PID調節的C語言及MATLAB實作方法

在模拟控制系統中，控制器最常用的控制規律就是PID控制，控制系統的原理框圖如圖所示：

在實作PID控制的過程中，關鍵的部分除了PID三個參數之外，就是system和系統回報了，需要能夠拿到系統目前的輸出，才能夠形成一個閉環的回報控制系統，而system就是系統本身的特性，表征系統自身的輸入與輸出之間的關系，進而能夠将PID控制器的輸出疊加到系統輸出上。将連續系統進行離散化處理，就能得到PID在離散系統中的表達。

C語言中建構PID控制器如下：

typedef struct {
	/* Controller gains */
	float Kp;
	float Ki;
	float Kd;
	/* Output limits */
	float limMin;
	float limMax;
	/* Integrator limits */
	float limMinInt;
	float limMaxInt;
	/* Sample time (in seconds) */
	float T;
	/* Controller "memory" */
	float integrator;
	float prevError;			/* Required for integrator */
	float differentiator;
	float prevMeasurement;		/* Required for differentiator */
	/* Controller output */
	float out;
} PIDController;

初始化PID控制器：

void PIDController_Init(PIDController *pid) {

	/* Clear controller variables */
	pid->integrator = 0.0f;
	pid->prevError  = 0.0f;

	pid->differentiator  = 0.0f;
	pid->prevMeasurement = 0.0f;

	pid->out = 0.0f;
}

控制方法的實作：

float PIDController_Update(PIDController *pid, float setpoint, float measurement) {

	/*Error signal*/
    float error = setpoint - measurement;

	/*Integral*/
	pid->integrator = pid->integrator + 0.5f *(error + pid->prevError);

	/* Anti-wind-up via integrator clamping */
    if (pid->integrator > pid->limMaxInt) {

        pid->integrator = pid->limMaxInt;

    } else if (pid->integrator < pid->limMinInt) {

        pid->integrator = pid->limMinInt;

    }

	/* Derivative (band-limited differentiator)*/
	pid->differentiator = (error - pid->prevError);	/* Note: derivative on measurement, therefore minus sign in front of equation! */

	/*Compute output and apply limits*/
    pid->out = (pid->Kp) * error + (pid->Ki)* (pid->integrator) + (pid->Kd) * (pid->differentiator);

    if (pid->out > pid->limMax) {

        pid->out = pid->limMax;

    } else if (pid->out < pid->limMin) {

        pid->out = pid->limMin;

    }

	/* Store error and measurement for later use */
    pid->prevError       = error;
    pid->prevMeasurement = measurement;

	/* Return controller output */
    return pid->out;
}

一個簡單的測試demo：

#define PID_KP  0.9f
#define PID_KI  0.5f
#define PID_KD  0.0f

#define PID_LIM_MIN -10.0f
#define PID_LIM_MAX  10.0f

#define PID_LIM_MIN_INT -5.0f
#define PID_LIM_MAX_INT  5.0f

#define SAMPLE_TIME_S 0.1f

/* Maximum runtime of simulation */
#define SIMULATION_TIME_MAX 50.0f

/* Simulated dynamical system (first order) */
float TestSystem_Update(float inp);

int main()
{
    /* Initialise PID controller */
    PIDController pid = { PID_KP, PID_KI, PID_KD,
                          PID_LIM_MIN, PID_LIM_MAX,
			  PID_LIM_MIN_INT, PID_LIM_MAX_INT,
                          SAMPLE_TIME_S };

    PIDController_Init(&pid);

    /* Simulate response using test system */
    float setpoint = 2.0f;

    printf("Time (s)\tsetpoint\tsystem output\r\n");
    for (float t = 0.0f; t <= SIMULATION_TIME_MAX; t += SAMPLE_TIME_S) {

        /* Get measurement from system */
        float measurement = TestSystem_Update(pid.out);

        /* Compute new control signal */
        PIDController_Update(&pid, setpoint, measurement);

        printf("%f\t%f\t%f\r\n", t, setpoint, measurement);

    }
    return 0;
}

float TestSystem_Update(float inp) {

    float output;
     output = (1.0f + inp);
    return output;
}

測試結果如下：

Time (s)        setpoint        system output
0.000000        2.000000        1.000000
0.100000        2.000000        2.150000
0.200000        2.000000        1.327500
0.300000        2.000000        2.198375
0.400000        2.000000        1.533118
0.500000        2.000000        2.198976
0.600000        2.000000        1.666681
0.700000        2.000000        2.179332
0.800000        2.000000        1.756443
0.900000        2.000000        2.153100
1.000000        2.000000        1.818723
1.100000        2.000000        2.126706
1.200000        2.000000        1.863164
1.300000        2.000000        2.102885
1.400000        2.000000        1.895624
1.500000        2.000000        2.082531
1.600000        2.000000        1.919776
1.700000        2.000000        2.065679
1.800000        2.000000        1.938002
1.900000        2.000000        2.051991
2.000000        2.000000        1.951903
2.100000        2.000000        2.041009
2.200000        2.000000        1.962585
2.300000        2.000000        2.032268
2.400000        2.000000        1.970840
2.500000        2.000000        2.025348
2.600000        2.000000        1.977244
2.700000        2.000000        2.019890
2.799999        2.000000        1.982225
2.899999        2.000000        2.015594
2.999999        2.000000        1.986107
3.099999        2.000000        2.012220
3.199999        2.000000        1.989137
3.299999        2.000000        2.009573
3.399999        2.000000        1.991503
3.499999        2.000000        2.007497
3.599999        2.000000        1.993352
3.699999        2.000000        2.005870
3.799999        2.000000        1.994798
3.899998        2.000000        2.004596
3.999998        2.000000        1.995930
4.099998        2.000000        2.003598
4.199998        2.000000        1.996815
4.299998        2.000000        2.002816
4.399998        2.000000        1.997507
4.499998        2.000000        2.002204
4.599998        2.000000        1.998049
4.699998        2.000000        2.001726
4.799998        2.000000        1.998473
4.899998        2.000000        2.001351
4.999998        2.000000        1.998805
5.099998        2.000000        2.001057
5.199997        2.000000        1.999065
5.299997        2.000000        2.000827
5.399997        2.000000        1.999268
5.499997        2.000000        2.000648
5.599997        2.000000        1.999427
5.699997        2.000000        2.000507
5.799997        2.000000        1.999552
5.899997        2.000000        2.000397
5.999997        2.000000        1.999649
6.099997        2.000000        2.000310
6.199996        2.000000        1.999725
6.299996        2.000000        2.000243
6.399996        2.000000        1.999785
6.499996        2.000000        2.000190
6.599996        2.000000        1.999832
6.699996        2.000000        2.000149
6.799996        2.000000        1.999868
6.899996        2.000000        2.000116
6.999996        2.000000        1.999897
7.099996        2.000000        2.000091
7.199996        2.000000        1.999919
7.299995        2.000000        2.000071
7.399995        2.000000        1.999937
7.499995        2.000000        2.000056
7.599995        2.000000        1.999951
7.699995        2.000000        2.000044
7.799995        2.000000        1.999961
7.899995        2.000000        2.000034
7.999995        2.000000        1.999970
8.099995        2.000000        2.000027
8.199995        2.000000        1.999976
8.299995        2.000000        2.000021
8.399996        2.000000        1.999981
8.499996        2.000000        2.000016
8.599997        2.000000        1.999986
8.699997        2.000000        2.000013
8.799997        2.000000        1.999989
8.899998        2.000000        2.000010
8.999998        2.000000        1.999991
9.099998        2.000000        2.000008
9.199999        2.000000        1.999993
9.299999        2.000000        2.000006
9.400000        2.000000        1.999995
9.500000        2.000000        2.000005
9.600000        2.000000        1.999996
9.700001        2.000000        2.000004
9.800001        2.000000        1.999997
9.900002        2.000000        2.000003
10.000002       2.000000        1.999998
10.100002       2.000000        2.000002
10.200003       2.000000        1.999998
10.300003       2.000000        2.000002
10.400003       2.000000        1.999999
10.500004       2.000000        2.000001
10.600004       2.000000        1.999999
10.700005       2.000000        2.000001
10.800005       2.000000        1.999999
10.900005       2.000000        2.000001
11.000006       2.000000        1.999999
11.100006       2.000000        2.000000
11.200006       2.000000        2.000000
11.300007       2.000000        2.000000
11.400007       2.000000        2.000000
11.500008       2.000000        2.000000
11.600008       2.000000        2.000000
11.700008       2.000000        2.000000
11.800009       2.000000        2.000000
11.900009       2.000000        2.000000
12.000010       2.000000        2.000000
12.100010       2.000000        2.000000
12.200010       2.000000        2.000000
12.300011       2.000000        2.000000
12.400011       2.000000        2.000000

經過一段時間後，系統達到了理想的輸出。接下來，在matlab中建立上面的模型：

PID調節的C語言及MATLAB實作方法

給定了一個較大的微分系數，系統在經過了一段時間的振蕩之後到達了穩定

PID調節的C語言及MATLAB實作方法

同樣的條件下，去掉微分項，結果如下：

PID調節的C語言及MATLAB實作方法

本文中，系統輸出與PID控制器輸出是一個非常簡單的線性關系，在PID控制的實際應用中通常需要系統特性的一個數學表達，将PID的輸出疊加到系統輸出中。

reference：

https://github.com/pms67/PID

PID調節的C語言及MATLAB實作方法

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