Mathematical Physics Vol 1

Chapter A. Fractional Calculus: A Survey of Useful Formulas

468

A.7 Transfer Functions

A.7.1 Discrete transfer function approximations These approximations are discrete transfer functions, i.e. transfer functions that depend on z − 1 , the inverse of the Z -transform variable of time, which can be identified with the delay operator. The Z -transform of a function sampled with sampling interval h is Z { f ( t ) } ( z )= z − k f ( kh ) . (A.82)

+ ∞ ∑ k = 0

Unless otherwise noted, the approximations below correspond to GL D α

0 + f ( x ) and are trun

cated after an arbitrary number N of terms. Euler or Griinwald-Letnikoff approximation, causal GL D α a + f ( x ) ≈ ∆ α h , a + h α , ( x > a ) . Therefore , s α ≈ 1 h α ⌊ x − a h ⌋ ∑ k = 0 ( − 1 ) k α k

(A.83)

z − k , ( x > a ) .

Euler or Grünwald-Letnikoff approximation, anti-causal

∆ α

h , b − h α

GL D α

f ( x ) ≈

( x < b ) .

,

b −

Therefore ,

(A.84)

( − 1 ) k

α k

⌊ b − x

h ⌋

1 h α

∑ k = 0

s α ≈

z k , ( x < b ) .

Tustin approximation (truncated Maclaurin series) s α ≈ 2 h α N ∑ k = 0 k ∑ n = 0

z − 1 ( − 1 ) n Γ ( α + 1 ) Γ ( − α + 1 ) Γ ( α − n + 1 ) Γ ( n + 1 ) Γ ( k − n + 1 ) Γ ( − α − k + n + 1 )

(A.85)

Tustin approximation (truncated continued fraction expansion) s α ≈ 2 h α " 1; 2 α − 1 z − 1 − α , α 2 − k 2 − 2 k + 1 z − 1 # N k = 1

(A.86)

First-order backwards finite difference approximation (truncated continued fraction expan sion) s α ≈ 1 h α   0; 1 1 , α z − 1 1 , − k ( k + α ) ( 2 k − 1 ) 2 k z − 1 1 , − k ( k − α ) 2 k , ( 2 k + 1 ) z − 1 1   N k = 1 (A.87) Impulse response approximation

h − α Γ ( 1 − α ) −

h − α − 1 Γ ( − α )

( kh ) − α − 1 Γ ( − α )

N ∑ k = 1

s α ≈

z − k

(A.88)

+

Step response approximation

h − α Γ ( 1 − α ) −

h − α − 1 Γ ( − α )

( kh ) − α Γ ( 1 − α ) −

N ∑ k = 1

k − 1 ∑ n = 0

s α ≈

k , a

a k z −

a n , k = 1 , 2 ,..., N . (A.89)

+

k =