Four technical notes on finite-frequency H∞ recoil control of deepwater drilling risers

Four technical notes on finite-frequency H∞ recoil control of deepwater drilling risers

Each note is self-contained: model, assumptions and results are stated in the note itself. Further details are 

available in the author's M.Sc. thesis [9], listed in the Iranian Scientific Information Database (IranDoc / Ganj).

Note 1. Finite-frequency H∞ state feedback for riser recoil suppression

Problem. After emergency disconnection of a deepwater drilling riser, the energy stored in the riser is released and the tensioner must damp the recoil motion. Delayed-feedback H∞ recoil control [1] has been reported, but full-band H∞ synthesis spends control effort at frequencies where the disturbance carries little energy.

Model and disturbance. A three-mass, spring–damper model of the axial riser–tensioner dynamics is written as ẋ = Ax + Bu + Dd, with the controlled output z = C_z x + B_z u + D_z d selecting the three mass displacements. The disturbance d combines platform heave (JONSWAP wave spectrum, effective support ≈ 0.25–5 rad/s [8]) and mud-discharge friction (≈ 0–0.3 rad/s). The performance index is therefore imposed only on Ω = [0, 5] rad/s: sup_{ω∈Ω} σ̄(G_zd(jω)) < γ.

Method. By the generalized KYP lemma [2, 3], this frequency-domain inequality is converted into linear matrix inequalities (LMIs) in the state-feedback gain, following the finite-frequency offshore-platform design of [4].

Result. Compared with two delayed H∞ controllers from [1] (artificial delays 0.02 and 0.163), the finite-frequency controller reduced the peak responses of the three masses by about 92 %/65 %, 95 %/79 % and 91 %/55 % (versus the 0.02/0.163 designs). It required a higher peak control force (4.01×10⁷ N versus 3.80×10⁷ and 1.62×10⁷ N) and its state-energy index was slightly higher than that of the 0.02-delay design (0.251 versus 0.236 m²).

Message. Restricting the H∞ index to the disturbance band lowers response peaks at a moderate increase in actuator effort. No claim of lower control energy is made.

Note 2. Non-fragile finite-frequency control with hard actuator and displacement constraints

Problem. Tensioner force is bounded and the relative displacements between riser segments must stay within allowed limits; the implemented gain also differs from the designed one because of ageing and tolerances.

Formulation. The applied law is u = (K + ΔK(t))x with ΔK(t) = M_k F(t) N_k and F(t)ᵀF(t) ≤ I. Hard constraints |u_i| ≤ u_i,max and |(Cx)_j| ≤ 1 (relative displacement normalised by ρ_max) are enforced with a quadratic Lyapunov function V = xᵀWx and an energy bound d_max on the admissible disturbance, which yields additional LMIs that are solved jointly with the finite-frequency condition of Note 1. The treatment of hard constraints in a finite-frequency KYP framework follows [4]; the KYP-based synthesis with parameter uncertainty follows [2, 3].

Simulation setting. Gain uncertainty with M_k = 0.1·1 and N_k = 0, actuator bound 10⁸ N, γ = 4.8.

Message. One convex program covers finite-frequency performance, gain fluctuation and hard constraints. Conditions are sufficient only; the conservatism of the level-set constraint is not quantified.

Note 3. Resilience to random denial-of-service on the actuation channel

Model. Transmission of the control signal is modelled as u(t) = η(t)(K + ΔK)x(t), with η(t) ∈ {0, 1} a Bernoulli sequence, Prob{η = 1} = η̄ (η̄ = 0.8 in the simulations), so the closed loop is ẋ = (A + η(t)B(K + ΔK))x + Dd. This is a stochastic counterpart of the deterministic duration/frequency DoS models of [5, 6]; mean-square stability replaces the average-duration conditions, which allows the finite-frequency LMIs of Note 1 to be retained.

Result. With DoS, gain uncertainty and hard constraints active, three designs were obtained by reducing the displacement bound ρ_max² from 1 to 0.5 to 0.2 m². The peak relative displacement M₁ fell 0.342 → 0.168 → 0.096 m, while the peak force did not decrease monotonically (1.91, 1.45, 2.16 ×10⁷ N).

Limitations to state. (i) Results are from a single realisation of η(t). (ii) The open-loop system already has small peak relative displacements (M₁ = 0.084 m, M₂ = 0.072 m), so the benefit is in decay of the response and cumulative energy, not in peaks, and is shown only for the two tighter designs. (iii) Reactive or intelligent jamming is not covered by a Bernoulli model.

Related finite-frequency work with imperfect actuation: reliable finite-frequency H∞ control with actuator faults [7].

Note 4. Trade-off between displacement limiting and actuator aggressiveness under DoS

Observation. Sweeping the single design parameter ρ_max produces a monotone reduction of peak relative displacement and of cumulative displacement energy (0.342 → 0.096 m; 9.6 → 2.6 ×10⁴ m²) while the control-energy index rises (0.81 → 0.89 ×10⁶ N²) and the tightest design shows a control spike when the channel returns after an outage. The middle design (ρ_max² = 0.5 m²) has the lowest peak force.

Use. The sweep yields a compact Pareto view (peak displacement, peak force, control energy) from which an operator can choose ρ_max according to available actuator headroom.

Suggested completion before submission. Repeat the sweep for the delayed-H∞ designs of [1] under identical DoS sequences; run a Monte-Carlo study over η(t); report the achieved γ for each design.

References

 [1]W. Zhang, B.-L. Zhang, Q.-L. Han, F.-B. Pang, Y.-T. Sun, X.-M. Zhang, "Recoil attenuation for deepwater drilling riser systems via delayed H∞ control," ISA Transactions, vol. 133, pp. 248–261, 2023. doi:10.1016/j.isatra.2022.07.003

[2] T. Iwasaki and S. Hara, "Generalized KYP lemma: unified frequency domain inequalities with design applications," IEEE Transactions on Automatic Control, vol. 50, no. 1, pp. 41–59, 2005.

[3] T. Iwasaki and S. Hara, "Feedback control synthesis of multiple frequency domain specifications via generalized KYP lemma," International Journal of Robust and Nonlinear Control, vol. 17, no. 5, pp. 415–434, 2007.

[4] A. Kazemy, J. Lam, and X. Li, "Finite-frequency H∞ control for offshore platforms subject to parametric model uncertainty and practical hard constraints," ISA Transactions, vol. 83, pp. 53–65, 2018. doi:10.1016/j.isatra.2018.08.007

[5] C. De Persis and P. Tesi, "Input-to-state stabilizing control under denial-of-service," IEEE Transactions on Automatic Control, vol. 60, no. 11, pp. 2930–2944, 2015. doi:10.1109/TAC.2015.2416924

[6] S. Feng and P. Tesi, "Resilient control under denial-of-service: robust design," Automatica, vol. 79, pp. 42–51, 2017.

[7] H. Quan, T. Zhang, X. Yan, and F. Jia, "Reliable finite-frequency H∞ control for switched systems with actuator faults and mode-dependent average dwell time," International Journal of Systems Science, vol. 55, no. 9, pp. 1907–1923, 2024.

[8] K. Hasselmann et al., "Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP)," Ergänzungsheft zur Deutschen Hydrographischen Zeitschrift, Reihe A, no. 12, 1973.

[9] S. A. Hosseini, "Design and simulation of a finite-frequency H∞ controller for riser recoil suppression in offshore drilling operations with practical constraints" (in Persian: «طراحی و شبیه سازی کنترل کننده H∞ فرکانس محدود برای سرکوب پس زدگی رایزر در عملیات حفاری دریایی با در نظر گرفتن قیود عملی»), M.Sc. thesis, Dept. Elect. Eng., Tafresh Univ., Tafresh, Iran, 2026 (supervisor: A. Kazemy). Available in the IranDoc (Ganj) database, 

https://ganj.irandoc.ac.ir (search by title).

Data availability. The complete derivations, simulation setup and numerical results are contained in the author's M.Sc. thesis [9], which can be found in the IranDoc (Ganj) database.