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Padé approximation of time delays

## Syntax

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

## Description

example

````[num,den] = padecoef(T,N)` returns the `N`th-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 `N`th-order polynomials.```

## Examples

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

`[a,b] = padecoef(2,2)`
```a = 1×3 1 -3 3 ```
```b = 1×3 1 3 3 ```

The result indicates that the second order approximation is

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

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 = 2.6717e-05 ```

## 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`

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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]

## References

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