pade
Padé approximation of model with time delays
Syntax
[num,den] = pade(T,N)
pade(T,N)
sysx = pade(sys,N)
sysx = pade(sys,NU,NY,NINT)
Description
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, 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 in Linear Systems 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
NUis the vector of approximation orders for the input channelNYis the vector of approximation orders for the output channelNINTis 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.
Examples
Third-Order Padé Approximation
Compute a third-order Padé approximation of a 0.1-second I/O delay.
s = tf('s');
sys = exp(-0.1*s);
sysx = pade(sys,3)sysx = -s^3 + 120 s^2 - 6000 s + 1.2e05 -------------------------------- s^3 + 120 s^2 + 6000 s + 1.2e05 Continuous-time transfer function.
Here, sys is a dynamic system representation of the exact time delay of 0.l s. sysx is a transfer function that approximates that delay.
Compare the time and frequency responses of the true delay and its approximation. Calling the pade command without output arguments generates the comparison plots. In this case the first argument to pade is just the magnitude of the exact time delay, rather than a dynamic system representing the time delay.
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.