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This paper presents a novel technique for improved voting by adaptively varying the membership boundaries of a fuzzy voter to achieve realistic consensus among inputs of redundant modules of a fault tolerant system. We demonstrate that suggested dynamic membership partitioning minimizes the number of occurrences of incorrect outputs of a voter as compared to the fixed membership partitioning voter implementations. Simulation results for the proposed voter for Triple Modular Redundancy (TMR) fault tolerant system indicate that our algorithm shows better safety and availability performance as compared to the existing one. However, our voter design is general and thus it can be potentially useful for improving safety and availability of critical fault tolerant systems.

The fault tolerant systems frequently use hardware redundancy along with a voter to achieve enhanced operational availability of critical mission oriented systems against predefined set of faults. One of the most commonly used methods is based on static redundancy technique incorporating Triple-Modular Redundancy (TMR) [

This paper presents an improved design of fuzzy voter based on adaptive fuzzy membership boundaries of the mod values of the differences between analog outputs of the redundant modules and its range. The availability and safety performance of the proposed voter design is evaluated through MATLAB simulation studies and it is shown that our voter design is potentially useful for handling larger mod differences (>1.5) between the outputs of redundant analog channels. It offers 20% error reduction in the voted output as compared to the fuzzy voter suggested by Shabgahi [

Fuzzy voters [

The fuzzification is achieved by transforming the numerical differences

Referring

Symmetry:

Equations (1)-(3) describe the functions for small, medium, and large membership grades respectively.

For each of the input

Where

& | ||||
---|---|---|---|---|

Small | Medium | Large | ||

Small | Vhigh | High | Medium | |

Medium | High | Low | Vlow | |

Large | Medium | Vlow | Vlow |

The fuzzified value of agreeability is converted in to numerical value of weight

The existing fuzzy voter considered fixed fuzzy partitioning parameter i.e. the values of parameter p, q and r are fixed. While the values of parameters p, q and r will not be optimum for all values of input ranges for example for very small magnitude of inputs, the value of p, q and r should be less as the accuracy tolerance between generated output and correct output will be less and for larger magnitude of input, the values of p, q and r could be high. so a modified fuzzy voting unit has been proposed in this paper where the value of output from voting unit is calculated with the same method as in the above fuzzy voting unit but the values of fuzzy partitioning will not be fixed, they will change themselves according to the input values and the numerical values of differences between them. The value of p is chosen considering the maximum and minimum of the input values and the minimum of the distance between input pairs.

The value of parameter p is calculated on the basis of fuzzy classification and values of q and r will be multiple of p, the fuzzy classification of p is achieved with three parameters as a, b, and c and their functions have been defined for module output ranging from {0 - 25}, while the partitioning of these parameters can be changed according to the application and the ranges of inputs.

The fuzzy membership a (

The fuzzy membership b (

The fuzzy membership c (

The fuzzy membership function for output variable p (

The fuzzy membership for output p is:

The value of p is categorized in four memberships values {small, medium, high, vhigh} which are governed by the fuzzy rules formulated as under.

1) if (c is small) and (b is small) then (p is small);

2) if (c is small) and (b is medium) then (p is medium);

3) if (c is small) and (b is large) then (p is medium);

4) if (a is small) and (c is medium) then (p is medium);

5) if (a is medium) and (b is small) and (c is medium) then (p is high);

6) if (a is medium) and (b is medium) and (c is medium) then (p is medium);

7) if (a is medium) and (b is large) and (c is medium) then (p is medium);

8) if (a is large) and (b is small) and (c is medium) then (p is high);

9) if (a is large) and (b is medium) and (c is medium) then (p is medium);

10) if (a is large) and (b is large) and (c is medium) then (p is high);

11) if (a is small) and (c is large) then (p is medium);

12) if (a is medium) and(b is small) and (c is large) then (p is medium);

13) if (a is medium) and(b is medium) and (c is large) then (p is medium);

14) if (a is medium) and (b is large) and (c is large) then (p is high);

15) if (a is large) and (c is large) then (p is vhigh).

Rule No. 1 to 3 deals with cases where difference between at least two inputs are small it implies that there is a reasonably high degree of agreeability therefore p is categorized as either small or medium depending upon maximum value of inputs. Rules 4 to 10 define conditions when minimum of differences in inputs is in medium range, here the value of p will be either medium or high depending on the distance between minimum and maximum of the inputs. Similarly rules 11 to 15 have been formulated to deal with poor agreeability between set of inputs i.e. when minimum of differences in inputs is in large range, here the value of p will be either medium, high or vhigh depending on the distance between minimum and maximum of the inputs.

Based on the above parameters and the set of fuzzy rules the value of p will be in range {0.2 - 0.9} and the values of q and r are arrived at as multiples of p i.e.

The defuzzification method used to calculate value of p is centroid method.

The output y will be calculated in same way as proposed in previous section but now the values of fuzzy partitioning parameters will not be fixed, they will adapt themselves according to inputs values and their differences.

The first experiment proves that the voted output generated by improved fuzzy voter is more closed to the actual output as compared to fuzzy voter proposed in [

To compare the performance of improved fuzzy voter with reference fuzzy voter, the parameters used for simulation experiments are listed below:

The input to modules: sinusoidal function

Case | Voter input | Improved fuzzy voter output (y) | Reference fuzzy voter output (y') | ||
---|---|---|---|---|---|

1 | [1.1 1.2 1.8] | 1.5 | 0.15 | 1.32 | 0.32 |

2 | [1.05 1.8 2.8] | 1.43 | 0.43 | 1.88 | 0.88 |

3 | [0.8 1.2 2.0] | 1.03 | 0.03 | 1.10 | 0.10 |

4 | [0.75 1.6 2.9] | 1.17 | 0.17 | 1.71 | 0.71 |

5 | [1.0 1.2 1.6] | 1.12 | 0.12 | 1.26 | 0.26 |

6 | [1.0 1.2 1.3] | 1.17 | 0.17 | 1.17 | 0.17 |

7 | [0.78 1.3 1.9] | 1.33 | 0.33 | 1.30 | 0.30 |

8 | [0.9 1.2 2.5] | 1.05 | 0.05 | 1.15 | 0.15 |

9 | [1.0 1.5 3.5] | 1.25 | 0.25 | 1.42 | 0.42 |

10 | [0.6 0.95 1.6] | 1.03 | 0.03 | 0.97 | 0.03 |

The errors has been injected in any two modules using a random generator with uniform distribution with amplitude from the interval {−emax +emax}.

Here the value of correct input will range from (0 - 20), the accuracy threshold value (ATV) will be different for different magnitudes of inputs, ATV is the max error allowed from the true output value and it is defined as:

if

else if

else if

where

Reference fuzzy voter has been designed by taking following values:

The improved fuzzy voter uses same values for u, v, and w i.e.

The voter output can be interpreted as correct, incorrect, or benign output. For each voter, the results of 104 voting cycles are performed. And

The availability A will be defined as:

And safety S will be defined as:

Simulation Results of Experiment 2:

A fuzzy voter proposed by Shabgahi has been studied for its performance in various conditions. The fuzzy voter

falls short of expectations with respect to safety and availability parameters for larger errors, so here we proposed an improvement to the fuzzy voter. The proposed modification considers dynamic partitioning parameter variation in a given range which can be chosen depending on the system requirements. The study shows that the

proposed improved fuzzy voter yields better results as compared to the existing voter. This scheme can be adapted to different operational conditions of the system by varying the accuracy requirements and fuzzy partitioning parameters. The future work envisaged in this area is to integrate various other techniques like history based module selection, TMR with spare etc. into the fuzzy voters to further improve the availability and reliability of the systems.