Bode's Amplitude/Phase relation for discrete time systems
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From: Tim Wescott on 22 May 2010 18:27 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 consulting www.wescottdesign.com
From: Tim Wescott on 22 May 2010 18:29 On 05/22/2010 03:07 PM, Demus wrote: > 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! I suspect it's a typo -- Jerry knows perfectly well he should have said "Bode plot" or "Laplace transform". -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
From: Tim Wescott on 22 May 2010 18:34 On 05/22/2010 03:18 PM, Demus wrote: >> 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'. Approximate methods that involve replacing s with some expression in z map poles and zeros identically. Exact methods (which take the zero-order hold formed by the usual DAC output) preserve pole locations (better than approximations do, as we would expect) but mangle zero locations pretty thoroughly. Since the zero-order-hold operation is one of pure delay, and is therefore not minimum phase, a sampled-time control system with one is not minimum phase. So the continuous-time system which incorporates the sampled-time system is known to not be minimum phase -- but I don't know if this implies that the exact model of the plant in the z domain is minimum phase because the original is, non-minimum phase because the overall system is, or varies from case to case because Math is Hard. -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
From: Demus on 22 May 2010 18:55 >I suspect it's a typo -- Jerry knows perfectly well he should have said >"Bode plot" or "Laplace transform". I'm sure Jerry does, but I didn't. I was honestly curious if there e.g. were other transforms perhaps 'specializing in the frequency response representations'. Didn't try to be knit-picky. >Approximate methods that involve replacing s with some expression in z >map poles and zeros identically. Exact methods (which take the >zero-order hold formed by the usual DAC output) preserve pole locations >(better than approximations do, as we would expect) but mangle zero >locations pretty thoroughly. Ok, I see the difference now. Thanks for clarifying.
From: Jerry Avins on 22 May 2010 20:26
On 5/22/2010 5:23 PM, HardySpicer wrote: ... > Indeed and I cannot understand why people still toute root-locus and > Nyquist. They are excellent teaching tools but practically Bode has > it. The more ways you have to look at a problem or procedure, the better able to understand it. Root-locus plots illuminate the effect of gain changes on many systems. Nyquist plots can be constructed from the information in a Bode plot, and if the Nyquist plot has frequency ticks along the curve, vice versa. The choice between the two is a matter of preference. Jerry -- "I view the progress of science as ... the slow erosion of the tendency to dichotomize." --Barbara Smuts, U. Mich. ����������������������������������������������������������������������� |