UKF python

UKF（Unscented Kalman Filter）是一种非线性滤波器，它通过使用无味卡尔曼滤波器（UKF）来逼近非线性函数的概率分布。下面是一个使用Python实现UKF的示例代码：

import numpy as np

def unscented_transform(sigma_points, weights):
    mean = np.average(sigma_points, axis=0, weights=weights)
    covariance = np.cov(sigma_points.T, ddof=0, aweights=weights)
    return mean, covariance

def unscented_kalman_filter(x, P, z, Q, R, f, h):
    n = len(x)
    m = len(z)
    alpha = 1e-3
    kappa = 0
    beta = 2

    lambda_ = alpha**2 * (n + kappa) - n
    c = n + lambda_

    Wm = np.full(2*n+1, 1.0 / (2*c))
    Wc = np.full(2*n+1, 1.0 / (2*c))
    Wm[0] = lambda_ / c
    Wc[0] = lambda_ / c + (1 - alpha**2 + beta)

    sigma_points = np.zeros((2*n+1, n))
    sigma_points[0] = x
    sqrt_P = np.linalg.cholesky(P)

    for i in range(n):
        sigma_points[i+1] = x + sqrt_P[i]
        sigma_points[n+i+1] = x - sqrt_P[i]
    predicted_sigma_points = np.zeros((2*n+1, n))
    for i in range(2*n+1):
        predicted_sigma_points[i] = f(sigma_points[i])

    predicted_x, predicted_P = unscented_transform(predicted_sigma_points, Wm)

    predicted_z_sigma_points = np.zeros((2*n+1, m))
    for i in range(2*n+1):
        predicted_z_sigma_points[i] = h(predicted_sigma_points[i])

    predicted_z, predicted_z_covariance = unscented_transform(predicted_z_sigma_points, Wm)

    Pzz = predicted_z_covariance + R
    Pxz = np.zeros((n, m))
    for i in range(2*n+1):
        Pxz += Wc[i] * np.outer(sigma_points[i] - x, predicted_z_sigma_points[i] - predicted_z)

    K = np.dot(Pxz, np.linalg.inv(Pzz))
    x = predicted_x + np.dot(K, z - predicted_z)
    P = predicted_P - np.dot(K, np.dot(Pzz, K.T))

    return x, P

# 定义状态转移函数
def f(x):
    return np.array([x[0] + x[1], x[1]])

# 定义观测函数
def h(x):
    return np.array([x[0]])

# 初始化状态和协方差矩阵
x = np.array([0, 0])
P = np.array([[1, 0], [0, 1]])

# 定义过程噪声和观测噪声的协方差矩阵
Q = np.array([[0.1, 0], [0, 0.1]])
R = np.array([[1]])

# 定义观测数据
z = np.array([1])

# 使用UKF进行滤波
x, P = unscented_kalman_filter(x, P, z, Q, R, f, h)

print("Filtered state:", x)
print("Filtered covariance:", P)