## 3 Proposed Control Methodology
### Learning Control Algorithm (LCA)  
The LCA scheme is formulated by incorporating model-reference tracking, dissipative, and anti-dissipative terms, which allows the control scheme to robustly suppress oscillations in nominal conditions while intensifying control efforts under severe disturbances, ensuring fast yet smooth recovery. The baseline LCA is formulated in Equation (24) [36]. \[\dot{u}(t)=-\beta u(t)+\gamma\Big{(}u(t)-u_{lqi}(t)\Big{)}+\delta\Big{(}e_{ \theta}(t)+\rho e_{\theta}^{3}(t)\Big{)}\text{sign}\big{(}\dot{e}_{\theta}(t) \big{)} \tag{24}\]  
such that, \(e_{\theta}(t)=\theta_{r}-\theta(t)\)  
where \(\theta_{r}\) is the reference pitch angle of the UAV, and \(e_{\theta}(t)\) is the error between the reference and the actual pitch angles of the UAV, \(u_{lqi}(t)\) represents the control effort generated by the LQI control law in (20), \(\beta\) represents the control-decay rate, \(\gamma\) represents the model-reference tracking rates, \(\delta\) represents the disturbance-rejection rate, and \(\rho\) is the predetermined weight of the error cube signal \(c_{\theta}^{3}(t)\). These parameters are optimized offline by using the tuning procedure discussed in Section 4. The LCA formulation comprises the three distinct control components presented in Table 1.  
Figure 2: Schematic of LQI control law for UAV’s longitudinal motion control. The model-reference tracking term is tasked to operate under moderate error conditions. The dissipative and anti-dissipative control terms manipulate the control system's adaptability under very small error (equilibrium) or very large error (disturbance) conditions, respectively. The following rationale is used to formulate the LCA [36]. The model-reference tracking term assists the LCA to track the baseline LQI controller, \(u_{lqi}(t)\), which acts as a reference control law under normal operating conditions. By aligning with the LQI baseline, the LCA ensures stability and optimal state regulation in the absence of major disturbances, resulting in smooth system operation and low control effort. Additionally, it also ensures smooth transitions between the dissipative and anti-dissipative terms. The dissipative component regulates the system's energy dissipation under small error conditions, ensuring a steady and controlled convergence to equilibrium while avoiding excessive servo control requirements. It exponentially reduces the applied control effort when the system's response is converging (or settling at) the desired reference or when the anti-dissipative term becomes fragile. The anti-dissipative component introduces robust corrective actions under large error conditions by appropriately amplifying (or attenuating) the applied control input in response to the system's error phase under major disturbances, effectively counteracting perturbations and accelerating transient recovery. The anti-dissipative component is realized by utilizing the error phase information of the system's state response under exogenous disturbances. Figure 3 illustrates the error profile of an arbitrary system subjected to an external disturbance, highlighting four distinct error phases (A to D) [37]. When the system response deviates from the reference (phases A and C), the product of the state error and its derivative becomes positive, indicating divergence and necessitating a stiff control action. In contrast, the said product becomes negative when the response is converging to the reference (phases B and D), and thus, necessitates a softer control application. The above rationale is mathematically formulated to define the phase-informed anti-dissipative term, \(h(t)\), as presented in Equation (25). \[h(t)=\Big{(}e_{\theta}(t)+\rho e_{\theta}^{3}(t)\Big{)}\text{ sign}\big{(}\dot{e}_{\theta}(t)\big{)} \tag{25}\]  
where \(\rho\) is the predefined weighting coefficient of \(e_{\theta}^{3}(t)\). It is optimized offline by using the tuning procedure discussed in Section 4. The signum function, \(\text{sign}(. )\), is expressed as follows: