A novel general stability criterion of time-delay fractional-order nonlinear systems based on WILL Deduction Method

A survey of some recent results in the stability and instability analyses of time-varying systems

Time delays are frequently appearing in many real-life phenomena and the presence of time delays in chaotic systems enriches its complexities. The analysis of fractional-order chaotic real nonlinear systems with time delays has a plenty of interesting results but the research on fractional-order chaotic complex nonlinear systems with time delays is in the primary stage. This paper studies the problem of hybrid projective synchronization (HPS) of fractional-order chaotic complex nonlinear systems with time delays. HPS is one of the extensions of projective synchronization, in which different state vectors can be synchronized up to different scaling factors. Based on Laplace transformation and the stability theory of linear fractional-order systems, a suitable nonlinear controller is designed to achieve synchronization between the master and slave fractional-order chaotic complex nonlinear systems with time delays in the sense of HPS with different scaling factors. Finally, the HPS between fractional-order delayed complex Lorenz system and fractional-order delayed complex Chen system and that of fractional-order delayed complex Lorenz system and fractional-order delayed complex Lu system are taken into account to demonstrate the effectiveness and feasibility of the proposed HPS techniques in the numerical example section.

This article considers the estimation problem for a class of uncertain fractional-order systems with time-varying delays and disturbances, in which the time-varying delays are present in the state, while the disturbances are present in the output. A new design method of functional interval observers is proposed. In contrast to interval observers currently available in the literature, which dealt with integer-order systems or fractional-order systems without time delays, the functional interval observers with fractional-order [Formula: see text] in this article can deal with fractional-order time-delay systems and integer-order time-delay systems (when [Formula: see text]). Existence conditions of functional interval observers for non-linear fractional-order systems with time-varying delays are established and they are formulated in terms of linear programmings. Two algorithms are provided for determining the functional interval observers. Illustrative examples of the functional interval observer application are presented.

Let f(x) be a continuous function on [a, b]. The Mean Value Theorem for Integrals asserts that there is a point c in (a, b) such that fab f(x) dx = f(c)(b a). Unlike the proof of the Mean Value Theorem for derivatives, the proof of the Mean Value Theorem for Integrals typically does not use Rolle's Theorem. In this note, we use Rolle's Theorem to introduce a generalization of the Mean Value Theorem for Integrals. Our generalization involves two functions instead of one, and has a very clear geometric explanation.

In this paper, impulsive control of nonlinear systems and its application to Chen's chaotic system are considered. A new impulsive control method for chaos suppression, using Chen's system as an example, is developed. Some new general criteria for exponential stability and asymptotical stability of nonlinear impulsive systems are established and, particularly, some simple conditions sufficient for driving the chaotic state of Chen's system to its zero equilibrium are presented.

The radioxenon from environmental samples is usually measured by the HPGe detector. The relative measurement method is used in the analysis process. However, the half-life of radioactive isotopes of xenon is too short to be used as a standard, so the calibration of the HPGe detector is difficult. In order to resolve this problem, the mixed disk sources are used to simulate the efficiencies of four radioactive xenon isotopes, and the mean value theorem of integrals is adopted, thereby improving the accuracy and simplifying the process. The samples are obtained by the xenon system under environment of high activity of radon, then the γ spectrum with regions of interest for 133Xe are obtained by the sample measurement. The activity of 133Xe is given by using the detection efficiency of disk source at a certain source-detector distance. The detection efficiency simulated by disk source is consistent with the detection efficiency of gas cell. Therefore, the simulated technology of disk source and mean value theorem of integrals are identified to be correct and effective.

Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem

Focusing on the high-precision motion control of time-delay nonlinear systems actuated by smart-material-based actuators and inspired by the flourishing development of fractional-order control theory, this article proposes an adaptive fractional-order command filter backstepping approximate inverse control scheme with the following key features. First, to the best of the authors’ knowledge, by combining the limited coverage lemma with the neural network, the unknown time-delay functions in the fractional-order nonlinear systems are precisely online estimated for the first time. This leads to the control problem of fractional-order nonlinear systems with time delays being overcome. Second, a new fractional-order filter is constructed to compute fractional differentiation of the designed virtual control laws, and simultaneously, the compensation signal is designed to eliminate the negative impact of filtering errors on the controller. Third, the approximate inverse model of asymmetric hysteresis for fractional-order asymmtric hysteretic nonlinear systems is derived, and the asymmetric hysteresis nonlinearities are precisely mitigated. Fourth, the independently developed giant magnetostrictive actuator-based motion platform is constructed, and a new fractional-order model of this platform is developed. Then, the motion control experiments are conducted to show the effectiveness of the proposed control scheme.

Stability of the Internet congestion control with diverse delays

Stability analysis of interconnected systems by means of aggregatre subsystem models

This paper presents new estimation methods for discrete fractional-order state-space systems with coloured measurement noise. A novel approach is proposed to convert a fractional system with coloured measurement noise to a system with white measurement noise in which the process and measurement noises are correlated with each other. In this paper, two new Kalman filter algorithms for fractional-order linear state-space systems with coloured measurement noise, as well as a new extended Kalman filter algorithm for state estimation in nonlinear fractional-order state-space systems with coloured measurement noise, are proposed. The accuracy of the equations and relations is confirmed in several theorems. The validity and effectiveness of the proposed algorithms are verified by simulation results and compared with previous work. Results show that for linear and nonlinear fractional-order systems with coloured noise, the proposed methods are more accurate than conventional methods regarding estimation error and estimation error covariance. Simulation results demonstrate that the proposed algorithms can accurately perform estimation in fractional-order systems with coloured measurement noise.

An integral split-step fourier method for digital back propagation to compensate fiber nonlinearity

In order to improve prediction accuracy of traffic accidents, an optimization GM (1, 1) power model is put forward based on the characteristics of S-type and a single traffic accident peak. Specifically, an intelligent optimization method is proposed to optimist the parameter α using PSO algorithm and the optimization formula of background value in GM (1, 1) power model is given based on the integral mean value theorem as well as the intelligent optimization algorithm of background value is proposed. The number of traffic fatalities during the period 1996–2009 is fitted and predicted using the optimization GM (1, 1) power model, whose results show that the optimization GM (1, 1) power model has higher fitting and prediction accuracy than the grey Verhaulst model.

Network Control.- Delay-Based Non-linear Observers for Congestion Control in Communication Networks.- On the Use of State Predictors in Networked Control Systems.- Networked Control Systems: Algorithms and Experiments.- Modeling and Closed Loop Control for Resource-Constrained Load Balancing with Time Delays in Parallel Computations.- Stability of Load Balancing Control.- Teleoperation.- Robust H ? Control of Bilateral Teleoperation Systems Under Communication Time-Delay.- Web Remote Control of Mechanical Systems: Delay Problems and Experimental Measurements of Round Trip Time.- PDE Approach for Time Delays in Robotized Teleoperation.- Emerging Methodologies.- From Time Delay to Distributed Parameter Systems in Communications.- Computing Maximum Delay Deviation Allowed to Retain Stability in Systems with Two Delays.- On Exact Controllability of Linear Time Delay Systems of Neutral Type.- New Computational Methods.- Applied Interval Computation: A New Approach for Time-Delays Systems Analysis.- Mathematical and Computational Tools for the Stability Analysis of Time-Varying Delay Systems and Applications in Mechanical Engineering.- Diffusive Representation for Operators Involving Delays.- OreModules: A Symbolic Package for the Study of Multidimensional Linear Systems.- Predictors, Inversion and Filtering.- Inversion and Tracking Problems for Time Delay Linear Systems.- Finite Impulse Response Systems for Almost Perfect Decoupling in Nonminimum-Phase Plants.- Linearization of the Power Amplifier in Mobile Telecommunications.- Merging Saturations and Input Delays.- Robust Sampled-Data Control: An Input Delay Approach.- Stabilization and Finite-Gain Stabilizability of Delay Linear Systems Subject to Input Saturation.- Global Asymptotic Stabilization of a PVTOL Aircraft Model with Delay in the Input.

This paper focuses on the stability analysis of linear fractional-order systems with fractional-order 0<α<2, in the presence of time-varying uncertainty. To obtain a robust stability condition, we first derive a new upper bound for the norm of Mittag-Leffler function associated with the nominal fractional-order system matrix. Then, by finding an upper bound for the norm of the uncertain fractional-order system solution, a sufficient non-Lyapunov robust stability condition is proposed. Unlike the previous methods for robust stability analysis of uncertain fractional-order systems, the proposed stability condition is applicable to systems with time-varying uncertainty. Moreover, the proposed condition depends on the fractional-order of the system and the upper bound of the uncertainty matrix norm. Finally, the offered stability criteria are examined on numerical uncertain linear fractional-order systems with 0<α<1 and 1<α<2 to verify the applicability of the proposed condition. Furthermore, the stability of an uncertain fractional-order Sallen–Key filter is checked via the offered condition.