Open loop Analysis

T he approach to the control system in this study has been to use the model to

understand the stability issues with the traction co ntrol problem and with this

knowledge develop a controller which is stable under all the conditions. This is to be achieved by using a PID algorithm and using gain scheduling to adapt the controller to changes in system dynamics. This should lead to a control system which is robust to

know n non-linearities. In this section, the powertrain m odel developed will be used to

explore the control and stability issues of the system.

In this section, the four key non-linear effects will b e explored. These are:-

i) The tyre force/slip relationship. ii) Torque converter

iii) The engine torque production characteristic. iv) Gear Ratio

T y g f o r e - s li p effects

Figure 2.13 shows a root locus of the open loop poles o f th e system for the powertrain,

excluding the torque converter characteristic as the slip to the driven wheels increases from pure rolling to approximately 20% slip. The p -slip characteristic o f the tyre is

shown also in Figure 2.13. The root-locus shows: i) The natural m odes for the wheels

ii) A n oscillatory mode associated with the axle complianceAvheel A engine inertias iii) The pole due to the vehicle inertia/vehicle road load.

The wheel modes are initially in the far left half o f d ie complex plane. This is not

show n for scaling reasons, but the modes begin at the p o in ts (—945,j0 ) A (—954,j0 ). As the tyre characteristic is traversed, these poles m ove t o th e right and, as the peak of

the p-slip curve is approached, the poles accelerate very rap id ly towards the imaginary axis.

The vehicle mode is situated slightly to the left o f the origin (due to the damping provided by the road load) and this mode also moves to the right as the peak o f the p-slip curve is traversed.

The oscillatory pole pair begin at the points (-1.55,± j2 1.8) and initially move to

the left. This mode is associated with the vehicle shuffle mode and clearly shows the

damping increasing as the slope o f the tyre characteristic decreases. This is consistent with work by Hrovat (51). At a critical tyre slope, the mode becomes very well damped at the (-3 4 .3 , ±J23.56) point. As the slope progressively decreases from this

critical value, a higher frequency mode emerges with the damping reducing eventually

to zero as the peak is traversed. This higher frequency mode is due to th e wheels beginning to oscillate independently o f the vehicle body mass.

At the peak o f the p-slip curve, all o f the poles are on the imaginary axis and, as

the negative p art o f the slope is reached, the poles m ove to the right half p lan e and the

system becomes unstable. It is worth noting that when a vehicle drives fro m a good

road surface onto ice, the above transition into instability occurs extrem ely quickly (i.e within several milliseconds) and therefore a control scheme requires high derivative action to catch the over-spin.

Torque converter effects

The effect o f introducing the torque converter can be seen in Figure 2.14. This

introduces one additional state. This is associated with the engine inertia n o w being

split into (i) the transmission inertia (including the torque converter turbine ) an d (ii) a reduced engine inertia (i.e. flex plate, torque converter impellor and gears). The torque converter generates an input torque which is related by the k facto r (Figure

2.12) to slip across the converter and an output torque multiplied by the to rq u e ratio

(Figure 2.12). This slip related torque is effectively a damper in series b etw een the engine and gearbox inertias.

The addition o f a torque converter does not significantly change the wheel modes

and vehicle mode described above. The change introduces a further pole at the origin and increases the damping o f the oscillatory mode. The oscillatory pole pair now start at (—8 .9 ,± /2 J .4). This represents the same shuffle mode described above, but with

considerably more damping. This is due to the significant damping provided by the

torque converter described above. This complex pole pair follow the same trend as

described above for the situation without the torque converter, except that for the same level o f slip, the high frequency oscillatory mode finishes at (-3 .1 4 , ±>6/.42) rather than crossing the imaginary axis - i.e. more stable. This illustrates how the torque

converter introduces m ore damping into the powertrain.

Engine torque production

A s was stated in section 2.3.1, the engine model used was a black box description (coded in Fortran with a variable delay and non-linear manifold) incorporated within

the code. This model represents both the non-linear steady-state and dynamic

behaviour o f the engine. With this structure, it was not possible to linearise the model and analyse the engine system dynamics. For this reason, the engine characteristics will be analysed directly.

The steady-state torque characteristic for the engine is shown in Figure 2.8. This

clearly shows a non-linear dependence o f torque on throttle angle. This is approximately logarithmic with low throttle angles giving a large change in torque

compared with high throttle angles giving a relatively small change in torque (i.e. at 1000 r/m in, 2% to 3% throttle gives a change in torque of approx 68Nm whereas 22%

Significant dynamics are associated with manifold Ailing. Powell et al (52) show

that the m anifold dynamics can be represented as a first order lag with a time constant given by:-

Hence the manifold time constant can be interpreted directly from the induction map

(Figure 2.9) as being inversely proportional to the angle subtended between the constant speed lines and the constant throttle lines. At high loads and high speeds this

angle is large, implying a small time constant; whereas at low speed and low load this angle is small, implying a large time constant. From the induction m ap for the engine

(Figure 2.9) the partial derivatives, and hence the manifold tim e constant, have been

calculated. This yields the characteristic shown in Figure 2.10.

The effect o f engine operating point upon both the steady state and dynamic torque production characteristics is large.