**Competitive ****Stability and Algebraic Criteria MCQs ( Control Systems ) MCQs – ****Control Systems ****MCQs**

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**Control Systems MCQs – Stability and Algebraic Criteria MCQs ( Control Systems ) MCQs**

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# Concept of Stability

**1. Stability of a system implies that :**

a) Small changes in the system input does not result in large change in system output

b) Small changes in the system parameters does not result in large change in system output

c) Small changes in the initial conditions does not result in large change in system output**d) All of the above mentioned****Answer: d****Explanation: Stability of the system implies that small changes in the system input, initial conditions, and system parameters does not result in large change in system output.****2. A linear time invariant system is stable if :**

a) System in excited by the bounded input, the output is also bounded

b) In the absence of input output tends zero**c) Both a and b**

d) System in excited by the bounded input, the output is not bounded**Answer: c****Explanation: A system is stable only if it is BIBO stable and asymptotic stable.****3. Asymptotic stability is concerned with:**

a) A system under influence of input**b) A system not under influence of input**

c) A system under influence of output

d) A system not under influence of output**Answer: b****Explanation: Asymptotic stability concerns a free system relative to its transient behavior.****4. Bounded input and Bounded output stability notion concerns with :****a) A system under influence of input**

b) A system not under influence of input

c) A system under influence of output

d) A system not under influence of output**Answer: a****Explanation: BIBO stability concerns with the system that has input present.****5. If a system is given unbounded input then the system is:**

a) Stable

b) Unstable**c) Not defined**

d) Linear**Answer: c****Explanation: If the system is given with the unbounded input then nothing can be clarified for the stability of the system.****6. Linear mathematical model applies to :**

a) Linear systems**b) Stable systems**

c) Unstable systems

d) Non-linear systems**Answer: b****Explanation: As the output exceeds certain magnitude then the linear mathematical model no longer applies.****7. For non-linear systems stability cannot be determined due to:**

a) Possible existence of multiple equilibrium states

b) No correspondence between bounded input and bounded output stability and asymptotic stability

c) Output may be bounded for the particular bounded input but may not be bounded for the bounded inputs**d) All of the mentioned****Answer: d****Explanation: For non-linear systems stability cannot be determined as asymptotic stability and BIBO stability concepts cannot be applied, existence of multiple states and unbounded output for many bounded inputs.****8. If the impulse response in absolutely integrable then the system is :****a) Absolutely stable**

b) Unstable

c) Linear

d) Stable**Answer: a****Explanation: The impulse response must be absolutely integrable for the system to absolutely stable.****9. The roots of the transfer function do not have any effect on the stability of the system.**

a) True**b) False****Answer: b****Explanation: The roots of transfer function also determine the stability of system as they may be real, complex and may have multiplicity of various order.****10. Roots with higher multiplicity on the imaginary axis makes the system :**

a) Absolutely stable**b) Unstable**

c) Linear

d) Stable**Answer: b****Explanation: Repetitive roots on the imaginary axis makes the system unstable.****11. Roots on the imaginary axis makes the system :**

a) Stable

b) Unstable**c) Marginally stable**

d) Linear**Answer: c****Explanation: Roots on the imaginary axis makes the system marginally stable.****12. If the roots of the have negative real parts then the response is ____________**

a) Stable

b) Unstable

c) Marginally stable**d) Bounded****Answer: d****Explanation: If the roots of the have negative real parts then the response is bounded and eventually decreases to zero.****13. If root of the characteristic equation has positive real part the system is :**

a) Stable**b) Unstable**

c) Marginally stable

d) Linear**Answer: b****Explanation: The impulse response of the system is infinite when the roots of the characteristic equation has positive real part.****14. A linear system can be classified as :**

a) Absolutely stable

b) Conditionally stable

c) Unstable**d) All of the mentioned****Answer: d****Explanation: A system can be stable, unstable and conditionally stable also.****15. ___________ is a quantitative measure of how fast the transients die out in the system.**

a) Absolutely stable

b) Conditionally stable

c) Unstable**d) Relative Stability****Answer: d****Explanation: Relative Stability may be measured by relative settling times of each root or pair of roots.**

**Necessary Conditions for Stability and Non-Linear Systems**

**1. The techniques of linear system can be used in the non-linear system entirely:****a) True**

b) False**Answer: a****Explanation: The techniques of the linear system cannot be entirely used in the non-linear system as they are differentiated by this way only.****2. The disadvantages of the linear system are:**

a) The constraints on the linear operation over wide range demands unnecessarily high quality.

b) The restriction to the linear theory may inhibit the designer’s curiosity to deliberately introduce the non-linear components.

c) Practically systems are non-linear**d) All of the mentioned****Answer: d****Explanation: Linear system impose certain restrictions as the components cost is very high and it will cause restriction to operate the otherwise linear components in non-linear region with a view to improve system response.**

Time Response Analysis, Design Specifications And Performance Indices MCQs

**3. System non-linearities are taken account by:**

a) Analytical

b) Graphical and numerical techniques**c) Both a and b**

d) None of the mentioned**Answer: c****Explanation: Systems non-linearities are taken into account by the analytical, graphical and numerical techniques.****4. The superposition theorem is :**

a) Homogeneity

b) Additivity**c) Combination of homogeneity and additivity**

d) Applied to non-linear systems**Answer: c****Explanation: Superposition theorem applies to linear system only and it refers to the additivity and homogeneity.****5. The standard test signal can be applied to give output to:****a) Linear systems**

b) Non-linear systems

c) Time variant systems

d) Time invariant systems**Answer: a****Explanation: For linear systems the standard test signals can be applied to give the desired output.****6. The amplitude of the standard test signal does not matter in linear systems:****a) True**

b) False**Answer: a****Explanation: The amplitude of the standard test signal is unimportant since any change in input signal amplitude results simply change in response scale with no change in the basic response characteristics.****7. The non-linear systems:**

a) Do not obey superposition theorem

b) May be highly sensitive to the input amplitude

c) Laplace and z transform are not applicable to the non-linear systems**d) All of the mentioned****Answer: d****Explanation: The non-linear systems do not obey superposition theorem and also may be highly sensitive to the input impedance and Laplace and z transform are only applicable to the linear systems.****8. The stability of the linear system:****a) Determined by the location of the poles**

b) Dependent entirely of whether or the system is driven

c) The stability of the undriven linear system is dependent on the magnitude of the final initial state.

d) Stability cannot be determined by the open loop poles**Answer: a****Explanation: Linear system’s stability can be determined by the location of poles and also it is independent entirely of whether or the system is driven and the stability of the undriven linear system is independent on the magnitude of the final initial state.****9. In non-linear system stability is :**

a) Dependent on the input

b) Independent on initial state

c) Independent on input**d) Dependent on input and initial state.****Answer: d****Explanation: In non-linear system the stability is dependent on the input and initial states.****10. Non-linear elements may exhibit___________**

a) Linear systems

b) Non-linear systems**c) Limit cycles**

d) Time invariant systems**Answer: c****Explanation: Non-linear elements may exhibit the limit cycles which are self-sustained oscillations of fixed frequency and amplitude. Determination of existence of limit cycles is not an easy task as these may depend upon both the type and amplitude of the excitation signal.****11. The necessary condition of stability are:**

a) Coefficient of characteristic equation must be real and have the same sign

b) Coefficient of characteristic equation must be non-zero**c) Both of the mentioned**

d) Coefficient of characteristic equation must be zero**Answer: c****Explanation: The necessary condition of stability are coefficient of characteristic equation must be real, non-zero and have the same sign.****12. None of the coefficients can be zero or negative unless one of the following occurs:**

a) One or more roots have positive real parts

b) A root at origin

c) Presence of root at the imaginary axis**d) All of the mentioned****Answer: d****Explanation: None of the coefficients can be zero or negative unless one or more roots have positive real parts, root at origin and presence of root at the imaginary axis.****13. The __________ of the coefficients of characteristic equation is necessary as well as sufficient condition for the stability of system of first and second order.**

a) Negativeness**b) Positiveness**

c) Positiveness and Negativeness

d) None of the mentioned**Answer: b****Explanation: The Positiveness of the coefficients of characteristic equation is necessary as well as sufficient condition for the stability of system of first and second order.****14. The Positiveness of the coefficients of characteristic equation is necessary as well as sufficient condition for:**

a) First order system

b) Second order system**c) Third order system**

d) None of the mentioned**Answer: c****Explanation: It does not ensure the negativeness of the real parts of the complex roots of the third or higher order systems.****15. Assertion (A): Routh criterion is in terms of array formulation, which is more convenient to handle.****Reason (R): This method is used to investigate the method of stability of higher order systems.**

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 but R is false

d) A is False but R is true**Answer: b****Explanation: Routh criterion is in terms of array formulation which is convenient to handle stability problems of higher order systems.**

**Routh-Hurwitz Stability Criterion**

**1. Routh Hurwitz criterion gives:****a) Number of roots in the right half of the s-plane**

b) Value of the roots

c) Number of roots in the left half of the s-plane

d) Number of roots in the top half of the s-plane**Answer: a****Explanation: Routh Hurwitz criterion gives number of roots in the right half of the s-plane.****2. Routh Hurwitz criterion cannot be applied when the characteristic equation of the system containing coefficient’s which is/are**

a) Exponential function of s

b) Sinusoidal function of s

c) Complex**d) Exponential and sinusoidal function of s and complex****Answer: d****Explanation: Routh Hurwitz criterion cannot be applied when the characteristic equation of the system containing coefficient/s which is/are exponential, sinusoidal and complex function of s.****3. Consider the following statement regarding Routh Hurwitz criterion:**

a) It gives absolute stability

b) It gives gain and phase margin

c) It gives the number of roots lying in RHS of the s-plane**d) It gives gain, phase margin and number of roots lying in RHS of the s-plane****Answer: d****Explanation: Routh Hurwitz gives the absolute stability and roots on the right of the s plane.****4. The order of the auxiliary polynomial is always:****a) Even**

b) Odd

c) May be even or odd

d) None of the mentioned**Answer: a****Explanation: Auxiliary polynomial denotes the derivative of the odd equation which is always even.****5. Which of the test signals are best utilized by the stability analysis.****a) Impulse**

b) Step

c) Ramp

d) Parabolic**Answer: a****Explanation: Computational task is reduced to much extent.****6. The characteristic equation of a system is given as3s4+10s3+5s2+2=0. This system is :**

a) Stable

b) Marginally stable**c) Unstable**

d) Linear**Answer: c****Explanation: There is a missing coefficient so the system is unstable.****7. The characteristic equation of a system is given ass3+25s2+10s+50=0. What is the number of the roots in the right half s-plane and the imaginary axis respectively?**

a) 1,1**b) 0,0**

c) 2,1

d) 1,2**Answer: b****Explanation: The characteristic equation has no sign changes so number of roots on the right half of s plane is zero.****8. Consider the following statement:****a) A system is said to be stable if its output is bounded for any input**

b) A system is said to be stable if all the roots of the characteristic equation lie on the left half of the s plane.

c) A system is said to be stable if all the roots of the characteristic equation have negative real parts.

d) A second order system is always stable for finite values of open loop gain**Answer: a****Explanation: A system is stable if its output is bounded for bounded input.****9. The necessary condition for the stability of the linear system is that all the coefficients of characteristic equation 1+G(s)H(s) =0, be real and have the :**

a) Positive sign

b) Negative sign**c) Same sign**

d) Both positive and negative**Answer: c****Explanation: The necessary condition for the stability of the linear system is that all the coefficients of characteristic equation 1+G(s)H(s) =0, is they must have same sign.****10. For making an unstable system stable:**

a) Gain of the system should be increased**b) Gain of the system should be decreased**

c) The number of zeroes to the loop transfer function should be increased

d) The number of poles to the loop transfer function should be increased**Answer: b****Explanation: For making an unstable system stable gain of the system should be decreased.**

**Relative Stability Analysis**

**1. A system with unity feedback having open loop transfer function as G(s) = K(s+1)/s3+as2+2s+1. What values of ‘K’ and ’a’ should be chosen so that the system oscillates ?**

a) K =2, a =1**b) K =2, a =0.75**

c) K =4, a =1

d) K =4, a =0.75**Answer: b****Explanation: Solving Routh Hurwitz table whenever row of zero occurs, the roots are located symmetrically on the imaginary axis then the system response oscillates, a =1+K/2+K. If K =2 is consider then a =0.75.****2. The open loop transfer functions with unity feedback are given below for different systems.****Among these systems the unstable system is**

a) G(s) =2/s+2

b) G(s) =2/s(s+2)**c) G(s) =2/(s+2)s ^{^}2**

d) G(s) =2(s+1)/s(s+2)

**Answer: c**

**Explanation: 1+2/s**

^{^}2(s+2) =0. The coefficient of‘s’ is missing. Hence the system is unstable.**3. Determine the stability of closed loop control system whose characteristic equation is**

**s**

^{5}+s^{4}+2s^{3}+2s^{2}+11s+10=0.a) Stable

**b) Marginally stable**

c) Unstable

d) None of the mentioned

**Answer: b**

**Explanation: By Routh array s =0 and s =+j. It is having a pair of conjugate root lying on imaginary axis. System is marginally stable.**

**4. Determine the condition for the stability of unity feedback control system whose open loop transfer function is given by**

**G(s) = 2e**

^{-st}/s(s+2)a) T >1

b) T <0

**c) T <1**

d) T >0

**Answer: c**

**Explanation: G(s) =2(1-sT)/s(s+2)**

**By Routh array analysis, for stable system, all the elements of first column need to be positive T<1.**

**5.Determine the value of K such that roots of characteristic equation given below lies to the left of the line s = -1. s**

^{3}+10s^{2}+18s+K.a) K>16 and K<9

b) K<16

**c) 9<K<16**

d) K<9

**Answer: c**

**Explanation: In Routh array analysis the first column must be positive and after solving K<16 and K>9.**

**6. Consider a negative feedback system where G(s) =1/(s+1) and H(s) =K/s(s+2). The closed loop system is stable for**

a) K>6

b) 0<K<2

c) 8<K<14

d) 0<K<6

**Answer: d**

**Explanation: Using Routh array, for stability k<6.**

**7. The characteristic equation of a feedback control system is s**

^{3}+Ks^{2}+9s+18. When the system is marginally stable, the frequency of the sustained oscillation:a) 1

b) 1.414

c) 1.732

**d) 3**

**Answer: d**

**Explanation: Solve using Routh array and for the system to be marginally stable, K = -2. Polynomial for sustained oscillation w = 3 rad/s.**

**8. Consider a characteristic equation, s**

^{4}+3s^{3}+5s^{2}+6s+k+10=0. The condition for stability isa) K>5

b) -10<K

c) K>-4

**d) -10<K<-4**

**Answer: d**

**Explanation: Solve Roth array for the system stable, -10<K<4.**

**9. The polynomial s**

^{4}+Ks^{3}+s^{2}+s+1=0 the range of K for stability is _____________a) K>5

b) -10<K

c) K>-4

**d) K-1>0**

**Answer: d**

**Explanation: Solving using Routh array we get K-1>0 and is always negative for K>1.**

**10. The characteristic equation of a system is given by3s**

^{4}+10s^{3}+5s^{2}+2=0. This system is:a) Stable

b) Marginally stable

**c) Unstable**

d) Linear

**Answer: c**

**Explanation: There is missing coefficient so system is unstable.**