Genetic-Algorithm-Based Strategies for Dynamic Identification of Nonlinear Systems with Noise-Corrupted Response

The main objective of this paper is to investigate efficiency and correctness of different real-coded genetic algorithms and identification criteria in nonlinear system identification within the framework of non-classical identification techniques. Two conventional genetic algorithms have been used, standard genetic algorithm and microgenetic algorithm. Moreover, an advanced multispecies genetic algorithm has been proposed: it combines an adaptive rebirth operator, a migration strategy, and a search space reduction technique. Initially, a critical analysis has been conducted on these soft computing strategies to provide some guidelines for similar engineering and physical applications. Therefore, the hysteretic Bouc-Wen model has been numerically investigated to achieve three main results. First, the computational effectiveness and accuracy of the proposed strategy are checked to show that the proposed optimizer outperforms the aforementioned conventional genetic algorithms. Secondarily, a comparative study is performed to show that an improved performance can be obtained by using the Hilbert transform-based acceleration envelope as objective function in the optimization problem (instead of the pure acceleration response). Finally, system identification is conducted by making use of the proposed optimizer to verify its substantial noise-insensitive property also in the presence of high noise-to-signal ratio.     

Modeling Nonlinear Systems by Volterra Series

The Volterra-series expansion is widely employed to represent the input-output relationship of nonlinear dynamical systems. This representation is based on the Volterra frequency-response functions (VFRFs), which can either be estimated from observed data or through a nonlinear governing equation, when the Volterra series is used to approximate an analytical model. In the latter case, the VFRFs are usually evaluated by the so-called harmonic probing method. This operation is quite straightforward for simple systems but may reach a level of such complexity, especially when dealing with high-order nonlinear systems or calculating high-order VFRFs, that it may loose its attractiveness. An alternative technique for the evaluation of VFRFs is presented here with the goal of simplifying and possibly automating the evaluation process. This scheme is based on first representing the given system by an assemblage of simple operators for which VFRFs are readily available, and subsequently constructing VFRFs of the target composite system by using appropriate assemblage rules. Examples of wind and wave-excited structures are employed to demonstrate the effectiveness of the proposed technique.     

Nonlinear Controller Synthesis for Single-Input Multiple-Output Systems with Application to Cruise Missile

The purpose of this paper is to present a new systematic procedure for synthesis of nonlinear controllers for single-input multiple-output systems. The procedure is based on several describing function models of the plant, and the nonlinear gains are obtained via an inverse describing function technique. The procedure and associated software are applied to control the bank angle of a cruise missile. The results are compared with a linear controller and another nonlinear controller that was previously reported in the open literature; it is shown that the developed procedure has resulted in a less conservative design (i.e., performance is not sacrificed to assure absolute stability).     

Stability and Accuracy of Weighted Four-Point Implicit Finite Difference Schemes for Open Channel Flow

The unsteady motion of streams, with a free surface, has been described by a system of equations that were proposed by Saint-Venant in 1871. These are universally known as “Saint-Venant’s equations.” Weighted four-point implicit finite difference schemes are largely used for their numerical solution in the case of one-dimensional flow. For these models, stability, dissipation, and dispersion are first investigated by looking at the truncation error and then by using Fourier’s classic linear analysis. Variations of the space and time weighting coefficients, the Courant number, the Froude number, and the frictional resistance term are examined in this paper. In particular, instabilities are analyzed that are due to the progressive accumulation of dispersion errors, which are the consequence of an increasing frictional resistance term. However, in this case the numerical scheme requires a dissipation mechanism. Computational suggestions are given for this mechanism.     

Stochastic Response and Stability of a Nonintegrable System

For determining the stochastic response and stability of a strongly nonlinear single-degree-of-freedom system using the stochastic averaging technique, the size of excitations should be small such that the response of the system converges weakly to a Markov process. This condition is not often met with practical problems, and therefore, application of this method for obtaining their responses becomes difficult. Further, for systems with nonlinearities that cannot be integrated in closed form, stability analysis by examining the conditions of the two boundaries of the problem is not possible. A semianalytical method along with a weighted residual technique is presented here to circumvent these difficulties and to determine the response and stability of a strongly nonlinear system subjected to sizable stochastic excitation. The weighted residual technique is employed to correct the errors in averaged drift and diffusion coefficients resulting due to the size of the stochastic excitation. Two example problems are solved as illustrations of the method.     

Time‐Series Models for Nonlinear Systems

Discrete time‐series models can be used for the dynamic response prediction of linear structures. When structural nonlinearities are present, it may be possible to modify the form of the discrete time‐series model to account for the nonlinearities. One approach is to allow the model parameters to become functions of state. This paper explores some possible forms of the parameter functions for various nonlinear structures. Numerical case studies using both a Duffing oscillator and a combined viscous and coulomb damped oscillator are presented. Also, experimental data from a highly nonlinear aircraft landing gear strut are used to evaluate different model forms. The results from these studies show the potential for future applications of nonlinear time‐series models.     

Updating Properties of Nonlinear Dynamical Systems with Uncertain Input

A spectral density approach for the identification of linear systems is extended to nonlinear dynamical systems using only incomplete noisy response measurements. A stochastic model is used for the uncertain input and a Bayesian probabilistic approach is used to quantify the uncertainties in the model parameters. The proposed spectral-based approach utilizes important statistical properties of the Fast Fourier Transform and their robustness with respect to the probability distribution of the response signal in order to calculate the updated probability density function for the parameters of a nonlinear model conditional on the measured response. This probabilistic approach is well suited for the identification of nonlinear systems and does not require huge amounts of dynamic data. The formulation is first presented for single-degree-of-freedom systems and then for multiple-degree-of freedom systems. Examples using simulated data for a Duffing oscillator, an elastoplastic system and a four-story inelastic structure are presented to illustrate the proposed approach.     

Analytical Solutions for Stability of Slurry Trench

Coulomb-type force equilibrium analyses are presented for general two- and three-dimensional stability of a slurry-supported trench. Analytical solutions are derived for the factor of safety and critical failure plane angle in each case for drained effective stress and undrained total stress conditions. The solutions can accommodate variable trench depth, trench length, slurry depth, groundwater table elevation, surcharge loading, tension crack depth, and level of fluid in the tension cracks. Drained analyses can account for c′??′ soil strength and the effect of soil suctions above the groundwater table using the total cohesion method. Undrained analyses can account for undrained shear strength that varies linearly with depth. The solutions reduce to previously published expressions for simplified cases. An example problem is provided to illustrate variations of the factor of safety and critical failure plane angle with the length of a three-dimensional slurry trench. Finally, the method shows excellent agreement with the results of a full-scale field experiment of the failure of a diaphragm wall slurry trench constructed in silty sand.     

Explicit Pseudodynamic Algorithm with Improved Stability Properties

An explicit pseudodynamic algorithm with an improved stability property is proposed herein. This algorithm is shown to be unconditionally stable for any linear elastic systems and any instantaneous stiffness softening systems. The most attracting stability property is that it can have unconditional stability for the instantaneous hardening systems with the instantaneous degree of nonlinearity less than or equal to 2. This property has never been found among the currently available explicit algorithms. Hence, it may be applied to perform a general pseudodynamic test without considering the stability problem since it is rare for a civil engineering structure whose instantaneous degree of nonlinearity is greater than 2. This explicit algorithm can be implemented as a common explicit pseudodynamic algorithm, such as the use of the Newmark explicit method, since it does not involve any iteration procedure. In addition, it possesses comparable accuracy as that of a general second-order accurate integration method such as the Newmark explicit method. Both numerical and error propagation properties are analytically studied and numerical experiments are used to confirm these properties. Actual pseudodynamic tests attested to the feasibility of this proposed explicit pseudodynamic algorithm.     

关于一类非线性微分方程非振动解的渐近性

本文研究了二阶非线性微分方程(a(t)φ(y'(t)))'+f(t,y(t))=0,t≥t0的非振动解的渐近性.     

Stability Analyses and Design Recommendations for Practical Shoring Systems during Construction

This paper presents stability analysis and design recommendations for the falsework of wood and metal post shores. Based on the setup of shoring systems used on actual construction sites, analysis models of the shoring system have been derived. The experimental test results indicate that the base stiffness to the ground of shores is 50 t-cm/rad for (490 kN-cm/rad) wood post shores and 70 t-cm/rad (686 kN-cm/rad) for metal post shores of which the joint stiffness between members is 750 t-cm/rad (7,355 kN-cm/rad). With these stiffnesses, factors of 0.75 for wood post shores and 0.85 for metal post shores are used to modify the critical loads of shoring systems calculated from individual shores. The critical loads of a shoring system increase with the number of fixed strong shores, but are not affected by the number of leaning columns. In simplifying shore design, the LeMessurier formula is used for the strength computation of shoring systems composed of wood post shores. The critical loads of shoring systems increase linearly with the number of strong shores, but they are invariant with the positions of strong shores. If the required number of strong shores is defined, the critical loads of shoring systems can be found by interpolation.     

Optimal Neurocontroller for Nonlinear Benchmark Structure

A neurocontrol method is applied to the nonlinear benchmark control problem. A neurocontroller is trained based on a reduced-order linear design model, then it is used to control a nonlinear evaluation model. In training the controller, a sensitivity evaluation scheme is used and weights are updated by minimizing a cost function. Absolute accelerations directly measured from sensors are used as the feedback signals for the controller. Not only the current step acceleration, but delay signals of sensor readings, are used to enhance the training capability. Numerical examples show that the controlled responses are considerably reduced compared with the uncontrolled case. In conclusion, the possibility of the proposed control algorithm as a candidate for the controller of nonlinear building is shown.     

Linking Nonlinear System Identification with Nonlinear Dynamic Simulation under OpenSees: Some Justifications and Implementations

This study seeks to bridge the gap between nonlinear system identification and nonlinear dynamic finite-element analysis. Motivated by the needs in earthquake simulation, it is first investigated under which conditions and to what degree the prediction of maximum lateral drift and base shear requires accurate nonlinear hysteretic moment-rotation joint models. A series of simulations is carried out using a simple but typical steel frame under two different earthquake ground motion time histories scaled up to various levels. As one of the two major classes of models in nonlinear system identification, nonparametric models are proposed to be implemented into OpenSees. A methodology with details is provided to effectively implement feedforward neural networks with one hidden layer as a new one-dimensional nonlinear smooth material model directly from a MATLAB environment to OpenSees. The same methodology can be applied to benefit the implementation of other parametric and nonparametric models with linear parameterization. Numerical examples are provided. Challenges are discussed and future work is identified.     

一类非线性控制系统的干扰解耦

讨论了一类非线性控制系统的干扰解耦问题.通过对系统正则型的研究给出了使系统可通过静态反馈达到干扰解耦控制的充分必要条件,并给予了严格的证明.     

Monte Carlo Methods for Estimating the Extreme Response of Dynamical Systems

The development of simple accurate, and efficient methods for estimation of the extreme response of dynamical systems subjected to random excitations is discussed in the present paper. The key quantity for calculating the statistical distribution of extreme response is the mean level upcrossing rate function. By exploiting the regularity of the tail behavior of this function, an efficient simulation based methodology for estimating the extreme response distribution function is developed. This makes it possible to avoid the commonly adopted assumption that the extreme value data follow an appropriate asymptotic extreme value distribution, which would be a Gumbel distribution for the models considered in this paper. It is demonstrated that the commonly quoted obstacle against using the standard Monte Carlo method for estimating extreme responses, i.e., excessive CPU time, can be circumvented, bringing the computational efforts down to quite acceptable levels.     

Spectral Approach to Equivalent Statistical Quadratization and Cubicization Methods for Nonlinear Oscillators

Random vibrations of nonlinear systems subjected to Gaussian input are investigated by a technique based on statistical quadratization, and cubicization. In this context, and depending on the nature of the given nonlinearity, statistics of the stationary response are obtained via an equivalent system with a polynomial nonlinearity of either quadratic or cubic order, which can be solved by the Volterra series method. The Volterra series response is expanded in a trigonometric Fourier series over an adequately long interval T, and exact expressions are derived for the Fourier coefficients of the second- and third-order response in terms of the Fourier coefficients of the first-order, Gaussian response. By using these expressions, statistics of the response are determined using the statistics of the Fourier coefficients of the first-order response, which can be readily computed since these coefficients are independent zero-mean Gaussian variables. In this manner, a significant reduction of the computational cost is achieved, as compared to alternative formulations of quadratization and cubicization methods where rather prohibitive multifold integrals in the frequency domain must be determined. Illustrative examples demonstrate the reliability of the proposed technique by comparison with data from pertinent Monte Carlo simulations.     

Three-Dimensional Lower Bound Solutions for Stability of Plate Anchors in Clay

Soil anchors are commonly used as foundation systems for structures that require uplift or lateral resistance. These types of structures include transmission towers, sheet pile walls, and buried pipelines. Although anchors are typically complex in shape (e.g., drag or helical anchors), many previous analyses idealize the anchor as a continuous strip under plane strain conditions. This assumption provides numerical advantages and the problem can be solved in two dimensions. In contrast to recent numerical studies, this paper applies three-dimensional numerical limit analysis to evaluate the effect of anchor shape on the pullout capacity of horizontal anchors in undrained clay. The anchor is idealized as either square, circular, or rectangular in shape. Estimates of the ultimate pullout load are obtained by using a newly developed three-dimensional numerical procedure based on a finite-element formulation of the lower bound theorem of limit analysis. This formulation assumes a perfectly plastic soil model with a Tresca yield criterion. Results are presented in the familiar form of break-out factors based on various anchor shapes and embedment depths, and are also compared with existing numerical and empirical solutions.     

Adaptive Fuzzy Backstepping Output Feedback Control of Nonlinear Time-delay Systems with Unknown High-frequency Gain Sign

In this paper, an adaptive fuzzy robust feedback control approach is proposed for a class of single-input and singleoutput (SISO) strict-feedback nonlinear systems with unknown nonlinear functions, time delays, unknown high-frequency gain sign, and without the measurements of the states. In the bazkstepping recursive design, fuzzy logic systems are employed to approximate the unknown smooth nonlinear functions, K-filters is dcsigncd to estimate the unmeasured states, and Nussbaum gain functions are introduced to solve the problem of unknown sign of high-frequency gain. By combining adaptive fuzzy control theory and adaptive backstepping design, a stable adaptive fuzzy output feedback control scheme is developed. It has been proven that the proposed adaptive fuzzy robust control approach can guarantee that all the signals of the closed-loop system are uniformly ultimately bounded and the tracking error can converge to a small neighborhood of the origin by appropriately choosing design parameters. Simulation results have shown the effectiveness of the proposed method.     

Delay-range-dependent Stability Criterion for Interval Time-delay Systems with Nonlinear Perturbations

In this paper, we consider the problem of robust stability for a class of linear systems with interval time-varying delay under nonlinear perturbations using Lyapunov-Krasovskii (LK) functional approach.By partitioning the delay-interval into two segments of equal length, and evaluating the time-derivative of a candidate LK functional in each segment of the delay-interval, a less conservative delay-dependent stability criterion is developed to compute the maximum allowable bound for the delay-range within which the system under consideration remains asymptotically stable.In addition to the delay-bi-segmentation analysis procedure, the reduction in conservatism of the proposed delay-dependent stability criterion over recently reported results is also attributed to the fact that the time-derivative of the LK functional is bounded tightly using a newly proposed bounding condition without neglecting any useful terms in the delay-dependent stability analysis.The analysis, subsequently, yields a stable condition in convex linear matrix inequality (LMI) framework that can be solved non-conservatively at boundary conditions using standard numerical packages.Furthermore,as the number of decision variables involved in the proposed stability criterion is less, the criterion is computationally more effective.The effectiveness of the proposed stability criterion is validated through some standard numerical examples.     

Equivalent Linearization for the Nonstationary Response Analysis of Nonlinear Systems with Random Parameters

This paper presents an approach for analyzing nonlinear systems with parameter uncertainty subjected to stochastic excitation. The uncertain parameters are modeled as time-independent random variables. A general solution procedure based on equivalent linearization is presented. The set of orthogonal polynomials associated with the probability density function is used as the solution basis for the response moments. In addition, the instantaneous equivalent stiffness and damping matrices are approximated as quadratic random functions. The resulting Liapunov system with explicit random coefficients can then be converted into a deterministic system using the method of weighted residuals. Applications to single-degree-of-freedom uncertain systems are given and the accuracy of the results is validated.