Questions: Robust Control and H-Infinity Synthesis

Question 1

An H-infinity controller guarantees that all closed-loop transfer functions have a maximum gain (H-infinity norm) below a specified bound. If you set a very tight bound (small γ), what typically happens?

• A. The resulting controller is more robust to disturbances and model uncertainty
• B. The controller becomes stiffer, may not exist (problem becomes infeasible), or requires very large control inputs that are impractical
• C. The closed-loop system becomes faster and more sensitive to noise
• D. The robust stability margin increases automatically

Answer: (B)

The H-infinity bound γ constrains the peak closed-loop gain; tightening it means the solver must reduce all input-to-output transfer function gains simultaneously. This is globally coupled: reducing disturbance sensitivity (S transfer function) while maintaining stability against uncertainty is difficult, often requiring large control effort. At some γ_min, the problem becomes infeasible — no controller can satisfy both the H-infinity bound and stability. The tradeoff is real: tighter bound means either the controller saturates actuators, or the problem has no solution. H-infinity synthesis automatically trades robustness and performance; the art is choosing appropriate weights that encode your priorities.

Question 2

In H-infinity synthesis, you define a weighted performance objective: minimize ||W_d·S + W_n·T||_∞, where S is sensitivity and T is complementary sensitivity. What does increasing W_d in the low-frequency band accomplish?

• A. It decreases the control bandwidth, making the controller slower
• B. It increases the penalty on disturbance sensitivity (S) at low frequencies, forcing the synthesis to improve disturbance rejection; the tradeoff is that T must grow, amplifying high-frequency noise
• C. It guarantees disturbance rejection at all frequencies
• D. It eliminates the need for integral action

Answer: (B)

The weighted norm ||W_d·S + W_n·T||_∞ sums two competing objectives: W_d penalizes S (the disturbance sensitivity), and W_n penalizes T (the noise sensitivity). Since S + T = 1 always (a universal law of feedback), improving one means worsening the other. By making W_d large at low frequencies and W_n large at high frequencies, you tell the synthesis 'I care most about disturbances at low frequencies and noise at high frequencies.' The optimizer will push S down at low frequency (good disturbance rejection) and push T down at high frequency (good noise attenuation), accepting increased T at low frequency (noise amplification in the disturbance band). This frequency-dependent prioritization is how H-infinity captures classical control intuition (low-gain at high frequencies to reject noise, high-gain at low frequencies to reject disturbances).

Question 3

Structured Singular Value (μ) analysis accounts for the 'structure' of model uncertainty. Why is the standard singular value (σ_max) inadequate for predicting robustness when the plant has repeated real uncertainties?

• T. True
• F. False

Answer: True

Standard singular value σ_max(M) gives the gain of M for an arbitrary perturbation, treating all perturbations as if they were full-rank. Structured uncertainty like two proportional parameter uncertainties (Δ = diag(δ₁, δ₂)) is constrained: δ₁ and δ₂ affect the system in correlated ways. μ(M) accounts for this structure, giving the smallest perturbation magnitude (in the constraint set) that destabilizes the system. μ is always ≤ σ_max, sometimes much smaller, capturing the fact that some perturbations are easier to tolerate because they act redundantly on the system state.

Question 4

You design a nominal LQR controller that achieves excellent setpoint tracking and disturbance rejection in simulation. When deployed to the real system with ±10% parameter variations, the controller becomes unstable. Why, and how does H-infinity design address this?

• T. True
• F. False

Answer: True

LQR is an optimal (locally) nominal design — it assumes the model is correct. Small model errors can result in poor closed-loop behavior (phase lag, reduced stability margins). H-infinity design explicitly includes uncertainty in the problem formulation: you specify an uncertainty model (e.g., |ΔG/G| ≤ 0.1 at each frequency), and the synthesis guarantees stability for all plants in the uncertainty set. The tradeoff: nominal performance is sacrificed (the LQR design is typically faster or uses less control energy), but the H-infinity controller remains stable under uncertainty. Modern industrial practice combines the two: use LQR for nominal performance, then use robust H-infinity analysis to identify margin deficiencies and iterate on the design.

Question 5

Explain the fundamental robustness-performance tradeoff in feedback control: why can you not simultaneously achieve very low sensitivity (S) and very low complementary sensitivity (T) across all frequencies?

Model answer: The constraint S + T = 1 (at each frequency) is a universal law of feedback control: the sensitivity function S = 1/(1+L) and complementary sensitivity T = L/(1+L), where L is the open-loop transfer function, always sum to the identity. This means you cannot push both S and T to zero simultaneously. At low frequency, you want S small (disturbance rejection and setpoint tracking), but this forces T close to 1, meaning the measurement noise passes through almost unfiltered. At high frequency, you want T small (noise rejection), but this forces S close to 1, meaning external disturbances are rejected poorly. The tradeoff is fundamental — not a limitation of controller design but a consequence of causality and linear feedback. Classical Bode design captured this intuitively: high gain at low frequency (low S) and low gain at high frequency (low T), with the transition determined by the controller bandwidth and stability margins.

This is why modern control design uses frequency-dependent weighting: you choose where to trade performance for robustness by specifying weights W_d(jω) and W_n(jω) that grow at frequencies where you care most. H-infinity synthesis respects the S + T = 1 constraint globally and finds the best achievable tradeoff given your weights.