Synchronization stability of general complex dynamical networks with time-varying delays: A piecewise analysis method

The synchronization problem of some general complex dynamical networks with time-varying delays is investigated. Both time-varying delays in the network couplings and time-varying delays in the dynamical nodes are considered. The delays considered in this paper are assumed to vary in an interval, where the lower and upper bounds are known. Based on a piecewise analysis method, the variation interval of the time delay is firstly divided into several subintervals, by checking the variation of the derivative of a Lyapunov function in every subinterval, then the convexity of matrix function method and the free weighting matrix method are fully used in this paper. Some new delay-dependent synchronization stability criteria are derived in the form of linear matrix inequalities. Two numerical examples show that our method can lead to much less conservative results than those in the existing references.     

Correlation propagation for uncertainty analysis of structures based on a non-probabilistic ellipsoidal model

Traditional non-probabilistic methods for uncertainty propagation problems evaluate only the lower and upper bounds of structural responses, lacking any analysis of the correlations among the structural multi-responses. In this paper, a new non-probabilistic correlation propagation method is proposed to effectively evaluate the intervals and non-probabilistic correlation matrix of the structural responses. The uncertainty propagation process with correlated parameters is first decomposed into an interval propagation problem and a correlation propagation problem. The ellipsoidal model is then utilized to describe the uncertainty domain of the correlated parameters. For the interval propagation problem, a subinterval decomposition analysis method is developed based on the ellipsoidal model to efficiently evaluate the intervals of responses with a low computational cost. More importantly, the non-probabilistic correlation propagation equations are newly derived for theoretically predicting the correlations among the uncertain responses. Finally, the multi-dimensional ellipsoidal model is adopted again to represent both uncertainties and correlations of multi-responses. Three examples are presented to examine the accuracy and effectiveness of the proposed method both numerically and experimentally.     

An interval uncertainty analysis method for structural response bounds using feedforward neural network differentiation

This paper proposes a new interval uncertainty analysis method for structural response bounds with uncertain‑but-bounded parameters by using feedforward neural network (FNN) differentiation. The information of partial derivative may be unavailable analytically for some complicated engineering problems. To overcome this drawback, the FNNs of real structural responses with respect to structure parameters are first constructed in this work. The first-order and second-order partial derivative formulas of FNN are derived via the backward chain rule of partial differentiation, thus the partial derivatives could be determined directly. Especially, the influences of structures of multilayer FNNs on the accuracy of the first-order and second-order partial derivatives are analyzed. A numerical example shows that an FNN with the appropriate structure parameters is capable of approximating the first-order and second-order partial derivatives of an arbitrary function. Based on the parameter perturbation method using these partial derivatives, the extrema of the FNN can be approximated without requiring much computational time. Moreover, the subinterval method is introduced to obtain more accurate and reliable results of structural response with relatively large interval uncertain parameters. Three specific examples, a cantilever tube, a Belleville spring, and a rigid-flexible coupling dynamic model, are employed to show the effectiveness and feasibility of the proposed interval uncertainty analysis method compared with other methods.     

An iteration method for predicting static response of nonlinear structural systems with non-deterministic parameters

Confident numerical method is a crucial issue in the field of structural health monitoring. This paper focuses on uncertainty propagation in nonlinear structural systems with non-deterministic parameters. An interval-based iteration method is proposed on the basis of interval analysis and Taylor series expansion. The proposed method aims to improve the bounds of static response calculated by the point-based iteration method. In the proposed method, the iterative interval of static response is updated by revising the lower and upper bounds, respectively, which is the essential difference in comparison with the previous point-based iteration method. In this paper, interval parameters are employed to quantify the non-deterministic parameters instead of random parameters in the case of insufficient sample data. Iterative scheme is established based on the first-order Taylor series expansion. For the implementation of interval-based iteration method, a general procedure is formulated. Moreover, the important source of the limitation of point-based iteration method is revealed profoundly, and the good performance of the proposed method is demonstrated by three numerical comparisons.     

Interval uncertainty analysis for static response of structures using radial basis functions

This paper proposes a new interval uncertainty analysis method for static response of structures with unknown-but-bounded parameters by using radial basis functions (RBFs). Recently, collocation methods (CM) which apply orthogonal polynomials are proposed to solve interval uncertainty quantification problems with high accuracy. These methods overcome the drawback of Taylor expansion based methods, which are prone to overestimate the response bounds. However, the form of orthogonal basis functions is very complicated in higher dimensions, which may restrict their application when there exist relatively more interval parameters. In contrast to orthogonal basis function, the form of radial basis function (RBF) is simple and stays the same in whatever dimension. This study introduces RBFs into interval analysis of structures and provides a relatively simple approach to solve structural response bounds accurately. A surrogate model of real structural response with respect to interval parameters is constructed with the RBFs. The extrema of the surrogate model can be calculated by some auxiliary methods. The static response bounds can be obtained accordingly. Two numerical examples are used to verify the proposed method. The engineering application of the proposed method is performed by a center wing-box. The results prove the effectiveness of the proposed method.     

New criteria for synchronization stability of continuous complex dynamical networks with non-delayed and delayed coupling

The synchronization stability problem of general complex dynamical networks with non-delayed and delayed coupling is investigated based on a piecewise analysis method, the variation interval of the time delay is firstly divided into several subintervals, by checking the variation of derivative of a Lyapunov functional in every subinterval, several new delay-dependent synchronization stability conditions are derived in the form of linear matrix inequalities, which are easy to be verified by the LMI toolbox. Some numerical examples show that, when the number of the divided subintervals increases, the corresponding criteria can provide much less conservative results.     

Exact bounds for the sensitivity analysis of structures with uncertain-but-bounded parameters

Based on interval mathematical theory, the interval analysis method for the sensitivity analysis of the structure is advanced in this paper. The interval analysis method deals with the upper and lower bounds on eigenvalues of structures with uncertain-but-bounded (or interval) parameters. The stiffness matrix and the mass matrix of the structure, whose elements have the initial errors, are unknown except for the fact that they belong to given bounded matrix sets. The set of possible matrices can be described by the interval matrix. In terms of structural parameters, the stiffness matrix and the mass matrix take the non-negative decomposition. By means of interval extension, the generalized interval eigenvalue problem of structures with uncertain-but-bounded parameters can be divided into two generalized eigenvalue problems of a pair of real symmetric matrix pair by the real analysis method. Unlike normal sensitivity analysis method, the interval analysis method obtains informations on the response of structures with structural parameters (or design variables) changing and without any partial differential operation. Low computational effort and wide application rang are the characteristic of the proposed method. Two illustrative numerical examples illustrate the efficiency of the interval analysis.     

Handling imprecise evaluations in multiple criteria decision aiding and robust ordinal regression by <Emphasis Type="Italic">n</Emphasis>-point intervals

We consider imprecise evaluation of alternatives in multiple criteria ranking problems. The imprecise evaluations are represented by n-point intervals which are defined by the largest interval of possible evaluations and by its subintervals sequentially nested one in another. This sequence of subintervals is associated with an increasing sequence of plausibility, such that the plausibility of a subinterval is greater than the plausibility of the subinterval containing it. We explain the intuition that stands behind this proposal, and we show the advantage of n-point intervals compared to other methods dealing with imprecise evaluations. Although n-point intervals can be applied in any multiple criteria decision aiding (MCDA) method, in this paper, we focus on their application in robust ordinal regression which, unlike other MCDA methods, takes into account all compatible instances of an adopted preference model, which reproduce an indirect preference information provided by the decision maker. An illustrative example shows how the method can be applied in practice.     

A collocation method for differential equations with one‐point Taylor initial conditions

This paper presents a new multistep collocation method for nth order differential equations. The interval of interest is first divided into N subintervals. By determining interval conditions in Taylor interpolation in each subinterval, Taylor polynomials are calculated with different step lengths. Then the obtained solutions in each subinterval are used as initial conditions in the next step. Optimal error is assessed using Peano lemma, which shows that the errors decay by decreasing the step length. In particular, for fixed step length h, the error is of O(m^?m), where m is the number of collocation points in each subinterval. Meanwhile, for fixed m, the error is of O(h^m). Numerical examples demonstrate the validity and high accuracy of the proposed method even for stiff problems.     

The inverse spectral problem for a Stieltjes eight-shaped string

We consider a spectral problem generated by the Stieltjes string equation on a metric figure-of-eight graph and the corresponding inverse problem which is stated as follows. Values of the point masses located on one of the loops and the lengths of subintervals on it are given together with the spectrum of the spectral problem on the whole graph, the total length of the second loop and the length of the first subinterval on it, and a certain constant. Values of the masses and the lengths of subintervals on the second loop are to be found. Conditions sufficient for such a problem to be solvable are given in the implicit form. An algorithm for recovering the masses and the subintervals on the second loop is proposed.     

A time-decomposition algorithm for a stiff linear two-point boundary-value problem and its application to a nonlinear optimal control problem

An algorithm is proposed to solve a stiff linear two-point boundary-value problem (TPBVP). In a stiff problem, since some particular solutions of the system equation increase and others decrease rapidly as the independent variable changes, the integration of the system equation suffers from numerical errors. In the proposed algorithm, first, the overall interval of integration is divided into several subintervals; then, in each subinterval a sub-TPBVP with arbitrarily chosen boundary values is solved. Second, the exact boundary values which guarantee the continuity of the solution are determined algebraically. Owing to the division of the integration interval, the numerical error is effectively reduced in spite of the stiffness of the system equation. It is also shown that the algorithm is successfully imbedded into an interaction-coordination algorithm for solving a nonlinear optimal control problem.The authors would like to thank Mr. T. Sera and Mr. H. Miyake for their help with the calculations.     

Robust asymptotic stability of fuzzy Markovian jumping genetic regulatory networks with time-varying delays by delay decomposition approach

In this paper, the robust asymptotic stability problem is considered for a class of fuzzy Markovian jumping genetic regulatory networks with uncertain parameters and switching probabilities by delay decomposition approach. The purpose of the addressed stability analysis problem is to establish an easy-to-verify condition under which the dynamics of the true concentrations of the messenger ribonucleic acid (mRNA) and protein is asymptotically stable irrespective of the norm-bounded modeling errors. A new Lyapunov–Krasovskii functional (LKF) is constructed by nonuniformly dividing the delay interval into multiple subinterval, and choosing proper functionals with different weighting matrices corresponding to different subintervals in the LKFs. Employing these new LKFs for the time-varying delays, a new delay-dependent stability criterion is established with Markovian jumping parameters by T–S fuzzy model. Note that the obtained results are formulated in terms of linear matrix inequality (LMI) that can efficiently solved by the LMI toolbox in Matlab. Numerical examples are exploited to illustrate the effectiveness of the proposed design procedures.     

An interval algorithm for sensitivity analysis of coupled vibro-acoustic systems

Sensitivity analysis is a vital part in the optimization design of coupled vibro-acoustic systems. A new interval sensitivity-analysis method for vibro-acoustic systems is proposed in this paper. This method relies on only interval perturbation analysis instead of partial derivatives and difference operations. For strongly nonlinear systems, in particular, this methodology requires parameter variation over narrower ranges in comparison with other methods. To implement sensitivity analysis based on this method, the interval ranges of the responses of the vibro-acoustic system with interval parameters should first be obtained. Therefore, an interval perturbation-analysis method is presented for obtaining the interval bounds of the sound-pressure responses of a coupled vibro-acoustic system with interval parameters. The interval perturbation method is then compared with the Monte Carlo method, which can be taken as the benchmark for comparative accuracy. Two numerical examples involving sensitivity analysis of vibro-acoustic systems illustrate the feasibility and effectiveness of the proposed interval-based method.     

Interval modal superposition method for impulsive response of structures with uncertain-but-bounded external loads

This paper is concerned with the problem of the dynamic response of structures with uncertain-but-bounded external loads. Based on the theory of complex modal analysis, and interval mathematics, a new non-probabilistic method-interval modal superposition method is proposed to find the least favorable impulsive response and the most favorable impulsive response of structures. Through mathematical analysis and numerical calculation, comparisons between interval modal superposition method and probabilistic approach are made. Instead of probabilistic density distribution or statistical quantities, in the presented method, only the bounds on uncertain parameters are needed, Numerical examples indicate that the width of the region of the dynamic response yielded by the interval modal superposition method is larger than those produced by probabilistic approach while the interval modal superposition method will required less computation effort.     

Automatic,guaranteed integration of analytic functions

Davis introduced a method for estimating linear functionals of analytic functions by using Cauchy's Integral Formula. This is used to construct methods for numerical integration which give rigorous error bounds. By combining these bounds with strategies for order and subinterval adaptation, a program is developed for automatic integration of analytic functions. Interval analysis is used to validate the bounds.     

Reduction of overestimation in interval arithmetic simulation of biological wastewater treatment processes

A novel interval arithmetic simulation approach is introduced in order to evaluate the performance of biological wastewater treatment processes. Such processes are typically modeled as dynamical systems where the reaction kinetics appears as additive nonlinearity in state. In the calculation of guaranteed bounds of state variables uncertain parameters and uncertain initial conditions are considered. The recursive evaluation of such systems of nonlinear state equations yields overestimation of the state variables that is accumulating over the simulation time. To cope with this wrapping effect, innovative splitting and merging criteria based on a recursive uncertain linear transformation of the state variables are discussed. Additionally, re-approximation strategies for regions in the state space calculated by interval arithmetic techniques using disjoint subintervals improve the simulation quality significantly if these regions are described by several overlapping subintervals. This simulation approach is used to find a practical compromise between computational effort and simulation quality. It is pointed out how these splitting and merging algorithms can be combined with other methods that aim at the reduction of overestimation by applying consistency techniques. Simulation results are presented for a simplified reduced-order model of the reduction of organic matter in the activated sludge process of biological wastewater treatment.     

Numerical Analysis of a High-Order Scheme for Nonlinear Fractional Differential Equations with Uniform Accuracy

We introduce a high-order numerical scheme for fractional ordinary differential equations with the Caputo derivative. The method is developed by dividing the domain into a number of subintervals, and applying the quadratic interpolation on each subinterval. The method is shown to be unconditionally stable, and for general nonlinear equations, the uniform sharp numerical order 3 − $ν$ can be rigorously proven for sufficiently smooth solutions at all time steps. The proof provides a general guide for proving the sharp order for higher-order schemes in the nonlinear case. Some numerical examples are given to validate our theoretical results.     

Constructive methods in convexC 2 interpolation using quartic splines

Using quartic splines on refined grids, we present a method for convexity preservingC ² interpolation which is successful for all strictly convex data sets. In the first stage, one suitable additional knot in each subinterval of the original data grid is fixed dependent on the given data values. In the second stage, a visually pleasant interpolant is selected by minimizing an appropriate choice functional.     

Adaptive alternating Lipschitz search method for structural analysis with unknown-but-bounded uncertainties

Multisource uncertainties, including property dispersibility of materials and fluctuating service environments, complicate structural design and reliability assessment. In this paper, a novel method named the adaptive alternating Lipschitz search method for structural analysis with unknown-but-bounded uncertainties (or interval uncertainties) is proposed. In contrast to traditional optimization methods that search twice to obtain response bounds, an adaptive alternate iteration strategy is proposed. By sampling step by step, two acquisition functions—named the Lipschitz upper bound and the Lipschitz lower bound—are defined. Structural response bounds can be simultaneously obtained by alternately optimizing the two acquisition functions. The parameter settings do not require adjustments for different types of problems. Additionally, the Bayesian Adaptive Direct Search method is adopted to improve the performance of the strategy. Numerical and experimental cases are presented to demonstrate the validity, accuracy, and efficiency of the proposed methodology. Detailed comparisons indicate that the proposed method is competitive when addressing complicated structural systems with different ranges of uncertainty.