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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 `N`th-order polynomials.

When invoked without output arguments,

pade(T,N)

plots the step and phase responses of the `N`th-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 `N`th-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

pade(0.1,3)

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.

`absorbDelay` | `c2d` | `thiran`

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