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Real plants differ from models due to unmodeled dynamics, parameter variation, and simplification. Uncertainty can be quantified as bounded multiplicative error ΔG(s) such that actual plant = nominal model × (1 + ΔG). Robust stability requires the loop gain to remain stable for all uncertainty within bounds. Gain and phase margins provide conservative robustness measures; more sophisticated μ-synthesis extends these concepts.
Every control system is designed using a model of the plant — a transfer function derived from differential equations, physical laws, or system identification. But the real plant is never exactly the model. Parameters vary with temperature, wear, or operating point. You simplified high-frequency dynamics to keep the model tractable. Sensors introduce noise and delays. Model uncertainty is the gap between your design model and the true system, and robust stability analysis asks: will my controller keep the system stable despite that gap?
The standard way to represent this gap is multiplicative uncertainty. You write the true plant as G_true(s) = G_nominal(s) × [1 + ΔG(s)], where ΔG(s) is an unknown perturbation satisfying some bound — typically |ΔG(jω)| ≤ W(ω) at each frequency ω, where W(ω) is a weighting function that describes how large the uncertainty is as a function of frequency. At low frequencies, your model is often quite accurate (W is small); at high frequencies, unmodeled resonances and delays make the model unreliable (W grows large). The multiplicative form is natural because most physical uncertainty scales with the size of the process — a 10% parameter error produces a 10% deviation in the transfer function magnitude.
Your study of gain and phase margins gave you the first robustness tools. Gain margin tells you how much the loop gain can increase before instability; phase margin tells you how much additional phase lag is tolerable. These are scalar measures of how close the nominal system is to the stability boundary — they implicitly bound how much uncertainty can be tolerated before the Nyquist curve encircles the -1 point. A system with 6 dB gain margin can tolerate a factor-of-two error in the plant gain before losing stability. These margins are easy to compute from a Bode plot and provide intuitive guarantees, but they measure robustness only at specific frequencies (the gain crossover and phase crossover frequencies).
The more complete picture comes from frequency-domain robustness conditions. For a multiplicative uncertainty model, the closed-loop system remains stable for all perturbations satisfying |ΔG(jω)| ≤ W(ω) if and only if |T(jω)| ≤ 1/W(ω) at all frequencies, where T(jω) = G(jω)C(jω)/[1 + G(jω)C(jω)] is the complementary sensitivity function (the closed-loop transfer function from setpoint to output). This condition makes the tradeoff explicit: to tolerate large high-frequency uncertainty, you must roll off the closed-loop gain at high frequencies — exactly what good control design already does by limiting bandwidth. Robust stability is not an exotic concern but the quantitative version of the engineering intuition that controllers should not be tuned too aggressively.