7/31/2019 ELEC 3002 HW Control Tutorial

1/2

Control Systems Tutorial System Response

1- A second order system is described byi0

0

2

0

2

66dt

d3

dt

d1.5 =++ . Find the % overshoot of

the response and the time taken to reach it when the system is subjected to a unit step input.

[16.3%, 1.814]

2- A second order position control system has a damping ratio of 0.5 and an undamped natural

frequency of 6 rad./s. Find for a unit step input:(i) Output response as a function of time. [1-1.1547exp(-3t)sin(5.196+600)

(ii) Value of the % peak overshoot [16.3%]

(iii) Settling time. [1.046s]

3- A flywheel driven by an electric motor is automatically controlled to follow the movement of

a hand wheel. The inclusive Moment of Inertia of the flywheel is 150 Kg-m 2 and the motor

torque applied to it is 2400 Nm/rad. of misalignment between the flywheel and the hand

wheel. The viscous friction is equivalent to a torque of 600 Nm/rad./s. If the hand wheel is

suddenly turned through 600 when the system is at rest, find an expression for the subsequent

angular position of the flywheel in relation to time.

Ans: 0=(/3)[1-exp(-2t){Cos3.46t+0.58Sin3.46t}]

4- A control system has a forward transfer function given by)1s2.0)(1s(s

24

++and the feedback

transfer function is s/6. Calculate (a) Undamped natural frequency, (b) Damped frequency, (c)

Damping Ratio, (d) Maximum Overshoot, (e) Time at which maximum overshoot occurs.

Assume negative feedback. [5 rad./s, 4 rad./s, 0.6, 0.095, 0.785s]

5- A unity feedback system has an open loop transfer function given by)8s(s

25

+. Determine the

damping ratio (0.8), peak overshoot (1.54%) and peak time (1.045). If a component having

transfer function s/10 is introduced in the forward path, what will be its effect?

6- A second order system has an overall transfer function of 250020ss

25002

++and the step input

is 40u(t). Calculate the (a) the % Overshoot, (b) the values of the first two response peaks and

their corresponding times, (c) the times at which the response equals the input for the first and

second times, (d) the settling time. [(a) 52.7%, (b) 61.1, 64.1 ms; 28.9, 128.2 ms, (c) 36.2 ms,

100.3 ms (d) 500 ms]

7- The angular position of a flywheel on a test rig is controlled by an error actuated closed loop

automatic system to follow the motion of an input lever. The lever is maintained in sinusoidal

oscillations through 600 with an angular frequency = 2 rad./s. The moment of inertia of the

flywheel is 150 Kgm2 and the stiffness of the control is 2400 Nm/rad. of misalignment.

Calculate (a) the viscous frictional torque, (b) the amplitude of the swing of the flywheel and

(c) the time lag between the flywheel and the control lever for critical damping. [1200Nm/rad./s, 480, 0.464s]

8- The angular position of an aerial is automatically controlled by an input wheel. To this wheel

a sinusoidal supply of sint4

is applied. The system has an inertia of 100Kgm2 and viscous

friction is 400Nm/rad./s. If the system is critically damped, calculate the swing of the aerial

and the time lag? [ 0.91srad,5

]

9- A servo system consists of a synchro unit, amplifier and a motor with connected load. J =

0.005 Kgm2, f = 0.01 Nm/rad./s, H = 0.05, combined amplifier and synchro gain (K) = 1000

V/rad, motor torque constant (Km) = 61050 Nm/Volt. Calculate the steady state error when

an input signal of 1 rad./s is applied? What is the effect of doubling the amplifier gain. [0.2rad., 0.1414 rad.].

7/31/2019 ELEC 3002 HW Control Tutorial

2/2

10- A servomechanism which is designed to control the angular position of a rotatable mass is

stabilized by means of acceleration feedback. The moment of inertia of the system is 10 -5

Kgm2, the viscous frictional torque is 10-4 Nm per radian per second and the motor torque Tmis given

Tm = 4 10-3[e + Ks

2o] Nm

Draw a block diagram of the system and develop the control equation. Hence determine the

value of K in order that the damping shall be critical. What is the steady state error for aninput signal of 1.26 rad./s. [2.34 10 -3, 0.0315 rad.].

11- A servomechanism is used to control the angular position o of a mass through a command

signal i. The moment of inertia of the moving parts referred to the load shaft is 200 Kg-m2

and the motor toque at the load shaft is 200 Kgm 2 and the motor torque at the load is 6.88104

Nm./rad. of error. The damping torque coefficient referred to the load shat is 5 103

Nm/rad./sec.

(a) Find the time response of the servomechanism to a step input of 1 rad. and

determine the frequency of transient oscillations, the peak time, and the maximum

overshoot? [1 - 1.355exp(-12.5t)sin(13.7t+0.828), 13.7 rad./s, 0.229s, 5.6%]

(b) Find the steady state error which exists when a steady torque of 1200 Nm isapplied to the load shaft? [1 degree]

(c) Find the steady state error when the input is a constant angular velocity of 1 revolution

per minute? [0.44 degree]

12- The open loop transfer function of a unity feedback system is given by)sT1(s

K)s(G

+

= ,

where K and T are positive constants. By what factor would the amplifier gain be reduced so

that the peak overshoot of a unit step response of the system is reduced from 75% to 25%?

[20]

13- A unity feedback system is characterised by an open loop transfer function)10s(s

K)s(G

+

=

Find the value of K so that the system will have a damping ratio of 0.5. For this value of K

find the settling time, peak overshoot, and the peak time? [100, 0.8s, 16.2%, 0.364 s]

14- A system response when subjected to a unit step input is

c(t)=1+0.2exp(-60t)-1.2exp(-10t)

Find (a) the closed loop transfer function, (b) the undamped natural frequency and the

damping ratio. [(a))60s)(10s(

600

++, (b) 24.5 rad./sec, 1.43]

CB/March 08