Documentation

Create sample open-loop system with an output delay.

s = tf('s');
P = exp(-2.6*s)/(s^2+0.9*s+1);

P is a second-order transfer function (tf) object with a time delay.

Compute the first-order Padé approximation of P.

Pnd1 = pade(P,1)

Pnd1 =
 
             -s + 0.7692
  ----------------------------------
  s^3 + 1.669 s^2 + 1.692 s + 0.7692
 
Continuous-time transfer function.

This command replaces all time delays in P with a first-order approximation. Therefore, Pnd1 is a third-order transfer function with no delays.

Compare the frequency response of the original and approximate models using bodeplot.

h = bodeoptions;
h.PhaseMatching = 'on';
bodeplot(P,'-b',Pnd1,'-.r',{0.1,10},h)
legend('Exact delay','First-Order Pade','Location','SouthWest')

The magnitude of P and Pnd1 match exactly. However, the phase of Pnd1 deviates from the phase of P beyond approximately 1 rad/s.

Increase the Padé approximation order to extend the frequency band in which the phase approximation is good.

Pnd3 = pade(P,3);

Compare the frequency response of P, Pnd1 and Pnd3.

bodeplot(P,'-b',Pnd3,'-.r',Pnd1,':k',{0.1 10},h)
legend('Exact delay','Third-Order Pade','First-Order Pade',...
       'Location','SouthWest')

The phase approximation error is reduced by using a third-order Padé approximation.

Compare the time domain responses of the original and approximated systems using stepplot.

stepplot(P,'-b',Pnd3,'-.r',Pnd1,':k')
legend('Exact delay','Third-Order Pade','First-Order Pade',...
       'Location','Southeast')

Using the Padé approximation introduces a nonminimum phase artifact (“wrong way” effect) in the initial transient response. The effect is quite pronounced in the first-order approximation, which dips significantly below zero before changing direction. The effect is reduced in the higher-order approximation, which far more closely matches the exact system’s response.