鉴于控制算法常于嵌入式平台使用，所以博主使用C语言实现模糊PID控制算法，该项目已上传至GitHub以及码云。实现的功能包括但不限于：
隶属度函数 Membership function
高斯隶属度函数 Gaussian membership function
广义钟形隶属度函数 Generalized bell-shaped membership function
S形隶属度函数 Sigmoidal membership function
梯形隶属度函数 Trapezoidal membership function
三角形隶属度函数 Triangular membership function
Z形隶属度函数 Z-shaped membership function
模糊算子 Fuzzy operator
并算子 Union operator
交算子 Intersection operator
平衡算子 Equilibrium operator
中心平均解模糊器 Center average defuzzifier
API使用的示例代码如下：
#include "fuzzyPID.h"
#define DOF 6
int main() {
// Default fuzzy rule base of delta kp/ki/kd
int rule_base[][qf_default] = {
//delta kp rule base
{PB, PB, PM, PM, PS, ZO, ZO},
{PB, PB, PM, PS, PS, ZO, NS},
{PM, PM, PM, PS, ZO, NS, NS},
{PM, PM, PS, ZO, NS, NM, NM},
{PS, PS, ZO, NS, NS, NM, NM},
{PS, ZO, NS, NM, NM, NM, NB},
{ZO, ZO, NM, NM, NM, NB, NB},
//delta ki rule base
{NB, NB, NM, NM, NS, ZO, ZO},
{NB, NB, NM, NS, NS, ZO, ZO},
{NB, NM, NS, NS, ZO, PS, PS},
{NM, NM, NS, ZO, PS, PM, PM},
{NM, NS, ZO, PS, PS, PM, PB},
{ZO, ZO, PS, PS, PM, PB, PB},
{ZO, ZO, PS, PM, PM, PB, PB},
//delta kd rule base
{PS, NS, NB, NB, NB, NM, PS},
{PS, NS, NB, NM, NM, NS, ZO},
{ZO, NS, NM, NM, NS, NS, ZO},
{ZO, NS, NS, NS, NS, NS, ZO},
{ZO, ZO, ZO, ZO, ZO, ZO, ZO},
{PB, PS, PS, PS, PS, PS, PB},
{PB, PM, PM, PM, PS, PS, PB}};
// Default parameters of membership function
int mf_params[4 * qf_default] = {-3, -3, -2, 0,
-3, -2, -1, 0,
-2, -1, 0, 0,
-1, 0, 1, 0,
0, 1, 2, 0,
1, 2, 3, 0,
2, 3, 3, 0};
// Default parameters of pid controller
float fuzzy_pid_params[DOF][pid_params_count] = {{0.65f, 0, 0, 0, 0.5f, -2.4f, 1},
{-0.34f, 0, 0, 0, 0.5f, 0.536f, 2.3f},
{-1.1f, 0, 0, 0, 0.5f, 0.00f, 1},
{-2.4f, 0, 0, 0, 0.5f, 2.3f, 1},
{1.2f, 0, 0, 0, 0.5f, -0.7f, 2.3f},
{1.2f, 0.05f, 0.1f, 0, 0, 0, 1}};
// Obtain the PID controller vector according to the parameters
struct PID **pid_vector = fuzzy_vector_pid_init(fuzzy_pid_params, 2.0f, 4, 1, 0, mf_params, rule_base, DOF);
printf("output:n");
bool direct[DOF] = {true, false, false, false, true, true};
float real = 0;
float idea = max_error * 0.9f;
for (int j = 0; j < 500; ++j) {
int out = fuzzy_pid_motor_pwd_output(real, idea, direct[5], pid_vector[5]);
real += (float) (out - middle_pwm_output) / (float) middle_pwm_output * (float) max_error * 0.1f;
printf("%d,%fn", out, real);
}
delete_pid_vector(pid_vector, DOF);
return 0;
}
下面进行详细讲解，一般的模糊控制流程图如下：
可以需要根据参考输入与被控对象的实际输出产生的误差 e ee 、误差变化量 Δ e Delta eΔe 以及整定得的模糊规则表获取最终的模糊控制量。本项目主要针对模糊PID控制算法，所以选择的模糊条件语句为双输入多输出，即：
IF e is A ∨ Δ e is B THEN Δ k p is C 1 and Δ k i is C 2 and Δ k d is C 3 ⋯ text{IF } e text{ is } A vee Delta e text{ is } B text{ THEN } Delta k_p text{ is } C_1 text{ and } Delta k_i text{ is } C_2 text{ and } Delta k_d text{ is } C_3 cdots
IF e is A∨Δe is B THEN Δk
p
is C
1
and Δk
i
is C
2
and Δk
d
is C
3
⋯
其中 ∨ vee∨ 代表并 (and) 操作，当然也可以用或 (or, ∧ wedge∧) 生成条件语句。同时由于模糊概念的存在，误差 e ee 、误差变化量 Δ e Delta eΔe 是 A AA 和 B BB 呈一定概率，该概率则使用隶属度函数计算得出。项目的源码实现如下：
int j = 0;
for (int i = 0; i < qf_default; ++i) {
float temp = mf(e, fuzzy_struct->mf_type[0], fuzzy_struct->mf_params + 4 * i);
if (temp > 1e-4) {
membership[j] = temp;
index[j++] = i;
}
}
count[0] = j;
for (int i = 0; i < qf_default; ++i) {
float temp = mf(de, fuzzy_struct->mf_type[1], fuzzy_struct->mf_params + 4 * i);
if (temp > 1e-4) {
membership[j] = temp;
index[j++] = i;
}
}
count[1] = j - count[0];
其中使用 fuzzy_struct->mf_type 存储隶属度函数类型， fuzzy_struct->mf_params 存储隶属度函数所需的参数，通过调用 mf 便可实现隶属度的计算，同时多输入的情况下需要使用交并函数（模糊算子）求出联合隶属度。项目的源码实现如下：
// Joint membership
float joint_membership[count[0] * count[1]];
for (int i = 0; i < count[0]; ++i) {
for (int j = 0; j < count[1]; ++j) {
joint_membership[i * count[1] + j] = fo(membership
, membership[count[0] + j], fuzzy_struct->fo_type);
}
}
其中使用 fuzzy_struct->fo_type 存储交并函数类型，通过调用 fo 实现联合隶属度的求取。在得到每个语言值的联合隶属度，便可以通过模糊推理求取。项目的源码实现如下：
// Mean of centers defuzzifier, only for two input multiple index
void moc(const float *joint_membership, const unsigned int *index, const unsigned int *count, struct fuzzy *fuzzy_struct) {
float denominator_count = 0;
float numerator_count[fuzzy_struct->output_num];
for (unsigned int l = 0; l < fuzzy_struct->output_num; ++l) {
numerator_count[l] = 0;
}
for (int i = 0; i < count[0]; ++i) {
for (int j = 0; j < count[1]; ++j) {
denominator_count += joint_membership[i * count[1] + j];
}
}
for (unsigned int k = 0; k < fuzzy_struct->output_num; ++k) {
for (unsigned int i = 0; i < count[0]; ++i) {
for (unsigned int j = 0; j < count[1]; ++j) {
numerator_count[k] += joint_membership[i * count[1] + j] *
fuzzy_struct->rule_base[k * qf_default * qf_default + index * qf_default +
index[count[0] + j]];
}
}
}
for (unsigned int l = 0; l < fuzzy_struct->output_num; ++l) {
fuzzy_struct->output[l] = numerator_count[l] / denominator_count;
}
}
最终便可以实现模糊PID控制算法了。同时项目中提供了便捷的模糊PID控制器的数组生成器，便于生成多个控制器服务于多自由度控制需求，调用接口如下：
struct PID **
fuzzy_pid_vector_init(float params[][pid_params_count], float delta_k, unsigned int mf_type, unsigned int fo_type,
unsigned int df_type, int *mf_params, int rule_base[][qf_default],
unsigned int count);
其中 count 用于设定控制器个数，rule_base 用于传递模糊规则库，delta_k 用于控制PID的三个参数在初始值的基础上可以调节的程度，params 用于传递基础的PID参数，mf_type、fo_type、df_type三个参数分别决定了隶属度函数类型、模糊算子类型以及解模糊器类型。
鉴于控制算法常于嵌入式平台使用，所以博主使用C语言实现模糊PID控制算法，该项目已上传至GitHub以及码云。实现的功能包括但不限于：
隶属度函数 Membership function
高斯隶属度函数 Gaussian membership function
广义钟形隶属度函数 Generalized bell-shaped membership function
S形隶属度函数 Sigmoidal membership function
梯形隶属度函数 Trapezoidal membership function
三角形隶属度函数 Triangular membership function
Z形隶属度函数 Z-shaped membership function
模糊算子 Fuzzy operator
并算子 Union operator
交算子 Intersection operator
平衡算子 Equilibrium operator
中心平均解模糊器 Center average defuzzifier
API使用的示例代码如下：
#include "fuzzyPID.h"
#define DOF 6
int main() {
// Default fuzzy rule base of delta kp/ki/kd
int rule_base[][qf_default] = {
//delta kp rule base
{PB, PB, PM, PM, PS, ZO, ZO},
{PB, PB, PM, PS, PS, ZO, NS},
{PM, PM, PM, PS, ZO, NS, NS},
{PM, PM, PS, ZO, NS, NM, NM},
{PS, PS, ZO, NS, NS, NM, NM},
{PS, ZO, NS, NM, NM, NM, NB},
{ZO, ZO, NM, NM, NM, NB, NB},
//delta ki rule base
{NB, NB, NM, NM, NS, ZO, ZO},
{NB, NB, NM, NS, NS, ZO, ZO},
{NB, NM, NS, NS, ZO, PS, PS},
{NM, NM, NS, ZO, PS, PM, PM},
{NM, NS, ZO, PS, PS, PM, PB},
{ZO, ZO, PS, PS, PM, PB, PB},
{ZO, ZO, PS, PM, PM, PB, PB},
//delta kd rule base
{PS, NS, NB, NB, NB, NM, PS},
{PS, NS, NB, NM, NM, NS, ZO},
{ZO, NS, NM, NM, NS, NS, ZO},
{ZO, NS, NS, NS, NS, NS, ZO},
{ZO, ZO, ZO, ZO, ZO, ZO, ZO},
{PB, PS, PS, PS, PS, PS, PB},
{PB, PM, PM, PM, PS, PS, PB}};
// Default parameters of membership function
int mf_params[4 * qf_default] = {-3, -3, -2, 0,
-3, -2, -1, 0,
-2, -1, 0, 0,
-1, 0, 1, 0,
0, 1, 2, 0,
1, 2, 3, 0,
2, 3, 3, 0};
// Default parameters of pid controller
float fuzzy_pid_params[DOF][pid_params_count] = {{0.65f, 0, 0, 0, 0.5f, -2.4f, 1},
{-0.34f, 0, 0, 0, 0.5f, 0.536f, 2.3f},
{-1.1f, 0, 0, 0, 0.5f, 0.00f, 1},
{-2.4f, 0, 0, 0, 0.5f, 2.3f, 1},
{1.2f, 0, 0, 0, 0.5f, -0.7f, 2.3f},
{1.2f, 0.05f, 0.1f, 0, 0, 0, 1}};
// Obtain the PID controller vector according to the parameters
struct PID **pid_vector = fuzzy_vector_pid_init(fuzzy_pid_params, 2.0f, 4, 1, 0, mf_params, rule_base, DOF);
printf("output:n");
bool direct[DOF] = {true, false, false, false, true, true};
float real = 0;
float idea = max_error * 0.9f;
for (int j = 0; j < 500; ++j) {
int out = fuzzy_pid_motor_pwd_output(real, idea, direct[5], pid_vector[5]);
real += (float) (out - middle_pwm_output) / (float) middle_pwm_output * (float) max_error * 0.1f;
printf("%d,%fn", out, real);
}
delete_pid_vector(pid_vector, DOF);
return 0;
}
下面进行详细讲解，一般的模糊控制流程图如下：
可以需要根据参考输入与被控对象的实际输出产生的误差 e ee 、误差变化量 Δ e Delta eΔe 以及整定得的模糊规则表获取最终的模糊控制量。本项目主要针对模糊PID控制算法，所以选择的模糊条件语句为双输入多输出，即：
IF e is A ∨ Δ e is B THEN Δ k p is C 1 and Δ k i is C 2 and Δ k d is C 3 ⋯ text{IF } e text{ is } A vee Delta e text{ is } B text{ THEN } Delta k_p text{ is } C_1 text{ and } Delta k_i text{ is } C_2 text{ and } Delta k_d text{ is } C_3 cdots
IF e is A∨Δe is B THEN Δk
p
is C
1
and Δk
i
is C
2
and Δk
d
is C
3
⋯
其中 ∨ vee∨ 代表并 (and) 操作，当然也可以用或 (or, ∧ wedge∧) 生成条件语句。同时由于模糊概念的存在，误差 e ee 、误差变化量 Δ e Delta eΔe 是 A AA 和 B BB 呈一定概率，该概率则使用隶属度函数计算得出。项目的源码实现如下：
int j = 0;
for (int i = 0; i < qf_default; ++i) {
float temp = mf(e, fuzzy_struct->mf_type[0], fuzzy_struct->mf_params + 4 * i);
if (temp > 1e-4) {
membership[j] = temp;
index[j++] = i;
}
}
count[0] = j;
for (int i = 0; i < qf_default; ++i) {
float temp = mf(de, fuzzy_struct->mf_type[1], fuzzy_struct->mf_params + 4 * i);
if (temp > 1e-4) {
membership[j] = temp;
index[j++] = i;
}
}
count[1] = j - count[0];
其中使用 fuzzy_struct->mf_type 存储隶属度函数类型， fuzzy_struct->mf_params 存储隶属度函数所需的参数，通过调用 mf 便可实现隶属度的计算，同时多输入的情况下需要使用交并函数（模糊算子）求出联合隶属度。项目的源码实现如下：
// Joint membership
float joint_membership[count[0] * count[1]];
for (int i = 0; i < count[0]; ++i) {
for (int j = 0; j < count[1]; ++j) {
joint_membership[i * count[1] + j] = fo(membership
, membership[count[0] + j], fuzzy_struct->fo_type);
}
}
其中使用 fuzzy_struct->fo_type 存储交并函数类型，通过调用 fo 实现联合隶属度的求取。在得到每个语言值的联合隶属度，便可以通过模糊推理求取。项目的源码实现如下：
// Mean of centers defuzzifier, only for two input multiple index
void moc(const float *joint_membership, const unsigned int *index, const unsigned int *count, struct fuzzy *fuzzy_struct) {
float denominator_count = 0;
float numerator_count[fuzzy_struct->output_num];
for (unsigned int l = 0; l < fuzzy_struct->output_num; ++l) {
numerator_count[l] = 0;
}
for (int i = 0; i < count[0]; ++i) {
for (int j = 0; j < count[1]; ++j) {
denominator_count += joint_membership[i * count[1] + j];
}
}
for (unsigned int k = 0; k < fuzzy_struct->output_num; ++k) {
for (unsigned int i = 0; i < count[0]; ++i) {
for (unsigned int j = 0; j < count[1]; ++j) {
numerator_count[k] += joint_membership[i * count[1] + j] *
fuzzy_struct->rule_base[k * qf_default * qf_default + index * qf_default +
index[count[0] + j]];
}
}
}
for (unsigned int l = 0; l < fuzzy_struct->output_num; ++l) {
fuzzy_struct->output[l] = numerator_count[l] / denominator_count;
}
}
最终便可以实现模糊PID控制算法了。同时项目中提供了便捷的模糊PID控制器的数组生成器，便于生成多个控制器服务于多自由度控制需求，调用接口如下：
struct PID **
fuzzy_pid_vector_init(float params[][pid_params_count], float delta_k, unsigned int mf_type, unsigned int fo_type,
unsigned int df_type, int *mf_params, int rule_base[][qf_default],
unsigned int count);
其中 count 用于设定控制器个数，rule_base 用于传递模糊规则库，delta_k 用于控制PID的三个参数在初始值的基础上可以调节的程度，params 用于传递基础的PID参数，mf_type、fo_type、df_type三个参数分别决定了隶属度函数类型、模糊算子类型以及解模糊器类型。
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