# A Stackelberg Game Approach to Model Reference Adaptive Control for Spacecraft Pursuit-Evasion
## 2 Problem Formulation
### Dynamic Stackelberg Pursuit-Evasion Game  
In the spacecraft PE problem, the evader acts as the leader in a Stackelberg game, and the pursuer is the follower. In this Stackelberg game, the information structure is sequential and asymmetric. The leader (evader) commits to a strategy first. The follower (pursuer) observes the leader's action and then chooses its own strategy to optimize its objective function. This does not imply direct real-time communication of the entire strategy, but rather that the pursuer's decision-making process at time t is based on knowledge of the evader's action at time t. The decision-making process is modeled as a dynamic game  
Figure 1: Schematic diagram of the pursuit and evasion process in the LVLH coordinate system. where each player optimizes a cost function. The cost functions for the pursuer (\(J_{p}\)) and evader (\(J_{e}\)) are defined as:  
\[J_{p}(\tilde{J}_{0},\mathbf{a}_{p,t},\mathbf{a}_{e,t})=\sum_{t=0}^{N-1}\{\tilde{J}_{t}^{ T}V_{p}\tilde{J}_{t}+\mathbf{a}_{p,t}^{\mathrm{T}}W_{1,p}\mathbf{a}_{p,t}-\mathbf{a}_{e,t}^{ \mathrm{T}}W_{1,e}\mathbf{a}_{e,t}\}+\tilde{J}_{N}^{\mathrm{T}}V_{N,p}\tilde{J}_{N} \tag{3}\]  
In this equation, \(\mathbf{V}_{p}\) and \(\mathbf{V}_{N,p}\) are \(6\times 6\) positive definite weighting matrices for the relative state error, while \(\mathbf{W}_{1,p}\) and \(\mathbf{W}_{1,e}\) are \(3\times 3\) positive definite weighting matrices for the control effort of the pursuer and evader, respectively. These matrices are design parameters used to tune the controller's performance. \[\mathbf{J}_{e}(\tilde{J}_{0},\mathbf{a}_{p,t},\mathbf{a}_{e,t})=-\sum_{t=0}^{N-1}\{\tilde{ J}_{t}^{T}\mathbf{V}_{e}\tilde{J}_{t}+\mathbf{a}_{p,t}^{\mathrm{T}}\mathbf{W}_{2,p}\mathbf{a}_{p,t}- \mathbf{a}_{e,t}^{\mathrm{T}}\mathbf{W}_{2,e}\mathbf{a}_{e,t}\}-\tilde{J}_{N}^{\mathrm{T}} \mathbf{V}_{N,e}\tilde{J}_{N} \tag{4}\]  
Here, \(V_{e}\),\(V_{N,e}\),\(W_{2,p}\), and \(W_{2,e}\) are similarly defined positive definite weighting matrices. A Stackelberg equilibrium is a pair of strategies (\(\mathbf{a}_{p,t}^{*}\),\(\mathbf{a}_{e,t}^{*}\)) that satisfies:  
\[ \begin{split}&\mathbf{J}_{e}\Big{(}\tilde{J}_{0},\mathbf{a}_{p,t}^{*}, \mathbf{a}_{e,t}^{*}\Big{)}\leq\mathbf{I}_{e}\Big{(}\tilde{J}_{0},\mathbf{a}_{p,t}^{*},\bm {a}_{e,t}\Big{)}\\ &\mathbf{J}_{p}\Big{(}\tilde{J}_{0},\mathbf{a}_{p,t}^{*},\mathbf{a}_{e,t}^{*} \Big{)}\leq\mathbf{I}_{p}\big{(}\tilde{J}_{0},\mathbf{a}_{p,t},\mathbf{a}_{e,t}^{*}\big{)} \end{split}  \tag{5}\]  
It is important to distinguish the Stackelberg equilibrium condition in (5) from [PERSON]'s Minimum Principle (PMP). PMP provides necessary conditions for a single control to be optimal for a given objective function. In contrast, a Stackelberg Equilibrium is a solution concept for a bi-level game, defining a pair of strategies (\(\mathbf{a}_{p}^{*}\), \(\mathbf{a}_{e}^{*}\)) where neither player has an incentive to unilaterally deviate, given the hierarchical decision structure. We model this dynamic interaction as the following two-stage Stackelberg optimization problem:  
\[ \begin{split}&\min_{\mathbf{a}_{e,t}}\mathbf{I}_{e}\big{(}\tilde{J}_{0}, \mathbf{a}_{e,t}^{*}(\mathbf{a}_{e,t}),\mathbf{a}_{e,t}\big{)}\\ \text{s.t. }&\mathbf{a}_{p,t}^{*}(\mathbf{a}_{e,t})=\arg \min_{\mathbf{a}_{p,t}}\mathbf{I}_{p}(\tilde{J}_{0},\mathbf{a}_{p,t},\mathbf{a}_{e,t})\\ &\text{and }\tilde{\mathbf{f}}_{t+1}=\mathbf{X}_{d}\tilde{\mathbf{f}}_{t}+\mathbf{Y}_{e}( \mathbf{a}_{e,t}-\mathbf{a}_{p,t})\end{split}  \tag{6}\]  
The hierarchical structure of the minimization problem itself acts as the primary constraint embodying the Stackelberg game logic; the inner minimization of \(\mathbf{J}_{p}\) with respect to \(\mathbf{a}_{p,t}\) is solved first for a given \(\mathbf{a}_{e,t}\), yielding an optimal response function for the pursuer, \(\mathbf{a}_{p,t}^{*}(\mathbf{a}_{e,t})\). The outer minimization of \(\mathbf{I}_{e}\) is then solved by the leader (evader), who anticipates the follower's rational response.