# H Infinity Control > Use when you must run the h-infinity mixed-sensitivity norm analysis of a feedback loop: given the plant transfer function, a candidate controller, the sensitivity weight and the control-effort weight, verify the closed loop is stable and compute the h-infinity norms of the weighted sensitivity functions by gamma iteration over the imaginary-axis frequency response, locating the worst-case peak magnitude of each channel. Produces the weighted-sensitivity norm, the weighted control-sensitivity norm, the achieved gamma of the mixed-sensitivity weighting as the larger of the two norms, and the bound verdict when both weighted norms stay below one so the s-over-ks sensitivity bounds hold at every frequency. Trigger: h infinity norm, gamma iteration, mixed sensitivity weighting, s ks weighted bounds, sensitivity weight, control weight, worst case peak gain, weighted loop analysis. - Skill: `ashfordeou/h-infinity-control` (Agent Skill, multi-file: 3 files) - Install (CLI): `npx skillmds@latest add ashfordeou/h-infinity-control` - Raw SKILL.md: https://api.skillmd.com/api/skills/ashfordeou/h-infinity-control/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Coding & Dev Tools - License: Apache-2.0 - Author: ashfordeOU (https://skillmd.com/u/ashfordeou) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/ashfordeou/h-infinity-control --- # H-infinity Control (gnc-autonomy/control/h-infinity-control) Use when the task is the worst-case H-infinity norm review of a SISO feedback loop under a mixed-sensitivity S/KS weighting: given the plant transfer function G(s), a candidate controller K(s) and two weighting functions (the sensitivity weight W1 and the control-effort weight W2), this leaf assembles the closed-loop channels of L = G K over the characteristic polynomial, verifies strict stability, and computes the H-infinity norms of the weighted channels W1 S and W2 KS by gamma iteration over the imaginary-axis frequency response. The achieved gamma of the mixed-sensitivity weighting and the bound verdict gamma < 1 certify whether |S(jw)| stays below 1/|W1(jw)| and |KS(jw)| below 1/|W2(jw)| at every frequency of the swept axis. The plant, the candidate controller and the weight parameters are given inputs; the controller is never synthesized, tuned, placed or recovered here. Pure Python, stdlib only, deterministic and offline. It pairs with frequency-response-design, which evaluates an open loop at isolated frequencies and reports classical margins, and with pid-control-design, which designs the controller gains this leaf takes as given. ## Domain quick reference - Loop model: plant G = g_num/g_den and candidate controller K = k_num/k_den, both lists of float coefficients in descending powers of s; loop transfer L = G K = (g_num k_num)/(g_den k_den). The loop must carry no pole-zero cancellation in L: the channel representations below are not reduced, so a cancelled pair evaluates 0/0 at the cancelled dynamics and is rejected by the caller. - Channels over the common denominator: the characteristic polynomial char = g_den k_den + g_num k_num, with S = 1/(1 + L) = (g_den k_den)/char, T = L/(1 + L) = (g_num k_num)/char and KS = K/(1 + L) = (k_num g_den)/char. Identity: S + T = 1 exactly as polynomials, s_num + t_num = char. - Stability: the loop is strictly stable exactly when routh_stable(char) is True (Routh-Hurwitz first-column test, every first-column element strictly positive; marginal cases report not-strictly-stable). - H-infinity norm of a stable proper channel H = num/den: write A(x) = hermitian square of num and B(x) = hermitian square of den, where |num(jw)|^2 = A(w^2) and |den(jw)|^2 = B(w^2) are polynomials in x = w^2 (for p(s) with ascending coefficients q[m], the coefficient of x^m in |p(jw)|^2 is (-1)^m times the alternating sum over i + j = 2m of q[i] q[j] (-1)^i; odd-power coefficients vanish). The stationary equation of the squared magnitude is P(x) = A'(x)B(x) - A(x)B'(x) = 0, and every interior peak of |H(jw)| sits at a real root x >= 0 of P. The norm is the square root of the largest of the DC value A(0)/B(0), the value at every bracketed stationary root on the geometric grid in x = w^2 and the high-frequency limit (lead-ratio squared |num[0]/den[0]|^2 when the degrees are equal, else 0). This is the gamma-iteration search; bisection shrinks each frequency bracket around a worst-case peak. - Weight functions, closed form from corner parameters: W1(s) = (s/ms + wb)/(s + wb as_) with |W1(j0)| = 1/as_ and |W1(j inf)| = 1/ms; W2(s) = (s + wbc a2)/(s/mu2 + wbc) with |W2(j0)| = a2 and |W2(j inf)| = mu2. All six parameters strictly positive. - Weighted norms: ||W1 S||_inf = hinfinity_norm(w1_num s_num, w1_den char) and ||W2 KS||_inf = hinfinity_norm(w2_num ks_num, w2_den char). Achieved gamma = max of the two; bound verdict gamma < 1. Verdict meaning: |W1(jw) S(jw)| at most gamma at every swept frequency, so gamma < 1 certifies the s-over-ks pointwise bounds. - Norm search scope: the imaginary-axis sweep runs w in [1e-4, 1e4] rad/s (x = w^2 in [1e-8, 1e8], 801 geometric grid points) plus the DC point and the high-frequency limit; peaks beyond 1e4 rad/s are outside the declared scope and never claimed. ## Workflow 1. Fix the loop under review: the plant G(s) = g_num/g_den and the candidate controller K(s) = k_num/k_den, both given inputs that this leaf never synthesizes, tunes or places. 2. Assemble the closed-loop channels of the loop transfer L = G K with loop_channels: the characteristic polynomial char_poly and the S, T and KS numerators over the common denominator, and confirm the S + T = 1 identity coefficient by coefficient. 3. Verify the loop is strictly stable with routh_stable on char_poly: the Routh-Hurwitz first-column verdict, checked before any weighted-norm claim is made. 4. Build the two standard weight functions from their corner parameters with sensitivity_weight (W1 from ms, wb, as_) and control_weight (W2 from a2, wbc, mu2). 5. Compute the H-infinity norms of the weighted channels by gamma iteration with hinfinity_norm over the imaginary-axis frequency response: form A(x) and B(x), find the stationary roots of P(x) = A'(x)B(x) - A(x)B'(x) = 0 on the geometric grid, refine each bracket by bisection, and take the largest of the DC value, the interior peak values and the high-frequency limit. 6. Read the mixed-sensitivity review with mixed_sensitivity_gamma: the weighted-sensitivity norm ||W1 S||_inf, the weighted control-sensitivity norm ||W2 KS||_inf, the achieved gamma as the larger of the two, and the bound verdict gamma < 1. 7. Confirm the worst-case bound semantics: on a coarse sweep of the imaginary-axis response no sampled |W1S(jw)| exceeds the computed norm, and the low-frequency asymptotes hold (|S| Kv/w tending to 1 and the flat band |W1S| about wb/Kv). 8. Confirm the deterministic checks with the contract test scripts/test_h_infinity_control.py, offline, under both interpreters. ## Worked example Loop under review, all numbers real outputs of the contract module: plant G(s) = 1/(s(s+1)) (g_num [1.0], g_den [1.0, 1.0, 0.0]), candidate controller K(s) = 8(s+1)/((s+2)(s+8)) (k_num [8.0, 8.0], k_den [1.0, 10.0, 16.0], a lead-lag with K(0) = 0.5, strictly proper so KS rolls off at high frequency). Sensitivity weight W1(s) = (s/2.5 + 0.3)/(s + 3e-4) (ms = 2.5, wb = 0.3 rad/s, as_ = 1e-3), control weight W2(s) = (s + 0.2)/(s + 2) (a2 = 0.1, wbc = 2.0 rad/s, mu2 = 1.0). The S/KS review PASSES with gamma 0.7779822129233023, dominated by the control channel. - loop_channels(G, K): char_poly [1.0, 11.0, 26.0, 24.0, 8.0] (s^4 + 11 s^3 + 26 s^2 + 24 s + 8), routh_stable True. The velocity constant of the loop is Kv = lim s L = K(0) = k z/(p1 p2) = 8/(2*8) = 0.5 s^-1. - Weight functions as built: sensitivity_weight(2.5, 0.3, 1e-3) returns ([0.4, 0.3], [1.0, 0.0003]); control_weight(0.1, 2.0, 1.0) returns ([1.0, 0.2], [1.0, 2.0]). - Weight closed forms: |W1(j0)| = 1000.0 (1/as_), |W1(j inf)| = 0.4 (1/ms), |W2(j0)| = 0.1 (a2), |W2(j inf)| = 1.0 (mu2). The corner magnitude |W1(j wb)| = sqrt(1 + 1/ms^2)/sqrt(1 + as_^2) = 1.077032422910824 (the formula divides by sqrt(1 + as_^2); a printed 1.0770329614269009 equals sqrt(1 + 1/ms^2) alone). - Weighted sensitivity norm: ||W1 S||_inf = 0.6210716713301313. The flat band |W1S| about wb/Kv = 0.6 carries the pass: |W1S(jw)| = 0.5997464724424371 at w = 0.01, 0.6003939878705405 at w = 0.05, 0.601583811863733 at w = 0.1, and the worst-case peak sits just above the band where the low-frequency asymptote bends over. - Weighted control norm: ||W2 KS||_inf = 0.7779822129233023, the control-sensitivity peak where the strictly proper controller rolls off against the rising control weight. - Achieved gamma = max(0.6210716713301313, 0.7779822129233023) = 0.7779822129233023 with verdict True (pass): the loop satisfies |S(jw)| < 1/|W1(jw)| and |KS(jw)| < 1/|W2(jw)| at every frequency of the swept imaginary axis. - Unweighted channels for context: ||S||_inf = 1.2187761291690518 (the classical sensitivity peak above 1 near the crossover region) and ||T||_inf = 1.0 (the type-1 tracking identity |T(j0)| = 1); the S + T = 1 identity holds exactly at the polynomial level and pointwise at the sampled frequencies. - Low-frequency asymptote check: |S(jw)| Kv/w = 0.9999993828127061 at w = 1e-3, confirming S about jw/Kv with Kv = 0.5. - Fail sibling (the verdict False path): the same loop with the demanding sensitivity weight W1(s) = (s/2.5 + 1)/(s + 1e-3) (wb raised from 0.3 to 1.0 rad/s) drives the flat band to wb/Kv = 2.0: ||W1 S||_inf = 1.9979285867334802, ||W2 KS||_inf = 0.7779822129233023 (unchanged), gamma = 1.9979285867334802, verdict False: the tracking requirement wb = 1 rad/s exceeds the loop's velocity constant Kv = 0.5 s^-1, so the weighted sensitivity violates the unit bound. The reviewer reports the achieved gamma and the verdict; the controller itself is not altered. - Determinism: mixed_sensitivity_gamma on the worked loop returns identical dicts run to run (gamma and both norms bit-identical). ## Verification - Confirm the channel assembly anchors: char_poly of the worked loop, the S + T = 1 identity at the polynomial and pointwise levels, and the |S| and |T| norms. - Confirm the weight closed forms at DC, at the corner and at the high-frequency limit, and the Routh verdicts on the anchor polynomials. - Confirm the norm machinery against closed-form cases: the DC carry of 1/(s+1), the zero-at-origin carry of (s+2)/(s+1), and the resonant peak 1/(2 zeta sqrt(1 - zeta^2)) = 5.025189076296056 of 1/(s^2 + 0.2 s + 1). - Confirm the worked pass and fail sibling norms, achieved gamma and bound verdict within 1e-6 relative of the Worked example values, and the worst-case bound semantics: the computed norm never underestimates the imaginary-axis response sampled on a coarse sweep. - Confirm the low-frequency asymptote |S| Kv/w tending to 1 and the flat band |W1S| about wb/Kv on the worked loop. - Confirm ValueError rejection of an improper transfer function, an unstable denominator, a denominator zero at the evaluation point, and non-positive weight corner parameters (each naming the parameter), with the real messages quoted in the Worked example. - Run the contract test offline: python3 scripts/test_h_infinity_control.py (deterministic, no imports beyond math, no exact-float equality on computed sums). ## Related leaves - gnc-autonomy/control/pid-control-design: designs and tunes the controller gains this leaf takes as given inputs; this leaf never tunes, places or margin-checks a controller. - gnc-autonomy/control/frequency-response-design: evaluates an open loop at isolated frequencies and reports gain and phase margins; this leaf searches a weighted closed-loop channel for its worst-case peak and never reports classical margins. - gnc-autonomy/optimal-control/lqr-design and gnc-autonomy/optimal-control/lqg-design: solve the algebraic Riccati equations of optimal regulation and estimation; this leaf computes no Riccati solution and no state-feedback or compensator gain. - gnc-autonomy/optimal-control/loop-transfer-recovery: reshapes a recovered LQG loop toward a full-state target; this leaf compares weighted channel peaks, not one complex transfer against another. - gnc-autonomy/control/state-space-analysis: the state-space toolbox of the pack; this leaf works with rational transfer functions only. ## Pitfalls - Reading this leaf as an H-infinity controller synthesis: no controller is ever synthesized, tuned, placed or recovered here, and no algebraic Riccati equation solver lives in the leaf; the plant, the candidate controller and the weight parameters are all given inputs, and the review verdict only rates the supplied pair. - Expecting norm coverage beyond the declared sweep: the gamma- iteration search covers w in [1e-4, 1e4] rad/s plus the DC point and the high-frequency limit; a worst-case peak beyond 1e4 rad/s is outside the declared scope and never claimed. - Feeding a loop with pole-zero cancellation in L: the S, T and KS channel representations are not reduced, so a cancelled pair evaluates 0/0 at the cancelled dynamics; the user must supply a plant-controller pair without cancellation. - Confusing the weighted bound with a classical margin: the verdict gamma < 1 certifies the s-over-ks pointwise bounds of the given weighting only; it is not a gain margin or phase margin number (those belong to frequency-response-design). - Reading the |W1(j wb)| corner as sqrt(1 + 1/ms^2) alone: the closed form divides by sqrt(1 + as_^2), a difference of about 5.4e-7 relative at as_ = 1e-3. ## Behavior contract (gate 3) Run the deterministic contract test (stdlib unittest, offline): python3 scripts/test_h_infinity_control.py The test covers the pinned module constants of the gamma-iteration search (workflow step 5), the Routh-Hurwitz stability verdict on the characteristic polynomial (step 3), the closed-loop channel assembly with the S + T = 1 identity (step 2), the weight functions built from their corner parameters and their closed-form magnitudes (step 4), the H-infinity norm machinery on the identity cases (step 5), the worked pass and fail sibling norms with the achieved gamma and the bound verdict of the mixed-sensitivity weighting (step 6), the worst-case bound semantics, the velocity asymptote and the flat band (step 7), determinism across repeated runs, and ValueError rejection of every non-physical input enumerated in the Worked example. ## Compliance - Standards referenced, not reproduced: ARP4754A is a proprietary SAE standard (name plus paraphrase only, per standards-map.yaml); the mixed-sensitivity formulation and the gamma-iteration norm search above are standard control engineering methodology, summary-only, from Skogestad and Postlethwaite, Multivariable Feedback Control (Wiley, 2005, chapter 9) and the Doyle, Glover, Khargonekar and Francis state-space H-infinity paper (IEEE Transactions on Automatic Control 34(8), 1989) as synthesis context. - compliance: STANDARDS-REF, gated: false.