Mathematical Physics Vol 1

A.7 Transfer Functions

469

R In every machine there will be a k max ∈ N which is the largest integer for which Γ ( k max ) does not yet return infinity. Because Γ ( x ) grows very fast, k max may be relatively small; if ⌊ | x − a | h ⌋ > k max , the summations in (A.83)-(A.84) will thereby be truncated. To avoid this, the following approximation can be used instead of (A.83), assuming that k max is even:

( − 1 ) k

α k

( − 1 ) k

α k

k max ∑ k = 0

k max ∑

m ∑ i = 2

1 h α

1 ( ih ) α

s α ≈

f ( t − kh )+

f ( t − kih )+

k max

k = ⌈

2 ⌉

(A.90)

⌊ t − c

( − 1 ) k

α k

( m + 1 ) h ⌋ ∑

1 [( m + 1 ) h ] α

f ( t − k ( m + 1 ) h )

+

k max

k = ⌈

2 ⌉

where

t − c

t − c ( m + 1 ) h ⌋

⌊ and m ∈ N . The expression (A.84) would be handled in a similar manner. mh ⌋ > k max ≥⌊

R Each k adds two terms to the continued fraction in (A.87). A truncated MacLaurin series of a first-order backwards finite difference returns the Euler approximation (A.83).

R A weighted average of approximations (A.83) and (A.85) is sometimes used [12]. The particular case of weights 3 4 for (A.83) and 1 4 for (A.85) is known as the Al-Alaoui operator [13] .

R Approximations (A.88)-(A.89) return the exact impulse and step responses at the interval h . From there on, either the impulse or the step response is followed; it is impossible to follow both. For x = 0, the output is always far from the exact value.

A.7.2 CRONE or Oustaloup approximation

1 + s 1 + s

j ω α C 1 +

N ∏ m = 1

ω z , m

s α ≈ C

where C =

(A.91)

j ω C ω z , m j ω C ω p , m

N ∏ m = 1

ω p , m

1 +

ω C ∈ [ ω ℓ , ω h ] , ω z , m = ω ℓ ω h ω ℓ

ω p , m = ω ℓ

ω h ω ℓ

2 m − 1 − α 2 N

2 m − 1 + α 2 N

(A.92)

R The N stable real poles and the N stable real zeros of (A.92) are recursively placed it [ ω ℓ , ω h ] , and verify ω z , m + 1 ω z , m = ω p , m + 1 ω p , m = ω h ω ℓ 1 N (A.93) It is advisable to make N ≥⌊ log 10 ω h ω ℓ ⌋ . Typically the approximation will be acceptable in [ 10 ω ℓ , ω h 10 ] . F]equency ω C , at which the gain will be exact, is arbitrary, but it is reasonable