Wave TUT
WAVE TUT
CONTROL ENGINEERING – EXAM STANDARD QUESTIONS
Instruction: Answer any four (4) questions. All questions carry equal marks.
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QUESTION ONE – ROUTH STABILITY CRITERION
(a)
A unity feedback control system has the characteristic equation:
S^4 + 3S^3 + 5S^2 + 7S + 10 = 0
Using the Routh Stability Criterion:
1. Construct the Routh array.
2. Determine the number of roots in the right-half plane.
3. State whether the system is stable or unstable.
4. Determine the range of gain for stability if the equation becomes:
S^4 + 3S^3 + 5S^2 + 7S + K = 0
(b)
The characteristic equation of a closed-loop system is given as:
S^5 + 2S^4 + 6S^3 + 10S^2 + 8S + 12 = 0
Using Routh Hurwitz Criterion:
1. Form the complete Routh table.
2. Determine the stability of the system.
3. Find the number of roots on the right-hand side of the s-plane.
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QUESTION TWO – ROUTH STABILITY WITH SPECIAL CASES
(a)
Given the characteristic equation:
S^4 + 2S^3 + 3S^2 + 6S + 9 = 0
1. Construct the Routh array.
2. Explain the special case encountered.
3. Determine the auxiliary equation.
4. Investigate the stability of the system.
(b)
A control system has the characteristic equation:
S^6 + 4S^5 + 8S^4 + 12S^3 + 15S^2 + 10S + 5 = 0
Using the Routh Stability Criterion:
1. Develop the Routh table.
2. Determine whether the system is absolutely stable.
3. State the number of roots in the left-half plane and right-half plane.
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QUESTION THREE – TRANSIENT AND STEADY STATE ANALYSIS
(a)
A second-order unity feedback control system has the transfer function:
\frac{C(s)}{R(s)} = \frac{25}{S^2 + 6S + 25}
Determine the following:
1. Damping ratio
2. Natural frequency
3. Peak time
4. Rise time
5. Settling time
6. Maximum overshoot
7. Nature of the system response
(b)
For a unity feedback system with open-loop transfer function:
G(s) = \frac{20}{S(S+4)}
Determine:
1. Position error constant
2. Velocity error constant
3. Acceleration error constant
4. Steady-state error for:
Unit step input
Unit ramp input
Unit parabolic input
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QUESTION FOUR – TRANSIENT RESPONSE ANALYSIS
(a)
The closed-loop transfer function of a system is given by:
T(s) = \frac{49}{S^2 + 10S + 49}
Calculate:
1. Damping ratio
2. Undamped natural frequency
3. Peak overshoot
4. Settling time
5. Peak time
6. Rise time
(b)
A unity feedback system has open-loop transfer function:
G(s)H(s) = \frac{K}{S(S+3)(S+5)}
1. Determine the type of system.
2. Determine the static error constants.
3. Find the steady-state error for a unit step and unit ramp input.
4. Determine the value of that gives a velocity error constant of 15.
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QUESTION FIVE – BLOCK DIAGRAM REDUCTION
(a)
Given a control system with forward path blocks:
And feedback blocks:
The system contains:
1. An inner feedback loop around
2. A series connection of , , and
3. An outer negative feedback loop through
Required:
1. Draw the block diagram.
2. Reduce the block diagram step-by-step.
3. Obtain the overall transfer function:
\frac{C(s)}{R(s)}
(b)
A control system consists of the following blocks:
G_1 = \frac{4}{S+1}, \quad G_2 = \frac{6}{S+2}, \quad G_3 = \frac{8}{S+5}
With feedback paths:
H_1 = 1, \quad H_2 = \frac{2}{S+4}
The system has:
One summing junction before
A take-off point after
Inner negative feedback through
Outer negative feedback through
Required:
1. Sketch the block diagram.
2. Reduce the diagram using block diagram reduction rules.
3. Determine the overall transfer function.
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QUESTION SIX – MIXED CONTROL SYSTEM ANALYSIS
(a)
For the characteristic equation:
S^4 + 5S^3 + 8S^2 + 6S + K = 0
1. Use Routh Criterion to determine the range of for stability.
2. Determine the condition for marginal stability.
(b)
A second-order system has the transfer function:
T(s) = \frac{36}{S^2 + 12S + 36}
Determine:
1. Damping ratio
2. Natural frequency
3. Rise time
4. Peak time
5. Settling time
6. Maximum overshoot
7. Whether the system is underdamped, overdamped, or critically damped.
(c)
Reduce the following block diagram to obtain the overall transfer function:
Forward path blocks:
G_1 = \frac{3}{S+1}, \quad G_2 = \frac{5}{S+3}, \quad G_3 = \frac{7}{S+4}
Feedback blocks:
H_1 = 2, \quad H_2 = \frac{1}{S+2}
The system contains both inner and outer feedback loops.
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END OF QUESTION PAPER
