Regularization method for sliding mode control based on sampling

I have the following problem: in a continuous time control system, I have proved that a sliding mode can be enforced by using a relay control. I would
like to show that if the command is sampled and held, the behavior of the discrete time system "converges" to the continuous one when the sampling time goes to zero. This is somehow related to show that there exists a regularization method for sliding mode control based on sampling the discontinuous control law (instead of using a relay with small hysteresis or a saturation with high gain to approximate the discontinuous element, as Utkin does in his papers). Is there anyone who can suggest me some bibliographic references related to this problem ? Many thanks (I apologize if I wasn't clear in the explanation of the problem).
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snipped-for-privacy@gmail.com wrote:

There is a very large body of theory and practice related to sampled-data systems. I'm unfamiliar with "sliding mode", so I can't point to the specifics of your application, but regardless of the details, it should be obvious that the continuous case is reached in the limit as the sampling interval decreases toward zero.
How familiar are you with digital signal processing? There is a bibliography of books at various levels (in English) at http://www.dspguru.com .
Jerry
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Discrete time representations in the z operator do not generally converge smoothly to the continuous time result which is why the delta operator was invented.
fred.
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snipped-for-privacy@gmail.com wrote:

The math may be bumpy, but there will be no discernible difference between the behavior of machines control system using a continuous signal, and those using a signal sampled at a GHz with enough bits to reach down into the noise.
Jerry
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Jerry Avins wrote:

Yes, I agree that in practice what you say is true, but I think the OP is looking for a mathematical proof of the concept that as T-> 0, the discrete time system will converge to the continuous one, if I interpreted the question correctly. If not, I apologize.
fred.

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Fred Stevens wrote:
...

What's to apologize for? I think you're right about what the OP wanted, but I also think he believes he found a new insight. I'm a mathematical philistine (I don't compare myself to Heavyside, though). I intended only to point out to the OP what I saw as the self-evident nature of his assertion, proved or not.
Jerry
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analog = sample rate higher than necessary.
Going into an area of thought here that always seems to cause controversy and is nearly useless for most applications but I seem to come back to it anyway.
Sample rate is only part of how completely a signal is analyzed. Each sample taken is an average of a period of time. The signal during that time has a minimum, maximum, average and mean value. Then there is that portion of the signal that does not get sampled, that dead zone in between samples. Quantum theory indicates that if the sample time is short enough, some pretty weird things will show up.
So what is my point here? Nothing on topic of this thread. You just reminded me of this in the way you characterized sample rate when you said "reach down into the noise" and I have an on going task for myself of defining "analog". I used to think I needed to define "digital" but have realized my problems are with analog.
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A. Paul Montgomery wrote:

I find that I get myself into the deepest trouble when I think I know something. As long as I'm a bit unsure, I usually do all right.
Jerry
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says...

I'm not like this. I like it when I have a very comfortable assurance that I know it down to where, in my head, I see how it works and all of the ramification thereof, it's ripples showing clearly in the Force.
But then I studied under Master Yoda.
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Do you have 'Sliding Mode Control in Electromechanical Systems' by Vadim Utkin, Jurgen Guldner and Jingxin Shi? Page 18 has what you want. It basically talks about the oscillations that can occur if the switching times are too long. Resolution of the feedback is not specifically mentioned, but it also plays a part in the 'imperfections' of digital circuits.
I would think analog types would know all about SMC because it is simple and relatively robust compared to PIDs due to the tolerance to gain changes.
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