Identifying the criterion: “The control system is unstable if the open-loop frequency response exhibits an amplitude ratio exceeding unity at the crossover frequency.” This statement corresponds to which stability criterion?

Introduction / Context:Control engineers use several classical criteria—Bode, Nyquist, Routh—to assess stability. Each articulates instability in distinct but related language. The quoted statement is a textbook description tied to gain/phase margins in Bode plots.

Given Data / Assumptions:

• “Amplitude ratio exceeding unity” refers to |L(jω)| > 1 (0 dB).
• “Crossover frequency” implies phase crossover (∠L = -180°) or gain crossover depending on context; in Bode phrasing it links |L| and phase margins.
• Single-loop negative feedback system.

Concept / Approach:The Bode criterion states that if at the phase crossover (∠L = -180°) the magnitude exceeds unity, the closed loop is unstable. Equivalently, at the gain crossover (|L| = 1), a negative phase margin (phase more negative than -180°) signals instability. This wording is characteristic of Bode analysis rather than the Nyquist encirclement rule or Routh’s algebraic test.

Step-by-Step Solution:

Verification / Alternative check:Nyquist gives an equivalent conclusion via encirclements of -1, but the phrasing with gain/phase margins is the hallmark of Bode plots.

Why Other Options Are Wrong:

• Nyquist: Same physics but different statement form (encirclement count).
• Routh: Uses characteristic equation coefficients, not frequency response.
• None/Lyapunov: Not aligned with the exact wording.

Common Pitfalls:Confusing the criteria because they are equivalent under certain assumptions; the language cues point to Bode.

Final Answer:Bode stability