Time-discretization of nonlinear control systems with state-delay via Taylor–Lie series

The state-delay is always existent in the practical systems. Analysis of the delay phenomenon in a continuous-time domain is sophisticated. It is appropriate to obtain its corresponding discrete-time model for implementation via digital computer. In this paper, we propose a new scheme for the discretization of nonlinear systems using Taylor series expansion and the zero-order hold assumption. This scheme is applied to the sample-data representation of a nonlinear system with constant state time-delay. The mathematical expressions of the discretization scheme are presented and the effect of the time-discretization method on equilibrium properties of nonlinear control system with state time-delay is examined. The proposed scheme provides a finite-dimensional representation for nonlinear systems with state time-delay enabling existing controller design techniques to be applied to them. The performance of the proposed discretization procedure is evaluated using a nonlinear system. For this nonlinear system, various sampling rates and time-delay values are considered.     

Discretization of nonlinear input-driven dynamical systems using the Adomian Decomposition Method

A numerical decomposition method proposed by Adomian provides solutions to nonlinear, or stochastic, continuous time systems without the usual restrictive restraints. It is applicable to differential, delay differential, integro-differential, and partial differential equations without the need for linearization or other restrictions. It also works through both uncoupled boundary conditions as well as delay systems. In the following paper, a new time discretization method for the development of a sample-data representation of a nonlinear continuous-time input-driven dynamical system is proposed. The proposed method is based on both the zero-order hold (ZOH) assumption as well as the Adomian Decomposition Method which exhibit unique algorithmic and computational advantages. To take advantage of this method, the following steps must be followed. First, the method is applied to a linear input-driven dynamical system to explicitly derive an exact sample-data representation, producing proper results. Second, the method is then applied to a nonlinear input-driven dynamical system, which thereby derives exact and approximate sample-data representations, the latter being most suited for practical applications. To evaluate the performance, the proposed discretization procedure was tested using simulations in a case study which involved an illustrative two-degree-of-freedom mechanical system that exhibited nonlinear behavior considering various control and input variable profiles. In conclusion, the suggested algorithm, in comparison to the results of a Taylor-Lie series expansion method, demonstrated increased performance and efficiency.     

An Ito–Taylor weak 3.0 method for stochastic dynamics of nonlinear systems

A new framework for development of order 3.0 weak Taylor scheme towards stochastic modeling and dynamics of coupled nonlinear systems is presented. The proposed method is derived by including third order multiple stochastic integral terms of Ito–Taylor expansion and developing them for a wide class of stochastic nonlinear systems. For computing the system responses of linear and a wide class of nonlinear structural systems, the use of lower order integration schemes is sufficient. But for highly non-linear stochastically driven systems like base isolated hysteretic systems and degrading stochastic systems the evaluation of higher order terms is necessary. Additionally, the use of higher order integration schemes for stochastic dynamics of higher dimensional nonlinear systems remains a challenge due to the arising mathematical complexities with the increase in the number of DOFs (degrees-of-freedom) which really necessitates the development of the proposed algorithm. The proposed algorithm is verified using a representative class of coupled nonlinear system in presence and absence of nonlinear degradation and hysteretic oscillators. The efficiency of the proposed numerical scheme over classical integration schemes is demonstrated through a practical engineering problem. Finally, an automated extension of the proposed algorithm is presented by generalizing it for a system of N-DOFs.     

Robustly complete synthesis of sampled-data control for continuous-time nonlinear systems with reach-and-stay objectives

Formal methods are becoming favorable for control and verification of safety-critical systems because of the rigorous model-based computation. Relying on an over-approximated model of the original system behaviors, formal control synthesis algorithms are not often complete, which means that a controller cannot necessarily be synthesized even if there exists one. The main result of this paper shows that, for continuous-time nonlinear systems, a sample-and-hold control strategy for a reach-and-stay specification can be synthesized whenever such a strategy exists for the same system with its dynamics perturbed by small disturbances. Control synthesis is carried out by a fixed-point algorithm that adaptively partitions the system state space into a finite number of cells. In each iteration, the reachable set from each cell after one sampling time is over-approximated within a precision determined by the bound of the disturbances. To meet such a requirement, we integrate validated high-order Taylor expansion of the system solution over one sampling period into every fixed-point iteration and provide a criterion for choosing the Taylor order and the partition precision. Two nonlinear system examples are given to illustrate the effectiveness of the proposed method.     

A High-Order Finite Difference Scheme for 3D Unsteady Convection Diffusion Reaction Equations北大核心CSCD

针对三维非稳态对流扩散反应方程,构造了一种高精度紧致有限差分格式,对空间的离散采用四阶紧致差分方法,对时间的离散采用Taylor级数展开和余项修正技术,所提格式在时间上的精度为二阶、在空间上的精度为四阶。利用Fourier稳定性分析法证明了该格式是无条件稳定的。最后给出数值算例验证了理论结果。     

Locally linearized fractional step methods for nonlinear parabolic problems

This work deals with the efficient numerical solution of a class of nonlinear time-dependent reaction-diffusion equations. Via the method of lines approach, we first perform the spatial discretization of the original problem by applying a mimetic finite difference scheme. The system of ordinary differential equations arising from that process is then integrated in time with a linearly implicit fractional step method. For that purpose, we locally decompose the discrete nonlinear diffusion operator using suitable Taylor expansions and a domain decomposition splitting technique. The totally discrete scheme considers implicit time integrations for the linear terms while explicitly handling the nonlinear ones. As a result, the original problem is reduced to the solution of several linear systems per time step which can be trivially decomposed into a set of uncoupled parallelizable linear subsystems. The convergence of the proposed methods is illustrated by numerical experiments.     

A new Taylor collocation method for nonlinear Fredholm‐Volterra integro‐differential equations

The aim of this article is to present an efficient numerical procedure for solving nonlinear integro‐differential equations. Our method depends mainly on a Taylor expansion approach. This method transforms the integro‐differential equation and the given conditions into the matrix equation which corresponds to a system of nonlinear algebraic equations with unkown Taylor coefficients. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer program written in Maple10. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A Chebyshev interval method for nonlinear dynamic systems under uncertainty

This paper proposes a new interval analysis method for the dynamic response of nonlinear systems with uncertain-but-bounded parameters using Chebyshev polynomial series. Interval model can be used to describe nonlinear dynamic systems under uncertainty with low-order Taylor series expansions. However, the Taylor series-based interval method can only suit problems with small uncertain levels. To account for larger uncertain levels, this study introduces Chebyshev series expansions into interval model to develop a new uncertain method for dynamic nonlinear systems. In contrast to the Taylor series, the Chebyshev series can offer a higher numerical accuracy in the approximation of solutions. The Chebyshev inclusion function is developed to control the overestimation in interval computations, based on the truncated Chevbyshev series expansion. The Mehler integral is used to calculate the coefficients of Chebyshev polynomials. With the proposed Chebyshev approximation, the set of ordinary differential equations (ODEs) with interval parameters can be transformed to a new set of ODEs with deterministic parameters, to which many numerical solvers for ODEs can be directly applied. Two numerical examples are applied to demonstrate the effectiveness of the proposed method, in particular its ability to effectively control the overestimation as a non-intrusive method.     

Output feedback control for MIMO nonlinear systems using factorization

A new output feedback adaptive control scheme for multi-input and multi-output (MIMO) nonlinear systems is presented based on the high frequency gain matrix factorization and the backstepping approach with vector form. The only required prior knowledge about the high frequency gain matrix of the linear part of the system is the signs of its leading principal minors. The proposed controller is a dynamic one that only needs the measurement of the system output, and the observer and the filters are introduced in order to construct a virtual estimate of the unmeasured system states. The global stability of the closed-loop systems is guaranteed through this control scheme, and the tracking error converges to zero. Finally, the numerical simulation results illustrate the effectiveness of the proposed scheme.     

An Improved Fuzzy Modeling Method for a Class of Multi-Input Non-affine Nonlinear Systems

In this paper, an improved fuzzy modeling method is developed for a class of non-affine nonlinear systems. The idea comes from the concepts of the optimization tools and the Takagi–Sugeno fuzzy modeling technique. Specifically, this method is suitable especially for non-affine nonlinear systems. Two benchmark single-input and one multi-input non-affine nonlinear systems are illustrated to show that the proposed modeling scheme is superior to existing modeling methods.     

A numerical method for the Cahn–Hilliard equation with a variable mobility

We consider a conservative nonlinear multigrid method for the Cahn–Hilliard equation with a variable mobility of a model for phase separation in a binary mixture. The method uses the standard finite difference approximation in spatial discretization and the Crank–Nicholson semi-implicit scheme in temporal discretization. And the resulting discretized equations are solved by an efficient nonlinear multigrid method. The continuous problem has the conservation of mass and the decrease of the total energy. It is proved that these properties hold for the discrete problem. Also, we show the proposed scheme has a second-order convergence in space and time numerically. For numerical experiments, we investigate the effects of a variable mobility.     

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.     

Robust digital controllers for uncertain chaotic systems: A digital redesign approach

In this paper, a new and systematic method for designing robust digital controllers for uncertain nonlinear systems with structured uncertainties is presented. In the proposed method, a controller is designed in terms of the optimal linear model representation of the nominal system around each operating point of the trajectory, while the uncertainties are decomposed such that the uncertain nonlinear system can be rewritten as a set of local linear models with disturbed inputs. Applying conventional robust control techniques, continuous-time robust controllers are first designed to eliminate the effects of the uncertainties on the underlying system. Then, a robust digital controller is obtained as the result of a digital redesign of the designed continuous-time robust controller using the state-matching technique. The effectiveness of the proposed controller design method is illustrated through some numerical examples on complex nonlinear systems––chaotic systems.     

New variants of Jarratt’s method with sixth-order convergence

In this paper, by using the two-variable Taylor expansion formula, we introduce some new variants of Jarratt’s method with sixth-order convergence for solving univariate nonlinear equations. The proposed methods contain some recent improvements of Jarratt’s method. Furthermore, a new variant of Jarratt’s method with sixth-order convergence for solving systems of nonlinear equations is proposed only with an additional evaluation for the involved function, and not requiring the computation of new inverse. Numerical comparisons are made to show the performance of the presented methods.     

A Taylor expansion approach for solving partial differential equations with random Neumann boundary conditions

Nonlinear partial differential equation with random Neumann boundary conditions are considered. A stochastic Taylor expansion method is derived to simulate these stochastic systems numerically. As examples, a nonlinear parabolic equation (the real Ginzburg-Landau equation) and a nonlinear hyperbolic equation (the sine-Gordon equation) with random Neumann boundary conditions are solved numerically using a stochastic Taylor expansion method. The impact of boundary noise on the system evolution is also discussed.     

A hierarchical structure of observer-based adaptive fuzzy-neural controller for MIMO systems

An observer-based adaptive controller developed from a hierarchical fuzzy-neural network (HFNN) is employed to solve the controller time-delay problem for a class of multi-input multi-output (MIMO) non-affine nonlinear systems under the constraint that only system outputs are available for measurement. By using the implicit function theorem and Taylor series expansion, the observer-based control law and the weight update law of the HFNN adaptive controller are derived. According to the design of the HFNN hierarchical fuzzy-neural network, the observer-based adaptive controller can alleviate the online computation burden. Moreover, the common adaptive controller is utilized to control all the MIMO subsystems. Hence, the number of adjusted parameters of the HFNN can be further reduced. In this paper, we prove that the proposed observer-based adaptive controller can guarantee that all signals involved are bounded and that the outputs of the closed-loop system track asymptotically the desired output trajectories.     

A polytopic strategy for improved non-asymptotic robust control via implicit Lyapunov functions

This paper is concerned with finite- and fixed-time robust stabilization of uncertain multi-input nonlinear systems via the implicit Lyapunov function method. Instead of splitting the system into a linear nominal model and an additive perturbation which gathers nonlinearities, parametric uncertainties, and exogenous disturbances, the methodology hereby proposed preserves some nonlinear terms in the nominal system via an exact polytopic representation which leads to design conditions in the form of linear matrix inequalities. As a result, feasible solutions are found where former approaches fail; these solutions have more accurate settling-time estimates with reduced control effort. The corresponding control law includes well-known high-order sliding modes as a particular case. Numerical simulations are provided to illustrate the advantages of the proposal.     

A linearized finite-difference scheme for the numerical solution of the nonlinear cubic Schrödinger equation

A linearized finite-difference scheme is used to transform the initial/boundary-value problem associated with the nonlinear Schrödinger equation into a linear algebraic system. This method is developed by re placing the time and the space partial derivatives by parametric finite-difference re placements and the nonlinear term by an appropriate parametric linearized scheme based on Taylor’s expansion. The resulting finite-difference method is analysed for stability and convergence. The results of a number of numerical experiments for the single-soliton wave are given.     

Optimal Control Formulation for Complementarity Dynamical Systems

In this paper, we show that the complementarity dynamical systems can be reformulated as optimal control problems. By using this reformulation, we present a pseudospectral scheme to discretize the complementarity dynamical systems. Applying this discretization, the complementarity dynamical system is reduced to a sequence of nonlinear programming problems. Numerical examples and comparison with two other methods are included to demonstrate the capability of the proposed method.     

非线性发展方程的Taylor展开方法

提出了积分非线性发展方程的新方法,即Taylor展开方法．标准的Galerkin方法可以看作0-阶Taylor展开方法,而非线性Galerkin方法可以看作1-阶修正Taylor展开方法A·D2此外,证明了数值解的存在性及其收敛性．结果表明,在关于严格解的一些正则性假设下,较高阶的Taylor展开方法具有较高阶的收敛速度．最后,给出了用Taylor展开方法求解二维具有非滑移边界条件Navier-Stokes方程的具体例子．