The Input signal is delayed by a given time instant, or more precisely:

y = u(time - delayTime) for time > time.start + delayTime = u(time.start) for time ≤ time.start + delayTime

The delay is approximated by a Pade approximation, i.e., by a transfer function

b[1]*s^m + b[2]*s^[m-1] + ... + b[m+1] y(s) = --------------------------------------------- * u(s) a[1]*s^n + a[2]*s^[n-1] + ... + a[n+1]

where the coefficients b[:] and a[:] are calculated such that the coefficients of the Taylor expansion of the delay exp(-T*s) around s=0 are identical up to order n+m.

The main advantage of this approach is that the delay is approximated by a linear differential equation system, which is continuous and continuously differentiable. For example, it is uncritical to linearize a system containing a Pade-approximated delay.

The standard text book version uses order "m=n", which is also the default setting of this block. The setting "m=n-1" may yield a better approximation in certain cases.

It is strongly recommended to always set parameter **balance** = true,
in order to arrive at a much better reliable numerical computation.
This is not the default, in order to be backwards compatible, so you have
to explicitly set it. Besides better numerics, also all states are initialized
with **balance** = true (in steady-state, so der(x)=0). Longer explanation:

By default the transfer function of the Pade approximation is implemented
in controller canonical form. This results in coefficients of the A-matrix in
the order of 1 up to the order of O(1/delayTime)^n. For already modest values
of delayTime and n, this gives largely varying coefficients (for example delayTime=0.001 and n=4
results in coefficients between 1 and 1e12). In turn, this results
in a large norm of the system matrix [A,B;C,D] and therefore in unreliable
numerical computations. When setting parameter **balance** = true, a state
transformation is performed that considerably reduces the norm of the system matrix.
This is performed without introducing round-off errors. For details see
function balanceABC.
As a result, both the simulation of the PadeDelay block, and especially
its linearization becomes more reliable.

Otto Foellinger: Regelungstechnik, 8. Auflage, chapter 11.9, page 412-414, Huethig Verlag Heidelberg, 1994

Date | Author | Comment |
---|---|---|

2015-01-05 | Martin Otter (DLR-SR) | Introduced parameter balance=true and a new implementation of the PadeDelay block with an optional, more reliable numerics |

Generated at 2024-04-14T18:15:55Z by OpenModelicaOpenModelica 1.22.3 using GenerateDoc.mos