## 2 Methodology
### Design of the Enhanced Error-Adaptive ESKF-MPC
By dynamically adjusting \(Q_{k}\), the prediction steps exhibiting larger errors are assigned higher weights, enabling the controller to prioritize optimizing the tracking performance of those steps. This selective prioritization enhances the overall tracking accuracy and robustness, particularly in challenging scenarios like load variations in coal-fired power plants. The mathematical formulation of this adaptive weighting, as presented in the accompanying equation, forms the foundation of the EASKF-MPC framework. This mechanism ensures that the controller adapts to real-time operational dynamics, improving performance over traditional MPC approaches. The basic EA-ESKF-MPC introduced here serves as the starting point for further refinements, as detailed in the subsequent sections. \[Y=[y(k+1|k),\ldots,y(k+Np|k)]^{T}=F\overline{x}(k)+\Phi\Delta U \tag{23}\]0  
To further enhance the adaptability of the dynamic weight \(Q_{k}\), an amplification factor is introduced to make \(Q_{k}\) more responsive to prediction errors, particularly when errors are significant. This amplification ensures that the controller can swiftly address large deviations, improving tracking performance in critical operating conditions, such as rapid load changes in power plant control. However, overly rapid adjustments to \(Q_{k}\) driven by transient error spikes could lead to excessive control actions, potentially causing instability or oscillations in the system. To address this, an exponential smoothing filter is applied to \(Q_{k}\), as shown in the corresponding equation. This filter smooths out abrupt changes in \(Q_{k}\), ensuring that the controller remains stable and responsive without overreacting to short-term fluctuations. The combination of amplification and smoothing allows the EA-ESKF-MPC to achieve a balance between sensitivity to errors and steady control performance, making it well suited to industrial applications, where both precision and reliability are paramount. \[Y=[y(k+1|k),\ldots,y(k+Np|k)]^{T}=F\overline{x}(k)+\Phi\Delta U \tag{23}\]1  
\[Y=[y(k+1|k),\ldots,y(k+Np|k)]^{T}=F\overline{x}(k)+\Phi\Delta U \tag{23}\]2  
To verify the stability of the EEA-ESKF-MPC, a Lyapunov-based stability analysis was performed. A Lyapunov candidate function is defined as follows:  
\[Y=[y(k+1|k),\ldots,y(k+Np|k)]^{T}=F\overline{x}(k)+\Phi\Delta U \tag{23}\]3  
where \(\mathrm{e}(k)=y(k)-y_{r}(k)\) represents the tracking error vector for the MIMO system outputs (steam pressure, unit load, and header temperature), and \(P\) is a positive definite matrix ensuring \(V(k)\geq 0\). The control law, incorporating the adaptive weights \(Q_{k}\) and exponential smoothing filter, minimizes the error dynamics. Stability is assessed by evaluating the time difference \(\Delta V(k)=V(k+1)-V(k)\), which must satisfy \(\Delta V(k)<0\) for asymptotic stability. The MPC optimization ensures that the control input \(\Delta U\) reduces the predicted error \(Y-Y_{r}\), while the ESKF updates provide accurate state estimation under disturbances,as validated in the disturbance tests (Section 3.3). The exponential smoothing filter prevents abrupt changes in \(Q_{k}\), avoiding oscillatory control actions that could destabilize the system. For bounded errors and disturbances, as observed in the load variation tests (70-320 MW, Section 3.2), the closed-loop system satisfies \(\Delta V(k)<0\), confirming asymptotic stability across the operating conditions. This analysis demonstrates that the adaptive error mechanism and robust ESKF estimation enhance the tracking performance while maintaining system stability, supporting the controller's reliability for supercritical power plant applications. Through an iterative testing process, the amplification factor and smoothing factor of the basic EA-ESKF-MPC were carefully tuned, with the amplification factor selected from a range of 0 to 2 to balance sensitivity to prediction errors and system stability.