26 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
where the transport delay,Td, and the combustion lag, Tc, are approximately,
Td=252
vand Tc=202
v.
Forthe purposesof our example we want to design an air/fuel ratio controller that
works for all enginesp eeds in therange 2,000 to 6,000 rpm. We will use a first order
Pad´eapproximation for the delay and express the Tdand Tcrelationships in an LFT
form with a normalized speed deviation,δv.
The dominant combustion lag can be expressed as an LFT on 1/Tcas follows,
0.9
1+T
c
s=F
u
(P
tc,T−1
c),
where
Ptc =
−1
s1
0.9
s0
.
Note that T−1
cis easily modeled in terms of δv,
1
Tc
=4000+ 2000δv
202 ,δ
v
∈R,|δ
v
|≤1.
The Pad´eapproximation (with delay time, Td)isgivenby
e
−T
d
s≈F
uP
delay,T−1
d,
where,
Pdelay =
−2
s1
4
s−1
.