Parameter identification of linear multi-delay systems via a hybrid of block-pulse functions and Taylor’s polynomials

In this paper, an efficient and effective procedure is successfully developed for parameter identification of linear time-invariant multi-delay systems. The proposed framework is based on a hybrid of block-pulse functions and Taylor’s polynomials. Two upper error bounds corresponding to hybrid functions are established. The excellent properties of these functions together with the associated operational matrices of integration and delay are utilised to transform the original problem into a system of linear algebraic equations. The least squares method is then implemented for estimation of the unknown parameters. Several numerical experiments are investigated to demonstrate the usefulness and effectiveness of the proposed procedure. Easy implementation, simple operations and accurate solutions are the main features of the suggested approximation scheme.     

Decoupled delay estimation in the identification of differential delay systems

Based on the variable projection functional in nonlinear least-squares theory, the estimation of pure time delay is decoupled from the determination of the remaining system parameters for a class of differential delay models. The approach utilizes input-output data on a fixed finite time interval and avoids estimating unknown initial conditions. Results of a simulation study are summarized for several examples.     

A hybrid analysis direct method in the calculus of variations

This paper presents a direct method for finding the solution of variational problems using a hybrid function. The properties of hybrid functions which consists of block-pulse functions plus Legendre polynomials are presented. An operational matrix of integration and the cross product of two hybrid function vectors are used to reduce a variational problem to the solution of algebraic equation. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

Adaptive parameter identification of linear SISO systems with unknown time-delay

An adaptive online parameter identification is proposed for linear single-input-single-output (SISO) time-delay systems to simultaneously estimate the unknown time-delay and other parameters. After representing the system as a parameterized form, a novel adaptive law is developed, which is driven by appropriate parameter estimation error information. Consequently, the identification error convergence can be proved under the conventional persistent excitation (PE) condition, which can be online tested in this paper. A finite-time (FT) identification scheme is further studied by incorporating the sliding mode scheme into the adaptation to achieve FT error convergence. The previously imposed constraint on the system relative degree is removed and the derivatives of the input and output are not required. Comparative simulation examples are provided to demonstrate the validity and efficacy of the proposed algorithms.     

Analysis of time-varying delay systems by hybrid of block-pulse functions and biorthogonal multiscaling functions

This paper proposes new effective methods for analysis of time-varying delay systems using new hybrid functions. The excellent properties of the hybrid functions which consist of block pulse functions and biorthogonal multiscaling functions are presented. We utilise operational matrices of product, delay and integration to reduce the solution of delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

Application of the hybrid functions to solve neutral delay functional differential equations

This paper presents a computational technique for the solution of the neutral delay differential equations with state-dependent and time-dependant delays. The properties of the hybrid functions which consist of block-pulse functions plus Legendre polynomials are presented. The approach uses these properties together with the collocation points to reduce the main problems to systems of nonlinear algebraic equations. An estimation of the error is given in the sense of Sobolev norms. The efficiency and accuracy of the proposed method are illustrated by several numerical examples.     

Numerical solution of optimal control problems governed by integro‐differential equations

In this paper, an efficient hybrid approximation scheme for solving optimal control problems governed by integro‐differential equations is proposed. The current approach is based on a generalization of the hybrid of block‐pulse functions and Legendre's polynomials. An upper bound for the generalized hybrid functions with respect to the maximum norm is acquired and its convergence is demonstrated. The optimal control problem under study is transcribed to a mathematical programming one. Two illustrative examples are considered to verify the capability and reliability of the proposed procedure.     

Modulating function-based system identification for a fractional-order system with a time delay involving measurement noise using least-squares method

This study proposes an identification algorithm for a time-delay fractional-order system with the measurement noise via the modulating function approach. The polynomial function is adopted as the modulating function, and the approach to determine the coefficients of the modulating function is provided. By the property of modulating function, the identified fractional-order equation is converted into an algebraic equation. By the recursive least squares estimation algorithm, the estimation method of coefficients is offered. Supposing that the measurement noise in the output signal is the Gauss white noise, a revised identification algorithm is proposed to compensate the effect of measurement noise. Finally, two examples are given to verify the effectiveness of the proposed method.     

A blind approach to the Hammerstein-Wiener model identification

In this paper, we propose a blind approach to the sampled Hammerstein-Wiener model identification. By using the blind approach, it is shown that all internal variables can be recovered solely based on the output measurements. Then, identification of linear and nonlinear parts can be carried out. No a priori structural knowledge about the input nonlinearity is assumed and no white noise assumption is imposed on the input.     

A hybrid computational approach to derive new ground-motion prediction equations

A novel hybrid method coupling genetic programming and orthogonal least squares, called GP/OLS, was employed to derive new ground-motion prediction equations (GMPEs). The principal ground-motion parameters formulated were peak ground acceleration (PGA), peak ground velocity (PGV) and peak ground displacement (PGD). The proposed GMPEs relate PGA, PGV and PGD to different seismic parameters including earthquake magnitude, earthquake source to site distance, average shear-wave velocity, and faulting mechanisms. The equations were established based on an extensive database of strong ground-motion recordings released by Pacific Earthquake Engineering Research Center (PEER). For more validity verification, the developed equations were employed to predict the ground-motion parameters of the Iranian plateau earthquakes. A sensitivity analysis was carried out to determine the contributions of the parameters affecting PGA, PGV and PGD. The sensitivity of the models to the variations of the influencing parameters was further evaluated through a parametric analysis. The obtained GMPEs are effectively capable of estimating the site ground-motion parameters. The equations provide a prediction performance better than or comparable with the attenuation relationships found in the literature. The derived GMPEs are remarkably simple and straightforward and can reliably be used for the pre-design purposes.     

A delay decomposition approach to delay‐dependent stability for linear systems with time‐varying delays

This paper is concerned with delay‐dependent stability for linear systems with time‐varying delays. By decomposing the delay interval into multiple equidistant subintervals, on which different Lyapunov functionals are chosen, and new Lyapunov‐Krasvskii functionals are then constructed. Employing these new Lyapunov‐Krasvskii functionals, some new delay‐dependent stability criteria are established. The numerical examples show that the obtained results are less conservative than some existing ones in the literature. Copyright © 2009 John Wiley & Sons, Ltd.     

A variational integrators approach to second order modeling and identification of linear mechanical systems

The theory of variational integration provides a systematic procedure to discretize the equations of motion of a mechanical system, preserving key properties of the continuous time flow. The discrete-time model obtained by variational integration theory inherits structural conditions which in general are not guaranteed under general discretization procedures. We discuss a simple class of variational integrators for linear second order mechanical systems and propose a constrained identification technique which employs simple linear transformation formulas to recover the continuous time parameters of the system from the discrete-time identified model. We test this approach on a simulated eight degrees of freedom system and show that the new procedure leads to an accurate identification of the continuous-time parameters of second-order mechanical systems starting from discrete measured data.     

Hopfield networks for identification of delay differential equations with an application to dengue fever epidemics in Cuba

This work is aimed at proposing an algorithm, based upon Hopfield networks, for estimating the parameters of delay differential equations. This neural estimator has been successfully applied to models described by Ordinary Differential Equations, whereas its application to systems with delays is a novel contribution. As a case in point, we present a model of dengue fever for the Cuban case, which is defined by a delay differential system. This epidemiological model is built upon the scheme of an SIR (susceptible, infected, recovered) population system, where both delays and time-varying parameters have been included. The latter are thus estimated by the proposed neural algorithm. Additionally, we obtain an expression of the Basic Reproduction Number for our model. Experimental results show the ability of the estimator to deal with systems with delays, providing plausible parameter estimations, which lead to predictions that are coherent with actual epidemiological data. Besides, when the Basic Reproduction Number is computed from the estimated parameter values, results suggest an evolution of the epidemic that is consistent with the observed infection. Hence the estimation could help health authorities to both predict the future trend of the epidemic and assess the efficiency of control measures.     

A general approach to hyperbolic partial differential equations by homotopy perturbation method

The hyperbolic partial differential equations (PDEs) have a wide range of applications in science and engineering. In this article, the exact solutions of some hyperbolic PDEs are presented by means of He's homotopy perturbation method (HPM). The results reveal that the HPM is very effective and convenient in solving nonlinear problems.     

A discrete delay decomposition approach to stability of linear retarded and neutral systems

This paper is concerned with stability of linear time-delay systems of both retarded and neutral types by using some new simple quadratic Lyapunov-Krasovskii functionals. These Lyapunov-Krasovskii functionals consist of two parts. One part comes from some existing Lyapunov-Krasovskii functionals employed in [Han, Q.-L. (2005a). Absolute stability of time-delay systems with sector-bounded nonlinearity. Automatica, 41, 2171-2176; Han, Q.-L. (2005b). A new delay-dependent stability criterion for linear neutral systems with norm-bounded uncertainties in all system matrices. International Journal of Systems Science, 36, 469-475]. The other part is constructed by uniformly dividing the discrete delay interval into multiple segments and choosing proper functionals with different weighted matrices corresponding to different segments. Then using these new simple quadratic Lyapunov-Krasovskii functionals, some new discrete delay-dependent stability criteria are derived for both retarded systems and neutral systems. It is shown that these criteria for retarded systems and neutral systems are always less conservative than the ones in Han (2005a) and Han (2005b) cited above, respectively. Numerical examples also show that the results obtained in this paper significantly improve the estimate of the discrete delay limit for stability over some other existing results.     

基于混合最小二乘支持向量机网络模型的非线性系统辨识

针对基于输入输出数据的非线性系统辨识问题,提出一种新的混合最小二乘支持向量机(LS-SVMs)网络模型及相应的学习算法.该算法将系统的辨识问题动态自适应的划分为若干子问题,将支持向量机(SVM)用于各子模块辨识;通过分析模型的统计学特性,给出基于整体框架优化的系统参数辨识方法.针对系统中参数相关联的特性,采用期望条件最大化(ECM)算法对其进行条件辨识,同时结合正则化理论和最小二乘法,保证各专家模块的结构风险最小化辨识原则.试验结果表明,该方法兼具良好的辨识精度和泛化性能.     

A delay-range-partition approach to analyse stability of linear systems with time-varying delays

In this paper, the stability analysis of linear systems with an interval time-varying delay is investigated. First, augmented Lyapunov–Krasovskii functionals are constructed, which include more information of the delay's range and the delay's derivative. Second, two improved integral inequalities, which are less conservative than Jensen's integral inequalities, and delay-range-partition approach are utilised to estimate the upper bounds of the derivatives of the augmented Lyapunov–Krasovskii functionals. Then, less conservative stability criteria are proposed no matter whether the lower bound of delay is zero or not. Finally, to illustrate the effectiveness of the stability criteria proposed in this paper, two numerical examples are given and their results are compared with the existing results.     

A convex relaxation approach to set-membership identification of LPV systems

Identification of linear parameter varying models is considered in this paper, under the assumption that both the output and the scheduling parameter measurements are affected by bounded noise. First, the problem of computing parameter uncertainty intervals is formulated in terms of nonconvex optimization. Then, on the basis of the analysis of the regressor structure, we present an ad hoc convex relaxation scheme for computing parameter bounds by means of semidefinite optimization.     

A generalized approach to stabilization of linear interconnected time‐delay systems

The objective of this paper is to propose a generalized approach to stabilization of systems which are composed of linear time‐delay subsystems coupled by linear time‐varying interconnections. The proposed algorithms, which are formulated within the convex optimization framework, provide decentralized solutions to the problem of delay‐dependent asymptotic stability with strict dissipativity. It is established that the new methodology can reproduce earlier results on passivity, positive realness and disturbance attenuation. Then a decentralized structure of dissipative state‐feedback controllers is designed to render the closed‐loop interconnected system delay‐dependent asymptotically stable with strict dissipativity. Numerical examples are presented to illustrate the applicability of the design method. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A neural differential evolution identification approach to nonlinear systems and modelling of shape memory alloy actuator

This paper proposes a hybrid modified differential evolution plus back‐propagation (MDE‐BP) algorithm to optimize the weights of the neural network model. In implementing the proposed training algorithm, the mutation phase of the differential evolution (DE) is modified by combining two mutation strategies rand/1 and best/1 to create trial vectors instead of only using one mutation operator or rand/1 or best/1 as the standard DE. The modification aims to balance the global exploration and local exploitation capacities of the algorithm in order to find potential global optimum solutions. Then the local searching ability of the back‐propagation (BP) algorithm is applied in that region so as to swiftly converge to the optimum solution. The performance and efficiency of the proposed method is tested by identifying some benchmark nonlinear systems and modeling the shape memory alloy actuator. The proposed training algorithm is compared with the other algorithms, such as the traditional DE and BP algorithm. As a result, the proposed method can improve the accuracy of the identification process.