Lidar is a remote sensing method for measuring wind speed that has gained attention in the wind energy industry in recent years. It enables additional wake measurements to be incorporated into wind turbine controllers, thereby facilitating the development of advanced control strategies . A detailed explanation of the lidar, the model, and the practical concerns can be found in Appendix . In this work, we adopt a continuous-wave lidar due to its uniform return time of all measurements, which simplifies controller design.

Figure  shows a three-dimensional view of the lidar setup: a lidar is mounted on the top of the wind turbine nacelle, orienting downwind. The lidar measures the wind speed information of a plane with the same diameter as the rotor disk with a focal distance of 1D. This distance is chosen as a trade-off between wake data quality, measurement feasibility, and the upper bandwidth limit of the controller imposed by output delay, further discussed in Sect. . Furthermore, the lidar captures the flow information across the entire rotor disk by simultaneously sampling 80 points uniformly distributed in the Cartesian coordinate system. The number of sampling points is determined to balance the spatial resolution and the computational cost. Lastly, the lidar is modeled to have a sampling frequency of 10Hz to ensure consistency with the QBlade simulation time step of 0.1s.

This section identifies a suitable controlled variable in the helical wake experimentally by analyzing the lidar-sampled data in simulations with the helix approach being activated.

Figure  presents lidar sampling snapshots in QBlade over a complete helix cycle Te. A counterclockwise (CCW) helix is applied with a Strouhal number of St=0.3, a uniform inflow wind speed of 10ms-1, and a helix amplitude of 3°. In the figure, wind speed is visualized through a color map, where higher velocities correspond to brighter shades. A distinct high-velocity region is observed rotating at the excitation frequency fe, corresponding to the hub vortex of the wind turbine. This structure extends along the streamwise direction and has been previously documented in . The findings of  further confirm the existence and rotational behavior of the hub vortex in helical wakes. Additionally, we analyzed sampled data under varying wind conditions, including vertical shear (exponential factor 0.2), turbulence (intensity 6%), and their combination, while maintaining an average wind speed of 10ms-1. In all cases, the hub vortex remains distinguishable and retains its rotational behavior.

Experiments were conducted by randomly varying the helix amplitude between 1 and 6° and the Strouhal number between 0.1 and 0.4 while recording the rotation magnitude and frequency of the hub vortex. These parameter ranges were selected as the regime in which the helix is most effective; see . The correlation between the hub vortex rotation magnitude and the helix amplitude was found to be 0.9987, and the correlation between the hub vortex rotation frequency and the Strouhal number was 0.9995. These results show that helix amplitude changes directly affect hub vortex rotation magnitude, while Strouhal number changes affect rotation frequency. This strong correlation indicates that controlling the hub vortex is equivalent to controlling the helical wake. Additionally, the hub vortex is readily distinguishable through signal filtering due to its elevated velocity. It is therefore selected as the controlled variable of the helical wake.

A noticeable characteristic of the hub vortex is its continuous rotation relative to the fixed frame at the helix frequency fe. To simplify the controller design, the helix frame transformation proposed by  is employed to map the hub vortex from the fixed frame to the helix frame, in which the vortex appears relatively stationary.

To achieve this, the principle of modulation–demodulation is applied, transforming the rotating helix from the rotating coordinate frame to the helix coordinate frame, shifting the 1P+fe helix rotation (assuming a CCW helix) to the DC gain. The summary of the frequency of the helix in different coordinate frames is shown in Table . The derivation of the helix frame transform resembles the MBC transform. The main difference is that the excitation frequency ωe is included: 5βcoleβtilteβyawe=Tcm(ωet)β1β2β3, where 6Tcm(ωet)=231/21/21/2cos⁡(ω1)cos⁡(ω2)cos⁡(ω3)sin⁡(ω1)sin⁡(ω2)sin⁡(ω3), and ωi=ψi+ωet, representing the CCW helix frequency. A detailed explanation of the helix frame transform can be found in Appendix . As a result, this transform maps the rotating input and output signals, (βtilt,βyaw) and (z,y), to a more static signal representation, (βtilte,βyawe) and (ze,ye), simplifying the subsequent controller design. Note that mean centering needs to be applied to signals to eliminate extra oscillating components.

As a result, the overall lidar-processing pipeline is developed as shown by Fig. , consisting of three parts:

Lidar sampling. Capture wind speed data uLOS at the specified focal distance of 1D downwind.

Figure  presents the output of the data-processing pipeline. The helix approach, configured with St=0.3 and u=10ms-1, is activated during two discrete intervals: from 50 to 150 s and from 250 to 350 s. Outside these periods, the wind turbine operates under baseline conditions. The result confirms that the pipeline effectively maps the rotating signals from the fixed frame to the helix frame. Although the signal in the helix frame is not fully static, likely due to residual noise that the low-pass filter cannot fully eliminate, a clear trend toward a constant value is still observable.

The presented controller follows the idea of internal model control, in which the difference between the actual system output and a predicted output is used within the controller to regulate the system . Therefore, a model G that describes the dynamics of the helical wake effect, or the dynamics of (βtilte,βyawe)→(ze,ye), is essential. In this work, the internal model G is acquired through system identification of the experimental data, focusing on wind speed uin=10ms-1. The primary motivation for deriving this model is to simplify the subsequent controller design and to ensure computational efficiency. Since the proposed framework is intended for real-time implementation, the surrogate model must maintain a low computational cost. Since both input and output signals are mapped to the helix frame, where they exhibit approximately linear behavior, building a linear model is an effective choice.

The detailed introduction of the system identification process can be found in Appendix . Consequently, the frequency domain response of the identified model is illustrated in Fig. , indicating several observations:

The identified system successfully captures the system dynamics within the frequency range.

Since this research employs a linear system identification method, G is linearly time invariant. Consequently, G is only applicable within a specific operating range. Although the model G exhibits steady-state decoupling, strong couplings at the bandwidth frequency, supported by the RGA matrix at the bandwidth frequency of -1.27802.27802.2780-1.2780, complicate the design of a decentralized controller combined with a pre-compensator, such as a diagonal control structure adopted in . A diagonal controller could be implemented by choosing a very low crossover frequency, but the slow reaction time would offer limited benefit. Consequently, an H∞ controller is adopted for its robustness to uncertainty and modeling errors in MIMO systems.

The H∞ controller synthesis uses the general control configuration, as shown in Fig. , where P is the generalized plant and K the generalized controller.

The idea of formulating a general control problem is to find a controller K to minimize the H∞ norm of the transfer function from input w to performance output z; see . Weight functions WU, WT, and WP are integrated to provide different weighted performance measures from outputs z1 to z3. The H∞ controller synthesis considers three criteria to design and evaluate the performance of the controller: the sensitivity S, the complementary sensitivity T, and the controller sensitivity U, defined as 9S=11+GK,T=GK1+GK,U=K1+GK. For the physical interpretation, S gives the transfer function from the disturbance to the system output, T is the transfer function from the reference to the output, and -U is the transfer function from the disturbance to the control signal.

As a result, the controller K is obtained by solving the mixed-sensitivity optimization problem, as defined in Eq. (): 10min⁡KWPSWTTWUU∞=min⁡KWP(1+GK)-1WTGK(1+GK)-1WUK(1+GK)-1∞. For the closed-loop system, a good disturbance rejection is desired, and therefore, S should be small for low frequencies. Furthermore, the control effort should be limited by having a roll-off in T after the bandwidth. Based on this, the weight functions are designed. WP(s) is designed as a low-pass filter with the form of 11WP(s)=s/M+ωcls+Aωcl, where ωcl denotes the desired closed-loop bandwidth, A is the desired disturbance attenuation inside the bandwidth, and M is the desired bound on ||S||∞ and ||T||∞. The upper bound M and lower bound A of S are set to M=10 and A=0.625, respectively. The desired closed-loop bandwidth ωcl is set to ωcl=0.02rads-1 due to the limitation introduced by the non-minimum-phase zeros of G(s). Specifically, the upper bound of ωcl is selected to remain below z/2, where z is the real part of the non-minimum-phase zeros in continuous time, as proven in . These non-minimum-phase zeros are likely introduced by unmodeled delays in the system: the residual pitch actuator delay that is not fully compensated for by the azimuth offset ψoff, as well as the inherent downwind sensing delay. It remains unclear whether the intrinsic dynamics of the helical wake also contribute to this non-minimum-phase behavior. Therefore, future work should investigate whether the helix-induced dynamics influence the emergence of non-minimum-phase zeros.

The non-minimum-phase zero indirectly limits the controller gains for the stability requirement. Consequently, the controller weight function WU(s) is designed as a band-limited high-pass filter to attenuate the control input in the high frequency as shown below: 12WU(s)=0.4B2⋅s2+2ωc+ωc2s2+B2ωcs+(Bωc)2. Parameter B scales the frequency at which control effort starts to be limited, and ωc is related to the cutoff frequency. In this study, B is selected as 10, and the crossover frequency ωc is set to 0.15rads-1 according to the pitch controller changing rate; a scaling factor of 0.4 is adopted to ensure the controller achieves the trade-off between performance and robustness without overly penalizing control effort. Finally, WT(z) is kept at 0 since we focused on tracking performance and disturbance rejection at low frequencies .

Under the shear condition, the closed-loop system effectively corrects the steady-state bias in the hub vortex trajectory, leading to a measurable increase in downstream power production. This improvement is driven by the redirection of the helical wake, which results in higher average inflow wind speeds at the 4D downstream position for the closed-loop case (CL2) compared to the open-loop baseline (OL2). However, these gains are accompanied by a trade-off between increased DELs for both turbines and a slight reduction in upstream power. These effects stem from the more aggressive control actions required for bias correction. Specifically, as shown in Fig. , the closed-loop controller superimposes oscillatory components onto the tilt and yaw pitch inputs (βtilte and βyawe). This contrasts with the constant inputs used in the OL2 case and likely contributes to the increased fatigue observed in the upstream turbine.

Under the turbulence condition, the closed-loop system performs according to its design limitations. Analysis of the hub vortex trajectory reveals no significant improvement in wake movement between the open-loop (OL3) and closed-loop (CL3) cases. This is primarily because the dominant turbulence frequency exceeds the controller’s roll-off frequency, preventing effective tracking of the rapid fluctuations. Consequently, the downstream power remains largely unchanged, as the wake recovery is dominated by natural turbulent mixing.

However, a slight increase in overall power production is observed, largely attributed to the upstream turbine. This can be explained by the comparison of the pitch input signals in Fig. , which shows that the closed-loop controller achieves a lower time-averaged magnitude despite more dynamic inputs. Specifically, the time-averaged magnitudes of βtilte and βyawe are 2.95 and 2.93, respectively, compared to the constant value of 3 in the open-loop case. As established by , this reduction in average pitch magnitude improves upstream power production. This demonstrates that incorporating wake measurements allows the controller to reduce unnecessary actuation compared to a fixed input, thereby enhancing overall performance and supporting the value of closed-loop wake control.

Regarding structural loads, the flapwise DEL increases for the downstream turbine and the farm overall, whereas the edgewise DEL remains nearly unchanged. This result is consistent with , who noted the lower sensitivity of edgewise DEL to variations in helix magnitude.

In the combined shear and turbulence scenario, the closed-loop system effectively corrects the shear-induced steady-state bias, though it does not fully mitigate the additional turbulence-induced oscillations. This correction facilitates enhanced wake mixing, leading to increased power production for the downstream turbine and the wind farm overall. However, these gains come at the cost of higher fatigue and a power reduction in the upstream turbine.

Compared to the open-loop baseline (OL4), the hub vortex in the closed-loop case (CL4) exhibits larger oscillations. This behavior can be explained by two primary reasons. The first reason is the higher time-averaged pitch actuation of 3.06 (β¯tilte) and 3.03 (β¯yawe) due to the increased control effort required to simultaneously correct the shear-induced bias and the attempt to stabilize turbulence-induced oscillations. The second reason is the close relationship between enhanced wake recovery and premature vortex breakdown, a phenomenon supported by the findings of  and . Consequently, the increased hub vortex oscillations in CL4 may be explained by this accelerated vortex breakdown. While these results are promising, further fluid dynamics studies are necessary to fully characterize this phenomenon. Finally, the power gain in this combined case is larger than that observed in the shear-only case (CL2), as the inherent natural mixing from the turbulence further accelerates the wake recovery process.

These observations demonstrate the closed-loop framework's efficacy in correcting shear-induced steady-state bias while simultaneously highlighting its performance boundaries under high-frequency turbulence. These results establish the technical feasibility of the controller but also show several practical trade-offs. To bridge the gap between simulations and field deployment, the following section provides a detailed discussion on the practical implementation and broader implications of the proposed framework.