Bode's Amplitude/Phase relation for discrete time systems
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From: Tim Wescott on 22 May 2010 13:14 On 05/22/2010 05:33 AM, Jerry Avins wrote: > On 5/21/2010 8:00 PM, Demus wrote: >> Could someone please show me what the discrete-time version of Bode's >> amplitude/phase relation looks like? > > I don't understand. A Bode plot consists of two graphs of a transfer > function. Both use a horizontal axis of log(frequency). One of them is > of log(amplitude), the other of phase. I suppose that it is necessary > that the transfer function be continuous, but that is not a severe > constraint. > > The properties of Bode plots that make them easy to draw and make it > possible to infer the phase plot from the amplitude plot apply to > minimum-phase systems only. Most transfer functions of systems of > discrete components are minimum phase. Most digital transfer functions > are not. Digital transfer functions derived from stable, minimum phase continuous-time plants by the exact method are, in my experience, minimum phase. I'm not sure, but they may be minimum phase by construction. Certainly it's foolish to cook up a digital controller transfer function that's not minimum phase unless one has very specific and peculiar specifications to satisfy. And Bode plot design does work if you're starting with a non-minumum phase system, as long as you start from a plot of a system that is known to be stable. Then the phase crossover points merely say that you are moving from safe to unsafe, and it is up to you to determine which side is which. -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
From: Mikolaj on 22 May 2010 13:39 Dnia 22-05-2010 o 10:05:02 HardySpicer <gyansorova(a)gmail.com> napisa³(a): (...) > If you sample fast enought you get back to analogue. Digital is just > an approximation to analogue, nothing more. (...) This is not precise and this is not true. You will never get analogue equivalent. (...) > For example, take an integrator K/s. Use the Bilinear transform and > sample high enough and it will have a slope of -20dB/decade just as > the analogue case and a phase of -90 degrees.. (...) Bilinear transform is a very special method, it preserves stability. There are plenty of ways to do it and every one of them gives different results. Different magnitude and phase shapes, deformation of freq. axis or deformation of time response. Some methods preserves time response for just one kind of signal and everything else is different. For example of sampling than we probably use step invariant transformation to get time response identical to continuous system. But freq. response will be affected :( and for every other input signal response will be diffrent from continues system. If we use ramp DAC or we use zero order holder methode will be modeling our discret equivalent in a different way. It's not so obvious which way should we do it. I'm not sure if there is even mathematical proof that infinit sampling in a limit gives us analogue equivalent and in what special conditions of sampling and number to physical value. Remember that system has input and output. I would not mix and compare digital to analogue. In practice ther are different worlds. -- Mikolaj
From: Mikolaj on 22 May 2010 13:42 On 22-05-2010 at 19:39:29 Mikolaj <sterowanie_komputerowe(a)poczta.onet.pl> wrote: Forgive me and please replace "For example of sampling than we probably" by "For example we will" -- Mikolaj
From: Jerry Avins on 22 May 2010 15:03 On 5/22/2010 1:14 PM, Tim Wescott wrote: > On 05/22/2010 05:33 AM, Jerry Avins wrote: >> On 5/21/2010 8:00 PM, Demus wrote: >>> Could someone please show me what the discrete-time version of Bode's >>> amplitude/phase relation looks like? >> >> I don't understand. A Bode plot consists of two graphs of a transfer >> function. Both use a horizontal axis of log(frequency). One of them is >> of log(amplitude), the other of phase. I suppose that it is necessary >> that the transfer function be continuous, but that is not a severe >> constraint. >> >> The properties of Bode plots that make them easy to draw and make it >> possible to infer the phase plot from the amplitude plot apply to >> minimum-phase systems only. Most transfer functions of systems of >> discrete components are minimum phase. Most digital transfer functions >> are not. > > Digital transfer functions derived from stable, minimum phase > continuous-time plants by the exact method are, in my experience, > minimum phase. I'm not sure, but they may be minimum phase by > construction. Certainly it's foolish to cook up a digital controller > transfer function that's not minimum phase unless one has very specific > and peculiar specifications to satisfy. > > And Bode plot design does work if you're starting with a non-minumum > phase system, as long as you start from a plot of a system that is known > to be stable. Then the phase crossover points merely say that you are > moving from safe to unsafe, and it is up to you to determine which side > is which. I didn't write that Bode plot design doesn't work if with non-minimum phase, but that the usual relations -- 45degree shift at a corner frequency, 27 degrees an octave away, and 5.7 degrees a decade away -- don't work, and one needs to calculate the numbers for each particular case. The Bode transform is indeed the way to go. Jerry -- "I view the progress of science as ... the slow erosion of the tendency to dichotomize." --Barbara Smuts, U. Mich. �����������������������������������������������������������������������
From: HardySpicer on 22 May 2010 17:22
On May 23, 5:39 am, Mikolaj <sterowanie_komputer...(a)poczta.onet.pl> wrote: > Dnia 22-05-2010 o 10:05:02 HardySpicer <gyansor...(a)gmail.com> napisa³(a): > > (...)> If you sample fast enought you get back to analogue. Digital is just > > an approximation to analogue, nothing more. > > (...) > > This is not precise and this is not true. > You will never get analogue equivalent. > > (...)> For example, take an integrator K/s. Use the Bilinear transform and > > sample high enough and it will have a slope of -20dB/decade just as > > the analogue case and a phase of -90 degrees.. > > (...) > > Bilinear transform is a very special method, > it preserves stability. > Of course we know that but there are at least two other methods we can use, forward and backward difference and one of these is also guaranteed stable. However, look to numerical integration which is all this is, we can use rectangular, trapezoidal or even better approximations. All of these are approximations - better and better and as you point out the Bilinear (Trapezoidal) method is a particular good one. However you cut it though they are all approximations to analogue coming from z=exp(sT). For example I can say z approx=1+sT and ignore higher order terms. So when you simulate ananalogue system, whether it be in state-space or using Bilinear transform etc they are all approximations. There is an argument to say that if I create an FIR filter with linear phase in digital then it has no counterpart in analogue, but that is working backwards! Hardy > Mikolaj |