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1 ADC采样模型 （本文为笔者早期所写，当时对卡尔曼滤波器理解尚未透彻，如今回顾，该模型还有所缺陷， 假设ADC采样的值已经为稳定状态，设 k + 1 k+1 k+1时刻ADC采样值为 X k + 1 X_{k+1} Xk+1,则 k k k时刻ADC实际值为 X k X_k Xk，假设 k + 1 k+1 k+1时刻的采样值为 Z k + 1 Z_{k+1} Zk+1，则有： { X k + 1 = X k + δ 1 , δ 1 为系统噪声 Z k + 1 = X k + 1 + δ 2 , δ 2 为测量噪声 \begin{cases} X_{k+1} = X_k+\delta_1, & \text{δ2为测量噪声} \end{cases} { Xk+1=Xk+δ1,Zk+1=Xk+1+δ2,δ1为系统噪声δ2为测量噪声 2 卡尔曼滤波算法我们知道卡尔曼滤波算法的公式如下：
由于相关系数都为1,于是可以得出如下公式： { P 0 , 0 = 0 P k , k − 1 = P k − 1 , k − 1 + Q G k = P k , k − 1 / ( P k , k − 1 + R ) P k , k = ( 1 − G k ) P k , k − 1 x 0 ∣ 0 = 0 x k ∣ k − 1 = x k − 1 ∣ k − 1 x k ∣ k = x k ∣ k − 1 + G k ( Z k − x k ∣ k − 1 ) \begin{cases} P_{0,0} = 0 \\P_{k,k-1}=P_{k-1,k-1} +Q \\ G_k = P_{k,k-1}/(P_{k,k-1}+R) \\ P_{k,k}=(1-G_k)P_{k,k-1} \\ x_{0|0} = 0 \\ x_{k|k-1} = x_{k-1|k-1} \\ x_{k|k}=x_{k|k-1}+G_k(Z_k-x_{k|k-1}) \end{cases} ⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧P0,0=0Pk,k−1=Pk−1,k−1+QGk=Pk,k−1/(Pk,k−1+R)Pk,k=(1−Gk)Pk,k−1x0∣0=0xk∣k−1=xk−1∣k−1xk∣k=xk∣k−1+Gk(Zk−xk∣k−1) 3 C语言代码用C实现的代码如下：
unsigned long kalman_filter(unsigned long ADC_Value){ float x_k1_k1,x_k_k1; static float ADC_OLD_Value; float Z_k; static float P_k1_k1; static float Q = 0.0001; static float R = 0.005; static float Kg = 0; static float P_k_k1 = 1; float kalman_adc; static float kalman_adc_old=0; Z_k = ADC_Value; x_k1_k1 = kalman_adc_old; x_k_k1 = x_k1_k1; P_k_k1 = P_k1_k1 + Q; Kg = P_k_k1/(P_k_k1 + R); kalman_adc = x_k_k1 + Kg * (Z_k - kalman_adc_old); P_k1_k1 = (1 - Kg)*P_k_k1; P_k_k1 = P_k1_k1; ADC_OLD_Value = ADC_Value; kalman_adc_old = kalman_adc; return kalman_adc;}
4 如何优化用以上的代码实现的滤波算法，通常要么滞后严重，要么滤波效果不明显，在这里给出两种优化方案。 方案一：在采样值与优化值相差大于某值时采用一阶滞后滤波算法，小于该值时采用卡尔曼滤波算法； 方案二：比较一段时间内的ADC采样值与优化值差值，若一直处于某个范围如（6~30），采用一阶滞后滤波算法，反之采用卡尔曼滤波算法。
unsigned long kalman_filter(unsigned long ADC_Value){float x_k1_k1,x_k_k1;static float ADC_OLD_Value;float Z_k;static float P_k1_k1;static float Q = 0.0001;static float R = 5;static float Kg = 0;static float P_k_k1 = 1;float kalman_adc;static float kalman_adc_old=0;Z_k = ADC_Value;if (abs(kalman_adc_old-ADC_Value)>=10){ x_k1_k1= ADC_Value*0.382 + kalman_adc_old*0.618;}else{ x_k1_k1 = kalman_adc_old;}x_k_k1 = x_k1_k1;P_k_k1 = P_k1_k1 + Q;Kg = P_k_k1/(P_k_k1 + R);kalman_adc = x_k_k1 + Kg * (Z_k - kalman_adc_old);P_k1_k1 = (1 - Kg)*P_k_k1;P_k_k1 = P_k1_k1;ADC_OLD_Value = ADC_Value;kalman_adc_old = kalman_adc;return get_int_num(kalman_adc);}
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