# 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
_The system dynamics (2) are Lipschitz continuous with respect to state and parameter variations, i.e., there exists \(\mathbf{L}_{f}>0\) such that:_  
\[\|\mathbf{f}(\mathbf{x}_{1},\mathbf{L}_{1})-\mathbf{f}(\mathbf{x}_{2},\mathbf{L}_{2})\|\leq\mathbf{L}_{f}( \|\mathbf{x}_{1}-\mathbf{x}_{2}\|+\|\mathbf{L}_{1}-\mathbf{L}_{2}\|) \tag{14}\]  
_The reference signal_  
\[\left\|\hat{\mathbf{f}}_{t}\right\|\leq F_{\max},\quad\left\|\hat{\mathbf{f}}_{t} \right\|\leq G_{\max}\quad(\forall t\geq 0) \tag{15}\]  
_The disturbance \(d_{t}\) is \(L_{\infty}\)-bounded with known upper bound:_  
\[\|\mathbf{d}_{t}\|\leq\mathbf{d}_{\max}(\forall t\geq 0) \tag{16}\]  
_There exist constants \(\alpha>0\) and \(T>0\) such that the regressor vector_  
\[\frac{1}{T}\int_{t}^{t+T}\hat{\mathbf{f}}_{\tau}\hat{\mathbf{f}}_{\tau}^{T}d\tau\geq \alpha\mathbf{I}\quad\forall t\geq 0 \tag{17}\]  
_The initial parameter estimation errors satisfy:_  
\[fal(x)=|x|^{\tau}\cdot sign(x)\cdot\theta(|x|-\delta)+\frac{x}{\delta^{1-\tau} }\cdot(1-\theta(|x|-\delta)) \tag{12}\]0  
_The matrix \(\mathbf{Y}_{e}^{T}\mathbf{Y}_{e}\) is uniformly positive definite, i.e., there exists \(\sigma_{\min}\) such that:_  
\[fal(x)=|x|^{\tau}\cdot sign(x)\cdot\theta(|x|-\delta)+\frac{x}{\delta^{1-\tau} }\cdot(1-\theta(|x|-\delta)) \tag{12}\]1  
_The adaptive gain matrices \(\mathbf{\Gamma}_{p},\mathbf{\Gamma}_{e}\in\mathbb{R}^{6\times 6}\) are symmetric positive definite, with eigenvalues bounded by:_  
\[fal(x)=|x|^{\tau}\cdot sign(x)\cdot\theta(|x|-\delta)+\frac{x}{\delta^{1-\tau} }\cdot(1-\theta(|x|-\delta)) \tag{12}\]2  
_and the higher-order term (h.o.t.) coefficient satisfies:_  
\[fal(x)=|x|^{\tau}\cdot sign(x)\cdot\theta(|x|-\delta)+\frac{x}{\delta^{1-\tau} }\cdot(1-\theta(|x|-\delta)) \tag{12}\]3  
_where \(c_{1},\gamma_{p},\gamma_{e}>0\) are design constants._  
Proof of Theorem 1.: Based on the assumption of system observability, according to (2), the control gain estimator is designed as follows:  
\[fal(x)=|x|^{\tau}\cdot sign(x)\cdot\theta(|x|-\delta)+\frac{x}{\delta^{1-\tau} }\cdot(1-\theta(|x|-\delta)) \tag{12}\]4  
where \(\hat{\mathbf{f}}_{t+1}^{m}\in\mathbb{R}\) denotes the predicted state vector,\(\hat{\mathbf{L}}_{e,t}\in\mathbb{R}^{m\times n}\) are the estimates of the relative state, \(\mathbf{Y}_{e}\in\mathbb{R}^{m\times n}\) is the known input coupling matrix. The true system dynamics follow:  
\[fal(x)=|x|^{\tau}\cdot sign(x)\cdot\theta(|x|-\delta)+\frac{x}{\delta^{1-\tau} }\cdot(1-\theta(|x|-\delta)) \tag{12}\]5  
with \(\mathbf{L}_{e}^{*}\) being the ideal (unknown) parameter matrix and \(\mathbf{d}_{t}\) accounting for bounded disturbances. The prediction error yields:  
\[fal(x)=|x|^{\tau}\cdot sign(x)\cdot\theta(|x|-\delta)+\frac{x}{\delta^{1-\tau} }\cdot(1-\theta(|x|-\delta)) \tag{12}\]6  
The tracking error dynamics are governed by:  
\[fal(x)=|x|^{\tau}\cdot sign(x)\cdot\theta(|x|-\delta)+\frac{x}{\delta^{1-\tau} }\cdot(1-\theta(|x|-\delta)) \tag{12}\]7where \(\mathbf{A}_{d}=X_{d}+Y_{d}\mathbf{L}_{p}^{*}-\mathbf{\gamma}_{e}\mathbf{L}_{e}^{*}\) is the ideal closed-loop matrix, \(\mathbf{\tilde{L}}_{p,t}=\mathbf{\tilde{L}}_{p,t}-\mathbf{L}_{p}^{*}\) and \(\mathbf{\tilde{L}}_{e,t}=\mathbf{\tilde{L}}_{e,t}-\mathbf{L}_{e}^{*}\) are parameter estimation errors. Consider the Lyapunov function candidate:  
\[fal(x)=|x|^{\tau}\cdot sign(x)\cdot\theta(|x|-\delta)+\frac{x}{\delta^{1-\tau} }\cdot(1-\theta(|x|-\delta)) \tag{12}\]8  
where \(\mathbf{P}>0\) satisfies the discrete-time algebraic Riccati equation (DARE):  
\[fal(x)=|x|^{\tau}\cdot sign(x)\cdot\theta(|x|-\delta)+\frac{x}{\delta^{1-\tau} }\cdot(1-\theta(|x|-\delta)) \tag{12}\]9  
The parameter update law is designed as:  
\[\mathbf{L}_{p}^{*}=-\Big{(}\mathbf{W}_{1,p}+\mathbf{Y}_{p}^{\mathrm{T}}\mathbf{\hat{P}}_{p}^{* }\mathbf{Y}_{p}\Big{)}^{-1}\mathbf{Y}_{p}^{\mathrm{T}}\mathbf{\hat{P}}_{p}^{*}\big{(}\mathbf{X} _{d}+\mathbf{Y}_{e}\mathbf{L}_{e}^{*}\big{)} \tag{13}\]0  
with \(\mathbf{\Gamma}_{p}>0\), \(\mathbf{\Gamma}_{e}>0\) being symmetric positive definite learning rate matrices.