Stability margin decreases when

 Q411. The state-space model represents the system in terms of:

A) Inputs and outputs only
B) Inputs, outputs, and states
C) Transfer functions
D) Laplace transforms

✅ Answer: B) Inputs, outputs, and states
Explanation:
State-space representation gives a time-domain model using first-order differential equations involving state variables.

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Q412. The state equation is generally written as:
A) $\dot{x} = A x + B u$
B) $y = Cx + Du$
C) $G(s) = \frac{Y(s)}{U(s)}$
D) $T(s) = \frac{1}{1 + G(s)}$

✅ Answer: A) $\dot{x} = A x + B u$
Explanation:
This is the state equation, describing system dynamics. $y = Cx + Du$ is the output equation.

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Q413. The controllability of a system ensures that:
A) All states can be observed
B) All states can be driven to any desired value using input
C) The output can be measured directly
D) System has stable poles

✅ Answer: B)
Explanation:
A system is controllable if input can move the system from any initial to any final state in finite time.

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Q414. The observability of a system ensures that:
A) The state can be controlled
B) The output can be made zero
C) The internal states can be determined from the output
D) System is stable

✅ Answer: C)
Explanation:
Observability means all internal states can be reconstructed by observing the system’s output and input.

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Q415. A system described by $\dot{x} = A x + B u$ is controllable if:
A) det[B AB A²B … Aⁿ⁻¹B] ≠ 0
B) det[A] ≠ 0
C) det[B] = 0
D) trace(A) = 0

✅ Answer: A)
Explanation:
This is the Kalman controllability condition — the controllability matrix must have full rank.

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Q416. In a state-space model, the matrix $A$ is known as:
A) Input matrix
B) System matrix
C) Output matrix
D) Feedthrough matrix

✅ Answer: B)
Explanation:
Matrix $A$ defines internal system dynamics → called system or state matrix.

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Q417. The characteristic equation of a system is given by:
A) det(sI − A) = 0
B) det(A − sI) = 0
C) det(sI + A) = 0
D) |B| = 0

✅ Answer: A)
Explanation:
Poles of the system are roots of the characteristic equation det(sI − A) = 0.

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Q418. The eigenvalues of matrix $A$ represent:
A) System zeros
B) System poles
C) Gain constants
D) Damping ratios

✅ Answer: B)
Explanation:
Eigenvalues of $A$ are the poles of the system in the state-space model.

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Q419. Stability in state-space is determined by:
A) Sign of eigenvalues of $A$
B) Determinant of $A$
C) Rank of $A$
D) Trace of $A$

✅ Answer: A)
Explanation:
If all eigenvalues of $A$ have negative real parts → the system is asymptotically stable.

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Q420. The transfer function from state-space model is:
A) $C(sI − A)^{-1}B + D$
B) $(sI − A)^{-1}B$
C) $G(s) = \frac{Y(s)}{U(s)} = C(sI − A)B$
D) $CB(sI − A)$

✅ Answer: A)
Explanation:
The standard relation between state-space and transfer function:
$G(s) = C(sI − A)^{-1}B + D$.

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Q421. In state-space analysis, the term “state” refers to:
A) Initial input
B) Memory of the system
C) Output signal
D) Transfer function

✅ Answer: B)
Explanation:
States store past information that influences the system’s future response.

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Q422. A linear time-invariant (LTI) system satisfies:
A) Superposition and scaling
B) Only causality
C) Non-linearity
D) Time-variance

✅ Answer: A)
Explanation:
LTI systems obey the principles of superposition and homogeneity (scaling).

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Q423. For a discrete system, stability requires all poles to:
A) Be inside the unit circle
B) Be on imaginary axis
C) Be outside unit circle
D) Have positive real parts

✅ Answer: A)
Explanation:
For discrete-time systems, poles inside unit circle → stable response.

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Q424. A lead compensator adds:
A) Negative phase
B) Positive phase
C) Zero phase
D) Constant delay

✅ Answer: B)
Explanation:
Lead compensator advances the phase → improves phase margin and speed of response.

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Q425. Lag compensator improves:
A) Steady-state accuracy
B) Phase margin
C) Speed
D) Overshoot

✅ Answer: A)
Explanation:
Lag adds low-frequency gain → improves steady-state accuracy but slows response.

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Q426. A PID controller combines:
A) Speed, stability, accuracy
B) Only damping
C) Only delay compensation
D) Only gain increase

✅ Answer: A)
Explanation:
Proportional → speed,
Integral → accuracy,
Derivative → damping/stability.

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Q427. Which control mode eliminates steady-state error?
A) Proportional
B) Integral
C) Derivative
D) None

✅ Answer: B)
Explanation:
Integral action forces the steady-state error to zero by accumulating error over time.

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Q428. Nyquist stability criterion is based on:
A) Open-loop frequency response
B) Closed-loop step response
C) Bode phase plot
D) Root locus

✅ Answer: A)
Explanation:
Nyquist uses open-loop frequency response to infer closed-loop stability.

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Q429. The number of encirclements in Nyquist plot equals:
A) Number of zeros
B) Poles + zeros
C) Difference between right-half-plane poles and zeros
D) Gain margin

✅ Answer: C)
Explanation:
Nyquist criterion: $N = Z − P$,
where $N$ = encirclements, $Z$ = RHP zeros, $P$ = RHP poles.

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Q430. The Bode magnitude plot slope for a single pole is:
A) -20 dB/decade
B) -40 dB/decade
C) +20 dB/decade
D) 0

✅ Answer: A)
Explanation:
Each pole adds -20 dB/decade to the slope.

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Q431. Gain margin is defined as:
A) Amount of gain increase before instability
B) Frequency of oscillation
C) Output/input ratio
D) Damping ratio

✅ Answer: A)
Explanation:
Gain margin → safety margin before oscillation; measured at phase = -180°.

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Q432. Phase margin indicates:
A) How much phase can decrease before instability
B) Gain increase margin
C) Steady-state error
D) Speed

✅ Answer: A)
Explanation:
It measures how close the system is to instability in phase.

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Q433. The root locus branches end at:
A) Zeros
B) Infinity
C) Both A and B
D) None

✅ Answer: C)
Explanation:
Each branch begins at poles and terminates at finite zeros or infinity.

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Q434. The centroid of asymptotes is given by:
A) (Sum of poles – Sum of zeros)/Number of asymptotes
B) (Sum of zeros – Sum of poles)/Number of poles
C) Sum of poles/zeros
D) None

✅ Answer: A)
Explanation:
Centroid = (Σpoles − Σzeros) / (n − m), where n = poles, m = zeros.

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Q435. A second-order system with ζ = 1 is:
A) Overdamped
B) Underdamped
C) Critically damped
D) Oscillatory

✅ Answer: C)
Explanation:
ζ = 1 ⇒ critical damping — fastest non-oscillatory response.

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Q436. Steady-state error constant for unit ramp in Type-1 system is:
A) $1/K_v$
B) $1/K_p$
C) $1/K_a$
D) Zero

✅ Answer: A)
Explanation:
Velocity error constant $K_v$ determines ramp steady-state error.

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Q437. If damping ratio decreases:
A) Overshoot increases
B) Settling time decreases
C) System becomes slower
D) Stability increases

✅ Answer: A)
Explanation:
Low damping → larger overshoot and oscillation.

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Q438. State-space approach is mainly used for:
A) Linear time-invariant systems
B) Nonlinear, multivariable systems
C) Frequency-domain analysis
D) Step-response only

✅ Answer: B)
Explanation:
State-space can easily handle multi-input, multi-output (MIMO) and nonlinear systems.

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Q439. The unit impulse response of a system is the:
A) Derivative of step response
B) Integral of step response
C) Product of input and output
D) Frequency response

✅ Answer: A)
Explanation:
Impulse response is the derivative of step response.

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Q440. Stability margin decreases when:
A) Gain increases
B) Damping increases
C) Bandwidth decreases
D) Phase margin increases

✅ Answer: A)
Explanation:
Higher gain → poles shift toward imaginary axis → reduced stability margin.