Deterministic and nondeterministic symbolic procedures for anisotropic elasticity

The formal symbolic approach is generalized to handle laminated and singly connected anisotropic thermoelasticity problems incorporating the possibility of both deterministic and nondeterministic exciting fields. For example, symbolic operators are developed for the mean as well as higher order statistical moments (variance, covariance, etc.). The generality of the results is such that statistically homogeneous as well as non-homogeneous ensembles of space and time “histories” can be handled.     

Analysis of Repeated Systems with Non- identical,Non-commutative Interactions Using Bond Graph Adapted Operator Perturbation

A technique is presented to study the dynamics of a class of repeated systems with non-commutative interactions. The system equations are formulated using tensor product algebra. A pre-analysis of the system equations is carried out to reduce the interactions to a small perturbing operator. The formal operator perturbation scheme is discussed. The procedure of pretreatment is adapted to bond graph techniques for the ease of representation of dynamic systems. It is shown that the method of operator perturbation may also be handled very conveniently by bond graph approach. The overall system dynamics can be studied from the analysis of the subsystems coupled through the perturbing operator. This results in a substantial reduction in the size of the problem to be considered. The procedure is illustrated by suitable examples.     

Output regulation for a class of positive switched systems

In this paper, a complete procedure for the study of the output regulation problem is established for a class of positive switched systems utilizing a multiple linear copositive Lyapunov functions scheme. The feature of the developed approach is that each subsystem is not required to has a solution to the problem. Moreover, two types of controllers and switching laws are devised. The first one depends on the state together with the external input and the other depends only the error. The conditions ensuring the solvability of the problem for positive switched systems are presented in the form of linear matrix equations plus linear inequalities under some mild constraints. Two examples are finally given to show the performance of the proposed control strategy.     

Modal analysis of turborotors using planar modes—theory

The generalized dynamic equations of motion have been obtained by the direct stiffness method for multimass flexible rotor bearing systems including the effects of gyroscopic moments, disc skew, and rotor acceleration. A set of undamped critical speed mode shapes calculated from the average horizontal and vertical bearing stiffness is used to transform the equations of motion into a set of coupled modal equations of motion. The modal equations are coupled by the generalized bearing coefficients and the gyroscopic moments. An analysis using only undamped critical speeds or decoupled modal analysis assuming proportional damping may lead to erroneous results. This paper presents a rapid method of calculating rotor resonance speeds with their corresponding amplification factors, stability and unbalance response of turborotors. Examples of the application of this modal approach are presented and results are compared to those of other methods such as matrix transfer analysis.     

A maximum principle approach to optimal control for one-dimensional hyperbolic systems with several state variables

The maximum principle developed by Sloss et al. [Optimal control of structural dynamic systems in one space dimension using a maximum principle, J. Vibr. Control 11 (2005) 245–261] is used to determine the optimal control functions for a class of one-dimensional distributed parameter structures. The distributed parameter structures are governed by systems of fourth order hyperbolic equations with constant coefficients. A quadratic performance index is formulated as the cost functional of the problem and can be used to represent the energy of the structure and the force spent in the control process. The developed maximum principle establishes a theoretical foundation for the solution of the optimal control problem and relates the optimal control vector to an adjoint variable vector. The method of solution is outlined which involves reducing the original problem to a system of ordinary differential equations. The solution of the general problem is given and a structural control problem is solved to illustrate the solution procedure. The effectiveness of the proposed control solution is shown by comparing the behavior of controlled and uncontrolled systems.     

The linear Legendre mother wavelets operational matrix of integration and its application

The linear Legendre mother wavelets operational matrix of integration P is derived. A general procedure of forming this matrix P is given. This matrix P can be used to solve problems such as calculus of variations, differential equations, optimal control and integral equations. Illustrative examples are included to demonstrate the validity and applicability of matrix P.     

On the matching equations of kinetic energy shaping in IDA-PBC

Interconnection and damping assignment passivity-based control scheme has been used to stabilize many physical systems such as underactuated mechanical systems through total energy shaping. In this method, some partial differential equations (PDEs) related to kinetic and potential energy shaping shall be solved analytically. Finding a suitable desired inertia matrix as the solution of nonlinear PDEs relevant to kinetic energy shaping is a challenging problem. In this paper, a systematic approach to solving this matching equation for systems with one degree of underactuation is proposed. A special structure for desired inertia matrix is proposed to simplify the solution of the corresponding PDE. It is shown that the proposed method is more general than that of some reported methods in the literature. In order to derive a suitable desired inertia matrix, a necessary condition is also derived. The proposed method is applied to three examples, including pendubot, VTOL aircraft, and 2D SpiderCrane.     

Optimal stabilization for differential systems with delays - Malkin’s approach

The paper considers a process controlled by a system of delayed differential equations. Under certain assumptions, a control function is determined such that the zero solution of the system is asymptotically stable and, for an arbitrary solution, the integral quality criterion with infinite upper limit exists and attains its minimum value in a given sense. To solve this problem, Malkin’s approach to ordinary differential systems is extended to delayed functional differential equations, and Lyapunov’s second method is applied. The results are illustrated by examples, and applied to some classes of delayed linear differential equations.     

Optimal control in unobservable integral Volterra systems

This paper presents solution of the optimal linear-quadratic controller problem for unobservable integral Volterra systems with continuous/discontinuous states under deterministic uncertainties, over continuous/discontinuous observations. Due to the separation principle for integral systems, the initial continuous problem is split into the optimal minmax filtering problem for integral Volterra systems with deterministic uncertainties over continuous/discontinuous observations and the optimal linear-quadratic control (regulator) problem for observable deterministic integral Volterra systems with continuous/discontinuous states. As a result, the system of the optimal controller equations are obtained, including the linear equation for the optimally controlled minmax estimate and two Riccati equations for its ellipsoid matrix (optimal gain matrix of the filter) and the optimal regulator gain matrix. Then, in the discontinuous problems, the equation for the optimal controller and the equations for the optimal filter and regulator gain matrices are obtained using the filtering procedure for deriving the filtering equations over discontinuous observations proceeding from the known filtering equations over continuous ones and the dual results in the optimal control problem for integral systems. The technical example illustrating application of the obtained results is finally given.     

Solution of Sub,Super, Harmonic and Combination Type Resonant Excitations in Nonlinear Lumped Parameter Systems

Through a systematic merging of the concepts of multiple time scales, strained coordinates and the imposition of naturally occurring constraints to bound the solution, a new perturbation theory is developed. The generality of the procedure is such that the effects of sub, super, harmonic and combination type resonant excitations of nonlinear vibratory systems can be simultaneously handled. Furthermore, both conservative and strongly 0(1) nonconservative responses can be treated. This is achieved with no a priori knowledge of the solution form to establish the existence and behavior of the various forms of resonance behavior. To illustrate the new scheme, analyses of several conservative and nonconservative nonlinear systems involving sub, super, harmonic and combination tone type excitations are presented in detail.     

Solution of linear state-space equations and parameter estimation in feedback systems using laguerre polynomial expansion

This paper analyzes the application of Laguerre polynomial expansion to linear systems. It can be applied to the solution of linear state equations by using an algebraic matrix to determine the coefficients of the Laguerre expansion. It also can be applied to system identification by using the expansion to determine the coefficients in the transfer function. Examples are given to demonstrate the accuracy of finite order expansion by Laguerre polynomials.     

Some novel necessary and sufficient conditions of exponential stability for discrete-time systems with multiple delays: A Lyapunov matrix approach

This paper aims at establishing necessary and sufficient conditions of exponential stability for linear discrete-time systems with multiple delays. Firstly, we introduce a new concept—Lyapunov matrix, and investigate its properties, existence and uniqueness by: (i) characterizing the solution of a boundary value problem of matrix difference equations; and (ii) constructing complete type Lyapunov–Krasovskii functionals with pre-specified forward difference. Secondly, a new constructive analysis methodology, named Lyapunov matrix approach, is proposed to establish necessary and sufficient exponential stability conditions for discrete-time systems with multiple delays. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results. It is worth emphasizing that, from a view of computation, the Lyapunov matrix approach proposed here is concerned with three key steps: (i) solve a systems of linear equations; (ii) check whether a constant matrix is of full-column-rank, and (iii) judge whether a constant matrix is positive definite. All of these can be easily realized by using the tool software—MATLAB.     

A numerical solution of a class of periodic coupled matrix equations

This paper studies the numerical solutions of a class of periodic coupled matrix equations. Based on the least square method, a finite iterative algorithm for a class of periodic coupled matrix equations is proposed, and the convergence of the algorithm is proved by theoretical derivation. For any initial value, the algorithm can converge to the solution in finite iterations. Since the equations considered in paper contain many variants, the proposed algorithm has a wide range of applications. Finally some numerical examples in practical systems are given to prove the effectiveness and efficiency of the algorithm.     

线性方程组求解的新方法

利用向量组的线性组合来讨论线性方程组的相容性,给出一种新的解法,即将方程组所确定的矩阵进行初等行变换以后,可以直接写出齐次线性方程组的基础解系和非齐次线性方程组的通解.它比通常所用的消元法简单明了,使用方便,容易掌握.     

Output regulation for matrix second order singular systems via measurement output feedback

This paper is concerned with the output regulation problem of matrix second order singular systems via measurement output feedback. It is shown that the solvability of output regulation problem of matrix second order singular systems is equivalent to the solvability of a set of matrix equations, which are called regulator equations. The solvability condition and solutions of the regulator equations are given, which can be used to solve the output regulation problem. A numerical example is provided to demonstrate the effectiveness of the proposed method.     

Search result diversification on attributed networks via nonnegative matrix factorization

Search result diversification is an effective way to tackle query ambiguity and enhance result novelty. In the context of large information networks, diversifying search result is also critical for further design of applications such as link prediction and citation recommendation. In previous work, this problem has mainly been tackled in a way of implicit query intent. To further enhance the performance on attributed networks, we propose a novel search result diversification approach via nonnegative matrix factorization. Our approach encodes latent query intents as well as nodes as representation vectors by a novel nonnegative matrix factorization model, and the diversity of the results accounts for the query relevance and the novelty w.r.t. these vectors. To learn the representation vectors of nodes, we derive the multiplicative updating rules to train the nonnegative matrix factorization model. We perform a comprehensive evaluation on our approach with various baselines. The results show the effectiveness of our proposed solution, and verify that attributes do help improve diversification performance.     

Physical parameter sensitivity of system eigenvalues and physical model reduction

The identification of subsystems and/or components that is related to a given eigenvalue of the overall system is a challenging and important topic. The use of special structure of the system matrices obtained busing bond graphs can result in identifying subsystems and/or components that affect a given eigenvalue of an overall system. This paper, by making use of a set of theorems and definitions proposes an efficient procedure for this purpose. The basic procedure is based upon the calculation of sensitivity of eigenvalues. The so-called “effect” matrices are produced that indicates the relative importance of physical parameters on a selected eigenvalue. In addition to the relative importance, the effect matrix is used for an efficient physical model reduction procedure. Furthermore, reasons of different dynamic behavior of a system can be explained. Use of effect matrices also improves the physical model reduction method based on decomposition procedures. Three examples are given to illustrate the approach and its consequences.     

Chebyshev polynomial solutions of systems of higher-order linear Fredholm-Volterra integro-differential equations

A Chebyshev collocation method, an expansion method, has been proposed in order to solve the systems of higher-order linear integro-differential equations. This method transforms the IDE system and the given conditions into the matrix equations via Chebyshev collocation points. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Chebyshev coefficients of the solution function. Some numerical results are also given to illustrate the efficiency of the method. Moreover, this method is valid for the systems of differential and integral equations.     

Robust sliding mode control for stochastic singular Markovian jump systems with unmatched one-sided Lipschitz nonlinearities

This paper investigates sliding mode control of stochastic singular Markovian jump systems with nonlinearity. The unmatched nonlinearity satisfies one-sided Lipschitz condition and quadratically inner-boundedness. In term of a new technical variable transformation, sufficient conditions are developed for nonlinear stochastic singular Markovian jump systems constrained on sliding manifold to guarantee stochastic admissibility and uniqueness of solution based on implicit function theorem. The sliding mode control law by which the trajectories of system can be compelled to the predefined sliding surface in finite time no matter what initial state value is, is synthesized. The derivative singular matrix is fully considered in the whole design process such that the derived conditions can be checked easily.The technical treatment of the nonlinear matrix term avoids the classification discussion of sliding mode controller design. Convex optimization problems subject to linear matrix inequalities are formulated to optimize the desired indexes of interest. Finally, the effectiveness of the proposed approach is illustrated by a numerical example and a practical example.     

A procedure to obtain the initial amplitude and phase for the Krylov–Bogoliubov method

Using the Krylov–Bogoliubov method for obtaining analytical solutions to systems with small non-linearities, a procedure is employed to determine the initial amplitude and phase in terms of the initial displacement and velocity. Equations representing the time rate of change of amplitude and phase are used directly. Whether the corresponding linear equations of the non-linear system has purely imaginary, complex conjugate or real roots, the same procedure can be applied.An example is given which demonstrates the initial amplitude and phase change for various higher order approximations.