Failure Detection in Linear Discrete Dynamical Systems and its Detectionability

Failure Detection of Linear Dynamical Systems by a Generalized-likelihood-ratio Method Based on a Detectability Analysis

This paper proposes a fast Fourier transforms (FFT)‐based spectral analysis method for the dynamic analysis of linear discrete dynamic systems which have non‐proportional viscous damping and are subjected to non‐zero initial conditions. To evaluate the proposed FFT‐based spectral analysis method, the forced vibration of a three degree‐of‐freedom (DOF) system is considered as an illustrative problem. The accuracy of the proposed FFT‐based spectral analysis method is evaluated by comparing the forced vibration responses obtained by the present FFT‐based spectral analysis method with those obtained by using the well‐known Runge‐Kutta method and modal analysis method.

In this article, we study discrete nonstationary linear dynamic systems of Volterra type. An essential feature of such kind of systems is that their current states depend on the previous states of this system. The formula Cauchy, which gives us the solution of linear Volterra systems with the control inputs, is obtained. The necessary and sufficient conditions of the pointwise controllability, pointwise output controllability, and observability are proven. Also the linear-quadratic optimization problem for the nonstationary Volterra control systems is studied.

In a presented research paper the problem of synthesis of the fault detection unit and failure occurrence locating in linear discrete dynamic time-invariant systems is considered. The result of synthesis is presented in the form of parallel type structure consisting of two independently functioning Kalman filter. The first of them calculates a system state vector estimation without taking note of faults, and the second - a degenerate type, creates a fault estimations. Linear combination of their exits forms the resulting state vector estimation. Both filters have dimensions smaller dimensions of the tested system and use the split procedure of an error differential signal. Splitting of the error signal is carried out before estimation process unlike Kitanidis filter. It allows to get a certain economy in computing costs, due to introduced restrictions and losses in accuracy. In general the obtained structure is suboptimal. Questions of stability and state vector estimation convergence of a dynamic system are briefly considered. Using of the computing resource of MatLab environment results of a functional methodcheck results are given.The article structure is constructed as follows. At first problem definition is executed and its resolvability from the mathematical point of view is analyzed. The following step is synthesis of the detection unit and localization of multiple faults then the convergence and stability of errors of estimation are analyzed. In final sections results of method operability check are given in the form of illustrative numerical example and the results of the performed research are summed up. In the conceptual plan research makes a generalization the known results for the continuous time systems.

Linear discrete time stochastic dynamical systems with switching parameters may represent a wide class of physical processes. Given the fact that, for non linear systems, the conditional density has no finite parametrization, optimal filters for this class of models are generally infinite dimensional. An exact recursive hybrid filter form has been recently found by Elliott and al [3]. The purpose of this note is to present a new simple approximation scheme of this exact version, which avoids the exponential growth memory problem of the original form. This new numerical scheme is shown to be computationnally efficient. Its implementation is discussed and a comparison of its performances with the Interacting Multiple Models (IMM) algorithm [2] shows its efficiency.

Linear dynamical systems are ubiquitous in hybrid systems, both as physical models or as software control modules. Therefore we need an unbounded-time reachability analysis that can cope with industrial-scale hybrid system models with hundreds of variables. Abstract acceleration is a method developed for the unbounded-time polyhedral reachability analysis of linear software loops that has made promising progress in recent years. The method relies on a relaxation of the solution of the linear recurrence equation, leading to a precise convex over-approximation of the set of reachable states. It has been shown to be competitive with alternative approaches using set-based simulation or constraint solving. This paper explains the basic concepts of the technique, surveys recent advances of the technique towards the application to hybrid discrete and continuous-time linear dynamical systems, and formulates challenges to be tackled.

This paper introduces the notion of absolutely distinguishable discrete dynamic systems, with particular applicability to linear time-invariant (LTI) systems. The motivation for this novel type of distinguishability stems, in particular, from the stability and performance requirements of worst-case adaptive control methodologies. The main results presented herein show that, in most practical cases, a persistence of excitation type of condition and a minimum number of iterations are required to properly distinguish two dynamic systems. We also demonstrate that the former constraint can be written as a lower bound on the intensity of the exogenous disturbances. The applicability of the developed theory is illustrated with a set of examples.

Information-theoretic concepts are utilized to develop a procedure for identifying a parameter of a stochastic linear discrete time dynamic scalar system based on noisy linear measurements of the system's state. After various simplifying approximations, the derived error entropy identification algorithm reduces to an on-line adaptive identification algorithm that is similar in many respects to well-established identification techniques. Conditions under which the developed on-line adaptive algorithm identifies the system with certainty are presented. Using an error entropy estimation lower bound, which is independent of any estimation procedure, conditions for which identification cannot be made with certainty are also presented. Examples involving non-Gaussian statistics are used to illustrate the efficiency of the error entropy adaptive identification algorithm as well as to compare it with several other identification procedures.

Chapter 2 - Structural and performance patterns

This paper presents a game theory approach to the constrained state estimation of linear discrete time dynamic systems. In the application of state estimators, there is often known model or signal information that is either ignored or dealt with heuristically. For example, constraints on the state values (which may be based on physical considerations) are often neglected because they do not easily fit into the structure of the state estimator. This paper develops a method for incorporating state equality constraints into a minimax state estimator. The algorithm is demonstrated on a simple vehicle tracking simulation.

A method is presented for constrained maximum likelihood identification of the (positive semi-definite) initial covariance of an otherwise known linear discrete time dynamical system, with guaranteed positive semi-definite estimate at each step. The technique is a modification of Newton-Raphson or Scoring procedures transformed linearly to the space of the Cholesky square root matrix. The required algorithm is specified completely, and numerical and analytic difficulties and their solutions are discussed. It is shown that in cases of interest this procedure can result in order of magnitude reduction in computational costs compared to other iterative ML schemes which guarantee a semi-definite covariance estimate at each step. Formal extension to the maximum likelihood identification of time constants and power spectral densities is presented.

IDENTIFICATION AND MINIMAL REALIZATION OF MULTIVARIABLE SYSTEMS

A new approach to solving discrete Lyapunov matrix algebraic equations is based on methods for spectral decomposition of their solutions. Assuming that all eigenvalues of the matrices on the left-hand side of the equation lie inside the unit disk, it is shown that the matrix of the solution to the equation can be calculated as a finite sum of matrix bilinear quadratic forms made up by products of Faddeev matrices obtained by decomposing the resolvents of the matrices of the Lyapunov equation. For a linear autonomous stochastic discrete dynamic system, analytical expressions are obtained for the decomposition of the asymptotic variance matrix of system’s states.

Traffic Signal Control Algorithms of a Traffic Network

Supply chain stability is a main feature of effective operation. In order to analyze the stability problem of supply chain system under the shock load, Bessel function has been used to simulate the stress wave propagation in supply chain system under the external shock load. The momentum properties of system's wave function are given by the Fourier transform, and it has been shown by superposition of supply chain inherent wave and shock load transfer wave. The supply chain stability conditions have been given by linear continuous system stable condition and discrete dynamic system stable condition. The stability critical state of supply chain system had been proposed by effective management span value through system's simulation analysis.