Problem 4. Consider a car, and let\'s call Hou (s) be the transfer function from

Question

Problem 4. Consider a car, and let's call Hou (s) be the transfer function from the throttle angle command to the speed u. The throttle angle varies between 0 and 43°. The car is taken onto a test track; the driver floors the gas pedal and holds it there. After a few seconds, the speed of the car reaches 120 mph and stays there. A table of speed vs time from the measurements during the test are shown below. Note that the throttle angle reaches its maximum value when the gas pedal is floored 1. What is the DC gain of the transfer function Ho (s)? 2. What is the time constant of the transfer function Ho (s)? time (sec.) speed (mph) 10 13 25 30 42 53 0 64.4 75.8 94.2 108.0 118.8 120

Explanation / Answer

DC Gain or Static Gain:

The DC gain of the system is defined as the ratio of output and input under steady state conditions.

If the input u = uo is constant and the system is stable then the output will reach the steady state value yo = G(0) uo

In the above given Car's transfer function , the input ( Throttle angle) is constant ( ie. = 43 ' ) and the system is stable and the poutput will reach the steady state value ( ie = 120 mph)

G(0) = yo / uo

= 120 / 43

= 2.790

ie the DC gain of the system is 2.790

Time constant:

The time constant of the system is defined as the time required for the system to decay to zero incase of the a decaying system

or    the time required for the system to reach the 1-1/e = 63.2% of its final (asymptotic ) value

The given system is a increasing system and so ,

the time constant is defined as the time taken by the system to reach 63.2% of its final value.

in given system , the final value of the system is 120 mph

therefore , 63.2 % of 120 = 75.84 mph

now the time taken for the system to reach 75.8 mph is known as the time constant of the system.

from the table given we can find that,

time taken for the system to reach 75.8 mph is 13 seconds

time constant = 13 s

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