After performing numerous simulations with the 20-state model to develop insights into the major dynamic interactions, a re- duced-order, linear differential equation model consisting of five state and two con- variables is formed to initiate the deter- mination of the controller gains. The state vector consists of perturbations on engine
speed, manifold pressure, throttle angle,
throttle motor rate, and spark advance. The two control variables correspond to throttle
rate command and spark advance command
perturbation. After linearization about a
nominal reference condition, a linear qua-
dratic problem (LQP) was defined to deter- mine the controller gains. The performance index emphasized small rpm deviations (for good set-point control) and small throttle rate deviations (for lower-cost throttle actuators).
Figure 2 shows the resultant LQP state
feedback controller when simulated with the
20-state simulation program. (In this simu-
lation, a load is placed on the engine at t =
0 and removed at t = 3 sec. Other control
policies shown are: no control, optimal throttle-only control, and optimal throttle/ spark state feedback with and without feed- forward control; the curve with the smallest amplitude oscillation is the feedforward
case.) The simulation indicates that coordi-
nated throttle and spark feedback gives a much improved transient response over the standard throttle-only control.
After simulating numerous hardware sys- tems, especially various types of throttle ac- tuators, a candidate hardware system was se-
lected. The digital control model structure
shown in Fig. 3 was then developed, and the Landau model reference identification tech-
nique was employed to obtain the model pa- rameters on an engine dynamometer. The re- sultant model was then employed with digital
control theory to define a coordinated throttle
and spark digital controller for vehicle im- plementation. Comparisons of actual vehicle
data and simulation data for the no-control,
throttle-only control, and throttle/spark con-
trol cases (all without feedforward) are
shown in Fig. 4.
The data shown in Fig. 4 are for a four-
cylinder engine, which means that during
each second approximately 20 combustion events occur. Thus, with throttle-only con- trol, approximately 2 sec are required for
recovery to the neighborhood of the set point
after a major disturbance at t = 0. Note that
the throttle-only controller is still lightly
damped beyond 2 sec. Alternatively, throt-
tle/spark control requires approximately 1 sec
to return to the set point, and the response
is relatively well damped in the neighbor-
hood of the set point. Also note that the throttle-only case has a speed droop of ap-
proximately 200 rpm, while the throttle/spark
case droops only 100 rpm.
Figure 4 indicates that the fidelity of the
model is good enough to allow considerable
"paper design" before vehicle implemen-
tation. The accuracy of the model is even better than Fig. 4 indicates in that only one vehicle test is shown in the figure. If an aver-
age vehicle response was displayed (instead
of a single response), then the vehicle and
model data would probably be in closer agreement. The heuristic reason why the
throttle/spark controller is better than the
throttle-only controller is due to the fact that
spark acts much more quickly than the throt-
tle (with its actuator and manifold delays).
Qualitatively, this is best represented by
comparing the root loci of the two cases in
the Z-plane. Figure 5 shows the closed-loop
poles for zero spark feedback as the throttle
feedback gain is increased. The system goes unstable when the magnitude of the throttle gain is equal to 0.4. Figure 6 shows the same
system with a fixed, nonzero spark feedback gain as the throttle feedback gain is in- creased. When the throttle gain magnitude reaches 0.4, the resultant closed-loop poles trol are well within the unit circle, and a stable, relatively insensitive design results.