This case study begins with model construction, starting from basic physics. Special consideration is given to incorporate wind resistance and include as a system *disturbance,* the onset of a hill. The laws of motion say that given a vehicle mass *m* and engine supplied force *cw*(*t*), where *c* is proportionality constant and 0 ≤ w(t) ≤ 1 represents the engine throttle, *F*_{tot}(*t*) = *mv*(*t*) = *cw*(*t*), *t* ≥ 0.

Air resistance, proportional to the velocity squared times constant ρ*,* produces drag on the vehicle. Additionally when the vehicle encounters a hill (the system disturbance described in the introduction) of angle *θ*, gravity creates a second counter force, *mg* sin(*θ*), where *g* is the gravitational constant of 9.8 m/s^{2}. A small angle is assumed in this case, so the following is valid.

The equation of motion is now

where the last line is divided through by the mass *m*. This is a nonlinear differential equation in terms of the vehicle velocity *v*(*t*) because of the appearance of *v*^{2}(*t*), the air resistance term.

## Observe the nonlinear differential equation

Although the nonlinear differential equation is not in a form suitable for design of a cruise control, there are some interesting observations that can be made. This understanding also helps with the linearizing process.

When the vehicle is on the flat (*θ* = 0) and at the maximum throttle value of 1.0, the nonlinear differential equation reduces to

The maximum velocity will be reached as *t* becomes large. At maximum velocity the acceleration must be zero, so

The equation further simplifies to

Don’t try this at home with your car!

A second observation is that the vehicle stalls when climbing a hill at full throttle for some critical angle *θ** _{s}*. You first need to know that stall means the vehicle velocity is zero and the acceleration is zero. From the full nonlinear differential equation

You now solve for *θ** _{s}* as sin

^{–1}[

*c*/(

*mg*)]. Note this analysis assumes that the vehicle stays in a fixed gear. In driving your own car you probably downshift to the lowest gear to avoid stalling.

The third observation is the solution to the nonlinear differential equation at maximum throttle and *θ* = 0. Begin by defining the constant

the differential equation with this *T* inserted becomes:

The exact solution to this simplified form is *v*(*t*) = *v** _{max}*tanh(

*t*/

*T*). You can verify this is a valid solution by inserting it back into the differential equation and see that the equation is satisfied. This solution gives you the velocity profile versus time with the accelerator held to the floor.

Note this doesn’t quite apply to a real car because the model does not take into account gear shifting with a transmission. By plotting *v*(*t*) for a given *v** _{max}* and

*T*you can see approximately how long it takes to accelerate to a particular cruising velocity, for example, 0 to 60 mph in 10 seconds.

## Linearize to a nominal cruising velocity

To linearize the differential equation for a nominal cruise velocity of *v*_{0} < *v*_{max} and corresponding throttle setting of 0 < *w*_{0} < 1, use a one-term *Taylor series **expansion.* In calculus, you learn that any function, say *y *=* f*(*x*), can be approximated near the point *x*_{0} using *f*(*x*_{0}) and its derivatives evaluated at *x*_{0}. A one term approximation is linear in *x* – *x*_{0}, that is

If *f*(*x*) = *x*^{2} the expansion becomes

because *f* '(*x*) = 2*x* evaluated at *x*_{0} is 2*x*_{0}.

For the problem at hand, you want to expand with respect to the nominal velocity *v*_{0} and throttle setting *w*_{0}. Make substitutions into the original differential equation as follows:

represent the deviation of velocity and throttle settings away from the nominal values. These are the new modeling parameters. In the original differential equation:

where a step function is included on the hill gravity term to model the hill onset to *t* = 0. Note in the last line

is zero because this is the nominal constant operating point, which also corresponds to the throttle setting *w*_{0}. If you define the time constant

the now linearized differential equation becomes