GATE EC 2025 Set 1 — Question 53
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Control Systems → State-Space Analysis → Solution of State Equations
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Question
Consider a system where , , and are three internal state signals and is the input signal. The differential equations governing the system are given byWhich of the following statements is/are TRUE?
Correct answer
(B) There exists a bounded input such that at least one of the signals x₁(t), x₂(t), and x₃(t) is unbounded
Solution
The system is decoupled into three first-order equations:
(A) False, because and can be unbounded for bounded inputs.
(B) True, for , and are unbounded.
(C) False, because is always bounded for any bounded input .
(D) False, for , all signals remain at zero (bounded).
1.: This has a pole at (RHP). It is unstable. For a bounded input like , grows exponentially and is unbounded.
2.: This has a pole at (LHP). It is BIBO stable. For any bounded input , remains bounded.
3.: This is an integrator with a pole at . For a bounded input like , , which is unbounded.
Evaluating options:(A) False, because and can be unbounded for bounded inputs.
(B) True, for , and are unbounded.
(C) False, because is always bounded for any bounded input .
(D) False, for , all signals remain at zero (bounded).
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