Stability analysis of distributed-order nonlinear dynamic systems

The problem of asymptotic stability analysis of equilibrium points in nonlinear distributed-order dynamic systems with non-negative weight functions is considered in this paper. The Lyapunov direct method is extended to be used for this stability analysis. To this end, at first, a discretisation scheme with convergence property is introduced for distributed-order dynamic systems. Then, on the basis of this tool, Lyapunov theorems are proved for asymptotic stability analysis of equilibrium points in distributed-order systems. As the order weight function assumed for the distributed-order systems is general enough, the results are applicable to a wide range of nonlinear distributed-order systems such as fractional-order systems with multiple fractional derivatives. To verify the applicability of the obtained results, these results are applied for the stability analysis of a distributed-order diffusion system and control of a fractional-order Lorenz system with multiple fractional derivatives.     

NASAB: a finite element software for the nonlinear aerostatic stability analysis of cable-supported bridges

This paper presents the development and applications of the finite element software, NASAB, which can be used for linear, geometrically nonlinear, and materially nonlinear analyses of structure and nonlinear aerostatic stability analysis of cable-supported bridges. The software program consists of two main parts: a programming part and a computational part. The windows programming part written in FORTRAN90 was designed mainly to present the NASAB software in a user-friendly environment. The computational part was written in FORTRAN77. The use of FORTRAN77 is to effectively take advantage of existing codes, thus speeding up code design and implementation. The usefulness of FORTRAN programming language to develop a user-friendly interface including pre-processing and post-processing has been demonstrated by the present version of the software.     

Nonlinear model predictive control of multivariable processes using block-structured models

Block-structured models, such as Wiener or Hammerstein models, have been used in nonlinear model predictive control to reduce the cost of identification and online computation. The solution of a nonlinear dynamic optimization problem has been avoided by inverting the nonlinear element and solving the resulting linear problem in the past. However, by exploiting the block structure for sensitivity calculation, the original nonlinear problem can also be solved at low computational cost. At the same time, greater modeling flexibility is achieved. Recently, a new Hammerstein model structure has been proposed for multivariable processes with input directionality, which exploits such increased modeling flexibility. This paper deals with nonlinear model predictive control constrained by models of Hammerstein or Uryson structure. A method for efficient calculation of sensitivity information is developed. In a simulation example, the method is shown to combine low computational cost with a significant reduction of the loss of optimality compared to the previous methods.     

Equilibrium analysis of multi-limbs walking and climbing robots

The paper discusses the complex problem of assessing online the static equilibrium of statically-indeterminate climbing and walking robots (CLAWARs) with quasi-static locomotion. The method proposed is general and works for whatever number of legs and ropes operated by actuated winches connecting the robot to the environment. The configuration of the robot is assigned. First, the compliance of the robot body, of the legs and the compliances of the ground and the ropes are modeled as localized elasticities. The static equilibrium problem of the resulting model is statically-determinate under the hypothesis that the foot and rope points (where the ropes are fixed to the robot body) are joined to the ground by bilateral constraints. Since these constraints are unilateral (the feet are contact points and can detach from the ground, and the ropes can become slack), it is necessary to apply an iterative solving procedure in order to solve the static equilibrium problem. The method presented in the paper is a fast and effective alternative to nonlinear analysis of a finite element model of the robot at any assigned configuration. As an example, we consider the case of the heavy-duty CLAWAR Roboclimber.     

Estimating the Attraction Domain for the Boost Inverter

This work presents an approach for estimating the domain of attraction for polynomial systems with state and control‐signal constraints, including saturation. In many problems, it is possible to derive global stability properties for such systems, neglecting constraints. Consideration of the constraints usually makes the problem much more complicated. In this paper, the stability analysis performed for the unconstrained case is used for the problem as a whole. For application of the method, there are powerful computational tools that can be employed in cases of polynomial systems. The technique is not only valid for the analysis of equilibrium points, but also for other attractors, such as limit cycles. As examples, the domain of attraction for given control laws is estimated for both a nonlinear DC‐DC boost converter and for a boost inverter.     

On the estimation of the equilibrium points of uncertain nonlinear systems

The analysis and control of nonlinear systems often require information about the location of their equilibrium points. This paper addresses the problem of estimating the set of equilibrium points of uncertain nonlinear systems, in particular, systems whose dynamics are described by a nonlinear function of the state depending polynomially on an uncertainty vector constrained in a polytope. It is shown that estimates of this set can be obtained by solving LMI problems, which are built through sum of squares techniques by introducing worst‐case truncations of the nonlinearities and by exploiting homogeneity of equivalent representations. In particular, the computation of estimates with fixed shape and the problem of establishing their tightness is firstly considered. Then, the paper shows how this methodology can be used to address the computation of the minimum volume estimate and the construction of the smallest convex estimate. Examples with random and real systems illustrate the proposed methodology. Copyright © 2011 John Wiley & Sons, Ltd.     

Theoretical analysis of unconstrained nonlinear model predictive control

Model predictive control (MPC) is a well-established controller design strategy for linear process models. Because many chemical and biological processes exhibit significant nonlinear behaviour, several MPC techniques based on nonlinear process models have recently been proposed. The most significant difference between these techniques is the computational approach used to solve the nonlinear model predictive control (NMPC) optimization problem. Consequently, analysis of NMPC techniques is often connected to the computational approach employed. In this paper, a theoretical analysis of unconstrained NMPC is presented that is independent of the computational approach. A nonlinear discrete-time, state-space model is used to predict the effects of future inputs on future process outputs. It is shown that model inverse, pole-placement, and steady-state controllers can be obtained by suitable selection of the control and prediction horizons. Moreover, the NMPC optimization problem can be modified to yield nonlinear internal model control (NIMC). The computational requirements of NIMC are considerably less than NMPC, but the NIMC approach is currently restricted to nonlinear models with well-defined and stable inverses. The NIMC controller is shown to provide superior servo and regulatory performance to a linear IMC controller for a continuous stirred tank reactor.     

Stability analysis for linear systems with input backlash through sufficient LMI conditions

This paper addresses the problem of stability analysis for a given class of nonlinear systems resulting from the connection of a linear system with an isolated backlash operator. Constructive conditions based on LMIs to ensure closed-loop stability are proposed by using some suitable Lyapunov functionals with quadratic terms and Lure type terms, and generalized sector conditions. Additionally, the boundary of the associated set of all the admissible equilibrium points is precisely defined.     

自治分段线性振荡系统的离散映射数值建模与稳定性分析

为研究自治分段线性振荡系统中出现的分叉及混沌动力学行为，系统建模及稳定性分析是一种必不可少的分析手段. 本文通过构造周期内离散映射解析模型，并结合边界条件的动态数值求解，提出了一种动态离散映射数值建模方法，进而推导了用以判定系统周期闭轨稳定性的Jacobian 矩阵求解模型. 最后，本文以电力电子系统中一种常用的5 维软开关逆变自治振荡电路作为实例，通过模型仿真观察到频率的分叉现象，并根据Jacobian 矩阵的特征乘数对系统的稳定性进行了研究. 基于该模型的仿真分析结果与实验系统中所观察到的现象一致，从而验证了该方法的有效性.     

Ocean software for modeling dynamics of gravity waves

This work is aimed at describing the Ocean software package developed for modeling the dynamics of nonlinear gravity waves and the process of breaking on the surface of a liquid of infinite depth. The modeling is carried out only for a simply connected domain. The used algorithms of Longuet-Higgins, Cokelet, and Krasil’nikov are described. The structure of the software package, computational methods, and indicative functions are described. Tactics for conducting the experiments are proposed. The results of modeling and their comparison with the experimental data are given.     

A general nonlinear analysis of concrete structures and comparison with frame tests

A computational technique is described for the general nonlinear analysis of large planar frames under static loading. The analysis accommodates both material and geometric sources of nonlinearity in a highly stable numerical procedure. For material behavior the moment-thrust-curvature relationships are reduced to polynomials made to represent common structural members. The nonlinear influence of axial shortening and P - Δ effects from displaced joints are accommodated by the analysis. Beam-column effects that magnify moments by the influence of axial force acting through deformed members can be approximated by introducing nodal points (or joints) at points of maximum deformation between the ends of members.Accuracy of the analytic procedure is demonstrated by comparing computed results with eight reinforced concrete frames tested at the University of Texas at Austin.     

适于估计OD矩阵的交通检测点的最优分布

讨论了适于估计起迄点出行分布矩阵(OD矩阵)的交通检测点的合理分布问题.根 据检测点应当满足的规则,建立了关于检测点分布的非线性规划模型.在已知极点问转移概 率的前提下,将检测点的分布问题描述成一个平均报酬Markov决策过程,并通过转化为一 个等价的整数线性规划问题来求解.最后实例结果表明该模型是有效的、合理的.     

On nonlinear dynamic analysis using substructuring and mode superposition

The solution of nonlinear dynamic equilibrium equations using mode superposition and substructuring is studied. The objective is to design schemes that in some analyses can significantly decrease the computational effort involved when compared to a complete direct integration solution. Specific schemes for mode superposition analysis and substructuring are proposed. These techniques have been implemented in ADINA. The results of a few sample analyses are presented and recommendations are given on the use of these procedures in practical analysis.     

Partitioned but strongly coupled iteration schemes for nonlinear fluid–structure interaction

We look at the computational procedure of computing the response of a coupled fluid–structure interaction problem. We use the so-called strong fluid–structure coupling––a totally implicit formulation. At each time step in an implicit formulation, new values for the solution variables have to be computed by solving a nonlinear system of equations, where we assume that we have solvers for the subproblems. This is often the case, when we have existing software to solve each subproblem separately, and want to couple both. We show how to solve the overall nonlinear system by using only the solvers for the subproblems. This is achieved not by considering the equilibrium equations, but the fixed-point problem resulting from the solution iteration for each of the subproblems.     

Development and Analysis of a Neural Dynamical Approach to Nonlinear Programming Problems

This technical note develops a neural dynamical approach to nonlinear programming (NP) problems, whose equilibrium points coincide with Karush-Kuhn-Tucker points of the NP problem. A rigorous analysis on the global convergence and the convergence rate of the proposed neural dynamical approach is carried out under the condition that the associated Lagrangian function is convex. Analysis results show that the proposed neural dynamical approach can solve general convex programming problems and a class of nonconvex programming problems. Two nonconvex programming examples are provided to demonstrate the performance of the developed neural dynamical approach.     

Performance analysis tools applied to a finite element adaptive mesh free boundary seepage parallel algorithm

A finite element, adaptive mesh, free surface seepage parallel algorithm is studied using performance analysis tools in order to optimize its performance. The physical problem being solved is a free boundary seepage problem which is nonlinear and whose free surface is unknown a priori. A fixed domain formulation of the problem is discretized and the parallel solution algorithm is of successive over-relaxation type. During the iteration process there is message-passing of data between the processors in order to update the calculations along the interfaces of the decomposed domains. A key theoretical aspect of the approach is the application of a projection operator onto the positive solution domain. This operation has to be applied at each iteration at each computational point.The VAMPIR and PARAVER performance analysis software are used to analyze and understand the execution behavior of the parallel algorithm such as: communication patterns, processor load balance, computation versus communication ratios, timing characteristics, and processor idle time. This is all done by displays of post-mortem trace-files. Performance bottlenecks can easily be identified at the appropriate level of detail. This will numerically be demonstrated using example test data and comparisons of software capabilities that will be made using the Blue Horizon parallel computer at the San Diego Supercomputer Center.     

Potential of minicomputer-array processor system for nonlinear finite-element analysis

A study is made of the potential of using a minicomputer-array processor system for efficient solution of large-scale nonlinear finite-element problems. A PRIME 750 is used as the host computer, and a software simulator residing on the PRIME is employed to assess the performance of the Floating Point Systems AP-120B array processor. Major hardware characteristics of the system such as virtual memory, parallel and pipeline processing are reviewed and the interplay between various hardware components is examined. Effective use of the minicomputer-array processor system for nonlinear analysis requires the following: (a) proper selection of the computational procedure and the capability to vectorize the numerical algorithms; (b) reduction of I/O operations; and (c) overlapping host and array-processor operations. A detailed discussion is given of techniques to accomplish each of these tasks. Two benchmark problems with 1715 and 3230 degrees of freedom, respectively, are selected to measure the anticipated gain in speed obtained by using the proposed algorithms on the array processor. Results of the study of the two benchmarks indicate that these two problems would run faster on a PRIME 750 coupled with the AP-120B than on the PRIME 750 alone. The 1715 degree-of-freedom problem would run about five times faster, and the 3230 degree-of-freedom problem would run about ten times faster. New advances in array-processor hardware are outlined, and possible improvements in the computational algorithms are discussed. The combination of the two can significantly enhance the effectiveness of the minicomputer-array processor system for large-scale nonlinear analysis.     

A generalized hybrid transfinite element computational approach for nonlinear/linear unified thermal-structural analysis

The paper describes the development of a new hybrid computational approach for nonlinear/linear thermal-structural analysis. The proposed transfinite element approach is a hybrid scheme as it combines the modeling versatility of contemporary finite elements with transform methods and the classical Bubnov-Galerkin schemes. Application of the proposed formulations for nonlinear analysis is also developed. Several test cases are presented to include nonlinear/linear unified thermal-stress and thermal-stress wave propagations. Comparative results validate the fundamental capabilities of the proposed hybrid transfmite element methodology.     

基于逆阵更新方法的局部非线性结构频响分析

针对含有非线性连接的大型局部非线性结构,采用描述函数表示其所连接的非线性内力,将非线性结构频响表示为拟线性动柔度矩阵(Quasilinear Receptance Matrix),提出一种逆阵更新(Inverse Matrix Updating Method,IMU)方法,将求解系统动柔度(频响特性)的高阶矩阵求逆转化为低阶矩阵的求逆,从而获得大型局部非线性结构主频响应的快速计算方法.仿真结果表明,本文的分析方法具有较好的稳定性,并能大幅提高大型局部非线性结构主频响应的计算效率.     

An efficient implementation of adjoint sensitivity analysis for optimal control problems

In this paper an efficient implementation of design sensitivity analysis techniques is presented for nonlinear optimal control problems using the adjoint variable method. Techniques for functionals (integrals) and dynamic response (or pointwise) constraints are developed. Emphasis is placed on the proper choice of numerical techniques which exploit the structure of the problem to achieve efficiency. Numerical results for two optimal control examples show great improvement over previous implementations. Unlike previous results the computational effort required for DSA is shown to increase only linearly with the number of discretization points used and is a much smaller percentage of total CPU time.