Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 676289, 10 pages

http://dx.doi.org/10.1155/2015/676289

## A Novel Strong Tracking Fault Prognosis Algorithm

Unit 302, Xi’an Research Institute of High-Tech, Xi’an 710025, China

Received 27 November 2014; Revised 21 December 2014; Accepted 21 December 2014

Academic Editor: Gang Li

Copyright © 2015 Qi Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Improving the ability to track abruptly changing states and resolving the degeneracy are two difficult problems to particle filter applied to fault prognosis. In this paper, a novel strong tracking fault prognosis algorithm is proposed to settle the above problems. In the proposed algorithm, the artificial immunity algorithm is first introduced to resolve the degeneracy problem, and then the strong tracking filter is introduced to enhance the ability to track abruptly changing states. The particles are updated by strong tracking filter, and better particles are selected by utilizing the artificial immune algorithm to estimate states. As a result, the degeneracy problem is resolved and the accuracy of the proposed fault prognosis algorithm is improved accordingly. The feasibility and validity of the proposed algorithm are demonstrated by the simulation results of the standard validation model and the DTS200 system.

#### 1. Introduction

Particle filter (PF) is a leading and powerful algorithm for estimating the states of nonlinear or non-Gaussian systems. The past decades have witnessed a wide range of applications, including target tracking [1–4], data detection [5], modeling [6, 7], price forecasting, and fault detection [8–11]. On the other hand, a great number of investigators are interested in enriching particle filtering framework, and many new particle filters are proposed in recent years [12–15]. In these studies, it is found that resolving the degeneracy problem and improving the ability to track abruptly changing states are two difficult problems to particle filter applied to fault prognosis [16–20]. The degeneracy problem means that most particles are assigned to zero weights. As a result, the performance of the particle filter deteriorates because most computational resource is wasted. It is noted that, however, degeneracy can be reduced by resampling or choosing good importance sampling functions. Along this line of research, many resampling algorithms have been proposed for reducing the degeneracy. In these resampling algorithms, sequential importance resampling (SIR) is the representation which largely copies the particles with larger weights to replace the particles with smaller weights [21, 22]. The degeneracy problem is partially addressed, but the sample impoverishment is not fully concerned. Sample impoverishment means that most particles are the same in the set of particles since the particles with larger weights are largely copied. In this circumstance, choosing good importance sampling functions deserves further studies, and many investigators are interested in this question. For example, extended Kalman filter (EKF) was introduced to propose extended particle filter (EPF) by De Freitas et al. [6], and unscented Kalman filter (UKF) was introduced to propose unscented particle filter (UPF) by van der Merwe et al. [23]. Both EPF and UPF can resolve the degeneracy problem, but they cannot track abruptly changing states due to the disadvantages of EKF and UKF.

Strong tracking filter (STF) has good performance for tracking abruptly changing states, and thus it can be used to update particles. In addition, it is known that artificial immunity (AI) can search for the best one from all the range, and thus it can be used to clone and vary particles. Therefore, in this paper STF and AI algorithms are utilized jointly to improve particle filter algorithm. As a result, a novel fault prognosis algorithm based on strong tracking artificial immunity particle filter (STAIPF) is proposed to settle the above discussed problems. In the proposed algorithm, the artificial immunity algorithm is first introduced to resolve the degeneracy problem, and then the strong tracking filter is introduced to enhance the ability to track abruptly changing states. More specifically, the particles are updated by strong tracking filter, and better particles for states estimation are selected by utilizing the artificial immune algorithm to enhance the diversity of samples. Therefore, the degeneracy problem and sample impoverishment are resolved simultaneously, and the accuracy of the proposed fault prognosis algorithm is improved as well. Finally, the feasibility and validity of the proposed algorithm are demonstrated by the simulation results of the standard validation model and the DTS200 system.

The remainder of this paper is structured as follows. In Section 2, the particle filter is introduced. Section 3 provides the artificial immune algorithm. In Section 4, we present a strong tracking filter. A novel strong tracking fault prognosis algorithm is proposed in Section 5. Section 6 provides simulation results. This paper is concluded in Section 7.

#### 2. Particle Filter

Actually, particle filter is a sequential Monte Carlo methodology. Its primary principle is to recursively compute relevant probability distributions by importance sampling and to approximate the probability distributions with discrete random variables. Detailed information of particle filter can be found in [21].

In general, the following state-space and observation equations are considered:where the subscript denotes time index, is an observations vector, is a state vector, is a system noise vector, is an observation noise vector, is a measurement function, and is a system transition function. The first equation is known as the state equation while the second one is known as the measurement equation. It is usually assumed that the analytical forms of the two functions and the distributions of the two noises in (1) are known. Then, the object is to recursively estimate based on the observations and the above assumptions.

Due to the general nature of model (1) and the impact of non-Gaussian noises, it is difficult to solve the above filtering problem in an analytical manner. In this case, particle filter is an effective alternative. In particle filter, the number of effective particles is commonly denoted by , which is used to weigh the degeneracy degree of the particles. In other words, the smaller implies the worse degeneracy degree. Here, is defined aswhere denotes the normalized weighting coefficient of particle at time , and denotes rounding to the nearest integer.

Based on the above descriptions, the SIR algorithm can be summarized as follows [21], which is one of the most popular particle filtering algorithms.

*A Procedure Description of SIR*

*Step 1 (initialization). *Draw particles according to initial importance function, and is assumed to denote the th particle, and then set .

*Step 2. *Update.where denotes the particle at time , which is updated according to formula (3).

*Step 3. *Assign the weighting coefficient of according to

*Step 4. *Normalize weighting coefficient by

*Step 5. *If , then run system resampling and assign the same weighting coefficient to all particles:End.

*Step 6. *Estimate the state at time according to

*Step 7. *Return to Step 2.

#### 3. Artificial Immune Algorithm

Artificial immune algorithm (AIA), heredity algorithm, and evolutionary algorithm are all bionic algorithms which simulate the behaviors of natural organisms. Artificial immune algorithm is characterized by its diversity, self-regulation, clone, and mutation, which is a global searching algorithm based on natural immune systems [24, 25].

Antigen and antibody correspond to target function and possible value of the associated optimization problem, respectively. Generally, affinity has two forms. One is the appetency which denotes the matching degree between antigen and antibody. The other is the repellency which denotes the similar degree of antigen and antibody, and the repellency ensures the diversity of the antibodies. Immune algorithm is based on the memory cell and clone number calculated according to affinity, which can converge to the optimal solution ensured by the above steps. The basic procedure of immune algorithm is summarized as Figure 1.