## 3 Overview of the Improved Model Reference Adaptive Control Framework
### State Predictor in Adaptive Control  
In the MRAC framework described above, high adaptation gains can lead to excessive high-frequency oscillations. The state predictor mitigates this problem by incorporating the prediction error in conjunction with the tracking error--effectively reducing noise in the control input. The architecture of the state predictor in adaptive control is shown in Figure 3. The following state predictor dynamics are presented to compute the prediction error [40]:  
\[\dot{\hat{x}}(t)=A_{prd}(\hat{x}(t)-x(t))+A_{m}x(t)+B_{m}u(t), \tag{31}\]Here, \(A_{prd}\in\mathbb{R}^{n\times n}\) is a Hurwitz matrix, and \(\hat{x}\in\mathbb{R}^{n}\) is the predicted state. The prediction error is defined as \(\hat{e}(t)=\hat{x}(t)-x(t)\). Consequently, the tracking-error and prediction-error dynamics can be represented as follows:  
\[\begin{cases}\hat{e}(t)=A_{m}e_{m}(t)+B_{p}\big{(}\Delta K_{x}^{T}x_{p}(t)+ \Delta K_{r}^{T}r(t)-\Delta W^{T}\beta(x(t))\big{)}\\ \hat{e}(t)=A_{prd}\hat{e}(t)-B_{p}\big{(}\Delta K_{x}^{T}x_{p}(t)+\Delta K_{r}^ {T}r(t)-\Delta W^{T}\beta(x(t))\big{)}\end{cases}, \tag{32}\]  
In order to regulate the parameter updates using both error signals, the training-error signal is defined as  
\[\overline{e}=\Big{(}e^{T}(t)P-\hat{e}^{T}(t)P_{prd}\Big{)}, \tag{33}\]  
where \(P_{prd}>0\) is the is a positive definite solution of the algebraic Lyapunov equation. This signal can be interpreted as a filtered version of the tracking error. While the conventional MRAC update law relies solely on \(e^{T}P\bar{B}_{P}\), the state predictor in the adaptive control structure augments this by subtracting the contribution of the prediction error. Consequently, the adaptive controller is less sensitive to high-frequency components in the tracking error and effectively serves as a low-pass filter. This adjustment makes the control input smoother and improves robustness against measurement noise and fast transients. The state predictor in adaptive control framework uses this prediction-error signal to define the following adaptive laws:  
\[\left\{\begin{array}{l}\dot{\hat{K}}_{x}=-\Gamma_{x}x(t)\overline{e}B_{p}\\ \dot{\hat{K}}_{r}=-\Gamma_{r}r(t)\overline{e}B_{p}\\ \dot{\hat{W}}=\Gamma_{\beta}\beta(x(t))\overline{e}B_{p}\end{array}\right., \tag{34}\]  
This structure enables the parameter update mechanism to account for both tracking and prediction performance, which ensures sufficient stability even under fast adaptation. The predictor, typically designed to be faster than the reference model, helps mitigate undesirable rapid parameter fluctuations and prevents control chattering.  
[PERSON] et al. [31] showed (through Lyapunov analysis) that all error signals in this framework are bounded, and both tracking and prediction errors asymptotically converge to zero. This shows that the predictor-based adaptive control system remains stable despite model uncertainties and parameter variations.  
In the context of HDVs, where the vehicle may experience large mass variations, steep road gradients, and actuator saturation, maintaining smooth and robust control is crucial. State predictors are particularly well-suited for such environments, as they maintain high-performance tracking while suppressing high-frequency oscillations. This ultimately  
Figure 3: State predictor in adaptive control. The red arrows represent the flow of updated gains based on the adaptation laws.  
improves ride comfort, reduces actuator wear, and enhances robustness in real-world driving conditions.