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Padé approximation of model with time delays


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


pade approximates time delays by rational 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 T seconds is exp(–sT). This exponential transfer function is approximated by a rational transfer function using Padé approximation formulas [1].

[num,den] = pade(T,N) returns the Padé approximation of order N of the continuous-time I/O delay exp(–sT) in transfer function form. The row vectors num and den contain the numerator and denominator coefficients in descending powers of s. Both are Nth-order polynomials.

When invoked without output arguments,


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 Models with Time Delays for more information about models with time 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.


Third-Order Padé Approximation

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



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.


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

See Also

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