Partial eigenvalue assignment and its stability in a time delayed system

Active vibration control strategy is an effective way to control dangerous vibrations in a structure, caused by resonance and to manipulate the dynamics of vibrational response. Implementation of this strategy requires real-time computations of two feedback control matrices such that a small amount of eigenvalues of the associated quadratic matrix pencil are replaced by suitably chosen ones while the remaining large number of eigenvalues and eigenvectors remain unchanged ensuring the no spill-over. This mathematical problem is referred to as the Quadratic Partial Eigenvalue Assignment problem. The greatest challenge there is to solve the problems using the knowledge of only a small number of eigenvalues and eigenvectors that are computable using state-of-the-art techniques. This paper generalizes the earlier work on partial assignment to constant time-delay systems. Furthermore, a posterior stability analysis is carried out to identify the ranges of the time-delay that maintains the closed-loop assignment while keeping the stability of the infinite number of eigenvalues for the time-delayed systems. The practical features of the proposed methods are that it is implemented in the second-order setting itself using only those small number of eigenvalues and the eigenvectors that are to be assigned and the no spill-over is established by means of mathematical results. The results of our numerical experiments support the validity of our proposed methods.     

A symmetry preserving alternating projection method for matrix model updating

The matrix model updating problem (MMUP), considered in this paper, concerns updating a symmetric second-order finite element model so that the updated model reproduces a given set of desired eigenvalues and eigenvectors by replacing the corresponding ones from the original model, and preserves the symmetry of the original model. In an optimization setting, this is a constrained nonlinear optimization problem. Taking advantage of the special structure of the constraint sets, it is first shown that the MMUP can be formulated as an optimization problem over the intersection of some special subspaces and linear varieties on the space of matrices. Using this formulation, an alternating projection method (APM) is then proposed and analyzed. The projections onto the involved subspaces and linear varieties are characterized. To the best of our knowledge, an alternating projection method for MMUP has not been proposed in the literature earlier. A distinct practical feature of the proposed method is that it is implementable using only a few measured eigenvalues and eigenvectors. No knowledge of the eigenvalues and eigenvectors of the associated quadratic matrix pencil is required. The results of our numerical experiments on both illustrative and benchmark problems show that the algorithm works well. The paper concludes with some future research problems.     

Robust pole assignment using velocity–acceleration feedback for second-order dynamical systems with singular mass matrix

In this paper the robust pole assignment problem using combined velocity and acceleration feedback for second-order linear systems with singular mass matrix is illustrated. This is promising for better applicability in several practical applications where the acceleration signals are easier to obtain than the proportional ones. First, the explicit parametric expressions of both the feedback gain controller and the eigenvector matrix are derived. The parametric solution involves manipulations only on the original second-order model. The available degrees of freedom offered by the velocity–acceleration feedback in selecting the associated eigenvectors are utilized to improve robustness of the closed-loop system. Straight-forward computational algorithms are introduced to demonstrate the effectiveness of the proposed approach. These algorithms are applicable for a dynamical system with mass matrices that can be either singular or nonsingular. Numerical examples are provided to illustrate the application of the proposed procedure.     

一维结构系统重频特征问题的摄动传递矩阵法

基于传递矩阵法，给出一维结构系统重频振动特征问题的二阶摄动计算方法，该方法适用于一般的一维结构系统的实数和复数重频振动特征问题的摄动分析，并且可避免模态截断误差的影响。只要知道某阶重频的特征值和它的特征向量，用该方法便可准确地计算出其特征值和特征向量的一阶与二阶摄动。算例对弹性支承梁的复数重频振动特征问题进行摄动分析，摄动计算结果和精确计算结果吻合良好。     

Optimal mass distribution in vibrating systems

The problem of determining the distribution of mass in vibrating systems that optimize the extreme eigenvalue is considered. We give explicit solutions for the fundamental discrete models of an axially vibrating rod and a Bernoulli–Euler beam.The problems are first recast as affine sum of matrices with linear total mass constraint. This formulation allows determination of the eigenvectors corresponding to the optimal eigenvalues by solving a set of quadratic equations recursively. The optimal mass distribution is then determined by the recovered eigenvectors.     

Robust pole placement in structures by the method of receptances

The problem of robust pole placement in structural vibration using receptance data is considered. Expressions are derived for the sensitivities of the eigenvalues to perturbations in the measurements and for robustness to measurement errors these sensitivity terms should be made as small as possible. A sequential multi-input state-feedback approach is described and by this procedure it is shown that a different eigenvalue may be assigned at each step without changing those eigenvalues assigned at previous steps. The columns of the force distribution matrix are chosen to excite easily the eigenvalue considered at the current step of the procedure. The sequential approach has the advantage of a characteristic equation that is linear in the control gains and is shown to be inherently more robust to measurement noise than the single-input method. The effects of sequential multi-input state feedback when combined with minimisation of eigenvalue sensitivity norms are investigated.     

振动系统特征值问题的矩阵灵敏度分析

应用Kronecker代数和矩阵微分理论，系统地发展了振动系统的特征值和特征向量的矩阵灵敏度分析方法，给出了向量值和矩阵值数的结构系统的特征值和特征向量的灵敏度，通过求导数排列成二维矩阵，所得的结果易形成计算机程序。该方法可以扩展延伸到具有一般矩阵和重根的特征值灵敏度分析的问题之中。     

Eigenvalue problems of rotor system with uncertain parameters

A general method for investigating the eigenvalue problems of a rotor system with uncertain parameters is presented in this paper. The recurrence perturbation formulas based on the Riccati transfer matrix method are derived and used for calculating the first- and secondorder perturbations of eigenvalues and their respective eigenvectors for the rotor system with uncertain parameters. In addition, these formulas can be used for investigating the independent, and repeated, as well as the complex eigenvalue problems. The general method is called the Riccati perturbation transfer matrix method (Riccati-PTMM). The formulas for calculating the mean value, variance, and covariance of the eigenvalues and eigenvectors of the rotor system with random parameters are also given. Riccati-PTMM is used for calculating the random eigenvalues of a simply supported Timoshenko beam and a test rotor supported by two oil bearings. The results show that the method is accurate and efficient.     

空间一般6R机械手的位置反解的新算法

使用D-H方法建立一般6R机械手的齐次坐标矩阵方程,引入酉矩阵将矩阵方程中关节角的三角函数转化成复指数函数,在复数域内找到10个形式相同的方程式。通过分离方程组中多项式的系数,构造一个16×16实数矩阵,计算矩阵的特征值和特征向量求出机械手的位置反解,并将该算法用C++来实现。     

Robust observer-based absolute stabilization for Lur’e singularly perturbed systems with state delay

This paper is concerned with the problem of robust observer-based absolute stabilization for Lur’e singularly perturbed time-delay systems. The aim is to design a suitable observer-based feedback control law such that the resulting closed-loop system is absolutely stable. First, a full-order state observer is constructed. Based on the linear matrix inequality (LMI) technique, a delay-dependent sufficient condition is presented such that the observer error system is absolutely stable. Then, for observer-based feedback control, by introducing some slack matrices, a sufficient condition for input-to-state stability (ISS) of the closed-loop system with regard to the observer error is presented. Thus, the absolute stabilization of the closed-loop system can be guaranteed based on the ISS property. In addition, the criteria presented are both independent of the small parameter and the upper bound for the absolute stability can be obtained in a workable algorithm. Finally, two numerical examples are provided to illustrate the effectiveness of the developed methods.     

闭环控制系统反馈增益向量的灵敏度分析

根据振动系统的模态控制方程,利用状态反馈控制和闭环控制系统的极点配置法,文中提出计算模态反馈增益向量的新方法,并在此基础上推导模态反馈增益向量灵敏度分析的方法.该分析方法涉及开环系统的特征值、特征向量以及模态控制矩阵的灵敏度, 由于在方法的推导过程中使用了模态控制方程,因此避免了非常复杂的数学推演.最后,将所提方法应用于一桁架结构的振动控制问题,以说明方法的有效性.     

Simulation and Control of an Active Tilting-Pad Journal Bearing

This study developed an active tilting-pad journal bearing with a feedback control system to regulate the orbit of a rotating shaft. The control is implemented by means of linear actuators installed behind the pivot of each pad, which allow the radial motion of the pads in real time. The control design uses the linear feedback of the state variables of the bearing-rotor system, with the feedback gains determined by the optimization of a quadratic performance index. The optimization is based on a linear spring-mass model that incorporates the direct stiffness and damping elements associated with each of the bearing pads. This linear model is found by the simulation of the system under small perturbations using a nonlinear Reynolds equation model. The nonlinear model is capable of simulating the radial motions of the pads by the actuators and is used to verify the effectiveness of the feedback control. It is shown that certain design parameters in the quadratic performance index may be used to determine both the stiffness and the damping of the closed-loop bearing system and that the shaft orbit can be thereby suitably regulated.     

Determination of spectral densities of dynamic characteristics of a nonlinear model of structure

A stochastic problem of the dynamics of a nonlinear model of structure is considered. The problem is solved with the use of an expansion of the solution in the truncated orthogonal basis of eigenvectors of the linear model. It is assumed that nonlinearities of the system do not result in a fundamental change in its dynamical behavior, but contribute a significant quantitative correction relative to the linear model. The nonlinear characteristics are statistically linearized. The coefficients of statistical linearization are formally interpreted as some variations in the corresponding elements of the stiffness matrix of the linear model. Spectral densities, displacement variances of the linearized system, as well as its eigenvalues and eigenvectors, are represented as a power series expansion in the above variations considering linear or square approximations. To find the coefficients in the expansion, the sensitivity theory is used. This allows one to form a set of algebraic transcendental equations in displacement variances, which is solved by the method of successive approximations. Then all the target spectral densities are found. The results are illustrated by examples.     

Partial eigenvalue assignment for high order system by multi-input control

An algorithm of partial eigenvalue assignment problem for high order systems is given that the spectrums are partially reassigned to predetermined locations and the remaining spectrums keep unchanged. The algorithm requires the knowledge of only a small number of eigenvalues and their corresponding eigenvectors. Numerical examples are done to illustrate the effect of the approach.     

Assignment of eigenvalue sensitivities from receptance measurements

The paper addresses the problem of eigenvalue assignment in active vibration control by the receptance method using single-input state feedback. It is demonstrated that not only can assignment be applied to the poles of the system but also to the sensitivities of the poles. The approach described has applications particularly in vibration absorption and the detuning of structures to avoid resonance. Insensitivity of the poles to perturbations in the system gains is desirable from the point of view of robust dynamical performance. The perturbation method is applied to a characteristic equation resulting in expressions for the derivatives of the pole eigenvalues with respect to the control gains. The method is based entirely upon measured vibration data in the form of receptances, there being no need to determine or to know the M, C, K matrices. The closed-loop sensitivities are expressed as linear functions of the control gains, thereby making accessible the assignment of the sensitivities to chosen values. For the case of the open-loop system, the method is shown to produce identical results to established eigenvalue sensitivity procedures. It is demonstrated that chosen eigenvalues may be rendered insensitive to all the control gain terms by choice of the distribution vector b. Numerical examples are provided to illustrate the working of the technique.     

Chaos synchronization of uncertain chaotic systems using composite nonlinear feedback based integral sliding mode control

This paper proposes a combination of composite nonlinear feedback and integral sliding mode techniques for fast and accurate chaos synchronization of uncertain chaotic systems with Lipschitz nonlinear functions, time-varying delays and disturbances. The composite nonlinear feedback method allows accurate following of the master chaotic system and the integral sliding mode control provides invariance property which rejects the perturbations and preserves the stability of the closed-loop system. Based on the Lyapunov- Krasovskii stability theory and linear matrix inequalities, a novel sufficient condition is offered for the chaos synchronization of uncertain chaotic systems. This method not only guarantees the robustness against perturbations and time-delays, but also eliminates reaching phase and avoids chattering problem. Simulation results demonstrate that the suggested procedure leads to a great control performance.     

A unique solution for principal component analysis-based multi-response optimization problems

A procedure to find a unique solution for multi-response optimization problems based on indexing is presented. The procedure utilizes principal component analysis to map the original data to a new vector of component scores, transforming the original response variables into uncorrelated principal components. This process involves loadings that are the elements of the eigenvectors corresponding to the eigenvalues of response variables in the correlation matrix. It is shown that for a given eigenvalue λ, its corresponding eigenvectors are not unique, which could lead to different “optimal” parametric (factor-level) settings and will further mislead the process or product improvement strategy. The proposed indexing method will determine a unique optimal solution in the presence of (2^p)(p!) combinations of eigenvectors.     

邻接矩阵在平面机构分析中的应用

研究了邻接矩阵在平面机构自由度计算、同构识别和速度瞬心求解中的应用。揭示了邻接矩阵与机构自由度的关系。给出了同构运动链邻接矩阵的特征向量及特征值关系的充要条件 ,据此提出了识别机构运动链同构的方法。最后 ,提出了应用邻接矩阵自乘求解机构全部瞬心及求解给定瞬心最短路经的方法     

SYMMETRIC INVERSE EIGENVALUE VIBRATION PROBLEM AND ITS APPLICATION

This paper summarises the authors' previous effort on inverse eigenvalue problem for linear vibrating systems described by a vector differential equation with constant coefficient matrices and non-proportional damping. The inverse problem of interest here is that of determining real symmetric coefficient matrices assumed to represent mass normalised velocity and position coefficient matrices, given a set of specified complex eigenvalues and eigenvectors. There are given two solutions of a symmetric inverse eigenvalue problem presented by Starek and Inman [1, 2].The theory of inverse eigenvalue problem is applied to the model updating problem. The goal of this paper is to recognise that the model updating problem is a subset of the inverse eigenvalue problem. The approach proposed here is to use the results of inverse eigenvalue problem to develop methods for model updating.Comments are made on how their procedure may be used to solve the damage detection problem.     

A direct method for model updating with incomplete measured data and without spurious modes

This paper presents a new method for finite element matrix updating problem in an undamped model. This method has a computationally convenient feature of handling the difficulty of the incomplete measured data in an algorithmic way without using standard modal expansion or reduction techniques. The completed eigenvector matrix is such that it is mass normalised with respect to the analytical mass matrix. The method is also capable of preserving the large number of eigenvalues and eigenvectors of the finite element model that are not affected by updating. The latter has an effect of preventing the appearance of spurious modes in the frequency range of interest, which is a common concern with most direct updating methods.