Online system identification using fractional-order Hammerstein model with noise cancellation

Slow convergence and low accuracy are two main drawbacks in nonlinear system identification methods. It becomes more complicated when time delay and noises are considered. In this paper, considering a fractional-order Hammerstein model, an online identification method is proposed. A combination of an evolutionary optimization method and recursive least square algorithm is used to estimate the system parameters and orders in the presence of unknown noises. Finally, simulation results are taken to prove the effectiveness of the proposed algorithm.

     

Stable fractional Chebyshev differentiation matrix for the numerical solution of multi-order fractional differential equations

This paper presents an algorithm to obtain numerically stable differentiation matrices for approximating the left- and right-sided Caputo-fractional derivatives. The proposed differentiation matrices named fractional Chebyshev differentiation matrices are obtained using stable recurrence relations at the Chebyshev–Gauss–Lobatto points. These stable recurrence relations overcome previous limitations of the conventional methods such as the size of fractional differentiation matrices due to the exponential growth of round-off errors. Fractional Chebyshev collocation method as a framework for solving fractional differential equations with multi-order Caputo derivatives is also presented. The numerical stability of spectral methods for linear fractional-order differential equations (FDEs) is studied by using the proposed framework. Furthermore, the proposed fractional Chebyshev differentiation matrices obtain the fractional-order derivative of a function with spectral convergence. Therefore, they can be used in various spectral collocation methods to solve a system of linear or nonlinear multi-ordered FDEs. To illustrate the true advantages of the proposed fractional Chebyshev differentiation matrices, the numerical solutions of a linear FDE with a highly oscillatory solution, a stiff nonlinear FDE, and a fractional chaotic system are given. In the first, second, and forth examples, a comparison is made with the solution obtained by the proposed method and the one obtained by the Adams–Bashforth–Moulton method. It is shown the proposed fractional differentiation matrices are highly efficient in solving all the aforementioned examples.     

Predicting solutions of the stochastic fractional order dynamical system using machine learning

The solution of fractional-order systems has been a complex problem for our research. Traditional methods like the predictor-corrector method and other solution steps are complicated and cumbersome to derive, which makes it more difficult for our solution efficiency. The development of machine learning and nonlinear dynamics has provided us with new ideas to solve some complex problems. Therefore, this study considers how to improve the accuracy and efficiency of the solution based on traditional methods. Finally, we propose an efficient and accurate nonlinear auto-regressive neural network for the fractional order dynamic system prediction model (FODS-NAR). First, we demonstrate by example that the FODS-NAR algorithm can predict the solution of a stochastic fractional order system. Second, we compare the FODS-NAR algorithm with the famous and good reservoir computing (RC) algorithms. We find that FODS-NAR gives more accurate predictions than the traditional RC algorithm with the same system parameters, and the residuals of the FODS-NAR algorithm are closer to 0. Consequently, we conclude that the FODS-NAR algorithm is a method with higher accuracy and prediction results closer to the state of fractional-order stochastic systems. In addition, we analyze the effects of the number of neurons and the order of delays in the FODS-NAR algorithm on the prediction results and derive a range of their optimal values.     

Robust synchronization of two different fractional-order chaotic systems with unknown parameters using adaptive sliding mode approach

In this paper, an adaptive sliding mode control method is introduced to ensure robust synchronization of two different fractional-order chaotic systems with fully unknown parameters and external disturbances. For this purpose, a fractional integral sliding surface is defined and an adaptive sliding mode controller is designed. In this method, no knowledge of the bounds of parameters and perturbation is required in advance and the parameters are updated through an adaptive control process. The proposed scheme is global and theoretically rigorous. Two examples are given to illustrate effectiveness of the scheme, in which the synchronizations between fractional-order chaotic Chen system and fractional-order chaotic Rössler system, between fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system, respectively, are successfully achieved. Corresponding numerical simulations are also given to verify the analytical results.

     

Adaptive synchronization control based on QPSO algorithm with interval estimation for fractional-order chaotic systems and its application in secret communication

In this paper, the synchronization problem and its application in secret communication are investigated for two fractional-order chaotic systems with unequal orders, different structures, parameter uncertainty and bounded external disturbance. On the basis of matrix theory, properties of fractional calculus and adaptive control theory, we design a feedback controller for realizing the synchronization. In addition, in order to make it better apply to secret communication, we design an optimal controller based on optimal control theory. In the meantime, we propose an improved quantum particle swarm optimization (QPSO) algorithm by introducing an interval estimation mechanism into QPSO algorithm. Further, we make use of QPSO algorithm with interval estimation to optimize the proposed controller according to some performance indicator. Finally, by comparison, numerical simulations show that the controller not only can achieve the synchronization and secret communization well, but also can estimate the unknown parameters of the systems and bounds of external disturbance, which verify the effectiveness and applicability of the proposed control scheme.     

Synchronization of nonidentical chaotic fractional-order systems with different orders of fractional derivatives

This paper addresses the problem of synchronization of chaotic fractional-order systems with different orders of fractional derivatives. Based on the stability theory of fractional-order linear systems and the idea of tracking control, suitable controllers are correspondingly proposed for two cases: the first is synchronization between two identical chaotic fractional-order systems with different fractional orders, and the other is synchronization between two nonidentical fractional-order chaotic systems with different fractional orders. Three numerical examples illustrate that fast synchronization can be achieved even between a chaotic fractional-order system and a hyperchaotic fractional-order system.     

Responses of Duffing oscillator with fractional damping and random phase

This paper aims to investigate dynamic responses of stochastic Duffing oscillator with fractional-order damping term, where random excitation is modeled as a harmonic function with random phase. Combining with Lindstedt–Poincaré (L–P) method and the multiple-scale approach, we propose a new technique to theoretically derive the second-order approximate solution of the stochastic fractional Duffing oscillator. Later, the frequency–amplitude response equation in deterministic case and the first- and second-order steady-state moments for the steady state in stochastic case are presented analytically. We also carry out numerical simulations to verify the effectiveness of the proposed method with good agreement. Stochastic jump and bifurcation can be found in the figures of random responses, and then we apply Monte Carlo simulations directly to obtain the probability density functions and time response diagrams to find the stochastic jump and bifurcation. The results intuitively show that the intensity of the noise can lead to stochastic jump and bifurcation.     

Parameter estimation,data compression and stochastic noise elimination in robotics: a wavelet domain-based integrated approach

Accurate parameter estimation in presence of stochastic noise is the essential part of almost all control hardware of the plants. However, the optimal design of the control hardware depends on processing power and installed memory. The proposed research investigation focuses on precise parameter estimation from compressed temporal data of error dynamics with exiguous susceptibility to the robot’s controllability. Instead of using Maximum Likelihood estimation (MLE) and Least Squares (LS) estimation in the time domain, the proposed method exploits the recursive wavelet domain’s properties to selectively store the error data coefficients negating the data related to noise. As a result, data compression is achieved. The proposed algorithm may be directly implemented on any scalable “Very Large Scale Integration” (VLSI) circuit due to the recursive implementation. For the evaluation of robustness, dynamic parameter variation is considered. The variation in scalar & vector-valued error is considered to evaluate the performance of the stochastic system. The proposed algorithm implementation is demonstrated experimentally on commercially available Omni Bundle robot.

     

A novel terminal sliding mode controller for a class of non-autonomous fractional-order systems

This paper introduces a finite-time control technique for control of a class of non-autonomous fractional-order nonlinear systems in the presence of system uncertainties and external noises. It is known that finite-time control methods demonstrate better robustness and disturbance rejection properties. Moreover, finite time control methods have optimal settling time. In order to design a robust finite-time controller, a new nonsingular terminal sliding manifold is proposed. The proposed sliding mode dynamics has the property of fast convergence to zero. Afterwards, a novel fractional sliding mode control law is introduced to guarantee the occurrence of the sliding motion in finite time. The convergence times of both reaching and sliding phases are estimated. The main characteristics of the proposed fractional sliding mode technique are (1) finite-time convergence to the origin; (2) the use of only one control input; (3) robustness against system uncertainties and external noises; and (4) the ability of control of non-autonomous fractional-order systems. At the end of this paper, some computer simulations are included to highlight the applicability and efficacy of the proposed fractional control method.     

Combined interval analysis - Monte Carlo simulation approach for the analysis of uncertainties in parallel manipulators

The quantification of the impact of uncertainties may increase the reliability and robustness of parallel manipulators. Monte Carlo simulation (MCS) and interval analysis are among the most common techniques used in uncertainty quantification. Interval analysis provides guaranteed performance since the interval evaluation of a function always contains the exact result. Nevertheless, interval analysis estimations are very conservative, frequently yielding overestimated results. Conversely, Monte Carlo Simulation avoids overestimation, but does not provide guaranteed performance. This paper proposes a novel hybrid algorithm combining the best features of interval analysis and Monte Carlo simulation for estimating probabilities of failure in the positioning error of parallel manipulators. A 3RRR manipulator is employed as case-study. The hybrid approach provides information on the bounds (minimum and maximum values) and estimated values of failure probabilities. The simulations herein compare the hybrid approach with pure interval analysis and pure Monte Carlo simulation. The results reveal that the hybrid technique may estimate not only the probability of failure in a continuous region, but also the worst and best case probabilities, much faster than interval analysis based approaches.

     

A novel secure image transmission scheme based on synchronization of fractional-order discrete-time hyperchaotic systems

In this paper, a secure image transmission scheme based on synchronization of fractional-order discrete-time hyperchaotic systems is proposed. In this scheme, a fractional-order modified-Hénon map is considered as a transmitter, the system parameters and fractional orders are considered as secret keys. As a receiver, a step-by-step delayed observer is used, and based on this one, an exact synchronization is established. To make the transmission scheme secure, an encryption function is used to cipher the original information using a key stream obtained from the chaotic map sequences. Moreover, to further enhance the scheme security, the ciphered information is inserted by inclusion method in the chaotic map dynamics. The first contribution of this paper is to propose new results on the observability and the observability matching condition of nonlinear discrete-time fractional-order systems. To the best of our knowledge, these features have not been addressed in the literature. In the second contribution, the design of delayed discrete observer, based on fractional-order discrete-time hyperchaotic system, is proposed. The feasibility of this realization is demonstrated. Finally, different analysis are introduced to test the proposed scheme security. Simulation results are presented to highlight the performances of our method. These results show that, our scheme can resist different kinds of attacks and it exhibits good performance.     

Statistical analysis for stochastic systems including fractional derivatives

An analytical scheme to determine the statistical behavior of a stochastic system including two terms of fractional derivative with real, arbitrary, fractional orders is proposed. In this approach, Green’s functions obtained are based on a Laplace transform approach and the weighted generalized Mittag–Leffler function. The responses of the system can be subsequently described as a Duhamel integral-type close-form expression. These expressions are applied to obtain the statistical behavior of a dynamical system excited by stationary stochastic processes. The numerical simulation based on the modified Euler method and Monte Carlo approach is developed. Three examples of single-degree-of-freedom system with fractional derivative damping under Gaussian white noise excitation are presented to illustrate application of the proposed method.     

Highly computationally efficient parameter estimation algorithms for a class of nonlinear multivariable systems by utilizing the state estimates

This paper investigates the parameter estimation issue for an input nonlinear multivariable state-space system. First, the canonical form of the input nonlinear multivariable state-space system is obtained through the linear transformation and the over-parameterization identification model of the considered system is derived. Second, by cutting down the redundant parameter estimates and extracting the unique parameter estimates from the parameter estimation vector in the least-squares identification method, we present an over-parameterization-based partially coupled average recursive extended least-squares parameter estimation algorithm to estimate the parameters. As for the unknown states in the parameter estimation algorithm, a new state estimator is designed to generate the state estimates. Third, in order to improve the computational efficiency of the parameter estimation algorithm, an over-parameterization-based multi-stage partially coupled average recursive extended least-squares algorithm is proposed. Finally, the computational efficiency analysis and the simulation examples are given to verify the effectiveness of the proposed algorithms.

     

An analysis of perturbation based methods for the treatment of parameter uncertainty in numerical groundwater models

Monte Carlo methods are robust approaches for the estimation of prediction uncertainty in groundwater flow and transport modelling under uncertain model parameters. However Monte Carlo procedures estimate the prediction statistics by generating a population of solutions from random realisations of the model parameters which are consistent with the parameter statistics, and as a result are computationally demanding. Taylor series based procedures offer an alternative to Monte Carlo methods for calculating the prediction statistics. Two such approaches, the first-order second moment and McLaughlin and Wood's perturbation method, are based on using a Taylor series to derive approximate expressions for the model predictions first and second statistical moments. In this paper the perturbation method presented by McLaughlin and Wood is rederived using Vetter matrix notation. This is compared with the first-order second moment (FOSM) method and while the steady state expressions for these two approaches are shown to be equivalent, the transient forms are considerably different. A new form of the FOSM is derived, which is simpler and has a lower computational burden. However, the transient McLaughlin and Wood expression is found to have a significantly lower computational overhead than either of the FOSM methods presented.     

Regular oscillations or chaos in a fractional order system with any effective dimension

This paper introduces a fractional order system which can generate regular oscillations or create chaos. It shows that this system is capable to create regular or nonregular oscillations under suitable conditions. These necessary conditions are achieved by violation of the no-chaos criteria. The effective dimension of the proposed system can be chosen any order less than three. Therefore, this system is a good example for limit cycle or chaos generation via fractional-order systems with low orders. Numerical simulations illustrate behavior of the proposed system in different situations.     

A novel statistical linearization solution for randomly excited coupled bending-torsional beams resting on non-linear supports

A novel statistical linearization technique is developed for computing stationary response statistics of randomly excited coupled bending-torsional beams resting on non-linear elastic supports. The key point of the proposed technique consists in representing the non-linear coupled response in terms of constrained linear modes. The resulting set of non-linear equations governing the modal amplitudes is then replaced by an equivalent linear one via a classical statistical error minimization procedure, which provides algebraic non-linear equations for the second-order statistics of the beam response, readily solved by a simple iterative scheme. Data from Monte Carlo simulations, generated by a pertinent boundary integral method in conjunction with a Newmark numerical integration scheme, are used as benchmark solutions to check accuracy and reliability of the proposed statistical linearization technique.

     

Chaos detection and parameter identification in fractional-order chaotic systems with delay

The paper first applies the 0–1 test for chaos to detecting chaos exhibited by fractional-order delayed systems. The results of the test reveal that there exists chaos in some fractional-order delayed systems with specific parameter values, which coincides with previous reports based on the phase portrait. In addition, it is very important to identify exactly the unknown specific parameters of fractional-order chaotic delayed systems in chaos control and synchronization. Thus, a method for parameter identification of fractional-order chaotic delayed systems based on particle swarm optimization (PSO) is presented. By treating the orders as parameters, the parameters and orders are identified through minimizing an objective function. PSO can efficiently find the optimal feasible solution of the objective function. Finally, numerical simulations on fractional-order chaotic logistic delayed system and fractional-order chaotic Chen delayed system show that the proposed method has effective performance of parameter identification.     

Observer-based projective synchronization of fractional systems via a scalar signal: application to hyperchaotic R?ssler systems

This paper introduces an observer-based approach to achieve projective synchronization in fractional-order chaotic systems using a scalar synchronizing signal. The proposed method, which enables a linear fractional error system to be obtained, exploits the Kalman decomposition and a proper stability criterion in order to stabilize the error dynamics at the origin. The approach combines three desirable features, that is, the theoretical foundation of the method, the adoption of a scalar synchronizing signal, and the exact analytical solution of the fractional error system written in terms of Mittag-Leffler function. Finally, the projective synchronization of the fractional-order hyperchaotic R?ssler systems is illustrated in detail.     

基于主动学习神经网络的转向架构架疲劳可靠性分析

为了提高转向架构架疲劳可靠性分析的精度与效率,提出一种主动学习BR-BP神经网络模型与Monte Carlo法相结合的可靠性分析方法。该方法针对BP神经网络的缺陷,使用贝叶斯正则BR(Bayesian regularization)算法作为训练算法,以提高神经网络的拟合精度与收敛速度,并考虑可靠性分析的固有特点,构造了一种适用于BP神经网络的主动学习函数,用于指导最佳样本点的选择。提出的学习函数不仅保证了样本点分布在极限状态函数附近,还考虑了样本点的预测误差以及样本点分布对失效概率计算的影响。转向架构架可靠性分析结果表明,本文方法在提高拟合精度的同时兼顾了计算效率,验证了所提方法的优越性与可行性。     

Finite-time command-filtered approximation-free attitude tracking control of rigid spacecraft

In this paper, a finite-time command-filtered approximation-free attitude tracking control strategy is proposed for rigid spacecraft. A novel finite-time prescribed performance function is first constructed to ensure that the attitude tracking errors converge to the predefined region in finite time. Then, a finite-time error compensation mechanism is constructed and incorporated into the backstepping control design, such that the differentiation of virtual control signals in recursive steps can be avoided to overcome the singularity issue. Compared with most of approximation-based attitude control methods, less computational burden and lower complexity are guaranteed by the proposed approximation-free control scheme due to the avoidance of using any function approximations. Simulations are given to illustrate the efficiency of the proposed method.