Multidimensional Dynamic System Research Articles

Application of neural networks for synthesizing flight control algorithms. I neural network inverse dynamics method for aircraft flight control

The application of artificial neural networks for aircraft motion control, in particular, for creation of nonlinear algorithms of the aircraft remote control system (RCS) is considered. Aircraft as a control object is represented as a multidimensional nonlinear dynamic system and nonlinear control methods are used to operate this system. The control loop is constructed using the method of inverse dynamics based on the feedback linearization principle. The nonlinear control law is represented as a neural network being learned (adjusted) by recorded or incoming measurements of motion parameters. Synthesis and testing of neural network control algorithms is performed with the fully nonlinear mathematical model of a maneuverable aircraft for three control channels. Simulation results of the closed system are presented.

한강수계에서의 다차원 시변화 유기물 및 영양상태 모델 연구

Multidimensional dynamic water quality model of organic matter and trophic state was applied to the Han River system. The model was calibrated using field measurement data obtained during the year of 2007. The model results showed reasonable performance in predicting temporal variations of TN, TP, Chl-a and BOD concentrations. The applied integrated modeling system can be effectively used to simulate water quality as well as hydrodynamic and water temperature for river-lake continuous system in the Han River. Utilizing the calibrated model, we analyzed the spatial and temporal distributions of TN, TP, Chl-a and BOD concentrations in the Han River system. The temporal variations of water quality at each river reach and lake were effectively simulated with the developed model and spatial distribution of water qualities in the Han River system could be compared. The multidimensional dynamic modeling system can simulate the water qualities of entire waterbody where Lake Paldang and the incoming flows are included using single modeling system. So it can be effectively used for integrated water quality management of the Han River system.

A Nonlinear Observer via Augmented Linear System based on Formal Linearization using Discrete Fourier Expansion

 Abstract—In order to expand a formal linearization method on designing an observer for nonlinear dynamic and measurement systems, we exploit the discrete Fourier expansion that can reduce computational complexity. A nonlinear multidimensional dynamic system is considered by a nonlinear ordinary differential equation, and a measurement equation is done by a nonlinear equation. Defining a linearization function which consists of the trigonometric functions considered up to the higher-order, the nonlinear dynamic system is transformed into an augmented linear one with respect to this linearization function by using the discrete Fourier expansion. Introducing an augmented measurement vector which consists of polynomials of measurement data, the measurement equation is transformed into an augmented linear one with respect to the linearization function in the same way. To these augmented linearized systems, a linear estimation theory is applied to design a nonlinear observer. Index Terms—nonlinear system, nonlinear observer, formal linearization, discrete Fourier expansion

Numerical methods for determine the equilibrium point's stability of multidimensional nonlinear dynamic system

SUMMARYThis work is concerned with the use of new nunumerical methods to establish some new criteria to determine the equilibrium point's stability of a multidimensional nonlinear dynamic system. Copyright © 2010 John Wiley & Sons, Ltd.

Achieving consensus and synchronization by adapting the network topology

AbstractIn this paper, we present a novel control strategy for achieving consensus via network evolution. Namely, we associate to each edge a multi-dimensional dynamical system forced by the state mismatch of its nodes. We consider the case in which the network evolution is driven by a multi-well potential, whose wells define the possible asymptotic levels of interactions between each pair of nodes. We show that the proposed network evolution supports asymptotic consensus while leading to an emerging steady-state weighted topology. Moreover, we illustrate how a proper tuning of the potential parameters can be used to tailor the properties of this emerging topology. Finally, numerical simulations confirm the theoretical predictions as well as the viability of this approach for synchronizing nonlinear dynamical systems.

УРОВЕНЬ ЭТНИЧЕСКОЙ ИНТЕГРИРОВАННОСТИ ОБРАЗНОЙ СФЕРЫ ЛИЧНОСТИ ПОТЕНЦИАЛЬНЫХ ЭМИГРАНТОВ

The article focuses on a level of ethnic integration of an image sphere of a potential immigrant personality. The concept of an «image sphere» is interpreted as a determinate multidimensional dynamic system of secondary psychic images and as a system that can fulfill integrating/differentiating function with/from an actual ethnosphere. The author introduces a coefficient of ethnic integration of an image sphere (Ie), the coefficient being determined by calculating the ratio of an observed number of positively and significantly experienced ethnically coordinated elements of an image sphere of a person to a number of ethnodifferentiating image elements.

Multi-dimensional particle swarm optimization in dynamic environments

Particle swarm optimization (PSO) was proposed as an optimization technique for static environments; however, many real problems are dynamic, meaning that the environment and the characteristics of the global optimum can change in time. In this paper, we adapt recent techniques, which successfully address several major problems of PSO and exhibit a significant performance over multi-modal and non-stationary environments. In order to address the pre-mature convergence problem and improve the rate of PSO’s convergence to the global optimum, Fractional Global Best Formation (FGBF) technique is used. FGBF basically collects all the best dimensional components and fractionally creates an artificial Global Best particle ( aGB) that has the potential to be a better “guide” than the PSO’s native gbest particle. To establish follow-up of local optima, we then introduce a novel multi-swarm algorithm, which enables each swarm to converge to a different optimum and use FGBF technique distinctively. Finally for the multi-dimensional dynamic environments where the optimum dimension also changes in time, we utilize a recent PSO technique, the multi-dimensional (MD) PSO, which re-forms the native structure of the swarm particles in such a way that they can make inter-dimensional passes with a dedicated dimensional PSO process. Therefore, in a multi-dimensional search space where the optimum dimension is unknown, swarm particles can seek for both positional and dimensional optima. This eventually pushes the frontier of the optimization problems in dynamic environments towards a global search in a multi-dimensional space, where there exists a multi-modal problem possibly in each dimension. We investigated both standalone and mutual applications of the proposed methods over the moving peaks benchmark (MPB), which originally simulates a dynamic environment in a unique (fixed) dimension. MPB is appropriately extended to accomplish the simulation of a multi-dimensional dynamic system, which contains dynamic environments active in several dimensions. An extensive set of experiments show that in traditional MPB application domain, FGBF technique applied with multi-swarms exhibits an impressive speed gain and tracks the global peak with the minimum error so far achieved with respect to the other competitive PSO-based methods. When applied over the extended MPB, MD PSO with FGBF can find optimum dimension and provide the (near-) optimal solution in this dimension.

Estimation of the matrices of parameters and covariations of the perturbation vectors in multidimensional discrete-time dynamic systems under special structure of the unknown covariance matrices

For the multidimensional linear dynamic system obeying a difference equation with an unknown covariance matrix of the vector of random perturbations having dependent components, consideration was given to estimation of the matrix of system parameters and the covariance matrix represented by a linear combination of the given symmetrical matrices. The family of joint probability densities of the observation vector was factorized, and the sufficient statistics was determined. For the estimates of the maximum likelihood of the system parameter matrices and the estimates of the coefficients of expansion of the covariance matrix, equations were presented. Developed was a recurrent procedure for joint estimation of the system parameter matrices and the covariance matrix with arrival of observations.

Localized Instability and Attraction along Invariant Manifolds

We derive a simple criterion for transverse instabilities along a general invariant manifold of a multidimensional dynamical system. The criterion requires an appropriately defined normal infinitesimal Lyapunov exponent (NILE) to be positive over regions of transverse instability on the manifold. Unlike classic Lyapunov-type numbers in the theory of normally hyperbolic invariant manifolds, the NILE can be computed analytically in applications. This enables us to locate, for example, regions of transient jumping along an invariant manifold that is otherwise globally attracting. To illustrate our results, we determine the locations of intermittent instabilities in bubble motion past a cylinder, in predator-prey interactions, and in soft-stiff structural systems.

Understanding neurodynamical systems via Fuzzy Symbolic Dynamics

Neurodynamical systems are characterized by a large number of signal streams, measuring activity of individual neurons, local field potentials, aggregated electrical (EEG) or magnetic potentials (MEG), oxygen use (fMRI) or activity of simulated neurons. Various basis set decomposition techniques are used to analyze such signals, trying to discover components that carry meaningful information, but these techniques tell us little about the global activity of the whole system. A novel technique called Fuzzy Symbolic Dynamics (FSD) is introduced to help in understanding of the multidimensional dynamical system's behavior. It is based on a fuzzy partitioning of the signal space that defines a non-linear mapping of the system's trajectory to the low-dimensional space of membership function activations. This allows for visualization of the trajectory showing various aspects of observed signals that may be difficult to discover looking at individual components, or to notice otherwise. FSD mapping can be applied to raw signals, transformed signals (for example, ICA components), or to signals defined in the time-frequency domain. To illustrate the method two FSD visualizations are presented: a model system with artificial radial oscillatory sources, and the output layer (50 neurons) of Respiratory Rhythm Generator (RRG) composed of 300 spiking neurons.

Further results on the boundedness of multidimensional systems

A sufficient condition is derived in this paper for the boundedness of a linear time invariant (LTI) multidimensional (MD) dynamic system over a special class of frequency domains. These frequency domains are able to include many practically adopted frequency domains, such as fans, rectangles, etc., as a special situation, as well as to give an approximation with a tolerable accuracy to many other physically significant frequency domains like ellipsoids and diamonds. This condition becomes also necessary when the prescribed frequency domain is path-connected. Moreover, it is expressed in a linear matrix inequality (LMI) form and can be directly applied to the optimization of system output matrix and direct transmission matrix, as well as the minimization of the frequency response bound. In addition, the dimension of the LMI is in the same order as that of the system matrices.

Reduction of the Rosenbrock matrix in analysis of invariant zeros of the linear MIMO-System

The problem is considered of reduction of the Rosenbrock matrix in analysis of invariant zeros of a linear multidimensional dynamical system with many inputs and many outputs. Approaches to the Rosenbrock matrix reduction are compared, which are carried out with the aid of the transformation of initial system equations to the Yokoyama canonical form and on the basis of zero divisors of numerical matrices. It is shown that the approach to the Rosenbrock matrix reduction on the basis of zero divisors is less arduous and offers the possibility of obtaining a well-posed problem in eigenvalues of the generalized linear bundle, while the reduction on the basis of subspaces of A.N. Krylov leads to the problem in eigenvalues of the nonlinear (polynomial) bundle of matrices, and its stipulation is generally rather poor. Various versions of the reduced matrices are presented, which are obtained in accordance with properties of the dynamical system. Properties of the Rosenbrock reduced matrices are analyzed.

Verification of the Mathematical Model of a Multidimensional Dynamic System for Adequacy Based on the Spectral Norm of a Residual Matrix

An approach is suggested to the operative verification for adequacy of the mathematical model of a multidimensional dynamic system on the basis of the statistical analysis of a sequence of residuals between output coordinates of the system and those of its mathematical model. Here, a new statistic, namely, the spectral norm of a normalized matrix of residuals, is introduced and a method of defining the upper confidence bound of this statistic is given.

Minimax bounds in nonparametric estimation of multidimensional deterministic dynamical systems

We consider the problem of estimating a multidimensional discrete deterministic dynamical system from the first n + 1 observations. We exhibit the optimal rate function rn and show that the nearest neighbor estimator achieves this optimal rate, extending a recent study carried out by Guerre and Maes [18] in the univariate case. Previous results strongly rely on the natural order structure of the set of real numbers. Here the optimal rate function is defined from multidimensional spacings which are edge lengths of simplices associated with a triangulation of the Voronoi cells built from the observations. A simulated example illustrates the theory.

General approach to the localization of unstable periodic orbits in chaotic dynamical systems

We present a method to detect the unstable periodic orbits of a multidimensional chaotic dynamical system. Our approach allows us to locate in an efficient way unstable cycles of, in principle, arbitrary length with a high accuracy. Based on a universal set of linear transformations the originally unstable periodic orbits are transformed into stable ones, and can consequently be detected and analyzed easily. This method is applicable to dynamical systems of any dimension, and requires no preknowledge with respect to the solutions of the original chaotic system. As an example of application of our method, we investigate the Ikeda attractor in some detail.

MODAL IDENTIFICATION OF WEAKLY NON-LINEAR MULTIDIMENSIONAL DYNAMICAL SYSTEMS USING A STOCHASTIC LINEARISATION METHOD WITH RANDOM COEFFICIENTS

It is known that an efficient approach for modal identification of a weakly non-linear multidimensional second-order dynamical system consists of using a model based on equivalent stochastic linearisation with constant coefficients. Such a model leads to a good identification of the total power of the stationary response but can give an incorrect identification of the matrix-valued spectral density functions. The objective of this paper is to present an identification procedure which is based on the use of a stochastic linearisation method with random coefficients. The model is then defined as a multidimensional linear second-order dynamical system with random coefficients. An optimisation procedure is developed to identify the parameters of the probability law of the random coefficients. The identification procedure is described step by step. Finally, an example is presented and shows the interest of the method proposed.

Detection of Sensor Faults by Means of Multivariable Calculation Methods

This paper deals with the detection and identification of abrupt changes (such as failures) in a multidimensional stochastic dynamic system which is representable by a vector autoregressive (AR) process, without any a priori knowledge about the process parameters. Two approaches are presented: In the first model the multidimensional system is considered an arrangement of multiple linearly coupled stochastic scalar autoregressive processes, and the residual is determined by inverse filtering. Alternatively, but using the AR approach, the system is modelled by a single stochastic process, and the analytical redundancy comprised in the multiple sensor signals can further contribute to residual generation. In both models the decision on an abrupt change in signal statistics is based on the generated residual signals and performed via the generalized likelihood ratio (GLR) approach, discovered by Willsky and Jones (1974). Finally some variations of the basic approach yielding a reduced number of numerical operations are presented, assuming either stationary state and/or using a bivariate calculation method.

CONSISTENCY OF LEAST SQUARES IDENTIFICATION

In this paper the least squares identification (LSI) of unknown coefficient matrices in the multi-dimensional dynamic system with uncorrelated noise is considered. Various conditions guaranteeing strong consistency of the LSI are put into comparison and the conditions proposed here and weaker than the previous one's are verified to be sufficient for the consistency of the LSI for which the asymptotic behaviors are also shown in the present paper. The main result is proved for the discrete-time system and its continuous-time analogue is also demonstrated.

An approximation method for estimation in linear systems with parameter uncertainty

Estimation of the state variables of a linear system with parameter uncertainties is performed using an asymptotically unbiased linear minimum-variance recursive estimator in continuous time. Estimates of the parameters can be obtained simultaneously, but are found to be biased. By augmenting additional linear dynamic equations which represent an asymptotic expansion in the unknown parameters, a linear structure is formed which approximates the original nonlinear system. However, the initial conditions and additive process noise are not Gaussian. The convergence properties of the state variance for this expansion are illustrated analytically by a scalar dynamic system. The numerical aspects of this example illustrate the behavior of the actual variance of the error in the state estimate and the predicted error variance as the order of the approximation increases. For the vector state problem, only the multidimensional dynamic system in canonical form with a single output is developed. For an n -dimensional system with n unknown constant parameters, a first-order approximation requires n additional linear equations. This approach can be extended to correlated parameter processes.