A perturbation method for estimation of dynamic systems

A novel framework called the Perturbed Jth Moment Extended Kalman Filter (PJMEKF), based on a classical perturbation technique is proposed for estimating the states of a nonlinear dynamical system from sensor measurements. This method falls under a class of architectures under investigation primarily to study the interplay of major issues in nonlinear estimation such as nonlinearity, measurement sparsity, and initial condition uncertainty in an environment with low levels of process noise. Taylor series expansion of the departure motion dynamics about the best estimate is used to derive a series representation of the unforced motion. It is found that such series representation evolves as a set of differential equations that force each other in a cascade manner, adding up to give the unforced motion (in a so-called “triangular” structure). This formal perturbation solution for the departure motion dynamics is used in deriving the differential equations governing the time evolution of the high order statistical moments of the estimation error. These tensor differential equations are found to possess a similar high order triangular structure in addition to being symmetric (in N tensorial dimensions and we appropriately term the evolution equations as Tensor Lyapunov Equations of statistical moment perturbations). Elegance of the tensor differential equations thus derived is accompanied by the computational advantages due to symmetry in all tensorial dimensions. A vector matrix representation of tensors is proposed with which the representation and solution of the tensor differential equations can be carried out effectively. Approximations are introduced to incorporate low levels of process noise forcing function in the propagation phase of the moment equations. The statistics thus propagated are used in a filtering framework to estimate the state vector of a nonlinear system from noisy measurements, within the traditional Kalman update paradigm. The Kalman gain thus determined is utilized in updating all high order moments in preparation for the subsequent propagation phase leading to improved estimation accuracy. The filter developed is applied to an orbit estimation problem and comparisons are presented with classical extended Kalman filter.     

Sensitivity Resonance and Attractor Morphing Quantified by Sensitivity Vector Fields for Parameter Reconstruction

A novel method of parameter variation reconstruction for systems exhibiting chaotic dynamics is presented. The algorithm reconstructs variations of system parameters without the need for explicit system equations of motion, or knowledge of the nominal parameter values. The concept of a sensitivity vector field (SVF) is developed. This construct captures geometrical deformations to the dynamical attractor of the system in state space. These fields are collected by means of a proposed unique approach referred to as point cloud averaging (PCA). PCA is applied to discrete time series data from the system with nominal parameter values (healthy) and the system with changed parameters. Test variations are reconstructed from an optimal basis of SVF snapshots which is generated by means of proper orthogonal decomposition. The method is applied to two system models, a magneto-elastic oscillator (MEO) and an atomic force microscope (AFM). The method is shown to be highly accurate, and capable of identifying multiple simultaneous variations. The success of the method as applied to an AFM and a MEO indicates a potential for highly accurate readings by exploiting the geometric features of observed chaotic vibrations. An exciting new phenomenon referred to as sensitivity resonance was also observed, and some implications regarding its use in further improving algorithm performance are discussed.An earlier version of this work has been presented at the 20-th Biennial Conference on Mechanical Vibration and Noise, Long Beach, 2005.     

Neural network reconstruction of fluid flows from tracer-particle displacements

We demonstrate some of the advantages of using artificial neural networks for the post-processing of particle-tracking velocimetry (PTV) data. This study is concerned with the data obtained after particle images have been matched and the obvious outliers have been removed. We show that it is easy to produce simple back-propagation neural networks that can filter the remaining random noise and interpolate between the measurements. They do so by performing a particular form of non-linear global regression that allows them to reconstruct the fluid flow for the entire field covered by the photographs. This is obtained by training these neural networks to learn the fluid dynamics function f that maps the position x of a fluid particle at time t to its position X at time t + Δt. They can do so with a high degree of precision when provided with pairs of matching particle positions (x, X) from only about 2 to 4 pairs of PTV photographs as exemplars. We show that whether they are trained on exact or on noisy data, they learn to interpolate with such a precision that their output is within one pixel of the theoretical output. We demonstrate their accuracy by using them to draw whole streamlines or flow profiles, by iteration from a single starting point. Received: 23 November 1998/Accepted: 14 July 2000     

Dynamics and Control of Initialized Fractional-Order Systems

Due to the importance of historical effects in fractional-order systems,this paper presents a general fractional-order system and control theorythat includes the time-varying initialization response. Previous studieshave not properly accounted for these historical effects. Theinitialization response, along with the forced response, forfractional-order systems is determined. The scalar fractional-orderimpulse response is determined, and is a generalization of theexponential function. Stability properties of fractional-order systemsare presented in the complex w-plane, which is a transformation of thes-plane. Time responses are discussed with respect to pole positions inthe complex w-plane and frequency response behavior is included. Afractional-order vector space representation, which is a generalizationof the state space concept, is presented including the initializationresponse. Control methods for vector representations of initializedfractional-order systems are shown. Finally, the fractional-orderdifferintegral is generalized to continuous order-distributions whichhave the possibility of including all fractional orders in a transferfunction.     

Using a Lumped Parameter Dynamic Model of a Rock Bolt to Produce Training Data for a Neural Network for Diagnosis of Real Data

Ground anchorage systems are used extensively throughout the world as supporting devices for civil engineering structures such as bridges and tunnels. The condition monitoring of ground anchorages is a new area of research, with the long term objective being a wholly automated or semi-automated condition monitoring system capable of repeatable and accurate diagnosis of faults and anchorage post-tension levels. The ground anchorage integrity testing (GRANIT) system operates by applying an impulse of known force by means of an impact device that is attached to the tendon of the anchorage. The vibration signals that arise from this impulse are complex in nature and require analysis to be undertaken in order to extract information from the vibrational response signatures that is relevant to the condition of the anchorage. Novel artificial intelligence techniques are used in order to learn the complicated relationship that exists between an anchorage and its response to an impulse. The system has a worldwide patent and is currently licensed commercially.A lumped parameter dynamic model has been developed which is capable of describing the general frequency relationship with increasing post-tension level as exhibited by the signals captured from real anchorages. The normal procedure with the system is to train a neural network on data that has been taken from an anchorage over a range of post-tension levels. Further data is needed in order to test the neural network. This process can be time consuming, and the lumped parameter dynamic model has the potential of producing data that could be used for training purposes, thereby reducing the amount of time needed on site, and reducing the overall cost of the system's operation.This paper presents data that has been produced by the lumped parameter dynamic model and compares it with data from a real anchorage. Noise is added to the results produced by the lumped parameter dynamic model in order to match more closely the experimental data. A neural network is trained on the data produced by the model, and the results of diagnosis of real data are presented. Problems are encountered with the diagnosis of the neural network with experimental data, and a new method for the training of the neural network is explored. The improved results of the neural network trained on data produced by the lumped parameter dynamic model to experimental data are shown. It is shown how the results from the lumped parameter dynamic model correspond well to the experimental results.     

On the use of neural networks for identification of linear and nonlinear systems

A procedure based on neural networks for the classification of linear and nonlinear systems is presented, using excitation and response data under swept sine excitation. Special attention is paid to the classification and identification of linear and bilinear systems, the latter being considered since they exhibit typical characteristics of cracked systems. The computer simulations show that: (1) using the procedure presented in this paper the trained classification network can reliably classify a linear system and different nonlinear systems; (2) the output of the trained identification neural network for a linear system and a bilinear system can be used as a quantitative indicator of characteristics of bilinear systems having different stiffness ratios (k _(x>0)/k _(x<0)) with respect to the bilinear system used in the training stage; (3) for two-degree-of-freedom systems, the trained network can not only determine the existence of a bilinear stiffness and the magnitude of its stiffness ratio, but also specify which stiffness is bilinear, i.e. indicate its position. These results provide a possibility of using the trained neural networks to detect and locate structural cracks which have the characteristics of bilinear systems.Visiting scholar, from People's Republic of China.     

Periodic orbits,basins of attraction and chaotic beats in two coupled Kerr oscillators

Kerr oscillators are model systems which have practical applications in nonlinear optics. Optical Kerr effect, i.e., interaction of optical waves with nonlinear medium with polarizability χ ⁽³⁾ is the basic phenomenon needed to explain, for example, the process of light transmission in fibers and optical couplers. In this paper, we analyze the two Kerr oscillators coupler and we show that there is a possibility to control the dynamics of this system, especially by switching its dynamics from periodic to chaotic motion and vice versa. Moreover, the switching between two different stable periodic states is investigated. The stability of the system is described by the so-called maps of Lyapunov exponents in parametric spaces. Comparison of basins of attractions between two Kerr couplers and a single Kerr system is also presented.     

Synchronization of chaotic system with uncertain variable parameters by linear coupling and pragmatical adaptive tracking

We study the synchronization of general chaotic systems which satisfy the Lipschitz condition only, with uncertain variable parameters by linear coupling and pragmatical adaptive tracking. The uncertain parameters of a system vary with time due to aging, environment, and disturbances. A?sufficient condition is given for the asymptotical stability of common zero solution of error dynamics and parameter update dynamics by the Ge?CYu?CChen pragmatical asymptotical stability theorem based on equal probability assumption. Numerical results are studied for a Lorenz system and a quantum cellular neural network oscillator to show the effectiveness of the proposed synchronization strategy.     

Identification of Fractional Systems Using an Output-Error Technique

An original method for modeling, simulation and identification of fractional systems in the time domain is presented in this article. The basic idea is to model the fractional system by a state-space representation, where conventional integration is replaced by a fractional one with the help of a non-integer integrator. This operator is itself approximated by a N-dimensional system composed of an integrator and of a phase-lead filter. An output-error technique is used in order to estimate the parameters of the model, including the fractional order N. Simulations exhibit the properties of the identification algorithm. Finally, this methodology is applied to the modeling of the dynamics of a real heat transfer system.     

Singularly perturbed feedback linearization with linear attitude deviation dynamics realization

A new approach for feedback linearization of attitude dynamics for rigid gas jet-actuated spacecraft control is introduced. The approach is aimed at providing global feedback linearization of the spacecraft dynamics while realizing a prescribed linear attitude deviation dynamics. The methodology is based on nonuniqueness representation of underdetermined linear algebraic equations solution via nullspace parametrization using generalized inversion. The procedure is to prespecify a stable second-order linear time-invariant differential equation in a norm measure of the spacecraft attitude variables deviations from their desired values. The evaluation of this equation along the trajectories defined by the spacecraft equations of motion yields a linear relation in the control variables. These control variables can be solved by utilizing the Moore–Penrose generalized inverse of the involved controls coefficient row vector. The resulting control law consists of auxiliary and particular parts, residing in the nullspace of the controls coefficient and the range space of its generalized inverse, respectively. The free null-control vector in the auxiliary part is projected onto the controls coefficient nullspace by a nullprojection matrix, and is designed to yield exponentially stable spacecraft internal dynamics, and singularly perturbed feedback linearization of the spacecraft attitude dynamics. The feedback control design utilizes the concept of damped generalized inverse to limit the growth of the Moore–Penrose generalized inverse, in addition to the concepts of singularly perturbed controls coefficient nullprojection and damped controls coefficient nullprojection to disencumber the nullprojection matrix from its rank deficiency, and to enhance the closed loop control system performance. The methodology yields desired linear attitude deviation dynamics realization with globally uniformly ultimately bounded trajectory tracking errors, and reveals a tradeoff between trajectory tracking accuracy and damped generalized inverse stability. The paper bridges a gap between the nonlinear control problem applied to spacecraft dynamics and some of the basic generalized inversion-related analytical dynamics principles.     

Generalizations of the Lotka-Volterra Population Ecology Model: Theory,Simulation, and Applications

Social scientists have attempted in vain to explain and predict the social phenomenon and particularly the behavior of the social system, with the unsatisfactory result that they were not so successful in terms of the accuracy of the prediction that they started to look into chaos theory. Several authors presented the ability of even the most simple predator-prey models to yield damped and explosive oscillations as well as stable limit cycles. Lotka and Volterra suggested models of population dynamics incorporating interpopulation competition. In this paper, biological population ecology model, especially Lotka-Volterra model is applied to organizations and social systems at large. This paper demonstrates the power of merging system dynamics with population ecology models to assess the sensitivity to initial conditions. The dynamical properties of the generalized Lotka-Volterra model were made by simulations using the Ithink software. The implications of using simulation in the analysis of chaotic behavior are presented.     

Applications of Nonlinear Dynamical Systems Theory in Developmental Psychology: Motor and Cognitive Development

Applications of nonlinear dynamical systems theory to psychology have led to recent advances in understanding neuromotor development and advances in theories of cognitive development. This article reviews published findings associated with a specific coherent and influential application from which a theory of adaptive, self-organized cognition has been derived and related to a theory of developmental dynamics of the neuromotor system. The review focuses on implications of the two theories for quantifying developmental phenomena, and suggests a method for quantifying the cognitive theory.     

Multibody graph transformations and analysis

This is the second part of a two-part paper that develops graph theoretic techniques for the topological transformation and analysis of multibody system dynamics. The first part focused on tree systems, and developed systematic and rigorous techniques for the partitioning, aggregation, and substructuring of multibody dynamics models. This second part, uses the aggregation techniques as the foundation to develop the constraint-embedding technique that enables the transformation of the nontree system graphs into tree graphs. This enables the application of a large family of analytical and computational techniques for trees to closed-chain systems. This is illustrated through an extension of the low-order articulated-body forward dynamics algorithm for tree systems to closed-chain systems.     

The adjoining cell mapping and its recursive unraveling,part I: Description of adaptive and recursive algorithms

A new type of cell mapping, referred to as an adjoining cell mapping, is developed in this paper for autonomous dynamical systems employing the cellular state space. It is based on an adaptive time integration employed to compute an associated cell mapping for the system. This technique overcomes the problem of determining an appropriate duration of integration time for the simple cell mapping method. Employing the adjoining mapping principle, the first type of algorithm developed here is an adaptive mapping unraveling algorithm to determine equilibria and limit cycles of the dynamical system in a way similar to that of the simple cell mapping. In addition, it is capable of providing useful information regarding the behavior of dynamical systems possessing pathological dynamics and of systems with rapidly changing vector field. The adjoining property inherent in the adjoining cell mapping method, in general, permits development of new recursive algorithms for unraveling dynamics. The required computer memory for a practical implementation of such algorithms is considerably less than that required by the simple cell mapping algorithm since they allow for a recursive partitioning of state space for trajectory analysis. The second type of algorithm developed in this paper is a recursive unraveling algorithm based on adaptive integration and recursive partitioning of state space into blocks of cells with a view toward its practical implementation. It can find equilibria of the system in the same manner as the simple cell mapping method but is more efficient in locating periodic solutions.     

Spatially localized vibrations in a rotor subjected to flutter

The current push toward lightweight structures in aerospace and aeronautical engineering is leading to slender design airfoils, which are more likely to undergo large deformation, hence experiencing geometrical nonlinearities. The problem of vibration localization in a rotor constituted by N coupled airfoils with plunge and pitch degrees of freedom subjected to flutter instability is considered. For a single airfoil, it is shown that depending on the system parameters, multiple static and dynamic equilibria coexist which may be a fixed point, a limit cycle, or irregular motion. By elastically coupling N airfoils, a simplified rotor model is obtained. The nonlinear dynamical response of the rotor is studied via time integration with particular attention to the emergence of localized vibrating solutions, which have been classified introducing a localization coefficient. Finally, the concept of basin stability is exploited to ascertain the likelihood of the system to converge to a certain localized state as a function of the airstream velocity. We found that homogeneous and slightly localized states are more likely to appear with respect to strongly localized states.

     

柔性关节-柔性臂空间机器人的神经网络自适应反演控制及双重柔性振动抑制

空间机器人系统的柔性主要体现在空间机器人的臂杆和连接各臂杆之间的铰关节。由于空间机器人系统结构的复杂性,以往研究人员对同时具有柔性关节和柔性臂的系统关注不够。为此探讨了参数未知柔性关节-柔性臂空间机器人系统的动力学模拟、轨迹跟踪控制算法设计和关节、臂杆双重柔性振动的主动抑制问题。首先,采用多体动力学建模方法并结合漂浮基空间机器人固有的线动量和角动量守恒动力学特性,推导了系统的动力学方程。以此为基础,考虑到空间机器人实际应用中各关节铰具有较强柔性的情况,引入一种关节柔性补偿控制器解决了传统奇异摄动法应用受关节柔性限制问题,导出了适用于控制系统算法设计的数学模型。然后,利用该模型,基于反演思想在慢时标子系统中设计神经网络自适应控制算法来补偿系统参数未知和柔性关节引起的转动误差,实现系统运动轨迹跟踪性能;针对快时标子系统,设计了鲁棒最优控制算法抑制因柔性关节及柔性臂引起的系统双重弹性振动,保证系统的稳定性。最后,通过仿真对比实验验证了所设计控制算法的有效性。     

Solution Structure of Multi-layer Neural Networks with Initial Condition

This paper studies the initial value problem of multi-layer cellular neural networks. We demonstrate that the mosaic solutions of such system is topologically conjugated to a new class in symbolic dynamical systems called the path set (Abram and Lagarias in Adv Appl Math 56:109–134, 2014). The topological entropies of the solution, output, and hidden spaces of a multi-layer cellular neural network with initial condition are formulated explicitly. Also, a sufficient condition for whether the mosaic solution space of a multi-layer cellular neural network is independent of initial conditions is addressed. Furthermore, two spaces exhibit identical topological entropy if and only if they are finitely equivalent.     

Fractional-order excitable neural system with bidirectional coupling

Fractional-order dynamics is applicable to biological excitable systems with strong interactions or systems with long-term memory effect. The activity of neural membrane voltage depends on the long-range correlations of ionic conductances. Such a behavior of the membrane voltage with long-range correlation can be better described with a fractional-order dynamics. A fractional-order coupled modified three-dimensional (3D) Morris–Lecar (M–L) neural system has been presented to show the variations in the firing patterns from resting state \( \rightarrow \) oscillatory pattern \( \rightarrow \) bursting and the synchronous behavior by designing a bidirectional coupling mechanism. The fractional exponents are lying between 0 and 1. The predominant controller of the changes of firing behavior is the fractional exponent. The stability of synchronization and nature of the fractional system dynamics have been analyzed. To make the investigations more convincing and biologically plausible, we consider a network of M–L oscillators with bidirectional synaptic coupling functions using global type connections and present the effectiveness of the coupling scheme.     

Application of Nonlinear Normal Mode Analysis to the Nonlinear and Coupled Dynamics of a Floating Offshore Platform with Damping

The nonlinear dynamics of ships and floating offshore platforms hasattracted much attention over the last several years. However the topicof multiple-degrees-of-freedom systems has almost been completely ignoredwith very few exceptions. This is probably due to the complexity ofanalyzing strongly nonlinear and coupled systems. It turns out thatcoupling may be particularly important for certain critical dynamicssuch as the dynamics of a floating offshore platform about its diagonalaxis. In a previous work, Kota et al. [1] applied the recently developed nonlinearnormal mode technique to analyze the coupled nonlinear dynamics of afloating offshore platform. Although this previous work was restrictedto unforced and undamped systems, in this work a comparison of the twoalternative nonlinear normal mode analysis techniques was completed.Considering the relative practical importance of damping versus externalforcing for this system, in the present work, we utilize just one of thetwo major techniques available [2] to analyze damped multiple-degrees-of-freedom nonlinear dynamics. Specifically, we investigate the effect ofnonlinearity, and non-proportionate damping. Our results show that thistechnique allows one to simply consider the effect of nonlinearity andgeneral damping on the resulting normal modes. This technique isparticularly powerful because it allows one to visualize the modes in ageometric fashion using the invariant manifold concept from dynamicalsystems.