Time-Delay Approximation
Many control design algorithms cannot handle time delays directly. For example, techniques such as root locus, LQG, and pole placement do not work properly if time delays are present. A common technique is to replace delays with all-pass filters that approximate the delays.
To approximate time delays in continuous-time LTI models, use
the pade command to compute
a Padé approximation. The Padé approximation
is valid only at low frequencies, and provides better frequency-domain
approximation than time-domain approximation. It is therefore important
to compare the true and approximate responses to choose the right
approximation order and check the approximation validity.
Time-Delay Approximation in Discrete-Time Models
For discrete-time models, use absorbDelay to
convert a time delay to factors of 1/z where the
time delay is an integer multiple of the sample time.
Use the thiran command to approximate a
time delay that is a fractional multiple of the sample time as a Thiran
all-pass filter.
For a time delay of tau and a sample time
of Ts, the syntax thiran(tau,Ts) creates
a discrete-time transfer function that is the product of two terms:
A term representing the integer portion of the time delay as a pure line delay, (1/z)N, where
N = ceil(tau/Ts).A term approximating the fractional portion of the time delay (
tau - NTs) as a Thiran all-pass filter.
Discretizing a Padé approximation does not guarantee good
phase matching between the continuous-time delay and its discrete
approximation. Using thiran to generate a discrete-time
approximation of a continuous-time delay can yield much better phase
matching. For example, the following figure shows the phase delay
of a 10.2-second time delay discretized with a sample time of 1 s,
approximated in three ways:
a first-order Padé approximation, discretized using the
tustinmethod ofc2dan 11th-order Padé approximation, discretized using the
tustinmethod ofc2dan 11th-order Thiran filter
The Thiran filter yields the closest approximation of the 10.2-second delay.
See the thiran reference
page for more information about Thiran filters.
See Also
absorbDelay | pade | thiran