National Aviation University

Institute of Aircraft Control Systems

Systems Engineering

Graphical – Calculation work

Subject: “Diagnostics and Complex Systems Checkup”

Created by:

Student of group 305

Kaliush Igor

Checked by:

Kozhohina E. V.

Kyiv 2011

Variant 7

The purpose of the graphical-calculation work is to work out the procedure of the construction of the conditional algorithm of the determination of upstate and finding of faults.

The general order of the construction of conditional algorithm.

Thus, for construction of the conditional algorithm it is necessary to perform the following operations:

1. to build the state matrix on FLM;

2. to calculate the probability of the elements faults on FLM;

3. to build the transposed matrix from the initial state matrix, in it S, - columns,

and y_t - raws;

4. to calculate according to the state matrix function W and W for each y_};

5. to select the first checking for which function W has minimum value;

6. to build two parts of the state matrix, into one of which the zero state of the first checking will enter, and another - single.

After the partition of initial matrix the value of checking y_} is selected from the condition of the states selecting. Thus, checking y_k is selected for the state s_kt checking y_s - for the state S, and so on.

The selected checking, which correspond to the minimum of function W or W\ at further partition of matrices do not participate;

7. to calculate for each raw of the obtained matrices of the value of function W or W^,­­ and according to their minimum values to appoint sequential checking; the operation of the partition of matrix to carry out until initial matrix would not be broken into the separate states;

8. to build the search scheme for faults (search graph).

The objective of this work is to construct a conditional algorithm for diagnosis of functional and logic model, which consists of elements that have equal probability of failure.

1. Construct the flm object and filling up of state matrices.

Fig.1. Functional-logical model of the object of diagnostics.

2. Construction of conditional algorithm of faults search.

The transposed matrix of states, obtained from the initial matrix by the replacement of the places of states S_t and checking y_t, is taken as the initial data.

If it is necessary to solve the task of determining of upstate and search of faults, then in the states matrix state S_o is included. If it is necessary to carry out only a search for faults, then states S_o is excluded from the matrix.

1. Case of the equal probabilities of the faults of elements.

If system is assigned by FLM with the equal probabilities of faults Q(S,), then the construction of conditional algorithm begins from the selection of the first checking. Depending on its outcome the set of possible states is divided into two subsets. In one of them will enter all states, for which according to the states matrix given checking has the positive outcome y_t = 1. The second subset will contain all states, for which outcome of given checking is negative y_t = 0.

Then the checking are selected, which divide the received subsets again into two subsets and so forth. The selection of checking continues until entire set of states will not be dividing into the separate states.

The first checking is selected from the condition of the minimum of function W, which is determined on the basis of one of the positions of information theory, which is consisted in the fact that checking, which divides all states of system into two equal parts, has maximum information about the system, i.e.

where "0" - is a sum of zeroes in raw y_j; ]T"V'- is a sum of ones in the same raw.

In this case it is necessary for each y_j to find the sum of the zero and ones state and to deduct from the absolute value of the sum of zero the sum of the one state

The minimum of function W determines testing of output y₃, which divides all states of the diagnosis object into two equal parts.

Checking y_j divides the states matrix into two parts. For these parts function W is calculated analogously and the checking y_j2 and y_j3 are determined in terms of its minimum value, each of which will divide the parts of the matrix on two.

The procedures of the determination of totality and priority of checking are continued until the set of the object states will not be divide into the separate states. Then the scheme of faults search (graph of searches) constructed and the value of the expected expenditures is determined.

The order of the construction of the conditional algorithm of the upstate determination and search for faults for the object of the diagnosis, which has the equal probabilities of the elements faults (table 2), it is shown on fig. 2.

	Y1	Y2	Y3	Y4	Y5	Y6	Y7	Y8	Y9
S0	1	1	1	1	1	1	1	1	1
S1	0	0	0	0	0	0	1	0	0
S2	1	0	0	1	1	1	1	1	1
S3	1	1	0	1	1	1	1	1	1
S4	1	1	1	0	0	0	1	0	0
S5	1	1	1	1	0	0	1	0	0
S6	1	1	1	1	1	0	1	1	1
S7	1	1	1	1	1	1	0	0	0
S8	1	1	1	1	1	1	1	0	0
S9	1	1	1	1	1	1	1	1	0

Transpose this matrix by replacing the states and places of significance tests.

	S0	S1	S2	S3	S4	S5	S6	S7	S8	S9
Y1	1	0	1	1	1	1	1	1	1	1
Y2	1	0	0	1	1	1	1	1	1	1
Y3	1	0	0	0	1	1	1	1	1	1
Y4	1	0	1	1	0	1	1	1	1	1
Y5	1	0	1	1	0	0	1	1	1	1
Y6	1	0	1	1	0	0	0	1	1	1
Y7	1	1	1	1	1	1	1	0	1	1
Y8	1	0	1	1	0	0	1	0	0	1
Y9	1	0	1	1	0	0	1	0	0	0

The first checking is selected from the condition of the minimum of function W, which is determined on the basis of one of the positions of information theory, which is consisted in the fact that checking, which divides all states of system into two equal parts, has maximum information about the system, i.e.

where "0" - is a sum of zeroes in raw y_j; ]T"V'- is a sum of ones in the same raw.

In this case it is necessary for each y_j to find the sum of the zero and ones state and to deduct from the absolute value of the sum of zero the sum of the one state

The minimum of function W determines testing of output y₃, which divides all states of the diagnosis object into two equal parts.

Checking y_j divides the states matrix into two parts. For these parts function W is calculated analogously and the checking y_j2 and y_j3 are determined in terms of its minimum value, each of which will divide the parts of the matrix on two.

The procedures of the determination of totality and priority of checking are continued until the set of the object states will not be divide into the separate states. Then the scheme of faults search (graph of searches) constructed and the value of the expected expenditures is determined.

The order of the construction of the conditional algorithm of the upstate determination and search for faults for the object of the diagnosis, which has the equal probabilities of the elements faults, see figure 2:

	S0	S1	S2	S3	S4	S5	S6	S7	S8	S9	W
Y1	1	0	1	1	1	1	1	1	1	1	8
Y2	1	0	0	1	1	1	1	1	1	1	6
Y3	1	0	0	0	1	1	1	1	1	1	4
Y4	1	0	1	1	0	1	1	1	1	1	6
Y5	1	0	1	1	0	0	1	1	1	1	4
Y6	1	0	1	1	0	0	0	1	1	1	2
0 Y7	1	1	1	1	1	1	1	0	1	1	1 8
Y8	1	0	1	1	0	0	1	0	0	1	0
Y9	1	0	1	1	0	0	1	0	0	0	2

	S1	S4	S5	S7	S8	W
Y1	0	1	1	1	1	3
Y4	0	0	1	1	1	1
Y5	0	0	0	1	1	1
Y7	1	1	1	0	1	2

	S0	S2	S3	S6	S9	W
Y2	1	0	0	0	1	1
Y3	1	1	0	0	0	1
Y6	1	1	0	1	1	3
Y9	1	1	0	0	1	1

­­

	S0	S6	S9	W
Y6	1	0	1	1
Y9	1	1	0	1

	S1	S4	S5	W
Y1	0	1	1	1
Y4	0	0	1	1

­­	S7	S8	W
Y7	0	1	0

S6

S8

S7

	S0	S9	W
Y9	1	0	0

	S1	S4	W
Y1	0	1	0

	S2	S3	W
Y2	0	1	0

S5

S0

S3

S9

S4

S1

S2

Fig. 2. Construction of conditional algorithm for case of equal probabilities of faults of functional elements.

It is necessary to keep in mind that the function W for checking of different outputs can have identical minimum values. Therefore checking of outputs can be equivalent, and diagnostics can be begun from any of them. One of the diagrams (graph) of the functional test and search for faults is shown in Fig. 3:

S0

Fig. 3 The scheme of the servisability checking and fault checking

Conclusion: Due to this work, I got skills for construction of the conditional algorithm of diagnosing, with help of theoretical knowledge, which I got on lectures for my practical lessons. For diagnosing by this method I use knowledge of construction FLM of the system and conditional matrix. For the construction of the diagnosing tree I used knowledge getting in the information theory, for determination advantage function W. All schemes and geometrical construction I performed in this calculated – graphical work.