepler’s problem, shows an equivalence between the computation of switching instants of the PWM and the eccentric anomaly of the planet orbit around the sun, giving a simple stability criterion and a sufficient condition for the absence of solutions of the harmonic balance equation and, therefore, the probable absence of limit cycles of a period of a multiple of that characteristic of the modulator. The derived stability criterion, by using the describing function method, is successively compared with the local stability of the closed-loop PWM system for first- and second-order plants. In the first case it has been formally proved that the proposed criterion ensures the local stability of an equilibrium point, while in the second one a Monte Carlo simulation has confirmed that the selection of the modulator parameters, according to the proposed criterion, gives an effective method to avoid limit cycles and to ensure the local stability.     

Robust Limit Cycle Suppression for Control Systems with Parametric Uncertainty and Nonlinearity

A systematic limit cycle suppression method is presented to predict and suppress the sustained limit cycles persisting in a class of uncertain control systems containing inherent nonlinearity. The aim is to determine the feasible controller parameter sets in the parameter plane for the entire uncertain system. A family of constant limit cycle amplitude loci is plotted. Successful application to pitch orientation control system with parametric uncertainties and nonlinearity is demonstrated. System output response is further investigated.     

基于改进等效平面法的结构系统可靠性分析

将复相关理论引入到结构系统可靠性分析中, 推导了并联体系等效功能函数 的递推公式, 提出了复相关等效平面法. 解决了等效平面法等效原则中不包含相关性信息的 问题, 克服了等效平面法计算并联体系误差大的缺点. 通过算例证明该方法有较高的计算精度, 优于传统结构系统可靠性分析方法, 适合大型结构系统的可靠性分析.     

Dynamical behavior analysis and bifurcation mechanism of a new 3-D nonlinear periodic switching system

This paper presents a new periodic switching chaotic system, which is topologically non-equivalent to the original sole chaotic systems. Of particular interest is that the periodic switching chaotic system can generate stable solution in a very wide parameter domain and has rich dynamic phenomena. The existence of a stable limit cycle with a suitable choice of the parameters is investigated. The complex dynamical evolutions of the switching system composed of the Rössler system and the Chua’s circuit are discussed, which is switched by equal period. Then the possible bifurcation behaviors of the system at the switching boundary are obtained. The mechanism of the different behaviors of the system is investigated. It is pointed out that the trajectories of the system have obvious switching points, which are decided by the periodic signal. Meanwhile, the system may be led to chaos via a period-doubling bifurcation, resulting in the switching collisions between the trajectories and the non-smooth boundary points. The complicated dynamics are studied by virtue of theoretical analysis and numerical simulation. Furthermore, the control methods of this periodic switching system are discussed. The results we have obtained clearly show that the nonlinear switching system includes different waveforms and frequencies and it deserves more detailed research.     

平面滑动型岩质边坡稳定性极限分析上限法

在极限分析理论框架体系的基础上,提出了平面滑动型岩质边坡极限分析上限法。该方法按照外力所做的功率等于内力所消耗的功率,即滑体处于极限状态时两功率相等的条件——虚功率方程,综合考虑作用在岩质边坡上的后缘裂缝静水压力、沿滑面扬压力、重力、水平地震作用力、锚固力等外力,按照强度折减法,推导得出了岩质边坡稳定性评价的极限分析上限解。锦屏一级水电站右岸泄洪洞引渠内侧工程边坡属于典型的多级台阶状岩质高边坡,该边坡岩体结构为层状结构,变形破坏模式为平面滑动。对引渠内侧岩质边坡稳定性分析结果表明,极限分析上限法对平面滑动型岩质边坡有较好的适用性。     

Steady state response analysis for a switched stiffness vibration control system based on vibration energy conversion

A novel semi-active vibration control concept with a serial-switch-stiffness-system was previously presented in our work. Differing from conventional vibration control systems, this system does not dissipate but converts vibration energy as potential energy stored in springs and then reacts against external disturbance. As a piecewise linear system, whether or not energy conversion limit happens is an interesting nonlinear dynamic issue related to the systems steady state response. This paper formulates this issue in depth using the approach called equivalence in control. The systems control force represented by the converted vibration energy is approximately decomposed into two portions. One is responsible for low-frequency free response and the other for high-frequency switching response. An equivalent linear system suffering from a decomposed high-frequency switching force is obtained instead of the original switched system. The steady state response of the disturbed system can be delivered through linear superposition as executed in a linear system. Energy conversion limit occurring in the system under a harmonic disturbance is numerically shown by means of fast Fourier transformation. Analytical formulation and numerical simulation for open- and closed-loop control of the system are further carried out, respectively. The results give that the proposed approach is capable of solving the stead state response of the switched system accurately, and meanwhile, energy conversion limit occurs in the vibration control system indeed. Experimental discussion is also executed.

     

A powerful force-based approach for the limit analysis of three-dimensional frames

For the estimation of the strength of a structure, one could avoid detailed elastoplastic analysis and resort, instead, to direct limit analysis methods that are formulated within linear programming. This work describes the application of the force method to the limit analysis of three-dimensional frames. For the limit analysis of a framed structure, the force method, being an equilibrium-based approach, is better suited than the displacement method and results, generally, to faster solutions. Nevertheless, the latter has been used mostly, since it has a better potential for automation. The difficulty for the direct computerization of the force method is to automatically pick up the structure’s redundant forces. Graph theory concepts may be used to accomplish this task, and a numerical procedure was proposed for the optimal plastic design of plane frames. An analogous approach is developed herein for the limit analysis of space frames which is computationally more cumbersome than the limit analysis of plane frames. The proposed procedure results in hypersparse matrices, and in conjunction with the kinematic upper bound linear program which is solved by a sparse solver, the proposed method appears computationally very efficient. It is also proved that it is much more effective than any displacement-based formulation. The robustness and efficiency of the approach are testified by numerical examples for grillages and multi-storey frames that are included.     

Solution of Fixed Final State Optimal Control Problems via Simple Cell Mapping

A strategy is proposed to solve the fixed final state optimalcontrol problem using the simple cell mapping method. A non-uniform timestep simple cell mapping is developed to create a general database fromwhich solutions of various optimal control problems can be obtained. Atwo-stage backward search algorithm is proposed to eliminate degeneratedpaths often associated with the simple cell mapping. The proposed methodcan accurately delineate the switching curves and eliminate false limitcycles in the solution. The method is applied to two optimal controlproblems with bang-bang control. The well-known minimum time controlproblem of moving a point mass from any initial condition to the originof the phase plane is studied first. This example has exact solutionsavailable which provide a yardstick to examine the accuracy of themethod. The cell size dependence of the solution accuracy is studiednumerically. The second example is a variable stiffness feedback controlproblem with tuning range saturation. The strategy proposed is able toprovide the switching curves in the phase plane. This result has notbeen obtained before.     

Robust adaptive intelligent sliding model control for a class of uncertain chaotic systems with unknown time-delay

In this paper, a robust adaptive intelligent sliding model control (RAISMC) scheme for a class of uncertain chaotic systems with unknown time-delay is proposed. A sliding surface dynamic is appropriately constructed to guarantee the reachability of the specified sliding surface. Within this scheme, neuro-fuzzy network (NFN) is utilized to approximate the unknown continuous function. The robust controller is an adaptive controller used to dispel the unknown uncertainty and approximation errors. The adaptive parameters of the control system are tuned on-line by the derived adaptive laws based on a Lyapunov stability analysis. Using appropriate Lyapunov–Krasovskii (L–K) functional in the Lyapunov function candidate, the uncertainty caused by unknown time delay is compensated and the global asymptotic stability of the error dynamics system in the specified switching surface is accomplished. Finally, the proposed RAISMC system is applied to control a Hopfield neural network, Cellular neural networks, Rössler system, and to achieve synchronization between the Chen system with two time delays with Rössler system without time delay. The results are representative of outperformance of the proposed method in all cases.     

Response analysis of randomly excited nonlinear systems with symmetric weighting Preisach hysteresis

An approximate method for analyzing the response of nonlinear systems with the Preisach hysteresis of the non-local memory under a stationary Gaussian excitation is presented based on the covariance and switching probability analysis. The covariance matrix equation of the Preisach hysteretic system response is derived. The cross correlation function of the Preisach hysteretic force and response in the covariance equation is evaluated by the switching probability analysis and the Gaussian approximation to the response process. Then an explicit expression of the correlation function is given for the case of symmetric Preisach weighting functions. The numerical result obtained is in good agreement with that from the digital simulation. The project supported by the National Natural Science Foundation of China (19972059) and Zhejiang Provincial Natural Science Foundation (101046)     

Upper-bound limit analysis based on the natural element method

The natural element method (NEM) is a newly-developed numerical method based on Voronoi diagram and Delaunay triangulation of scattered points, which adopts natural neighbour interpolation to construct trial functions in the framework of Galerkin method. Owing to its distinctive advantages, the NEM is used widely in many problems of computational mechanics. Utilizing the NEM, this paper deals with numerical limit analysis of structures made up of perfectly rigid-plastic material. According to kinematic theorem of plastic limit analysis, a mathematical programming natural element formulation is established for determining the upper bound multiplier of plane problems, and a direct iteration algorithm is proposed accordingly to solve it. In this algorithm, the plastic incompressibility condition is handled by two different treatments, and the nonlinearity and nonsmoothness of the goal function are overcome by distinguishing the rigid zones from the plastic zones at each iteration. The procedure implementation of iterative process is quite simple and effective because each iteration is equivalent to solving an associated elastic problem. The obtained limit load multiplier is proved to monotonically converge to the upper bound of true solution. Several benchmark examples are investigated to validate the significant performance of the NEM in the application field of limit analysis.     

Computational limit analysis of rigid-plastic bodies in plane strain

This paper deals with the computational methods for limit analysis of plane strain problems. The finite element mathematical programming formula (FE-MPF) for determining the upper bound load multiplier established by the authors earlier is adopted and modified for plane strain problems. The penalty method is used to impose the incompressibility constraint. The FE-MPF is solved by a direct iteration procedure without the need of a searching process. This algorithm is not sensitive to the volumetric locking effect. And it can be easily extended to the limit analysis of three dimensional problems. The results of numerical examples are satisfactory and show the stable convergency of the present algorithm.     

控制存在时滞的线性系统主动控制的滑移模态方法

本文研究控制存在时滞的线性系统的滑模控制方法。在控制时滞量为采样周期的整数倍和非整数倍的两种情况下，通过采用零阶保持器，将包含时滞的连续系统转化为形式上不包含时滞的标准离散线性系统，然后分别进行切换函数和控制律的设计。所得出的切换面和控制律表达式中，除了含有当前的状态反馈外，还包含有前若干步控制项的线形组合。最后对某三自由度结构模型进行仿真计算，验证了所给控制方法的有效性。     

An Eigenvalue Method for Calculation of Stability and Limit Cycles in Nonlinear Systems

In the calculation of periodic oscillations of nonlinear systems –so-called limit cycles – approximative and systematic engineeringmethods of linear system analysis are known. The techniques, working inthe frequency domain, perform a quasi-linearization of the nonlinear system,replacing nonlinearities by amplitude-dependent describing functions.Frequently, the resulting equations for the amplitude and frequency ofpresumed limit cycles are solved directly by a graphical procedure in aNyquist plane or by solving the nonlinear equations or a parameteroptimization problem. In this paper, an indirect numerical approach isdescribed which shows that, for a system of nonlinear differentialequations, the eigenvalues of the quasi-linear system simply indicateall limit cycles and, additionally, yield stability regions for thelinearized case. The method is applicable to systems with multiplenonlinearities which may be static or dynamic. It is demonstrated foran example of aircraft nose gear shimmy dynamics in the presence ofdifferent nonlinearities and the results are compared with those fromsimulation.     

基于降维积分的结构系统可靠性分析研究

复杂结构系统一般具有多个失效模式. 传统系统可靠性分析模型是在假设各失效模式相互独立的条件下建立的. 而在工程实际问题中,由于结构系统的组成单元之间紧密联系,系统的失效模式大多是相互耦合的. 简单地在失效模式相互独立的假设条件下进行系统可靠性分析与评价常常会导致过大的误差,甚至得出错误的结论. 提出一种相关失效模式结构系统可靠性分析方法. 利用降维法和Gauss-Hermite数值积分技术计算随机参数结构系统极限状态函数的统计矩,采用极限状态函数的前四阶累积量拟合其累积量生成函数,通过鞍点逼近方法拟合结构系统极限状态函数的概率密度函数和累积分布函数,进而获取结构系统的可靠度(或失效概率).数值算例表明该方法具有较高的计算精度和效率,通用性强.     

动力调谐陀螺仪脉冲数字再平衡回路中的量程转换

本文从理论上对回路的极限环进行了分析，并作了相应的仿真，结果表明，对于Ｉ型系统来说，采样频率低于８ＫＨｚ时，系统不存在极限环。本文还在设计双刻度开关逻辑的基础上，通过对回路的数字仿真，确定出量程转换时所选用的合适阈值。     

Stability of Nonstationary Automatic-Control Systems of Variable Structure in Forced Motion

Conditions of stability in sliding mode are obtained for nonstationary dynamic automatic-control processes of variable structure. The nonstationary parameters of a control process subjected to external perturbations vary in given ranges, whereas the parameters of the switching plane remain constant for the appropriately chosen parameters of discontinuous logic control laws. The corresponding system of differential equations contains coefficients that vary stepwise in time together with the parameters of the controlled process. The general stability criteria for the solutions of variable-structure systems are used to analyze the stability of motion in sliding mode of an automatic variable-structure control system of the second order     

Periodic solution of the system with impulsive state feedback control

The order-1 periodic solution of the system with impulsive state feedback control is investigated. We get the sufficient condition for the existence of the order-1 periodic solution by differential equation geometry theory and successor function. Further, we obtain a new judgement method for the stability of the order-1 periodic solution of the semi-continuous systems by referencing the stability analysis for limit cycles of continuous systems, which is different from the previous method of analog of Poincarè criterion. Finally, we analyze numerically the theoretical results obtained.     

A new technique to generate independent periodic attractors in the state space of nonlinear dynamic systems

This paper proposes a method to generate several independent periodic attractors, in continuous-time nonchaotic systems (with an equilibrium point or a limit cycle), based on a switching piecewise-constant controller. We demonstrate here that the state space equidistant repartition of these attractors is on a precise zone of a precise curve that depends on the parameters of the system. We determine the state space domains where the attractors are generated from different initial conditions. A mathematical formula giving their maximal number in function of the controller piecewise-constant values is then deduced. Throughout this study, the proposed methodology is illustrated with several examples.     

Tracking control of uncertain switched nonlinear cascade systems: a nonlinear H ∞ sliding mode control method

This paper considers the tracking control problem for a class of uncertain switched nonlinear cascade systems via the multiple Lyapunov functions (MLFs) method. Each subsystem under consideration is composed of two cascade-connected parts: the null space dynamics part and the range space dynamics part. The two main robust control strategies, nonlinear H _∞ control (NHC) and the sliding mode control (SMC), are integrated to function in a complementary manner for tracking control tasks. Furthermore, sufficient conditions for the solvability of the tracking control problem of the switched system and design of both switching laws and controllers are presented. Finally, a simulation example is provided to demonstrate the feasibility of the developed method.