No idea what you mean by "first principle" but basically traditional BLT transform (takes the transfer function of a continuous-time LTI system and substitutes s<-2/T*(z-1)/(z+1), then simplifies resulting discrete-time transfer function to obtain a rational that we can implement as a direct form.So you're saying that ZDF/TPT is the theoretical first principle for non-LTI?
If the system is not LTI, then "transfer function" is ill-defined concept, because "transfer functions" in the conventional sense only apply to LTI systems. However, what we can do is observe that the subsitution s<-2/T*(z-1)/(z+1) or it's reciprocal 1/s<-T/2*(z+1)/(z-1) is in fact the trapezoidal rule of numerical integration and we can use the trapezoidal method to integrate (basically) any continuous-time system, LTI or not.
So rather than evaluating the continuous-time transfer function and substituting the trapezoidal rule there, we can take the (time-domain) differential equations and use the same trapezoidal method to integrate those instead. If the system "happens to be" LTI then we end up with the exact same discrete-time transfer function either way... but when the system is not LTI, doing the subsitution in time-domain allows us to preserve the original state variables: each integrator remains an integrator after the transform.
Finally, since in both cases we're really using the same approximation (the trapezoidal rule) for Laplace s, we can also take the "prewarping" commonly applied to conventional bilinear transform and apply it the same way to our time-domain trapezoidal rule, which gives us full equivalence in the LTI case and in fact if we evaluate the digital transfer function (for an LTI system) then simplify we end up with the same direct form coefficients we would have obtained by substituting directly into the continuous-time transfer function.
So.. in a sense, the bilinear transform as applied to transfer functions to obtain direct form coefficients is a mathematical shortcut to trapezoidal integration that applies when the system is LTI. If the system is not LTI, you can still do trapezoidal "the hard way" and this what is known as ZDF/TPT.
Statistics: Posted by mystran — Sun Mar 24, 2024 10:39 pm