Bode's Amplitude/Phase relation for discrete time systems
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From: Demus on 23 May 2010 12:59 >For any _one_ frequency. Also, a high band edge is sharpened. > >Jerry >-- Thanks Jerry, you confirmed what I thought but didn't dare to ask!
From: Jerry Avins on 23 May 2010 13:03 On 5/23/2010 12:59 PM, Demus wrote: >> For any _one_ frequency. Also, a high band edge is sharpened. >> >> Jerry >> -- > > Thanks Jerry, you confirmed what I thought but didn't dare to ask! Didn't dare? Why not? Answers come more quickly when you pose questions. Jerry -- "I view the progress of science as ... the slow erosion of the tendency to dichotomize." --Barbara Smuts, U. Mich. �����������������������������������������������������������������������
From: Tim Wescott on 23 May 2010 13:28 On 05/23/2010 03:50 AM, Rune Allnor wrote: > On 23 Mai, 00:06, Tim Wescott<t...(a)seemywebsite.now> wrote: > >> 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. > > Pre-warping has to do with countering that 'after the fact' warping > of the BLT. > > The end target of the BLT-based techniques is a discrete-time (DT) > domain filter, and the continuous time (CT) is only a convenient > domain to do the computations. > > Now, since the end target is DT the up front spec is also DT. > But one does need a CT spec to do the core computations of the > CT filter. > > Because of all this, one accounts for the BLT warps already > at the spec stage, so that the CT filter is designed from an > already warped - pre warped - spec. Done correctly, the end > DT filter woks out exactly right. True. I don't see it applying much to this particular one in the context that the author intended, but true. From what I get reading the book, the author is presenting this particular technique not as a means for filter design, but as a means of translating z-domain systems descriptions into descriptions that are amenable to stability analysis by all the tests familiar to people trained and experienced with Laplace-domain analysis. It never got popular in large part because with a bit of effort most of the Laplace-domain techniques have cognates in the z domain (Bode and Nyquist plots, in particular, translate straight across). But there are some that don't translate. I mentioned it in the first place because I've struggled to make a Bode's sensitivity integral work for the z domain -- for some reason Demus's post made me think of Houpis & Lamont's w plane, for which the sensitivity integral construction for sampled time is almost trivial -- which is why I mentioned it. With this transform, telling someone how to translate Bode's sensitivity integral into sampled time is just a matter of "use this here variable transformation" and I'm done -- I'm not going to spit on that! -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
From: HardySpicer on 23 May 2010 22:12 On May 23, 10:27 am, Tim Wescott <t...(a)seemywebsite.now> wrote: > On 05/22/2010 01:05 AM, HardySpicer wrote: > > > > > On May 22, 4:41 pm, Tim Wescott<t...(a)seemywebsite.now> wrote: > >> On 05/21/2010 08:03 PM, HardySpicer wrote: > > >>> On May 22, 12:00 pm, "Demus"<sodemus(a)n_o_s_p_a_m.hotmail.com> wrote: > >>>> Could someone please show me what the discrete-time version of Bode's > >>>> amplitude/phase relation looks like? > > >>> Discrete-time is an approximation to analogue so Bodes Theorem is the > >>> same. > > >> Eh? Aside from being completely wrong that's a very sensible and wise > >> statement. > > >> Discrete time is discrete time, and within discrete time the z transform > >> is exact. The Bode theorem is exact within discrete time control > >> because the z transform is exact. > > >> There's absolutely no reason at all that one has to make any > >> approximations when doing discrete time control of a continuous-time > >> plant (ask me how before you start your flame). Sometimes the > >> approximations are convenient, but they are not necessary. > > -- snip -- > > > Actually no. > > Actually yes. Thanks for following directions and asking for > clarification before you said something stupid -- it really helps. > > > There is nothing special because you sample a system or a > > signal. > > To quote a famous USENET line: Actually no. The act of sampling the > signal renders a linear time invariant system fundamentally different -- > it makes it time variant, and completely changes the types of analysis > you need to do. > > > If you sample fast enought you get back to analogue. > > _If_ you make the right constraints on your sampled-time system _and_ > you make those constraints vary correctly with the sampling rate _and_ > you choose a suitable reconstruction scheme, yes, you can make a > sampled-time system whose behavior, in the limit as the sampling rate > reaches infinity, becomes continuous time. > > > Digital is just an approximation to analogue, nothing more. > > Digital is no more an approximation to analog than sampled time is to > continuous time. And yes, I know what you meant when you chose to play > fast and loose with terminology. > > > 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.. However, if you are > > thinking of special contrived digital systems eg FIR then I might > > agree. > > All of which is completely irrelevant to the statement you're > disagreeing with. Perhaps you want to review it before you make another > attempt. > > When doing discrete-time control of a continuous-time, linear time > invariant plant, you can adopt a design flow that goes: > > 1: Model the plant in the Laplace domain. > 2: Define your output scheme (usually a zero-order hold) > 3: Define your input scheme (usually straight sampling, although > there is a lot of utility in doing an integrate-and-dump for > exactly one sampling interval). > 4: Use 1, 2 and 3 to model the input/output behavior of the > plant _exactly_ in the sampled time domain. > 5: Proceed with your design as normal inside the z domain. > 6: Do any necessary verification of what you've done taking > sampling and reconstruction into account, to see that your > system works correctly out here in the continuous-time > domain. > > Other than any necessary approximations adopted in steps 1-3 (find a > real-world plant that is really linear!), the method is exact. Step 4 > -- the one that you would disagree with if you understood the assertion > that I made -- is dead exact. It's mathematically exact. Not only > that, it is really easy math if you know your state-space systems, and > it can be done on a transfer function using the definition of the z > transform and a minimum of hocus-pocus and hand waving if you can't wrap > your head around state-space forms. > > If you do not understand how to do 4 given 1, 2 and 3, if you do not > understand _why_ 4 is possible given 1, 2 and 3, if you do not > understand why you need 2 and 3 to do 4, then you are not qualified to > say "Actually no" and anything you _do_ say on this subject is totally > irrelevant and I'll ignore it. > > If you truly wish to achieve wisdom, see problem 2.5-12 on page 174 of > "Linear Systems", be Thomas Kailath, Prentice-Hall, 1980. That > demonstrates 1, 2, 3 and 4 when your input is defined as straight sampling. > > -- > Tim Wescott > Control system and signal processing consultingwww.wescottdesign.com You're making a meal of something that is essentially very simple. I expect it is an ego thing since you make your living via DSP and are at odds to think that sampled and analogue are so closely related as to be one of the same thing. Don't confuse things with zero-order holds, the facts remain that digital samples waveforms or systems are approximations to real-world analogue. This is not my assertion, it is explained in nearly every book on the subject that follows from numerical integration. Now to take your example of the ZOH, it dammed nearly disappears if you sample fast enough. In the old-days and in old text books they used to show you how a ZOH effected a closed-loop system. This it does by introducing extra delay and hence reducing phase-margin. If you sample high enough the delay is insignificant and you get back to near analogue. The state-space method is yet another approximation to analogue. It does include a hold of course whereas other approximations do not. So what? It doesn't add some magical thing that wasn't there before now does it. The ZOH also does a slight amount of attenuating but this too gets smaller at faster sample rates. Now of course you can design directly in the z-domain and do things differently that in continuous time. My assertion however is that if you have an analogue system G(s) that its equivalent G(z) is an approximation and that there are many ways to do this approximation. I will give you another example. Suppose we have an algorithm such as LMS or RLS Such (weight) equations normally have the form W(k+1) = W(k) + gain X error(k) You can recognise this as an approximate multivariable integrator. Straight away you will see that there must be a continuous-time equivalent of LMS (which there is). The continuous-time version you may not wish to use but it has the advantage that unlike a digital integrator (with feedback) where the pole drifts outside the unit- circle for high gain (step size) that this does not happen in the continuous-time case and it is always stable. So you could start from continuous-time steepest descent and show that LMS is a sampled approximation! Hardy
From: Tim Wescott on 24 May 2010 00:14
On 05/23/2010 07:12 PM, HardySpicer wrote: > On May 23, 10:27 am, Tim Wescott<t...(a)seemywebsite.now> wrote: >> On 05/22/2010 01:05 AM, HardySpicer wrote: >> >> >> >>> On May 22, 4:41 pm, Tim Wescott<t...(a)seemywebsite.now> wrote: >>>> On 05/21/2010 08:03 PM, HardySpicer wrote: >> >>>>> On May 22, 12:00 pm, "Demus"<sodemus(a)n_o_s_p_a_m.hotmail.com> wrote: >>>>>> Could someone please show me what the discrete-time version of Bode's >>>>>> amplitude/phase relation looks like? >> >>>>> Discrete-time is an approximation to analogue so Bodes Theorem is the >>>>> same. >> >>>> Eh? Aside from being completely wrong that's a very sensible and wise >>>> statement. >> >>>> Discrete time is discrete time, and within discrete time the z transform >>>> is exact. The Bode theorem is exact within discrete time control >>>> because the z transform is exact. >> >>>> There's absolutely no reason at all that one has to make any >>>> approximations when doing discrete time control of a continuous-time >>>> plant (ask me how before you start your flame). Sometimes the >>>> approximations are convenient, but they are not necessary. >> >> -- snip -- >> >>> Actually no. >> >> Actually yes. Thanks for following directions and asking for >> clarification before you said something stupid -- it really helps. >> >>> There is nothing special because you sample a system or a >>> signal. >> >> To quote a famous USENET line: Actually no. The act of sampling the >> signal renders a linear time invariant system fundamentally different -- >> it makes it time variant, and completely changes the types of analysis >> you need to do. >> >>> If you sample fast enought you get back to analogue. >> >> _If_ you make the right constraints on your sampled-time system _and_ >> you make those constraints vary correctly with the sampling rate _and_ >> you choose a suitable reconstruction scheme, yes, you can make a >> sampled-time system whose behavior, in the limit as the sampling rate >> reaches infinity, becomes continuous time. >> >>> Digital is just an approximation to analogue, nothing more. >> >> Digital is no more an approximation to analog than sampled time is to >> continuous time. And yes, I know what you meant when you chose to play >> fast and loose with terminology. >> >>> 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.. However, if you are >>> thinking of special contrived digital systems eg FIR then I might >>> agree. >> >> All of which is completely irrelevant to the statement you're >> disagreeing with. Perhaps you want to review it before you make another >> attempt. >> >> When doing discrete-time control of a continuous-time, linear time >> invariant plant, you can adopt a design flow that goes: >> >> 1: Model the plant in the Laplace domain. >> 2: Define your output scheme (usually a zero-order hold) >> 3: Define your input scheme (usually straight sampling, although >> there is a lot of utility in doing an integrate-and-dump for >> exactly one sampling interval). >> 4: Use 1, 2 and 3 to model the input/output behavior of the >> plant _exactly_ in the sampled time domain. >> 5: Proceed with your design as normal inside the z domain. >> 6: Do any necessary verification of what you've done taking >> sampling and reconstruction into account, to see that your >> system works correctly out here in the continuous-time >> domain. >> >> Other than any necessary approximations adopted in steps 1-3 (find a >> real-world plant that is really linear!), the method is exact. Step 4 >> -- the one that you would disagree with if you understood the assertion >> that I made -- is dead exact. It's mathematically exact. Not only >> that, it is really easy math if you know your state-space systems, and >> it can be done on a transfer function using the definition of the z >> transform and a minimum of hocus-pocus and hand waving if you can't wrap >> your head around state-space forms. >> >> If you do not understand how to do 4 given 1, 2 and 3, if you do not >> understand _why_ 4 is possible given 1, 2 and 3, if you do not >> understand why you need 2 and 3 to do 4, then you are not qualified to >> say "Actually no" and anything you _do_ say on this subject is totally >> irrelevant and I'll ignore it. >> >> If you truly wish to achieve wisdom, see problem 2.5-12 on page 174 of >> "Linear Systems", be Thomas Kailath, Prentice-Hall, 1980. That >> demonstrates 1, 2, 3 and 4 when your input is defined as straight sampling. >> >> -- >> Tim Wescott >> Control system and signal processing consultingwww.wescottdesign.com > > You're making a meal of something that is essentially very simple. I > expect it is an ego thing since you make your living via DSP and are > at odds to think that sampled and analogue are > so closely related as to be one of the same thing. Don't confuse > things with zero-order holds, the facts remain that digital samples > waveforms or systems are approximations to real-world analogue. Bulls**t. What I am doing, apparently, is feeding a troll. > This is not my assertion, Then you'd better run a poll on all your alternate personalities to find out who had control of your hands when you wrote that. > it is explained in nearly every book on the > subject that follows from numerical integration. (a) Then where is your extensive citation list? Did you check the citations that I provided? (b) Sampled time (which is what you mean when you misuse the term "digital") can be used to approximate continuous time (which is what you mean when you misuse the term "analog"), and does quite well. But that doesn't change the fact that they are two different mathematical beasts. If you were a farmer would you try to convince me that deer and cows are exactly the same, and require the same care and feeding? After all, they're both herbivores, and they both have quite tasty meat -- would you then claim that they are exactly the same thing, and that a successful cattle farmer could transition to raising deer without a hitch? That's essentially what you're doing here with sampled time vs. continuous time. > Now to take your > example of the ZOH, it dammed nearly disappears if you sample fast > enough. So you are going to restrict all sampled-time endevours to ones that "sample fast enough"? That's going to cut a lot of commercially viable systems from the world. > In the old-days and in old text books they used to show you > how a ZOH effected a closed-loop system. This it does by introducing > extra delay and hence reducing phase-margin. That is the simplified, hand-waving explanation, yes. > If you sample high enough > the delay is insignificant and you get back to near analogue. The > state-space method is yet another approximation to analogue. It does > include a hold of course whereas other approximations do not. The state space method that I cite is _not_ an approximation. You are entirely wrong in your assertion. Either you are too damn lazy to do the math for yourself, or you misunderstand the framework in which the problem is posed. In either case you are entirely, completely, unquestionably, 100% wrong! > So what? So, if you can be so blatantly wrong with that assertion, how can anything else you say be believed in any way shape or form? > It doesn't add some magical thing that wasn't there before now does > it. The ZOH also does a slight amount of attenuating but this too gets > smaller at faster sample rates. > > Now of course you can design directly in the z-domain and do things > differently that in continuous time. My assertion however is that if > you have an analogue system G(s) that its equivalent G(z) is an > approximation and that there are many ways to do this approximation. And my assertion is that under specific conditions -- notably the ones that the OP was working within -- you don't have to approximate at all. Not At All. So your "G(z) is an approximation" assertion is false, and therefor any conclusions that you draw from that assertion are themselves false. > I > will give you another example. Suppose we have an algorithm such as > LMS or RLS > > Such (weight) equations normally have the form W(k+1) = W(k) + gain X > error(k) > > You can recognise this as an approximate multivariable integrator. > Straight away you will see that there must be a continuous-time > equivalent of LMS (which there is). The continuous-time version you > may not wish to use but it has the advantage that unlike a digital > integrator (with feedback) where the pole drifts outside the unit- > circle for high gain (step size) that this does not happen in the > continuous-time case and it is always stable. Oh please. There isn't a real system in the universe that behaves sufficiently like a pure integrator that you can put it into a feedback loop and jack up the gain without limit while keeping the system stable. Show me a system model with a finite number of well defined states and you'll be showing me a system model that's leaving behavior out. _All_ systems have an infinite number of states. _All_ systems have nonlinearities. _All_ systems are time varying. If you claim that there is a _real_ system with an integrator that is pure enough to remain stable with unbounded feedback gain then you are completely wrong once again. > So you could start from > continuous-time steepest descent and show that LMS is a sampled > approximation! So what? You've found a case where one particular sampled time system approximates a continuous-time system. That means damn near almost nothing. I never said you couldn't _use_ a sampled time system as an approximation for a continuous time system. My only assertions are that: (a) sampled time is different than continuous time, and (b) there are situations in designing feedback control systems where you can model the behavior of a continuous-time system, as seen by a system that is sampling it, exactly. -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com |