Découvrez notre Chaîne YouTube "Ingénierie et Projets"


  • Prendre conscience des problèmes qui sont liés à l’instabilité d’un système, saturation, divergence, etc.
  • Savoir les caractéristiques de la fonction de saturation, la tangente hyperbolique
  • Savoir adapter la fonction tangente hyperbolique aux valeurs crêtes de son correcteur
  • Etc.

Importance de la tangente hyperbolique

La tengente hyperbolique est une fonction de saturation par excellence. Elle est caractérisée par son allure linaire lorsque x tant vers 0 (petits signaux), et converge vers [-1, 1] lorsque l’amplitude du signal converge vers l’infini.

  • S(t) petit : tanh(s(t)) => s(t)
  • S(t) grand : tanh(s(t)) => [-1, 1]

tangente hyperbolique

Voir le tuto analyse de la tangente hyperbolique avec Matlab pour plus de détails.

Comment adapter les valeurs crêtes de son correcteur ?

Le tuto aborde le problème lié à la divergence de la commande du correcteur d’une part, et la saturation d’une variable par la carte Arduino. Il est primordial de limiter la dynamique de sa commande durant l’implémentation d’un correcteur PID numérique. La tangente hyperbolique est une fonction rapide et efficace dédiée à cet effet contrairement à l’instruction IF (longue et non linaire). De plus la tangente hyperbolique dispose des caractéristiques linéaires et adoucit le signal durant les transitions (voir le tuto Matlab). Ci-dessous un extrait du code de l’adaptation des niveaux : [-1, 1] vers [-Vmax, Vmax].

const double Vmax=20.0;


Code Arduino

#define   Fn      5.00
#define   Zeta    0.05 //0.70710678118
#define   K       1.0
#define   T_ms    2

#define   A_step  10.0    // Amplitude
#define   c_step  500     // 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[2];        // "0" Non corrigé, "1": Corrigé
double eps_n[2];      // Erreur
double y_capt[2];    // Sortie du capteur   
double y_corr[2];    // Sortie du correcteur   

// Variables internes des systèmes
double x1[2], y1[3]; // Système Non Corrigé
double x2[2], y2[3]; // Système Corrigé

// Variables internes du correcteur 
double x_c[2], y_c[2]; 

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

void setup()
  // Port série de la réponse du système 

void loop()
  // 1. La consigne (l'entrée) x(n) pour les deux systèmes  
  c++; c=c%c_step; 
  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
  // 3. Soustracteur: Calcul de l'erreur eps(n) 
  // 4.1  Correcteur
  y_corr[0]=eps_n[0];     // Système non Corrigé

  // 4.2 Correcteur PID (voir le lien en commentaire) 
  double k0=10.00; 
  double N_fil=250.0;
  double z_PID=1.5;   
  double w_PID=Wn; //sqrt(1-Zeta*Zeta); 
  y_corr[1]= PID(eps_n[1], x_c, y_c,  N_fil,  z_PID,  w_PID,  k0, T_s);

  // 4.3 Ajout du bloc de la saturation 
  const double Vmax=20.0; 

  // 5. Calcul de la sortie: Système non corrigé 
  y_n[0]=Sys2All(y_corr[0], x1, y1, Zeta, Wn, K, T_s);

  // 5. Calcul de la sortie: Système corrigé  
  y_n[1]=Sys2All(y_corr[1], x2, y2, Zeta, Wn, K, T_s);

  // Affichage des signaux 
  Serial.print(x_nn); Serial.print(","); 
  Serial.print(y_n[0]); Serial.print(",");

  // Période d'échantillonnage 

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)
  // Mise à jour de la sortie 

  // Mise à jour de la sortie 
  // Renvoie du résultat 
  return y_nn;

double CorrPI(double x_nn, double *xpi, double *ypi, double kp, double ki, double T)
  // Variables de l'entrée et la sortie 
  double y_nn=0.0;
  // Calcul de la nouvelle sortie 
  y_nn=ypi[1] + kp*x_nn + (2.0*T*ki)*xpi[0] -kp*xpi[1]; 
  // y(n)=y(n-2)+ k1*x(n) + 2*T*k2*x(n-1) - k1*x(n-2)
  // Mise à jour de la sortie 

  // Mise à jour de la sortie 
  // Renvoie du résultat 
  return y_nn;

double CorrPD_AP(double x_nn, double *x_cc, double *y_cc, double a, double t0, double k0, double T)
  // Variables de l'entrée et la sortie 
  double y_nn=0.0;

  // Paramètres du correcteur
  double alfa=a*t0;
  double beta=t0;  
  // Calcul de la nouvelle sortie
  // Modèle Analogique: C(p)= k*(1+aTp)/(1+Tp)
  // Modèle numérique:  y(n)=[alfa*k 2kT -alfa*k]*[x(n) x(n-1) x(n-2)]'
  //                        -[2T -beta]*[y(n-1) y(n-2)]'
  //                    y(n)=y(n)/beta 
  y_nn=(alfa*k0*x_nn)+(2*k0*T*x_cc[0]) - (alfa*k0*x_cc[1]);
  y_nn=y_nn- (2*T*y_cc[0]) - (beta*y_cc[1]); 
  // Mise à jour de la sortie 

  // Mise à jour de la sortie 
  // Renvoie du résultat 
  return y_nn;

double PID(double x_nn, double *x_cc, double *y_cc, double N_fil, double z, double w, double k, double T)
  double t1=1.0/(w*(z+sqrt(z*z-1.0))); 
  double t2=1.0/(w*(z-sqrt(z*z-1.0)));

  double td=t1+t2; 
  double ti=t1*t2/td; 

  double a1=2.0*z/w; 
  double a2=1.0/(w*w); 
  double b1=ti; 
  double b2=ti*t2/N_fil; 

  double b[3]; 
  b[0]= a1/(2*T)+a2/(T*T); 
  b[1]= 1.0-2.0*a2/(T*T); 

  double c[3];
  c[0]= b1/(2*T)+b2/(T*T);
  c[1]= -2.0*b2/(T*T);
  c[2]= -b1/(2*T)+b2/(T*T);
  // Variables de l'entrée et la sortie 
  double y_nn=0.0;

  // Calcul de la nouvelle sortie
  y_nn=k*(b[0]*x_nn + b[1]*x_cc[0] + b[2]*x_cc[1]);  
  y_nn=y_nn-(c[1]*y_cc[0] + c[2]*y_cc[1]);  
  // Mise à jour de la sortie 

  // Mise à jour de la sortie 
  // Renvoie du résultat 
  return y_nn;

Accueil Asservissement avec Arduino

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