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Objectifs

  1. Implémentation d’un système du 2nd ordre en boucle fermée (BF)
  2. Analyse d’un système en BF
  3. La réponse indicielle/rampe en BF
  4. Avantage et inconvénient d’un système en boucle fermée à retour unitaire sans correcteur  La réponse à une rampe d’un système en 2nd ordre
  5. Etc.

Voir le tuto d’avant pour plus des détails

Fonctionnement

Le tuto a pour objectif d’introduire l’architecture d’asservissement avec Arduino. En particulier la boucle fermée avec le retour unitaire et sans correcteur. On va analyser la réponse indicielle et une rampe du système en BF en comparaison avec la boucle ouverte et en déduire les avantages et incontinents des deux stratégies de commande. L’architecture introduite dans ce tuto sera utilisé dans les tutos prochains avec les divers types des correcteurs numériques (P, PI, PD, PID, …). Voir le tuto pour plus des détails. Ci-dessous le code Arduino commenté.

système asservissement retour unitaire

Les paramètres du système

#define   Fn      10.00

#define   Zeta    0.70710678118

#define   K       1.0

#define   T_ms    2

Les paramètres de l’échelon (ou rampe)

#define   A_step  10.0 // Amplitude

#define   c_step  200  // Période = 2*c_step*T_ms

unsigned long c=0; // Compteur (période) 

bool Step=false;

Les variables en BO et BF

double x_nn=0.0;    // Consigne (entrée)

double y_n_BF=0.0;  // Sortie en BF

double y_n_BO=0.0;  // Sortie en BO

double eps_n=0.0;   // Erreur

double y_capt=0.0;  // Sortie du capteur  

double y_corr=0.0;  // Sortie du correcteur

Les variables internes des systèmes

double x1[2], y1[3]; // Système en BF

double x2[2], y2[3]; // Système en BO

Les étapes de l’implémentation de la boucle fermée avec Arduino

 La consigne (l’entrée) x(n)

  c++; c=c%c_step;

  if(!c)

  {

    Step=!Step;

    c=0;

  }

  x_nn=A_step*(double)Step; // Réponse à un échelon x(n)=cte

  //x_nn=(double)c;           // Réponse à une rampe x(n)=n

Sortie du capteur: Retour unitaire

  y_capt=y_n_BF;

Soustracteur: Calcul de l’erreur eps(n)

  eps_n=x_nn-y_capt;

Correcteur

  y_corr=eps_n;

Calcul de la sortie en BF

  y_n_BF=Sys2All(y_corr, x1, y1, Zeta, Wn, K, T_s);

Calcul de la sortie en BO

  y_n_BO=Sys2All(x_nn, x2, y2, Zeta, Wn, K, T_s);

Affichage des signaux

  Serial.print(x_nn); Serial.print(",");

  Serial.print(y_n_BO); Serial.print(",");

  Serial.println(y_n_BF); 

Implémentation d’un système du second ordre

double Sys2All(double x_nn, double *x, double *y, double zeta, double wn, double k, double T)

{

  // Paramètre du système

  double a1=2.0*zeta/wn;

  double a2=1.0/(wn*wn);




  const double b0=(a1/(2.0*T))+(a2/(T*T));

  const double b1=-2.0*a2/(T*T);

  const double b2=(-1.0*a1/(2.0*T))+(a2/(T*T));

  const double b[3]={b0,b1,b2};




  // Variables de l'entrée et la sortie

  double y_nn=0.0;




  // Calcul de la nouvelle sortie

  y_nn= -(y[0]*(1.0+b[1]))-(y[1]*b[2])+(k*x[0]); // y[1]: y(n-2), y[0]: y(n-1)

  y_nn/=b[0];




  // Mise à jour de la sortie

  y[1]=y[0];

  y[0]=y_nn;




  // Mise à jour de la sortie

  x[0]=x_nn;




  // Renvoie du résultat

  return y_nn;

}

Asservissement Arduino Architecture d’un Système en boucle fermée 1

Le code complet


/*
 * 1. Implémentation d'un système du 2nd ordre en boucle fermée (BF)  
 * 2. Analyse d'un système en BF
 * 3. La réponse indicielle/rampe en BF 
 * 4. La réponse à une rampe d'un système en 2nd ordre 
 * 5. Etc.
 * 
                 -------------
                 -           -  
 x(n) ------->---   SYS2     ------->-- y(n): BO 
                 -           -  
                 -------------  
         
                 -------------  -------------
                 -           -  -           -
 x(n) --[-]--->---   CORR    ----   SYS2    ------->-- y(n): BF  
         -       -           -  -           -   -   
         -       -------------  -------------   -
         -       -------------                  -
         -       -           -                  - 
         ---<-----   CAPT    ----------<---------
                 -           -
                 -------------            
*/

 
#define   Fn      10.00
#define   Zeta    0.70710678118
#define   K       1.0
#define   T_ms    2

#define   A_step  10.0 // Amplitude
#define   c_step  200  // Période = 2*c_step*T_ms

double Wn=2.0*PI*Fn;
double T_s=(double)T_ms/1000.0;


double x_nn=0.0;    // Consigne (entrée) 
double y_n_BF=0.0;  // Sortie en BF 
double y_n_BO=0.0;  // Sortie en BO
double eps_n=0.0;   // Erreur
double y_capt=0.0;  // Sortie du capteur   
double y_corr=0.0;  // Sortie du correcteur   

// Variables internes des systèmes
double x1[2], y1[3]; // Système en BF 
double x2[2], y2[3]; // Système en BO


// Paramètres de l'échelon
unsigned long c=0; // Compteur (période)  
bool Step=false; 

void setup()
{
  // Port série de la réponse du système 
  Serial.begin(9600); 
}

void loop()
{ 
  // 1. La consigne (l'entrée) x(n) 
  c++; c=c%c_step; 
  if(!c) 
  {
    Step=!Step;
    c=0; 
  }
  x_nn=A_step*(double)Step; // Réponse à un échelon x(n)=cte
  //x_nn=(double)c;           // Réponse à une rampe x(n)=n
  
  // 2. Sortie du capteur: Retour unitaire
  y_capt=y_n_BF; 
  
  // 3. Soustracteur: Calcul de l'erreur eps(n) 
  eps_n=x_nn-y_capt; 

  // 4. Correcteur 
  y_corr=eps_n;
  
  // 5. Calcul de la sortie en BF 
  y_n_BF=Sys2All(y_corr, x1, y1, Zeta, Wn, K, T_s);

  // 6. Calcul de la sortie en BO 
  y_n_BO=Sys2All(x_nn, x2, y2, Zeta, Wn, K, T_s);
  
  // Affichage des signaux 
  Serial.print(x_nn); Serial.print(","); 
  Serial.print(y_n_BO); Serial.print(",");
  Serial.println(y_n_BF);  

  // Période d'échantillonnage 
  delay(T_ms);
}



double Sys2All(double x_nn, double *x, double *y, double zeta, double wn, double k, double T)
{
  // Paramètre du système 
  double a1=2.0*zeta/wn;
  double a2=1.0/(wn*wn);

  const double b0=(a1/(2.0*T))+(a2/(T*T)); 
  const double b1=-2.0*a2/(T*T); 
  const double b2=(-1.0*a1/(2.0*T))+(a2/(T*T)); 
  const double b[3]={b0,b1,b2};

  // Variables de l'entrée et la sortie 
  double y_nn=0.0;
  
  // Calcul de la nouvelle sortie 
  y_nn= -(y[0]*(1.0+b[1]))-(y[1]*b[2])+(k*x[0]); // y[1]: y(n-2), y[0]: y(n-1)
  y_nn/=b[0];
  
  // Mise à jour de la sortie 
  y[1]=y[0];
  y[0]=y_nn;

  // Mise à jour de la sortie 
  x[0]=x_nn;
  
  // Renvoie du résultat 
  return y_nn;
}

Accueil Asservissement avec Arduino

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