Bode's Amplitude/Phase relation for discrete time systems
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From: HardySpicer on 22 May 2010 17:23 On May 23, 7:03 am, Jerry Avins <j...(a)ieee.org> wrote: > 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. > Indeed and I cannot understand why people still toute root-locus and Nyquist. They are excellent teaching tools but practically Bode has it. Hardy
From: Demus on 22 May 2010 17:53 >> but why bother >> when you can't change the input (nor observe the system) between the >> samples anyway (assuming zero order hold)? > >For analyzing stability there is no reason. For a system that is being >sampled slowly compared to the speed at which the plant reacts, you may >need to know if your plant behavior is acceptable between samples. > Sorry I didn't remember that. I have actually come across this when I designed a discrete time "optimal controller" for a continuous system. I remember having to take this into account for the weighting matrices, which among other things gave rise to a cross term in the minimization criterion. >There is a transform, something like w = 0.5 * (z - 1) / (z + 1). Note >that it's _a_ bilinear transform, but it's not _the_ bilinear transform >that folks use to approximate s in the z domain. At any rate, the w >transform maps the z plane onto the w plane, and maps the unit circle on >the z plane onto the imaginary axis of the w plane (if the above >equation I give doesn't have this property then I got it wrong -- I'm >away from my books right now, and frankly I'm too lazy to go look). I don't think I get this... it looks like Tustin with prewarping at with exactness at a particular frequency. What's w? Not just Im(s) since it's domain is a plane...? Or rather maybe, what's the whole thing called so that I can look it up on my own if it would get to lengthy. Thanks for all the info, Tim!
From: Tim Wescott on 22 May 2010 18:06 On 05/22/2010 02:53 PM, Demus wrote: >>> but why bother >>> when you can't change the input (nor observe the system) between the >>> samples anyway (assuming zero order hold)? >> >> For analyzing stability there is no reason. For a system that is being >> sampled slowly compared to the speed at which the plant reacts, you may >> need to know if your plant behavior is acceptable between samples. >> > > Sorry I didn't remember that. I have actually come across this when I > designed a discrete time "optimal controller" for a continuous system. I > remember having to take this into account for the weighting matrices, which > among other things gave rise to a cross term in the minimization > criterion. > >> There is a transform, something like w = 0.5 * (z - 1) / (z + 1). Note >> that it's _a_ bilinear transform, but it's not _the_ bilinear transform >> that folks use to approximate s in the z domain. At any rate, the w >> transform maps the z plane onto the w plane, and maps the unit circle on >> the z plane onto the imaginary axis of the w plane (if the above >> equation I give doesn't have this property then I got it wrong -- I'm >> away from my books right now, and frankly I'm too lazy to go look). > > I don't think I get this... it looks like Tustin with prewarping at with > exactness at a particular frequency. What's w? Not just Im(s) since it's > domain is a plane...? Or rather maybe, what's the whole thing called so > that I can look it up on my own if it would get to lengthy. > > Thanks for all the info, Tim! OK. I'm downstairs now, with book in hand. I'm still lazy, but since it's easy... The book is "Digital Control Systems: Theory, Hardware, Software", Constantine H. Houpis & Gary B. Lamont, McGraw-Hill, 1985. Their discussion starts on page 216 under the heading "Bilinear Transformations", and they just call it the 'w plane'. z = (w + 1) / (-w + 1), from which one derives that w = (z - 1)/(z + 1). There's no prewarping -- in fact there's significant warping after the fact. I was unclear about the nice thing about it. The nice thing about it is that it maps the unit _disk_ onto the left half plane, with the unit _circle_ mapped onto the imaginary axis. Since all the math in the Laplace domain involves clever ways to analyze polynomials, much of the same analysis can then be used in the w plane with the same effect. -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
From: Demus on 22 May 2010 18:07 Sorry if this is a stupid question, but what is the "bode transform"? Is it just any transform that aims to preserve the frequency response of the continuous system? >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 Yeah, I know, sorry again I wasn't clear on what bode-thingy I was referring to!
From: Demus on 22 May 2010 18:18
>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. What am I missing now? Since the inverse transfer function is obtained by the same operation in both continuous time and discrete time (exchanging the numerator and denominator) any approximation method that maps stable-stable should also map minphase - minphase, no? I just mean, I think I agree, it is 'by construction'. |