## 3 Proposed Control Methodology
### Reconfigurable Learning Control Algorithm (RLCA)  
To further improve the system's adaptability to parametric uncertainties, the aforementioned LCA employs a state-error-driven gain modulation mechanism that regulates the inclusion of dissipative, model reference tracking, and anti-dissipative control terms. The proposed RLCA dynamically reconfigures the weights applied to its constituent control components based on state error variations, ensuring efficient disturbance rejection while preventing excessive control effort. This arrangement provides a gradual transition between control phases, minimizing chattering and actuator wear. The adaptive weight modulation of the three control terms constituting the LCA is governed by a pre-calibrated hyperbolic secant function (HSF). It dynamically adjusts the weighting of the model-reference tracking, dissipative, and anti-dissipative contributions based on state error magnitudes. The HSF is chosen for its smooth, even-symmetric waveform, which normalizes the input variables within the range of zero to one [38]. The HSF-based adaptive weighting functions are formulated as per the following metarules. 1. For small errors, the adaptation law intensifies the dissipative control action, applying minimal corrective control. 2. For moderate errors, a combination of dissipative and model-reference tracking terms is deployed to ensure optimal state regulation without excessive control input. 3. For large errors, the anti-dissipative term is activated to intensify the (phase-informed) control actions for robust disturbance rejection, followed by a gradual transition back to nominal conditions. The aforementioned rationale is diagrammatically expressed in Figure 5.  
The following HSF-based weighting functions are thus constituted to comply with the aforementioned materules. \[\alpha_{S}(t)=\text{sech}(\mu\ e_{\theta}(t)),\ \alpha_{L}(t)=1-\alpha_{S}(t),\ \alpha_{M}(t)=2\alpha_{L}(t)\alpha_{S}(t) \tag{27}\]  
Figure 4: Variation pattern of the error cube polynomial with respect to error variations. where \(\text{sech}(. )\) represents the HSF, and \(\mu\) is the preset positive variation rate of the HSF. The waveforms of these hyperbolic gain scaling functions are shown in Figure 6.  
The dissipative term is weighted by \(\alpha_{S}(t)\) to direct the control law under small error conditions, the anti-dissipative term is weighted by \(\alpha_{L}(t)\) to direct the control law under large error conditions, and the model-reference tracking term is weighted by \(\alpha_{M}(t)\) to ensure the applied control actions track the nominal LQI trajectory. Parameter \(\mu\) is optimized offline by using the tuning procedure discussed in Section 4. The proposed RLCA is formulated in Equation (28). \[\dot{u}(t)=-\alpha_{S}(t)\beta u(t)+\alpha_{M}(t)\gamma\Big{(}u(t)-u_{lqi}(t) \Big{)}+\alpha_{L}(t)\delta\Big{(}e_{\theta}(t)+\rho e_{\theta}^{3}(t)\Big{)} \text{sign}\big{(}\dot{e}_{\theta}(t)\big{)} \tag{28}\]  
If \(u_{m}(t)=u(t)-u_{lqi}(t)\), the RLCA control law can be rewritten as follows. \[\dot{u}(t)=-\alpha_{S}(t)\beta\ u(t)+\alpha_{M}(t)\gamma\ u_{m}(t)+\alpha_{L} (t)\delta\ h(t) \tag{29}\]  
The system continuously refines its control actions based on the formulated algebraic law. This differential equation is numerically integrated at each sampling interval, as shown in Equation (30), to ensure real-time adaptability. \[u(t)=e^{(-\alpha_{S}(t)\beta t)}\ u(0)+\int\limits_{0}^{t}\Big{(}e^{(-\alpha_ {S}(t)\beta(t-p))}[\alpha_{M}(p)\gamma\ u_{m}(p)+\alpha_{L}(p)\delta\ h(p)] \Big{)}dp \tag{30}\]  
The control actions are refined online, after every sampling instant, based on the system's state and error variations. The control adaptation process begins by utilizing the nominal control signals \(u(0)\) generated by the LQI controller. It is to be noted that as long as  
Figure 5: Rationale of the proposed RLCA framework. Figure 6: Waveforms of the hyperbolic gain scaling functions expressed in (27).