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Padé approximation of time delays


[num,den] = padecoef(T,N)



[num,den] = padecoef(T,N) returns the Nth-order Padé Approximation of the continuous-time delay exp(-T*s) 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.


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Use padecoef to estimate the value of $e^{-2s}$ to second order.

[a,b] = padecoef(2,2)
a =

     1    -3     3

b =

     1     3     3

The result indicates that the second order approximation is

$$f(s) \approx \frac{a}{b} = \frac{s^2 - 3s+3}{s^2 + 3s + 3}.$$

Compare the approximation to the actual value at $s = 0.25$.

f_approx = @(s) (s^2 - 3*s+3)/(s^2 + 3*s + 3);
f_actual = @(s) exp(-2*s);
abs(f_approx(0.25) - f_actual(0.25))
ans =


Input Arguments

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Time delay, specified as a real numeric scalar.

Data Types: single | double

Order of approximation, specified as a real numeric scalar.

Data Types: single | double

More About

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Padé Approximation

The Laplace transform of a time delay of T seconds is exp(-Ts). The padecoef function approximates this exponential transfer function by a rational transfer function using Padé approximation formulas. [1]


[1] Golub, G. H. and C. F. Van Loan. Matrix Computations. 4th ed. Johns Hopkins University Press, Baltimore: 2013, pp. 530–532.

See Also

Introduced in R2008a

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