PID控制器的离散化推导及其C语言实现

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PID控制器的离散化推导及其C语言实现

2024-07-11 02:38| 来源: 网络整理| 查看: 265

一、理论

下图是采用了PID控制器的闭环控制策略。

PID控制器的传递函数：

$\frac{Y\left ( s \right )}{U\left ( s \right )}=K_p+\frac{K_i}{s}+K_ds$

上式中，Y是控制器的输出，U是控制器的输入。

有时候，在Matlab仿真中已经调好了Kp和Ki、Kd参数，但是离散化后，系数和离散时间有关。因此需要重新计算系数。

方法一、用公式替代s算子

用Trapezoid(Tustin)方法离散化PID控制器。另外更多的传递函数离散化方法请浏览：https://blog.csdn.net/qq_27158179/article/details/82739641#2%E3%80%81%E7%A6%BB%E6%95%A3%E5%8C%96%E9%A2%84%E5%A4%87%E7%9F%A5%E8%AF%86

$\frac{Y\left ( s \right )}{U\left ( s \right )}=K_p+\frac{K_i}{s}+K_ds$

把 $s=\frac{2}{T_s}\frac{z-1}{z+1}$ 代入上式，得到

$\frac{Y\left ( s \right )}{U\left ( s \right )}=K_p+\frac{K_i}{\frac{2}{T_s}\frac{z-1}{z+1}}+K_d\frac{2}{T_s}\frac{z-1}{z+1}$

化简过程略，得到：

$\frac{Y\left ( z \right )}{U\left ( z \right )}= \frac{\frac{\left ( 2T_sK_p+T_s^2K_i+4K_d \right )}{2T_s}+\frac{\left ( 2T_s^2K_i-8K_d \right )}{2T_s}z^{-1}+\frac{\left ( -2T_sK_p+T_s^2K_i+4K_d \right )}{2T_s}z^{-2}}{1-z^{-2}}$

故离散化公式为：

$y\left ( k \right )-y\left ( k-2 \right )=\left ( K_p+\frac{T_s}{2}K_i+2K_d \right )u\left ( k \right )+\left ( T_sK_i-\frac{4}{T_s}K_d \right )u\left ( k-1 \right )+\left ( -K_p+\frac{T_s}{2}K_i+\frac{2}{T_s}K_d \right )u\left ( k-2 \right )$

方法二、传统计算方法

PID控制器分为比例环节、积分环节、微分环节。如下：

$y\left ( k \right )=K_pu\left ( k \right )+K_iT_s\sum_{i=1}^{k}u\left ( i \right )+K_d\frac{u\left ( k \right )-u\left ( k-1 \right )}{T_s}$

上式也是位置式PID。

如果要转换为增量式，过程如下：

我们把PID控制器分为比例环节、积分环节、微分环节。由于控制器的输出就是三个环节的叠加，因此可以分别计算每个环节：

比例部分：

$P\left ( k \right )=K_pu\left ( k \right )$

积分部分：

$I\left ( k \right )=K_iT_s\sum_{i=1}^{k}u\left ( i \right )$

由于

$I\left ( k-1 \right )=K_iT_s\sum_{i=1}^{k-1}u\left ( i \right )$

因此，可以用I(k-1)替代冗长的u(i)，i=1,2,...,k的累加。

$I\left ( k \right )=K_iT_su\left ( k \right )+I\left ( k-1 \right )$

微分部分：

$D\left ( k \right )=K_d\frac{u\left ( k \right )-u\left ( k-1 \right )}{T_s}$

把比例、积分、微分三部分叠加，可以得到公式：

$y\left ( k \right )=K_pu\left ( k \right )+K_iT_su\left ( k \right )+I\left ( k-1 \right )+K_d\frac{u\left ( k \right )-u\left ( k-1 \right )}{T_s}$

化简：

$y\left ( k \right )=\left ( K_p+K_iT_s+\frac{K_d}{T_s} \right )u\left ( k \right ) -\frac{K_d}{T_s}u\left ( k-1 \right )+I\left ( k-1 \right )$

至此，只要需要了解离散时间，也就是执行本控制器算法的间隔Ts，即可把已经在上位机仿真模型中调好的Kp、Ki、Kd、通过计算转换成u(k)和u(k-1)的系数。

二、实践

上机，以下代码主要来自XMC™ Control Library。Infineon提供的library中有PI控制器，我在这个基础上增加了微分环节。

1、PID.c

/* * PID.c * * Created on: 2018年10月1日 * Author: xxJian */ #include "PID.h" /* Initializes a PID controller with anti-windup scheme and programmable output limit saturation. */ void XMC_CONTROL_LIB_PIDFloatInit(XMC_CONTROL_LIB_PID_DATA_FLOAT_t* handle_ptr, volatile float* ref_ptr, volatile float* fdbk_ptr, float kp, float ki, float kd, float limit_min, float limit_max, volatile uint32_t* out) { //memset((void *)handle_ptr, 0, sizeof(*handle_ptr)); handle_ptr->fdbk_ptr = fdbk_ptr; /**< pointer to ADC register which is used for feedback */ handle_ptr->ref_ptr = ref_ptr; /**< ADC ref */ handle_ptr->kp = kp; /*coefficient filter current input*/ handle_ptr->ki = ki; /*coefficient filter input one previous*/ handle_ptr->kd = kd; handle_ptr->out_limit_max = limit_max; /*maximum output saturation*/ handle_ptr->out_limit_min = limit_min; /*output saturation min for filter implementation*/ handle_ptr->out = out; } void XMC_CONTROL_LIB_PIDFloat(XMC_CONTROL_LIB_PID_DATA_FLOAT_t* handle_ptr) { /* * I(k) = Ki * error + I(k-1) * U(k) = Kp * error + I(k) + Kd * (error - error_1) * if(min < U(k) < max){I(k-1) = I(k);}//Anti wind-up */ handle_ptr->error = *(handle_ptr->ref_ptr) - *(handle_ptr->fdbk_ptr); handle_ptr->ik = (handle_ptr->error * handle_ptr->ki) + handle_ptr->ik_1; handle_ptr->uk = (handle_ptr->error * handle_ptr->kp) + handle_ptr->ik + handle_ptr->kd*(handle_ptr->error - handle_ptr->error_1); if (handle_ptr->uk > handle_ptr->out_limit_max) { handle_ptr->uk = handle_ptr->out_limit_max; /*Disable anti wind up filter*/ } else if (handle_ptr->uk < handle_ptr->out_limit_min) { handle_ptr->uk = handle_ptr->out_limit_min; /*Disable anti wind up filter*/ } else { handle_ptr->ik_1 = handle_ptr->ik; /*Enable anti wind up filter*/ } handle_ptr->error_1 = handle_ptr->error; *(handle_ptr->out) = (uint32_t)handle_ptr->uk; }

2、PID.h

/* * PID.h * * Created on: 2018年10月1日 * Author: xxJian */ #ifndef PID_H_ #define PID_H_ #include "main.h" #include "stm32f4xx_hal.h" /** * @brief PI floating point data & coefficients structure */ typedef struct XMC_CONTROL_LIB_PID_DATA_FLOAT { volatile float* fdbk_ptr; /**< Pointer to feedback, generally ADC result register*/ volatile float* ref_ptr; /**< Pointer to reference */ float error; /*!< PID error signal (reference value - feedback value), error[k] */ float error_1; /*!< PID error signal (reference value - feedback value), error[k-1] */ float kp; /*!< Proportional gain Kp constant */ float ki; /*!< Integral gain Ki constant*/ float kd; /*!< Differential gain Ki constant*/ float out_limit_max; /*!< Maximum value of the PI output */ float out_limit_min; /*!< Minimum value of the PI output */ float uk; /*!< PI output U[k] */ float ik; /*!< Integral result I[k] */ float ik_1; /*!< Integral result I[k-1] */ volatile uint32_t* out; /*!< PI Output after anti-windup scheme */ } XMC_CONTROL_LIB_PID_DATA_FLOAT_t; /*******************************************************************************************************************/ /** * @brief Initializes the data structure of XMC_CONTROL_LIB_PIFloat() API. * @param handle_ptr pointer to XMC_CONTROL_LIB_PI_DATA_FLOAT_t structure * @param ref_ptr Pointer to reference voltage variable * @param fdbk_ptr pointer to feedback variable * @param kp Proportional gain constant * @param ki Integral gain constant * @param ki Differential gain constant * @param limit_max maximum output limit for anti-windup scheme * @param limit_min minimum output limit for anti-windup scheme * @param out Pointer to PI output variable * * \parDescription: * The function can be used to initialize the data structure XMC_CONTROL_LIB_PI_DATA_FLOAT_t used by * XMC_CONTROL_LIB_PIFloat() API. * * Example Usage: * @code #include int main(void) { uint32_t ref = 3247.1304f; //reference voltage 3.3 volts in steps of ADC uint32_t vout = 2702.5139f; //measured feedback voltage 3 volts in steps of ADC uint32_t compare_value; //CCU4/8 compare value for PWM generation float kp = 0.001f; //Proportional gain constant float ki = 0.01f; //Integral gain constant float kd = 0.0f; //Integral gain constant float limit_max = 1023.0f; //maximum output for anti-windup scheme float limit_min = 155.0f; //minimum output for anti-windup scheme XMC_CONTROL_LIB_PID_DATA_FLOAT_t handle; //control data structure XMC_CONTROL_LIB_PIDFloatInit(&handle, &ref, &vout, kp, ki, kd, limit_min, limit_max, &compare_value ); XMC_CONTROL_LIB_PIDFloat(&handle); //compare_value will holds the output while(1) { } return 0; } * @endcode * */ void XMC_CONTROL_LIB_PIDFloatInit(XMC_CONTROL_LIB_PID_DATA_FLOAT_t* handle_ptr, volatile float* ref_ptr, volatile float* fdbk_ptr, float kp, float ki, float kd, float limit_min, float limit_max, volatile uint32_t* out); /** * @brief Calculates a proportional integral controller output with programmable output limits for anti-windup scheme. * @param handle_ptr XMC_CONTROL_LIB_PI_DATA_FLOAT_t structure with proper input values. * * \parDescription: * The function can be used to implement control loop feedback mechanism which commonly used in industrial applications. * A PI controller continuously calculates an error value as the difference between a reference and a feedback. * * I[k] = Ki * error + I[k-1] * U[k] = Kp * error + I[k] + Kd* (error[k]-error[k-1]) * if(min < U[k] < max){I[k-1] = I[k];}//Anti wind-up * * Example Usage: * @code #include int main(void) { uint32_t ref = 3247.1304f; //reference voltage 3.3 volts in steps of ADC uint32_t vout = 2702.5139f; //measured feedback voltage 3 volts in steps of ADC uint32_t compare_value; //CCU4/8 compare value for PWM generation float kp = 0.001f; //Proportional gain constant float ki = 0.01f; //Integral gain constant float kd = 0.0f; //Integral gain constant float limit_max = 1023.0f; //maximum output for anti-windup scheme float limit_min = 155.0f; //minimum output for anti-windup scheme XMC_CONTROL_LIB_PI_DATA_FLOAT_t handle; //control data structure XMC_CONTROL_LIB_PIDFloatInit(&handle, &ref, &vout, kp, ki, kd, limit_min, limit_max, &compare_value ); XMC_CONTROL_LIB_PIDFloat(&handle); //compare_value will holds the output while(1) { } return 0; } * @endcode * */ void XMC_CONTROL_LIB_PIDFloat(XMC_CONTROL_LIB_PID_DATA_FLOAT_t* handle_ptr); #endif /* PID_H_ */

在STM32开发板上和直流有刷电机连接，电机上的霍尔编码器和单片机的TIM3连接。对电机进行PID闭环控制。由于我不喜欢微分环节（某些地方不喜欢振荡，微分环节会引入振荡，即闭环控制的输出参数会围绕着参考值上下波动），因此把Kd整为0.把Kp和Ki调节到合适的参数，可以让电机稳定在一个转速。我想把电机控制到1.6转每秒。把实时的转速通过USB转UART上传到电脑，使用STLINK可以直接改变Kp和Ki的值。Kp=100，Ki=0.5，可以让系统有微微的超调量。让电机运动有较好的快速性和稳定性。

现在TI的资料，controlSuite里面的软件，都用上汇编语言了，十分不方便移植到别的单片机中。看来围绕着Arduino、树莓派等开源活动，已经动摇电子界了。工程师们的知识产权有了提高，在开源所带来的声誉，和闭源所带来的安全，有了一个更好的平衡点。

特别鸣谢：www.infineon.com/xmc。

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PID控制器的离散化推导及其C语言实现

PID控制器的离散化推导及其C语言实现

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