absorbDelay

Replace time delays by poles at z = 0 or phase shift

Syntax

sysnd = absorbDelay(sysd)
[sysnd,G] = absorbDelay(sysd)


Description

sysnd = absorbDelay(sysd) absorbs all time delays of the dynamic system model sysd into the system dynamics or the frequency response data.

For discrete-time models (other than frequency response data models), a delay of k sampling periods is replaced by k poles at z = 0. For continuous-time models (other than frequency response data models), time delays have no exact representation with a finite number of poles and zeros. Therefore, use pade (Control System Toolbox) to compute a rational approximation of the time delay.

For frequency response data models in both continuous and discrete time, absorbDelay absorbs all time delays into the frequency response data as a phase shift.

[sysnd,G] = absorbDelay(sysd) returns the matrix G that maps the initial states of the ss (Control System Toolbox) model sysd to the initial states of the sysnd.

Examples

Absorb Time Delay into System Dynamics

Create a discrete-time transfer function that has a time delay.

z = tf('z',-1);
sysd = (-0.4*z -0.1)/(z^2 + 1.05*z + 0.08);
sysd.InputDelay = 3

sysd =
 
              -0.4 z - 0.1
  z^(-3) * -------------------
           z^2 + 1.05 z + 0.08
 
Sample time: unspecified
Discrete-time transfer function.

The display of sysd represents the InputDelay as a factor of z^(-3), separate from the system poles that appear in the transfer function denominator.

Absorb the time delay into the system dynamics as poles at z= 0.

sysnd = absorbDelay(sysd)

sysnd =
 
        -0.4 z - 0.1
  -------------------------
  z^5 + 1.05 z^4 + 0.08 z^3
 
Sample time: unspecified
Discrete-time transfer function.

The display of sysnd shows that the factor of z^(-3) has been absorbed as additional poles in the denominator.

Verify that sysnd has no input delay.

sysnd.InputDelay

ans = 0

Convert Leading Structural Zeros of Polynomial Model to Regular Coefficients

Create a discrete-time polynomial model.

m = idpoly(1,[0 0 0 2 3]);

Convert m to a transfer function model.

sys = tf(m)

sys =
 
  z^(-2) * (2 z^-1 + 3 z^-2)
 
Sample time: unspecified
Discrete-time transfer function.

The numerator of the transfer function, sys, is [0 2 3] and the transport delay, sys.IODelay, is 2. This is because the value of the B polynomial, m.B, has 3 leading zeros. The first fixed zero shows lack of feedthrough in the model. The two zeros after that are treated as input-output delays.

Use absorbDelay to treat the leading zeros as regular B coefficients.

m2 = absorbDelay(m);
sys2 = tf(m2)

sys2 =
 
  2 z^-3 + 3 z^-4
 
Sample time: unspecified
Discrete-time transfer function.

The numerator of sys2 is [0 0 0 2 3] and transport delay is 0. The model m2 treats the leading zeros as regular coefficients by freeing their values. m2.Structure.B.Free(2:3) is TRUE while m.Structure.B.Free(2:3) is FALSE.

Version History

Introduced in R2012a