General Nonlinear Dynamical Systems Research Articles

A new real-time suboptimum filtering and prediction scheme for general nonlinear discrete dynamic systems with Gaussian or non-Gaussian noise

A new suboptimum state filtering and prediction scheme is proposed for nonlinear discrete dynamic systems with Gaussian or non-Gaussian disturbance and observation noises. This scheme is an online estimation scheme for real-time applications. Furthermore, this scheme is very suitable for state estimation under either constraints imposed on estimates or missing observations. State and observation models can be any nonlinear functions of the states, disturbance and observation noises as long as noise samples are independent, and the density functions of noise samples and conditional density functions of the observations given the states are available. State models are used to calculate transition probabilities from gates to gates. If these transition probabilities are known or can be estimated, state models are not needed for estimation. The proposed scheme (PR) is based upon state quantisation and multiple hypothesis testing. Monte Carlo simulations have shown that the performance of the PR, sampling importance resampling (SIR) particle filter and extended Kalman (EK) filter are all model-dependent, and that the performance of the PR is better than both the SIR particle filter and EK filter for some nonlinear models, simulation results of three of which are given in this article.

Stochastic Averaging in Continuous Time and Its Applications to Extremum Seeking

We investigate stochastic averaging theory in continuous time for locally Lipschitz systems and the applications of this theory to stability analysis of stochastic extremum seeking algorithms. First, we establish a general stochastic averaging principle and some related stability theorems for a class of continuous-time nonlinear systems with stochastic perturbations and remove or weaken several significant restrictions present in existing results: global Lipschitzness of the nonlinear vector field, equilibrium preservation under the stochastic perturbation, global exponential stability of the average system, and compactness of the state space of the perturbation process. Then, we propose a continuous-time extremum seeking algorithm with stochastic excitation signals instead of deterministic periodic signals. We analyze the stability of stochastic extremum seeking for static maps and for general nonlinear dynamic systems.

Multi-objective PID control for non-Gaussian stochastic distribution system based on two-step intelligent models

A new method for controlling the shape of the conditional output probability density function (PDF) for general nonlinear dynamic stochastic systems is proposed based on B-spline neural network (NN) model and T-S fuzzy model. Applying NN approximation to the measured PDFs, we transform the concerned problem into the tracking of given weights. Meanwhile, the complex multi-delay T-S fuzzy model with exogenous disturbances, parametric uncertainties and state constraints is used to represent the nonlinear weight dynamics. Moreover, instead of the non-convex design algorithms and PI control, the improved convex linear matrix inequality (LMI) algorithms and the generalized PID controller are proposed such that the multiple control objectives including stability, robustness, tracking performance and state constraint can be guaranteed simultaneously. Simulations are performed to demonstrate the efficiency of the proposed approach.

On Constrained Steady-State Regulation: Dynamic KKT Controllers

This technical note presents a solution to the problem of regulating a general nonlinear dynamical system to an economically optimal operating point. The system is characterized by a set of exogenous inputs as an abstraction of time-varying loads and disturbances. The economically optimal operating point is implicitly defined as a solution to a given constrained convex optimization problem, which is related to steady-state operation. The system outputs and the exogenous inputs represent respectively the decision variables and the parameters in the optimization problem. The proposed solution is based on a specific dynamic extension of the Karush-Kuhn-Tucker optimality conditions for the steady-state related optimization problem, which is conceptually related to the continuous-time Arrow-Hurwicz-Uzawa algorithm. Furthermore, it can be interpreted as a generalization of the standard output regulation problem with respect to a constant reference signal.

Growing Structure Multiple Model Systems for Anomaly Detection and Fault Diagnosis

A new anomaly detection scheme based on growing structure multiple model system (GSMMS) is proposed in this paper to detect and quantify the effects of anomalies. The GSMMS algorithm combines the advantages of growing self-organizing networks with efficient local model parameter estimation into an integrated framework for modeling and identification of general nonlinear dynamic systems. The identified model then serves as a foundation for building an effective anomaly detection and fault diagnosis system. By utilizing the information about system operation region provided by the GSMMS, the residual errors can be analyzed locally within each operation region. This local decision making scheme can accommodate for unequally distributed residual errors across different operational regions. The performance of the newly proposed method is evaluated through anomaly detection and quantification in an electronically controlled throttle system, which is simulated using a high-fidelity engine simulation software package provided by a major automotive manufacturer for control system development.

Constrained PI Tracking Control for Output Probability Distributions Based on Two-Step Neural Networks

In this paper, a new method for the control of the shape of the conditional output probability density function (pdf) for general nonlinear dynamic stochastic systems is presented using two-step neural networks (NNs). Following the square-root B-spline NN approximation to the measured output pdf, the problem is transferred into the tracking of dynamic weights. Different from the previous related works, time-delay dynamic NNs with undetermined parameters are employed to identify the nonlinear relationships between the control input and the weighting vectors. In order to achieve the required control objective and satisfy the state constraints due to the property of output pdfs, a constrained PI tracking controller is designed by solving a class of linear matrix inequalities and algebraic equations. With the proposed tracking controller and adaptive projection algorithms, both identification and tracking errors can be made to converge to zero, and the state constraints can also be simultaneously guaranteed. Finally, two simulated examples are given, which effectively demonstrate the use of the proposed control algorithm.

Listening to the noise: random fluctuations reveal gene network parameters

The cellular environment is abuzz with noise originating from the inherent random motion of reacting molecules in the living cell. In this noisy environment, clonal cell populations show cell-to-cell variability that can manifest significant phenotypic differences. Noise-induced stochastic fluctuations in cellular constituents can be measured and their statistics quantified. We show that these random fluctuations carry within them valuable information about the underlying genetic network. Far from being a nuisance, the ever-present cellular noise acts as a rich source of excitation that, when processed through a gene network, carries its distinctive fingerprint that encodes a wealth of information about that network. We show that in some cases the analysis of these random fluctuations enables the full identification of network parameters, including those that may otherwise be difficult to measure. This establishes a potentially powerful approach for the identification of gene networks and offers a new window into the workings of these networks.

Integrated control for high accuracy complex system

For high performance outputs, industrial or technical systems development rests upon better performing components with on top a controller acting with its specific features. As ultimately a larger frequency domain than presently covered by actuators is used, robustification of controllers is required as with modern ones improved over classical PID. They usually provide more efficient splitting of frequency band into lower range for power input with high gain for performance, and higher range with small gain for robustness. However, obtained stability is only simple, and the limit is often reached for moderate performance even with high quality components for partly known and/or intrinsically too noisy systems, where performance level and power requirement cannot be simultaneously met. This effect is the larger as the system size becomes smaller, and its structure more complex as in modern micro-systems. The problem has to be reassessed, and another controller giving asymptotic stability is proposed, and further improvement by distributing the controllers location in an optimum design view of the system is analyzed on a general nonlinear split n-dimensional dynamical system.

Optimal tracking control of nonlinear dynamical systems

This paper presents a simple methodology for obtaining the entire set of continuous controllers that cause a nonlinear dynamical system to exactly track a given trajectory. The trajectory is provided as a set of algebraic and/or differential equations that may or may not be explicitly dependent on time. Closed-form results are also provided for the real-time optimal control of such systems when the control cost to be minimized is any given weighted norm of the control, and the minimization is done not just of the integral of this norm over a span of time but also at each instant of time. The method provided is inspired by results from analytical dynamics and the close connection between nonlinear control and analytical dynamics is explored. The paper progressively moves from mechanical systems that are described by the second-order differential equations of Newton and/or Lagrange to the first-order equations of Poincaré, and then on to general first-order nonlinear dynamical systems. A numerical example illustrates the methodology.

A generalised time-varying output feedback controller for non-linear systems

This paper presents a solution for the design of a time-varying and parametric output feedback controller for known general non-linear dynamic systems subjected to linear output structure. The controller consists of two parts: an adaptive observer and a time-varying controller. Under a certain assumption on the non-linear dynamics of the system, a stable and non-linear adaptive observer can be firstly built. This observer can then be logically combined with any pre-specified smooth non-linear controller that uses the estimated state feedback with a tuning parameter. A combined parameter dependent Lyapunov function is constructed and used to formulate an effective tuning rule for the time-varying parameters involved in both the observer and the controller. With this selection of the tuning rule, it has been shown that the closed loop system is stable and the output can be controlled to converge to zero. An example is included to illustrate the use of the proposed methods and encouraging results have been obtained.

Generalized multiscale radial basis function networks

A novel modelling framework is proposed for constructing parsimonious and flexible multiscale radial basis function networks (RBF). Unlike a conventional standard single scale RBF network, where all the basis functions have a common kernel width, the new network structure adopts multiscale Gaussian functions as the bases, where each selected centre has multiple kernel widths, to provide more flexible representations with better generalization properties for general nonlinear dynamical systems. As a direct extension of the traditional single scale Gaussian networks, the new multiscale network is easy to implement and is quick to learn using standard learning algorithms. A k-means clustering algorithm and an improved orthogonal least squares (OLS) algorithm are used to determine the unknown parameters in the network model including the centres and widths of the basis functions, and the weights between the basis functions. It is demonstrated that the new network can lead to a parsimonious model with much better generalization property compared with the traditional single width RBF networks.

Standard errors and confidence intervals in inverse problems: sensitivity and associated pitfalls

We review the asymptotic theory for standard errors in classical ordinary least squares (OLS) inverse or parameter estimation problems involving general nonlinear dynamical systems where sensitivity matrices can be used to compute the asymptotic covariance matrices. We discuss possible pitfalls in computing standard errors in regions of low parameter sensitivity and/or near a steady state solution of the underlying dynamical system.

A new method for transient stability analysis

AbstractThis paper proposes a brand‐new method for transient stability analysis in power systems. The proposed method directly computes the critical trajectory for a given contingency to obtain the critical condition of the studied system. Although the method may be useful for general nonlinear dynamic systems, it is applied to the problem of obtaining a controlling UEP, unstable equilibrium point, which provides inevitable information for the energy function methods to assess transient stability. Namely, the proposed method effectively yields a critical trajectory on PEBS together with the controlling UEP, thus improving the conventional BCU method. The effectiveness of the proposed method is demonstrated in 3‐machine 9‐bus and 6‐machine 30‐bus systems. © 2007 Wiley Periodicals, Inc. Electr Eng Jpn, 159(3): 26–33, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20245

Equilibrium through self-organization and a stability condition derived from generalized self-organization theory

The equilibria and stability of the least-dissipated relaxed state are derived from a generalized theory of self-organization for open or closed general nonlinear and dissipative dynamical systems. An application of the general theory is presented for the two-fluid model of plasmas. The concept of selective decay. together with helicity invariance, is analytically shown to have no connection with relaxed states derived from helicity invariance.

Quasi-Monte Carlo filtering in nonlinear dynamic systems

We develop a new framework for Bayesian filtering in general nonlinear dynamic systems based on the quasi-Monte Carlo (QMC) numerical techniques. We first propose a general approach to deterministic filtering called the quasi-Monte Carlo Kalman filter (QMC-KF), which unifies several existing advanced filtering methods in the literature, such as the unscented Kalman filter (UKF) and the quadrature Kalman filter (QKF). The computationally expensive step of calculating the Jacobian matrix involved in the extended Kalman filter (EKF) is avoided in the proposed QMC-KF approach. We also propose sequential quasi-Monte Carlo (SQMC) filtering techniques which is analogous to the sequential Monte Carlo (SMC) or particle filtering methods in the literature. We show in particular how to effectively combine deterministic filtering and adaptive importance sampling schemes, which lead to powerful SQMC filtering strategies. The properties of the proposed SQMC and SQMC/IS methods in terms of almost sure convergence and numerical error propagation behavior are analyzed. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed new QMC-based filtering algorithms.

On the zeros of the Abelian integrals for a class of Liénard systems

A planar polynomial differential system has a finite number of limit cycles. However, finding the upper bound of the number of limit cycles is an open problem for the general nonlinear dynamical systems. In this Letter, we investigated a class of Liénard systems of the form x ˙ = y , y ˙ = f ( x ) + y g ( x ) with deg f = 5 and deg g = 4 . We proved that the related elliptic integrals of the Liénard systems have at most three zeros including multiple zeros, which implies that the number of limit cycles bifurcated from the periodic orbits of the unperturbed system is less than or equal to 3.

“Assumed inherent sensor” inversion based ANN dynamic soft-sensing method and its application in erythromycin fermentation process

An artificial neural network (ANN) soft-sensing method, based on the “assumed inherent sensor” and its inversion concepts, is proposed and used to estimate some crucial process variables which would be very difficult to be measured directly. For a real biochemical process whose mathematical model is a general nonlinear dynamic system, one may assume that, in its interior, there exists an “inherent sensor” subsystem whose inputs are exactly the process variables to be estimated while whose outputs are the directly measurable ones. To verify this assumption, this paper presents an algorithm to construct the mathematical model of the “assumed inherent sensor” and furthermore presents a global invertibility condition of the “assumed inherent sensor” which guarantees the existence of the inversion of such an “assumed inherent sensor” in theory. The “assumed inherent sensor” inversion consists of a set of nonlinear functions and a series of differentiators and could be treated as the dynamic soft-sensing model because its outputs are capable of reproducing the input variables of the “assumed inherent sensor”, or the process variables to be estimated. To overcome the difficulty in constructing the above “assumed inherent sensor” inversion in an analytic manner, a static ANN is used to approximate the nonlinear function so that the ANN-inversion dynamic soft-sensing model or the desired soft-sensor is finally completed. This makes the proposed ANN-inversion soft-sensor stricter in construction principle and more credible in practical use than most proposed soft-sensors. The soft-sensor consisting of a static ANN and a set of differentiators has been put into use of estimating such crucial biochemical variables as mycelia concentration, sugar concentration and chemical potency in erythromycin fermentation process. The field results show that the soft-sensing values approximately coincide with the offline analyzing ones sampled from the production process.

The simplest normal form and its application to bifurcation control

This paper is concerned with the computation of the simplest normal forms with perturbation parameters, associated with codimension-one singularities, and applications to control systems. First, an efficient method is presented to compute the normal forms for general semi-simple cases, which combines center manifold theory and normal form theory in one unified procedure. The efficient approach is then applied to find the explicit simplest normal forms of general n-dimensional nonlinear dynamical systems whose Jacobian matrices evaluated at an equilibrium point contain either single zero or a purely imaginary pair. In addition to near-identity nonlinear transformation, time and parameter rescalings are used to obtain the simplest normal forms. It is shown that, unlike the classical normal forms, the simplest normal forms for single zero and Hopf singularities are finite up to an arbitrary order, which greatly simplify stability and bifurcation analysis. The new method is applied to consider controlling bifurcations of the Lorenz system and a nonlinear electrical circuit. Symbolic programs have been developed using Maple, which greatly facilitates applications.

COMPETITIVE MODES AND THEIR APPLICATION

We investigate nonlinear dynamical systems from the mode competition point of view, and propose the necessary conditions for a system to be chaotic. We conjecture that a chaotic system has at least two competitive modes (CM's). For a general nonlinear dynamical system, we give a simple, dynamically motivated definition of mode suitable for this concept. Since for most chaotic systems it is difficult to obtain the form of a CM, we focus on the competition between the corresponding modulated frequency components of the CM's. Some direct applications result from the explicit form of the frequency functions. One application is to estimate parameter regimes which may lead to chaos. It is shown that chaos may be found by analyzing the frequency function of the CM's without applying a numerical integration scheme. Another application is to create new chaotic systems using custom-designed CM's. Several new chaotic systems are reported.

Linear estimate of the number of limit cycles for a class of non-linear systems

A dynamic system has a finite number of limit cycles. However, finding the upper bound of the number of limit cycles is an open problem for general non-linear dynamical systems. In this paper, we investigated a class of non-linear systems under perturbations. We proved that the upper bound of the number of zeros of the related elliptic integrals of the given system is 7n + 5 including multiple zeros, which also gives the upper bound of the number of limit cycles for the given system.