Stationarity and stochastic differential equations

A linear differential equation of order N with stochastic process coefficients and excitation are studied. The objective of this paper is to demonstrate that, by using an expansion method, when the coefficients and excitation are strict sense stationary processes, the response is also a strict sense stationary process. Such problems occur frequently in the engineering sciences and are very important. Applications include parametric random vibrations, turbulent environment rotorcraft dynamics, and dynamics of axially loaded structural members, among others. An example application is provided.     

The theory of parametric oscillations

The correctness of the existing definitions of the parametric oscillations of linear and non-linear systems is discussed. The possibility of an erroneous choice of the mathematical parametric model instead of the autooscillatory model, connected with the existence in such systems of the same periodic solutions, is pointed out. Some non-local properties of parametric oscillations in Hamiltonian systems are established. It is shown, in particular, that stability regions are convex with respect to the frequency of the parametric excitation (i.e., all the points between the boundaries of neighbouring instability regions correspond to stable solutions). At the critical frequencies of parametric resonance the well-known Rayleigh and Zhuravlev theorems on the behaviour of the frequencies of natural oscillations when the stiffness and inertia changes are generalized. Some additional assertions on the limits of the first instability region for the Hill vector equations are established.     

Stabilization via parametric excitation of multi-dof statically unstable systems

The problem of re-stabilization via parametric excitation of statically unstable linear Hamiltonian systems is addressed. An n-degree-of-freedom dynamical system is considered, at rest in a critical equilibrium position, possessing a pair of zero-eigenvalues and n − 1 pairs of distinct purely imaginary conjugate eigenvalues. The response of the system to a small static load, making the zero eigenvalues real and opposite, simultaneous to a harmonic parametric excitation of small amplitude, is studied by the Multiple Scale perturbation method, and the stability of the equilibrium position is investigated. Several cases of resonance between the excitation frequency and the natural non-zero frequencies are studied, calling for standard and non-standard applications of the method. It is found that the parametric excitation is able to re-stabilize the equilibrium for any value of the excitation frequencies, except for frequencies close to resonant values, provided a sufficiently large excitation amplitude is enforced. Results are compared with those provided by a purely numerical approach grounded on the Floquet theory.     

Closure method and asymptotic expansions for linear stochastic systems

A closure procedure for the hierarchy of moment equations related to linear systems of ordinary differential equations with a random parametric excitation is introduced. A generalization of Pringsheim's theorem for continued fractions is used in a proof of the procedure convergence. The boundary function method for singular perturbation problems is applied to obtain asymptotic expansions for the moments of the solutions of such systems.     

参数不确定非线性组合大系统的鲁棒稳定性及参数鲁棒域的估计(英)

本文研究了参数不确定非线性组合大系统．首先利用现代微分几何理论将系统化为由线性子系统互联而成的组合大系统，然后给出了参数不确定非线性组合大系统鲁棒稳定性的若干判据．最后通过一个例子说明了本文的结论及参数鲁棒域的估计方法．     

参数激励Duffing-VanderPol振子的动力学响应及反馈控制

研究了Duffing-Van der Pol振子的主参数共振响应及其时滞反馈控制问题．依平均法和对时滞反馈控制项Taylor展开的截断得到的平均方程表明,除参数激励的幅值和频率外,零解的稳定性只与原方程中线性项的系数和线性反馈有关,但周期解的稳定性还与原方程中非线性项的系数和非线性反馈有关．通过调整反馈增益和时滞,可以使不稳定的零解变得稳定．非零周期解可能通过鞍结分岔和Hopf分岔失去稳定性,但选择合适的反馈增益和时滞,可以避免鞍结分岔和Hopf分岔的发生．数值仿真的结果验证了理论分析的正确性．     

On the equivalence of parametric contexts for linear inequality systems

In this paper we analyze the connections among different parametric settings in which the stability theory for linear inequality systems may be developed. Our discussion is focussed on the existence, or not, of an index set (possibly infinite). For some stability approaches it is not convenient to have a fixed set indexing the constraints. This is the case, for example, of discretization techniques viewed as approximation strategies (i.e., discretization regarded as data perturbation). The absence of a fixed index set is also a key point in the stability analysis of parametrized convex systems via standard linearization. In other frameworks the index set is very useful, for example if the constraints are perturbed one by one, even to measure the global perturbation size. This paper shows to what extent an index set may be introduced or removed in relation to stability.     

Nonlinear vibrations of offshore structures under random wave excitation

An account is given of some mathematical methods which have been used to analyse the response of offshore structures to random wave excitation. The analysis of nonlinear phenomena and the assessment of related nongaussian probability distributions are emphasized. The following problems are analysed in some depth: probability distributions for Morison-type wave loading; response near resonance of nonlinearly damped systems to random excitation; parametric resonance and instability of nonlinearly damped systems with randomly fluctuating restoring force coefficient. Solutions to each of these problem areas are illustrated by practical examples.     

Analysis of stability and synthesis of stable control systems for a class of nonlinear nonstationary systems

On the basis of sufficient condition of the stability of linear nonstationary discrete systems stated in this paper, we carry out an analysis of stability for a special class of discrete systems with nonstationary linear part, whose parameters satisfy constraints in the form of sets, and with a scalar nonlinear function satisfying linear or nonlinear restrictions. The problem of parametric synthesis of robustly stable control systems is solved for the same class of objects. The obtained results are generalized to the class of nonstationary systems with many nonlinearities of the same type.     

Robust dynamical compensator design for discrete-time linear periodic systems

This paper considers dynamical compensators design for purpose of pole assignment for discrete-time linear periodic systems. Similar to linear time-invariant systems, it is pointed out that the design of a periodic dynamical compensator can be converted into the design of a periodic output feedback controller for an augmented system. Utilizing the recent result on output feedback pole assignment, parametric solutions for this problem are obtained. The design approach can be used as a basis for the robust dynamical compensator design for this type of systems. Combined with a robustness index presented in this paper, robust dynamical compensator design problem is converted into a constrainted optimization problem. A numerical example is employed to illustrate the validity and feasibility of the methods.     

Consistent Initialization of Sensitivity Matrices for a Class of Parametric DAE Systems

A class of parametric semi-explicit differential algebraic equation (DAE) systems up to index 2 is considered. It is well known that initial value problems with DAE systems do not have a solution for every initial value. The initial value has to be consistent. Therefore, a method for the calculation of consistent initial values for this class of systems is introduced. In addition, various applications need information about the dependency of the solution of an initial value problem with respect to given parameters. This question leads to a linear matrix DAE system, the sensitivity DAE system, for which consistent initial values have to be provided as well. An appropriate consistent initialization method based on the solution differentiability of parametric nonlinear optimization problems in combination with Newton's method is developed. An illustrative example shows the capability of the method.     

Parametric computation of a fuzzy set solution to a class of fuzzy linear fractional optimization problems

The class of fuzzy linear fractional optimization problems with fuzzy coefficients in the objective function is considered in this paper. We propose a parametric method for computing the membership values of the extreme points in the fuzzy set solution to such problems. We replace the exhaustive computation of the membership values—found in the literature for solving the same class of problems—by a parametric analysis of the efficiency of the feasible basic solutions to the bi-objective linear fractional programming problem through the optimality test in a related linear programming problem, thus simplifying the computation. An illustrative example from the field of production planning is included in the paper to complete the theoretical presentation of the solving approach, but also to emphasize how many real life problems may be modelled mathematically using fuzzy linear fractional optimization.     

Effect of fast parametric viscous damping excitation on vibration isolation in sdof systems

In this work, we investigate analytically the effect of cubic nonlinear parametric viscous damping on vibration isolation in sdof systems. Attention is focused on the case of a fast parametric damping excitation. The method of direct partition of motion is used to derive the slow dynamic and steady-state solutions of this slow dynamic are analyzed to study the influence of the fast nonlinear parametric damping on the vibration isolation. This study shows that adding periodic nonlinear damping variation to the vibration isolation device can reduce transmissibility over the whole frequency range. The results also reveals that this nonlinear parametric viscous damping enhances vibration isolation comparing to the case where the cubic nonlinear damping is time-independent.     

Interpolatory model reduction of parameterized bilinear dynamical systems

Interpolatory projection methods for model reduction of nonparametric linear dynamical systems have been successfully extended to nonparametric bilinear dynamical systems. However, this has not yet occurred for parametric bilinear systems. In this work, we aim to close this gap by providing a natural extension of interpolatory projections to model reduction of parametric bilinear dynamical systems. We introduce necessary conditions that the projection subspaces must satisfy to obtain parametric tangential interpolation of each subsystem transfer function. These conditions also guarantee that the parameter sensitivities (Jacobian) of each subsystem transfer function are matched tangentially by those of the corresponding reduced-order model transfer function. Similarly, we obtain conditions for interpolating the parameter Hessian of the transfer function by including additional vectors in the projection subspaces. As in the parametric linear case, the basis construction for two-sided projections does not require computing the Jacobian or the Hessian.     

Features of the behavior of solutions to a nonlinear dynamical system in the case of two-frequency parametric resonance

Two-frequency parametric resonance in nonlinear dynamical systems is studied by analyzing a delay differential equation with the delay obeying a two-frequency law, which arises in the mathematical simulation of some physical processes. It is shown that the system can exhibit chaotic oscillations (strange attractors) when the parametric excitation frequencies are both close to the doubled eigenfrequency of the system (degenerate case). The formation mechanisms of chaotic attractors are discussed, and the Lyapunov exponents and the Lyapunov dimension are calculated for them. If only one of the parametric excitation frequencies is close to the double eigenfrequency, a two-frequency regime occurs in the system.     

Necessary and sufficient constraint qualifications for solvability of systems of infinite convex inequalities

In this paper we present constraint qualifications which completely characterize the Farkas–Minkowski and the locally Farkas–Minkowski convex (possibly infinite) inequality systems posed in topological vector spaces. The number of constraints and the dimension of the linear space are arbitrary (possibly infinite). The constraint qualifications considered in this paper are expressed in terms of the solvability of certain parametric convex (linear) systems and the uniform strong duality or the uniform min–max duality relative to the Lagrange (Haar) dual problems of suitable convex (linear) parametric optimization problems.     

Influence of internal parametric and kinematic excitation on nonlinear gearbox vibration

This paper deals with mathematical modelling of nonlinear vibration of large rotating shaft systems with gears and rollingelement bearings. Gearing and bearing couplings bring into the system nonlinear phenomena like impact motions due to the possibility of the mesh interruption. The motion of the system is influenced by the internal kinematic excitation in gearing and by the parametric excitation caused by periodic change of number of teeth in gear meshing. The influence of simultaneous internal kinematic and parametric excitation is investigated in dependence on revolutions of the driving shaft of a test-gearbox. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Based on Walsh packet parametric identification technique for dynamic systems

The article presents an application of wavelet packet analysis in the parametric identification of discrete dynamic systems. The Walsh packet basis and their operational matrixes enables reduction of row derivatives, which are necessary in the procedure of identification of signal derivatives for systems, which are analyzed, and also enable algebraization their mathematical describe, what in combination with wavelet packet analysis with Haar filter properties decided on a great efficiency of parametric identification of linear systems procedure, which is built on wavelet packet analysis. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parametric and Autoparametric Resonance

A large number of mathematical questions are related to problems of parametric and autoparametric resonance in engineering models. The linearized problems generally produce systems of differential equations with periodic coefficients with special stability and genericity questions. We start by reviewing linear systems while discussing normal form techniques and bifurcation results. The linear and nonlinear analysis is illustrated in three cases: rotor dynamics, autoparametric resonance of a parametric oscillator and autoparametric resonance of a self-excited oscillator. In all cases bifurcations, symmetry considerations and attraction to nonclassical limit sets play a part.     

Bilinear Control and Application to Flexible a.c. Transmission Systems

The purpose of this paper is an integrated overview of bilinear systems (BLS) research which has evolved over the past few decades, and a new result on control of flexible a.c. transmission systems (FACTS) is presented. BLS may be derived in many cases from principles of physics, chemistry, biology, socioeconomics, and engineering. In other cases, BLS are more accurate approximations to nonlinear systems than are traditional linear systems, as shown for example by the added bilinear terms (in state and control) for the Taylor series.While an appropriately designed linear control system may be optimum relative to some quadratic performance index without added constraints, bilinear or parametric control can be designed to improve more global performance and indeed to increase the region of attainable states. Such controllability and stabilization of BLS and of a series line-capacitor controlled FACTS is presented.