Identification of Friction Coefficients in the Hinge of a Controlled Physical Pendulum Using the Amplitudes of Steady-State Oscillations

A dynamic model of a controlled physical pendulum is considered. The Pontryagin method of searching for the periodic solutions to near-Hamiltonian systems is used to formulate a programmed law of pendulum oscillations such that the test modes of oscillations become steady and orbitally stable. An approach to identify the friction parameters in the hinge of the pendulum is proposed for the case of the active motor mode. This approach is based on the data available about the integral characteristics of motion. The motion of the system under consideration is numerically simulated.     

Response and bifurcation analysis of a MDOF rotor system with a strong nonlinearity

A new HB (Harmonic Balance)/AFT (Alternating Frequency Time) method is further developed to obtain synchronous and subsynchronous whirling response of nonlinear MDOF rotor systems. Using the HBM, the nonlinear differential equations of a rotor system can be transformed to algebraic equations with unknown harmonic coefficients. A technique is applied to reduce the algebraic equations to only those of the nonlinear coordinates. Stability analysis of the periodic solutions is performed via perturbation of the solutions. To further reduce the computational time for the stability analysis, the reduced system parameters (mass, damping, and stiffness) are calculated in terms of the already known harmonic coefficients. For illustration, a simple MDOF rotor system with a piecewise-linear bearing clearance is used to demonstrate the accuracy of the calculated steady-state solutions and their bifurcation boundaries. Employing ideas from modern dynamics theory, the example MDOF nonlinear rotor system is shown to exhibit subsynchronous, quasi-periodic and chaotic whirling motions.     

Simplified global fault accommodation control design of uncertain nonlinear systems with unknown time-varying powers

This paper addresses the problem of global robust fault accommodation tracking for a class of uncertain nonlinear systems with unknown powers and actuator faults. It is assumed that the powers of the concerned system are unknown time-varying functions, all system nonlinearities are unknown, and unknown actuator faults depend on the time-varying power of a control input. A fault accommodation state-feedback controller is explicitly constructed based on the nonlinear error transformation technique using time-varying performance functions. Global tracking with the preselected performance bounds is established in the presence of unknown time-varying powers and unexpected actuator faults. Different from the previous results dealing with the problem of unknown time-varying powers, the proposed tracking strategy does not require the knowledge of the bounds of the time-varying powers and the nonlinear bounding functions of system nonlinearities. An underactuated mechanical system is simulated to validate the effectiveness of the proposed theoretical approach.     

On the identification of non-linear mechanical systems using orthogonal functions

Real world mechanical systems present non-linear behavior and in many cases simple linearization in modeling the system would not lead to satisfactory results. Coulomb damping and cubic stiffness are typical examples of system parameters currently used in non-linear models of mechanical systems. This paper uses orthogonal functions to represent input and output signals. These functions are easily integrated by using a so-called operational matrix of integration. Consequently, it is possible to transform the non-linear differential equations of motion into algebraic equations. After mathematical manipulation the unknown linear and non-linear parameters are determined. Numerical simulations, involving single and two degree-of-freedom mechanical systems, confirm the efficiency of the above methodology.     

Sliding mode control with adaptive fuzzy dead-zone compensation for uncertain chaotic systems

The dead-zone nonlinearity is frequently encountered in many industrial automation equipments and its presence can severely compromise control system performance. In this work, an adaptive variable structure controller is proposed to deal with a class of uncertain nonlinear systems subject to an unknown dead-zone input. The adopted approach is primarily based on the sliding mode control methodology but enhanced by an adaptive fuzzy algorithm to compensate the dead-zone. Using Lyapunov stability theory and Barbalat??s lemma, the convergence properties of the closed-loop system are analytically proven. In order to illustrate the controller design methodology, an application of the proposed scheme to a chaotic pendulum is introduced. A comparison between the stabilization of general orbits and unstable periodic orbits embedded in chaotic attractor is carried out showing that the chaos control can confer flexibility to the system by changing the response with low power consumption.     

Stabilization of the cart pole system: by sliding mode control

This paper presents a control strategy designed as a combination of a PD controller and a twisting-like algorithm to stabilize the damped cart pole system, provided that the pendulum is initially placed within the upper-half plane. To develop the strategy, the original system is transformed into a four-order chain of integrator form, where the damping force is included through an additional nonlinear perturbation. The strategy consists of simultaneously bringing the position and velocity of the pendulum to within a compact region by applying the PD controller. Meanwhile, the system state variables are brought to the origin by the twisting-like algorithm. The corresponding convergence analysis is done using several Lyapunov functions. The control strategy is illustrated with numerical simulations.     

An optimal type II fuzzy sliding mode control design for a class of nonlinear systems

This study deals with the problem of controlling a class of uncertain nonlinear systems in the presence of external disturbances. To achieve this goal, a new Optimal Type-2 Fuzzy Sliding Mode Controller (OT2FSMC) is introduced. In the proposed controller, a novel heuristic algorithm, namely particle swarm optimization with random inertia weight (RNW–PSO), is employed. To achieve an optimal performance, the parameters of the proposed controller as well as the input and output membership functions are optimized simultaneously by RNW–PSO. The globally asymptotic stability of the closed-loop system is mathematically proved. Finally, this method of control is applied to the inverted pendulum system as a case study. Simulation results show the system performance is desirable.     

基于荷载形函数的大跨桥梁结构移动荷载识别

大跨桥梁结构中的移动荷载识别是结构健康监测的重要组成部分之一,其作为动力学中的反问题,存在唯一性及稳定性等病态问题。本文首先推导出移动荷载作用下结构响应的卷积形式的离散公式,然后利用有限元理论中的形函数拟合移动荷载,推导出基于荷载形函数的移动荷载识别表达式。将移动荷载的识别转化为荷载形函数拟合系数的识别,降低了需要识别的未知量,减小了逆运算的计算量,消除或减弱逆问题病态性,并提高了计算效率。利用某大跨刚构-连续预应力混凝土桥梁修正后的有限元模型进行数值仿真,考虑路面粗糙度,由模态叠加法计算移动荷载作用下的响应;然后采用荷载形函数方法识别移动荷载,证明该方法在5%以下的噪声时能快速并精确识别复杂桥梁结构的移动荷载。     

Identification for disturbed MIMO Wiener systems

The identification of Multi-input Multi-output (MIMO) Wiener systems is concerned in this paper. The system presented is comprised of a multi-dimensional linear subsystem and a memory-less nonlinear block which is made of discontinuous asymmetric piece-wise linear functions. A recursive algorithm is proposed to estimate all the unknown parameters of the system with interference noises. It is shown that the recursive algorithm for the disturbed MIMO Wiener system is convergent. Finally, some simulation results illustrate the identification accuracy and the convergence rate.     

具有稳定数值解的三维谐振子

谐振子广泛应用于物理系统的描述和物理现象的数值模拟。由于二维或三维谐振子对于系统参数、初始条件和边界条件的高度敏感性,很多物理过程的动力学模拟都会出现数值解不稳定的现象。近年来发展的无网格法、物质点法和近场动力学法等数值模拟方法均绕开了对固体材料固有构形的量化描述。本文引入了定常耗散项和弹簧耗散项,考虑随机微扰效应,提出了一种三维耗散谐振子,构建了基于蛙跳法和边界松弛技术的数值积分算法。应用三维谐振子构建了耗散型弹簧摆、简化弦和简化梁三个模型,设定了13个定解问题进行动力学模拟。数值试验结果表明,三维谐振子是稳定的。基于简化弦模型,模拟了拉弦、放弦和重弦三个有界弦振动问题;其中,拉弦和放弦问题成功模拟了有界弦的三维振形;重弦问题模拟再现了悬链线在水平向的微幅振荡现象。基于简化梁模型,模拟了三维梁的拉伸、剪切和扭转行为,验证了三维谐振子对于非线性大变形问题动力学模拟的描述能力,及其对外部作用的高速响应能力。本文方法可以为弦振动问题和材料力学非线性大变形问题的动力学模拟提供一条可行的实现途径。     

On damping properties of a frictionless physical pendulum with a moving mass

The damping effects are generated in a frictionless oscillating physical pendulum by a continuous motion of an auxiliary mass. The main parameters affecting the damping properties of the pendulum-mass system are identified. In particular, the effective damping ratio for a cycle is introduced and derived in a closed form from the energy considerations and then independently from Mathieu's equation. It is shown that a continuous damping can be achieved if the mass motion is synchronized with the pendulum rotation. Otherwise the system becomes prone to ‘beating’ phenomenon. The results presented may be useful for design of active control strategy of autonomous systems with negligible passive damping.     

Property and evolution of the running-in attractor in an actual dynamic system

In this paper, the issue of adaptive neural tracking control for uncertain switched multi-input multi-output (MIMO) nonstrict-feedback nonlinear systems with average dwell time is studied. The system under consideration includes unknown dead-zone inputs and output constraints. The uncertain nonlinear functions are identified via neural networks. Also, neural networks-based switched observer is constructed to approximate all unmeasurable states. By means of the information for dead-zone slopes and barrier Lyapunov function (BLF), the problems of dead-zone inputs and output constraints are tackled. Furthermore, dynamic surface control (DSC) scheme is employed to ensure that the computation burden is greatly reduced. Then, an observer-based adaptive neural control strategy is developed on the basis of backstepping technique and multiple Lyapunov functions approach. Under the designed controller, all the signals existing in switched closed-loop system are bounded, and system outputs can track the target trajectories within small bounded errors. Finally, the feasibility of the presented control algorithm is proved via simulation results.

     

Controlling the inverted pendulum by means of a nested saturation function

A nonlinear controller is presented for the stabilization of the underactuated inverted pendulum mounted on a cart. The fact that this system can be expressed as a chain of integrators, with an additionally nonlinear perturbation, allows us to use a nested saturation control technique to bring the pendulum to the top position, with zero displacement of the cart. The obtained closed-loop system is semiglobal, asymptotically stable, and locally exponentially stable, under the assumption that the position of the angle is initialized above the upper half plane.     

Analysis of time-periodic nonlinear dynamical systems undergoing bifurcations

In this study a new procedure for analysis of nonlinear dynamical systems with periodically varying parameters under critical conditions is presented through an application of the Liapunov-Floquet (L-F) transformation. The L-F transformation is obtained by computing the state transition matrix associated with the linear part of the problem. The elements of the state transition matrix are expressed in terms of Chebyshev polynomials in timet which is suitable for algebraic manipulations. Application of Floquet theory and the eigen-analysis of the state transition matrix at the end of one principal period provides the L-F transformation matrix in terms of the Chebyshev polynomials. Since this is a periodic matrix, the L-F transformation matrix has a Fourier representation. It is well known that such a transformation converts a linear periodic system into a linear time-invariant one. When applied to quasi-linear equations with periodic coefficients, a dynamically similar system is obtained whose linear part is time-invariant and the nonlinear part consists of coefficients which are periodic. Due to this property of the L-F transformation, a periodic orbit in original coordinates will have a fixed point representation in the transformed coordinates. In this study, the bifurcation analysis of the transformed equations, obtained after the application of the L-F transformation, is conducted by employingtime-dependent center manifold reduction andtime-dependent normal form theory. The above procedures are analogous to existing methods that are employed in the study of bifurcations of autonomous systems. For the two physical examples considered, the three generic codimension one bifurcations namely, Hopf, flip and fold bifurcations are analyzed. In the first example, the primary bifurcations of a parametrically excited single degree of freedom pendulum is studied. As a second example, a double inverted pendulum subjected to a periodic loading which undergoes Hopf or flip bifurcation is analyzed. The methodology is semi-analytic in nature and provides quantitative measure of stability when compared to point mappings method. Furthermore, the technique is applicable also to those systems where the periodic term of the linear part does not contain a small parameter which is certainly not the case with perturbation or averaging methods. The conclusions of the study are substantiated by numerical simulations. It is believed that analysis of this nature has been reported for the first time for this class of systems.     

A logarithmic complexity floating frame of reference formulation with interpolating splines for articulated multi-flexible-body dynamics

An interpolating spline-based approach is presented for modeling multi-flexible-body systems in the divide-and-conquer (DCA) scheme. This algorithm uses the floating frame of reference formulation and piecewise spline functions to construct and solve the non-linear equations of motion of the multi-flexible-body system undergoing large rotations and translations. The new approach is compared with the flexible DCA (FDCA) that uses the assumed modes method [1]. The FDCA, in many cases, must resort to sub-structuring to accurately model the deformation of the system. We demonstrate, through numerical examples, that the interpolating spline-based approach is comparable in accuracy and superior in efficiency to the FDCA. The present approach is appropriate for modeling flexible mechanisms with thin 1D bodies undergoing large rotations and translations, including those with irregular shapes. As such, the present approach extends the current capability of the DCA to model deformable systems. The algorithm retains the theoretical logarithmic complexity inherent in the DCA when implemented in parallel.     

Global exponential stability of switched systems

This paper proposes a method for the stability analysis of deterministic switched systems. Two motivational examples are introduced (nonholonomic system and constrained pendulum). The finite collection of models consists of nonlinear models, and a switching sequence is arbitrary. It is supposed that there is no jump in the state at switching instants, and there is no Zeno behavior, i.e., there is a finite number of switches on every bounded interval. For the analysis of deterministic switched systems, the multiple Lyapunov functions are used, and the global exponential stability is proved. The exponentially stable equilibrium of systems is relevant for practice because such systems are robust to perturbations.     

Lyapunov-Based Controller for the Inverted Pendulum Cart System

A nonlinear control force is presented to stabilize the under-actuated inverted pendulum mounted on a cart. The control strategy is based on partial feedback linearization, in a first stage, to linearize only the actuated coordinate of the inverted pendulum, and then, a suitable Lyapunov function is formed to obtain a stabilizing feedback controller. The obtained closed-loop system is locally asymptotically stable around its unstable equilibrium point. Additionally, it has a very large attraction domain.Contributed by Prof. F. Pfeiffer.     

A New Methodology for the Analysis of Periodic Systems

A new numerical-analytical method for the combined global-localanalysis of nonlinear periodic systems referred to as an ExpandedPoint Mapping (EPM) is presented. This methodology combines thecell to cell mapping and point mapping methods toinvestigate the basins of attraction and stability characteristics ofequilibrium points and periodic solutions of nonlinear periodicsystems. The proposed method is applicable to multi-degrees-of-freedomsystems, multi-parameter systems, and allows analytical studies oflocal stability characteristics of steady state solutions. Inaddition, the EPM approach allows the study of stabilitycharacteristics as function of system parameters to obtain analyticalconditions for bifurcation. In the paper, the theoretical basis forthe EPM method is formulated and a procedure for the analysis ofnonlinear dynamical systems is presented. Analysis of a pendulum witha periodically excited support in the plane is used to illustrate themethod. The results demonstrate the efficiency and accuracy of theproposed approach in analyzing nonlinear periodic systems.     

Directly adaptive fuzzy control of discrete-time chaotic systems by least squares algorithm with dead-zone

A new design scheme of directly adaptive fuzzy control for a class of discrete-time chaotic systems is proposed in this paper. The T-S fuzzy model is employed to represent the discrete-time chaotic systems. Then a fuzzy controller is designed and the unknown coefficients of the controller are identified by least squares algorithm with dead-zone. By Lyapunov method, all the signals involved in the closed-loop systems are shown to be bounded and the error between the system output and the reference output is proved to converge to a small neighborhood of zero. Simulation results demonstrate the effectiveness of the theoretical results.     

Full- and reduced-order synchronization of a class of time-varying systems containing uncertainties

Investigation on chaos synchronization of autonomous dynamical systems has been largely reported in the literature. However, synchronization of time-varying, or nonautonomous, uncertain dynamical systems has received less attention. The present contribution addresses full- and reduced-order synchronization of a class of nonlinear time-varying chaotic systems containing uncertain parameters. A unified framework is established for both the full-order synchronization between two completely identical time-varying uncertain systems and the reduced-order synchronization between two strictly different time-varying uncertain systems. The synchronization is successfully achieved by adjusting the determined algorithms for the estimates of unknown parameters and the linear feedback gain, which is rigorously proved by means of the Lyapunov stability theorem for nonautonomous differential equations together with Barbalat’s lemma. Moreover, the synchronization result is robust against the disturbance of noise. We illustrate the applicability for full-order synchronization using two identical parametrically driven pendulum oscillators and for reduced-order synchronization using the parametrically driven second-order pendulum oscillator and an additionally driven third-order Rossler oscillator.