Identify the fundamental nature of a lag compensator in control engineering: does it behave closest to a low-pass, high-pass, or band-pass filter?

Introduction / Context:
Compensators shape loop gain to meet stability and performance targets. A lag compensator is routinely used to boost low-frequency gain (improving steady-state accuracy) while minimally affecting high-frequency phase margin.

Given Data / Assumptions:

• Lag compensator transfer form: Gc(s) = (s/ζ + 1) / (s/ζβ + 1), with β > 1 and pole closer to origin than the zero.
• Focus on qualitative frequency response.

Concept / Approach:
The pole is at lower frequency than the zero (|p| < |z|). Magnitude response rises at very low frequencies and then rolls off as frequency increases, which is characteristic of a low-pass filter. Phase lag is introduced mainly around the pole–zero break frequencies, hence the name “lag”.

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