3. CRUISE CONTROL
Cruise control (speed control, auto-cruise
or tempomat) is a system that automatically
controls the speed of a motor vehicle. The system
takes over the throttle of the car to maintain a
steady speed as set by the driver.
5. SYSTEM MODELLING
feedback control system
purpose is to maintain a constant vehicle
speed despite external disturbances, such
as changes in wind or road grade.
accomplished by
i. measuring the vehicle speed
ii. comparing it to the desired or reference speed
iii. automatically adjusting the throttle according
to a control law
6. PHYSICAL SETUP: FBD
bv u
Mass m
Control force u
Resistive forces bv
Vehicle velocity v
u = force generated at the road/tire
interface
we will assume that :
i. u can be controlled directly
ii. the dynamics of the
powertrain, tires, etc are 0
iii. bv, due to rolling resistance and wind
drag varies linearly with the vehicle
velocity, v, and act in the direction
opposite the vehicle's motion
7. FIRST ORDER EQUATION
We are considering a first order mass-damper
system.
Summing forces in the x-direction and applying
Newton's 2nd law, we arrive at the following
system equation:
m(dv/dt)+bv=u
Since v is the required output:
y = v
8. TRANSFER FUNCTION
Taking the Laplace transform and assuming
zero initial conditions, we find the transfer
function of the cruise control system to be:
P(s) = V(s)/U(s)
= 1/(ms+b)
10. PARAMETERS
m vehicle mass 1000 kg
b damping coefficient 50 N.s/m
r reference speed 10 m/s
Rise time < 5 s
Overshoot < 10%
Steady-state error < 2%
12. PROPORTIONAL
CONTROL
The root-locus plot shows the locations of
all possible closed-loop poles when a single
gain is varied from zero to infinity.
Only a proportional controller Kp will be
considered to solve this problem. The
closed-loop transfer function becomes:
Y(s)/R(s) = Kp/(ms + ( b + Kp ) )
13. PROPORTIONAL
CONTROL
MATLAB command sgrid
Used to display an acceptable region of the
root-locus plot
Damping ratio (zeta) and the natural
frequency (Wn) need to be determined
15. PROPORTIONAL
CONTROL
We can then find a gain to place the closed-
loop poles in the desired region by
employing the rlocfind command
specific loop gain
[Kp,poles]=rlocfind(P_cruise)
In between the dotted lines (zeta > 0.6) and
outside the semi-ellipse (wn > 0.36)
16. LAG CONTROLLER
With the gain Kp being the only functional
gain and Ki and Kd being zero, the rise time
and the overshoot criteria have been met
A steady-state error of more than 10%
remains
To reduce the steady-state error, a lag
controller is added to the system
17. LAG CONTROLLER
To reduce the steady-state error, a lag
controller will be added to the system.
A pole and a zero, not too distant spacing-
wise are introduced i.e:
18. LAG CONTROLLER
With the gain Kp being the only functional
gain and Ki and Kd being zero, the rise time
and the overshoot criteria have been met
A steady-state error of more than 10%
remains
To reduce the steady-state error, a lag
controller is added to the system
the steady-state error will be reduced by a
factor of zo/po
19. LAG CONTROLLER
With the gain Kp excluded for the moment, the
transfer function of PID becomes:
Adding Kp to the equation, the transfer function of
PID becomes:
20. LEAD CONTROLLER
The lead controller is basically added to
improve the transient response of the
system i.e. Ts and Tp mainly
Not used here as it is not needed and the
desired parameters are already being
achieved
Editor's Notes
The two dotted lines in an angle indicate the locations of constant damping ratio (zeta=0.6); the damping ratio is greater than 0.6 in between these lines and less than 0.6 outside the lines. The semi-ellipse indicates the locations of constant natural frequency (Wn=0.36); the natural frequency is greater than 0.36 outside the semi-ellipse, and smaller than 0.36 inside.