Bayesian inference of linear time-varying systems based on Hilbert-Huang transform

This paper proposes an identification method for general linear time-varying (LTV) MDOF systems (including chainlike and non-chainlike systems) based on the Hilbert-Huang Transform (HHT). First, by using Bayesian Inference and a Transitional Markov Chain Monte Carlo (TMCMC) algorithm [1], initial knowledge about the system responses and the white noise in system responses is updated based on measured system responses, which yields the posterior distributions of the noise parameters. Second, each sample system responses are obtained from the posterior distribution and are processed by HHT in order to obtain intrinsic mode functions (IMFs) and the residue as well as the corresponding analytical IMFs and the analytical residue for system responses of each DOF. Finally, the above analytical signals for each DOF are summed respectively to form new analytical responses for each set of sample system responses, which are then used in the identification equations [2] to identify the distributions of system parameters. The proposed method is applied to chainlike and non-chainlike LTV systems with three types of stiffness variations: smooth, abrupt and periodical variations. The effectiveness and accuracy of the proposed method on 1DOF and 2DOF systems is discussed in numerical simulations. System responses are perturbed by white noise, and the identified results demonstrate the robustness of the method. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new method of dynamic analysis for linear and nonlinear systems

A new method of dynamic analysis is proposed which involves the superposition of vectors which are not exact eigenvectors. The efficiency and accuracy of the approach has been demonstrated for linear systems. The purpose of this paper is to summarize the method and to indicate that the technique can be extended to multilevel substructures and to structural systems with a limited number of nonlinear members.     

Stabilization algorithm for linear time-varying systems

We consider the state feedback stabilization problem for a linear time-varying system. Attention is mainly paid to the reduction of the system to canonical form; to this end, we suggest an algorithm for constructing the transformation matrix. This algorithm is based on the solution of a hybrid system and, in contrast to the classical approach, does not require the multiple differentiability of the system parameters.     

On stability of linear and weakly nonlinear switched systems with time delay

This paper studies time-delayed switched systems that include both stable and unstable modes. By using multiple Lyapunov-functions technique and a dwell-time approach, several criteria on exponential stability for both linear and nonlinear systems are established. It is shown that by suitably controlling the switching between the stable and unstable modes, exponential stabilization of the switched system can be achieved. Some examples and numerical simulations are provided to illustrate our results.     

Relations between Bohl exponents and general exponent of discrete linear time-varying systems

In the paper, properties of the upper Bohl exponents and senior upper general exponent of discrete linear time-varying systems are investigated. The relation of these exponents to uniform exponential stability is discussed. Moreover, an example of system, which is not uniformly exponentially stable but each trajectory tends uniformly and exponentially to zero is provided.     

Polynomial matrix-fraction representations for linear time-varying systems

This paper deals with the existence and associated realization theory of skew polynomial fraction representations for linear time-varying discrete-time systems. It is shown that the time-varying analog of the polynomial model always has a free state-space. Necessary and sufficient conditions for the existence of Bezout factorizations are obtained in terms system theoretic properties of state-space realizatios.     

Piecewise homotopy perturbation method for solving linear and nonlinear weakly singular VIE of second kind

The purpose of this paper is to obtain the approximation solution of linear and strong nonlinear weakly singular Volterra integral equation of the second kind, especially for such a situation that the equation is of nonsmooth solution and the situation that the problem is a strong nonlinear problem. For this purpose, we firstly make a transform to the equation such that the solution of the new equation is as smooth as we like. Through modifying homotopy perturbation method, an algorithm is successfully established to solve the linear and nonlinear weakly singular Volterra integral equation of the second kind. And the convergence of the algorithm is proved strictly. Comparisons are made between our method and other methods, and the results reveal that the modified homotopy perturbation is effective.     

Gradient based iterative parameter identification for Wiener nonlinear systems

This paper focuses on the identification problem of Wiener nonlinear output error systems. The application of the key-term decomposition technique provides a special form of the Wiener model with polynomials, where all the model parameters to be estimated are separated. To solve the identification problem of Wiener nonlinear output error systems with the unmeasurable variables in the information vector, an auxiliary model-based gradient iterative algorithm is presented by replacing the unmeasurable variables with their corresponding iterative estimates. The performances of the proposed algorithm are analyzed and compared by using numerical examples.     

The Newton-Kleinman method for the optimal control of stable weakly regular linear systems

We give an algorithm to find the approximate solution of the linear-quadratic optimal control problem for stable weakly regular linear systems. This algorithm can be understood as a generalization of the Newton-Kleinman method known from the finite-dimensional theory. The central characteristic of our approach is the possibility to solve problems with unbounded control and observation operators, which is motivated by partial differential equations with boundary control and observation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On HSS-based iteration methods for weakly nonlinear systems

Based on separable property of the linear and the nonlinear terms and on the Hermitian and skew-Hermitian splitting of the coefficient matrix, we present the Picard-HSS and the nonlinear HSS-like iteration methods for solving a class of large scale systems of weakly nonlinear equations. The advantage of these methods over the Newton and the Newton-HSS iteration methods is that they do not require explicit construction and accurate computation of the Jacobian matrix, and only need to solve linear sub-systems of constant coefficient matrices. Hence, computational workloads and computer memory may be saved in actual implementations. Under suitable conditions, we establish local convergence theorems for both Picard-HSS and nonlinear HSS-like iteration methods. Numerical implementations show that both Picard-HSS and nonlinear HSS-like iteration methods are feasible, effective, and robust nonlinear solvers for this class of large scale systems of weakly nonlinear equations.     

Stability,stabilization and duality for linear time-varying systems

In this article, we study stability properties of linear continuous time-varying systems. Based on a time-varying version of the Lyapunov stability theorem, we obtain stabilizability, stability and duality properties of associated systems.     

Non-integrability for general nonlinear systems

In this paper, we give some simple criteria for the non-existence of analytic integrals of general nonlinear systems.     

Analytical approximation of weakly nonlinear continuous systems using renormalization group method

A direct method based on renormalization group method (RGM) is proposed for determining the analytical approximation of weakly nonlinear continuous systems. To demonstrate the application of the method, we use it to analyze some examples. First, we analyze the vibration of a beam resting on a nonlinear elastic foundation with distributed quadratic and cubic nonlinearities in the cases of primary and subharmonic resonances of the nth mode. We apply the RGM to the discretized governing equation and also directly to the governing partial differential equations (PDE). The results are in full agreement with those previously obtained with multiple scales method. Second, we obtain higher order approximation for free vibrations of a beam resting on a nonlinear elastic foundation with distributed cubic nonlinearities. The method is applied to the discretized governing equation as well as directly to the governing PDE. The proposed method is capable of producing directly higher order approximation of weakly nonlinear continuous systems. It is shown that the higher order approximation of discretization and direct methods are not in general equal. Finally, we analyze the previous problem in the case that the governing differential equation expressed in complex-variable form. The results of second order form and complex-variable form are not in agreement. We observe that in use of RGM in higher order approximation of continuous systems, the equations must not be treated in second order form.     

A framework for analyzing nonlinear eigenproblems and parametrized linear systems

Associated with an n×n matrix polynomial of degree , are the eigenvalue problem P(λ)x=0 and the linear system problem P(ω)x=b, where in the latter case x is to be computed for many values of the parameter ω. Both problems can be solved by conversion to an equivalent problem L(λ)z=0 or L(ω)z=c that is linear in the parameter λ or ω. This linearization process has received much attention in recent years for the eigenvalue problem, but it is less well understood for the linear system problem. We develop a framework in which more general versions of both problems can be analyzed, based on one-sided factorizations connecting a general nonlinear matrix function N(λ) to a simpler function M(λ), typically a polynomial of degree 1 or 2. Our analysis relates the solutions of the original and lower degree problems and in the linear system case indicates how to choose the right-hand side c and recover the solution x from z. For the eigenvalue problem this framework includes many special cases studied in the literature, including the vector spaces of pencils L₁(P) and L₂(P) recently introduced by Mackey, Mackey, Mehl, and Mehrmann and a class of rational problems. We use the framework to investigate the conditioning and stability of the parametrized linear system P(ω)x=b and thereby study the effect of scaling, both of the original polynomial and of the pencil L. Our results identify situations in which scaling can potentially greatly improve the conditioning and stability and our numerical results show that dramatic improvements can be achieved in practice.     

Asymptotic stability of switched linear time-varying systems based on top-floor function

This paper considers the problem of asymptotic stability for switched linear time-varying (SLTV) systems. First, some stability conditions for SLTV systems are given by using infinite integrals. Then, based on the results obtained, two stability conditions are proposed by combining the methods of top-floor function and average dwell time. Moreover, using strict top-floor function, a stability condition is also provided when some subsystems are unstable. With the help of top-floor function, the stability problem of SLTV systems can be simplified and solved by using the technique of linear matrix inequalities (LMIs). Finally, a numerical example is given to illustrate the theoretical results.