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.
CRUISE CONTROL Cruise control (speed control, auto-cruiseor tempomat) is a system that automaticallycontrols the speed of a motor vehicle. The systemtakes over the throttle of the car to maintain asteady speed as set by the driver.
SYSTEM MODELLING feedback control system purpose is to maintain a constant vehiclespeed despite external disturbances, suchas changes in wind or road grade. accomplished byi. measuring the vehicle speedii. comparing it to the desired or reference speediii. automatically adjusting the throttle accordingto a control law
PHYSICAL SETUP: FBDbv u Mass m Control force u Resistive forces bv Vehicle velocity v u = force generated at the road/tireinterface we will assume that :i. u can be controlled directlyii. the dynamics of thepowertrain, tires, etc are 0iii. bv, due to rolling resistance and winddrag varies linearly with the vehiclevelocity, v, and act in the directionopposite the vehicles motion
FIRST ORDER EQUATIONWe are considering a first order mass-dampersystem.Summing forces in the x-direction and applyingNewtons 2nd law, we arrive at the followingsystem equation:m(dv/dt)+bv=uSince v is the required output:y = v
TRANSFER FUNCTIONTaking the Laplace transform and assumingzero initial conditions, we find the transferfunction of the cruise control system to be:P(s) = V(s)/U(s)= 1/(ms+b)
PROPORTIONALCONTROL The root-locus plot shows the locations ofall possible closed-loop poles when a singlegain is varied from zero to infinity. Only a proportional controller Kp will beconsidered to solve this problem. Theclosed-loop transfer function becomes:Y(s)/R(s) = Kp/(ms + ( b + Kp ) )
PROPORTIONALCONTROL MATLAB command sgrid Used to display an acceptable region of theroot-locus plot Damping ratio (zeta) and the naturalfrequency (Wn) need to be determined
PROPORTIONALCONTROL We can then find a gain to place the closed-loop poles in the desired region byemploying the rlocfind command specific loop gain[Kp,poles]=rlocfind(P_cruise) In between the dotted lines (zeta > 0.6) andoutside the semi-ellipse (wn > 0.36)
LAG CONTROLLER With the gain Kp being the only functionalgain and Ki and Kd being zero, the rise timeand the overshoot criteria have been met A steady-state error of more than 10%remains To reduce the steady-state error, a lagcontroller is added to the system
LAG CONTROLLER To reduce the steady-state error, a lagcontroller will be added to the system. A pole and a zero, not too distant spacing-wise are introduced i.e:
LAG CONTROLLER With the gain Kp being the only functionalgain and Ki and Kd being zero, the rise timeand the overshoot criteria have been met A steady-state error of more than 10%remains To reduce the steady-state error, a lagcontroller is added to the system the steady-state error will be reduced by afactor of zo/po
LAG CONTROLLER With the gain Kp excluded for the moment, thetransfer function of PID becomes: Adding Kp to the equation, the transfer function ofPID becomes:
LEAD CONTROLLER The lead controller is basically added toimprove the transient response of thesystem i.e. Ts and Tp mainly Not used here as it is not needed and thedesired parameters are already beingachieved