101 lines
4.0 KiB
Python
101 lines
4.0 KiB
Python
import numpy as np
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from scipy.linalg import solve_discrete_lyapunov
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import sys
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def generate_ground_truth_system(dx=2, du=1, dy=1, rng_seed=None):
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"""
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生成一个 "真实" 的、稳定的、可控的、可观测的 LTI 系统。
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该过程遵循论文 6.2 节中描述的方法。
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参数:
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dx (int): 状态维度 (state dimension)
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du (int): 输入维度 (input dimension)
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dy (int): 输出维度 (output dimension)
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rng_seed (int, optional): 用于复现的随机种子
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返回:
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tuple: (A, B, C, D) 矩阵
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"""
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# 初始化随机数生成器
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if rng_seed is None:
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rng = np.random.default_rng()
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else:
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rng = np.random.default_rng(rng_seed)
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# 尝试生成一个良态的系统，最多重试 100 次
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max_retries = 100
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for attempt in range(max_retries):
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try:
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# --- 1. 生成稳定的 A 矩阵 (dx x dx) ---
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# 论文图 2(b) 显示了复特征值，遵循 6.2 节的极坐标法
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# 在极坐标下采样一对共轭复特征值
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r_squared = rng.uniform(0, 1.0) # 采样 r^2，确保在单位圆内
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r = np.sqrt(r_squared)
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theta = rng.uniform(0, np.pi) # 仅在上半平面采样角度
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lambda1 = r * (np.cos(theta) + 1j * np.sin(theta))
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lambda2 = np.conjugate(lambda1)
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# 创建对应的实数块对角矩阵
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alpha, beta = lambda1.real, lambda1.imag
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lambda_block = np.array([[alpha, beta],
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[-beta, alpha]])
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# 生成一个随机正交矩阵 V (dx x dx) [cite: 511, 584]
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Z = rng.standard_normal(size=(dx, dx))
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V, _ = np.linalg.qr(Z)
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# 组装 A = V * Lambda_block * V^T
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A = V @ lambda_block @ V.T
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# --- 2. 生成 B (dx x du) 和 C (dy x dx) 矩阵 ---
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# 元素从 N(0, 1) 独立采样
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B = rng.standard_normal(size=(dx, du))
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C = rng.standard_normal(size=(dy, dx))
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# --- 3. 检查可控性和可观测性 ---
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# 求解离散时间李雅普诺夫方程 (Lyapunov equation)
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Wc = solve_discrete_lyapunov(A, B @ B.T) # Controllability Gramian
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Wo = solve_discrete_lyapunov(A.T, C.T @ C) # Observability Gramian
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# 检查 Gramian 矩阵的条件
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# "拒绝主要特征值占总能量 99% 以上的系统"
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eig_Wc = np.linalg.eigvalsh(Wc)
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eig_Wo = np.linalg.eigvalsh(Wo)
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cond_c = np.max(eig_Wc) / np.sum(eig_Wc)
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cond_o = np.max(eig_Wo) / np.sum(eig_Wo)
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# 如果系统是良态的，则跳出循环
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if cond_c < 0.99 and cond_o < 0.99:
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# --- 4. 定义 D 矩阵 (dy x du) ---
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# 算例 6.3 中 D=0
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D = np.zeros((dy, du))
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print(f"--- 成功生成 Ground Truth 系统 (尝试次数: {attempt + 1}) ---")
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print(f"特征值: {lambda1:.4f}, {lambda2:.4f}")
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print(f"Gramian 能量占比: Wc={cond_c:.4f}, Wo={cond_o:.4f}")
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print("A = \n", A)
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print("B = \n", B)
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print("C = \n", C)
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print("D = \n", D)
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return A, B, C, D
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except np.linalg.LinAlgError:
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# 李雅普诺夫方程求解器可能失败（例如，A 矩阵数值上不稳定）
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print(f"尝试 {attempt + 1} 失败 (LinAlgError)。正在重试...", file=sys.stderr)
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continue
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# 如果循环结束仍未成功
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raise RuntimeError(f"在 {max_retries} 次尝试后未能生成一个良态的系统。")
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if __name__ == '__main__':
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# --- 运行示例 ---
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# 设置一个随机种子，以便每次运行时都能得到相同的结果
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A_true, B_true, C_true, D_true = generate_ground_truth_system(rng_seed=42)
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