Game Theoretic Consensus Control

---
name: game-theoretic-consensus-control
description: "H-infinity leader-following consensus control using coalitional zero-sum game theory and differential games. Uses GARE decomposition and dynamic average consensus for distributed implementation. Activation: game-theoretic consensus, H-infinity control, differential games, 博弈论一致性控制."
---

# Game-Theoretic Consensus Control

## Description
使用联盟零和博弈和微分博弈理论的H∞领导者-跟随者一致性控制。通过将鲁棒控制问题建模为全局联盟极小极大零和博弈，设计对抗攻击下的分布式一致性控制策略。

## Activation Keywords
- game-theoretic consensus
- H-infinity control
- differential games
- 博弈论一致性控制
- coalitional zero-sum games
- GARE decomposition
- robust consensus
- adversarial attack control

## Problem Statement

### 对抗攻击场景

考虑一类受对抗攻击类外部输入影响的多智能体系统：
- 恶意干扰试图破坏系统一致性
- 需要设计鲁棒控制策略
- 传统方法难以处理对抗性扰动

### 博弈论解决方案

使用微分博弈理论将鲁棒领导者-跟随者控制问题建模为：
- **全局联盟极小极大零和博弈**
- 智能体控制输入形成联盟最小化全局成本
- 攻击形成对立联盟最大化成本

## Mathematical Framework

### 1. 多智能体系统模型

```python
import numpy as np
from scipy.linalg import solve_continuous_are

class MultiAgentSystem:
    """多智能体系统模型"""
    
    def __init__(self, n_agents, state_dim, dynamics_type='linear'):
        self.n = n_agents
        self.nx = state_dim
        self.dynamics_type = dynamics_type
        
        # 系统矩阵
        self.A = np.eye(state_dim)  # 状态矩阵
        self.B = np.eye(state_dim)  # 控制输入矩阵
        self.C = np.eye(state_dim)  # 扰动输入矩阵
        
        # 通信拓扑
        self.L = None  # 拉普拉斯矩阵
        self.adjacency = None
    
    def set_communication_topology(self, adjacency_matrix):
        """设置通信拓扑"""
        self.adjacency = adjacency_matrix
        # 计算拉普拉斯矩阵
        degree = np.sum(adjacency_matrix, axis=1)
        self.L = np.diag(degree) - adjacency_matrix
    
    def linearized_dynamics(self, state, control, disturbance):
        """
        反馈线性化动力学
        
        x_dot = Ax + Bu + Cw
        
        Args:
            state: 状态向量
            control: 控制输入
            disturbance: 扰动输入
        
        Returns:
            state_derivative: 状态导数
        """
        return self.A @ state + self.B @ control + self.C @ disturbance
    
    def feedback_linearization(self, nonlinear_state, nonlinear_control):
        """
        反馈线性化变换
        
        将非线性动力学转换为线性形式
        """
        # 适用于车辆编队等系统
        if self.dynamics_type == 'vehicle':
            # 车辆模型线性化
            theta = nonlinear_state[2]  # 朝向角
            v = nonlinear_state[3]      # 速度
            
            # 线性化矩阵
            A_lin = np.array([
                [0, 0, -v*np.sin(theta), np.cos(theta)],
                [0, 0, v*np.cos(theta), np.sin(theta)],
                [0, 0, 0, 0],
                [0, 0, 0, 0]
            ])
            
            B_lin = np.array([
                [0, 0],
                [0, 0],
                [1, 0],
                [0, 1]
            ])
            
            return A_lin, B_lin
```

### 2. 联盟零和博弈

```python
class CoalitionalZeroSumGame:
    """联盟零和博弈"""
    
    def __init__(self, n_agents, cost_weights):
        self.n = n_agents
        self.Q = cost_weights['state']  # 状态代价权重
        self.R = cost_weights['control']  # 控制代价权重
        self.Gamma = cost_weights['disturbance']  # 扰动代价权重
        
        # 博弈参与者
        self.control_coalition = list(range(n_agents))  # 控制联盟
        self.disturbance_coalition = ['attacker']  # 扰动联盟
    
    def compute_cost(self, states, controls, disturbances):
        """
        计算全局成本函数
        
        J = ∫(x^T Q x + u^T R u - w^T Gamma w) dt
        
        Args:
            states: 状态轨迹
            controls: 控制轨迹
            disturbances: 扰动轨迹
        
        Returns:
            cost: 成本值
        """
        state_cost = np.sum([s.T @ self.Q @ s for s in states])
        control_cost = np.sum([u.T @ self.R @ u for u in controls])
        disturbance_cost = np.sum([w.T @ self.Gamma @ w for w in disturbances])
        
        return state_cost + control_cost - disturbance_cost
    
    def minimax_principle(self, system, time_horizon):
        """
        极小极大原理
        
        min_u max_w J(x, u, w)
        
        Args:
            system: 多智能体系统
            time_horizon: 时间范围
        
        Returns:
            optimal_control: 最优控制策略
            worst_disturbance: 最坏扰动
        """
        # 求解微分博弈的鞍点
        # 使用动态规划或迭代方法
        
        return self._solve_saddle_point(system, time_horizon)
    
    def _solve_saddle_point(self, system, T):
        """求解鞍点"""
        # 简化的鞍点求解
        # 实际使用需要求解HJB方程
        
        # 对于线性二次博弈，有解析解
        if system.dynamics_type == 'linear':
            # 求解GARE
            P = self._solve_gare(system)
            
            # 最优控制
            K = np.linalg.inv(self.R) @ system.B.T @ P
            
            # 最坏扰动增益
            L = np.linalg.inv(self.Gamma) @ system.C.T @ P
            
            return K, L
        
        return None, None
```

### 3. 广义代数Riccati方程(GARE)

```python
class GARESolver:
    """广义代数Riccati方程求解器"""
    
    def __init__(self, max_iter=1000, tol=1e-8):
        self.max_iter = max_iter
        self.tol = tol
    
    def solve_gare(self, A, B, C, Q, R, Gamma):
        """
        求解GARE
        
        A^T P + P A - P B R^{-1} B^T P + P C Gamma^{-1} C^T P + Q = 0
        
        Args:
            A, B, C: 系统矩阵
            Q: 状态权重
            R: 控制权重
            Gamma: 扰动权重
        
        Returns:
            P: GARE解
        """
        nx = A.shape[0]
        P = np.eye(nx)  # 初始猜测
        
        for i in range(self.max_iter):
            P_prev = P.copy()
            
            # 迭代求解 (Kleinman迭代)
            A_cl = A - B @ np.linalg.inv(R) @ B.T @ P + C @ np.linalg.inv(Gamma) @ C.T @ P
            
            # 求解Lyapunov方程
            P = self._solve_lyapunov(A_cl.T, Q + P @ B @ np.linalg.inv(R) @ B.T @ P)
            
            if np.linalg.norm(P - P_prev) < self.tol:
                break
        
        return P
    
    def _solve_lyapunov(self, A, Q):
        """求解Lyapunov方程"""
        from scipy.linalg import solve_continuous_lyapunov
        return solve_continuous_lyapunov(A, -Q)
    
    def decompose_gare(self, P, n_subsystems):
        """
        GARE分解
        
        将高维GARE分解为多个低维GARE
        
        Args:
            P: 原GARE解
            n_subsystems: 子系统数量
        
        Returns:
            P_decomposed: 分解后的GARE解列表
        """
        nx = P.shape[0]
        nx_sub = nx // n_subsystems
        
        P_decomposed = []
        for i in range(n_subsystems):
            start = i * nx_sub
            end = (i + 1) * nx_sub
            P_sub = P[start:end, start:end]
            P_decomposed.append(P_sub)
        
        return P_decomposed
```

### 4. 分布式计算策略

```python
class DecentralizedComputation:
    """分布式计算策略"""
    
    def __init__(self, n_agents, communication_graph):
        self.n = n_agents
        self.graph = communication_graph
        self.neighbors = self._compute_neighbors()
    
    def _compute_neighbors(self):
        """计算每个智能体的邻居"""
        neighbors = {}
        for i in range(self.n):
            neighbors[i] = [j for j in range(self.n) if self.graph[i, j] > 0]
        return neighbors
    
    def decentralized_gare_solve(self, local_system, global_gare):
        """
        分布式GARE求解
        
        每个智能体求解本地低维GARE
        
        Args:
            local_system: 本地系统模型
            global_gare: 全局GARE参数
        
        Returns:
            local_solution: 本地GARE解
        """
        # 提取本地参数
        A_local = local_system.A
        B_local = local_system.B
        
        # 求解本地GARE
        solver = GARESolver()
        P_local = solver.solve_gare(
            A_local, B_local, local_system.C,
            global_gare['Q_local'],
            global_gare['R_local'],
            global_gare['Gamma_local']
        )
        
        return P_local
    
    def dynamic_average_consensus(self, initial_values, max_iter=1000):
        """
        动态平均共识算法
        
        用于解耦控制律的分布式实现
        
        Args:
            initial_values: 初始值列表
            max_iter: 最大迭代次数
        
        Returns:
            consensus_values: 共识值
        """
        values = initial_values.copy()
        epsilon = 0.1  # 步长
        
        for t in range(max_iter):
            new_values = values.copy()
            
            for i in range(self.n):
                # 邻居信息融合
                neighbor_sum = sum(values[j] for j in self.neighbors[i])
                degree = len(self.neighbors[i])
                
                # 共识更新
                new_values[i] = values[i] + epsilon * (neighbor_sum - degree * values[i])
            
            values = new_values
            
            # 检查收敛
            if np.std(values) < 1e-6:
                break
        
        return values
```

### 5. H∞控制律

```python
class HInfinityController:
    """H∞控制器"""
    
    def __init__(self, gamma_bound):
        """
        Args:
            gamma_bound: H∞性能界
        """
        self.gamma = gamma_bound
        self.P = None  # GARE解
        self.K = None  # 控制增益
        self.L = None  # 扰动增益
    
    def design(self, system, game):
        """
        设计H∞控制器
        
        Args:
            system: 多智能体系统
            game: 联盟博弈
        
        Returns:
            controller: 控制器参数
        """
        # 求解GARE
        solver = GARESolver()
        self.P = solver.solve_gare(
            system.A, system.B, system.C,
            game.Q, game.R, game.Gamma
        )
        
        # 计算控制增益
        R_inv = np.linalg.inv(game.R)
        self.K = R_inv @ system.B.T @ self.P
        
        # 计算扰动增益
        Gamma_inv = np.linalg.inv(game.Gamma)
        self.L = Gamma_inv @ system.C.T @ self.P
        
        return {
            'P': self.P,
            'K': self.K,
            'L': self.L
        }
    
    def control_law(self, state, consensus_error):
        """
        H∞控制律
        
        u = -K * (x - x_leader)
        
        Args:
            state: 当前状态
            consensus_error: 一致性误差
        
        Returns:
            control: 控制输入
        """
        return -self.K @ consensus_error
    
    def worst_disturbance(self, state):
        """最坏扰动"""
        return self.L @ state
    
    def robustness_analysis(self, system, disturbances):
        """
        鲁棒性分析
        
        验证H∞性能界
        
        Args:
            system: 系统
            disturbances: 扰动集合
        
        Returns:
            robust: 是否满足鲁棒性要求
        """
        # 计算H∞范数
        # ||T_zw||_∞ ≤ γ
        
        max_gain = 0
        for w in disturbances:
            # 计算从扰动到输出的增益
            gain = self._compute_gain(system, w)
            max_gain = max(max_gain, gain)
        
        return max_gain <= self.gamma
```

## Complete System Implementation

```python
class GameTheoreticConsensusControl:
    """博弈论一致性控制系统"""
    
    def __init__(self, n_agents, is_leader=None):
        """
        Args:
            n_agents: 智能体数量
            is_leader: 领导者标志列表（None表示第一个为领导者）
        """
        self.n = n_agents
        self.is_leader = is_leader if is_leader else [True] + [False] * (n_agents - 1)
        
        # 初始化系统
        self.system = MultiAgentSystem(n_agents, state_dim=2)
        
        # 初始化博弈
        self.game = CoalitionalZeroSumGame(n_agents, {
            'state': np.eye(2),
            'control': 0.1 * np.eye(2),
            'disturbance': np.eye(2)
        })
        
        # 初始化控制器
        self.controller = None
        self.decentralized = DecentralizedComputation(
            n_agents, 
            np.ones((n_agents, n_agents)) - np.eye(n_agents)
        )
    
    def setup_communication(self, adjacency):
        """设置通信拓扑"""
        self.system.set_communication_topology(adjacency)
    
    def design_controller(self, gamma=1.0):
        """
        设计控制器
        
        Args:
            gamma: H∞性能界
        """
        self.controller = HInfinityController(gamma)
        self.controller.design(self.system, self.game)
    
    def compute_consensus_error(self, states, leader_state):
        """计算一致性误差"""
        errors = []
        for i, (state, is_leader) in enumerate(zip(states, self.is_leader)):
            if is_leader:
                errors.append(np.zeros_like(state))
            else:
                # 与领导者的一致性误差
                error = state - leader_state
                errors.append(error)
        return errors
    
    def simulate_step(self, states, leader_state, disturbances, dt):
        """
        单步仿真
        
        Args:
            states: 当前状态
            leader_state: 领导者状态
            disturbances: 扰动
            dt: 时间步长
        
        Returns:
            new_states: 新状态
            controls: 控制输入
        """
        # 计算一致性误差
        errors = self.compute_consensus_error(states, leader_state)
        
        # 计算控制
        controls = []
        new_states = []
        
        for i, (state, error, is_leader) in enumerate(zip(states, errors, self.is_leader)):
            if is_leader:
                # 领导者按预定轨迹运动
                control = np.zeros_like(state)
                new_state = state + dt * self.system.A @ state
            else:
                # 跟随者使用H∞控制
                control = self.controller.control_law(state, error)
                
                # 状态更新
                state_deriv = self.system.linearized_dynamics(
                    state, control, disturbances[i]
                )
                new_state = state + dt * state_deriv
            
            controls.append(control)
            new_states.append(new_state)
        
        return new_states, controls
    
    def run_formation_control(self, initial_states, leader_trajectory, T, dt):
        """
        运行编队控制仿真
        
        Args:
            initial_states: 初始状态
            leader_trajectory: 领导者轨迹
            T: 总时间
            dt: 时间步长
        
        Returns:
            trajectory: 轨迹
            controls: 控制历史
        """
        states = initial_states
        trajectory = [states.copy()]
        control_history = []
        
        n_steps = int(T / dt)
        
        for t in range(n_steps):
            leader_state = leader_trajectory[t]
            
            # 生成扰动
            disturbances = [np.random.randn(2) * 0.1 for _ in range(self.n)]
            
            # 单步仿真
            states, controls = self.simulate_step(
                states, leader_state, disturbances, dt
            )
            
            trajectory.append(states.copy())
            control_history.append(controls)
        
        return np.array(trajectory), control_history
```

## Numerical Validation

```python
# 创建编队控制系统
n_agents = 5
control_system = GameTheoreticConsensusControl(n_agents)

# 设置通信拓扑 (链式)
adjacency = np.array([
    [0, 1, 0, 0, 0],
    [1, 0, 1, 0, 0],
    [0, 1, 0, 1, 0],
    [0, 0, 1, 0, 1],
    [0, 0, 0, 1, 0]
])
control_system.setup_communication(adjacency)

# 设计控制器
control_system.design_controller(gamma=1.5)

# 初始状态
initial_states = [
    np.array([0, 0]),     # 领导者
    np.array([-10, 0]),
    np.array([-20, 0]),
    np.array([-30, 0]),
    np.array([-40, 0])
]

# 领导者轨迹 (圆形)
t = np.linspace(0, 20, 2000)
leader_trajectory = [np.array([10*np.cos(0.1*ti), 10*np.sin(0.1*ti)]) for ti in t]

# 运行仿真
trajectory, controls = control_system.run_formation_control(
    initial_states, leader_trajectory, T=20, dt=0.01
)

print(f"编队控制完成")
print(f"最终编队误差: {np.linalg.norm(trajectory[-1][0] - trajectory[-1][-1]):.2f}")
```

## Tools Used

- **python/numpy**: 数值计算
- **scipy.linalg**: Riccati和Lyapunov方程求解
- **networkx**: 通信拓扑分析
- **matplotlib**: 轨迹可视化

## Best Practices

### 1. GARE求解调优
```python
def tune_gare_solver(system, game):
    """GARE求解器参数调优"""
    best_tol = None
    best_time = float('inf')
    
    for tol in [1e-6, 1e-8, 1e-10]:
        solver = GARESolver(tol=tol)
        # 测试求解时间
        # ...
    
    return best_tol
```

### 2. γ选择
```python
def select_gamma_bound(system, desired_performance):
    """
    选择H∞性能界
    
    使用二分搜索
    """
    gamma_low = 0.1
    gamma_high = 100
    
    while gamma_high - gamma_low > 0.01:
        gamma_mid = (gamma_low + gamma_high) / 2
        
        controller = HInfinityController(gamma_mid)
        controller.design(system, game)
        
        if controller.robustness_analysis(system, test_disturbances):
            gamma_high = gamma_mid
        else:
            gamma_low = gamma_mid
    
    return gamma_high
```

## References

- Paper: "Coalitional Zero-Sum Games for H-infinity Leader-Following Consensus Control" (arXiv:2604.06089v1, 2026)
- Authors: Yunxiao Ren, Dingguo Liang, Yuezu Lv, et al.
- Category: eess.SY

## Related Skills

- `event-triggered-formation-control`: 事件触发编队控制
- `distributed-bilevel-macroscopic-optimization`: 分布式双层优化
- `risk-averse-stochastic-mpc`: 风险规避随机MPC
- `distributed-optimal-consensus`: 分布式最优一致性

## Limitations

- 需要反馈线性化
- 高维GARE求解计算密集
- 通信拓扑影响收敛速度
- 对抗攻击模型假设较简单

---

_Last updated: 2026-04-14_