Research Progress of the Algebraic and Geometric Signal Processing

The investigation of novel signal processing tools is one of the hottest research topics in modern signal processing community. Among them, the algebraic and geometric signal processing methods are the most powerful tools for the representation of the classical signal processing method. In this paper, we provide an overview of recent contributions to the algebraic and geometric signal processing. Specifically, the paper focuses on the mathematical structures behind the signal processing by emphasizing the algebraic and geometric structure of signal processing. The two major topics are discussed. First, the classical signal processing concepts are related to the algebraic structures, and the recent results associated with the algebraic signal processing theory are introduced. Second, the recent progress of the geometric signal and information processing representations associated with the geometric structure are discussed. From these discussions, it is concluded that the research on the algebraic and geometric structure of signal processing can help the researchers to understand the signal processing tools deeply, and also help us to find novel signal processing methods in signal processing community. Its practical applications are expected to grow significantly in years to come, given that the algebraic and geometric structure of signal processing offer many advantages over the traditional signal processing.

A Geometric Process Repair Model for the Repairable System Consisting of One Component

This paper considers the optimal replacement problem of a repairable system consisting of one component and a single repairman, assume that the system after repair is not 'as good as new', by using the geometric process, we consider a placement policy T based on the age of the system. The problem is to determine the optimal replacement policy T * such that the long_run expected benefit per unit time is maximized. Also, the explicit expression of the long_run expected benefit per unit time can be found. In some conditions, the existence and uniqueness of the optimal policy T * can be proved, finally, we prove that the policy T * is better than the policy T * in .

Improved Quality Prediction Model for Multistage Machining Process Based on Geometric Constraint Equation

Product variation reduction is critical to improve process efficiency and product quality, especially for multistage machining process（MMP）. However, due to the variation accumulation and propagation, it becomes quite difficult to predict and reduce product variation for MMP. While the method of statistical process control can be used to control product quality, it is used mainly to monitor the process change rather than to analyze the cause of product variation. In this paper, based on a differential description of the contact kinematics of locators and part surfaces, and the geometric constraints equation defined by the locating scheme, an improved analytical variation propagation model for MMP is presented. In which the influence of both locator position and machining error on part quality is considered while, in traditional model, it usually focuses on datum error and fixture error. Coordinate transformation theory is used to reflect the generation and transmission laws of error in the establishment of the model. The concept of deviation matrix is heavily applied to establish an explicit mapping between the geometric deviation of part and the process error sources. In each machining stage, the part deviation is formulized as three separated components corresponding to three different kinds of error sources, which can be further applied to fault identification and design optimization for complicated machining process. An example part for MMP is given out to validate the effectiveness of the methodology. The experiment results show that the model prediction and the actual measurement match well. This paper provides a method to predict part deviation under the influence of fixture error, datum error and machining error, and it enriches the way of quality prediction for MMP.

A generalized geometric process based repairable system model with bivariate policy

The maintenance model of simple repairable system is studied.We assume that there are two types of failure,namely type Ⅰ failure(repairable failure)and type Ⅱ failure(irrepairable failure).As long as the type Ⅰ failure occurs,the system will be repaired immediately,which is failure repair(FR).Between the(n-1)th and the nth FR,the system is supposed to be preventively repaired(PR)as the consecutive working time of the system reaches λ^(n-1) T,where λ and T are specified values.Further,we assume that the system will go on working when the repair is finished and will be replaced at the occurrence of the Nth type Ⅰ failure or the occurrence of the first type Ⅱ failure,whichever occurs first.In practice,the system will degrade with the increasing number of repairs.That is,the consecutive working time of the system forms a decreasing generalized geometric process(GGP)whereas the successive repair time forms an increasing GGP.A simple bivariate policy(T,N)repairable model is introduced based on GGP.The alternative searching method is used to minimize the cost rate function C(N,T),and the optimal(T,N)^(*) is obtained.Finally,numerical cases are applied to demonstrate the reasonability of this model.

Condition-based maintenance policy for gamma deteriorating systems

A condition-based maintenance model for gamma deteriorating system under continuous inspection is studied. This methodology uses a gamma distribution to model the material degradation, and the impact of imperfect maintenance actions on the system reliability is investigated. The state of a degrading system immediately after the imperfect maintenance action is assumed as a random variable and the maintenance time follows a geometric process. Furthermore, the explicit expressions for the long-run average cost and availability per unit time of the system are evaluated, an optimal policy （ε^＊） could be determined numeri- cally or analytically according to the optimization model. At last, a numerical example for a degrading system modeled by a gamma process is presented to demonstrate the use of this policy in practical applications.

Optimal Replacement Policy for Cold Standby System

Investigators are attracted by the complexity and significance of preventive maintenance problem,and there are hundreds of maintenance models and methods to solve the maintenance problems of companies and army,going with a lot of investigative harvests. However,one-component system or series system is focused by most of the literature.The problem of preventive maintenance（PM） on cold standby repairable system does not attach importance despite the fact that the cold standby repairable system is ubiquitous in engineering systems.In this paper,an optimal replacement model for gamma deteriorating system is studied.This methodology presented uses a gamma distribution to model the material degradation,and the impact of imperfect maintenance actions on system reliability is investigated.After an imperfect maintenance action,the state of a degrading system is assumed as a random variable and the maintenance time follows a geometric process.A maintenance policy（N）is applied by which the system will be repaired whenever it experiences the Nth PM,and an optimal policy（N~＊） could be determined numerically or analytically for minimizing the long-run average cost per unit time.A numerical example about how to confirm the optimal maintenance time by the inspecting information of liquid coupling device is given to demonstrate the use of this policy.This paper presents a condition-based replacement policy for cold standby repairable system under continuous monitoring.Its contribution embody in two aspects,relaxing the restrictions of hypothesis and investigating the condition-based maintenance policy of the cold standby repairable system which is ignored by others.

Optimal maintenance decisions for gamma deteriorating systems

An optimal replacement model for gamma deteriorating systems is studied. This methodology uses a gamma distribution to model the material degradation, and the impact of imperfect maintenance actions on the system reliability is investigated. The state of a degrading system immediately after the imperfect maintenance action is assumed as a random variable and the maintenance time follows a geometric process. A maintenance policy （N） is applied by which the system will be repaired whenever it experiences Nth preventive maintenance （PM）, and an optimal policy （N＊） could be determined numerically or analytically for minimizing the long-run average cost per unit time. Finally, a numerical example is presented to demonstrate the use of this policy.

Analysis of Data from a Series of Events by a Geometric Process Model

Geometric process was first introduced by Lam.A stochastic process {X_i,i=1,2,...} iscalled a geometric process (GP) if,for some a>0,{a^(i-1)X_i,i=1,2,...} forms a renewal process.In thispaper,the GP is used to analyze the data from a series of events.A nonparametric method is introduced forthe estimation of the three parameters in the GP.The limiting distributions of the three estimators are studied.Through the analysis of some real data sets,the GP model is compared with other three homogeneous andnonhomogeneous Poisson models.It seems that on average the GP model is the best model among these fourmodels in analyzing the data from a series of events.

Some Limit Theorems in Geometric Processes

Geometric process (GP) was introduced by Lam[4,5], it is defined as a stochastic process {X n , n = 1, 2, · · ·} for which there exists a real number a 】 0, such that {a n−1 X n , n = 1, 2, · · ·} forms a renewal process (RP). In this paper, we study some limit theorems in GP. We first derive the Wald equation for GP and then obtain the limit theorems of the age, residual life and the total life at t for a GP. A general limit theorem for S n with a 】 1 is also studied. Furthermore, we make a comparison between GP and RP, including the comparison of their limit distributions of the age, residual life and the total life at t.

ANALYSIS OF A PARALLEL SYSTEM WITH TWO DIFFERENT UNITS

In this Paper, a parallel repairable model consisting of two units and one repairman isstudied. The working time and the repair time of the two units are all exponeotially distributed.Assume that one unit after repair will be 'as good as new', but the other one not. By introducingthe geometric process and using the method of supplementary variable, some importaDt reliabilityindlces are determined.