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Bandwidth is the frequency at which magnitude drops to −3 dB (0.707 times the DC value). It indicates how fast a system can respond to changing reference inputs. Resonant peaks indicate underdamped modes; peak height increases as damping decreases.
From your study of frequency response and Bode plots, you know how to compute and plot a system's gain and phase as a function of frequency — the magnitude plot showing |G(jω)| and the phase plot showing ∠G(jω). Bandwidth and resonance are the two most important features to read off those plots when assessing how a system will perform in closed-loop operation.
Bandwidth (ω_BW) is the frequency at which the closed-loop magnitude response drops to −3 dB, which corresponds to a gain of 0.707 (or 1/√2) relative to the DC value. The −3 dB criterion is not arbitrary: it is the frequency at which output power has dropped to half of its DC value (since power is proportional to amplitude squared). Below the bandwidth, the system tracks reference inputs faithfully — a sinusoidal command at frequency ω < ω_BW produces a nearly full-amplitude output. Above the bandwidth, the system cannot keep up: output amplitude shrinks and phase lag increases, meaning fast reference changes are partially or fully filtered out. Bandwidth is therefore a direct measure of speed: a wider bandwidth means the system can track faster-changing references. Practically, doubling the bandwidth roughly halves the rise time of the step response.
Resonant peaks in the frequency response appear when the system has underdamped poles — complex conjugate poles whose real part is small relative to their imaginary part. A second-order system with natural frequency ω_n and damping ratio ζ has a closed-loop frequency response that peaks near ω_n when ζ < 1/√2 ≈ 0.707. The peak magnitude is M_p = 1/(2ζ√(1−ζ²)), which grows without bound as ζ → 0. A resonant peak in the frequency response translates directly into overshoot in the step response: a system whose magnitude peaks at +6 dB will produce roughly 30% overshoot on a step input. This is the critical link between the frequency domain (where controller design happens) and the time domain (where performance specifications are written).
The tradeoff between bandwidth and resonance is central to control design. Increasing controller gain generally pushes the closed-loop bandwidth higher — faster tracking — but also brings the closed-loop poles closer to the imaginary axis, increasing the resonant peak and overshoot. Aggressive bandwidth comes at the cost of oscillatory transient behavior, noise sensitivity (high-frequency disturbances are amplified near the resonant peak), and eventually instability. The practical design goal is to achieve the bandwidth required by the speed specification while keeping M_p below about 3–6 dB (corresponding to ζ ≥ 0.35–0.5), ensuring acceptable damping. Resonant peaks at or above 0 dB — meaning the magnitude response exceeds DC gain at some frequency — indicate very underdamped behavior and often signal that the design is approaching instability margins.