
114 Synthetic Gene Network: Modeling, Analysis, and Robust Design Methods
In other words,
()
()
()()
0
,0
f
t
T
EV X c E v t v t dt+
∫
is the upper bound
of the tracking error energy
() ()
0
,,
f
t
E Y ct QY ct dt
∫
. Since the suboptimal
tracking design is to minimize its upper bound instead of the tracking error
energy, the suboptimal tracking design for a synthetic gene network is to
select a promoter set to minimize
()
()
()(
0
,0
f
T
EV X c E v t v t
+
∫
dt. Since
the environmental disturbance term
()(
0
f
t
T
Evtvt
∫
dt is independent on
the choice of promoter set. Therefore the suboptimal tracking design of a
synthetic gene network is reduced to
()
()
1, ,
min