Automated Upgraded Generalized Full-Discretization Method …
91
p H =
1
(−q + 1)!( p c + q − 1)!((t) p c +1 rep(P H ((t),
+
,
),
(26)
and rep(P(t),
+
,
) is a vector-valued function which produces a vector by replacing any plus sign in the coefficients of a polynomial P(t) with a single void space.
For example, rep(−t
4
+ t
2
− t,
+
,
) = {−t
4 0t
3 t
2
− t}
T . The operator is a
special form of dot product operator that multiplies every element of a vector with the
correspondingly subscripted sub-matrix of a concatenation and sums the products.
The polynomials in t, represented as P L ((t) and P H ((t), are given as
P L ((t) =
0
j=1− p c
(− jt + 1),
for q = 1,
(27)
P L ((t) = (−1) −q ((t − 1)
0
j=1− p c , j =q
(− jt + 1),
for q = 0, − 1, − 2, . . . , 1 − p c , (28)
P H ((t) = ((t − 1)P L ((t),
for q = 1, 0, − 1, − 2, . . . , 1 − p c .
(29)
The terms of P L ((t) and P H ((t) are arranged from the lowest to the highest powers of t from left to right. Since the products B(t)x(t) and B(t)x(t − τ ) in the
integrand of Eq. (16) are power polynomials in t, Eq. (16) simply requires the
execution of the items
t i+1
t i
t
0 e
A(t i+1 −t) dt,
t i+1
t i
t
1 e
A(t i+1 −t) dt,
t i+1
t i
t
2 e
A(t i+1 −t) dt,. . .,
t i+1
t i
t
p c +1 e
A(t i+1 −t) dt for B(t)x(t) and the execution of the items
t i+1
t i
t
0 e
A(t i+1 −t) dt,
t i+1
t i
t
1 e
A(t i+1 −t) dt,
t i+1
t i
t
2 e
A(t i+1 −t) dt, . . . ,
t i+1
t i
t
p d +1 e
A(t i+1 −t) dt for B(t)x(t − τ ).
The integration terms associated with B(t)x(t) are the F matrices which are concatenated in F L and F H as
F L = {F p c +2 F p c +1 · · · F p c +3−size(F L ) }
T
(30)
and
F H = {F p c +2 F p c +1 · · · F p c +3−size(F H ) }
T
,
(31)
where
F 1 = (F 0 − I)A
−1
,
(32)
F p c +2−l = [( p c + 1 − l)F p c +1−l − ((t)
p c +1−l I]A
−1
,
for p c , p c − 1, . . . , 0,
(33)
and F 0 is the matrix exponential function evaluated at t with
F 0 = e
At
.
(34)