Q.1 In state‑space representation, which matrix relates the input vector to the state derivative?
A matrix
B matrix
C matrix
D matrix
Explanation - The B matrix (input matrix) multiplies the input vector u(t) to produce its contribution to the state derivative: \(\dot{x}=Ax+Bu\).
Correct answer is: B matrix
Q.2 For a continuous‑time linear system \(\dot{x}=Ax+Bu\), the system is asymptotically stable if:
All eigenvalues of A have negative real parts
All eigenvalues of A are real and positive
The determinant of A is zero
The rank of B equals the number of states
Explanation - A continuous‑time system is asymptotically stable when every eigenvalue λ of A satisfies Re(λ) < 0.
Correct answer is: All eigenvalues of A have negative real parts
Q.3 The controllability matrix of a system with state matrix A and input matrix B is given by:
[B AB A^2B … A^{n-1}B]
[C CA CA^2 … CA^{n-1}]
[A B C D]
[B^T A^T B^T A^T]
Explanation - The controllability matrix \(\mathcal{C}= [B\; AB\; A^2B\; \dots\; A^{n-1}B]\) tests whether the state can be driven anywhere using the input.
Correct answer is: [B AB A^2B … A^{n-1}B]
Q.4 If the controllability matrix of a system is rank deficient, what does that imply?
The system is not completely controllable
The system is not observable
The system is unstable
The system has no poles
Explanation - A rank‑deficient controllability matrix means some states cannot be influenced by the input; the system is not fully controllable.
Correct answer is: The system is not completely controllable
Q.5 Which of the following statements about the observability matrix is true?
It is constructed as [C^T (CA)^T … (CA^{n-1})^T]^T
It is identical to the controllability matrix
It depends on the B matrix
It determines the pole locations directly
Explanation - The observability matrix \(\mathcal{O}=\begin{bmatrix} C \; CA \; CA^2 \; \dots \; CA^{n-1} \end{bmatrix}\) tests if the states can be reconstructed from the output.
Correct answer is: It is constructed as [C^T (CA)^T … (CA^{n-1})^T]^T
Q.6 A system with state matrix A has eigenvalues at -2, -5, and 3. What can be said about its stability?
It is asymptotically stable
It is marginally stable
It is unstable
Stability cannot be determined without B and C
Explanation - The presence of an eigenvalue with positive real part (3) makes the system unstable.
Correct answer is: It is unstable
Q.7 In discrete‑time state‑space form \(x[k+1]=Ax[k]+Bu[k]\), stability requires:
All eigenvalues of A lie inside the unit circle
All eigenvalues of A have magnitude greater than 1
det(A) = 0
B must be full rank
Explanation - A discrete‑time linear system is asymptotically stable if every eigenvalue λ of A satisfies |λ| < 1.
Correct answer is: All eigenvalues of A lie inside the unit circle
Q.8 Pole placement by state feedback \(u = -Kx\) modifies which matrix?
A matrix
B matrix
C matrix
D matrix
Explanation - With state feedback, the closed‑loop A matrix becomes \(A_{cl}=A-BK\). The poles (eigenvalues) are those of \(A_{cl}\).
Correct answer is: A matrix
Q.9 The Linear‑Quadratic Regulator (LQR) cost function typically includes terms:
x^TQx + u^TRu
x^TCx + u^TDu
Ax + Bu
det(Q) + det(R)
Explanation - LQR minimizes the integral of \(J=\int (x^T Q x + u^T R u) dt\), where Q ≥ 0 and R > 0 weight states and inputs.
Correct answer is: x^TQx + u^TRu
Q.10 For a single‑input system, the controllability index equals:
The smallest integer ν such that rank([B AB … A^{ν-1}B]) = n
The number of outputs
The dimension of B
The number of poles
Explanation - The controllability index ν is the minimum number of columns needed from the controllability matrix to achieve full rank n.
Correct answer is: The smallest integer ν such that rank([B AB … A^{ν-1}B]) = n
Q.11 Which transformation keeps the input‑output behavior unchanged while altering the state representation?
Similarity transformation
Fourier transform
Laplace transform
Z‑transform
Explanation - A similarity transformation \(x = Tz\) with nonsingular T yields new matrices \(A' = T^{-1}AT, B' = T^{-1}B, C' = C T\) that describe the same I/O behavior.
Correct answer is: Similarity transformation
Q.12 If a system is both controllable and observable, which canonical form can be used for controller design?
Controllable canonical form
Observable canonical form
Both A and B
Neither, a different form is needed
Explanation - A completely controllable and observable system can be represented in either controllable or observable canonical form; both are valid for design.
Correct answer is: Both A and B
Q.13 The transfer function of a state‑space system \(\dot{x}=Ax+Bu,\; y=Cx+Du\) is:
C(sI-A)^{-1}B + D
B(sI-C)^{-1}A + D
A(sI-B)^{-1}C + D
D(sI-A)^{-1}C + B
Explanation - Applying Laplace transform and eliminating states gives \(G(s)=C(sI-A)^{-1}B + D\).
Correct answer is: C(sI-A)^{-1}B + D
Q.14 A system with state matrix \(A = \begin{bmatrix}0 & 1\\-2 & -3\end{bmatrix}\) has natural frequency (ω_n) of:
√2 rad/s
√3 rad/s
2 rad/s
3 rad/s
Explanation - The characteristic equation is \(s^2+3s+2=0\). Roots: -1 and -2 → ζ = (3)/(2√2), ω_n = √2 rad/s.
Correct answer is: √2 rad/s
Q.15 Which condition guarantees that a continuous‑time system can be stabilized by state feedback?
The system is controllable
The system is observable
The system has distinct poles
The D matrix is zero
Explanation - If the pair (A,B) is controllable, any set of stable poles can be assigned using state feedback.
Correct answer is: The system is controllable
Q.16 The eigenvalues of the closed‑loop matrix \(A-BK\) are called:
Closed‑loop poles
Open‑loop zeros
System zeros
Observer poles
Explanation - State feedback moves the poles of the system; the eigenvalues of \(A-BK\) are the closed‑loop poles.
Correct answer is: Closed‑loop poles
Q.17 In observer design, the error dynamics matrix is:
A-LC
A+LC
A-BK
A+BK
Explanation - The observer error \(e = x - \hat{x}\) evolves as \(\dot{e} = (A-LC)e\).
Correct answer is: A-LC
Q.18 Which of the following is NOT a requirement for a system to be observable?
rank([C; CA; …; CA^{n-1}]) = n
The pair (A,B) is controllable
All states affect the output over time
The observability matrix has full rank
Explanation - Observability depends on (A,C), not on controllability (A,B). The other statements are part of the observability condition.
Correct answer is: The pair (A,B) is controllable
Q.19 A state‑space model with A = [[0,1],[ -4, -5]], B = [[0],[1]], C = [1 0] has a transfer function numerator of:
s
1
s+5
4
Explanation - G(s)=C(sI-A)^{-1}B = [1 0]·(sI-A)^{-1}·[0;1] = s/(s^2+5s+4). Numerator is s.
Correct answer is: s
Q.20 For a SISO system, the minimal realization is the representation with:
Smallest possible state dimension
Largest possible state dimension
Zero D matrix
Diagonal A matrix
Explanation - A minimal realization uses the lowest order (fewest states) while preserving the input‑output behavior.
Correct answer is: Smallest possible state dimension
Q.21 If a system matrix A is nilpotent (A^k = 0 for some k), what are its eigenvalues?
All zero
All one
All negative
All positive
Explanation - A nilpotent matrix has only the eigenvalue 0 (with algebraic multiplicity equal to its size).
Correct answer is: All zero
Q.22 In the context of state‑space, the term "modal decomposition" refers to:
Expressing the system in terms of its eigenvectors
Separating controllable and uncontrollable modes
Transforming to the frequency domain
Applying a Fourier series expansion
Explanation - Modal decomposition rewrites the system using a similarity transformation that diagonalizes A (if possible), revealing modal contributions.
Correct answer is: Expressing the system in terms of its eigenvectors
Q.23 Which matrix equation is solved to compute the LQR gain K?
Riccati equation
Lyapunov equation
Sylvester equation
Eigenvalue equation
Explanation - The continuous‑time algebraic Riccati equation (CARE) yields the optimal state‑feedback gain K = R^{-1}B^TP.
Correct answer is: Riccati equation
Q.24 A system with A = [[0,1],[ -1,0]] represents:
A simple harmonic oscillator
A first‑order lag
An unstable integrator
A dead‑time element
Explanation - The eigenvalues are ±j, giving undamped oscillations – the classic harmonic oscillator.
Correct answer is: A simple harmonic oscillator
Q.25 The Kalman filter is primarily used for:
Optimal state estimation
Pole placement
Root locus plotting
Bode plot generation
Explanation - Kalman filtering combines model predictions with noisy measurements to produce the best linear unbiased estimate of the state.
Correct answer is: Optimal state estimation
Q.26 In discrete‑time LQR, the cost function is summed over:
k = 0 to ∞
t = 0 to ∞
s = -∞ to ∞
Only at k = 0
Explanation - Discrete‑time LQR minimizes J = Σ_{k=0}^{∞} (x_k^T Q x_k + u_k^T R u_k).
Correct answer is: k = 0 to ∞
Q.27 For a second‑order system, the damping ratio ζ is defined as:
ζ = -Re(λ)/|λ|
ζ = Im(λ)/|λ|
ζ = |λ|/Re(λ)
ζ = Re(λ)·Im(λ)
Explanation - For complex conjugate poles λ = -ζω_n ± j ω_n√(1-ζ^2), ζ = -Re(λ)/|λ|.
Correct answer is: ζ = -Re(λ)/|λ|
Q.28 Which property does a controllable canonical form guarantee?
The controllability matrix is the identity
The A matrix is diagonal
The B matrix contains a single 1 in the last row
C matrix is zero
Explanation - In controllable canonical form, B = [0 … 0 1]^T, making the controllability matrix trivially full rank.
Correct answer is: The B matrix contains a single 1 in the last row
Q.29 If a system has a zero at s = -2, which of the following statements is true?
The transfer function numerator becomes zero at s = -2
The system is unstable
The pole at s = -2 is cancelled
The D matrix must be non‑zero
Explanation - A zero means the numerator polynomial equals zero at that s‑value, causing a dip in the frequency response.
Correct answer is: The transfer function numerator becomes zero at s = -2
Q.30 The steady‑state error of a type‑1 system to a ramp input is:
Finite (non‑zero)
Zero
Infinite
Depends on the D matrix
Explanation - A type‑1 system (one integrator) tracks a step with zero error but has a constant finite error for a ramp.
Correct answer is: Finite (non‑zero)
Q.31 Which matrix must be positive definite for the LQR problem to have a unique solution?
R matrix
A matrix
B matrix
C matrix
Explanation - R must be positive definite (R > 0) to guarantee a unique stabilizing solution of the Riccati equation.
Correct answer is: R matrix
Q.32 In a minimal realization, the number of poles equals:
The order of the denominator polynomial
The number of zeros
The number of inputs
The size of D matrix
Explanation - A minimal realization has a state dimension equal to the degree of the denominator, i.e., the number of poles.
Correct answer is: The order of the denominator polynomial
Q.33 The concept of "dual" in control theory refers to the symmetry between:
Controllability and observability
Stability and instability
Poles and zeros
Continuous‑time and discrete‑time
Explanation - Duality states that results for controllability of (A,B) have counterparts for observability of (A^T,C^T).
Correct answer is: Controllability and observability
Q.34 A system with A = [[-2, 0],[0, -3]] and B = [[1],[0]] is:
Fully controllable
Not controllable
Partially controllable
Unobservable
Explanation - The second state (associated with eigenvalue -3) cannot be influenced because the second row of B is zero, so rank(Ctr) = 1 < 2.
Correct answer is: Partially controllable
Q.35 When designing an observer, the eigenvalues of A‑LC are typically placed:
Faster than the closed‑loop poles
At the same locations as the closed‑loop poles
Slower than the closed‑loop poles
Exactly on the imaginary axis
Explanation - Observer poles are chosen more negative (faster) so that the estimation error converges quickly relative to the plant dynamics.
Correct answer is: Faster than the closed‑loop poles
Q.36 A system described by \(\dot{x}=Ax\) (no input) is called:
Autonomous
Linear time‑varying
Non‑linear
Stochastic
Explanation - An autonomous system has no external input; its dynamics are governed solely by the state equation.
Correct answer is: Autonomous
Q.37 If the D matrix in a state‑space model is non‑zero, what does it represent?
Direct feed‑through from input to output
State‑dependent output
Integral action
System delay
Explanation - y = Cx + Du; D maps the input directly to the output without state mediation.
Correct answer is: Direct feed‑through from input to output
Q.38 The matrix exponential \(e^{At}\) is used to compute:
The state transition matrix
The controllability matrix
The observability matrix
The Laplace transform of B
Explanation - Φ(t) = e^{At} describes how the state evolves from an initial condition in the homogeneous solution.
Correct answer is: The state transition matrix
Q.39 For a discrete‑time system, the state transition matrix over one sample period is:
A_d = e^{A T}
A_d = A T
A_d = A^{-1}
A_d = B T
Explanation - Discretizing a continuous‑time system with sample time T gives \(A_d = e^{A T}\) (zero‑order hold assumption).
Correct answer is: A_d = e^{A T}
Q.40 Which of the following statements about the eigenvectors of A is correct?
They form a basis if A is diagonalizable
They are always orthogonal
They are equal to the rows of B
They determine the size of D
Explanation - If A has n linearly independent eigenvectors, they span the state space, allowing diagonalization.
Correct answer is: They form a basis if A is diagonalizable
Q.41 In the context of model reduction, a balanced realization is obtained by:
Equalizing controllability and observability Gramians
Setting A to a diagonal matrix
Zeroing the D matrix
Using a high‑order Taylor series
Explanation - Balanced truncation makes the controllability and observability Gramians equal and diagonal, ordering states by their energy contribution.
Correct answer is: Equalizing controllability and observability Gramians
Q.42 The Lyapunov equation \(A^TP + PA = -Q\) is used to:
Test stability of A
Compute the transfer function
Design a Kalman filter
Place poles via state feedback
Explanation - If a positive definite P exists for a given Q > 0, then A is stable (Lyapunov’s direct method).
Correct answer is: Test stability of A
Q.43 The term "zero‑input response" refers to:
The response due only to initial conditions
The response due only to the input
The steady‑state output
The response when D ≠ 0
Explanation - Zero‑input response (homogeneous solution) is caused by the system's initial state with u(t)=0.
Correct answer is: The response due only to initial conditions
Q.44 A system whose A matrix has a pair of repeated eigenvalues but only one linearly independent eigenvector is:
Defective
Diagonalizable
Controllable
Observable
Explanation - A defective matrix lacks a full set of linearly independent eigenvectors; it cannot be diagonalized.
Correct answer is: Defective
Q.45 In a state‑feedback law \(u = -Kx + r\), the term r is:
Reference input
Disturbance
Measurement noise
Observer gain
Explanation - r is added to the feedback to make the system follow a desired reference signal.
Correct answer is: Reference input
Q.46 The controllable subspace of a system is:
The span of the columns of the controllability matrix
The nullspace of B
The set of eigenvectors of A
The set of outputs reachable from zero input
Explanation - All states reachable from the origin via admissible inputs are linear combinations of the columns of \([B\; AB\; …]\).
Correct answer is: The span of the columns of the controllability matrix
Q.47 If a system has a transfer function with a pole at the origin, it is said to be:
Type‑1
Type‑0
Over‑damped
Non‑minimum phase
Explanation - A pole at s=0 indicates an integrator, defining the system type (number of poles at the origin).
Correct answer is: Type‑1
Q.48 For a continuous‑time system, the state transition matrix satisfies:
dΦ/dt = AΦ, Φ(0)=I
Φ = A^{-1}
Φ = B
Φ = C^T
Explanation - The definition of the state transition matrix Φ(t) is the solution of \(\dot{Φ}=AΦ\) with Φ(0)=I.
Correct answer is: dΦ/dt = AΦ, Φ(0)=I
Q.49 In a state‑space model, the term "strictly proper" means:
D = 0
C = 0
B = 0
A is nilpotent
Explanation - A strictly proper system has no direct feed‑through from input to output, i.e., D matrix is zero.
Correct answer is: D = 0
Q.50 Which of the following is true for a system that is both controllable and observable?
It can be transformed into a minimal realization of order n
It must have a diagonal A matrix
Its B matrix is always invertible
Its D matrix must be identity
Explanation - Full controllability and observability guarantee that a realization of order equal to the system order is minimal.
Correct answer is: It can be transformed into a minimal realization of order n
Q.51 A system with A = [[0, 1],[-2, -3]], B = [[0],[1]], and C = [1 0] has a natural frequency (ω_n) of:
√2 rad/s
√5 rad/s
2 rad/s
3 rad/s
Explanation - Characteristic equation: s^2+3s+2=0 → ω_n = √(2).
Correct answer is: √2 rad/s
Q.52 In the context of pole‑zero cancellation, which of the following is generally undesirable?
Exact cancellation of a pole by a zero
Cancellation of a non‑minimum phase zero
Cancellation of a stable pole
Cancellation of a pole at the origin
Explanation - Cancelling a non‑minimum phase zero (right‑half‑plane) can hide unstable dynamics, leading to hidden instability.
Correct answer is: Cancellation of a non‑minimum phase zero
Q.53 The observability Gramian over [0,∞) exists if:
A is stable
A is controllable
B is full rank
C is zero
Explanation - The integral defining the observability Gramian converges only when the system is asymptotically stable.
Correct answer is: A is stable
Q.54 A minimal realization of a transfer function has which of the following properties?
No uncontrollable or unobservable modes
A diagonal C matrix
Zero D matrix
B is identity
Explanation - Minimality means the realization contains only the necessary dynamic modes; all are both controllable and observable.
Correct answer is: No uncontrollable or unobservable modes
Q.55 When designing a state feedback controller, the desired closed‑loop poles are placed at -4±j3. The required characteristic polynomial is:
s^2 + 8s + 25
s^2 + 8s + 16
s^2 + 6s + 9
s^2 + 4s + 9
Explanation - Roots at -4±j3 give polynomial (s+4−j3)(s+4+j3)=s^2+8s+25.
Correct answer is: s^2 + 8s + 25
Q.56 The term "zero‑state response" refers to:
Response due solely to the input with zero initial conditions
Response due solely to initial conditions
Steady‑state error
Response when D ≠ 0
Explanation - Zero‑state response is the forced response with x(0)=0, i.e., the particular solution.
Correct answer is: Response due solely to the input with zero initial conditions
Q.57 If a system matrix A is Hurwitz, what does that imply?
All eigenvalues have negative real parts
All eigenvalues are positive
A is symmetric
A has full rank
Explanation - A Hurwitz matrix defines a stable continuous‑time system.
Correct answer is: All eigenvalues have negative real parts
Q.58 The rank condition for observability of an n‑state system is:
rank(\mathcal{O}) = n
rank(\mathcal{C}) = n
det(A) ≠ 0
det(B) ≠ 0
Explanation - Observability requires the observability matrix \(\mathcal{O}\) to have full row rank n.
Correct answer is: rank(\mathcal{O}) = n
Q.59 In a discrete‑time LQR problem, the Riccati recursion is performed:
Backward in time
Forward in time
Simultaneously forward and backward
Only at steady‑state
Explanation - The discrete‑time Riccati equation is solved backwards from a terminal condition to obtain the optimal gain sequence.
Correct answer is: Backward in time
Q.60 A state‑space system with B = 0 is:
Uncontrollable
Unobservable
Both controllable and observable
Stable by definition
Explanation - If B = 0, the input cannot affect the states, so the system is not controllable.
Correct answer is: Uncontrollable
Q.61 The term "modal damping ratio" refers to:
Damping associated with each eigenmode
Overall system damping
Damping of the B matrix
Damping of the output only
Explanation - Each eigenvalue (mode) has its own damping ratio; modal damping quantifies energy dissipation per mode.
Correct answer is: Damping associated with each eigenmode
Q.62 Which of the following is a necessary condition for a system to be stabilizable?
All uncontrollable modes are stable
All observable modes are stable
B must be invertible
D must be zero
Explanation - Stabilizability requires that any eigenvalues associated with uncontrollable states lie in the stable region.
Correct answer is: All uncontrollable modes are stable
Q.63 If the controllability Gramian W_c satisfies \(W_c = \int_0^{\infty} e^{A t} B B^T e^{A^T t} dt\), which of the following is true?
W_c is positive definite if (A,B) is controllable
W_c equals the observability Gramian
W_c is always diagonal
W_c does not depend on B
Explanation - A positive definite controllability Gramian indicates full controllability.
Correct answer is: W_c is positive definite if (A,B) is controllable
Q.64 A state‑feedback gain K that places the closed‑loop poles at desired locations can be computed using:
Ackermann’s formula
Fourier series
Bode plot
Nyquist criterion
Explanation - Ackermann’s formula provides a direct method to compute K for single‑input controllable systems.
Correct answer is: Ackermann’s formula
Q.65 In a second‑order system, the quality factor Q is related to the damping ratio ζ by:
Q = 1/(2ζ)
Q = 2ζ
Q = ζ^2
Q = √(1-ζ^2)
Explanation - The quality factor Q = 1/(2ζ) for under‑damped second‑order systems.
Correct answer is: Q = 1/(2ζ)
Q.66 When converting a continuous‑time model to discrete‑time using zero‑order hold, which matrix is directly affected?
A_d
B_d
C_d
D_d
Explanation - A_d = e^{A T} changes, while B_d = \int_0^{T} e^{A τ} B dτ, both depend on the sample time, but the primary structural change is in A_d.
Correct answer is: A_d
Q.67 A system with transfer function G(s)= (s+2)/(s^2+4s+5) has a zero at:
s = -2
s = -5
s = -4
s = 0
Explanation - The numerator s+2 becomes zero at s = -2, which is the system zero.
Correct answer is: s = -2
Q.68 In a Luenberger observer, the matrix L is chosen to:
Place observer poles
Cancel system poles
Increase system order
Reduce input dimension
Explanation - L determines the eigenvalues of (A−LC), which are the observer poles.
Correct answer is: Place observer poles
Q.69 The controllability index of a system with n = 4 states and rank of controllability matrix = 3 is:
3
4
Undefined
1
Explanation - If the controllability matrix does not have full rank, the controllability index is not defined for the whole system (the system is not completely controllable).
Correct answer is: Undefined
Q.70 In a state‑space model, a non‑zero D matrix can cause:
A direct feed‑through term in the transfer function
Unstable poles
Loss of controllability
Loss of observability
Explanation - D appears in the transfer function as the constant term, representing instantaneous input‑output coupling.
Correct answer is: A direct feed‑through term in the transfer function
Q.71 Which of the following statements is true about a system that is both controllable and observable?
It is minimal and can be realized with the smallest possible order
Its A matrix must be diagonal
Its B matrix must be invertible
Its C matrix must be full rank
Explanation - Full controllability and observability guarantee a minimal realization; no further order reduction is possible.
Correct answer is: It is minimal and can be realized with the smallest possible order
Q.72 The term "state transition matrix" Φ(t) satisfies Φ(t1+t2) = Φ(t1)Φ(t2). This property is known as:
Semigroup property
Commutativity
Linearity
Time‑invariance
Explanation - The state transition matrix forms a continuous‑time semigroup, reflecting the composition of time intervals.
Correct answer is: Semigroup property
Q.73 When the pair (A,B) is not controllable, which method can still be used to stabilize the controllable subsystem?
Pole placement on the controllable subspace
LQR on the full system
Observer design only
Increase the order of the system
Explanation - One can design feedback for the controllable part; uncontrollable modes must already be stable.
Correct answer is: Pole placement on the controllable subspace
Q.74 A discrete‑time system is marginally stable if:
All eigenvalues lie on or within the unit circle and those on the circle are simple
All eigenvalues have magnitude greater than 1
All eigenvalues are zero
Determinant of A is 1
Explanation - Marginal stability permits eigenvalues on the unit circle provided they are not repeated, preventing growth.
Correct answer is: All eigenvalues lie on or within the unit circle and those on the circle are simple
Q.75 In state‑space, the term "zero dynamics" refers to:
Internal dynamics when the output is constrained to zero
Dynamics of the input matrix
Dynamics of the D matrix
The dynamics of the eigenvectors
Explanation - Zero dynamics are the behavior of the internal states when the output is forced to be zero, important for non‑minimum phase analysis.
Correct answer is: Internal dynamics when the output is constrained to zero
Q.76 For a SISO system, the relative degree is defined as:
Number of poles minus number of zeros
Number of zeros minus number of poles
Number of integrators in the system
Degree of the denominator polynomial
Explanation - Relative degree = n_poles – n_zeros; it indicates the number of times the output must be differentiated before the input appears.
Correct answer is: Number of poles minus number of zeros
Q.77 When applying state feedback, the closed‑loop characteristic equation is:
det(sI - (A - BK)) = 0
det(sI - A) + BK = 0
det(sI - A) = 0
det(sI - B) = 0
Explanation - Closed‑loop poles are the roots of the characteristic equation of \(A_{cl}=A-BK\).
Correct answer is: det(sI - (A - BK)) = 0
Q.78 A system with a repeated pole at s = -2 and a single eigenvector is said to have:
A Jordan block of size 2
Two independent eigenvectors
Diagonal A matrix
No controllable states
Explanation - A repeated eigenvalue with deficient eigenvectors leads to a Jordan block in the canonical form.
Correct answer is: A Jordan block of size 2
Q.79 Which of the following is a sufficient condition for the existence of a stabilizing solution to the continuous‑time algebraic Riccati equation (CARE)?
(A,B) controllable and (A,Q^{1/2}) observable
(A,B) observable and (A,R^{1/2}) controllable
A is diagonal
B is zero
Explanation - These conditions ensure a unique positive semidefinite solution to the CARE that yields a stabilizing K.
Correct answer is: (A,B) controllable and (A,Q^{1/2}) observable
Q.80 If the transfer function of a system has more poles than zeros, the system is said to be:
Strictly proper
Improper
Biproper
Non‑causal
Explanation - Strictly proper means denominator degree > numerator degree.
Correct answer is: Strictly proper
Q.81 In a state‑space model, a direct transmission term D ≠ 0 influences which of the following frequency‑domain characteristics?
High‑frequency gain
Low‑frequency gain
Phase margin only
Poles only
Explanation - At high frequencies, the contribution of the dynamics diminishes, leaving the D term as the dominant gain.
Correct answer is: High‑frequency gain
Q.82 Which of the following methods can be used to compute the state‑space matrices from a given transfer function?
Realization algorithms (e.g., Ho‑Kalman)
FFT
Monte‑Carlo simulation
Laplace inversion only
Explanation - Realization methods construct minimal state‑space models from input‑output data.
Correct answer is: Realization algorithms (e.g., Ho‑Kalman)
Q.83 A system whose A matrix has eigenvalues at -1±j2 is:
Underdamped with natural frequency √5 rad/s
Overdamped
Critically damped
Unstable
Explanation - Complex conjugate poles with negative real part indicate under‑damped behavior; ω_n = √((-1)^2+2^2)=√5.
Correct answer is: Underdamped with natural frequency √5 rad/s
Q.84 In the context of linear systems, the term "controllability Gramian" is defined as:
W_c = ∫_0^∞ e^{A t} B B^T e^{A^T t} dt
W_o = ∫_0^∞ e^{A^T t} C^T C e^{A t} dt
det(A) + det(B)
C A B
Explanation - The controllability Gramian quantifies the energy needed to move the state, defined by that integral.
Correct answer is: W_c = ∫_0^∞ e^{A t} B B^T e^{A^T t} dt
Q.85 When using pole placement, why is it important that the desired poles are not placed at the same locations as the system's zeros?
Pole‑zero cancellation can hide undesirable dynamics
It leads to a singular B matrix
The C matrix becomes non‑invertible
It reduces the system order
Explanation - Cancelling a pole with a zero can remove observable dynamics, potentially leaving hidden unstable modes.
Correct answer is: Pole‑zero cancellation can hide undesirable dynamics
Q.86 The eigenvalues of the matrix (A‑BK) can be arbitrarily assigned if and only if:
(A,B) is controllable
(A,C) is observable
D = 0
A is diagonal
Explanation - Full controllability is the necessary and sufficient condition for arbitrary pole placement via state feedback.
Correct answer is: (A,B) is controllable
Q.87 In a minimal state‑space representation, the number of states equals:
The order of the denominator polynomial of the transfer function
The number of inputs
The number of outputs
The number of zeros
Explanation - A minimal realization has a state dimension equal to the degree of the denominator (i.e., number of poles).
Correct answer is: The order of the denominator polynomial of the transfer function
Q.88 If a system is not observable, which of the following is true?
There exists at least one state that cannot be reconstructed from the output
All poles are unstable
The B matrix is zero
The D matrix must be identity
Explanation - Lack of observability means some internal states have no effect on the measured output.
Correct answer is: There exists at least one state that cannot be reconstructed from the output
Q.89 The term "state feedback" refers to a control law where:
The input is a linear combination of the states
The output is fed back to the input directly
The input is integrated
The system matrix A is modified
Explanation - State feedback uses u = -Kx (plus possibly a reference) to shape closed‑loop dynamics.
Correct answer is: The input is a linear combination of the states
Q.90 The matrix (A‑LC) in an observer design is analogous to which matrix in state‑feedback design?
(A‑BK)
(A+B)
(A+C)
(B‑K)
Explanation - Both (A‑BK) and (A‑LC) define the closed‑loop dynamics for the plant and observer respectively.
Correct answer is: (A‑BK)
Q.91 A system with A = [[0, 1],[-1, 0]] is marginally stable because:
Its eigenvalues lie on the imaginary axis
Its eigenvalues are negative
Its B matrix is zero
Its D matrix is non‑zero
Explanation - The eigenvalues are ±j, giving purely oscillatory (undamped) behavior, which is marginally stable.
Correct answer is: Its eigenvalues lie on the imaginary axis
Q.92 The term "dual system" of (A,B,C,D) is:
(A^T, C^T, B^T, D^T)
(A, B, C, D)
(A^{-1}, B^{-1}, C^{-1}, D^{-1})
(A, C, B, D)
Explanation - The dual system swaps the roles of inputs and outputs, transposing the system matrices.
Correct answer is: (A^T, C^T, B^T, D^T)
Q.93 If the determinant of the controllability matrix is non‑zero, the system is:
Completely controllable
Completely observable
Unstable
Non‑minimum phase
Explanation - A non‑zero determinant implies full rank, which means the controllability matrix spans the state space.
Correct answer is: Completely controllable
Q.94 A discrete‑time system with eigenvalues 0.8 and -0.9 is:
Asymptotically stable
Unstable
Marginally stable
Critically damped
Explanation - Both eigenvalues have magnitude < 1, satisfying discrete‑time stability.
Correct answer is: Asymptotically stable
Q.95 The Kalman filter gain is computed using:
Riccati equation
Ackermann’s formula
Bode plot
Root locus
Explanation - The steady‑state Kalman gain is obtained from the solution of the discrete or continuous Riccati equation.
Correct answer is: Riccati equation
Q.96 In a state‑space model, the term "strictly proper" indicates that:
D = 0
C = 0
B = 0
A is singular
Explanation - Strictly proper systems have no direct feed‑through, i.e., D matrix is zero.
Correct answer is: D = 0
Q.97 When applying a similarity transformation x = Tz, which of the following stays unchanged?
Transfer function G(s)
A matrix
B matrix
C matrix
Explanation - A similarity transformation changes the internal representation but leaves the input‑output behavior (transfer function) invariant.
Correct answer is: Transfer function G(s)
Q.98 The order of a minimal realization equals:
The number of poles of the transfer function
The number of zeros
The number of inputs
The number of outputs
Explanation - Minimal realization order is the degree of the denominator polynomial (number of poles).
Correct answer is: The number of poles of the transfer function
Q.99 If a system is stabilizable but not controllable, which statement is correct?
All uncontrollable modes are already stable
All controllable modes are unstable
The system cannot be observed
The D matrix must be non‑zero
Explanation - Stabilizability means any eigenvalues that cannot be moved by control are already in the stable region.
Correct answer is: All uncontrollable modes are already stable
Q.100 In a second‑order system, the damping ratio ζ = 0.7 corresponds to which type of response?
Underdamped
Overdamped
Critically damped
Unstable
Explanation - 0 < ζ < 1 indicates underdamped behavior with oscillatory decay.
Correct answer is: Underdamped
Q.101 Which matrix equation is used to verify observability?
rank(\mathcal{O}) = n
det(A) ≠ 0
rank(\mathcal{C}) = n
A = A^T
Explanation - Observability requires the observability matrix to have full rank equal to the number of states.
Correct answer is: rank(\mathcal{O}) = n
Q.102 A system with A = [[-1, 0],[0, -2]], B = [[1],[0]], C = [0 1] is:
Partially observable
Fully observable
Unobservable
Uncontrollable
Explanation - Only the second state appears in the output (C selects second state), so the first state cannot be observed → not fully observable.
Correct answer is: Partially observable
Q.103 The concept of "dual" in control theory connects controllability of (A,B) with observability of:
(A^T, C^T)
(A, C)
(B^T, A^T)
(C, B)
Explanation - Duality states that controllability of (A,B) ⇔ observability of (A^T, C^T).
Correct answer is: (A^T, C^T)
Q.104 In state‑space, the term "zero‑state response" is also known as:
Forced response
Natural response
Transient response
Steady‑state response
Explanation - Zero‑state response results from the input acting on the system with zero initial conditions.
Correct answer is: Forced response
Q.105 For a continuous‑time system, the Lyapunov equation \(A^TP + PA = -Q\) with Q>0 has a unique solution P>0 if and only if:
A is Hurwitz
A is symmetric
B is invertible
C is zero
Explanation - A Hurwitz matrix guarantees existence of a unique positive definite P solving the Lyapunov equation.
Correct answer is: A is Hurwitz
Q.106 In the design of an LQR controller, the weighting matrix Q penalizes:
State deviations
Control effort
Measurement noise
Direct feed‑through
Explanation - Q weights the state vector in the cost function, encouraging small deviations.
Correct answer is: State deviations
Q.107 A system with a pole at s = 0 has a type:
Type‑1
Type‑0
Type‑2
Non‑minimum phase
Explanation - Each pole at the origin adds one to the system type; a single pole at s=0 gives type‑1.
Correct answer is: Type‑1
Q.108 The matrix that maps the state vector to the output in a state‑space model is:
C matrix
B matrix
A matrix
D matrix
Explanation - y = Cx + Du; C defines how states affect the output.
Correct answer is: C matrix
Q.109 If a system is both controllable and observable, which of the following is guaranteed?
A minimal realization exists
A is diagonalizable
B is invertible
D = 0
Explanation - Full controllability and observability ensure that the system can be represented with the smallest possible state dimension.
Correct answer is: A minimal realization exists
Q.110 Which of the following is true for a strictly proper transfer function?
The degree of the denominator > degree of the numerator
The degree of the numerator > degree of the denominator
Denominator and numerator have equal degree
There is a direct feed‑through term
Explanation - Strictly proper means the numerator order is lower than the denominator order, so no direct term.
Correct answer is: The degree of the denominator > degree of the numerator
Q.111 The state‑space representation is said to be in "controllable canonical form" when:
B = [0 … 0 1]^T
C = [1 0 … 0]
A is diagonal
D = I
Explanation - In controllable canonical form, the B vector has a 1 in the last entry and zeros elsewhere.
Correct answer is: B = [0 … 0 1]^T
Q.112 For a discrete‑time system, the state transition matrix over k steps is:
Φ(k) = (A_d)^k
Φ(k) = e^{A k}
Φ(k) = A^k T
Φ(k) = B k
Explanation - In discrete time, the transition over k steps is the k‑th power of the discrete‑time A matrix.
Correct answer is: Φ(k) = (A_d)^k
Q.113 When the D matrix is non‑zero, the system is said to be:
Proper (or non‑strictly proper)
Improper
Unobservable
Uncontrollable
Explanation - A non‑zero D introduces a direct term, making the system proper but not strictly proper.
Correct answer is: Proper (or non‑strictly proper)
Q.114 A system with A = [[0,1],[-5,-6]] has poles at:
-3 ± j1
-2 ± j2
-5 ± j6
-1 ± j5
Explanation - Characteristic equation: s^2+6s+5=0 → poles s = -3 ± √(9-5) = -3 ± j1.
Correct answer is: -3 ± j1
Q.115 The concept of "zero dynamics" is most important when:
Designing tracking controllers for non‑minimum phase systems
Computing the controllability Gramian
Choosing a similarity transformation
Evaluating the D matrix
Explanation - Zero dynamics dictate internal behavior when the output is forced to follow a trajectory; non‑minimum phase zeros lead to unstable zero dynamics.
Correct answer is: Designing tracking controllers for non‑minimum phase systems
Q.116 In LQR design, if R is chosen very small, the resulting controller will:
Use large control effort
Minimize control effort aggressively
Become unstable
Ignore state deviations
Explanation - A small R reduces the penalty on u, so the optimizer tends to use larger control actions to reduce state error.
Correct answer is: Use large control effort
Q.117 Which of the following statements is correct for a system with a zero at the right‑half plane?
It is non‑minimum phase
It is always unstable
It has no effect on time response
It guarantees zero steady‑state error
Explanation - Zeros in the right‑half plane make the system non‑minimum phase, leading to phase lag and possible undershoot.
Correct answer is: It is non‑minimum phase
Q.118 The matrix (A‑BK) is called the:
Closed‑loop system matrix
Open‑loop system matrix
Observer matrix
Transfer matrix
Explanation - Applying state feedback modifies A to A‑BK, which governs the closed‑loop dynamics.
Correct answer is: Closed‑loop system matrix
Q.119 When the pair (A,C) is observable, the observability matrix has:
Full rank equal to number of states
Rank zero
Rank equal to number of inputs
Rank equal to number of outputs
Explanation - Observability requires the observability matrix to span the entire state space.
Correct answer is: Full rank equal to number of states
Q.120 The steady‑state error to a step input for a type‑0 system is:
Non‑zero
Zero
Infinite
Depends on D matrix
Explanation - A type‑0 system has no integrator, so it cannot eliminate steady‑state error to a step.
Correct answer is: Non‑zero
Q.121 In the state‑space equation \(\dot{x}=Ax+Bu\), the term "autonomous" refers to the case when:
u(t)=0
B=0
A=0
C=0
Explanation - An autonomous system has no external input; its dynamics are governed solely by \(\dot{x}=Ax\).
Correct answer is: u(t)=0
Q.122 If the eigenvalues of A are repeated and the geometric multiplicity is less than the algebraic multiplicity, the system matrix A is:
Defective
Diagonalizable
Symmetric
Positive definite
Explanation - Insufficient independent eigenvectors make A defective, requiring Jordan blocks.
Correct answer is: Defective
Q.123 Which of the following is true about the matrix exponential e^{At} for a time‑invariant A?
It satisfies the differential equation dΦ/dt = AΦ with Φ(0)=I
It equals A^t
It is always diagonal
It does not exist for defective A
Explanation - The matrix exponential defines the state transition matrix and meets that differential equation.
Correct answer is: It satisfies the differential equation dΦ/dt = AΦ with Φ(0)=I
Q.124 In a state‑feedback design, the number of independent gains in K for an n‑state, m‑input system is:
n×m
n+m
n
m
Explanation - K is an m×n matrix; each element is a gain, giving n×m independent parameters.
Correct answer is: n×m
Q.125 The term "relative degree" of a transfer function is equal to:
Number of poles minus number of zeros
Number of zeros minus number of poles
Number of integrators
Order of the numerator
Explanation - Relative degree indicates how many times the output must be differentiated before the input appears.
Correct answer is: Number of poles minus number of zeros
Q.126 A system with A = [[0,1],[-2,-3]] has a natural frequency ω_n of:
√2 rad/s
√5 rad/s
2 rad/s
3 rad/s
Explanation - Characteristic equation: s^2+3s+2=0 → ω_n = √2.
Correct answer is: √2 rad/s
Q.127 The controllability Gramian W_c is positive definite if:
(A,B) is controllable
(A,C) is observable
D = 0
A is symmetric
Explanation - Full controllability ensures that the Gramian is positive definite, indicating reachable states with finite energy.
Correct answer is: (A,B) is controllable
Q.128 In the context of Luenberger observers, the gain L is chosen to:
Place the observer poles at desired locations
Cancel the plant poles
Increase the system order
Make D zero
Explanation - L determines the eigenvalues of (A‑LC), shaping the convergence speed of state estimates.
Correct answer is: Place the observer poles at desired locations
Q.129 A system is said to be "minimum phase" if:
All its zeros lie in the left‑half plane
All its poles lie in the right‑half plane
It has no zeros
Its D matrix is zero
Explanation - Minimum‑phase systems have zeros in the stable region, guaranteeing favorable phase characteristics.
Correct answer is: All its zeros lie in the left‑half plane