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Model Order Reduction By Mixed...
amplitude (P a ) and peak time (T p ). the reduced second order approximant maintains the characteristics of the
original system. This method tries to minimize the Integral Square Error (ISE) and the reduced second order
model behaviour to the input signal almost matches with the original system behaviour.
Padé Approximation:
An asymptotic expansion or a Taylor expansion can often be accelerated quite dramatically by being re-arranged
into a ratio of two such expansions.
A Padé approximation
G s =
M k =0 a k s k
N
k =0 b k s k
(i)
(normalized by b 0 = 1) generalizes the Taylor expansion with the same total number of coefficients:
T M+N (s) =
M+N
k=0 c k s k
(ii)
(the two expansions being the same in the special case of M = 0). From a truncated Taylor expansion (ii), one
determines the corresponding Padé coefficients by requiring that if (i) is Taylor expanded; the result shall match
all the terms given in (ii).
Such that:
G s =
M k =0 a k s k
N
k =0 b k s k
The coefficients of the power series can be calculated as:
= c 0 + c 1 s + c 2 s 2 +…..
c 0 = a 0
k
c k = 1/b 0 [a k - j=1 b j c k−j ]; k > 0
a k = 0 k > n-1
Hence, Padé Approximants can be given as:
a 0 = b 0 c 0
a 1 = b 0 c 1 + b 1 c 0
.
.
. a k-1 = b 0 c k-1 + b 1 c k-2 +…+ b k-1 c 1 + b k c 0
The reduced model’s transfer function can be formed by using Padé Approximants in the numerator and the
denominator is derived by the basic characteristics of higher order systems as:
Consider a fourth order system:
G s =
a 0 + a 1 s
s 2 + 2ξω n s + ω n
2
III. NUMERICAL EXAMPLE
Characteristics:
Rise Time: 2.2609
Settling Time: 3.9312
Settling Min: 0.9062
Settling Max: 1.0000
Overshoot: 0
Undershoot: 0
Peak: 1.0000
Peak Time: 10.4907
G s =
s 3 + 7s 2 + 24s + 24
s 4 + 10s 3 + 35s 2 + 50s + 24
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