Suggested Error Evaluation

The usual dividing of the integration period into two equal intervals, during adaptive integrations, when the quadrature local error exceeds its limit (Gonnet 2009), means that performing two integrations one of them can be unnecessary. Searching for the appropriate interval length successively to assure that integration local error is preserved can be another possible approach. In the next sections, this is going to analysed and evaluated for fuzzy type reduction purposes. The trapezoidal rule error proportional nature, where integration error is zero for straight segments and high for highly curved segments, facilitates using the successive search by evaluating the local integration error on a variable integration period. It is required to find the maximum interval length, which still has an integration error within the required limit. In the successive search technique (McNeill et al. 2011); if the required criteria level is not exceeded, then the search step size is doubled, otherwise, it is halved. This algorithm has been used frequently for high speed Analog to Digital Converters (ADC) because of its

76

simplicity and ability to be realized using simple hardware, consuming low power (Saberi and Lotfi 2014).

This is going to be used here to find the correct interval length in the successive adaptive quadrature algorithm while assuming that the fuzzy membership is defined using discrete equally spaced points as: 𝑓(𝑦_𝑛) = {𝜇0, 𝜇1, 𝜇2, ….,𝜇𝑁}, which it is one of the simple and most common definitions that’s being use for digital systems. The goal is to find the maximum possible interval length successively within a convex fuzzy set. This can be done by using the well-known trapezoidal-rule error over a period [𝑎, 𝑏], which is defined by equation ( 4-15), then re-formulating this for a temporary segment section 𝑆𝑠𝑐 by using its first point (𝑦_{0𝑆𝑒𝑐}, 𝜇_{0𝑆𝑒𝑐}) and its last point (𝑦_{𝑛𝑆𝑒𝑐}, 𝜇_{𝑛𝑆𝑒𝑐}). This re-shaping begins from the definition of the slope at the start point of the temporary section 𝑆𝑠𝑐, as follows: 𝑆1𝑆𝑒𝑐 = (𝜇_{1𝑆𝑒𝑐}− 𝜇_{0𝑆𝑒𝑐}) (𝑦1𝑆𝑒𝑐− 𝑦0𝑆𝑒𝑐) =(𝜇1𝑆𝑒𝑐− 𝜇0𝑆𝑒𝑐) ∆𝑦 ( 4-17)

This section can be defined in different lengths as multiplicands of the system discretisation level (𝑛_𝑆𝑒𝑐. 𝑑𝑦). Its slope at the end-point is equal to:

𝑆𝑛𝑆𝑒𝑔 =

(𝜇_{𝑛𝑆𝑒𝑐}− 𝜇_{(𝑛−1)𝑆𝑒𝑐}) (𝑦𝑛𝑆𝑒𝑐 − 𝑦(𝑛−1)𝑆𝑒𝑐)

= (𝜇𝑛𝑆𝑒𝑐− 𝜇(𝑛−1)𝑆𝑒𝑐)

∆𝑦 ( 4-18)

77

Then, as shown in Figure 4-2, from the average second-derivation of the curved function over this section:

𝑓′′_{(æ) =} 𝑆𝑛𝑆𝑒𝑐 − 𝑆1𝑆𝑒𝑐

(𝑛 − 0)_𝑆𝑒𝑐. ∆𝑦 ( 4-19)

Can get the average slope of the straight line that joining the start and end points of the curved section to be;

𝑆_{𝑇𝑆𝑒𝑔} = (𝜇𝑛𝑆𝑒𝑐 − 𝜇0𝑆𝑒𝑐) (𝑦_{𝑛𝑆𝑒𝑐}− 𝑦_{0𝑆𝑒𝑐}) =

(𝜇_{𝑛𝑆𝑒𝑐}− 𝜇_{0𝑆𝑒𝑐})

𝑛_𝑆𝑒𝑐. ∆𝑦 ( 4-20)

This slope, according to the mid-point interpolation theorem (Rao 2007), can be approximated by: 𝑆_{𝑇𝑆𝑒𝑐} ≈𝑆𝑛𝑆𝑒𝑐 + 𝑆1𝑆𝑒𝑐 2 ( 4-21) Getting: 𝑆_{𝑛𝑆𝑒𝑐} ≈ 2𝑆_{𝑇𝑆𝑒𝑐}− 𝑆_{1𝑆𝑒𝑐 ( 4-22)} Substitute in ( 4-19) to get; 𝑓′′(æ) ≈2(𝑆𝑇𝑆𝑒𝑐− 𝑆1𝑆𝑒𝑐) 𝑛_𝑆𝑒𝑐. ∆𝑦 ( 4-23)

Re-write using the start and

end points: _𝑓′′_{(æ) ≈}2[(𝜇𝑛𝑆𝑒𝑐− 𝜇0𝑆𝑒𝑐) − 𝑛(𝜇1𝑆𝑒𝑐 − 𝜇0𝑆𝑒𝑐)]

𝑛2_{. (∆𝑦)}2 ( 4-24) Using equation ( 4-15) to get: _{𝑒𝑇𝑛 =}−𝑛3. (∆𝑦)3

12 𝑓

′′_(æ)

( 4-25) Re-write, using the section terminal points:

𝑒_𝑇𝑛 ≈−𝑛. ∆𝑦 [𝑛(𝜇1𝑆𝑒𝑐− 𝜇0𝐼) − (𝜇𝑛𝑆𝑒𝑐− 𝜇0𝑆𝑒𝑐)

6 ( 4-26)

The local error limit is the unit-length error multiplied by the local interval length. The unit-length error is the global error divided by the equally spaced total integration divisions, as follows:

𝑒_{𝑢𝑛𝑖𝑡𝐿} = 𝑒𝑇𝑟𝑎𝑝𝑒𝑧

78

Thus, error threshold for an interval of length 𝑛. ∆𝑦 should be bounded as:

𝑒𝑇𝑛 ≤ 𝑒𝑢𝑛𝑖𝑡𝐿. 𝑛. ∆𝑦 _{( 4-28)}

Substituting in equation ( 4-26) to get the local integration error constraints as:

|𝑛(𝜇_{1𝑆𝑒𝑐}− 𝜇_{0𝑆𝑒𝑐}) − (𝜇_{𝑛𝑆𝑒𝑐}− 𝜇_{0𝑆𝑒𝑐})| ≤6 . |𝑒𝑇𝑟𝑎𝑝𝑒𝑧|

𝑁. ∆𝑦 ( 4-29)

𝑛(𝜇1𝑆𝑒𝑐− 𝜇0𝑆𝑒𝑐) − (𝜇𝑛𝑆𝑒𝑐− 𝜇0𝑆𝑒𝑐) ≤ |6. 𝑒𝑇𝑝.𝑢.𝑙| ; 𝑓𝑜𝑟 ∆𝑦 = 1 ( 4-30)

This error constraint is correct for functions that have gradual slope change over their domain. However, this is not the case with fuzzy sets that have horizontal cuts, which caused by different fuzzy threshold levels. The error, in this case, is bounded as shown in Figure 4-2 by the dotted area, which has to be considered in order to keep the total error under the required limit. The maximum possible integration error for such shapes with horizontal cuts can be in its maximum value when the point 𝑦_𝑐 falls in the middle distance between 𝑦0 and 𝑦𝑛. The integration error for this case can be evaluated as:

𝑒𝑆𝑒𝑔 = (𝑦𝐶𝑆𝑒𝑐− 𝑦0𝑆𝑒𝑐) (𝜇_{𝐶𝑆𝑒𝑐}− 𝜇_{0𝑆𝑒𝑐}) 2 + (𝑦_{𝑛𝑆𝑒𝑐}− 𝑦_{𝐶𝑆𝑒𝑐})(𝜇𝑛𝑆𝑒𝑐− 𝜇0𝑆𝑒𝑐) + (𝜇𝐶𝑆𝑒𝑐− 𝜇0𝑆𝑒𝑐) 2 −(𝑦𝑛𝑆𝑒𝑐 − 𝑦0𝑆𝑒𝑐) (𝜇_{𝐶𝑆𝑒𝑐} − 𝜇_{0𝑆𝑒𝑐}) 2 ( 4-31)

Re-formulating and using the slope of the start and end points:

𝑆0 = (𝜇_𝑐− 𝜇₀) (𝑦𝑐 − 𝑦0) = (𝜇𝑐− 𝜇0) 0.5(𝑦𝑛− 𝑦0) ( 4-32) 𝑆𝑛 = (𝜇_𝑛− 𝜇_𝑐) (𝑦𝑛− 𝑦𝑐) = (𝜇𝑛− 𝜇𝑐) 0.5(𝑦𝑛− 𝑦0) ( 4-33)

79 𝑆₀

𝑆_𝑛 =

(𝜇_𝑐− 𝜇₀)

(𝜇_𝑛− 𝜇_𝑐) ( 4-34)

To get point 𝜇𝑐 definition as: 𝜇𝑐 =

𝑆_𝑛 . 𝜇₀+ 𝑆₀ . 𝜇_𝑖

𝑆0 + 𝑆𝑛 ( 4-35)

Using equation ( 4-21) to get 𝜇_𝑐 in term of total slope 𝑆_𝑇 as below:

𝜇_𝑐 ≈ 2𝑆𝑇 . 𝜇0− 𝑆0 . 𝜇0+ 𝑆0 . 𝜇𝑛

2𝑆_𝑇 ( 4-36)

This can be simplified to: 𝜇_𝑐 ≈ 𝜇₀+ 𝑆0

2𝑆_𝑇(𝜇𝑛− 𝜇0) = 𝜇0+ 𝑛

2(𝜇1− 𝜇0) ( 4-37)

Substitute in equation ( 4-31) to get: 𝑒_𝑆𝑒𝑔 ≈𝑛∆𝑦

4 [𝑛(𝜇1− 𝜇0) − (𝜇𝑛− 𝜇0)] ( 4-38) Thus, the error will stay under the required boundaries, if the following relation is held:

𝑛(𝜇1− 𝜇0) − (𝜇𝑛− 𝜇0) ≤

4 . |𝑒_{𝑇𝑟𝑎𝑝𝑒𝑧}|

𝑁. ∆𝑦 ( 4-39)

Or, 𝑛(𝜇₁− 𝜇₀) − (𝜇_𝑛− 𝜇₀) ≤ |4. 𝑒_{𝑇𝑝.𝑢.𝑙}| ; 𝑓𝑜𝑟 ∆𝑦 = 1 _{( 4-40)}

This relation defines the Trapezoidal error in stricter form if compared to relation ( 4-30), which it is the result of a traditional trapezoidal error equation ( 4-15).

80