Environmental Engineering Reference
In-Depth Information
!
a
þ
b
c
T
i
ð
1
þ
1
Þ
P
1
V
1
;
1
V
23
;
1
þ
P
1
=
1
c3
C
1
c
V
1
;
P
1
V
1
;
1
V
23
;
1
ð
1
þ
1
Þ
"
#
þ
þ
TDC
23
;
¼
(4
:
2)
1
c
P
1
V
1
;
1
V
23
;
1
ð
1
þ
1
Þð
1
þ
Þ
c
T
i
ð
1
þ
1
Þ
P
1
V
1
;
1
V
23
;
1
E
a
R
V
TDC
V
1
;
exp
þ
P
1
=
1
c3
such that
23
;
¼
f
2
ð
system states; system inputs
Þ
¼
f
2
ð
P
1
;
23
;
1
;
;
V
1
;
;
1
;
V
1
;
1
Þ
The constant C
1
is a linear function of the engine speed and inversely
dependent on the equivalence ratio
(again, a metric for the amount of fuel
per unit amount of air). Together, the dynamic equations (Eqs. 4.1 and 4.2) for
peak pressure and combustion timing complete the physics-based control
model of residual-affected LTC under single-cylinder constant engine speed
conditions. Like the simulation model, the control model's physically oriented
formulation is extendable to other conditions, including residual trapping [51]
and multi-cylinder, variable engine speed operation.
Once formulated, the control model was validated against both experimental
data and the more complex simulation model during both steady-state and
transient operating conditions. The control model was then used as a launching
point for the development of the several physics-based control strategies, result-
ing in the first generalizable, validated, and experimentally implemented con-
trol approach for residual-affected LTC engines.
4.3.4 Synthesis and Implementation of Controllers
from Control Models
By using the control model described in Section 4.3.3, the author and colleagues
developed the very first physics-based, experimentally validated control strat-
egy for LTC [40]. This approach relies on the ability to vary the inducted gas
composition with the VVA system and the existence of an operating manifold
with nearly constant combustion timing. Specifically, the control model was
used to synthesize a strategy capable of cycle-to-cycle control of peak pressure
through modulation of the inducted gas composition. Here a linear control law
was synthesized from a linearized version of the nonlinear peak pressure
dynamics. The self-stabilizing nature of the process is used to maintain nearly
constant combustion timing without direct control of the timing. The stability
of this linear controller, in closed-loop with the full nonlinear peak pressure
dynamics, was formulated in [52]. In this work, a Lyapunov-based analysis
utilizing sum of squares decomposition and a theorem from real algebraic
