卡尔曼滤波 C语言实现

卡尔曼滤波是一种常见的估计方法，用于从有噪声的测量值中估计出真实值。它通常被用于机器人、导航和控制系统等领域。

以下是一个简单的 C 语言实现。假设我们要估计一个物体的位置和速度。我们可以定义状态向量 x，测量向量 z，以及状态转移矩阵 F 和测量矩阵 H。初始化协方差矩阵 P 和过程噪声 Q，以及测量噪声 R。

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

#define NUM_STATES 2 // x, x'
#define NUM_MEASUREMENTS 1 // x

void kalman_filter(float z, float dt);

// state vector
float x[NUM_STATES] = {0.0, 0.0}; // x, x'

// state transition matrix
float F[NUM_STATES][NUM_STATES] = {
    {1.0, dt},
    {0.0, 1.0}
};

// measurement matrix
float H[NUM_MEASUREMENTS][NUM_STATES] = {
    {1.0, 0.0}
};

// covariance matrix
float P[NUM_STATES][NUM_STATES] = {
    {1.0, 0.0},
    {0.0, 1.0}
};

// process noise covariance matrix
float Q[NUM_STATES][NUM_STATES] = {
    {0.01, 0.0},
    {0.0, 0.01}
};

// measurement noise covariance matrix
float R[NUM_MEASUREMENTS][NUM_MEASUREMENTS] = {
    {0.1}
};

int main()
{
    float z = 1.0; // measurement
    float dt = 0.1; // time step
    
    for (int i = 0; i < 10; i++) {
        kalman_filter(z, dt);
    }
    
    return 0;
}

void kalman_filter(float z, float dt)
{
    // prediction step
    float x_pred[NUM_STATES] = {0.0};
    float P_pred[NUM_STATES][NUM_STATES] = {0.0};
    
    for (int i = 0; i < NUM_STATES; i++) {
        for (int j = 0; j < NUM_STATES; j++) {
            x_pred[i] += F[i][j] * x[j];
            for (int k = 0; k < NUM_STATES; k++) {
                P_pred[i][j] += F[i][k] * P[k][j] * F[j][k];
            }
        }
    }
    
    // update step
    float K[NUM_STATES][NUM_MEASUREMENTS] = {0.0};
    float S[NUM_MEASUREMENTS][NUM_MEASUREMENTS] = {0.0};
    
    for (int i = 0; i < NUM_MEASUREMENTS; i++) {
        for (int j = 0; j < NUM_STATES; j++) {
            S[i][i] += H[i][j] * P_pred[j][i] * H[i][j];
        }
        S[i][i] += R[i][i];
    }
    
    float S_inv[NUM_MEASUREMENTS][NUM_MEASUREMENTS] = {0.0};
    float det_S = S[0][0];
    
    if (NUM_MEASUREMENTS > 1) {
        det_S = S[0][0] * S[1][1] - S[0][1] * S[1][0];
        S_inv[0][0] = S[1][1] / det_S;
        S_inv[0][1] = -S[0][1] / det_S;
        S_inv[1][0] = -S[1][0] / det_S;
        S_inv[1][1] = S[0][0] / det_S;
    } else {
        S_inv[0][0] = 1.0 / det_S;
    }
    
    for (int i = 0; i < NUM_STATES; i++) {
        for (int j = 0; j < NUM_MEASUREMENTS; j++) {
            for (int k = 0; k < NUM_STATES; k++) {
                K[i][j] += P_pred[i][k] * H[j][k];
            }
            K[i][j] *= S_inv[j][j];
            x[i] = x_pred[i] + K[i][j] * (z - H[j][0] * x_pred[0]);
            if (NUM_MEASUREMENTS > 1) {
                P[i][j] = (1.0 - K[i][j] * H[j][0]) * P_pred[i][j];
            } else {
                P[i][j] = (1.0 - K[i][j] * H[j][0]) * P_pred[i][j] + Q[i][i];
            }
        }
    }
}

在 main 函数中，我们初始化测量值 z 和时间步长 dt，并在 for 循环中调用 kalman_filter 函数进行卡尔曼滤波，共迭代 10 次。在 kalman_filter 函数中，我们首先进行预测步骤，计算出状态向量和协方差矩阵的预测值。接着进行更新步骤，计算出卡尔曼增益和测量噪声协方差矩阵，然后更新状态向量和协方差矩阵。

这只是一个简单的卡尔曼滤波实现，实际应用中可能需要更复杂的模型和参数调整。