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Electrical Engineering › Control Systems

State-Space Analysis MCQ [Free PDF] – 5 Objective Questions with Answers

Every State-Space Analysis question asked in GPSC DEE Electrical — solve, check instantly, then read the formula and exam tip.

5 PYQs4 subtopics1 examUpdated 23 Oct 2026
Topic practice0/5✓ 0✗ 0

State Variables (2)

Q1Moderate
Asked in: GPSC DEE Electrical · DEE Master paper · 2026
Consider the following statements regarding State Variable Analysis:
1. State variables define system dynamics completely.
2. State-space model can represent MIMO systems.
3. Transfer function can always be uniquely obtained from state model.
4. State model includes initial conditions.
Which of the above statements is/are correct?
View answer & solution
Correct answer: D. 1, 2 and 4 only
Quick explanation

A state-space model represents internal dynamics using a minimal set of state variables. It naturally handles MIMO systems and incorporates initial conditions. Key relation: ẋ=Ax+Bu; y=Cx+Du.

Exam focus: Eigenvalues of A are the state-model characteristic roots. Common trap: Output is not generally independent of state variables.

ẋ=Ax+Bu; y=Cx+Du.
Verified · SRO editorial deskSolution page → Report
Q2Moderate
Asked in: GPSC DEE Electrical · DEE Master paper · 2026
A system is represented in state-space form with two state variables and stable eigenvalues, then
1. The system order is two.
2. The system is stable since eigenvalues lie in left half-plane.
3. State variables completely describe system dynamics.
4. Output depends only on input and not on state variables.
Which of the above statements is/are correct?
View answer & solution
Correct answer: C. 1, 2 and 3 only
Quick explanation

A state-space model represents internal dynamics using a minimal set of state variables. Output generally depends on both state and direct input through y=Cx+Du. Key relation: ẋ=Ax+Bu; y=Cx+Du.

Exam focus: Eigenvalues of A are the state-model characteristic roots. Common trap: Output is not generally independent of state variables.

ẋ=Ax+Bu; y=Cx+Du.
Verified · SRO editorial deskSolution page → Report

Controllability and Observability (1)

Q3Moderate
Asked in: GPSC DEE Electrical · DEE Electrical · 2024
Read the following statements:
Statement 1: A system is said to be completely controllable if it is possible to transfer the system state from an initial state X(t_0) to any desired state X(t) in a specified finite time by a control vector u(t).
Statement 2: A system is said to be completely observable, if every state X(t_0) can be completely identified by measurement of the outputs y(t) over the time interval 0 ≤ t ≤ ∞.
Choose the correct options from the following:
View answer & solution
Correct answer: A. Both statement 1 and statement 2 are correct.
Quick explanation

Statement 1 gives the standard meaning of complete controllability: the input can drive the state to any desired state in finite time. Statement 2 gives the observability idea: the initial state can be reconstructed from output measurements. For an n-state LTI model, the algebraic checks are rank(𝒞)=n and rank(𝒪)=n. Hence option A.

𝒞 = [ B AB A²B … An−1B ] Complete controllability: rank(𝒞) = n 𝒪 = [ C ; CA ; CA² ; … ; CAn−1 ] Complete observability: rank(𝒪) = n 𝒞 = [ B AB A²B … An−1B ] 𝒪 = [ C ; CA ; CA² ; … ; CAn−1 ] rank(𝒞) = n ⇒ completely controllable rank(𝒪) = n ⇒ completely observable
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Stability (1)

Q4Moderate
Asked in: GPSC DEE Electrical · DEE Master paper · 2026
Consider the following statements regarding Stability in State-Space Analysis:
1. Eigenvalues of the system matrix A define the characteristic roots that determine the system's stability.
2. A system is asymptotically stable if and only if, all eigenvalues of the system matrix A have strictly negative real parts (located in the Left-Half Plane).
3. The presence of at least one eigenvalue with a positive real part (located in the Right-Half Plane) implies that the system is unstable.
4. Stability is a property of the system itself and is determined by the internal system parameters and the A matrix.
Which of the above statements is/are correct?
View answer & solution
Correct answer: D. All of the above
Quick explanation

A state-space model represents internal dynamics using a minimal set of state variables. Stability of a continuous-time LTI model is tied to eigenvalues of A: strictly negative real parts imply asymptotic stability. Key relation: ẋ=Ax+Bu; y=Cx+Du.

Exam focus: Eigenvalues of A are the state-model characteristic roots. Common trap: Output is not generally independent of state variables.

ẋ=Ax+Bu; y=Cx+Du.
Verified · SRO editorial deskSolution page → Report

State-Space to Transfer Function (1)

Q5Hard
Asked in: GPSC DEE Electrical · DEE Electrical · 2024
A certain system has a state space model as
[Ẋ_1 ; Ẋ_2] = [−2 −3 ; 4 2][X_1 ; X_2] + [3 ; 5]U
Y = [1 1][X_1 ; X_2], D = 0
What will be the transfer function of this system?
View answer & solution
Correct answer: A. (8s + 1)/(s^2 + 8)
Quick explanation

Use G(s)=C(sI−A)^−1B+D. Here det(sI−A)=s^2+8. Multiplying C·adj(sI−A)·B gives 8s+1; therefore G(s)=(8s+1)/(s^2+8).

[Ẋ1 ; Ẋ2] = [−2 −3 ; 4 2][X1 ; X2] + [3 ; 5]U Y = [1 1][X1 ; X2], D = 0 G(s) = C(sI − A)−1B + D sI − A = [s + 2 3 ; −4 s − 2] det(sI − A) = (s + 2)(s − 2) − 3(−4) = s² + 8 (sI − A)−1 = adj(sI − A)/(s² + 8) A = [−2 −3 ; 4 2] B = [3 ; 5] C = [1 1] D = 0 G(s) = C(sI − A)−1B + D sI = [s 0 ; 0 s] sI − A = [s + 2 3 ; −4 s − 2] det(sI − A) = (s + 2)(s − 2) − (3)(−4) det(sI − A) = s² − 4 + 12 det(sI − A) = s² + 8 adj(sI − A) = [s − 2 −3 ; 4 s + 2] (sI − A)−1 = [s − 2 −3 ; 4 s + 2]/(s² + 8) [s − 2 −3 ; 4 s + 2][3 ; 5] = [3(s − 2) − 15 ; 12 + 5(s + 2)] = [3s − 21 ; 5s + 22] C·adj(sI − A)·B = [1 1][3s − 21 ; 5s + 22] = (3s − 21) + (5s + 22) = 8s + 1 G(s) = (8s + 1)/(s² + 8) + 0 Characteristic polynomial = det(sI − A) = s² + 8 All four options have denominator s² + 8
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