**Most Latest Control Systems ****MCQs – New ****Digital Control Systems MCQs ( Control Systems ) MCQs**

**Latest Control Systems MCQs**

By practicing these MCQs of ** Digital Control Systems MCQs ( Control Systems ) MCQs – Latest Competitive MCQs **, an individual for exams performs better than before. This post comprising of objective questions and answers related to **“** Digital Control Systems MCQs ( Control Systems ) Mcqs **“.** As wise people believe “Perfect Practice make a Man Perfect”. It is therefore practice these mcqs of Control Systems to approach the success. Tab this page to check **” Digital Control Systems MCQs ( Control Systems )”** for the preparation of **competitive mcqs, FPSC mcqs, PPSC mcqs, SPSC mcqs, KPPSC mcqs, AJKPSC mcqs, BPSC mcqs, NTS mcqs, PTS mcqs, OTS mcqs, Atomic Energy mcqs, Pak Army mcqs, Pak Navy mcqs, CTS mcqs, ETEA mcqs** and others.

**Control Systems MCQs – Digital Control Systems MCQs ( Control Systems ) MCQs**

The most occurred **mcqs of Digital Control Systems MCQs ( Control Systems )** in past papers. **Past papers of Digital Control Systems MCQs ( Control Systems )** Mcqs. **Past papers of** ** Digital Control Systems MCQs ( Control Systems )** Mcqs . Mcqs are the necessary part of any competitive / job related exams. The Mcqs having specific numbers in any written test. It is therefore everyone have to learn / remember the related ** Digital Control Systems MCQs ( Control Systems )** Mcqs. The Important series of ** Digital Control Systems MCQs ( Control Systems ) Mcqs** are given below:

# Spectrum Analysis of Sampling Process

**1. Statement (I): Aliasing occurs when the sampling frequency is less than twice the maximum frequency in the signal.****Statement (II): Aliasing is a reversible process.**

a) Both statement (I) and Statement (II) are individually true and Statement (II) is the correct explanation of Statement (I).

b) Both Statement (I) and Statement (II) are individually true but Statement (II) is not correct explanation of Statement (I)**c) Statement (I) is true but Statement (II) is false**

d) Statement (I) is False but Statement (II) is true**Answer: c****Explanation: Aliasing is an irreversible process. Once aliasing has occurred then signal can-not be recovered back.****2. A band limited signal with a maximum frequency of 5 KHz to be sampled. According to the sampling theorem, the sampling frequency which is not valid is:****a) 5 KHz**

b) 12 KHz

c) 15 KHz

d) 20 KHz**Answer: a****Explanation: fs (min) =2fm****fs (min) =2*5 =10 KHz****So, fs >=1o KHz.****3. Let x(t) be a continuous-time, real valued signal band-limited to F Hz. The Nyquist sampling rate in Hz, For y(t) =x(0.5t) +x(t)-x(2t) is**

a) F

b) 2F**c) 4F**

d) 8F**Answer: c****Explanation: Expansion in time domain in compression in frequency domain and vice-versa. So, the maximum frequency component in given signal is 2F Hz. And according to sampling theorem.****Nyquist rate =2fm =4F Hz.****4. Increased pulse-width in the flat-top sampling leads to:****a) Attenuation of high frequencies in reproduction**

b) Attenuation of low frequencies in reproduction

c) Greater aliasing errors in reproduction

d) No harmful effects in reproduction**Answer: a****Explanation: As pulse width is increased, the width of the first lobe of the spectrum is decreased. Hence, increased pulse-width in the flat-top sampling, leads to attenuation of high frequencies in reproduction.****5. A bandpass sampling extends from 4-6 kHz. What is the smallest sampling frequency required to retain all the information in the signal.**

a) 1 kHz

b) 2 kHz

c) 3 kHz**d) 4 kHz****Answer: d****Explanation: fh =6 kHz****Bandwidth = 2 kHz****Fs =4 kHz.****6. A signal represented by x(t) =5cos 400πt is sampled at a rate 300 samples/sec. The resulting samples are passed through an ideal low pass filter of cut-off frequency 150 Hz. Which of the following will be contained in the output of the LPF?****a) 100 Hz**

b) 100 Hz, 150 Hz

c) 50 Hz, 100 Hz

d) 50 Hz, 100 Hz, 150 Hz**Answer: a****Explanation: x (t) =5cos400πt****fm =200 Hz****The output of the LPF will contain frequencies which are less than fc =150 Hz.****So, fs-fm =300-200 =100 Hz is the only component present in the output of LPF.****7. A signal m(t) with bandwidth 500 Hz is first multiplied by a signal g(t). The resulting signal is passed through an ideal low pass filter with bandwidth 1 kHz. The output of the low pass filter would be :**

a) Impulse

b) m(t)**c) 0**

d) m(t)del(t)**Answer: c****Explanation: m (t) g (t)->M (f)*G (f)****After low pass filtering with fc =1 kHz, hence the output is zero.****8. An LTI system having transfer function s ^{2}+1/s^{2}+2s+1 and input x(t) =sin(t+1) is in steady state. The output is sampled at ws rad/s to obtain the final output {y (k)}. Which of the following is true?**

**a) Y is zero for all sampling frequencies ws**

b) Y is non zero for all sampling frequencies ws

c) Y is non zero for ws>2 but zero for ws<2

d) Y is zero for ws>2 but non zero for ws<2

**Answer: a**

**Explanation: x (t) =sin (t+1)**

**w = 1 rad/s**

**X(s) = e**

^{s}/s^{2}+1**Y(s) = e**

^{s}/s^{2}+2s+1**Y (∞) =0.**

**9. A digital measuring instrument employs a sampling rate of 100 samples/second. The sampled input x(n) is averaged using the difference equation:**

**Y (n) =[x (n)+x (n-1)+x(n-2)+x(n-4)/4] For a step input, the maximum time taken for the output to reach the final value after the input transition is**

a) 20 ms

**b) 40 ms**

c) 80 ms

d) ∞

**Answer: b**

**Explanation: Since output y depends on input, such as no delay, delay by 1 unit, and delay by 2 unit, delay by 4 unit, so it will sum all the samples after 4 Ts (maximum delay), to get one sample of y[n].**

**T =40 msec.**

**10. The sinusoid x(t) =6cos10πt is sampled at the rate of 15 Hz and applied to ideal rectangular LPF with cut-off frequency of 10 Hz, then the output of filter contains:**

a) Only 10π rad/sec component

**b) 10π rad/sec component**

c) 10π rad/sec and 20π rad/sec components

d) +10π rad/sec and +20π rad/sec components

**Answer: b**

**Explanation: Output filter is the device that is used to filter some frequencies and make some frequencies in the response and in this case it contains 10π rad/sec.**

**Signal Reconstruction**

**1. Sampling can be done by:**

a) Impulse train sampling

b) Natural sampling

c) Flat-top sampling**d) All of the mentioned****Answer: d****Explanation: Sampling is the process in which the continuous systems are sampled by the application of the zero order hold and can be done by all the three methods.****2. The first step required to convert analog signal to digital is :****a) Sampling**

b) Holding

c) Reconstruction

d) Quantization**Answer: a****Explanation: Sampling is the process in which the continuous systems are sampled by the application of the zero order hold and is the first step in the conversion of analog to digital signals.****3. Sampling is necessary :**

a) In complex control systems**b) Where high accuracy is required**

c) Non automated control systems

d) Automated control system**Answer: b****Explanation: Sampling is the process where the continuous systems are converted into discrete time systems with the help of zero order hold of the signal and sampling is necessary where high accuracy is needed.****4. Sampled data technique is appropriate as:**

a) For long distance data transmission

b) Pulses are transferred by little loss of accuracy

c) More than one channel of information is sequentially sampled and transmitted.**d) All of the mentioned****Answer: d****Explanation: Sampled data technique refers to the data which is sampled and is appropriate as for long distance communication, for accurate transmission and multi channel transmission.****5. Signal sampling reduces the power demand made on the signal.****a) True**

b) False**Answer: a****Explanation: Signal sampling refers to the sampling of the signal and reduces the power demand that is the power required by the signal and is therefore helpful for signals of weak origin.****6. The use of sampled data control system are:**

a) For using analog components as the part of the control loop

b) For time division of control components**c) Whenever a transmission channel forms a part of closed loop**

d) None of the mentioned**Answer: c****Explanation: Sampled data control system is the system where the data used is sampled and is used whenever a transmission channel forms a part of closed loop system.****7. _______________ is a sampling pattern which is repeated periodically**

a) Single order sampling**b) Multi order sampling**

c) Zero order sampling

d) Unordered sampling**Answer: b****Explanation: Multi-order sampling is a sampling pattern in which the sampling is of different signals and which is repeated periodically.****8. For the successful reconstruction of signals :**

a) Sampling frequency must be equal to the message signal

b) Sampling frequency must be greater to the message signal

c) Sampling frequency must be less to the message signal**d) Sampling frequency must be greater than or equal to the message signal****Answer: d****Explanation: Reconstruction of signals refers to the conversion of the discrete time signals into continuous tiem signals and for the successful reconstruction of signals sampling frequency must be greater than or equal to the message signal but ideally it is always preferred to be greater.****9. The signal is reconstructed back with the help of**

a) Zero order hold circuits

b) Extrapolations**c) Signal is reconstructed with zero order holds and extrapolations**

d) Signal is not reconstructed**Answer: c****Explanation: The signal is reconstructed that is the process of converting the discrete time signals into the continuous time signals and this can be done with the help of hold circuits or extrapolations.****10. Aliasing is caused when:**

a) Sampling frequency must be equal to the message signal

b) Sampling frequency must be greater to the message signal**c) Sampling frequency must be less to the message signal**

d) Sampling frequency must be greater than or equal to the message signal**Answer: c****Explanation: Aliasing refers to the process when the discrete time signal is reconstructed back then due to the error some part of the signal is lost and is caused when sampling frequency must be less than frequency of message signal.**

**Difference Equations**

**1. Difference equation model results in:**

a) Sampled-data systems

b) Numerical analysis of continuous time systems

c) Continuous time feedback systems**d) Both a and b****Answer: d****Explanation: Equation in discrete time systems can be difference equation which are similar to the differentiation in the continuous time systems and difference equation results in sampled data system and numerical analysis of continuous system.****2. Difference equation is used in :****a) Discrete time analysis**

b) Continuous time analysis

c) Digital analysis

d) None of the mentioned**Answer: a****Explanation: Difference equation are similar to the differentiation in the continuous systems and they have same function in discrete time systems and is used in discrete time analysis.****3. Difference equation in discrete systems is similar to the _____________ in continuous systems.**

a) Difference equation**b) Differential equation**

c) Quadratic equation

d) None of the mentioned**Answer: b****Explanation: Difference equation are the equations used in discrete time systems and difference equations are similar to the differential equation in continuous systems.****4. Difference equation solution yields at the sampling instants only:****a) True**

b) False**Answer: a****Explanation: Difference equation are the equations used in discrete time systems and difference equations are similar to the differential equation in continuous systems solution yields at the sampling instants only.****5. Difference equation technique for higher order systems is used in:**

a) Laplace transform

b) Fourier transform**c) Z-transform**

d) None of the mentioned**Answer: c****Explanation: For higher order systems Z- transform which is transforming discrete time domain into z domain and is technique is convenient for analysis and design of linear sampled data systems.****6. Assertion (A): An LTI discrete system represented by the difference equation. y (n+2)-5y(n+1)+6y(n) =x(n) is unstable.****Reason (R): A system is unstable if the roots of the characteristic equation lie outside the unit circle.****a) Both A and R are true and R is the correct explanation of A**

b) Both A and R are true but R is NOT the correct explanation of A

c) A is true but R is false

d) A is false but R is false**Answer: a****Explanation: Difference equation is y (n+2)-5 y (n+1) + 6 y (n) =x (n).****Taking z-transform, H (z) =1/ (z-2) (z-3).****The characteristic equation has roots z =2, 3. Since, the characteristic equation has roots outside the unit circle, hence the system is unstable.****7. The poles of a digital filter with linear phase response can lie**

a) Only at z =0**b) Only on the unit circle**

c) Only inside the unit circle but not at z =0

d) On the left side of Real (z) =0 line**Answer: b****Explanation: For stable systems the poles in z plane must lie inside or on the unit circle and Minimum number of delay elements = (Maximum power of z)-(minimum power of z).****8. If X(z) =(z+z ^{-3})/(z+z^{-1}), then x(n) series has:**

**a) Alternate 0s**

b) Alternate 1s

c) Alternate 2s

d) Alternate -1s

**Answer: a**

**Explanation: Using the long division method divide from denominator. So, x(n) series has alternate zeros n =1,3,5……**

**9. Assertion (A): The stability of the system is assured if the ROC includes the unit circle in z-plane.**

**Reason (R): For a causal stable system all the poles should be outside the unit circle in the z-plane.**

a) Both A and R are true and R is the correct explanation of A

b) Both A and R are true bit R is NOT the correct explanation of A

**c) A is true but R is false**

d) A is false but R is true

**Answer: c**

**Explanation: A causal system LTI system is stable if and only if all of poles of H(z) lie inside the unit circle.**

**10. Assertion (A): For the rational transfer function H(z) to be causal, stable and causally invertible, both the zeroes and the poles should lie within the unit circle in the z-plane.**

Reason (R): For a rational system, ROC is bounded by poles

a) Both A and R are true and R is the correct explanation of A**b) Both A and R are true bit R is NOT the correct explanation of A**

c) A is true but R is false

d) A is false but R is true**Answer: b****Explanation: For the rational transfer function H (z) to be causal, stable and causally invertible, both the zeroes and the poles should lie within the unit circle in the z-plane. For a rational system, ROC is bounded by poles.**

**The Z-Transform**

**1. The discrete-time signal x (n) = (-1) ^{n} is periodic with fundamental period**

a) 6

b) 4

**c) 2**

d) 0

**Answer: c**

**Explanation: Period of the signal refers to the instant of time at which the signal repeats itself and for this Period =2 of the given discrete time signal.**

**2. The frequency of a continuous time signal x (t) changes on transformation from x (t) to x (α t), α > 0 by a factor**

**a) α**

b) 1/α

c) α

^{2}

d) α

**Answer: a**

**Explanation: x(t)->x(αt), α > 0**

**α > 1 compression in t, expansion in f by α.**

**α < 1 expansion in t, compression in f by α.**

**3. Two sequences x1 (n) and x2 (n) are related by x2 (n) = x1 (- n). In the z- domain, their ROC’s are**

a) The same

**b) Reciprocal of each other**

c) Negative of each other

d) Complements of each other

**Answer: b**

**Explanation: x1(n) is the signal in the discrete domain and X1(z) is the signal in the z domain, RoC Rx**

**z Reciprocals x2(n) = x1(-n) X1(1/z), RoC 1/ Rx**

**4. The ROC of z-transform of the discrete time sequence x(n) = is:**

**a) 1/3>|z|<1/2**

b) |z|>1/2

c) |z|<1/3

d) 2>|z|<3

**Answer: a**

**Explanation: One part of the equation is the right sided signal and other part is the left sided signal hence the ROC of the system will be 1/3>|z|<1/2.**

**5. Which one of the following is the correct statement? The region of convergence of z-transform of x[n] consists of the values of z for which x[n] is:**

a) Absolutely integrable

**b) Absolutely summable**

c) Unity

d) <1

**Answer: b**

**Explanation: The region of convergence of z-transform of x[n] consists of the values of z for which x[n]r**

^{-n}is absolutely summable.**6. The region of convergence of the z-transform of a unit step function is:**

**a) |z|>1**

b) |z|<1

c) (Real part of z)>0

d) (Real part of z)<0

**Answer: a**

**Explanation: h[n] =u[n] Hence, Region of Convergence is the region for which the values of the roots in z transform are lying in the function and is the range of values of z for which |z|>1.**

**7. If the region of convergence of x1[n]+x2[n] is 1/>|z|<2/3, the region of convergence of x1[n]-x2[n] includes:**

a) 1/3>|z|<3

b) 2/3>|z|<3

c) 3/2>|z|<3

**d) 1/3>|z|<2/3**

**Answer: d**

**Explanation: Region of Convergence is the region for which the values of the roots in z transform are lying in the function and ROC remains the same for addition and subtraction in z-domain.**

**8. A sequence x (n) with the z-transform X (z) = Z**

^{4}+ Z^{2}– 2z + 2 – 3Z^{-4}is applied to an input to a linear time invariant system with the impulse response h (n) = 2δ (n-3). The output at n = 4 will be:a) -6

**b) Zero**

c) 2

d) -4

**Answer: b**

**Explanation: H (z) = 2z**

^{-3}**Then taking the inverse Laplace transform of the equation of Y (z) at n=4 y(n) =0.**

**9. H (z) is discrete rational transfer function. To ensure that both H(z) and its inverse are stable:**

a) Poles must be inside the unit circle and zeros must be outside the unit circle

**b) Poles and zeroes must be inside the unit circle**

c) Poles and zeroes must be outside the unit circle

d) Poles must be outside the unit circle and zeros must be inside the unit circle

**Answer: b**

**Explanation: For H(z) to be stable the poles must be inside the unit circle and for the inverse of H(z) to be stable the poles of it must be inside the unit circle.**

**10. Z and Laplace transform are related by:**

a) s = ln z

**b) s =ln z/T**

c) s =z

d) s= T/ln z

**Answer: b**

**Explanation: z = e**

^{st}**s =ln z/T.**

**The Z-Transfer Function**

**1. Consider the following statements regarding a linear discrete-time system:****H (z) = z ^{2}+1/(z+0.5)(z-0.5)**

**1. The system is stable**

**2. The initial value of h(0) of the impulse response is -4**

**3. The steady-state output is zero for a sinusoidal discrete time input of frequency equal to one-fourth the sampling frequency**

**Which of these statements are correct?**

a) 1,2 and 3

b) 1 and 2

**c) 1 and 3**

d) 2 and 3

**Answer: c**

**Explanation: Characteristic equation is (z+0.5) (z-0.5) =0**

**Its root are z =0.5, -0.5**

**Since both roots are inside the unit circle, hence the system is stable.**

**2. The minimum number of delay elements required realizing a digital filter with transfer function**

H (z) =

a) 2

**b) 3**

c) 4

d) 5

**Answer: b**

**Explanation: H (z) =**

**Minimum number of delay elements= (Maximum power of z-minimum power of z)**

**Minimum number of delay elements = 3.**

**3. A system can be represented in the form of state equations as:**

**S (n+1) =A S (n) +B x (n)**

**Y (n) = C S (n) +D x (n)**

**Where, A, B, C, D are the matrices , S(n) is the state vector , x(n) is the input and y(n) is the output . The transfer function of the system.**

**H (z) =Y (z)/X (z) is given by:**

a) A(ZI – B)

^{-1}C + D

b) B(ZI – C)

^{-1}D + A

**c) C(ZI – A)**

^{-1}B + Dd) D(ZI – A)

^{-1}C + B

**Answer: c**

**Explanation: Solving both the equations and substituting the value of the output equation into the state equation we get the value of the transfer function as obtained.**

**4. Assertion (A): The signals a**

^{n}u(n) and a^{n}u(-n-1) have the same Z transform, z/(z-a)**Reason (R): the region of convergence of a**

^{n}u(n) is |z|>|a|, whereas the ROC for a^{n}u(-n-1) is |z|<|a|.a) Both A and R are true and R is correct explanation of A

b) Both A and R are true but R is not correct explanation of A

c) A is true but R is false

**d) A is false but R is true**

**Answer: d**

**Explanation: Both have the ROC as given in the reason is true but the z transform for the second is with a minus sign.**

**5. What is the number of roots of the polynomial F(z) = 4z**

^{3}-8z^{2}-z+2, lying outside the unit circle?a) 0

**b) 1**

c) 2

d) 3

**Answer: b**

**Explanation: Factorizing F (z) and then the factors are the roots which here come out to be 3.**

**6. Assertion (A): The discrete time system described by y[n] =2x[n] +4x[n-1] is unstable**

**Reason (R): It has an impulse response with a finite number of non-zero samples**

a) Both A and R are true and R is correct explanation of A

b) Both A and R are true but R is not correct explanation of A

c) A is true but R is false

**d) A is false but R is true**

**Answer: d**

**Explanation: For the system to be stable the value of the transfer function in the discrete time domain must be summable and H[n] calculated is summable hence the system is stable.**

**7. What is the z-transform of the signal x[n] = a**

^{n}u(n)?a) X(z) =1/z-1

b) X(z) = 1/1-z

**c) X(z) = z/z-a**

d) X(z) = 1/z-a

**Answer: c**

**Explanation: By definition this is the basic example of the z-transform and the Z-Transform of the equation is calculated is z/z-a.**

**8. Which one of the following rules determine the mapping of s-plane to z-plane?**

a) Right side of the s-plane maps into outside of the unit circle in z-plane

b) Left half of s-plane maps into inside of the unit circle

c) Imaginary axis in s-plane maps into the circumference of the unit circle

**d) All of the mentioned**

**Answer: d**

**Explanation: S- plane can be mapped into the z plane with certain rules than right side maps into the outside, left side maps into the inside and imaginary axis maps on the unit circle of the z plane.**

**9. Assertion (A): The z-transform of the output of the sampler is given by the series.**

**Reason (R): The relationship is the result of the application of z = e**

^{-sT}, where T stands for the time gap between the samples.a) Both A and R are true and R is correct explanation of A

b) Both A and R are true but R is not correct explanation of A

**c) A is true but R is false**

d) A is false but R is true

**Answer: c**

**Explanation: T is termed as the time of the sampling instant and z transform is always defined for the instant of the sampling event and this can be as desired by the user.**

**10. Convolution of two sequences X1[n] and X2[n] are represented by:**

**a) X1(z)*X2(z)**

b) X1(z)X2(z)

c) X1(z)+X2(z)

d) X1(z)/X2(z)

**Answer: a**

**Explanation: Convolution of the two sequences is the combination of multiplication and addition of the two sequences at each instant and convolution in time domain is multiplication in the frequency domain.**

**The inverse z-transform and Response of Linear Discrete Systems**

**1. Unit step response of the system described by the equation y(n) +y(n-1) =x(n) is:****a) z ^{2}/(z+1)(z-1)**

b) z/(z+1)(z-1)

c) z+1/z-1

d) z(z-1)/z+1

**Answer: a**

**Explanation: Response of the system is calculated by taking the z-transform of the equation and input to the transfer function in the step input.**

**2. Inverse z-transform of the system can be calculated using:**

a) Partial fraction method

b) Long division method

c) Basic formula of the z-transform

**d) All of the mentioned**

**Answer: d**

**Explanation: Inverse z-transform is the opposite method of converting the transfer function in Z domain to the discrete time domain and this can be calculated using all the above formulas.**

**3. Assertion (A): The system function**

**H(z) = z**

^{3}-2z^{2}+z/z^{2}+1/4z+1/s is not causal**Reason (R): If the numerator of H (z) is of lower order than the denominator, the system may be causal.**

**a) Both A and R are true and R is correct explanation of A**

b) Both A and R are true and R is not correct Explanation of A

c) A is True and R is false

d) A is False and R is true

**Answer: a**

**Explanation: The transfer function is not causal as for causality the numerator of H (z) is of lower order than the denominator, the system may be causal.**

**4. Assertion (A): Z-transform is used to analyze discrete time systems and it is also called pulsed transfer function approach.**

**Reason(R): The sampled signal is assumed to be a train of impulses whose strengths, or areas, are equal to the continuous time signal at the sampling instants.**

**a) Both A and R are true and R is correct explanation of A**

b) Both A and R are true and R is not correct Explanation of A

c) A is True and R is false

d) A is False and R is true

**Answer: a**

**Explanation: Z-transform is used to convert the discrete time systems into the z domain and it is also called pulsed transfer function approach that is justified only at the sampling instants.**

**5. The z-transform corresponding to the Laplace transform G(s) =10/s(s+5) is**

**Answer: b**

**Explanation: Laplace transform is the technique of relating continuous time to the frequency domain while the z transform is relating discrete time hence Laplace transform is first converted into time domain and then the z transform is calculated.**

**6. Homogeneous solution of: y(n) -9/16y(n-2) = x(n-1)**

**a) C1(3/4)**

^{n}+C2(3/4)^{-n}b) C1-(3/4)

^{n-1}+C2(3/4)

^{n-1}

c) C1(3/4)

^{n}

d) C1-(3/4)

^{n}

**Answer: a**

**Explanation: Taking the z-transform of the given difference equation and solving the homogeneous equation and finding the solution using complimentary function.**

**7. If the z transform of x(n) is X(z) =z(8z-7)/4z**

^{2}-7z+3, then the final value theorem is :**a) 1**

b) 2

c) ∞

d) 0

**Answer: a**

**Explanation: Final value theorem is calculated for the transfer function by equating the value of z as 1 and this can be calculated only for stable systems.**

**8. Final value theorem is used for:**

a) All type of systems

**b) Stable systems**

c) Unstable systems

d) Marginally stable systems

**Answer: b**

**Explanation: Final value theorem is used to calculate the final value as for time infinite and for z = 1 the final value theorem can be calculated and final value theorem is for for stable systems.**

**9. If the z-transform of the system is given by**

**H (z) = a+z**

^{-1}/1+az^{-1}**Where a is real valued:**

a) A low pass filter

b) A high pass filter

**c) An all pass filter**

d) A bandpass filter

**Answer: c**

**Explanation: The discrete time frequency response will be aperiodic and does not depend on the frequency and the transfer function will be representing the all pass filter.**

**10. The system is stable if the pole of the z-transform lies inside the unit circle**

**a) True**

b) False

**Answer: a**

**Explanation: For the system to be stable in Z domain the pole in the this domain must lie inside the unit circle and for the causal stable region must be outside the circle and hence the locus will be a ring.**

**The Z-transform Analysis of Sampled-Data Control Systems**

**1. The DFT of a signal x(n) of length N is X(k). When X(k) is given and x(n) is computed from it, the length of x(n)****a) Is increased to infinity**

b) Remains N

c) Becomes 2N – 1

d) Becomes N2**Answer: a****Explanation: When X (k) is given and x (n) is computed from it, the length of x (n) is increased to infinity.****2. The system having input x (n) related to output y(n) as y (n) = log |x (n)| is:****a) Nonlinear, causal, stable**

b) Linear, noncausal, stable

c) Nonlinear, causal, not stable

d) Linear, noncausal, not stable**Answer: a****Explanation: As y (n) is the function of x (n) hence it is nonlinear but it is bounded and also depends upon past and present values therefore it is stable and causal respectively.****3. Zero-order hold used in practical reconstruction of continuous-time signals is mathematically represented as a weighted-sum of rectangular pulses shifted by:**

a) Any multiples of the sampling interval**b) Integer multiples of the sampling interval**

c) One sampling interval

d) 1 second intervals**Answer: b****Explanation: Zero-order hold is used to reconstruct the continuous-time signal which is represented as a weighted sum of rectangular pulses shifted by integer multiples of the sampling interval.****4. When two honest coins are simultaneously tossed, the probability of two heads on any given trial is:**

a) 1

b) 3/4

c) 1/2**d) ¼****Answer: d****Explanation: Total outcomes =4****Favorable outcomes =1****Hence probability =1/4.****5. A continuous-time periodic signal x(t), having a period T, is convolved with itself. The resulting signal is**

a) Not periodic**b) Periodic having a period T**

c) Periodic having a period 2T

d) Periodic having a period T/2**Answer: b****Explanation: Periodic having a period T Convolution of a periodic signal (period T) with itself will give the same period T.**

Frequency Response Analysis MCQs

**6. If the Fourier series coefficients of a signal are periodic then the signal must be**

a) Continuous-time, periodic**b) Discrete-time, periodic**

c) Continuous-time, non-periodic

d) Discrete-time, non-periodic**Answer: b****Explanation: Discrete-time, periodic these are the properties of the discrete-time periodic signal.****7. The region of convergence of a causal finite duration discrete-time signal is****a) The entire z-plane except z = 0**

b) The entire z-plane except z = ∞

c) The entire z-plane

d) A strip in z-plane enclosing jω–axis**Answer: a****Explanation: For discrete time signal for any causal system the region of convergence is always entire z-plane but except z =0.****8. The probability cumulative distribution function must be monotone and**

a) Increasing

b) Decreasing

c) Non-increasing**d) Non-decreasing****Answer: d****Explanation: The cumulative distribution function that is monotone and non-decreasing or can be increasing.****9. Convolution is used to find:**

a) The impulse response of an LTI System

b) Frequency response of a System**c) The time response of a LTI system**

d) The phase response of a LTI system**Answer: c****Explanation: Convolution is the combination of addition and multiplication of the signals and used to find the impulse response of the LTI system.****10. The Fourier Transform of a rectangular pulse is**

a) Another rectangular pulse

b) Triangular pulse**c) Sinc function**

d) Impulse**Answer: c****Explanation: The Fourier Transform of a rectangular pulse that is the sinc function.**

**The z- and s-Domain Relationship**

**1. The auto-correlation function of a rectangular pulse of duration T is**

a) A rectangular pulse of duration T

b) A rectangular pulse of duration 2T

c) A triangular pulse of duration T**d) A triangular pulse of duration 2T****Answer: d****Explanation: The auto-correlation function is the method of correlating the various instants of the signal with itself and that of a rectangular pulse of duration T is a triangular pulse of duration 2T.****2. The FT of a rectangular pulse existing between t = − T 2/ to t = T / 2 is a**

a) Sinc squared function**b) Sinc function**

c) Sine squared function

d) Sine function**Answer: b****Explanation: By definition the Fourier transform is the transformation of time domain of signal to frequency domain and that of a rectangular pulse is a sinc function.****3. The system characterized by the equation y(t) = ax(t) + b is****a) Linear for any value of b**

b) Linear if b > 0

c) Linear if b < 0**d) Non-linear****Answer: d****Explanation: The system is non-linear because x(t) = 0 does not lead to y (t) = 0, which is a violation of the principle of homogeneity.****4. The continuous time system described by 2 y(t) = x (t ^{2}) is**

a) Causal, linear and time varying

b) Causal, non-linear and time varying

c) Non causal, non-linear and time-invariant

**d) Non causal, linear and time-invariant**

**Answer: d**

**Explanation: Y (t) depends upon the future value therefore the system is anticipative and hence is not causal. But as it follows the superposition theorem so it is linear.**

**5. If G( f) represents the Fourier Transform of a signal g (t) which is real and odd symmetric in time, then G (f) is**

a) Complex

**b) Imaginary**

c) Real

d) Real and non-negative

**Answer: b**

**Explanation: For the real and odd symmetric signal in time domain on the Fourier transform the resulting signal is always imaginary.**

**6. If a periodic function f(t) of period T satisfies f(t) = −f (t + T/2) , then in its Fourier series expansion**

a) The constant term will be zero

**b) There will be no cosine terms**

c) There will be no sine terms

d) There will be no even harmonics

**Answer: b**

**Explanation: The fourier series will contain the cosine terms if the periodic function f (t) of period T satisfies f (t) = -f(t+T/2), and this can be proved by the basic definition of the fourier transform.**

**7. Given a unit step function u (t), its time-derivative is:**

**a) A unit impulse**

b) Another step function

c) A unit ramp function

d) A sine function

**Answer: a**

**Explanation: Unit step function is one of the test signals and for the basic standard signals they are interrelated as the function of differentiation and integration as unit step function is the integral of impulse function.**

**8. The order of a linear constant-coefficient differential equation representing a system refers to the number of**

a) Active devices

b) Elements including sources

c) Passive devices

**d) None of the mentioned**

**Answer: d**

**Explanation: The order of the differential equation is the power of the highest order of the differential term and which refers to the number of poles in the transfer function and practically it refers to the number of components that are energy storing elements .**

**9. Z-transform converts convolution of time-signals to**

a) Addition

b) Subtraction

**c) Multiplication**

d) Division

**Answer: c**

**Explanation: Convolution is the combination of addition and multiplication that is between the same signal or the different signals and convolution in time domain is always multiplication in z domain.**

**10. Region of convergence of a causal LTI system**

a) Is the entire s-plane

**b) Is the right-half of s-plane**

c) Is the left-half of s-plane

d) Does not exist

**Answer: b**

**Explanation: Causal system refers to the system that is only defined for the positive time system and for positive values and therefore region of convergence of a causal LTI system is right half of s-plane.**

**Stability Analysis**

**1. First column elements of the Routh’s tabulation are 3, 5, -3/4, ½, 2. It means that there are:**

a) Is one root in the left half of s-plane

b) Are two roots in the left half of s-plane**c) Are two roots in the right half of the s-plane**

d) Is one root in the right half of s-plane**Answer: c****Explanation: Routh hurwitz criteria is used to find the stability of the system and this is determined by the number of roots in which the number of roots is equal to the number of sign changes.****2. Assertion (A): Feedback control system offer more accurate control over open-loop systems.****Reason (R): The feedback path establishes a link for input and output comparison and subsequent error correction.****a) Both A and R are true and R is correct explanation of A**

b) Both A and R are true and R is not correct Explanation of A

c) A is True and R is false

d) A is False and R is true**Answer: a****Explanation: Feedback control system offers more accuracy and also reduces the gain of the system and establishes the link for input and output comparison and subsequent error correction.****3. Consider the following statements:**

a) The effect of feedback is to reduce the system error

b) Feedback increases the gain of the system in one frequency range but decreases in the other

c) Feedback can cause a system originally stable to become unstable**d) Both a and c****Answer: d****Explanation: Feedback reduces error and can cause stable system to become unstable and also can make unstable system stable.****4. The Routh-Hurwitz criterion cannot be applied when the characteristic equation of the system contains any coefficients which is :**

a) Negative real and exponential function**b) Negative real, both exponential and sinusoidal function of s**

c) Both exponential and sinusoidal function of s

d) Complex, both exponential and sinusoidal function of s**Answer: b****Explanation: The Routh-Hurwitz criterion cannot be applied when the characteristic equation of the system contains any coefficients which is negative real, both exponential and sinusoidal function of s.****5. The following characteristic equation results in stable operation of the feedback system s ^{3}+4s^{2}+10s+11=0**

**a) True**

b) False

**Answer: a**

**Explanation: Stable operation can be checked using the Routh-Hurwitz criterion where the first row of the the table is checked and with that.**

**6. Consider the following statements:**

**Routh-Hurwitz criterion gives:**

**1. Absolute stability**

**2. The number of roots lying on the right half of the s-plane**

**3. The gain margin and the phase margin**

a) 1,2 and3

**b) 1 and 2**

c) 2 and 3

d) 1 and 3

**Answer: b**

**Explanation: Routh-Hurwitz criterion gives absolute stability and number of roots lying on the right half of the s-plane.**

**7. The given characteristic equation s**

^{4}+s^{3}+2s^{2}+2s+3=0 has:**a) Zero root in the s-plane**

b) One root in the RHS of s-plane

**c) Two root in the RHS of s-plane**

d) Three root in the RHS of s-plane

**Answer: c**

**Explanation: The stability analysis is done using Routh-Hurwitz criterion and hence the number of roots on the right is calculated.**

**8. Which of the following techniques is utilized to determine at the actual point at which the root locus crosses the imaginary axis?**

a) Nyquist technique

**b) Routh-Hurwitz technique**

c) Nichol’s technique

d) Bode technique

**Answer: b**

**Explanation: Routh-Hurwitz technique is utilized to determine at the actual point at which the root locus crosses the imaginary axis.**

**9. The characteristic equation of a control system is given by s**

^{6}+2s^{5}+8s^{4}+12s^{3}+20s^{2}+16s+16=0 . The number of the roots of the equation which lie on the imaginary axis of s-plane:a) 0

b) 2

**c) 4**

d) 6

**Answer: c**

**Explanation: The stability analysis is done using Routh-Hurwitz criterion and hence the number of roots on the right is calculated.**

**10. Assertion (A): A linear, negative feedback control system is invariable stable if its open loop configuration is stable**

Reason (R): the negative feedback reduces the overall gain of the feedback.

a) Both A and R are true and R is correct explanation of A

b) Both A and R are true and R is not correct Explanation of A

c) A is True and R is false

**d) A is False and R is true**

**Answer: d**

**Explanation: A linear, negative feedback control system is not necessarily stable if its open loop configuration is stable.**

**Compensation Techniques**

**1. Assertion (A): The closed loop stability can be determined from the poles of an open loop system and the polar plot of the frequency response.**

Reason (R): Unstable system has right half poles.

a) Both A and R are true and R is correct explanation of A**b) Both A and R are true and R is not correct Explanation of A**

c) A is True and R is false

d) A is False and R is true**Answer: b****Explanation: Closed loop system can be made stable if the poles of the closed loop system are all lying on the left half of the plane and this can be determined from the poles of an open loop system.****2. The open loop control transfer function of a unity feedback system is given by :****G(s) =K/(s+2)(s+4)(s^2+6s+25)****Which is the value of K which causes sustained oscillations in the closed loop system?****a) 590**

b) 790

c) 990

d) 1190**Answer: a****Explanation: The value of the sustained oscillations is calculated from the Rout-Hurwitz table by equation the quadratic equation=0.****3. The characteristic equation of the control system is s ^{5}+15s^{4}+85s^{3}+225s^{2}+274s+120=0 . What are the number of roots of the equation which lie to the left of the line s+1 = 0?**

a) 2

b) 3

**c) 4**

d) 5

**Answer: c**

**Explanation: The solution is obtained with the help of the Routh-Hurwitz table and in this the solution is obtained by equating the value as s-1 in the given equation.**

**4. The characteristic equation of a system is 2s**

^{5}+s^{4}+4s^{3}+2s^{2}+2s+1=0 . Which one of the following is correct?a) Stable

b) Marginally stable

**c) Unstable**

d) Oscillatory

**Answer: c**

**Explanation: The solution is obtained by using the Routh-Hurwitz table in which with the conventional method the number of sign changes are seen in the first row of the table and number of sign changes is equal to the number of the roots on the right half of s plane.**

**5. The characteristic equation of a control system is given by s**

^{5}+s^{4}+2s^{3}+2s^{2}+4s+6=0. The number of the roots of the equation which lie on the right half of s-plane:a) 0

b) 1

**c) 2**

d) 3

**Answer: c**

**Explanation: The solution is obtained by using the Routh-Hurwitz table in which with the conventional method the number of sign changes are seen in the first row of the table and number of sign changes is equal to the number of the roots on the right half of s plane.**

**6. Which of the following may result in instability problem?**

a) Large error

b) High selectivity

**c) High gain**

d) Noise

**Answer: c**

**Explanation: High gain results in instability problem and this is due to the low damping factor and damping factor is inversely proportional to the gain of the system.**

**7. For what values of K does the polynomial s**

^{4}+8s^{3}+24s^{2}+32s+K=0 have roots with zero real parts?a) 10

b) 20

c) 40

**d) 80**

**Answer: d**

**Explanation: The solution is obtained by using the Routh-Hurwitz table and in which with the conventional method the number of sign changes are seen in the first row of the table and number of sign changes is equal to the number of the roots on the right half of s plane.**

**8. How many roots with positive real parts do the equation s**

^{3}+s^{2}-s+1=0 have?a) 0

b) 1

**c) 2**

d) 3

**Answer: c**

**Explanation: The roots with positive real parts are calculated with the help of the routh-hurwitz table and since the sign changes two times in first column therefore have two roots have positive parts.**

**9. Phase margin is always positive for stable feedback system?**

**a) True**

b) False

**Answer: a**

**Explanation: Phase margin is calculated at the gain cross over frequency at which the gain is 1 and this is calculated for stability and is always positive for stable feedback system.**

**10. In closed loop control system, what is the sensitivity of the gain of the overall system, M to the variation in G?**

**a) 1/1+GH**

b) 1/1+G

c) G/1GH

d) G/1+G

**Answer: a**

**Explanation: The sensitivity of the control system is defined as the change in the output to the change in the input and sensitivity of the gain of the overall system, M to the variation in G is 1/1+GH.**