Time-Delay Approximation

许多控制设计算法不能处理时间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 thepadecommand 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, useabsorbDelayto convert a time delay to factors of 1/zwhere the time delay is an integer multiple of the sample time.

Use thethirancommand to approximate a time delay that is a fractional multiple of the sample time as a Thiran all-pass filter.

For a time delay oftauand a sample time ofTs, the syntaxthiran(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, whereN = 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. Usingthiranto 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 thetustinmethod ofc2d
an 11th-order Padé approximation, discretized using thetustinmethod ofc2d
an 11th-order Thiran filter

The Thiran filter yields the closest approximation of the 10.2-second delay.

See thethiranreference page for more information about Thiran filters.

Time-Delay Approximation

Time-Delay Approximation in Discrete-Time Models

See Also

Related Topics

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