GATE EC 2017 Set 1 — Question 48
Go beyond PYQs with Success TrackerAI-powered personalised practice and doubt support. Unlimited practice on eligible plans; AI usage limits apply.MCQ+2 / -0.67MediumRouth-Hurwitz CriterionStability AnalysisControl Systems
Control Systems → Stability Analysis → Routh-Hurwitz Criterion
Last updated
Question
Which one of the following options correctly describes the locations of the roots of the equation on the complex plane?
Correct answer
(C) Two RHP roots and two LHP roots
Solution
The given characteristic equation is .Method 1: Direct Solution
Let . The equation becomes a quadratic in :
The roots for are:
Now, solve for :
Roots: (Right Half Plane) and (Left Half Plane).
Roots: (Right Half Plane) and (Left Half Plane).In total, there are two roots in the Right Half Plane (RHP) and two roots in the Left Half Plane (LHP).Method 2: Routh-Hurwitz Criterion
Construct the Routh array for :
: 1 1 1
: 0 (Row of zeros)
Form the auxiliary equation from the row: .
Take the derivative: .
Replace the row with coefficients [4, 2]:
: 1 1 1
: 4 2
: 1
:
: 1There are two sign changes in the first column (0.5 to -6 and -6 to 1), indicating 2 roots in the RHP. Since the polynomial is even, the roots are symmetric about the origin, implying 2 roots in the LHP. No roots are on the imaginary axis as the coefficient is non-zero and the Routh array completed with sign changes.Thus, option (C) is correct.
Let . The equation becomes a quadratic in :
The roots for are:
Now, solve for :
1.For :
Roots: (Right Half Plane) and (Left Half Plane).
2.For :
Roots: (Right Half Plane) and (Left Half Plane).In total, there are two roots in the Right Half Plane (RHP) and two roots in the Left Half Plane (LHP).Method 2: Routh-Hurwitz Criterion
Construct the Routh array for :
: 1 1 1
: 0 (Row of zeros)
Form the auxiliary equation from the row: .
Take the derivative: .
Replace the row with coefficients [4, 2]:
: 1 1 1
: 4 2
: 1
:
: 1There are two sign changes in the first column (0.5 to -6 and -6 to 1), indicating 2 roots in the RHP. Since the polynomial is even, the roots are symmetric about the origin, implying 2 roots in the LHP. No roots are on the imaginary axis as the coefficient is non-zero and the Routh array completed with sign changes.Thus, option (C) is correct.
Continue learning with Success Tracker
A step still unclear? Work through it with support
Use Success Tracker to ask about the reasoning, then try another GATE EC question to check your understanding.
AI-powered practice· Unlimited practice on eligible plans
- PYQs with solutions
- Attempt available previous-year questions, then compare your reasoning with the worked solution. Coverage varies by stream.
- Practice that adapts
- Choose a topic, work on weaker areas and bookmark questions to revisit. Your attempts feed your progress tracking.
- AI doubt support
- Ask follow-up questions about a step or concept while practising, instead of stopping at the final answer.
Unlimited practice is available on eligible plans. Free practice and AI usage have limits; check the current plan allowances before choosing.
This page stays readable without an account. AI responses can be wrong; check them against the solution and source material.
More questions on Stability Analysis
2026 Set 1 Q11Consider the differential equation , with…2026 Set 1 Q19A control system is shown in the Figure. Which option represents the correct transfer function of…2026 Set 1 Q40For the control system shown in the Figure, the transfer function of a plant,…2026 Set 1 Q41The state and output equations for a control system are:…2026 Set 1 Q59Consider the unity negative feedback control system shown in the Figure. The value of gain …