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Friday, October 1, 2010

Modeling of Physical Systems and Study of their Open Loop Response

Objective:

The objective of this experiment is the modeling of physical systems and study of their open loop response.

Simulation of a Cruise Control System

Physical Interpretation & System Equations:

Let us assume a car that travels only in one direction. Control to the car was applied in such away that it has a smooth start up, along with a constant-speed ride.

If it is assumed that friction is opposing the motion of the car, then the modeling equations become,

mv’+bv=u

y=v

Mathematical Analysis:

Taking the Laplace transform of the equations, we find

msV(s)+bV(s)=U(s)

Y(s)=V(s)

Substituting V(s) in terms of Y(s)

msY(s)+bY(s)=U(s)

The transfer function of the system becomes

MatLab Code:

clear all;

close all;

b=100;

u=500;

figure;

for i=1:5

m=300*i;

num=[1];

den=[m b];

sys=tf(num,den);

step(u*sys);

hold on

grid

title('Step Response of the Cruise Control when Friction is Fixed');

xlabel('Time');

ylabel('Velocity (m/s)');

end

figure;

m=1500;

for j=1:5

b=20*j;

num=[1];

den=[m b];

sys=tf(num,den);

step(u*sys);

hold on

grid

title('Step Response of the Cruise Control when Mass is Fixed');

xlabel('time')

ylabel('Velocity (m/s)');

end

MatLab Simulation:

MatLab Data:

Mass (m)

Friction (b)

Rise Time (Tr)

Settling Time (Ts)

300

600

900

1200

1500

100

100

100

100

100

06.588

13.163

19.746

26.430

32.810

11.8

23.6

35.3

46.6

58.5

1500

1500

1500

1500

1500

20

40

60

80

100

166.10

83.87

54.37

41.23

33.25

289

144

97.3

72.2

58.1

Report:

Define rise time, settling time, percentage overshoot and steady state error of a system for step input.

Rise Time, Tr:

The time required for the response to go from 10% of the final value to 90% of the final value is known as rise time. It is denoted by Tr. For a step input, the output is naturally expected to reach at the final value gradually instead of impulsive response.

Settling Time, Ts:

The time required for the transient’s damped oscillations to reach and stay within ±2% of the steady state value is known as settling time. It is denoted by Ts.

Percentage Overshoot, %OS:

The amount that, the response overshoots, the steady state or final value at the peak time, expressed as a percentage of the steady state value. It is expressed in percentage form as,

For a unit step input, Mpt is the peak value of the time response, and fv is the final value of the response.

Steady State Error:

Steady state error is the difference in level between the response desired for the input command and steady state final response.

From the plot, observe the steady-state speed of the vehicle and the time it takes to reach the value.

The plot has been attached with the report. From the plot, following observations are made:

Mass (m)

Friction (b)

Settling Time (Ts)

Steady State Speed (m/s)

300

600

900

1200

1500

100

100

100

100

100

11.8

23.6

35.3

46.6

58.5

1500

1500

1500

1500

1500

20

40

60

80

100

289

144

97.3

72.2

58.1

How steady state speed, rise time and overshoot of the output response vary with the variation of m and b?

When Mass was Varied:

· Steady state speed of the system remains unchanged with the variation of the mass of the system. So, steady state speed is immune to mass change of the system.

· Rise time reacts to mass change. When the mass increases, rise time of the system increases and when the mass of the system decreases, rise time of the system decreases. So a system with small mass will rise to the steady state faster than the system with a large mass.

· No overshoot was seen in the given system.

When Friction was Varied:

· Steady state speed increases with decrease in friction & decreases with increase in friction.

· Rise time increases with decrease in b & decreases with increase in b.

· No overshoot was observed in the given system.

Justify the steady state speed of the vehicle depends only on the friction coefficient b.

In the cruise control system, it has been assumed that the vehicle travels only in one direction. If the inertia of the wheels is neglected, then the steady state speed of the vehicle depends on the friction coefficient b and the mass of the body m. According to the classical theory of mechanics, mass of a body is a constant quantity and it cannot be changed when it is running. So, from the equation it is clear that the steady state speed of the vehicle depends only on the friction coefficient b. It decreases when b increases when b decreases, cause friction or damping always works against motion. It is also seen from the plots obtained by software simulation that any change in m doesn’t change the steady state speed. So, it is decidable that the steady state speed of the vehicle depends only on the friction coefficient b, not on the mass of the system, m.

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