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26 CHAPTER 2. OVERVIEW OF THE UNDERLYING THEORY
where the transport delay, T
d
, and the combustion lag, T
c
, are approximately,
T
d
=
252
v
and T
c
=
202
v
.
For the purposes of our example we want to design an air/fuel ratio controller that
works for all engine speeds in the range 2,000 to 6,000 rpm. We will use a first order
Pad´e approximation for the delay and express the T
d
and T
c
relationships in an LFT
form with a normalized speed deviation, δ
v
.
The dominant combustion lag can be expressed as an LFT on 1/T
c
as follows,
0.9
1+T
c
s
= F
u
(P
tc
,T
−1
c
),
where
P
tc
=
−1
s
1
0.9
s
0
.
Note that T
−1
c
is easily modeled in terms of δ
v
,
1
T
c
=
4000 + 2000δ
v
202
,δ
v
∈R,|δ
v
|≤1.
The Pad´e approximation (with delay time, T
d
)isgivenby
e
−T
d
s
≈F
u
P
delay
,T
−1
d
,
where,
P
delay
=
−2
s
1
4
s
−1
.