The driver–vehicle system: stability and control

Same envelope, different lap times

In Lesson 1.1 the g-g diagram told us what a car can do. But imagine two cars with identical envelopes. In one, the driver can sit right at the edge of the envelope lap after lap. The other twitches and snaps whenever it gets near the limit, so the driver backs off to keep it on the road. On paper they are equal. On the stopwatch, the first car wins every time.

Bill Milliken spent his early career in aircraft flight testing, where the question was never just “how fast can it fly?” but also “can a pilot fly it safely and precisely?” He brought the same two ideas to race cars: stability and control. This lesson introduces both, and treats the driver as part of the system.

Open loop and closed loop

A driver does not steer by memorizing a sequence of inputs. They watch where the car is going, feel how it is rotating, and continuously correct. Engineers call this a closed-loop system: the output (the car's motion) is fed back to the input (the driver's hands and feet).

The driver–vehicle loop. Cut the dashed feedback path and you have an open-loop test: the car's response with the driver's input held fixed.

Vehicle engineers study the car in both ways:

Open-loop tests fix the input and watch the car respond: hands off the wheel after a bump, or a sudden fixed step of steering. These reveal properties that belong to the car alone.
Closed-loop tests give the driver a task, like a lane change or a lap, and measure how well it is done and how hard the driver had to work. These reveal how the car and driver perform together.

Stability: does the car settle down by itself?

An equilibrium is a condition the car can hold steadily, such as driving straight at constant speed. Stability asks what happens after a small disturbance, like a gust of wind or a bump, with the driver's inputs held fixed.

The classic picture of stability. For a car, the “nudge” is a disturbance in yaw and the question is whether the rotation dies out or grows.

Here is the physical picture for a car. Suppose a gust swings the nose slightly to the left. Now both axles are running at a small slip angle, and both tires push sideways. The front tires push ahead of the center of gravity; the rear tires push behind it. It is a tug-of-war in yaw:

If the rear tires win, the car straightens itself out, like the feathers on an arrow or a weathervane turning into the wind.
If the front tires win, the yaw is made worse, and the car keeps rotating further.

A car whose front tires “win” tends to oversteer; one whose rear tires win tends to understeer (Module 5 makes these definitions precise). There is a second effect: as the car rotates, its own yaw rate also creates slip angles that resist the rotation — a kind of damping. That damping gets weaker as speed rises. So an oversteering car can be perfectly stable in town and become unstable above a critical speed:

\[ U_{\text{crit}} = \sqrt{\frac{L}{|K|}} \]

where \(L\) is the wheelbase and \(K\) is the understeer gradient, a single number that summarizes the car's balance (negative for oversteer). You will derive this in Module 5; for now, just notice the trend: the more oversteer (larger \(|K|\)), the lower the speed at which the car becomes unstable. An understeering car (\(K > 0\)) has no critical speed at all.

Start with a 50% understeer car. Raise the speed from 15 to 45 m/s. Does it ever become unstable? What changes in the yaw-rate trace?
Set the balance to “Oversteer 50%”. Read the critical speed, then test speeds just below and just above it.
Keep a ghost of a stable run and compare the Heading change of an understeer car and a neutral car. Which one wanders further off its original line?
Notice that even the stable cars end up on a new heading. Who is responsible for getting the car back to its original path?
Run an unstable case in Slow motion. Compare where the car's nose points with the purple arrow showing where it is actually going. The growing gap between them is the sideslip angle \(\beta\).

“An unstable car is one that has run out of grip.” Not necessarily. The gust in the test above is small, and the tires start well inside their linear range. Instability is about whether the tire forces restore or amplify a disturbance, and that can happen far below the limit. Running out of grip makes things worse, but it is a separate effect.

Control: can the driver put the car where they want it?

Stability is about the car alone. Control (or controllability) is about the car and driver together: how easily and precisely can the driver make the car follow the path they choose? Two things matter:

Response: how much the car turns for a given steering input, and how quickly. A heavily understeering car can be stable yet sluggish. It needs a lot of steering and arrives late at the new direction.
Predictability: whether the response stays consistent as speed and grip change, so the driver's learned “feel” still applies near the limit.

The driver as a feedback controller

If you have taken a controls course, you know this loop. If not, here is all you need for now. A driver behaves roughly like a controller with three properties:

Preview: the driver looks ahead, typically about a second down the road, and aims the car at a point there.
Gain: how strongly they correct an error. Too little and the car drifts off line; too much and they overcorrect.
Delay: a human takes roughly 0.15–0.3 s to perceive an error and start correcting it. During that time the car keeps moving on old information.

Drivers also feel yaw rate and lateral acceleration through their body and the steering wheel, usually before they can see the path error. That “seat-of-the-pants” feedback lets a skilled driver catch a car that is starting to rotate, which is exactly what an unstable car needs.

A driver can drive a car that is slightly unstable, but only by constantly catching it. The car's stability and response decide how much of the driver's attention and skill go into simply keeping it on the road, and how much is left for going fast.

Run the defaults (slight understeer, 30 m/s). Keep the run as a ghost. Then set the balance to “Understeer 100%” and run again. Which car follows the line more closely?
Set the balance to “Oversteer 50%” at 35 m/s. The readout says the car alone is unstable. Can the model driver still complete the maneuver? Try reaction times of 0.15 s and 0.3 s.
With that same unstable car, set the yaw feel to zero. What does this tell you about why drivers value feedback through the seat and steering wheel?
Go back to a stable car and turn the steering gain up to 3 with a 0.4 s reaction time. What goes wrong, even though the car itself is stable?
Switch to You drive at half speed. Compare your steering trace and reversal count for an understeer car and a neutral car.

Why the fastest car is not always the easiest to drive

A strongly understeering car is stable and forgiving, but it is reluctant to turn. The driver must wait for it at corner entry, and it scrubs speed with its front tires. A car close to neutral turns in eagerly and uses both axles fully at the limit, which is good for the g-g envelope. The cost is a smaller stability margin, so the driver must work harder and react faster to keep it there.

Race engineers therefore tune the balance to sit near neutral, but on the side that this driver can exploit confidently, for this race distance. Fatigue, tire wear, changing fuel load and track conditions all shift the best compromise. This is the engineering judgment that Module 9's capstone asks you to practice.

“Understeer is safe, so more understeer is always better.” More understeer adds stability, but the car becomes sluggish and imprecise, as you saw in the lane change at higher speed. It also wastes rear-tire grip at the limit. Too much understeer is slow, and it can be dangerous in an emergency maneuver.

Check your understanding

A driver approaching a braking zone at 180 km/h has a reaction time of 0.25 s. How far does the car travel before the driver even begins to respond to something they see?

Convert the speed to m/s first. Distance = speed × time.

\(180\ \text{km/h} \div 3.6 = 50\ \text{m/s}\), so \(d = 50 \times 0.25 = 12.5\ \text{m}\): about three car lengths of driving on old information.

A car has a wheelbase of 2.5 m and an understeer gradient of \(K = -0.0040\ \text{rad/(m/s}^2)\) (it oversteers). Above what speed is it unstable without driver correction? Give your answer in km/h.

Use \(U_{\text{crit}} = \sqrt{L/|K|}\), which gives m/s.

\(U_{\text{crit}} = \sqrt{2.5 / 0.0040} = \sqrt{625} = 25\ \text{m/s} = 90\ \text{km/h}\). A car like this would be fine in a parking lot and treacherous on a fast straight.

A car drives straight with the steering wheel clamped. A small bump gives it a slight yaw, and the yaw rate keeps growing until the car spins. What does this tell you?

The tires had no grip left.Not necessarily: instability can begin well inside the tires' linear range. Growth from a small disturbance is the signature of instability, not of saturation.
The car is unstable at that speed: its tire forces amplify a yaw disturbance instead of restoring it.With no driver input (open loop), a disturbance that grows by itself means the equilibrium is unstable.
The driver overcorrected.The wheel was clamped. This was an open-loop test with no driver corrections at all.

An oversteering car is stable at 60 km/h but unstable at 150 km/h. Why does speed matter?

The damping effect of the tires on yaw rotation weakens as speed rises, so above a critical speed the oversteer tendency wins.The slip angles created by a given yaw rate scale with \(r/U\), so the same rotation produces less restoring force at higher speed.
The tires have less grip at high speed.Grip may change slightly with speed, but that is not the reason. The car becomes unstable even with constant tire properties.
The driver's reaction time is longer at high speed.Stability is an open-loop property of the car alone. No driver is involved.

In the lane change, a model driver with a high steering gain and a long reaction time weaves increasingly and spins a car that is stable on its own. What is happening?

The car became unstable because of the lane change.The open-loop stability of this car does not change. The problem is in the loop, not in the car alone.
The driver is correcting hard based on old information, so each correction arrives too late and overshoots. The closed loop is unstable even though the car is not.Gain and delay together can destabilize a loop. That is why calm, smooth inputs are often faster.
The steering is too slow to respond.The steering responds instantly in the model; the delay is in the driver's perception and reaction.

Going further with Milliken

Read: RCVD Chapter 1, The Problem Imposed by Racing, focusing on how the driver and vehicle are treated as one system. Then skim the opening pages of Chapters 5 and 6, Simplified Steady-State and Simplified Transient Stability and Control, to see where this lesson is heading.

As you read, look for answers to these questions:

Which ideas from aircraft stability and control do the authors carry over to the automobile, and which do not fit?
How do the authors distinguish stability from control (or “controllability”)?
What do they say a driver needs from a car in order to drive it at the limit?

Summary

Idea	In one line
Open loop	The car's response with the inputs held fixed. Reveals properties of the car alone.
Closed loop	The driver observes and corrects. Reveals how car and driver perform together.
Stability	Does a small disturbance die out (stable) or grow (unstable)?
Critical speed	\(U_{\text{crit}} = \sqrt{L/\|K\|}\) for an oversteering car; understeering cars have none.
Control	How readily and predictably the car follows the driver's intended path.
Driver as controller	Preview, gain and delay. Too much gain with too much delay causes overcorrection.

Next: Module 2 opens up the tire, the source of every force in this lesson, to see how slip angle creates those restoring (or destabilizing) forces.