A feedback control system has a gain margin of 15 dB and a phase margin of 65°, both considered excellent by classical standards. An engineer claims this system is robustly stable against all reasonable plant perturbations. What is the flaw in this reasoning?
AGain and phase margins above 10 dB and 45° guarantee robust stability by the Nyquist criterion
BClassical gain and phase margins only test robustness in specific perturbation directions; simultaneous gain and phase shifts can destabilize a system with otherwise excellent margins
CRobust stability requires gain margin above 20 dB, so 15 dB is insufficient
DPhase margin is irrelevant for systems with multiplicative uncertainty
Gain margin measures how much loop gain can increase before instability; phase margin measures how much phase can lag. Each tests a single perturbation direction. A system can have large margins in both individual directions yet be fragile to simultaneous gain-and-phase perturbations — a gap that disk margins or structured singular value (μ) analysis are designed to fill. Classical scalar margins are not a complete robustness certificate.
Question 2 Multiple Choice
The robust stability condition for multiplicative uncertainty states |T(jω)W(jω)| < 1 for all ω, where W(jω) is large at high frequencies. What does this imply for controller design at high frequencies?
AThe loop gain L(jω) must be increased at high frequencies to dominate the uncertainty
BThe sensitivity function S(jω) must be made small at high frequencies to reject disturbances
CThe complementary sensitivity T(jω) must be made small at high frequencies where model uncertainty is large
DController bandwidth must be extended into the high-frequency uncertainty region
Where W(jω) is large (model uncertainty is large), the condition |T(jω)W(jω)| < 1 requires |T(jω)| < 1/|W(jω)|, which is small. Since T = 1 − S and S + T = 1, making T small at high frequencies means S is close to 1 there — less disturbance rejection, but necessary to ensure the nominal controller doesn't destabilize under the uncertain true plant. This is the quantitative form of the robustness-performance tradeoff.
Question 3 True / False
In the H∞ framework, the designer's primary tool for encoding knowledge about uncertainty magnitude and performance requirements is the choice of weighting functions on the sensitivity and complementary sensitivity functions.
TTrue
FFalse
Answer: True
H∞ synthesis minimizes the peak gain from exogenous inputs to performance outputs, but how that objective is specified — which signals matter, at which frequencies, and how much — is entirely encoded in the weighting functions. W_S shapes disturbance rejection requirements, W_T encodes the uncertainty profile, W_U limits control effort. The resulting controller is only as good as the designer's weighting choices, which is why 'choosing weights is the designer's art.'
Question 4 True / False
An H∞ controller that achieves the minimum possible ||T_zw||∞ for a given plant and problem formulation is the unique optimal solution to the robust control problem.
TTrue
FFalse
Answer: False
H∞ synthesis generally produces a set of controllers that achieve the optimal bound, not a unique solution — and the bound itself depends on the weighting functions chosen. Different valid weighting functions yield different H∞ controllers, all 'optimal' within their respective formulations. The synthesis is not a single objective fact about the plant; it reflects both the plant dynamics and the designer's encoded knowledge about uncertainty and performance priorities.
Question 5 Short Answer
Why do classical gain and phase margins fail to capture all robustness concerns, and what does the robust stability condition |T(jω)W(jω)| < 1 add?
Think about your answer, then reveal below.
Model answer: Gain margin asks how much loop gain can grow before instability; phase margin asks how much phase can lag — each probes only one perturbation direction. A system can pass both tests yet be fragile to perturbations that combine gain and phase changes simultaneously. The robust stability condition generalizes this by asking: for all possible plants within the uncertainty set (defined by the weighting function W), does any perturbation cause instability? If |T||W| < 1 at every frequency, no perturbation of magnitude bounded by |W| can push the Nyquist plot to encircle −1. This is a global certificate over all perturbation directions, not just two.
The small gain theorem is the key: two stable systems in feedback are jointly stable if the product of their frequency-domain gains is less than one everywhere. Applying this to the uncertain plant loop gives the condition directly. Classical margins are special cases — they check two scalar directions — while the full condition checks the entire uncertainty ball at each frequency.