Optimal Mathematical Models for Nonlinear Dynamical Systems

We propose a new method for the optimal causal representation of nonlinear systems. The proposed approach is based on the best constrained approximation of mappings in probability spaces by operators constructed from matrices of special form so that the approximant preserves the causality property. It is supposed that the observable input is contaminated with noise. The approximant minimises the mean-square difference between a desired output signal and the output signal of the approximating model. The method provides a numerically realisable mathematical model of the system. An analysis is given of the error associated with this representation.     

Stochastic resonance in a linear static system driven by correlated multiplicative and additive noises

In this work, we investigate the signal transmission in a linear static system driven by correlated multiplicative and additive noises. When the input signal is periodic, we depict the stochastic resonance (SR) phenomenon by employing the signal-to-noise ratio (SNR) theory; while the input signal is aperiodic, we describe the SR phenomenon by using the input–output cross correlation theory. And the exact analytic expressions of the output SNR and the normalized time averaged cross covariance between input and output are obtained. The results show: under the condition of negative correlated noises, SR arises; while with positive correlated or uncorrelated noises, there is no SR. This result may extend the SR theory to a common linear static system.     

“Chaos” in superregenerative receivers

The superregenerative principle has been known since the early 1920s. The circuit is extremely simple and extremely sensitive. Today, superheterodyne receivers generally supplant superregenerative receivers in most applications because there are several undesirable characteristics: poor selectivity, reradiation, etc. Superregenerative receivers undergo a revival in recent papers for wireless systems, where low cost and very low power consumption are relevant: house/building meters (such as water, energy, gas counter), personal computer environment (keyboard, mouse), etc. Another drawback is the noise level which is higher than that of a well-designed superheterodyne receiver; without an antenna input signal, the output of the receiver hears in an earphone as a waterfall noise; this sound principally is the inherent input noise amplified and detected by the circuit; however, when the input noise is negligible with respect of an antenna input signal, we are faced to an other source of “noise” self-generated by the superregenerative working. The main objective of this paper concerns this self-generated noise coming from an exponential growing followed by a re-injection process for which the final state is a function of the phase of the input signal.     

Stochastic resonance with tuning system parameters: the application of bistable systems in signal processing

A thorough evaluation of stochastic resonance with tuning system parameters in bistable systems is presented as a nonlinear signal processor. It is shown that the output signal-to-noise ratio obtained by adjusting systems parameters can exceed that by tuning noise intensity, especially when the input noise intensity is already beyond the resonance region. It is demonstrated that the theory and the method presented here can markedly improve the output signal-to-noise ratio, and minimize phase lag as well as the distortion of the system output signal with multi-frequency.     

Blind Channel Identification Based on Noisy Observation by Stochastic Approximation Method

A stochastic approximation algorithm for estimating multichannel coefficients is proposed, and the estimate is proved to converge to the true parameters a.s. up-to a constant scaling factor. The estimate is updated after receiving each new observation, so the output data need not be collected in advance. The input signal is allowed to be dependent and the observation is allowed to be corrupted by noise, but no noise statistics are used in the estimation algorithm.     

External perturbation manifested system size resonance in a mesoscopic hormone signaling model

In mesoscopic reaction systems that contain only finite number of reactants, molecular fluctuation (or the so called internal noise), which can be characterized by the system size, plays an important role in nonlinear systems. In this work, the effect of internal noise is studied in a mesoscopic hormone signaling model with the presence of external perturbations (noise or signal). Simulation results reveal that the internal noise can play a constructive role to optimize the regularity of the noise induced internal signal only when external perturbation is present. This is a novel external perturbations induced system size resonance, which indicates that in complex mesoscopic systems, the system can automatically avoid the destructive environmental effects by tuning its size. Such kind of nontrivial cooperative effect may attribute to the fact that external perturbations broadened the regulation scope of the key mutual feedback. Therefore, current finding is of practical significance for similar physiological systems where the intercellular regulation plays the dominate role for signaling.     

Enhanced coding for exponentially distributed signals using suprathreshold stochastic resonance

Our previous work on stochastic resonance (SR) in threshold based systems proved that the SR effect is dependent on the nature of the input signal distribution; more specifically, for certain types of signal distribution SR is not observed [Das A, Stocks NG, Nikitin A, Hines EL. Quantifying stochastic resonance in a single threshold detector for random aperiodic signals. Fluctuation Noise Lett 2004;4:L247–65]. Here we show that suprathreshold stochastic resonance (SSR) – a novel and distinct form of SR – removes this limitation and hence leads to the conclusion that SSR can probably enhance the transmission of signals of any distribution and amplitude. SSR effects are studied in a parallel array of identical nonlinear threshold based devices. A double exponential signal distribution is chosen because this distribution did not demonstrate conventional SR effects in a single threshold device [Das A, Stocks NG, Nikitin A., Hines EL. Quantifying Stochastic resonance in a single threshold detector for random aperiodic signals. Fluctuation and Noise Letters 2004;4:L247-L265.]. SSR as a possible mechanism for enhancing transmission of speech signals in the human ear is also discussed.     

Natural gradient algorithm for stochastic distribution systems with output feedback

In this paper, we use natural gradient algorithm to control the shape of the conditional output probability density function for the stochastic distribution systems from the viewpoint of information geometry. The considered system here is of multi-input and single output with an output feedback and a stochastic noise. Based on the assumption that the probability density function of the stochastic noise is known, we obtain the conditional output probability density function whose shape is only determined by the control input vector under the condition that the output feedback is known at any sample time. The set of all the conditional output probability density functions forms a statistical manifold (M), and the control input vector and the output feedback are considered as the coordinate system. The Kullback divergence acts as the distance between the conditional output probability density function and the target probability density function. Thus, an iterative formula for the control input vector is proposed in the sense of information geometry. Meanwhile, we consider the convergence of the presented algorithm. At last, an illustrative example is utilized to demonstrate the effectiveness of the algorithm.     

Determination of the pulse transfer function of composite plants using orthogonal functions

Methods are described for determination of the dynamic characteristics of composite plants with input and output noise. The methods utilize the smoothing effect of orthogonal polynomials. This effect is considered for stationary plants and is extended to nonlinear and nonstationary plants. For the last two, the analysis is restricted to the case with a test signal input.Berdichev Teachers' College. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 130–134, 1989.     

A philosophy for the modelling of realistic nonlinear systems

A nonlinear dynamical system is modelled as a nonlinear mapping from a set of input signals into a corresponding set of output signals. Each signal is specified by a set of real number parameters, but such sets may be uncountably infinite. For numerical simulation of the system each signal must be represented by a finite parameter set and the mapping must be defined by a finite arithmetical process. Nevertheless the numerical simulation should be a good approximation to the mathematical model. We discuss the representation of realistic dynamical systems and establish a stable approximation theorem for numerical simulation of such systems.

     

基于小波域的多尺度非参数图像分割方法

主要介绍了一种基于信息熵理论及图像多尺度信息来对图像进行非参数主动轮廓模型分割的有效方法.由于小波多分辨率特性的引入,可以最大程度地利用图像多尺度信息以确保分割的准确性和完整性.又由于小波变换的特性,低频信息的使用更是进一步降低了噪声影响.文中把图像分割问题定义为在分割区域边缘长度满足一定约束条件下,图像标记场与各个尺度图像像素值之间的互信息熵最大化过程.该方法可以有效地降低噪声对于分割的影响,及确保分割的准确性和完整性.     

Design of RLS Wiener fixed-lag smoother using covariance information in linear discrete stochastic systems

In this paper, we propose a new design for the recursive least-squares (RLS) Wiener fixed-lag smoother and filter in linear discrete-time wide-sense stationary stochastic systems. It is assumed that the signal is observed with additive white observation noise. The signal is uncorrelated with the observation noise. The estimators require knowledge of the system matrix, the observation matrix and the variance of the state vector. These quantities can be obtained from the auto-covariance function of the signal. In the estimation algorithms, moreover, the variance of the observation noise is assumed to be known, as a priori information.     

Inversion of linear delay dynamical systems

We consider the problem of inversion of a dynamical system, that is, the problem of real-time reconstruction of the unknown input of the system on the basis of measurements of its output. We consider linear systems of functional-differential equations in the case of commensurable delays. We obtain necessary conditions for the invertibility of this class of systems and suggest an inversion algorithm that permits one to obtain an estimate of the unknown input signal with given accuracy.     

Based on interval type-2 fuzzy-neural network direct adaptive sliding mode control for SISO nonlinear systems

In this paper, a novel direct adaptive interval type-2 fuzzy-neural tracking control equipped with sliding mode and Lyapunov synthesis approach is proposed to handle the training data corrupted by noise or rule uncertainties for nonlinear SISO nonlinear systems involving external disturbances. By employing adaptive fuzzy-neural control theory, the update laws will be derived for approximating the uncertain nonlinear dynamical system. In the meantime, the sliding mode control method and the Lyapunov stability criterion are incorporated into the adaptive fuzzy-neural control scheme such that the derived controller is robust with respect to unmodeled dynamics, external disturbance and approximation errors. In comparison with conventional methods, the advocated approach not only guarantees closed-loop stability but also the output tracking error of the overall system will converge to zero asymptotically without prior knowledge on the upper bound of the lumped uncertainty. Furthermore, chattering effect of the control input will be substantially reduced by the proposed technique. To illustrate the performance of the proposed method, finally simulation example will be given.     

A new probabilistic fuzzy model: Fuzzification–Maximization (FM) approach

Over the past few decades, fuzzy logic systems have been used for nonlinear modeling and approximation in many fields ranging from engineering to science. In this paper, a new fuzzy model is developed from the probabilistic and statistical point of view. The proposed model decomposes the input–output characteristics into noise-free part and probabilistic noise part and identifies them simultaneously. The noise-free model recovers the nominal input–output characteristics of the target system and the noise model gives approximation to the probabilistic nature of the added noise. To identify the two submodels simultaneously, we propose the Fuzzification–Maximization (FM). Finally, some simulations are conducted and the effectiveness of the proposed method is demonstrated through the comparison with the previous methods.     

Explicit formulas for Hankel norm approximations of infinite-dimensional systems

Recently Hankel norm approximation for finite dimensional systems has been studied extensively, both for continuous time systems and discrete time systems. One of the approaches is to combine the work of J.A. Ball and J.W. Helton with results on Wiener-Hopf factorization to produce explicit formulas for the solutions. This has been done by J.A. Ball and A.C.M. Ran for the finite-dimensional case and for a class of infinite-dimensional systems by A.C.M. Ran. Here we show that this approach can be extended to a much larger class of infinite-dimensional systems; those with an exponentially stable semigroup and unbounded input and output operators which satisfy a smoothness condition. This class is larger than the nuclear class for which this problem was solved by K. Glover, R.F. Curtain and J.R. Partington by an approximation approach.     

Improved Hessian approximation with modified secant equations for symmetric rank-one method

Symmetric rank-one (SR1) is one of the competitive formulas among the quasi-Newton (QN) methods. In this paper, we propose some modified SR1 updates based on the modified secant equations, which use both gradient and function information. Furthermore, to avoid the loss of positive definiteness and zero denominators of the new SR1 updates, we apply a restart procedure to this update. Three new algorithms are given to improve the Hessian approximation with modified secant equations for the SR1 method. Numerical results show that the proposed algorithms are very encouraging and the advantage of the proposed algorithms over the standard SR1 and BFGS updates is clearly observed.     

Information theory approach in efficiency measurement

The current non–parametric method of measuring productive efficiency of input–output systems is generalized here in the stochastic case in terms of an information theory approach based on the concept of entropy. Use of maximum entropy as a method of finding the most probable distribution of the input–output data set and as a predictive criterion is illustrated for production systems with multiple inputs and outputs.     

Stability of bilinear time-delay systems

In this paper, the stability of the differential bilinear time-delaysystems is first studied. We consider time-varying bilineartime-delay systems with output feedback. The input or controlu(t)is not only a signal but also an input with output feedback.The analysis is given by using norm-transformation methods.     

Diagnostics of stochastic resonance using Poincaré recurrence time distribution

In the paper we calculate the distribution density of Poincaré recurrence times for a one-dimensional nonhyperbolic cubic map subjected to white noise and a harmonic signal. It is established that for small vicinities of recurrence the distribution density is not described by an exponential law and is periodically modulated with the external signal frequency. It is shown that the Fourier spectrum of the distribution density exhibits a well-distinguishable peak at the external signal frequency. The peak amplitude achieves its maximal value in the regime of stochastic resonance (SR), that can be used for detecting the SR effect.