pade

Control System Toolbox

Pade approximation of time delays.

[num,den] = pade(Td,N)
pade(Td,N)

sysx = pade(sys,N)
sysx = pade(sys,[N1,...,Nm])

Description

pade approximates time delays by rational LTI models. Such approximations are useful to model time delay effects such as transport and computation delays within the context of continuous-time systems. The Laplace transform of a time delay of

seconds is

. This exponential transfer function is approximated by a rational transfer function using the Pade approximation formulas [1].

[num,den] = pade(Td,N) returns the Nth-order (diagonal) Pade approximation of the continuous-time delay exp(-s*Td) in transfer function form. The row vectors num and den contain the numerator and denominator coefficients in descending powers of

. Both are Nth-order polynomials.

When invoked without left-hand argument,

pade(Td,N)

plots the step and phase responses of the Nth-order Pade approximation and compares them with the exact responses of the time delay Td. Note that the Pade approximation has unit gain at all frequencies.

sysx = pade(sys,N) produces a delay-free approximation sysx of the continuous delay system sys. All input delays are replaced by their Nth-order Pade approximation. If sys is a multi-input system with m inputs, you can also specify an independent approximation order for each input channel by

sysx = pade(sys,[N1,...,Nm])

Example

Compute a third-order Pade approximation of a 0.1 second time delay and compare the time and frequency responses of the true delay and its approximation:

» pade(0.1,3)

Limitations

High-order Pade approximations produce transfer functions with clustered poles. Because such pole configurations tend to be very sensitive to perturbations, Pade approximations with order N>10 should be avoided.