Wave-to-wire robust control for a point-absorber wave energy converter

1 Introduction

Ocean wave energy represents a significant yet under-utilised renewable resource, offering higher energy density and greater temporal predictability than either solar or wind power (Guo and Ringwood 2021). With an estimated global exploitable potential of approximately 2 TW (Gunn and Stock-Williams 2012), wave energy has attracted sustained research investment as a complementary technology for decarbonised power systems. Nevertheless, despite decades of development, the commercial viability of wave energy conversion (WEC) technology remains challenged by issues of conversion efficiency, device survivability under extreme sea conditions, and economic competitiveness relative to mature offshore wind technology (Faedo et al., 2017). A comprehensive review of recent advances across WEC hydrodynamic modelling, power take-off (PTO) mechanisms, and control strategies is provided in Guo et al. (2022), which identifies control system development as a main impactful pathway to reducing the levelised cost of energy (LCoE) of wave energy systems.

WEC devices are broadly classified, according to their operating principle, installation location, and PTO mechanism, into oscillating water columns (OWCs), overtopping devices, attenuators, terminators, and point absorbers, among others (Drew et al., 2009). Among these, the Point-Absorber Wave Energy Converter (PAWEC) class has consistently attracted the largest global share of research and development effort (Guo et al., 2022; Drew et al., 2009), owing to four principal features: (i) its compact size relative to the dominant wavelength (typically 1 / 10th to 1 / 20th) makes it omnidirectional and independent of wave heading; (ii) its comparatively simple structure is straightforward to manufacture and deploy; (iii) its small footprint allows array deployment to enhance collective power absorption; and (iv) its survivability is enhanced by reduced structural complexity compared with multi-body attenuator or terminator configurations.

A heaving PAWEC converts the oscillatory motion of a floating buoy into electrical energy via a power take-off (PTO) mechanism. The two principal PTO technologies are hydraulic systems and direct-drive tubular permanent-magnet linear generators (TPMLG); a detailed mathematical description of both the hydrodynamic and TPMLG dynamics is given in Section 2.

The control of WECs has evolved significantly over the past 2 decades, progressing from simple passive loading to sophisticated model-based and data-driven approaches (Ringwood et al., 2014; Faedo et al., 2017). Table 1 provides a structured comparative summary of the main control strategies applied to WEC systems in recent literature, including their respective advantages, disadvantages, and key references.

TABLE 1. Comparative summary of WEC control strategies.

Abbrev: PAWEC: Point Absorber WEC; TPMLG: Tubular PM, linear generator; W2W: Wave-to-Wire; MPC: model predictive control; SMC: sliding mode control; CCC: Complex-Conjugate Control.

One can categorize this as follows:

Resistive Loading and Reactive Control: Passive Resistive Loading (RL) extracts power via a damping force proportional to buoy velocity, providing simplicity but sub-optimal energy capture at off-resonance frequencies. Reactive control, also known as complex-conjugate control (CCC) or impedance matching, augments the PTO force with a stiffness term to tune the buoy toward resonance, potentially doubling energy absorption in monochromatic waves (Hals et al., 2011). However, reactive strategies require bi-directional PTO operation that induces large oscillatory stator currents, significantly amplifying copper losses in DDLG systems (de la Villa Jaén et al., 2013).

Model Predictive Control: MPC has received considerable attention due to its ability to handle physical constraints, incorporate short-term wave forecasts, and optimise multi-objective cost functions simultaneously (Cretel et al., 2011; Richter et al., 2013). Montoya Andrade et al. (2014) proposed an MPC strategy that explicitly accounts for copper losses in a TPMLG, demonstrating significant improvements in net electrical power. An adaptive MPC framework accommodating PTO losses and physical constraints for irregular sea conditions was presented by Zhan et al. (2018). More recently, an improved MPC combined with a sliding-mode inner loop to reduce model mismatch sensitivity was proposed in Wang et al. (2021). A hybrid MPC strategy for a two-body WEC with mechanical PTO, showing up to 22.5% gain over uncontrolled operation, was reported in Zhang et al. (2023). A computationally efficient nonlinear MPC scheme based on input move-blocking, designed to avoid the need for accurate wave forecasting, is presented in Zhan et al. (2025). Haider et al. (2023) compared nonlinear MPC against deep reinforcement learning for a multi-DOF WEC, demonstrating that MPC retains clear advantages in constraint satisfaction and interpretability.

Sliding Mode Control: SMC offers guaranteed robustness to matched model uncertainties and external disturbances, making it attractive for WEC systems operating in variable and harsh sea states (Abdelrahman and Patton 2017; Richter et al., 2014; Fusco and Ringwood 2013). Zou et al. (2023) proposed a sliding mode controller explicitly handling unknown noise and hydrodynamic nonlinearities, demonstrating robust energy absorption under irregular waves. Moghaddam and Mohamadie (2023) developed a fast terminal nonsingular SMC for PAWEC energy maximisation with finite-time convergence guarantees. A super-twisting second-order SMC for heaving point absorbers, combining chattering reduction with robustness to parametric uncertainty, was presented in Li et al. (2025). The combination of backstepping and SMC has demonstrated particular promise for systems with unmatched uncertainties arising from cascaded multi-subsystem models ? However, its application within a full wave-to-wire framework that includes TPMLG dynamics and copper losses remains limited in the open literature.

Data-Driven and Reinforcement Learning Control: Data-driven approaches, including Gaussian process MPC (Liu et al., 2022) and deep reinforcement learning (DRL), have emerged as promising alternatives that can avoid the need for explicit physics-based models and adapt online to changing sea conditions. A soft actor-critic DRL agent was shown to outperform linear MPC at high wave heights in Anderlini et al. (2017). More recent RL implementations demonstrate performance approaching that of optimal controllers under realistic irregular sea states (Bani Hani et al., 2026). Neural network backstepping has also been applied to OWC systems to estimate and compensate disturbances under dynamic wave conditions (Nath et al., 2025). However, data-driven methods generally lack the formal stability and constraint-satisfaction guarantees required for deployment in harsh marine environments.

Moment-Based and H ∞ Control: A theoretically rigorous moment-based framework treats the WEC control problem through the lens of system theory, enabling real-time optimal energy extraction that naturally accommodates nonlinear Froude–Krylov effects and PTO nonlinearities (Faedo et al., 2022; Faedo and Ringwood 2024). H ∞ robust control provides formal frequency-domain robustness guarantees against structured uncertainty (Richter et al., 2014), and has been applied to WEC systems as a convex alternative to Lyapunov-based designs.

Direct measurement of the WEF in operational sea conditions is impractical, rendering online estimation an essential enabling step for model-based WEC controllers (Fusco and Ringwood 2010). Kalman filter (KF) based estimators are widely used owing to their recursive formulation, but require representation of the WEF dynamics via harmonic oscillator models, which can introduce modelling error and increase estimator order (Nguyen and Tona 2017). Abdelrahman and Patton (2019) demonstrated that treating the WEF as an unknown input to a Luenberger observer avoids these approximations and improves accuracy under parametric uncertainty. Faedo et al. (2020) proposed a moment-based framework for real-time WEF estimation via KF, demonstrated on a full-scale WEC array. An extended Kalman filter (KF) approach for nonlinear PAWEC models is presented in Zhang et al. (2024). Data-based estimators using recurrent and LSTM neural networks, benchmarked against KF and unknown input observer methods using experimental Wavestar data, are evaluated in Saavedra et al. (2025).

The integration of robust WEF estimation with energy-maximising control within a unified closed-loop framework, as pursued in this work, remains an open research challenge.

The wave-to-wire (W2W) modelling paradigm integrates the complete energy conversion chain, from incident wave excitation through hydrodynamic energy absorption to electrical power generation and grid interface (Penalba and Ringwood 2016). W2W models enable co-design optimisation of hydrodynamic, structural, PTO, and control subsystems to maximise overall system performance and minimise (García-Violini et al., 2020; Tan et al., 2023; Umeda et al., 2023) developed a model-based reactive control scheme that considers copper losses and displacement constraints for a floating PAWEC, demonstrating improved net power output over conventional CCC. ? presented a fixed-time disturbance observer combined with a backstepping SMC for current tracking in a direct-drive WEC, explicitly accounting for environmental disturbances. Despite these contributions, the following critical gaps persist in the literature:

• Most existing control strategies either ignore generator electrical dynamics entirely or adopt oversimplified copper loss models, limiting their practical relevance to real TPMLG-based systems where loss minimisation is fundamental to net power maximisation.
• Few works address the unmatched parametric uncertainties that arise naturally from the coupling of buoy hydrodynamics and generator electrical dynamics within a single unified state-space representation.
• The integration of robust WEF estimation with energy-maximising robust control within a complete W2W framework, with rigorous stochastic Monte-Carlo validation of robustness under parametric scatter, remains underexplored.

To address these challenges, this paper proposes a unified wave-to-wire co-design framework for a point-absorber WEC equipped with a TPMLG. The main contributions are:

• A restructured W2W state-space model that integrates existing hydrodynamic buoy dynamics with detailed TPMLG dynamics, including copper losses and electromagnetic coupling. The model is reorganised using feedback linearisation to compensate nonlinear coupling terms and decouple the system into tractable subsystems, enabling robust energy-maximising controller design using a combination of sliding-mode and backstepping techniques that handles both matched and unmatched uncertainties.
• An H ∞-based unknown input observer for robust real-time estimation of buoy velocity and wave excitation force using only position and q-axis current measurements, reducing sensing requirements while maintaining estimation accuracy under model uncertainties.
• Comprehensive simulation studies demonstrating higher total conversion efficiency and constrained buoy motion compared to approximate complex-conjugate control (ACC) and the loss-modified ACC (ACCM) benchmark methods, with extensive Monte-Carlo validation of robustness under parametric uncertainties. All results are obtained in simulation using a validated hydrodynamic model; experimental validation of the controller is reserved for future work.

The remainder of this paper is organised as follows. Section 2 presents the W2W mathematical model for the PAWEC and the unknown input observer design for WEF and velocity estimation. Section 3 develops the sliding-mode backstepping (SMBS) controller for energy maximisation and the PI controller for d-axis current regulation. Section 4 presents simulation results, including performance comparisons with ACC and ACCM benchmark methods under regular-wave conditions only (Section 4.3), robustness assessment through Monte-Carlo analysis under regular waves (Section 4.1), and an additional evaluation under irregular wave excitation (Section 4.2). Section 5 provides concluding remarks and directions for future research.

2 System model

2.1 Dynamic modelling and power take-off mechanisms of PAWECs

A heaving PAWEC consists of a floating buoy that responds to incident waves primarily in the heave (vertical) degree of freedom, connected via a PTO mechanism to a fixed or compliant sea-bed reaction structure. The incident wave exerts a wave excitation force (WEF) on the buoy, driving it into oscillatory motion against the hydrostatic restoring force and the radiation damping of the surrounding fluid. The hydrodynamic behaviour of this interaction is described by Cummins’ equation (Falnes, 2002), given in Equation 1:where z, z ̇, z ̈ are the heave displacement, velocity, and acceleration of the buoy; m is the buoy mass; m ∞ is the infinite-frequency added mass; h r ( t ) is the radiation impulse-response function capturing fluid memory effects; K s is the hydrostatic stiffness coefficient (corresponding to K m in the numerical model of Section 2.2); F e ( t ) is the WEF; F PTO ( t ) is the PTO force; and F v ( t ) represents nonlinear viscous drag. Maximum power transfer occurs when the buoy operates near hydrodynamic resonance, where the effective PTO damping matches the radiation damping (Falnes 2002). Since ocean waves are irregular and broadband, advanced control strategies are required to dynamically retune the device impedance toward resonance across varying sea states. Recent work has further highlighted the importance of nonlinear hydrodynamic effects, particularly nonlinear Froude–Krylov forces, in accurately describing large-amplitude PAWEC motion under energy-maximising control (Faedo et al., 2022).

m + m ∞ z ̈ + ∫ 0 t h r t − τ z ̇ τ d τ + K s z = F e t + F PTO t + F v t , (1)

The PTO system is the subsystem responsible for converting oscillatory buoy motion into electrical energy. For PAWECs, PTO technology must accommodate the slow (0.05–0.5 Hz), bi-directional, large-stroke oscillatory motion imposed by ocean waves, while supporting active, real-time force control for energy maximisation. A detailed review of PTO options and their optimisation for point absorbers is given in Prasad et al. (2022). The two principal PTO technologies for PAWECs are as follows.

Hydraulic PTO: The buoy drives a hydraulic cylinder, pressurising fluid that is stored in a hydraulic accumulator before driving a rotary motor-generator pair. The accumulator provides short-term energy buffering that smooths the inherently pulsating wave-power input. Hydraulic systems offer high force density and a mechanical decoupling between the wave-absorbing and electrical sub-systems, but carry risks of fluid leakage, sealing degradation, and efficiency losses through the hydraulic circuit.

Direct-Drive Linear Generator (DDLG) PTO: The DDLG couples the translator of a linear electrical machine directly to the buoy, eliminating intermediate conversion stages. The most common realisation for PAWECs is the Tubular Permanent Magnet Linear Generator (TPMLG), in which an array of permanent magnets on the translator interacts with the armature windings of the stator to directly induce an electromotive force proportional to buoy velocity (Polinder et al., 2004; Eriksson et al., 2005). The PTO force and stator voltage equations governing the TPMLG are given in Equations 2, 3: where K f is the electromagnetic force constant, R s and L s are the stator winding resistance and inductance respectively, i s is the stator current, V s is the converter-applied terminal voltage, and K f z ̇ is the back-EMF. A power electronics converter rectifies and inverts the variable-frequency, variable-amplitude AC output for grid connection or storage. A wave-to-wire analysis of a PAWEC with a TPMLG, including detailed efficiency mapping across sea states, is presented in Tan et al. (2023). The DDLG PTO enables rapid, precise, and bidirectional force control, making it ideally suited to advanced real-time control strategies (Polinder et al., 2004). A critical drawback, however, is the copper (Ohmic) loss P C u = R s i s 2, which can consume a substantial fraction of the absorbed mechanical power, particularly under reactive control strategies that demand large oscillatory stator currents (de la Villa Jaén et al., 2013; de la Villa Jaén et al., 2014). This motivates the explicit inclusion of TPMLG electrical dynamics in the control design.

F PTO t = K f i s t , (2)

V s t = R s i s t + L s i ̇ s t + K f z ̇ t , (3)

This work considers a co-design technique for joint control and estimation for the Hull University Point Absorber Wave Energy Converter (PAWEC) model, with both the heaving buoy-water interaction hydrodynamics and the linear generator dynamics in a single state-space model. This complete model gives a wider picture about the behaviour of the PAWEC system, especially under control. In addition, consideration of this complete model is important during the PAWEC design stage as this gives an idea about the ratings of the linear generator required by the energy-maximising control and also gives an idea about the amount of expected copper losses inside the generator winding. Accordingly, the designer can choose suitable specifications of the buoy geometry and/or the generator ratings in such a way to reduce the cost along with enhancing the conversion efficiency. This idea of co-design comes from an attempt to make a more central design of a complete system, taking into account various components: energy-maximising robust control, WEF estimation, PTO dynamics, and heaving buoy dynamics.

Combining the heaving buoy and the linear generator dynamics into one model description brings nonlinear coupling between the subsystems of the model dynamics. Hence, a new organisation of the PAWEC mathematical model is introduced to overcome this coupling where the nonlinear coupling terms can be hidden from the dynamics through the use of Feedback Linearisation (Khalil, 2002). Two linear subsystems result from the application of Feedback Linearisation which facilitate the use of linear control and estimation design methods. Moreover, the modelling uncertainty becomes unmatched with the control input after considering the linear generator dynamics in the control model. So, the SM robust control techniques that do not include the linear generator dynamics in the state space model, such as the techniques proposed in Abdelrahman and Patton (2017) cannot be used, as it is only able to handle efficiently the matched uncertainty. Hence, the SM is combined with Backstepping control to handle this arising unmatched uncertainty, alongside the originally existing matched uncertainty.

As part of the WEF estimation, the buoy heaving velocity is also obtained robustly using an H ∞-based observer, see (Abdelrahman and Patton 2019; 2017). Obtaining the velocity is particularly useful from an economic point of view as it removes the need for a dedicated velocity sensor, which may be expensive to include in the PAWEC system.

2.2 Hull University PAWEC

This section presents the University of Hull 1/50th scale 1-DOF PAWEC system comprising a heaving buoy with a linear generator PTO. The prototype, shown in Figure 1, is equipped with position, acceleration, pressure, and wave height sensors, with data acquisition implemented in LabView. Extensive Computational fluid dynamics (CFD) simulations and two rounds of tank tests in the DEEP aquarium at Hull University produced a validated high-fidelity (partly) nonlinear state-space model. The layout of the experimental setup, including the measurement locations and the positions of the different elements (wave maker, wave gauges, model, and absorption beach), is shown in Figure 2. Full details of the device geometry, tank test setup, and model identification are reported in (Guo, 2017; Guo et al., 2016), which form the basis of the control and estimation designs presented in this paper.

The key physical and hydrodynamic parameters of the 1/50 scale PAWEC are summarized in Table 2.

TABLE 2. Physical and hydrodynamic parameters of the Hull University 1/50 scale PAWEC.

Assuming heave-only motion and linear wave theory, the buoy dynamics are (Falnes, 2002; Guo, 2017):where f b, f v, f r, f e, and f PTO are the buoyancy, viscous, radiation, wave excitation, and PTO forces, respectively, v ( t ) is the buoy velocity, and M is the total moving mass. The hydrostatic buoyancy force is:with hydrostatic stiffness K m = 710 N/m (Guo, 2017). Since the PAWEC is a slender structure (diameter-to-wavelength ratio < 0.2), the viscous drag is (Guo, 2017):where v w ( t ) is the water surface velocity and K v = 50.96 N⋅ s 2/m² (Guo, 2017). The radiation force is:with added mass M ∞ = 6.5 kg (Guo, 2017). The convolution term f ̄ r ( t ) = K r ( t ) ∗ v ( t ) is approximated by a third-order state-space subsystem identified from the Boundary Element Method (BEM) software NEMOH (Guo, 2017; Guo et al., 2016):where the identified radiation subsystem matrices are given in Equations 9, 10: Combining (Equations 4–8), the full PAWEC state-space model is (Guo, 2017):with state x ( t ) = [ p ( t ) v ( t ) x r ( t ) ] T ∈ R 5, control input u ( t ) = f PTO ( t ) ∈ R, output y ( t ) ∈ R 2, M t = M + M ∞, and the system matrices given in Equation 12:with A ∈ R 5 × 5, B ∈ R 5 × 1, C ∈ R 2 × 5. The term f u l ( t ) = f e ( t ) + f v ( t ) lumps the unknown wave excitation and nonlinear viscous forces, with B u l = B ∈ R 5 × 1.

M v ̇ t = f b t + f v t + f r t + f e t + f PTO t , (4)

f b t = − K m p t , (5)

f v t = − K v | v t − v w t | v t − v w t , (6)

f r t = − M ∞ v ̇ t − K r t ∗ v t , (7)

x ̇ r t = A r x r t + B r v t , f ̄ r t = C r x r t , (8)

A r = − 3.1848 − 4.3372 − 3.1009 4.3372 − 0.0875 − 0.3882 3.1009 − 0.3882 − 2.8499 , (9)

B r = − 40.6964 5.9737 16.2722 T , C r = − 0.4070 − 0.0597 − 0.1627 . (10)

x ̇ t = A x t + B u t + B u l f u l t , y t = C x t , (11)

A = 0 1 0 1 × 3 − K m M t 0 − C r M t 0 3 × 1 B r A r , B = 1 M t 0 1 0 3 × 1 , C = 1 0 0 1 × 3 0 1 0 1 × 3 , (12)

The above (nonlinear) state-space model of the 1/50 scale Hull University PAWEC has been experimentally and CFD-validated, see (Guo, 2017; Jin et al., 2018). In this model, the radiation force is approximated by a finite-order state-space representation identified from NEMOH BEM results, achieving a goodness-of-fit of G f = 0.9991. Furthermore, the nonlinear viscous drag force is explicitly incorporated via the Morison equation, see (Equation 6). This term has been confirmed, through wave tank experiments and Navier–Stokes CFD simulations, to be the dominant nonlinearity (Jin et al., 2018). It is primarily responsible for the significant discrepancy between the linear BEM response amplitude operator (RAO), approximately 5.2, and the experimentally measured RAO, approximately 1.06, near resonance (Jin et al., 2018).

Nonlinear Froude-Krylov (NLFK) forces are important for accurate modeling of wave energy converters, especially when the waterplane area varies with motion (Giorgi and Ringwood 2017; 2018; Faedo et al., 2022). However, for simplicity, they are not considered here because the WEC under study is a vertical cylindrical buoy moving in a single degree of freedom. For this geometry (radius r = 0.15 m, uniform cross-section), the waterplane area S w s = π r 2 remains constant with heave displacement. Consequently, the hydrostatic restoring force is exactly linear, and the nonlinear static FK correction (which arises from variation of the instantaneous wetted surface) is zero (Giorgi and Ringwood 2017). The dynamic FK correction is secondary for the moderate wave conditions considered here, compared to the viscous drag nonlinearity which is already explicitly included and experimentally validated by Guo (2017); Jin et al. (2018). Extensions to non-cylindrical geometries, where NLFK corrections become significant, are left for future work.

2.2.1 Tubular Permanent Magnet Linear Generator

The mathematical model of the TPMLG, in d-q axes coordinates, is given by Kundur et al. (1994); DelliColli et al. (2006); Krishnan (2010): where U d and U q are the d-axis and q-axis voltage components, respectively, and I d and I q are the corresponding current components; I ̇ d ( t ) and I ̇ q ( t ) are their respective time derivatives. K e is the back Electromotive Force (EMF) constant, R s is the generator winding synchronous resistance, and L q and L d are the q-axis and d-axis synchronous inductances. Numerical values of K e , R s , L q and L d are taken from Guo et al. (2016). T 1 ( t ) and T 2 ( t ) are (nonlinear) coupling terms given by: with L f = π / τ, where τ is the TPMLG pole pitch. The PTO force is given by Guo et al. (2016):where K f is the generator force constant. Equation 17 represents a link between the mechanical (buoy) and electrical (TPMLG) dynamic components of the PAWEC system. The numerical values of all physical parameters of the TPMLG in (Equations 13–17) are given in Table 3.

I ̇ d t = − R s L d I d t + 1 L d U d t + T 1 t , (13)

I ̇ q t = − R s L q I q t − K e L q v t + 1 L q U q t + T 2 t , (14)

T 1 = L q L f I q t v t , (15)

T 2 = − L d L f I d t v t , (16)

f PTO t = − K f I q t , (17)

TABLE 3. The TPMLG physical parameters (Guo et al., 2016).

2.2.2 Overall PAWEC mathematical model description

By combining the buoy dynamics, see (Abdelrahman and Patton 2019), with the TPMLG dynamics, (Equations 13, 14, 17), the overall mathematical model of the WEC dynamics can be expressed as: where R m ( ω ) is the radiation damping coefficient (Guo 2017). This model can be separated into two subsystems: Subsystem I contains the d-axis current dynamics, (Equation 18), while Subsystem II contains the dynamics for position, velocity, and q-axis current, (Equation 19). Subsystem I is a linear first-order state equation used only to design a regulator for I d ( t ) (see Section 3.2). Subsystem II is used for energy maximisation and WEF estimation, to generate estimates for f e ( t ) and v ( t ), using the technique described in Section 2.3.

I : I ̇ d t = − R s L d I d t + 1 L d U d t + T 1 t , (18)

I I : I ̇ q t = − R s L q I q t − K e L q v t + 1 L q U q t + T 2 t , v ̇ t = − K m M t ω p t − R m ω M t ω v t + K f M t ω I q t − 1 M t ω f e t , p ̇ t = v t , (19)

The Subsystem II dynamics can be represented in compact form as Equations 20–22: where x = [ x 1 x 2 x 3 ] T = [ p v I q ] T,and V₂(t) is defined in Equation 23:The term d ( t ) lumps all uncertainties due to model-parameter variations and un-modelled dynamics (such as the viscous force, see Equation 6), considered as un-matched uncertainty. Therefore, the SM control approach presented in Abdelrahman and Patton (2017) cannot be used. Section 3 proposes a robust SM Backstepping control scheme for this purpose.

x ̇ 1 t = x 2 t , (20)

x ̇ 2 t = − a 1 x 1 t − a 2 x 2 t + a 3 x 3 t + a 4 f e t + d t , (21)

x ̇ 3 t = − b 2 x 2 t − b 3 x 3 t + 1 L q V 2 t , (22)

a 1 = K m M t ω , a 2 = R m ω M t ω , a 3 = K f M t ω , a 4 = 1 M t ω , b 2 = K e L q , b 3 = R s L q ,

V 2 t = U q t − L q L f I d t v t . (23)

Note that T 1 and T 2 act as coupling terms. Decoupling is achieved by considering them as disturbances and subtracting them from the control inputs (see Figures 3, 4), which is a standard Feedback Linearisation method (Isidori, 2013).

As shown above, the complete PAWEC model is decoupled, via feedback linearisation, into two sub-models, I and II. First, the nonlinear coupling terms T 1 and T 2 are cancelled, leaving simple linear sub-models suitable for linear estimation and control. Second, Submodel II is structured in the Strict Feedback Form (SFF), which is necessary for applying the sliding-Mode Back-Stepping (SMBS) robust control in Section 3.

2.3 Observer-based wave excitation force estimator

To enable model-based energy-maximising control, real-time estimates of the WEF f e ( t ) and buoy velocity v ( t ) are required. Since direct measurement of f e ( t ) is impractical in operational conditions (Fusco and Ringwood, 2010), an unknown input observer (UIO) is designed for the PAWEC state-space model Equation 11. The approach follows (Abdelrahman and Patton, 2019), extended here to account for parametric uncertainty and measurement noise within a robust H ∞ framework (Lan and Patton, 2016).

For observer design, R m and M t are assumed constant at the PAWEC resonance frequency ω r, i.e., R m = R m ( ω r ) and M add = M add ( ω r ). The frequency-dependent uncertainty in a 1 , a 2 and a 3 is modelled as a disturbance in Equation 24:so that the system can be written as in Equation 25:where with a i r , i = 1,2,3,4 denoting the values of a i at resonance, and f n s ( t ) representing measurement noise on p ( t ).

d t = δ a 1 x 1 t + δ a 2 x 2 t + δ a 3 x 3 t , (24)

x ̇ t = A x t + B u t + F a f e t + D d t , x , y t = C x t + F n s f n s t , (25)

A = 0 1 0 − a 1 r − a 2 r a 3 r 0 − b 2 − b 3 , B = 0 0 1 / L q , C = 1 0 0 0 0 1 ,

D = 0 1 0 , F a = 0 a 4 r 0 , F n s = 1 0 ,

2.3.1 Augmented system formulation

To simultaneously estimate the system states, WEF, and account for sensor noise, an augmented state vector is formed as:where Here, n = 3 is the number of states, r = 1 is the number of unknown inputs, q = 2 is the number of outputs (position p and current I q), and s = 1 is the number of noisy measurements.

x ̄ ̇ t = A ̄ x ̄ t + B ̄ u t + D ̄ d ̄ t , x , y t = C ¯ x ¯ t , (26)

x ̄ t = x t f e t f n s t , A ̄ = A F a 0 0 0 0 0 0 0 , B ̄ = B 0 0 ,

C ̄ = C 0 F n s , D ̄ = D 0 0 0 1 0 0 0 1 , d ̄ = d t f ̇ e t f ̇ n s t .

2.3.2 Observer design

The following observer, inspired by (Lan and Patton, 2016), is proposed to estimate the augmented state x ̄ ( t ):where z ( t ) ∈ R n + r + s is the observer state, x ̄ ̂ ( t ) ∈ R n + r + s is the estimate of x ̄ ( t ), and M ∈ R ( n + r + s ) × ( n + r + s ), L ∈ R ( n + r + s ) × q, G ∈ R ( n + r + s ) × ( n + r + s ), H ∈ R ( n + r + s ) × q are design matrices.

z ̇ t = M z t + G B ̄ u t + L y t , x ̄ ̂ t = z t + H y t , (27)

Using (Equations 26, 27), the estimation error dynamics are:where e ( t ) = x ̄ ( t ) − x ̄ ̂ ( t ), A = E A ̄ − L 1 C ̄, E = I n + r + s − H C ̄, and L = L 1 + L 2.

e ̇ t = A e t + A − M z t + E − G B ̄ u t + A H − L 2 y t + E D ¯ d ¯ t , (28)

Given the augmented system (Equation 26) and observer (Equation 27), if there exist matrices L 1 , L 2 , H , and G such that:

• M = A = E A ̄ − L 1 C ̄ is Hurwitz,
• E − G = 0 ,
• A H − L 2 = 0 ,

and the H ∞ attenuation conditionholds for a prescribed level γ > 0, then the estimation errors e ( t ) are uniformly ultimately bounded and the WEF estimate f ̂ e ( t ) converges to a neighbourhood of f e ( t ).

sup d ̄ ≠ 0 ‖ e t ‖ 2 ‖ d ̄ t ‖ 2 < γ , (29)

Proof Under conditions 1–3, the error dynamics (Equation 28) reduce to Equation 30:

Consider the Lyapunov function V ( t ) = e T ( t ) P e ( t ) with P = P T > 0. The time derivative along trajectories of Equation 30 satisfies Equation 31:To enforce the H ∞ attenuation (Equation 29), the following Linear Matrix Inequalities (LMI) condition is required (Lan and Patton, 2016):Since A is Hurwitz by condition 1, and (32) is feasible, V ̇ ( t ) < 0 outside a compact set, guaranteeing uniform ultimate boundedness of e ( t ). The WEF estimate is then recovered as f ̂ e ( t ) = [ x ̄ ̂ ( t ) ] n + 1.

e ̇ t = A e t + E D ¯ d ¯ t . (30)

V ̇ t = e T t h e P A e t + 2 e T t P E D ¯ d ¯ t . (31)

h e P A + C T C P E D ̄ ⋆ − γ 2 I < 0 . (32)

The LMI (Equation 32) is solved using the Matlab LMI toolbox. The designer provides the attenuation level γ and the toolbox returns the symmetric positive definite matrix P and the observer gains L 1, L 2, H, and G. A smaller γ enforces tighter disturbance rejection at the cost of feasibility; γ should be increased gradually until the LMI becomes feasible.

The measurement noise f n s ( t ) on p ( t ) is included explicitly in the augmented formulation (Equation 26), so the observer (Equation 27) simultaneously estimates the WEF and rejects sensor noise without requiring a separate filter. The water surface velocity v w ( t ), needed to separate f e ( t ) from f u l ( t ), can be obtained by filtered differentiation of the water surface elevation measurement or via an Inertial Measurement Unit (IMU) (Fusco and Ringwood, 2010).

The observer performance is governed by the choice of γ. Small values of γ yield a high-gain observer with fast response to high-frequency WEF variations, but increased sensitivity to noise. Large values of γ produce a smoother, more noise-robust observer at the cost of slower convergence. The designer should iterate on γ, starting from a large value and reducing it until a satisfactory balance between disturbance rejection and noise sensitivity is achieved.

3 Controller design for PAWEC

In this section, two separate controllers are designed for each of the PAWEC subsystems using linear techniques. Sub-model II is controlled by a combination of Sliding Mode (SM) and Backstepping approaches to overcome the resulting unmatched uncertainty arising from the dynamics of the linear generator coupled with the buoy hydrodynamics. Sub-model I is controlled using a simple PI regulator for the d-axis current of the linear generator.

Backstepping (Kokotovic, 1992; Krstic et al., 1995) is a control method based on Lyapunov’s second theorem. It has the advantage of flexible design because of its recursive use of Lyapunov functions. To design a Backstepping controller, the model should be represented in so-called triangular form and then the whole system is divided into subsystems, where each subsystem is related to one of the system states. Then, the regulatory or error variable is defined and used to set a Lyapunov function to ensure the stability of each subsystem. After that, a virtual control law is defined so as to make the derivative of the Lyapunov function negative definite, hence achieve stability. These three steps are repeated until the last subsystem, where the actual control input is designed in the same way to ensure the stability of the system via an augmented Lyapunov function. Considering the features of SM and Backstepping approaches, several studies, e.g., (Koshkouei et al., 2004; Lu et al., 2011; Dong et al., 2011; Kwan and Lewis, 2000; Zinober and Liu, 1996), attempt to use a combination between both methods in order to benefit from both the simplicity and flexibility of Backstepping and the robustness and fast response of the SM.

3.1 Robust SM with backstepping control for subsystem II

This subsection presents a control technique to track an optimal reference position for the PAWEC system using adaptive Backstepping combined with SM control (Koshkouei et al., 2004). This method ensures stability and robustness against uncertainties such as un-modelled dynamics and parameter variations. The optimal reference position is generated using the impedance matching algorithm of Guo et al. (2016) to maximize absorbed wave energy. Consider the state-space representation of the q-axis subsystem (Subsystem II) from (19) and let the output be the buoy position, defined in Equation 33:We aim to design a controller to track the reference p ref ( t ):For simplicity, the ( t ) will be dropped from signal notation, e.g., x i denotes x i ( t ).

y t = x 1 t = p t . (33)

y r t = p ref t .

STEP 1: Define the error variable as in Equation 34 Defineso that Equation 36 holds: where α 1 is a design variable to ensure subsystem stability via the Lyapunov function in Equation 37:Then Equation 38 follows:Choosing α ₁ as in Equation 39:makes T ̇ l 1 negative definite if the term z 1 z 2 is cancelled in Step 3.

z 1 = x 1 − y r ⇒ z ̇ 1 = x 2 − y ̇ r . (34)

z 2 = x 2 − α 1 (35)

z ̇ 1 = z 2 + α 1 − y ̇ r , (36)

T l 1 = 1 2 z 1 2 . (37)

T ̇ l 1 = z 1 z ̇ 1 = z 1 z 2 + α 1 − y ̇ r . (38)

α 1 = − c 1 z 1 + y ̇ r (39)

STEP 2: Differentiating (Equation 35) yields Equation 40:Substituting x ̇ 2 from (Equation 21):Consider the extended Lyapunov function:Define z ₃ as in Equation 43:and substitute into (Equation 41) to obtain Equation 44:Choosewhere D bounds the disturbance d. Substitute (Equation 45) into (Equation 42) to get Equation 46:

z ̇ 2 = x ̇ 2 − α ̇ 1 . (40)

z ̇ 2 = − a 1 x 1 − a 2 x 2 + a 3 x 3 + a 4 f e + d − α ̇ 1 . (41)

T l 2 = T l 1 + 1 2 z 2 2 ⇒ T ̇ l 2 = z 1 z 2 − c 1 z 1 2 + z 2 z ̇ 2 . (42)

z 3 = x 3 − α 2 (43)

z ̇ 2 = − a 1 x 1 − a 2 x 2 + a 3 z 3 + a 3 α 2 + a 4 f e + d − α ̇ 1 . (44)

a 3 α 2 = a 1 x 1 + a 2 x 2 − a 4 f e − D + α ̇ 1 − c 2 z 2 , (45)

T ̇ l 2 = − c 1 z 1 2 − c 2 z 2 2 + a 3 z 2 z 3 . (46)

STEP 3: Introduce the sliding surface for SM control, defined in Equation 47:and the Lyapunov function in Equation 48: Choose the input V 2 to cancel cross terms and ensure negative definiteness of T ̇ l 3:where K and W are positive constants andThe actual PAWEC input isThe controller includes frequency-dependent parameters a 1 , a 2 , a 3 , a 4, which vary with time in irregular sea conditions. Instantaneous values are calculated using Guo et al. (2016) based on online estimation of ω and pre-identified functions R m ( ω ) and M add ( ω ). Proof of stability of the control law (Equation 50) is provided in Supplementary Appendix 6. In practical implementation, the discontinuous sign ( σ ) term in (Equations 49, 50) is replaced by the boundary-layer saturation function sat ( σ , ϕ ) defined in Remark 1 below.

σ = k 1 z 1 + k 2 z 2 + z 3 (47)

T l 3 = T l 2 + 1 2 σ 2 . (48)

1 l q V 2 = − k 1 z ̇ 1 − k 2 z ̇ 2 + b 2 x 2 + b 3 x 3 + α ̇ 2 − a 3 z 2 − W σ − K sign σ , (49)

sign σ = − 1 σ < 0 1 σ ≥ 0 .

U q ( t ) = V 2 ( t ) + L q L f I d ( t ) v ( t ) = L q − k 1 z ̇ 1 − k 2 z ̇ 2 + b 2 x 2 + b 3 x 3 + α ̇ 2 − a 3 z 2 − W σ − K sign ( σ ) + L q L f I d x 2 . (50)

3.2 d-axis current regulation

Subsystem I is controlled by a simple PI regulator for the d-axis current. Regulation of I d ≈ 0 maximizes the force-to-current ratio (Guo, 2017). From (Equation 13), Equation 51 follows:where V 1 ( t ) = U d ( t ) + T 1 ( t ) and the nonlinear term T 1 = L q L f I q ( t ) v ( t ) is cancelled by decoupling. Figure 3 shows the PI regulation block.

I ̇ d t = − R s L d I d t + 1 L d V 1 t , (51)

Remark 1 Two additional physical mechanisms further suppress any residual high-frequency switching content in practical implementation. First, the generator winding presents a natural first-order low-pass filter with electrical time constant τ e = L q / R s ≈ 24 ms , which substantially attenuates high-frequency voltage components before they can produce significant current perturbations; since the PTO force is proportional to I q via f PTO = − K f I q , force chattering is attenuated by the same factor. Second, in any practical implementation of the controller on the physical linear generator, a power-electronics converter will be employed. Any real Pulse-Width Modulation (PWM) converter operates at a finite carrier frequency, typically 5–20 kHz. This hardware characteristic means that the very high switching frequency implied by the ideal sign ( σ ) function cannot occur in practice: the converter inherently limits switching to its maximum carrier frequency. The PWM averaging also acts as a second low-pass stage, ensuring that the voltage actually applied to the generator windings is a smooth average of the demanded control signal rather than a high-frequency switched waveform. From a practical implementation standpoint, the combined effect of (i) the boundary-layer saturation with ϕ = 0.05 , (ii) the winding inductance low-pass filter ( τ e ≈ 24 ms ) , and (iii) the PWM carrier-frequency averaging ensures that chattering does not produce harmful high-frequency voltage spikes, excessive current ripple, or mechanical vibration in the linear generator translator assembly.

4 Simulation results

The inclusion of PTO dynamics, specifically the linear generator, with the water-buoy interaction dynamics allows consideration of losses such as copper losses due to winding resistance. Most WEC control studies use only water-buoy interaction dynamics. However, de la Villa Jaén et al. (2014) shows that copper losses in linear generators may consume a large portion of absorbed energy, leaving only reduced power to be delivered to the grid. They proposed modified RL and ACC methods that account for copper losses and are more efficient than their original forms.

To verify this, simulations calculate average values of: (a) incident power on the buoy, representing maximum power that can be absorbed mechanically; (b) absorbed (mechanical) power; and (c) electrical power reaching the linear generator output for transfer to the grid. The difference between incident and absorbed powers represents mechanical losses (friction, thermal), while the difference between absorbed and electrical powers represents copper loss in the TPMLG, determining generator efficiency.

All performance comparisons between ACC, ACCM, and SMBS presented in Section 4.3 are conducted exclusively under regular (monochromatic) wave excitation. The irregular-wave evaluation in Section 4.2 assesses the robustness of the SMBS controller only; because ACC and ACCM are not evaluated under the same irregular-wave conditions, no benchmark comparison is drawn for that section.

The incident wave power on the buoy is given by Equation 52:wwhere, for regular (monochromatic) waves, the wave energy flux per unit crest width is given by Equation 53 (Falnes, 2002; Guo, 2017)with ρ = 1025 kg/m 3 the seawater density, g = 9.81 m/s 2 the gravitational acceleration, T = 2 π / ω the monochromatic (regular) wave period, r b the buoy radius, and H the regular-wave height. This expression is valid only for monochromatic regular waves; for irregular sea states a spectral formulation based on the significant wave height and energy period would be required (Falnes, 2002).

P Wave t = 2 r b J , (52)

J = ρ g 2 32 π T H 2 , (53)

The mechanical power captured by the buoy is given by Equation 54:The electrical power delivered at generator terminals is given by Equation 55:where I phase ( t ) and V phase ( t ) are phase current and voltage, respectively.

P Mech t = f PTO t v t . (54)

P Elect t = 3 I phase t V phase t , (55)

The control and estimation architecture is simulated in MATLAB with incoming regular waves at different frequencies, corresponding to different sea states. A wave height of H = 0.07 m and wave frequencies between 2.5 and 7.5 rad/s are considered, with the device resonance at approximately ω r ≈ 5.2 rad/s (from the BEM radiation state-space model of Guo (2017)). A 200-s simulation for each of the ACC, ACCM, and proposed SMBS methods is performed using MATLAB/Simulink.

Note: The proposed SMBS controller faces two simultaneous challenges: modelling uncertainty due to the unmodelled viscous force (treated as the lumped disturbance d ( t ) in (21)) and parametric uncertainty from the deliberate variation of K m and M t with wave frequency. The Monte-Carlo study is used to quantify the controller’s robustness under both conditions, a capability not expected from ACC or ACCM, as further illustrated by the simulation results in this section.

4.1 SMBS robustness assessment under regular waves

This simulation studies SMBS robustness against parametric uncertainty in the PAWEC model under regular (monochromatic) wave excitation, where uncertainty arises from unmodelled dynamics, component aging, or extreme wave states. Parametric uncertainties are intentionally introduced in K m and M t, sampled from normal distributions with mean equal to nominal value and 10% standard deviation. Monte-Carlo simulation with 100 iterations (1000 s each) provides consistent outcomes. Mean values for conversion efficiency and maxima of buoy position, velocity, and PTO force are computed.

Figure 5 illustrates the parametric perturbations applied during the Monte-Carlo simulations, where panel (a) shows the hydrodynamic stiffness perturbation and panel (b) shows the mass perturbation. The mean value and standard deviation of the position tracking error under SMBS control, shown in Figure 6a, remain very small, demonstrating the low sensitivity of the proposed controller to parameter uncertainties. Consequently, only limited variations in the total power conversion efficiency are observed, as shown in Figure 6b. The efficiency variability increases at lower frequencies, reaching a maximum standard deviation of approximately ± 4 %.

Figure 7 shows average and standard deviation of PTO force during Monte-Carlo simulation. The controller autonomously adapts to uncertainty by generating suitable PTO force to guarantee good position tracking; the mean maximum PTO force reaches approximately 90–115 N at low wave frequencies and decreases toward resonance, with a standard deviation of approximately 7–15 N reflecting the spread across Monte-Carlo iterations.

Position and velocity variation during Monte-Carlo simulation (Figures 8, 9) show nearly zero standard deviation, meaning position and velocity remain unchanged with uncertainty, following reference signals despite model discrepancy, consistent with small tracking error. In Figures 6–9, the centre lines indicate the mean values, while the shaded regions represent the variability, with a width equal to twice the standard deviation.

4.2 Irregular wave robustness assessment

The reference position signal under irregular excitation was generated by applying the impedance-matching algorithm of Guo et al. (2016) using the estimated wave excitation force f ̂ e ( t ) produced by the OBUIE observer (Section 2.3). Consequently, the reference generator relies on the real-time estimated excitation force rather than perfect non-causal future knowledge of f e ( t ). Any estimation error in f ̂ e ( t ) therefore directly affects the quality of impedance matching and hence the reported efficiency figures. Readers should interpret the efficiency values below as achievable under causal, real-time estimation rather than under an oracle wave forecast; translating these results to a practical implementation does not require additional non-causal wave prediction.

The SMBS + OBUIE architecture was evaluated under irregular wave excitation using the Monte Carlo framework described above. Figure 10 shows the mean mechanical, electrical, and total conversion efficiencies across the frequency sweep, with shaded regions indicating ± 1 σ variability. The SMBS controller achieves a peak total efficiency of approximately 10.8% near hydrodynamic resonance (ω p ≈ 5.2 rad/s), which is significantly lower than the 31% peak observed under regular waves (Section 4.1). This reduction is physically consistent, as irregular waves introduce broadband spectral content and stochastic phase relationships that degrade impedance matching, while the OBUIE observer must estimate excitation forces and separate viscous effects using only position and current measurements. Nevertheless, the efficiency variability remains modest (± 2–4%), confirming that the controller maintains consistent performance despite parametric uncertainty. As shown in Figure 11, attempts to exploit displacement margin by exaggerating motion lead to reduced PTO efficiency, which may become negative, indicating that under irregular excitation the controller should operate with relatively small excursions to ensure positive net energy conversion.

Figure 11 presents the Monte Carlo statistics for maximum buoy position, velocity, PTO force, and tracking error. The SMBS controller respects all physical constraints (| z | ≤ 0.30 m, | v | ≤ 2.0 m/s) across all iterations, with mean maximum position and velocity remaining well below their limits. The PTO force exhibits larger variability (standard deviation 5–15 N) than motion signals, reflecting the controller’s autonomous adaptation to maintain tracking under model mismatch. Compared to the regular wave case, the irregular wave results show slightly elevated motion amplitudes at low frequencies (ω p < 3.5 rad/s), where the broadband wave spectrum amplifies the effect of observer estimation error. Overall, the irregular wave Monte Carlo analysis provides simulation-based evidence of the robustness of the proposed wave-to-wire co-design: the SMBS controller achieves robust energy maximisation, implicit copper loss reduction, and constraint satisfaction under realistic sea conditions and parametric uncertainty. Since ACC and ACCM were not evaluated under the same irregular-wave conditions, no direct benchmark comparison is drawn here.

4.3 Comparison with ACC and ACCM under regular waves

The proposed SMBS controller is compared, under regular (monochromatic) wave conditions, against the benchmark controllers ACC and ACCM. Mean mechanical, electrical, and total conversion efficiencies from Monte-Carlo simulation under ACC, ACCM, and SMBS with 10% parametric uncertainties are shown in Figures 12–14. ACC mechanical efficiency is relatively low, with a peak of 15% at 3 rad/s. ACCM is better, especially below 5 rad/s, peaking at 35% at 2.5 rad/s. SMBS is highest at all frequencies except 2.5 rad/s (about 5% lower than ACCM). At resonance (ω r ≈ 5.2 rad/s), SMBS achieves 28% mechanical efficiency, about 21% higher than ACC and ACCM.

ACC electrical efficiency deteriorates quickly off-resonance. Outside 4–6.5 rad/s, it becomes negative, meaning power is drawn from grid to sea. ACCM significantly improves electrical efficiency (Figure 13), ranging from 50% at highest frequency to nearly 100% at 4.5 rad/s. SMBS averages best in electrical efficiency, very close to ACCM at resonance and better off-resonance.

SMBS superiority is clear in total conversion efficiency (Figure 14), higher than ACC and ACCM at all frequencies, reaching 31% peak compared to only 6% for ACC and 21% for ACCM. ACCM outperforms ACC due to explicit copper loss reduction. SMBS reduces PTO copper losses implicitly.

PTO force maxima (Figure 15) reveal an important trade-off of the SMBS controller. Below resonance ( ω < ω r ), SMBS requires the highest PTO force among the three methods, reaching approximately 115 N at 2.5 rad/s and remaining above 90 N across the sub-resonance range. This is the cost of the SMBS achieving higher energy capture and robustness: the controller applies a larger force to drive the buoy onto the impedance-matched reference trajectory despite model uncertainties and unmodelled viscous drag. ACC produces a moderately high force at low frequencies (approximately 63 N at 2.5 rad/s), which decreases steeply toward resonance. ACCM maintains the lowest and flattest force profile throughout (≈ 13–15 N), owing to its explicit copper-loss minimisation. Near and above resonance ( ω ≥ ω r ), SMBS force drops sharply to values comparable to ACCM, while ACC rises again at high frequencies (exceeding SMBS above approximately 6.5 rad/s). These results confirm that the improved total conversion efficiency of SMBS (Section 4.3) is achieved at the expense of higher PTO force demands below resonance, a design trade-off inherent to energy-maximising control under parametric uncertainty.

Motion signals (Figures 16, 17) show ACC produces increasing motion at lower frequencies, potentially violating limitations, indicating large parametric uncertainty effects. ACCM improves motion, keeping position within ± 0.15 m and velocity within ± 0.5 m/s. SMBS achieves improved conversion efficiency with smallest motion average, advantageous as it respects limitations and enhances safety.

Finally, it is important to emphasise that the reported differences in conversion efficiency are not generic statements, but are directly supported by the presented results and consistent with the underlying control design assumptions. In particular, the low average efficiency obtained with ACC (approximately 6 %) arises from two compounding factors: (i) the method is formulated to maximise mechanical power absorption only, without accounting for copper losses, and (ii) it lacks robustness to modelling uncertainties, including unmodelled viscous forces and parametric perturbations in K m and M t. When these effects are introduced in the Monte-Carlo simulations, a significant degradation in total conversion efficiency is observed, which is consistent with previously reported findings (de la Villa Jaén et al., 2014; Montoya Andrade et al., 2014), which highlighted that ACC performance considerably drops if copper losses were not explicitly considered in the energy maximisation control. A more appropriate and fair comparison is therefore between ACCM and the proposed SMBS method. ACCM, which explicitly accounts for copper losses, achieves a peak total conversion efficiency of approximately 21 %, whereas the proposed SMBS method reaches approximately 31 % (see Figure 14). This improvement can be directly attributed to the fact that, in addition to accounting for copper losses, SMBS explicitly incorporates robustness against parametric uncertainty and unmodelled dynamics. As a result, the observed performance ordering—ACC ( ∼ 6 % ), ACCM ( ∼ 21 % ), and SMBS ( ∼ 31 % )—is physically justified, systematically reproduced across the Monte-Carlo simulations under regular wave conditions, and fully supported by the evidence presented in this section.

5 Conclusion

An architecture for the control and estimation of a 1 Degree of Freedom (DOF) heaving buoy with a linear generator PTO has been presented in this paper. In the literature of PAWEC control, the PTO dynamics are usually ignored under the assumption of ideal PTO, i.e., with no losses. However, this could be misleading as some control strategies, e.g., the ACC, under certain sea states, may indirectly increase the PTO losses in such a way that all absorbed energy is consumed, leaving nothing to deliver to the grid or even drawing energy back from the grid to the waves. Hence, the inclusion of the PTO dynamics in the control design allows the consideration of copper losses in the linear generator, providing a more practical assessment of the useful power that can be delivered by the PAWEC to the grid.

The architecture includes a state observer dedicated to robustly estimating the wave excitation force. The second part of the architecture is a sliding-mode backstepping (SMBS) robust controller to track an optimal position reference computed based on the complex-conjugate control concept. In addition, the regulation of d-axis current is achieved using a simple proportional-integral (PI) controller. Simulations have been performed with regular wave input at wave frequencies 2.5–7.5 rad/s and a wave height of 0.07 m, with additional Monte Carlo evaluation under irregular (Pierson-Moskowitz, PM) wave excitation. All results are obtained through simulation using a hydrodynamic model that has been experimentally and CFD-validated (Guo, 2017; Jin et al., 2018); the proposed controller itself has not yet been experimentally validated, and the findings constitute model-based evidence of feasibility rather than experimentally confirmed performance.

The simulation results show that the proposed architecture achieves a total power conversion efficiency higher than the modified ACC method proposed by Montoya Andrade et al. (2014), which improves the deteriorated performance of the ACC method when PTO copper losses are considered. The SMBS has the advantage of implicitly reducing the copper losses in the linear generator PTO mechanism. Moreover, the SMBS achieves this improved power conversion efficiency with constrained buoy motion, respecting motion constraints and improving operational safety.

Future work includes experimental tank test validation of the proposed method, and extension of the architecture to multi-degree-of-freedom wave energy converters and other types of WECs with more challenging dynamics, including nonlinearities due to Froude–Krylov (NLFK) forces. Although the proposed techniques for both WEF estimation and energy-maximising control are robust against model uncertainties, the architecture still incorporates a reference generation block (Fusco and Ringwood 2013; Guo et al., 2016), as shown in Figure 18, which assumes accurate knowledge of the WEF (or, in the irregular-wave case, a sufficiently accurate real-time estimate; as noted in Section 4.2, the reference generator uses f ̂ e ( t ) from the OBUIE observer, so estimation errors directly affect energy maximisation performance). This represents a limitation of the proposed architecture and highlights a bottleneck for impedance-matching energy-maximising control strategies, where energy is maximized by ensuring the reference velocity of the point absorber is in phase with the WEF. Addressing this challenge (developing reference generation methods that are robust to WEF estimation errors or that can operate without explicit WEF knowledge) remains an important direction for the research community and is posed here as future work.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fenrg.2026.1785204/full#supplementary-material

Proof of SMBS Controller Stability. Supplementary material at the source

Supplementary material Supplementary material at the source

Footnotes

Edited by: Mohammed Hannan, University of British Columbia, Canada

Reviewed by: Carlos Pérez-Collazo, University of Vigo, Spain

Siyang Tang, Loughborough University, United Kingdom