# A Stackelberg Game Approach to Model Reference Adaptive Control for Spacecraft Pursuit-Evasion
## 3 Stackelberg Equilibrium-Based Model Reference Adaptive Control Algorithm
### Stackelberg Equilibrium-Based Model Reference Adaptive Control Algorithm  
After obtaining the optimal control strategy of the evader in the Stackelberg game, \(\mathbf{L}_{e}^{*}\), we design a model reference adaptive controller for the pursuer. The controller uses the evader's optimal strategy as a reference, allowing the pursuer to dynamically adjust its own controller parameters via observation and feedback to minimize error and achieve an efficient pursuit. The adaptive update law for the pursuer's estimate of the evader's gain, \(\mathbf{\hat{L}}_{e,m}\), is designed as a robust gradient-descent-type law derived from the estimation error dynamics. \[\mathbf{\hat{L}}_{e,m+1}=\mathbf{\hat{L}}_{e,m}-fal(\varepsilon_{t}^{m})\mathbf{\hat{f}}_ {t}^{\mathrm{T}}\,\Gamma(M+\mathbf{\hat{f}}_{t}^{\mathrm{T}}\,\mathbf{\hat{f}}_{t})^{ -1} \tag{11}\]  
where \(\varepsilon_{t}^{m}\) is the estimation error. The nonlinear function \(fal(\cdot)\) is used in place of a linear error term to achieve better performance. It provides high gain when the error is small, leading to faster convergence, and avoids excessive gain when the error is large, which enhances robustness to noise. It is defined as:  
\[fal(x)=|x|^{\tau}\cdot sign(x)\cdot\theta(|x|-\delta)+\frac{x}{\delta^{1-\tau} }\cdot(1-\theta(|x|-\delta)) \tag{12}\]  
Once the pursuer obtains a converged estimate of the evader's gain, \(\mathbf{\hat{L}}_{e}\), it calculates its own optimal control gain using:  
\[\mathbf{L}_{p}^{*}=-\Big{(}\mathbf{W}_{1,p}+\mathbf{Y}_{p}^{\mathrm{T}}\mathbf{\hat{P}}_{p}^{* }\mathbf{Y}_{p}\Big{)}^{-1}\mathbf{Y}_{p}^{\mathrm{T}}\mathbf{\hat{P}}_{p}^{*}\big{(}\mathbf{X} _{d}+\mathbf{Y}_{e}\mathbf{L}_{e}^{*}\big{)} \tag{13}\]  
The stability of this adaptive system is proven using a Lyapunov-based analysis. By selecting an appropriate Lyapunov function candidate and designing the parameter update laws, it can be shown that the tracking and parameter estimation errors are uniformly ultimately bounded. The overall control architecture is depicted in the flowchart in Figure 2, and a timeline of the process is shown in Figure 3.  
Figure 2 illustrates the overall control architecture. The process is divided into two main parts: the Model Reference Part and the Adaptive Control Part. In the Model Reference Part (top), the system calculates the optimal reference gain for the evader, \(L_{e}^{*}\). This involves initializing parameters, building the mathematical model, and iteratively solving the Riccati Equations (7) and (8) to compute the optimal strategies. The resulting optimal evader gain \(L_{e}^{*}\) is then fed as a reference to the Adaptive Control Part (bottom). Here, the pursuer observes the system state and uses an adaptive law to update its own strategy \(L_{p}^{*}\) to effectively track the evader's behavior. Figure 3 provides a timeline perspective of a single iteration cycle of the pursuer's adaptive strategy generation. It begins with the 'Pursuer Observation and Estimation' phase, where the pursuer acquires the state of the evader. This is followed by 'Error Feedback and Optimization', where the error is calculated and parameters are updated based on the reference model. This leads to 'Adaptive Strategy Generation' and finally 'Strategy execution and closure'. The cycle repeats, allowing for continuous adaptation. **Theorem 1**. : _Under assumptions 1-7, the following laws apply [25]:_  
Figure 3: Stackelberg equilibrium–based model reference adaptive control for spacecraft Pursuit–Evasion game timeline flowchart. Figure 2: Stackelberg equilibrium–based model reference adaptive control for spacecraft Pursuit–Evasion game flowchart.