Fai Ma

The exact steady-state solution of a class of non-linear stochastic systems

Abstract In this paper exact steady state solutions are constructed for a class of non-linear systems subjected to stochastic excitation. The results are then applied to both classical and non-classical oscillator problems.

Extension of second moment analysis to vector-valued and matrix-valued functions

Abstract In this paper vector and matrix transformations of random variables are considered. Techniques from matrix calculus and Kronecker algebra are employed to systematically develop generalized formulae for second moment analysis. The results derived are easily amenable to computational procedures.

Constrained Kalman Filter for Nonlinear Structural Identification

It is well known that hysteresis is a principal source of internal energy dissipation in structural systems. One of the widely accepted hysteretic models for structures is a differential model, called the modified Bouc-Wen model. In this paper, a constrained Kalman filter with a global weighted iteration strategy has been used to estimate the parameters in the modified Bouc-Wen model. An actual seismic signal was used to excite a nonlinear structural system described by the modified Bouc-Wen model that has both degradation and pinching characteristics. The convergence results show that, for both clean and noise-corrupted data, this algorithm is effective in estimating all the 11 parameters with satisfactory precision and fast convergence. As this algorithm has modest computing requirements, it should be acceptable as a basic tool for estimating hysteretic parameters in engineering design.

On the stability of stochastic difference systems

Abstract In this paper the stability of linear stochastic difference equations and a class of weakly non-linear stochastic difference equations is considered. For the linear systems explicit criteria are derived for the stability of the moments of any order. We also show how the moments of a linear stochastic difference system can be computed when a certain Lie-algebraic condition is satisfied.

The transformation of second-order linear systems into independent equations

The class of second-order linear dynamical systems is considered. A method and algorithm are presented to transform any system with n degrees of freedom into n independent second-order equations. The conversion utilizes a real, invertible but nonlinear mapping and is applicable to practically every linear system. Two examples from earthquake engineering are provided to indicate the utility of this approach.

Mean stability of stochastic difference systems

Abstract In this paper the mean stability of linear and non-linear stochastic difference systems is considered. For linear systems the relationship between mean stability and other stability definitions is explored. For the non-linear system explicit criteria for mean stability are derived when the non-linear term satisfies a certain realistic condition.

Stability Theory of Stochastic Difference Systems

Publisher Summary This chapter discusses the stability theory of stochastic difference systems. A stochastic difference system is one in which one or more variables can change stochastically at discrete instants of time. Stochastic difference systems are the stochastic versions of deterministic discrete time systems. The class of stochastic difference systems includes most modern industrial and military control systems, for they invariably include some elements whose inputs or outputs are discrete in time. Examples of such elements are digital computers, pulsed radar units, and coding units in most communication systems. One of the most important qualitative properties of stochastic difference systems is the stability of such systems. The chapter presents criteria for stability of linear and nonlinear stochastic difference systems. It highlights the relationship between various stability definitions is shown. The chapter also discusses properties peculiar to linear systems are also discussed. Various special methods have been devised for the study of the stability of stochastic differential equations.

On the arithmetic complexity of matrix Kronecker powers

Abstract In this paper we study the arithmetic complexity of computing the p th Kronecker power of an n × n matrix. We first analyze a straightforward inductive computation which requires an asymptotic average of p multiplications and p – 1 additions per computed output. We then apply efficient methods for matrix multiplication to obtain an algorithm that achieves the optimal rate of one multiplication per output at the expense of increasing the number of additions, and an algorithm that requires O(log p) multiplications and O(log 2 p) additions per output.

Stationary Response of a Class of Nonlinear Stochastic Systems Undergoing Markovian Jumps

Rong-Hua Huan Department of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China Wei-qiu Zhu 1 Department of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China e-mail: wqzhu@zju.edu.cn Fai Ma Department of Mechanical Engineering, University of California, Berkeley, CA 94720 Zu-guang Ying Department of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China Stationary Response of a Class of Nonlinear Stochastic Systems Undergoing Markovian Jumps Systems whose specifications change abruptly and statistically, referred to as Markovian- jump systems, are considered in this paper. An approximate method is presented to assess the stationary response of multidegree, nonlinear, Markovian-jump, quasi-nonintegrable Hamiltonian systems subjected to stochastic excitation. Using stochastic averaging, the quasi-nonintegrable Hamiltonian equations are first reduced to a one-dimensional It^ o equation governing the energy envelope. The associated Fokker–Planck–Kolmogorov equation is then set up, from which approximate stationary probabilities of the original system are obtained for different jump rules. The validity of this technique is demon- strated by using a nonlinear two-degree oscillator that is stochastically driven and capa- ble of Markovian jumps. [DOI: 10.1115/1.4029954] Keywords: Markovian jumps quasi-nonintegrable Hamiltonian system, stochastic excitation, stochastic averaging Introduction Markovian-jump systems represent a class of stochastic and hybrid systems whose operational rules can change in accordance with a Markov process. Complex dynamical systems are often Markovian-jump systems because abrupt changes in their configurations may occur due to component or interconnection failure or sudden environmental disturbances. Indeed, important examples of such systems include industrial plants and economic systems. Markovian-jump systems were first introduced by Kra- sovskii and Lidskii [1–3] in 1961, and they have since constituted an area of continuing research. In the past few decades, issues concerning stability, optimal control, filtering, and robustness have been examined in the literature. However, most of the pub- lished results are only applicable to linear Markovian-jump sys- tems. The reader is referred to Mariton [4], Kushner [5], Luo [6], Boukas and Liu [7,8], Farias et al. [9], Souza and Fragoso [10], Kats and Martynyuk [11], Sworder [12], Ghosh et al. [13], Fragoso and Hemerly [14], Ji and Chizeck [15], Natache and Vilma [16], and the references therein. Far less is known about nonlinear Markovian-jump systems, particularly for multi-degree- of-freedom (MDOF) systems. Development of methodology for the analysis of MDOF nonlinear Markovian-jump systems is thus much deserving. The purpose of this paper is to present an approximate method for evaluating the stationary response of MDOF, nonlinear, Markovian-jump, quasi-nonintegrable Hamiltonian systems sub- jected to stochastic excitation. The organization of this paper is as follows. In Sec. 2, the equations of Markovian-jump, quasi- nonintegrable, Hamiltonian systems are examined. Stochastic aver- aging [17–19] is applied to these systems in Sec. 3, which permits the reduction of the Hamiltonian equations to a one-dimensional It^ o equation governing the approximate energy envelope of the original system. The Fokker–Planck–Kolmogorov equation associated with Corresponding author. Manuscript received January 28, 2015; final manuscript received February 28, 2015; published online March 31, 2015. Editor: Yonggang Huang. Journal of Applied Mechanics the It^ o equation of energy envelope is then set up in Sec. 4. By solving the stationary Fokker–Planck– Kolmogorov equation, sta- tionary probabilities for assessing the long-term behavior of the original system are obtained. In Sec. 5, validity and accuracy of the method thus developed are demonstrated by using a two-degree-of- freedom nonlinear oscillator driven by Gaussian white noise, wherein comparison with direct system simulation is made and detailed calculations are provided. A summary of findings is given in Sec. 6. Throughout the paper, an effort is made to clarify the the- oretical development in practical terms. Problem Statement The equations of motion of an n-degree-of-freedom dynamical system are composed of n second-order equations in the general- ized displacements. These second-order equations can always be recast as 2n first-order equations, usually in the state space or in the Hamiltonian phase space. Consider an n-degree-of-freedom, stochastically driven, nonlinear Hamiltonian system with Marko- vian jumps governed by q _ i ¼ p _ i ¼ @H @p i @H @H ec ij ðq; p; sðtÞÞ þ e 1=2 f ik ðq; sðtÞÞW k ðtÞ @q i @p j where i; j ¼ 1; 2; …; n; k ¼ 1; 2; …; m; q i , p i are, respectively, the generalized displacements and momenta; q ¼ ðq 1 ; …; q n Þ and p ¼ ðp 1 ; …; p n Þ. In accordance with the summation convention, the repeated indices j and k in Eq. (2) are summed over their respective ranges. In the above equations, H ¼ Hðq; pÞ is the Hamiltonian, sðtÞ is a continuous-time Markov jump process, e is a small parameter, ec ij ðq; p; sðtÞÞ denote the jump coefficients of quasi-linear damping, and e 1=2 f ik ðq; sðtÞÞ denote the jump amplitudes of excitations. The sto- chastic excitations W k ðtÞ are independent zero-mean Gaussian white C 2015 by ASME Copyright V MAY 2015, Vol. 82 / 051008-1 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 04/07/2015 Terms of Use: http://asme.org/terms

An Orthogonal-Beam Tunnel-Effect MEMS Gyroscope

Tunnel-effect MEMS gyroscopes have broad applications in astronautics because they have high sensitivity, low measurement ranges, and small volumes. This paper describes the design of a novel orthogonal-beam gyroscope based on the principle of tunnel effect. The mathematical model of this class of gyroscopes is set up and the associated performance is obtained with ANSYS simulation software. Related MEMS technology for the construction of these orthogonal-beam tunnel-effect gyroscopes is also described.Copyright