Connection between Volterra series and perturbation method in nonlinear systems analyses

This paper is concerned with the connection between the Volterra series and the regular perturbation method in nonlinear systems analyses. It is revealed for the first time that, for a forced polynomial nonlinear system, if its derived linear system is a damped dissipative system, the steady response obtained through the regular perturbation method is exactly identical to the response given by the Volterra series. On the other hand, if the derived linear system is an undamped conservative system, then the Volterra series is incapable of modeling the forced polynomial nonlinear system. Numerical examples are further presented to illustrate these points. The results provide a new criterion for quickly judging whether the Volterra series is applicable for modeling a given polynomial nonlinear system.

MODELING OF NONLINEAR SYSTEMS BY MULTIPLE LINEARIZED MODELS

In order to design linear controller for nonlinear systems,a simple but efficient method of modeling a nonlinear system was proposed by means of multiple linearized models at different operating points in the entire range of the expected changes of the operating points.The original nonlinear system was described by linear combination of these multiple linearized models,with the linear combination parameters being identified on line based on least squares method.Model Predictive Control,an optimization based technique,was used to design the linear controller.A sufficient condition for ensuring the existence of a linear controller for the original nonlinear system was also given.Good performance indicated by two simulated examples confirms the usefulness of the proposed method.

Generalized hyperbolic perturbation method for homoclinic solutions of strongly nonlinear autonomous systems

A generalized hyperbolic perturbation method is presented for homoclinic solutions of strongly nonlinear autonomous oscillators, in which the perturbation proce- dure is improved for those systems whose exact homoclinic generating solutions cannot be explicitly derived. The generalized hyperbolic functions are employed as the basis functions in the present procedure to extend the validity of the hyperbolic perturbation method. Several strongly nonlinear oscillators with quadratic, cubic, and quartic nonlinearity are studied in detail to illustrate the efficiency and accuracy of the present method.

Robust Stability Criteria for Neutral Systems with Interval Time-Varying Delays and Nonlinear Perturbations

This paper investigates the problem of delay-dependent robust stability analysis for a class of neutral systems with interval time-varying delays and nonlinear perturbations. Such nonlinear perturbations are with time-varying but norm-bounded characteristics. Based on a new Lyapunov-Krasovskii functional together with a free-weighting matrices technique,improved delay-dependent stability criteria are established. It is shown that less conservative results can be obtained in terms of linear matrix inequalities( LMIs). Numerical examples are provided to demonstrate the effectiveness and less conservatism of the proposed approach.

Improved Exponential Stability Criteria for Uncertain Neutral System with Nonlinear Parameter Perturbations

This paper investigates the problem of robust exponential stability for neutral systems with time-varying delays and nonlinear perturbations. Based on a novel Lyapunov functional approach and linear matrix inequality technique, a new delay-dependent stability condition is derived. Since the model transformation and bounding techniques for cross terms are avoided, the criteria proposed in this paper are less conservative than some previous approaches by using the free-weighting matrices. One numerical example is presented to illustrate the effectiveness of the proposed results.

Synchronization of perturbed chaotic systems via nonlinear control

Chaos synchronization of systems with perturbations was investigated.A generic nonlinear control scheme to realize chaos synchronization of systems was proposed.This control scheme is flexible and practicable,and gives more freedom in designing controllers in order to achieve some desired performance.With the aid of Lyapunov stability theorem and partial stability theory,two cases were presented:1) Chaos synchronization of the system without perturbation or with vanishing perturbations;2) The boundness of the error state for the system with nonvanishing perturbations satisfying some conditions.Finally,several simulations for Lorenz system were provided to verify the effectiveness and feasibility of our method.Compared numerically with the existing results of linear feedback control scheme,the results are sharper than the existing ones.

奇异摄动区间变时滞不确定非线性系统稳定性分析

奇异摄动系统是实际问题中普遍存在的一类系统,为了使此类系统在应用上达到一定的稳定,利用矩阵分块方法对含有不确定结构的奇异摄动区间变时滞非线性系统稳定性进行分析,通过构造新的Lyapunov函数,基于Lyapunov稳定性理论,结合舒尔补引理、积分不等式和线性矩阵不等式等方法推出奇异摄动系统在摄动参数界内时滞相关的稳定性判据.

分段线性系统经典数学模型的修正与动力学分析

由于间隙的存在,很多机械系统可以简化为分段线性模型,而简化后的模型中副簧系统通常包含阻尼.在大多数文献建立的数学模型中,主、副系统的接触点与分离点固定在间隙处.文章研究发现,由于副簧系统中弹簧与阻尼的力学特性不同,主、副系统的接触点与分离点位置实际上是随着系统参数和运动状态而变化的.若忽视这一点,后续的包括分岔和混沌在内的动力学分析就会出现误差甚至错误.首先基于经典的数学模型,用数值方法揭示了在简谐激励下主系统未返回到间隙处就与副簧系统提前分离,证明了经典数学模型的不当之处.进一步研究发现主系统不仅会出现提前分离,还会出现接触滞后的现象.因此对系统的接触与分离条件提出了修正,得到了更合理的数学模型.研究发现修正后的接触点、分离点位置与修正前相差较大,修正后的幅频响应曲线与修正前存在一定差别;在复杂运动中,修正前后的运动性质也可能发生改变,证明了修正后的模型更加合理,更能反映工程实际.然后,采用平均法并对平均法的积分区间进行推广,求得了修正模型的幅频响应的解析解,并通过龙格库塔法验证了解析解的正确性.对解析解进行稳定性分析,得到了解析解的稳定性判别式.最后,探究了修正模型的副簧系统参数对主系统幅频响应的影响.

在线优化参数的神经网络预测监督控制

使用具有辅助变量的全格式动态线性化方法逼近系统,构建了神经网络监督控制预测模型。利用线性跟踪-微分器建立过渡过程,应用线性扩张状态观测器估计输出预测值及其微分,给出了线性PID控制算法。根据对角回归神经网络构成直接逆控制,提出了改进的控制目标函数。依据非线性递推最小二乘法在线优化了PID控制参数和对角回归神经网络的连接权。当系统控制误差大于一定值时,重置PID控制参数。最后提出了在线优化参数的神经网络预测监督控制,克服了已有的神经网络监督控制存在建模难的问题。仿真研究结果表明控制算法的响应具有理想性能。

RANDOM RESPONSES OF A TWO-DEGREE-OF-FREEDOM STRUCTURE WITH STRONGLY NONLINEAR COUPLING AND PARAMETRIC INTERACTION

This paper investigates the random responses of a TDOF structure with strongly nonlinear coupling and parametric vibration. With the nonlinear cou- pling of inertia in the equations of motion of the system being removed by successive elimination, the non-Gaussian moment equation method (NGM) is applied and 69 moment equations are integrated with central cumulative truncation technique. The stochastic central difference-cum-statistical linearization method(SCD-SL) and the digital simulation method(DSM) are also used. A comparison of results by different methods are given and the SCD-SL method is the most efficient method.

Robust entry guidance using multi-segment linear pseudospectral model predictive control

This paper presents a robust entry guidance algorithm for the high lift-to-drag ratio entry vehicle that employs the recently developed pseudospectral model predict control in a segmented manner. Here, the guidance commands are the longitudinal lift-to-drag (L/D) and bank reversal commands, which are calculated by successively solving multiple segment linear algebraic equations. These equations are derived using the linear pseudospectral method, control parametrization and calculus of variations. The method uses orthogonal polynomials and computes the updates through a series of analytical formulae, which makes it accurate and computationally effective. Moreover, it is able to adjust the number of bank reversals by providing the precise bank reversal point so as to fully exploit the potential of lateral maneuver. The method also employs proportional navigation and polynomial guidance after the last bank reversal to meet multiple terminal constraints. High-fidelity numerical simulations with various destinations are carried out to demonstrate its applicability. Furthermore, Monte Carlo simulations are also conducted to show that the proposed algorithm consistently offers very stable and robust performances and has superior performances in computational efficiency, guidance accuracy and lateral trajectory shaping capability in comparison with other typical methods. © 1990-2011 Beijing Institute of Aerospace Information.

Numerical Solution of Nonlinear Mixed Integral Equation with a Generalized Cauchy Kernel

In this article, we present approximate solution of the two-dimensional singular nonlinear mixed Volterra-Fredholm integral equations (V-FIE), which is deduced by using new strategy (combined Laplace homotopy perturbation method (LHPM)). Here we consider the V-FIE with Cauchy kernel. Solved examples illustrate that the proposed strategy is powerful, effective and very simple.

Adomian Decomposition Method for Solving Goursat's Problems

In this paper, Goursat’s problems for: linear and nonlinear hyperbolic equations of second-order, systems of nonlinear hyperbolic equations and fourth-order linear hyperbolic equations in which the attached conditions are given on the characteristics curves are transformed in such a manner that the Adomian decomposition method (ADM) can be applied. Some examples with closed-form solutions are studied in detail to further illustrate the proposed technique, and the results obtained indicate this approach is indeed practical and efficient.

基于复合模型的铅酸蓄电池数据驱动自适应控制

针对复杂非线性时变动态铅酸蓄电池系统,现有控制方法的控制器设计,要么需要已知其精确数学模型或结构信息,要么依赖人工经验,增加了不确定性。为避免该问题,笔者采用基于控制器的动态线性化方法,并利用参数自适应更新算法,基于复合模型给出铅酸蓄电池的数据驱动自适应控制设计过程。仿真结果表明,与经典PID方法相比,所采用方法的电压跟踪过程平稳度可提高48.21%,电流变化平滑度可提高35.56%。所采用方法在仅需铅酸蓄电池输入和输出数据情况下,仍能获得良好控制效果。

基于动态模式分解的精馏吸收状态变量非设计条件下的重构与预测

精馏和吸收作为典型的非线性过程,其操作过程中存在大量描述系统特征的状态变量。为了对这些状态变量进行重构和预测,实现精馏吸收过程的实时数字孪生,通过动态模式分解算法(DMD)获取非线性系统的近似线性化模型,用于快速获取精馏吸收过程中各级浓度、流量、温度和持料量等状态变量。在此基础上,应用Kalman滤波器对DMD生成的线性模型进行实时校正,使得在非设计和有限测量条件下,也可以有效地预测吸收或精馏的状态变量,而无须重新训练模型。

An incomplete generalized minimum backward perturbation algorithm for large nonsymmetric linear systems

This paper gives the truncated version of the generalized minimum backward error algorithm(GMBACK)—the incomplete generalized minimum backward perturbation algorithm(IGMBACK)for large nonsymmetric linear systems.It is based on an incomplete orthogonalization of the Krylov vectors in question,and gives an approximate or quasi-minimum backward perturbation solution over the Krylov subspace.Theoretical properties of IGMBACK including finite termination,existence and uniqueness are discussed in details,and practical implementation issues associated with the IGMBACK algorithm are considered.Numerical experiments show that,the IGMBACK method is usually more efficient than GMBACK and GMRES,and IMBACK,GMBACK often have better convergence performance than GMRES.Specially,for sensitive matrices and right-hand sides being parallel to the left singular vectors corresponding to the smallest singular values of the coefficient matrices,GMRES does not necessarily converge,and IGMBACK,GMBACK usually converge and outperform GMRES.

A LOW-COST OPTIMIZATION APPROACH FOR SOLVING MINIMUM NORM LINEAR SYSTEMS AND LINEAR LEAST-SQUARES PROBLEMS

Recently,the authors proposed a low-cost approach,named Optimization Approach for Linear Systems(OPALS)for solving any kind of a consistent linear system regarding the structure,characteristics,and dimension of the coefficient matrix A.The results obtained by this approach for matrices with no structure and with indefinite symmetric part were encouraging when compare with other recent and well-known techniques.In this work,we proposed to extend the OPALS approach for solving the Linear Least-Squares Problem(LLSP)and the Minimum Norm Linear System Problem(MNLSP)using any iterative low-cost gradient-type method,avoiding the construction of the matrices AT A or AAT,and taking full advantage of the structure and form of the gradient of the proposed nonlinear objective function in the gradient direction.The combination of those conditions together with the choice of the initial iterate allow us to produce a novel and efficient low-cost numerical scheme for solving both problems.Moreover,the scheme presented in this work can also be used and extended for the weighted minimum norm linear systems and minimum norm linear least-squares problems.We include encouraging numerical results to illustrate the practical behavior of the proposed schemes.

基于逆系统方法的广义非线性系统控制及电力系统应用

对一般广义非线性系统的反馈线性化问题,通过将代数变量视为虚拟输入,给出了构造性递归求逆算法,实现了系统的精确反馈线性化.作为该算法的应用,讨论了带有非线性负荷的三节点电力系统的励磁控制问题,设计了非线性励磁控制器.仿真结果表明了方法的有效性.

伺服系统线性特性和非线性摩擦的解耦辨识方法研究

针对伺服进给机构包含线性结构和非线性摩擦特性的特点,提出一种将线性模型和非线性摩擦特性进行分步解耦辨识的方法。首先分别确定伺服系统的线性模型和非线性模型,将伺服系统结构模型转化为线性模型加非线性摩擦反馈的结构。为了消除非线性摩擦力对线性模型的影响,采用2路同向非过零速信号对系统激励,利用2组系统输入和输出信号的差值作为线性参数的辨识数据对线性参数进行估计。获得线性模型后进一步利用系统稳态输出实现对非线性Coulomb摩擦幅值特性的估计。系统仿真和实验都证明了该辨识方法对提高伺服系统的辨识精度及控制器设计的有效性和可靠性。