Documentation

Create sample continuous-time closed-loop system with an internal delay.

Construct a model Tcl of the closed-loop transfer function from r to y.

s = tf('s');
G = (s+1)/(s^2+.68*s+1)*exp(-4.2*s);
C = pid(0.06,0.15,0.006);
Tcl = feedback(G*C,1);

Examine the internal delay of Tcl.

Tcl.InternalDelay

ans = 4.2000

Compute the first-order Padé approximation of Tcl.

Tnd1 = pade(Tcl,1);

Tnd1 is a state-space (ss) model with no delays.

Compare the frequency response of the original and approximate models.

h = bodeoptions;
h.PhaseMatching = 'on';
bodeplot(Tcl,'-b',Tnd1,'-.r',{.1,10},h);
legend('Exact delay','First-Order Pade','Location','SouthWest');

The magnitude and phase approximation errors are significant beyond 1 rad/s.

Compare the time domain response of Tcl and Tnd1 using stepplot.

stepplot(Tcl,'-b',Tnd1,'-.r');
legend('Exact delay','First-Order Pade','Location','SouthEast');

Using the Padé approximation introduces a nonminimum phase artifact (“wrong way” effect) in the initial transient response.

Increase the Padé approximation order to see if this will extend the frequency with good phase and magnitude approximation.

Tnd3 = pade(Tcl,3);

Observe the behavior of the third-order Padé approximation of Tcl. Compare the frequency response of Tcl and Tnd3.

bodeplot(Tcl,'-b',Tnd3,'-.r',Tnd1,'--k',{.1,10},h);
legend('Exact delay','Third-Order Pade','First-Order Pade',...
       'Location','SouthWest');

The magnitude and phase approximation errors are reduced when a third-order Padé approximation is used.