State space analysis numerical problems solved

State Space Model Numerical PYQs with Solutions

Below are important numerical problems on State Space Model frequently asked in GATE, SSC JE, BEL, ESE, and Polytechnic Lecturer (Electronics) exams.

Numerical 1: State Space Model from Differential Equation

Question:
Obtain the state space model of the system:

d²y/dt² + 4 dy/dt + 3y = u(t)

Solution:

Let the state variables be:

x₁ = y
x₂ = dy/dt

Then,

dx₁/dt = x₂
dx₂/dt = −3x₁ − 4x₂ + u

State equation:

[ dx/dt ] = [ 0 1 ; −3 −4 ] [ x ] + [ 0 ; 1 ] u

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Numerical 2: Transfer Function to State Space Model

Question:
Obtain the state space representation of:

G(s) = 2 / (s² + 5s + 6)

Solution:

Choose phase variables:

dx₁/dt = x₂
dx₂/dt = −6x₁ − 5x₂ + 2u

Output equation:

y = x₁

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Numerical 3: Controllability Test

Question:
Check the controllability of the system:

A = [ 0 1 ; −2 −3 ]
B = [ 0 ; 1 ]

Solution:

AB = [ 1 ; −3 ]

Controllability matrix:

C = [ B AB ] = [ 0 1 ; 1 −3 ]

det(C) ≠ 0

Conclusion: The system is completely controllable.

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Numerical 4: Observability Test

Question:
Check observability of the system:

A = [ 0 1 ; −4 −5 ]
C = [ 1 0 ]

Solution:

CA = [ 0 1 ]

Observability matrix:

O = [ 1 0 ; 0 1 ]

Rank(O) = 2

Conclusion: The system is observable.

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Numerical 5: Stability of State Space System

Question:
Determine stability of the system:

A = [ 0 1 ; −6 −5 ]

Solution:

Characteristic equation:

s² + 5s + 6 = 0

Roots: s = −2, −3

Conclusion: The system is asymptotically stable.

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Exam Tip

For Polytechnic Lecturer / SSC JE, focus on:

• Differential equation to state model
• Controllability and observability numericals
• Stability using eigenvalues

For GATE / BEL / ESE, practice advanced rank-based numericals.