Towards a general test presentation in the test sequencing problem

A. Z? uz?ek, A. Biasizzo, F. Novak

Joz?ef Stefan Institute

Jamova 39, 1000 Ljubljana, Slovenia

email: [email protected]

Abstract

In this paper we propose an approach to a general test presentation that covers the situations with multiple test outcomes, noiseless tests and unreliable (noisy) tests as the input data for the solutions of the test sequencing problem. Symptomatic information and modular diagnosis are also easily included within the same frame. An example is given to illustrate the proposed approach.

1 Introduction

Most of the earlier papers on sequential diagnosis assumed symmetrical tests with only two possible outcomes: pass or fail. However, diagnostic tests used in the practical field maintenance may have quite different properties. In reality, a test may have an arbitrary number of possible outcomes. Its application may yield a deterministic response dividing the set of candidate faulty system states into two or more distinguishable sets, or may exhibit some degree of unreliability (i.e., for a given failure the outcome of the test is unpredictable). The first alternative, also referred to as noiseless tests has ben recently addressed by Yeung, [9]. On the other hand, the problem of unreliable nature of tests has been mentioned in the work of Pattipati and Alexandridis [6] with some general ideas indicated for possible further work in this direction. In the followingwe use the notion of noisy tests to characterize the tests with unpredictable outcomes.

The goal of this paper is to define in a unique way the variety of test properties that can be met in practice such that all available information provided by the outcomes of the tests can be used when deriving the system diagnosis.

We focus on the following two practical aspects of test properties: (i) the number of possible outcomes of a test and (ii) the determinism of the test outcomes for a given fault situation.

(i) Because of a different nature tests may differ in the number of possible outcomes. For example, Go/NoGo test is a binary test with two possible outcomes (Pass/Fail), while a voltage test can be a ternary test - (nominal/high/low voltage), or even with a number of outcomes

corresponding to the successive intervals of the entire voltage range. Hence, a concept of a multi-outcome test [9] should be considered in order to reflect realistic situations in practice. Furthermore, the number of possible responses for each test may be arbitrary.

(ii) Applicationof a test may result either in a deterministic response for a given fault situation (noiseless tests), or can be unpredictable (noisy tests).

Each response of a multi-outcome test is associated with a list of candidate system failure states. If each failure state corresponds exactly to one list of the candidate states, then test is noiseless, otherwise it is a noisy test.

In the case of a binary test the term of symmetric test is used for the noiseless test and asymmetric test for the noisy test, respectively.

1.1 General test presentation

A general test presentation including situations with multiple test outcomes, noiseless tests and unreliable (noisy) tests is given by the following definition.

Definition: Let a test t have k different outcomes R = fr jg, 1 <= j <= k,which are used to diagnose a system with possible states presented by the set S = fs0; s1; ::; smg. For each test outcome r j we can define the outcome vector r j = [pi(r j)], <= i <= m, <= pi(r j) <= 1, where pi(r j ) is
the probability that the outcome of test t is r j, if the system is in the state si. In the case of noiseless tests probabilities pi(r j ) is equal to or 1.

For any given state si the sum of probabilities over all outcomes of a test t must be equal to 1. Notice that any vector r j corresponding to the outcome r j is determined by all the other outcome vectors. As a consequence, in the presentation of test diagnostic capabilities a vector for one outcome (e.g. Pass response of a binary test) can be omitted.

The paper is organized as follows. In Section 2 a general test sequencing problem is formulated. The proposed approach is illustrated by a simple example in Section 3. Finally, in Section 4 some concluding remarks are drawn.