pade - Padé approximation of model with time delays

Syntax

[num,den] = pade(T,N) sysx = pade(sys,N) sysx = pade(sys,NU,NY,NINT)

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 an time delay of seconds is . This exponential transfer function is approximated by a rational transfer function using Padé approximation formulas [1].

[num,den] = pade(T,N) returns the Nth-order Padé approximation of the continuous-time I/O delay 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 output arguments,

pade(T,N)

plots the step and phase responses of the Nth-order Padé approximation and compares them with the exact responses of the model with I/O delay T. Note that the Padé approximation has unit gain at all frequencies.

sysx = pade(sys,N) produces a delay-free approximation sysx of the continuous delay system sys. All delays are replaced by their Nth-order Padé approximation. See Time Delays for details on LTI models with delays.

sysx = pade(sys,NU,NY,NINT) specifies independent approximation orders for each input, output, and I/O or internal delay. Here NU, NY, and NINT are integer arrays such that

NU is the vector of approximation orders for the input channel
NY is the vector of approximation orders for the output channel
NINT is the approximation order for I/O delays (TF or ZPK models) or internal delays (state-space models)

You can use scalar values for NU, NY, or NINT to specify a uniform approximation order. You can also set some entries of NU, NY, or NINT to Inf to prevent approximation of the corresponding delays.

Example

Compute a third-order Padé approximation of a 0.1 second I/O delay and compare the time and frequency responses of the true delay and its approximation. To do this, type

pade(0.1,3)

Limitations

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

References

[1] Golub, G. H. and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, 1989, pp. 557-558.