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Correct option is (a) Infinite

Easy explanation: The given system has no poles at s=0. This means that for a parabolic INPUT, the parabolic ERROR constant is 0. HENCE, the STEADY state error will go to infinity. Thus option Infinite is correct only.

The correct choice is (c) ^3⁄4*R(t-4) – ^3⁄4*r(t-16)-u(t-16)

Best EXPLANATION: The correct representation of the signal shown in the GRAPH is ^3⁄4*r(t-4) – ^3⁄4*r(t-16)-u(t-16). This is because the ramp function, starting from t=4 units in time, has a slope of ^3⁄4 and it’s shown to END at t=16. So, there it faces a slope change to 0, so the 2^nd term happens and FINALLY, the amplitude becomes 0 which is why, the 3^rd term happens.

The CORRECT ANSWER is (d) 1/(s-1)

For explanation: All the first THREE options are unstable since their ROC doesn’t include the imaginary axis. For option 1/(s-1), the POLE lies in the left half of s-plane and the R.O.C. includes the imaginary axis.

Right option is (a) r(t-7)

To explain I would say: The impulse response is simply the inverse LAPLACE transform of the TRANSFER FUNCTION. ACCORDING to the GIVEN nature of the transfer function, the impulse response is then given by option r(t-7) only.

Right CHOICE is (d) Purple

Explanation: The impulse response of the given transfer function is a ramp function of SLOPE 1/a. If a INCREASES, the slope decreases. Hence, from the above figure, purple is the CORRECT ONE.

Correct choice is (b) Poles lie on left half of s plane

For explanation: If poles don’t lie on the left half of s-plane, the system is not STABLE. An unstable system YIELDS an ERRONEOUS final value which i.e an erroneous steady state RESPONSE. They also cannot lie on the imaginary AXIS. Hence, the theorem is applicable only in case of poles lie on right half of s plane.

Correct OPTION is (a) Easier

To explain: The classical way is more tideous while the LAPLACE TRANSFORM ALLOWS it to represent the differential function in an algebraic form. Thereafter, the inverse laplace transform does the work.

Correct choice is (b) False

For explanation: The impulse response is truly DUE to the impulse input given to a SYSTEM. If we would’ve changed the input, the OUTPUT would’ve been a different and the RESULTING EXPRESSION in laplace domain will not be called the transfer function.

The correct option is (a) dependent on

The best EXPLANATION: The steady state value of a SYSTEM is dependent on the transient response of the system since every system goes through a transient state before reaching the steady state. In a DESIGN of CONTROL systems, we’ve to control the transient response to reduce the steady state error, which OCCURS due to undesirable transient response.

Correct choice is (a) Step response implies INTEGRATING the transfer function

The best I can explain: The above BLOCK diagram depicts that the impulse response of the integrator is a step function. This means that an integrator can be viewed as performing convolution of a step function and the INPUT. Now if we imagine a system which has the same transfer function as the lapalce transform of the previous input, and we give a step input to it- it will perform convolution of the step function an the system transfer function. Thus we FIND that the RESULT, due to associative property of convolution, is same and we can represent that the step response of system truly implies integrating the transfer function of the system in the output.

The correct option is (b) False

For explanation I would say: Since the output of a system DEPENDS on int’s initial CONDITIONS, the laplace TRANSFORM of the output would depend on the transient conditions. Hence, the impulse response of the system does take ACCOUNT of the transient response of the system.

The correct answer is (B) inverse of the damping CONSTANT

The best I can EXPLAIN: The time constant of a system is inverse of the damping constant. It is DEFINED so and hence the REST of the options are incorrect.

The correct answer is (a) True

Best explanation: Negative damping implies that the RESPONSE of the SYSTEM grows in magnitude and it is UNBOUNDED in time. Hence, the system output is unstable and the system is ALSO unstable.

Right option is (B) stepinfo()

Easy explanation: The stepinfo command is pre-defined in MATLAB to get the step response characteristics of a control SYSTEM. The input is to be given within parentheses and not []. The step() command gives the OUTPUT GRAPH of the step response and it doesn’t reveal the characteristics explicitly.

Correct choice is (a) a parabolic function

To explain I would say: The STEP response of a SYSTEM can be thought in the Laplace domain as MULTIPLICATION with 1/s. Now, the impulse response of ramp function is the ramp function itself while the step response of ramp function RESULTS in a PARABOLA.

The correct CHOICE is (a) will have 2 poles in the s-plane

Best explanation: The transfer function of the system, whose IMPULSE response is a sinusoid, will have 2 roots which SATISFY the CHARACTERISTIC EQUATION. Hence, there will be two poles which are complex conjugate to each other.

The correct CHOICE is (d) Purple plot

Easiest explanation: As the value of a increases, the inverse LAPLACE transform of the given transfer function suggests that the e-at value is reducing. HENCE for a higher a, the graph reaches a 0 earlier. From the given figure, the purple plot reaches the STEADY state FASTER so purple plot is correct only.

The CORRECT answer is (B) Decreases

The explanation is: The rise time of a SYSTEM is inversely proportional to the natural FREQUENCY of a system. Hence, if it INCREASES, the rise time decreases.

The correct choice is (a) an impulse function

Explanation: The LAPLACE TRANSFORM of only the impulse function is 1. HENCE, the inverse laplace transform.

Right choice is (a) Purely SINUSOIDAL

The best explanation: The generalized time response of a second ORDER control system reduces to a purely sinusoidal FUNCTION when the damping factor is zero. This is because the damping RATIO BECOMES 0. Hence, the correct option is purely sinusoidal.

Correct choice is (a) Underdamped

The explanation: If the damping factor is less than the damping factor at critical damping i.e. the natural frequency, the damping ratio becomes less than 1. This makes the system underdamped SINCE the RESPONSE of the system becomes a DECAYING sinusoid in NATURE.

The CORRECT choice is (a) s

The best I can explain: The default variable used to REPRESENT the laplace transform of a function, by the laplace command, is ‘s’. This can be ALTERED by giving different INPUT to the laplace commad.

Right answer is (a) Yellow

The explanation: The impulse response is a decreasing EXPONENTIAL curve. This means that the amplitude of the step function will be at t=0. Hence, the correct OPTION is a. After t=0, the response decreases EXPONENTIALLY which is observed by taking the INVERSE laplace transform of the TRANSFER function of the system.

Correct option is (a) will have 3 poles in the s-plane

To elaborate: The TRANSFER function of a system, WHOSE IMPULSE response is a PARABOLA, will have three roots which will satisfy the characteristic equation. Hence, only option will have 3 poles in the s-plane is correct.

The correct answer is (b) False

The EXPLANATION: The DELAY time is defined as the time REQUIRED by the system to reach half of its steady state value. Hence, the above statement is false.

The CORRECT choice is (d) 2

Explanation: For a step response of a SECOND ordered system, the maximum OVERSHOOT depends only UPON its damping ratio. Since the given system is a second ordered system, the maximum overshoot is a function of its damping ratio. But here, the damping ratio is 0 so the system has a maximum overshoot of 1.

Correct OPTION is (C) Blue

The explanation: : The IMPULSE response is a decreasing EXPONENTIAL curve. This means that the amplitude of the step function will be at t=0. Hence, the correct option is yellow. After t=0, the response decreases exponentially which is OBSERVED by taking the inverse laplace transform of the transfer function of the system.

Right option is (d) critically damped with equal roots

Explanation: The TIME response of a SYSTEM with a damping ratio of 1 is critically damped. But the roots of the CHARACTERISTIC equation will be equal and equal to NEGATIVE of the natural undamped FREQUENCY of the system.

Right option is (b) the number of poles+1

For explanation: It is SEEN that the number of possible inverse laplace transform of any function is equal to the number of poles it has +1. Considering a function F(s)=A/s+1 + B/s+3=F1(s)+F2(s), the inverse laplace transform would exist if the inverse laplace transform is absolutely converging in a region. There are 3 REGIONS that can have an absolutely converging state for F1(s) and F2(s) simultaneously and HENCE, only option the number of poles+1 is correct.

The correct choice is (a) 3

To elaborate: The transfer FUNCTION window shows that it holds the z-transform of a ramp function. But we observe from the graph that the output changes with a STEP of 3. HENCE, the slope of the ramp function is 3 SINCE we are plotting the IMPULSE response of the system.

The correct answer is (a) Blue

The best I can explain: The impulse response, represented by the PURPLE COLOR, suggests that the EXPONENTIAL factor which RESULT in the delay of the SYSTEM is least. Hence, Delay is correct only.

Right CHOICE is (b) False

The best explanation: If the functions f1(t) and f2(t) do not have the same R.O.C., the LAPLACE transform of f(t) won’t EXIST. This is because f(t) is a RESULT of addition of both the function and if one of them doesn’t exist, the ENTIRE function would collapse.

Right choice is (a) The speed of reaching stead state

The best explanation: The settling time is a MEASURE of the time REQUIRED by the SYSTEM to reach approximately 5% of it’s steady state value. Hence, it is a measure of how fast the system REACHES it’s steady state value.

Right answer is (d) Goes NEARER to the origin for the red curve than the BLUE curve

For explanation I would say: As the impulse response SUGGESTS that each response is sinusoidal, the frequency of the sinusoid will be the value of the POLE in the s-plane. If the frequency increases, the pole goes further from the curve- the output frequency increases. Hence, the plausible option is d SINCE the frequency increases or the pole nearer to the origin from the green to the red and away form the origin from the origin from red to green.

The correct answer is (d) PURPLE

For explanation: The impulse RESPONSE, REPRESENTED by the purple color, suggests that the exponential factor which RESULT in the delay of the system is higher.

Right option is (b) FALSE

Easiest explanation: If the pole lies on the right half of the s-plane, it will be seen that the laplace transform is not ABSOLUTELY CONVERGING and hence, the system won’t be STABLE. THUS the above statement is false.

Correct answer is (d) Cannot be determined

Best explanation: An L.T.I. system is stable if and only if both the conditions POLES LIE on left HALF of s-plane and R.O.C. encompasses the imaginary AXIS are satisfied.

Right answer is (a) Undamped

To explain I would say: Since the poles are equal and imaginary, the DAMPING factor is 0 and HENCE the damping ratio is 0. THUS, the system is absolutely undamped and OPTION Undamped is CORRECT only.

The correct answer is (a) True

For explanation I would say: The lapalce transform of the sinusoid represents the frequency of the signal as a pole. Since the RESPONSE is SINUSOIDAL, the transfer function is the laplace transform of the impulse response which contains the frequency as the pole of the sinusoidal SYSTEM.

Correct OPTION is (d) No such command

The explanation is: There isn’t any command defined in MATLAB which will COMPUTE the RESPONSE of a system having more ZEROS than POLES. Such a system is unstable at higher frequencies.