## 2 Materials and Methods
### Control Design
#### 2.8.2 Adaptive Robust PI Controller  
To withstand disturbances of realistic, persistent but slow wind velocity, let the extended error be  
\[S=\dot{q}-\dot{q}_{r}\equiv\Delta q+\alpha\Delta q \tag{33}\]  
where \(\dot{q}_{r}=\dot{q}_{d}-\alpha\Delta q\) is a velocity nominal reference, for \(\alpha\in\mathbb{R}_{+}\). Then, adding the state bounded function \(Y_{r}(\cdot)=H(q)\dot{q}_{r}+C(q,\dot{q})\dot{q}_{r}+g(q)-\tau_{fr}\) to (29); for \(\tau_{fr}=B\dot{q}_{r}+K\tanh(\lambda q_{r})\), with _tanh_ the piecewise hyperbolic tangent function and \(\lambda\) a constant diagonal matrix, one obtains the following open loop error equation in a \(S\)-coordinate system  
\[H(q)\dot{S}+C(q,\dot{q})S=\tau+\tau_{D}-Y_{r}(\cdot) \tag{34}\]  
We have the following result.  
**Proposition 1**.: _Consider the decentralized Model-free Adaptive Proportional-Integral (A-PI) robust controller_  
\[\tau =-K_{D}S-K_{I}I \tag{35a}\] \[\dot{I} =S \tag{35b}\]  
_in closed loop with (34). Then, the closed-loop system yields asymptotically robust regulation provided that disturbance can be parameterized as constant in the time window before the next desired via point._  
Proof.: Substituting (35) into (34), one obtains the closed loop system  
\[H(q)\dot{S}+C(q,\dot{q})S+K_{D}S+K_{I}\Delta I=-Y_{r}(\cdot) \tag{36}\]  
for \(\Delta I=\hat{I}-K_{I}^{-1}\tau_{D}\) and \(\hat{I}=K_{I}\int_{t_{0}}^{t_{f}}Sdt\). Now, consider the Lyapunov function \(V=\frac{1}{2}S^{T}H(q)S+\frac{1}{2}\Delta I^{T}K_{I}\Delta I\), whose time derivative becomes \(\dot{V}=S^{T}H(q)\dot{S}+\frac{1}{2}S^{T}\dot{H}(q)S+\dot{\Delta I}^{T}K_{I} \Delta I\). Assuming that disturbance is slowly time varying in \(\Delta t=t_{f}-t_{0}\) and that the settle time of the controller is less than \(\Delta t\), then \(q(t_{f})\approx q_{d}(t_{f})\); thus, \(\frac{d}{dt}\tau_{D}(q_{d}(t_{f}))\approx 0\) given that \(q_{d}(t_{f})\) is a constant, and \(\tau_{D}(q_{d}(t_{f}))\) approximates to a constant.  
This chain of arguments leads to \(\Delta I=\hat{I}-K_{I}^{-1}\tau_{D}\equiv\hat{I}\), and then one obtains \(\dot{V}=S^{T}H(q)\dot{S}+\frac{1}{2}S^{T}\dot{H}(q)S+\Delta I^{T}K_{I}\hat{I}\). Evaluating \(\dot{V}\) along solution (36) yields \(\dot{V}=-S^{T}K_{D}S-S^{T}Y_{r}(\cdot)-S^{T}K_{I}\Delta I+\Delta I^{T}K_{I} \hat{I}=-S^{T}K_{D}S-Y_{r}^{T}S\), where we have used the skew-symmetric property of Coriolis and inertia matrices \(C(q,\dot{q})+C^{T}(q,\dot{q})=\dot{H}(q)\)[31] and Equation (35b).  
Since \(Y_{r}(\cdot)\) is bounded by a functional in terms of boundedness of system dynamics [33], there exists a \(\epsilon\) such that \(\|Y_{r}\|\leq\epsilon\); thus, \(\dot{V}=-S^{T}K_{D}S-S^{T}Y_{r}\leq-S^{T}K_{D}S+\epsilon\|S\|\). Henceforth, \(\dot{V}=-(\lambda_{min}(K_{D})-\epsilon)\|S\|\). Since the system is autonomous in \(\Delta t\), using the maximum invariance principle leads to the boundedness of \(S\) into a small vicinity \(\Omega\) of its origin \(S=0\). Given (33), then errors \((\Delta q,\Delta q)\) converge asymptotically to a small vicinity of \(\Omega=0\) for high enough damping \(\lambda_{min}(K_{D})>\epsilon\).  
**Remark 3** (How to Implement the Adaptive Robust PI Controller in Generalized Coordinates).: _Notice that (33) qualifies for a manifold composed of independent generalized coordinates, then controller (35) shows a decentralized model-free PID structure for \(K_{p}=\alpha K_{D}+K_{I}>0\), \(K_{d}=K_{D}>0\) and \(K_{i}=\alpha K_{I}>0\). This PID structure can be implemented in industrial COTS actuators throughout tuning accordingly these feedback gains, representing arguably also a major advantage over nonlinear controllers that require unfeasible open architecture systems to implement them. Finally, a rightful comparison of our proposal versus conventional PID controller should show advantages to claim its usefulness in practice, rather than to become simply another proposed controller in the literature._